R13alphabeta[x_]=x^13+(-13s)*x^12+a11*x^11+a10*x^10+a9*x^9+a8*x^8+a7*x^7+a6*x^6+a5 x^5+a4 x^4+a3 x^3+a2 x^2+a1 x+a0;
a0=Coefficient[R13alphabeta[x],x,0]//Simplify
1/326998425600 (-479001600 \[Beta]+12510201600 \[Beta]^3-21501043200 \[Beta]^5+2877922944 \[Beta]^7-3901540500 \[Beta]^9+8073245297 \[Beta]^11-2435333421 \[Beta]^13+7340688973975552 s^11 (1+\[Beta]^2)+136247800 \[Alpha]^11 (1+\[Beta]^2)-1859 \[Alpha]^10 \[Beta] (3945361+4127359 \[Beta]^2)-18590 \[Alpha]^9 (-210384-339577 \[Beta]^2+1601265 \[Beta]^4)+1859 \[Alpha]^8 \[Beta] (6998436+67403311 \[Beta]^2+18403453 \[Beta]^4)+1144 \[Alpha]^7 (-6913080-109627416 \[Beta]^2-67989967 \[Beta]^4+180738803 \[Beta]^6)+26 \[Alpha]^6 \[Beta] (1455237504+772325928 \[Beta]^2-26864130991 \[Beta]^4+4813669719 \[Beta]^6)-52 \[Alpha]^5 (23408640-699525936 \[Beta]^2-15431756256 \[Beta]^4+4993138735 \[Beta]^6+3940230937 \[Beta]^8)-26 \[Alpha]^4 \[Beta] (764415360+15698096064 \[Beta]^2-6474288852 \[Beta]^4-30009456373 \[Beta]^6+9328839641 \[Beta]^8)-26 \[Alpha] \[Beta]^2 (-257299200+5472829440 \[Beta]^2-7052454432 \[Beta]^4+695493936 \[Beta]^6+412883581 \[Beta]^8+95061835 \[Beta]^10)-208 \[Alpha]^3 (-12744000-435486240 \[Beta]^2+60413148 \[Beta]^4+4808474604 \[Beta]^6-2910193618 \[Beta]^8+248843331 \[Beta]^10)+13 \[Alpha]^2 \[Beta] (-563155200-1687092480 \[Beta]^2+43592389632 \[Beta]^4-40341471408 \[Beta]^6+4610912317 \[Beta]^8+906183619 \[Beta]^10)-3105676104374272 s^10 (5 \[Alpha] (1+\[Beta]^2)+\[Beta] (3+7 \[Beta]^2))+298622702343680 s^9 (-12-23 \[Beta]^2+93 \[Beta]^4+41 \[Alpha]^2 (1+\[Beta]^2)+2 \[Alpha] \[Beta] (37+73 \[Beta]^2))-68912931310080 s^8 (52 \[Alpha]^3 (1+\[Beta]^2)+\[Alpha]^2 \[Beta] (295+481 \[Beta]^2)+\[Beta] (-36-309 \[Beta]^2+289 \[Beta]^4)+\[Alpha] (-80-66 \[Beta]^2+762 \[Beta]^4))-530099471616 s^7 (-1136-6804 \[Beta]^2+34737 \[Beta]^4-16435 \[Beta]^6+1105 \[Alpha]^4 (1+\[Beta]^2)-100 \[Alpha]^3 \[Beta] (161+195 \[Beta]^2)+\[Alpha]^2 (5212-9114 \[Beta]^2-71886 \[Beta]^4)+\[Alpha] (9848 \[Beta]+54268 \[Beta]^3-66580 \[Beta]^5))+142719088512 s^6 (4849 \[Alpha]^5 (1+\[Beta]^2)+65 \[Alpha]^4 \[Beta] (-109+23 \[Beta]^2)-2 \[Alpha]^3 (-604+21383 \[Beta]^2+42547 \[Beta]^4)+\[Alpha]^2 (26480 \[Beta]+82766 \[Beta]^3-169370 \[Beta]^5)-\[Beta] (864+37280 \[Beta]^2-58399 \[Beta]^4+16461 \[Beta]^6)-\[Alpha] (4160+15496 \[Beta]^2-175485 \[Beta]^4+100795 \[Beta]^6))-130695136 s^5 (247040+3954080 \[Beta]^2-21352932 \[Beta]^4+16497201 \[Beta]^6-2826579 \[Beta]^8+1328145 \[Alpha]^6 (1+\[Beta]^2)+546 \[Alpha]^5 \[Beta] (6599+14531 \[Beta]^2)-105 \[Alpha]^4 (21700+169625 \[Beta]^2+54653 \[Beta]^4)-4 \[Alpha]^3 \[Beta] (-1980732+376223 \[Beta]^2+14535359 \[Beta]^4)+\[Alpha]^2 (-1037280+5825832 \[Beta]^2+90141279 \[Beta]^4-70251321 \[Beta]^6)-6 \[Alpha] \[Beta] (443040+6225368 \[Beta]^2-13747133 \[Beta]^4+4587975 \[Beta]^6))+50267360 s^4 (191841 \[Alpha]^7 (1+\[Beta]^2)+546 \[Alpha]^6 \[Beta] (7094+9721 \[Beta]^2)+273 \[Alpha]^5 (-7184-20321 \[Beta]^2+41131 \[Beta]^4)-2 \[Alpha]^4 \[Beta] (165426+13415347 \[Beta]^2+8025352 \[Beta]^4)+\[Alpha]^3 (694032+16548576 \[Beta]^2+34643055 \[Beta]^4-54735673 \[Beta]^6)-6 \[Alpha]^2 \[Beta] (663560+3622780 \[Beta]^2-16492028 \[Beta]^4+6997865 \[Beta]^6)+\[Alpha] (350720+4265840 \[Beta]^2-51185328 \[Beta]^4+51520257 \[Beta]^6-10438659 \[Beta]^8)-2 \[Beta] (56064-3827480 \[Beta]^2+7265226 \[Beta]^4-3049497 \[Beta]^6+280326 \[Beta]^8))+193336 s^3 (3313152+101184704 \[Beta]^2-559706160 \[Beta]^4+494634708 \[Beta]^6-101722095 \[Beta]^8+3401685 \[Beta]^10+15081105 \[Alpha]^8 (1+\[Beta]^2)-1560 \[Alpha]^7 \[Beta] (82790+68411 \[Beta]^2)-5460 \[Alpha]^6 (-8285+35709 \[Beta]^2+142932 \[Beta]^4)-40 \[Alpha]^5 \[Beta] (-7951119-42572366 \[Beta]^2+16074664 \[Beta]^4)+8 \[Alpha]^3 \[Beta] (18910968-105441726 \[Beta]^2-578579734 \[Beta]^4+375577395 \[Beta]^6)+2 \[Alpha]^4 (-46332696-478792890 \[Beta]^2+669503347 \[Beta]^4+883988641 \[Beta]^6)+8 \[Alpha] \[Beta] (-4239184-138579976 \[Beta]^2+408441843 \[Beta]^4-218173470 \[Beta]^6+24134310 \[Beta]^8)+4 \[Alpha]^2 (-102992+58080760 \[Beta]^2+924266055 \[Beta]^4-1430747937 \[Beta]^6+361311930 \[Beta]^8))-7436 s^2 (67553785 \[Alpha]^9 (1+\[Beta]^2)+975 \[Alpha]^8 \[Beta] (-38531+194305 \[Beta]^2)-156 \[Alpha]^7 (603566+10519973 \[Beta]^2+10713827 \[Beta]^4)-52 \[Alpha]^6 \[Beta] (-26231028-54292223 \[Beta]^2+99025765 \[Beta]^4)-6 \[Alpha]^5 (42901744+109331748 \[Beta]^2-2255298053 \[Beta]^4+88429503 \[Beta]^6)+6 \[Alpha]^4 \[Beta] (-84866112-1828559440 \[Beta]^2-342161409 \[Beta]^4+1590837171 \[Beta]^6)+12 \[Alpha]^2 \[Beta] (-34502400-217018784 \[Beta]^2+1901926284 \[Beta]^4-1428876851 \[Beta]^6+203415129 \[Beta]^8)+4 \[Alpha]^3 (37677024+882687312 \[Beta]^2+1096268754 \[Beta]^4-6489450415 \[Beta]^6+2269760435 \[Beta]^8)+3 \[Alpha] (5226240+179723136 \[Beta]^2-2634405088 \[Beta]^4+3335969416 \[Beta]^6-885305413 \[Beta]^8+19210755 \[Beta]^10)+\[Beta] (-28825344+902969856 \[Beta]^2-1773719040 \[Beta]^4+684328896 \[Beta]^6-40692341 \[Beta]^8+20685655 \[Beta]^10))+13 s (-90547200-10947628800 \[Beta]^2+57630329856 \[Beta]^4-43827610176 \[Beta]^6+3128766036 \[Beta]^8-4354940469 \[Beta]^10+3026665631 \[Beta]^12+1655350411 \[Alpha]^10 (1+\[Beta]^2)+1430 \[Alpha]^9 \[Beta] (8625609+14155717 \[Beta]^2)+429 \[Alpha]^8 (-24379324-152353877 \[Beta]^2+5293807 \[Beta]^4)-1144 \[Alpha]^7 \[Beta] (-25664460+89044393 \[Beta]^2+189285025 \[Beta]^4)+12 \[Alpha]^5 \[Beta] (-6514145376-45275589832 \[Beta]^2+70858725507 \[Beta]^4+13843842591 \[Beta]^6)-22 \[Alpha]^6 (-101030112-7278993816 \[Beta]^2-28471586841 \[Beta]^4+14397544847 \[Beta]^6)+8 \[Alpha]^3 \[Beta] (-1484270208+39087888480 \[Beta]^2+127177509468 \[Beta]^4-178925461239 \[Beta]^6+35025913241 \[Beta]^8)+2 \[Alpha]^4 (5292063360+85380147360 \[Beta]^2-392670797268 \[Beta]^4-384942833069 \[Beta]^6+267588993727 \[Beta]^8)-2 \[Alpha] \[Beta] (-1727458560-66333526272 \[Beta]^2+235775317824 \[Beta]^4-132006023280 \[Beta]^6+5794390525 \[Beta]^8+1557482905 \[Beta]^10)+3 \[Alpha]^2 (-543448320-17158900992 \[Beta]^2-187895722176 \[Beta]^4+431987914608 \[Beta]^6-162227553851 \[Beta]^8+7147816885 \[Beta]^10)))
a1=Coefficient[R13alphabeta[x],x,1]//Simplify
1/14863564800 (-564668382613504 s^10+977310662215680 s^9 (\[Alpha]+\[Beta])-6264811937280 s^8 (-44+93 \[Alpha]^2+278 \[Alpha] \[Beta]+93 \[Beta]^2)+963817221120 s^7 (-336 \[Alpha]+85 \[Alpha]^3-336 \[Beta]+1175 \[Alpha]^2 \[Beta]+1175 \[Alpha] \[Beta]^2+85 \[Beta]^3)+25948925184 s^6 (-1648+2145 \[Alpha]^4-10440 \[Alpha]^3 \[Beta]+3996 \[Beta]^2+2145 \[Beta]^4+\[Alpha]^2 (3996-31618 \[Beta]^2)-8 \[Alpha] \[Beta] (-2023+1305 \[Beta]^2))-998035584 s^5 (25779 \[Alpha]^5+27455 \[Alpha]^4 \[Beta]-22 \[Alpha]^3 (596+11747 \[Beta]^2)+\[Alpha]^2 (174872 \[Beta]-258434 \[Beta]^3)+\[Beta] (-29280-13112 \[Beta]^2+25779 \[Beta]^4)+\[Alpha] (-29280+174872 \[Beta]^2+27455 \[Beta]^4))+4569760 s^4 (462848+717171 \[Alpha]^6+5782686 \[Alpha]^5 \[Beta]-486432 \[Beta]^2-2886156 \[Beta]^4+717171 \[Beta]^6-21 \[Alpha]^4 (137436+213287 \[Beta]^2)-4 \[Alpha]^3 \[Beta] (-754380+5641967 \[Beta]^2)+6 \[Alpha] \[Beta] (-956320+502920 \[Beta]^2+963781 \[Beta]^4)-3 \[Alpha]^2 (162144-6819752 \[Beta]^2+1493009 \[Beta]^4))+1054560 s^3 (186927 \[Alpha]^7-4267263 \[Alpha]^6 \[Beta]-7 \[Alpha]^5 (-287664+976315 \[Beta]^2)+\[Alpha]^4 \[Beta] (7147760+15918061 \[Beta]^2)+\[Alpha]^3 (-1921456-16594496 \[Beta]^2+15918061 \[Beta]^4)+\[Alpha]^2 (5546992 \[Beta]-16594496 \[Beta]^3-6834205 \[Beta]^5)+\[Alpha] (-611328+5546992 \[Beta]^2+7147760 \[Beta]^4-4267263 \[Beta]^6)+\[Beta] (-611328-1921456 \[Beta]^2+2013648 \[Beta]^4+186927 \[Beta]^6))-4056 s^2 (7417344+18917925 \[Alpha]^8-35925500 \[Alpha]^7 \[Beta]+18469824 \[Beta]^2-101697264 \[Beta]^4-402220 \[Beta]^6+18917925 \[Beta]^8-23660 \[Alpha]^6 (17+17677 \[Beta]^2)+20 \[Alpha]^5 \[Beta] (20966566+2707027 \[Beta]^2)+4 \[Alpha]^3 \[Beta] (-16430688-305991676 \[Beta]^2+13535135 \[Beta]^4)+2 \[Alpha]^4 (-50848632+13980182 \[Beta]^2+468706007 \[Beta]^4)+\[Alpha]^2 (18469824+555246816 \[Beta]^2+27960364 \[Beta]^4-418237820 \[Beta]^6)-4 \[Alpha] \[Beta] (24722976+16430688 \[Beta]^2-104832830 \[Beta]^4+8981375 \[Beta]^6))-9 (-4838400+1708759 \[Alpha]^10+147755010 \[Alpha]^9 \[Beta]-47174400 \[Beta]^2+89107200 \[Beta]^4+130052416 \[Beta]^6-92070524 \[Beta]^8+1708759 \[Beta]^10+169 \[Alpha]^8 (-544796+50803 \[Beta]^2)-104 \[Alpha]^7 \[Beta] (1964864+11664457 \[Beta]^2)-26 \[Alpha]^6 (-5002016-69185528 \[Beta]^2+1807141 \[Beta]^4)+52 \[Alpha]^5 \[Beta] (-15468416+8868928 \[Beta]^2+42386143 \[Beta]^4)-26 \[Alpha]^4 (-3427200+23952416 \[Beta]^2+151045508 \[Beta]^4+1807141 \[Beta]^6)-104 \[Alpha]^3 \[Beta] (-2721600-24943616 \[Beta]^2-4434464 \[Beta]^4+11664457 \[Beta]^6)+13 \[Alpha]^2 (-3628800-57254400 \[Beta]^2-47904832 \[Beta]^4+138371056 \[Beta]^6+660439 \[Beta]^8)+26 \[Alpha] \[Beta] (3628800+10886400 \[Beta]^2-30936832 \[Beta]^4-7859456 \[Beta]^6+5682885 \[Beta]^8))+52 s (89864255 \[Alpha]^9+456315015 \[Alpha]^8 \[Beta]-1404 \[Alpha]^7 (281074+1385283 \[Beta]^2)-4 \[Alpha]^6 \[Beta] (-399153294+903185105 \[Beta]^2)+6 \[Alpha]^4 \[Beta] (-509498736-1236247836 \[Beta]^2+866215507 \[Beta]^4)+6 \[Alpha]^5 (-33002544+964308636 \[Beta]^2+866215507 \[Beta]^4)-108 \[Alpha]^2 \[Beta] (6105312-33561680 \[Beta]^2-53572702 \[Beta]^4+18008679 \[Beta]^6)-4 \[Alpha]^3 (-127950624-906165360 \[Beta]^2+1854371754 \[Beta]^4+903185105 \[Beta]^6)+\[Beta] (32659200+511802496 \[Beta]^2-198015264 \[Beta]^4-394627896 \[Beta]^6+89864255 \[Beta]^8)+3 \[Alpha] (10886400-219791232 \[Beta]^2-1018997472 \[Beta]^4+532204392 \[Beta]^6+152105005 \[Beta]^8)))
a2=Coefficient[R13alphabeta[x],x,2]//Simplify
-1/163499212800 13 (564668382613504 s^11-1194490809374720 s^10 (\[Alpha]+\[Beta])+22970977103360 s^9 (-28+41 \[Alpha]^2+110 \[Alpha] \[Beta]+41 \[Beta]^2)-21203978864640 s^8 (-44 \[Alpha]+13 \[Alpha]^3-44 \[Beta]+97 \[Alpha]^2 \[Beta]+97 \[Alpha] \[Beta]^2+13 \[Beta]^3)-40776882432 s^7 (-5296+1105 \[Alpha]^4-17800 \[Alpha]^3 \[Beta]+10492 \[Beta]^2+1105 \[Beta]^4+\[Alpha]^2 (10492-43106 \[Beta]^2)-8 \[Alpha] \[Beta] (-4391+2225 \[Beta]^2))+10978391424 s^6 (4849 \[Alpha]^5-2795 \[Alpha]^4 \[Beta]+\[Alpha]^3 (888-64534 \[Beta]^2)+\[Alpha]^2 (69352 \[Beta]-64534 \[Beta]^3)+\[Alpha] (-17600+69352 \[Beta]^2-2795 \[Beta]^4)+\[Beta] (-17600+888 \[Beta]^2+4849 \[Beta]^4))-50267360 s^5 (506368+265629 \[Alpha]^6+1153698 \[Alpha]^5 \[Beta]-699360 \[Beta]^2-973812 \[Beta]^4+265629 \[Beta]^6-21 \[Alpha]^4 (46372+101289 \[Beta]^2)+\[Alpha]^3 (2412624 \[Beta]-6536644 \[Beta]^3)+6 \[Alpha] \[Beta] (-685792+402104 \[Beta]^2+192283 \[Beta]^4)-3 \[Alpha]^2 (233120-3162968 \[Beta]^2+709023 \[Beta]^4))+3866720 s^4 (191841 \[Alpha]^7+4590495 \[Alpha]^6 \[Beta]+273 \[Alpha]^5 (-13136+13997 \[Beta]^2)-\[Alpha]^4 \[Beta] (7484208+18389117 \[Beta]^2)+\[Alpha]^3 (3296016+30774528 \[Beta]^2-18389117 \[Beta]^4)+3 \[Alpha]^2 \[Beta] (-5463920+10258176 \[Beta]^2+1273727 \[Beta]^4)+\[Beta] (2995712+3296016 \[Beta]^2-3586128 \[Beta]^4+191841 \[Beta]^6)+\[Alpha] (2995712-16391760 \[Beta]^2-7484208 \[Beta]^4+4590495 \[Beta]^6))+14872 s^3 (64916992+15081105 \[Alpha]^8-117936780 \[Alpha]^7 \[Beta]+30116032 \[Beta]^2-330029232 \[Beta]^4+59301060 \[Beta]^6+15081105 \[Beta]^8-5460 \[Alpha]^6 (-10861+93463 \[Beta]^2)+20 \[Alpha]^5 \[Beta] (42186510+12505639 \[Beta]^2)+4 \[Alpha]^3 \[Beta] (4916256-685340428 \[Beta]^2+62528195 \[Beta]^4)+2 \[Alpha]^4 (-165014616-135730914 \[Beta]^2+659861371 \[Beta]^4)-4 \[Alpha] \[Beta] (142336928-4916256 \[Beta]^2-210932550 \[Beta]^4+29484195 \[Beta]^6)-4 \[Alpha]^2 (-7529008-486475032 \[Beta]^2+67865457 \[Beta]^4+127576995 \[Beta]^6))+40 (\[Alpha]-\[Beta])^2 (262015 \[Alpha]^9-13905957 \[Alpha]^8 \[Beta]+52 \[Alpha]^6 \[Beta] (2100659+547550 \[Beta]^2)-52 \[Alpha]^7 (-198517+1045908 \[Beta]^2)+6 \[Alpha]^4 \[Beta] (-3234204-123405694 \[Beta]^2+45300293 \[Beta]^4)+6 \[Alpha]^5 (-8807436-4040498 \[Beta]^2+45300293 \[Beta]^4)-12 \[Alpha]^2 \[Beta] (27440544-61443540 \[Beta]^2+2020249 \[Beta]^4+4532268 \[Beta]^6)+4 \[Alpha]^3 (4575456+184330620 \[Beta]^2-185108541 \[Beta]^4+7118150 \[Beta]^6)+\[Alpha] (62780544-329286528 \[Beta]^2-19405224 \[Beta]^4+109234268 \[Beta]^6-13905957 \[Beta]^8)+\[Beta] (62780544+18301824 \[Beta]^2-52844616 \[Beta]^4+10322884 \[Beta]^6+262015 \[Beta]^8))-572 s^2 (67553785 \[Alpha]^9+75939825 \[Alpha]^8 \[Beta]-156 \[Alpha]^7 (1586606+10315117 \[Beta]^2)-52 \[Alpha]^6 \[Beta] (-43643178+35435875 \[Beta]^2)+6 \[Alpha]^4 \[Beta] (-715297456-1426851076 \[Beta]^2+633840837 \[Beta]^4)+6 \[Alpha]^5 (-106378864+844316996 \[Beta]^2+633840837 \[Beta]^4)-12 \[Alpha]^2 \[Beta] (173497888-550130320 \[Beta]^2-422158498 \[Beta]^4+134096521 \[Beta]^6)-4 \[Alpha]^3 (-293163744-1650390960 \[Beta]^2+2140276614 \[Beta]^4+460666375 \[Beta]^6)+3 \[Alpha] (83009280-693991552 \[Beta]^2-1430594912 \[Beta]^4+756481752 \[Beta]^6+25313275 \[Beta]^8)+\[Beta] (249027840+1172654976 \[Beta]^2-638273184 \[Beta]^4-247510536 \[Beta]^6+67553785 \[Beta]^8))+s (-7880279040+1655350411 \[Alpha]^10+16288648090 \[Alpha]^9 \[Beta]-28077315840 \[Beta]^2+56296846080 \[Beta]^4+23022272064 \[Beta]^6-18966932556 \[Beta]^8+1655350411 \[Beta]^10-429 \[Alpha]^8 (44211964+61340373 \[Beta]^2)-1144 \[Alpha]^7 \[Beta] (-1566624+136818943 \[Beta]^2)+22 \[Alpha]^6 (1046466912+17175405096 \[Beta]^2+1200249593 \[Beta]^4)+12 \[Alpha]^5 \[Beta] (-22826388224+1533761792 \[Beta]^2+24319697277 \[Beta]^4)+8 \[Alpha]^3 \[Beta] (12839227200+101701283712 \[Beta]^2+2300642688 \[Beta]^4-19565108849 \[Beta]^6)+2 \[Alpha]^4 (28148423040-60220103712 \[Beta]^2-404894138916 \[Beta]^4+13202745523 \[Beta]^6)+2 \[Alpha] \[Beta] (38122202880+51356908800 \[Beta]^2-136958329344 \[Beta]^4+896108928 \[Beta]^6+8144324045 \[Beta]^8)-3 \[Alpha]^2 (9359105280+122592960000 \[Beta]^2+40146735808 \[Beta]^4-125952970704 \[Beta]^6+8771673339 \[Beta]^8)))
a3=Coefficient[R13alphabeta[x],x,3]//Simplify
1/14863564800 13 (43436029431808 s^10-75177743247360 s^9 (\[Alpha]+\[Beta])+481908610560 s^8 (-92+93 \[Alpha]^2+278 \[Alpha] \[Beta]+93 \[Beta]^2)-74139786240 s^7 (-672 \[Alpha]+85 \[Alpha]^3-672 \[Beta]+1175 \[Alpha]^2 \[Beta]+1175 \[Alpha] \[Beta]^2+85 \[Beta]^3)-1996071168 s^6 (-6448+2145 \[Alpha]^4-10440 \[Alpha]^3 \[Beta]+7356 \[Beta]^2+2145 \[Beta]^4+\[Alpha]^2 (7356-31618 \[Beta]^2)-8 \[Alpha] \[Beta] (-3943+1305 \[Beta]^2))+76771968 s^5 (25779 \[Alpha]^5+27455 \[Alpha]^4 \[Beta]-34 \[Alpha]^3 (908+7601 \[Beta]^2)+\[Alpha]^2 (327032 \[Beta]-258434 \[Beta]^3)+\[Beta] (-106080-30872 \[Beta]^2+25779 \[Beta]^4)+\[Alpha] (-106080+327032 \[Beta]^2+27455 \[Beta]^4))-351520 s^4 (3478784+717171 \[Alpha]^6+5782686 \[Alpha]^5 \[Beta]-1155744 \[Beta]^2-5436396 \[Beta]^4+717171 \[Beta]^6-21 \[Alpha]^4 (258876+213287 \[Beta]^2)+\[Alpha]^3 (4630320 \[Beta]-22567868 \[Beta]^3)+6 \[Alpha] \[Beta] (-3351328+771720 \[Beta]^2+963781 \[Beta]^4)-3 \[Alpha]^2 (385248-12643304 \[Beta]^2+1493009 \[Beta]^4))-81120 s^3 (186927 \[Alpha]^7-4267263 \[Alpha]^6 \[Beta]-7 \[Alpha]^5 (-473808+976315 \[Beta]^2)+\[Alpha]^4 \[Beta] (13904720+15918061 \[Beta]^2)+\[Alpha]^3 (-6910384-29697152 \[Beta]^2+15918061 \[Beta]^4)+\[Alpha]^2 (18040816 \[Beta]-29697152 \[Beta]^3-6834205 \[Beta]^5)+\[Alpha] (-4041216+18040816 \[Beta]^2+13904720 \[Beta]^4-4267263 \[Beta]^6)+\[Beta] (-4041216-6910384 \[Beta]^2+3316656 \[Beta]^4+186927 \[Beta]^6))+312 s^2 (102342144+18917925 \[Alpha]^8-35925500 \[Alpha]^7 \[Beta]+140912064 \[Beta]^2-305649264 \[Beta]^4-18835180 \[Beta]^6+18917925 \[Beta]^8-1820 \[Alpha]^6 (10349+229801 \[Beta]^2)+20 \[Alpha]^5 \[Beta] (37031398+2707027 \[Beta]^2)+4 \[Alpha]^3 \[Beta] (-72657888-551603836 \[Beta]^2+13535135 \[Beta]^4)+2 \[Alpha]^4 (-152824632+57979382 \[Beta]^2+468706007 \[Beta]^4)+4 \[Alpha]^2 (35228016+453143544 \[Beta]^2+28989691 \[Beta]^4-104559455 \[Beta]^6)-4 \[Alpha] \[Beta] (159395616+72657888 \[Beta]^2-185156990 \[Beta]^4+8981375 \[Beta]^6))+9 (-11289600+131443 \[Alpha]^10+11365770 \[Alpha]^9 \[Beta]-42336000 \[Beta]^2+25401600 \[Beta]^4+36319552 \[Beta]^6-11071788 \[Beta]^8+131443 \[Beta]^10+13 \[Alpha]^8 (-851676+50803 \[Beta]^2)-8 \[Alpha]^7 \[Beta] (4586784+11664457 \[Beta]^2)+\[Alpha]^6 (36319552+235793136 \[Beta]^2-3614282 \[Beta]^4)+4 \[Alpha]^5 \[Beta] (-42334976+19757568 \[Beta]^2+42386143 \[Beta]^4)+8 \[Alpha]^3 \[Beta] (17640000+77614976 \[Beta]^2+9878784 \[Beta]^4-11664457 \[Beta]^6)-2 \[Alpha]^4 (-12700800+88719776 \[Beta]^2+267057348 \[Beta]^4+1807141 \[Beta]^6)+\[Alpha]^2 (-42336000-333043200 \[Beta]^2-177439552 \[Beta]^4+235793136 \[Beta]^6+660439 \[Beta]^8)+2 \[Alpha] \[Beta] (42336000+70560000 \[Beta]^2-84669952 \[Beta]^4-18347136 \[Beta]^6+5682885 \[Beta]^8))-4 s (89864255 \[Alpha]^9+456315015 \[Alpha]^8 \[Beta]-468 \[Alpha]^7 (1588742+4155849 \[Beta]^2)-4 \[Alpha]^6 \[Beta] (-590920254+903185105 \[Beta]^2)+6 \[Alpha]^4 \[Beta] (-1654517616-2157913596 \[Beta]^2+866215507 \[Beta]^4)+6 \[Alpha]^5 (-27787824+1777573116 \[Beta]^2+866215507 \[Beta]^4)-36 \[Alpha]^2 \[Beta] (103936416-306780400 \[Beta]^2-296262186 \[Beta]^4+54026037 \[Beta]^6)-4 \[Alpha]^3 (-753382944-2761023600 \[Beta]^2+3236870394 \[Beta]^4+903185105 \[Beta]^6)+\[Beta] (381024000+3013531776 \[Beta]^2-166726944 \[Beta]^4-743531256 \[Beta]^6+89864255 \[Beta]^8)+3 \[Alpha] (127008000-1247236992 \[Beta]^2-3309035232 \[Beta]^4+787893672 \[Beta]^6+152105005 \[Beta]^8)))
a4=Coefficient[R13alphabeta[x],x,4]//Simplify
-1/46448640 13 (104413532288 s^9-144572583168 s^8 (\[Alpha]+\[Beta])+1853494656 s^7 (-50+33 \[Alpha]^2+116 \[Alpha] \[Beta]+33 \[Beta]^2)+998035584 s^6 (78 \[Alpha]+\[Alpha]^3+78 \[Beta]-109 \[Alpha]^2 \[Beta]-109 \[Alpha] \[Beta]^2+\[Beta]^3)-9596496 s^5 (-2312+771 \[Alpha]^4-1560 \[Alpha]^3 \[Beta]+1236 \[Beta]^2+771 \[Beta]^4+\[Alpha]^2 (1236-6974 \[Beta]^2)+\[Alpha] (8392 \[Beta]-1560 \[Beta]^3))+1476384 s^4 (1209 \[Alpha]^5+3175 \[Alpha]^4 \[Beta]-4 \[Alpha]^2 \[Beta] (-3913+2698 \[Beta]^2)-4 \[Alpha]^3 (919+2698 \[Beta]^2)+\[Beta] (-6288-3676 \[Beta]^2+1209 \[Beta]^4)+\[Alpha] (-6288+15652 \[Beta]^2+3175 \[Beta]^4))-1352 s^3 (1156160+43953 \[Alpha]^6+1265208 \[Alpha]^5 \[Beta]+400824 \[Beta]^2-1286082 \[Beta]^4+43953 \[Beta]^6-21 \[Alpha]^4 (61242+10829 \[Beta]^2)-8 \[Alpha]^3 \[Beta] (30951+506686 \[Beta]^2)+\[Alpha]^2 (400824+7761588 \[Beta]^2-227409 \[Beta]^4)+24 \[Alpha] \[Beta] (-203458-10317 \[Beta]^2+52717 \[Beta]^4))-624 s^2 (39936 \[Alpha]^7-236418 \[Alpha]^6 \[Beta]-7 \[Alpha]^5 (-20235+95522 \[Beta]^2)+7 \[Alpha]^4 \[Beta] (210031+144464 \[Beta]^2)-2 \[Alpha]^2 \[Beta] (-681598+1031819 \[Beta]^2+334327 \[Beta]^4)+2 \[Alpha]^3 (-408670-1031819 \[Beta]^2+505624 \[Beta]^4)+\[Alpha] (-330912+1363196 \[Beta]^2+1470217 \[Beta]^4-236418 \[Beta]^6)+\[Beta] (-330912-817340 \[Beta]^2+141645 \[Beta]^4+39936 \[Beta]^6))+3 s (7658880+805701 \[Alpha]^8+966316 \[Alpha]^7 \[Beta]+14039424 \[Beta]^2-11191320 \[Beta]^4-3868564 \[Beta]^6+805701 \[Beta]^8-28 \[Alpha]^6 (138163+576611 \[Beta]^2)-4 \[Alpha]^5 \[Beta] (-7326818+612739 \[Beta]^2)-4 \[Alpha]^3 \[Beta] (7087464+23520388 \[Beta]^2+612739 \[Beta]^4)+2 \[Alpha]^4 (-5595660+9218954 \[Beta]^2+17684207 \[Beta]^4)+4 \[Alpha]^2 (3509856+22178268 \[Beta]^2+4609477 \[Beta]^4-4036277 \[Beta]^6)+4 \[Alpha] \[Beta] (-9131712-7087464 \[Beta]^2+7326818 \[Beta]^4+241579 \[Beta]^6))-(\[Alpha]-\[Beta])^2 (31837 \[Alpha]^7+586859 \[Alpha]^6 \[Beta]+3 \[Alpha]^5 (-236664+66395 \[Beta]^2)-3 \[Alpha]^4 \[Beta] (339912+1132787 \[Beta]^2)+\[Alpha]^3 (984528+8849088 \[Beta]^2-3398361 \[Beta]^4)+3 \[Alpha]^2 \[Beta] (-2735856+2949696 \[Beta]^2+66395 \[Beta]^4)+\[Beta] (3168000+984528 \[Beta]^2-709992 \[Beta]^4+31837 \[Beta]^6)+\[Alpha] (3168000-8207568 \[Beta]^2-1019736 \[Beta]^4+586859 \[Beta]^6)))
a5=Coefficient[R13alphabeta[x],x,5]//Simplify
1/5160960 13 (8031810176 s^8-8649641728 s^7 (\[Alpha]+\[Beta])+332678528 s^6 (-18+7 \[Alpha]^2+32 \[Alpha] \[Beta]+7 \[Beta]^2)+12795328 s^5 (280 \[Alpha]+37 \[Alpha]^3+280 \[Beta]-317 \[Alpha]^2 \[Beta]-317 \[Alpha] \[Beta]^2+37 \[Beta]^3)-246064 s^4 (-4536+1265 \[Alpha]^4-800 \[Alpha]^3 \[Beta]+452 \[Beta]^2+1265 \[Beta]^4+\[Alpha]^2 (452-8666 \[Beta]^2)+\[Alpha] (12328 \[Beta]-800 \[Beta]^3))+18928 s^3 (1939 \[Alpha]^5+10055 \[Alpha]^4 \[Beta]-2 \[Alpha]^3 (6452+9749 \[Beta]^2)+\[Alpha]^2 (31272 \[Beta]-19498 \[Beta]^3)+\[Beta] (-14704-12904 \[Beta]^2+1939 \[Beta]^4)+\[Alpha] (-14704+31272 \[Beta]^2+10055 \[Beta]^4))+104 s^2 (-498944+19201 \[Alpha]^6-334684 \[Alpha]^5 \[Beta]-385256 \[Beta]^2+345170 \[Beta]^4+19201 \[Beta]^6-35 \[Alpha]^4 (-9862+2619 \[Beta]^2)+8 \[Alpha]^3 \[Beta] (53435+127923 \[Beta]^2)+4 \[Alpha] \[Beta] (400292+106870 \[Beta]^2-83671 \[Beta]^4)-\[Alpha]^2 (385256+2207060 \[Beta]^2+91665 \[Beta]^4))+7 (40320+1781 \[Alpha]^8+9364 \[Alpha]^7 \[Beta]+80640 \[Beta]^2-16920 \[Beta]^4-19948 \[Beta]^6+1781 \[Beta]^8-4 \[Alpha]^6 (4987+10873 \[Beta]^2)+\[Alpha]^5 (74936 \[Beta]-19444 \[Beta]^3)-4 \[Alpha]^3 \[Beta] (37800+77788 \[Beta]^2+4861 \[Beta]^4)+2 \[Alpha]^4 (-8460+50294 \[Beta]^2+51791 \[Beta]^4)+\[Alpha]^2 (80640+336240 \[Beta]^2+100588 \[Beta]^4-43492 \[Beta]^6)+4 \[Alpha] \[Beta] (-40320-37800 \[Beta]^2+18734 \[Beta]^4+2341 \[Beta]^6))-4 s (121147 \[Alpha]^7-266343 \[Alpha]^6 \[Beta]-7 \[Alpha]^5 (11296+242043 \[Beta]^2)+\[Alpha]^4 \[Beta] (3997696+1920137 \[Beta]^2)+\[Alpha]^3 (-2380432-4227744 \[Beta]^2+1920137 \[Beta]^4)+\[Alpha]^2 (2891152 \[Beta]-4227744 \[Beta]^3-1694301 \[Beta]^5)+\[Alpha] (-564480+2891152 \[Beta]^2+3997696 \[Beta]^4-266343 \[Beta]^6)+\[Beta] (-564480-2380432 \[Beta]^2-79072 \[Beta]^4+121147 \[Beta]^6)))
a6=Coefficient[R13alphabeta[x],x,6]//Simplify
-1/645120 13 (617831552 s^7-499017792 s^6 (\[Alpha]+\[Beta])+3198832 s^5 (-116+21 \[Alpha]^2+158 \[Alpha] \[Beta]+21 \[Beta]^2)+1230320 s^4 (120 \[Alpha]+29 \[Alpha]^3+120 \[Beta]-113 \[Alpha]^2 \[Beta]-113 \[Alpha] \[Beta]^2+29 \[Beta]^3)-18928 s^3 (-2636+540 \[Alpha]^4+195 \[Alpha]^3 \[Beta]-543 \[Beta]^2+540 \[Beta]^4+\[Alpha] \[Beta] (5458+195 \[Beta]^2)-\[Alpha]^2 (543+3326 \[Beta]^2))-3 (\[Alpha]-\[Beta])^2 (1307 \[Alpha]^5+2775 \[Alpha]^4 \[Beta]-2 \[Alpha]^3 (3024+7801 \[Beta]^2)+\[Alpha]^2 (37152 \[Beta]-15602 \[Beta]^3)+\[Beta] (-29104-6048 \[Beta]^2+1307 \[Beta]^4)+\[Alpha] (-29104+37152 \[Beta]^2+2775 \[Beta]^4))+364 s^2 (1011 \[Alpha]^5+15335 \[Alpha]^4 \[Beta]-2 \[Alpha]^3 (10762+10189 \[Beta]^2)+\[Alpha]^2 (34628 \[Beta]-20378 \[Beta]^3)+\[Beta] (-16992-21524 \[Beta]^2+1011 \[Beta]^4)+\[Alpha] (-16992+34628 \[Beta]^2+15335 \[Beta]^4))+s (-1264128+79121 \[Alpha]^6-483154 \[Alpha]^5 \[Beta]-1284336 \[Beta]^2+436296 \[Beta]^4+79121 \[Beta]^6-371 \[Alpha]^4 (-1176+1003 \[Beta]^2)+4 \[Alpha]^3 \[Beta] (341412+411113 \[Beta]^2)-\[Alpha]^2 (1284336+3977136 \[Beta]^2+372113 \[Beta]^4)+\[Alpha] (3267168 \[Beta]+1365648 \[Beta]^3-483154 \[Beta]^5)))
a7=Coefficient[R13alphabeta[x],x,7]//Simplify
1/92160 13 (47525504 s^6-27418560 s^5 (\[Alpha]+\[Beta])+175760 s^4 (-124+3 \[Alpha]^2+130 \[Alpha] \[Beta]+3 \[Beta]^2)+13520 s^3 (384 \[Alpha]+137 \[Alpha]^3+384 \[Beta]-317 \[Alpha]^2 \[Beta]-317 \[Alpha] \[Beta]^2+137 \[Beta]^3)-208 s^2 (-9092+1185 \[Alpha]^4+1725 \[Alpha]^3 \[Beta]-3741 \[Beta]^2+1185 \[Beta]^4+\[Alpha] \[Beta] (14806+1725 \[Beta]^2)-\[Alpha]^2 (3741+8012 \[Beta]^2))-8 s (609 \[Alpha]^5-14395 \[Alpha]^4 \[Beta]+2 \[Alpha]^2 \[Beta] (-12689+7373 \[Beta]^2)+2 \[Alpha]^3 (10589+7373 \[Beta]^2)+\[Alpha] (9720-25378 \[Beta]^2-14395 \[Beta]^4)+\[Beta] (9720+21178 \[Beta]^2+609 \[Beta]^4))+5 (-3456+205 \[Alpha]^6-674 \[Alpha]^5 \[Beta]-3888 \[Beta]^2+288 \[Beta]^4+205 \[Beta]^6+\[Alpha]^4 (288-1069 \[Beta]^2)+4 \[Alpha]^3 \[Beta] (972+769 \[Beta]^2)-\[Alpha]^2 (3888+8352 \[Beta]^2+1069 \[Beta]^4)+\[Alpha] (7776 \[Beta]+3888 \[Beta]^3-674 \[Beta]^5)))
a8=Coefficient[R13alphabeta[x],x,8]//Simplify
-1/3840 13 (913952 s^5-351520 s^4 (\[Alpha]+\[Beta])-27040 s^3 (11+\[Alpha]^2-9 \[Alpha] \[Beta]+\[Beta]^2)-2 (\[Alpha]-\[Beta])^2 (236 \[Alpha]+23 \[Alpha]^3+236 \[Beta]-119 \[Alpha]^2 \[Beta]-119 \[Alpha] \[Beta]^2+23 \[Beta]^3)+520 s^2 (68 \[Alpha]+35 \[Alpha]^3+68 \[Beta]-55 \[Alpha]^2 \[Beta]-55 \[Alpha] \[Beta]^2+35 \[Beta]^3)+s (13808-895 \[Alpha]^4-3260 \[Alpha]^3 \[Beta]+7424 \[Beta]^2-895 \[Beta]^4-4 \[Alpha] \[Beta] (4656+815 \[Beta]^2)+\[Alpha]^2 (7424+9078 \[Beta]^2)))
a9=Coefficient[R13alphabeta[x],x,9]//Simplify
13/768 (70304 s^4-16224 s^3 (\[Alpha]+\[Beta])-416 s^2 (35+6 \[Alpha]^2-23 \[Alpha] \[Beta]+6 \[Beta]^2)-3 (-80+\[Alpha]^4+20 \[Alpha]^3 \[Beta]-48 \[Beta]^2+\[Beta]^4+4 \[Alpha] \[Beta] (24+5 \[Beta]^2)-6 \[Alpha]^2 (8+7 \[Beta]^2))+32 s (16 \[Alpha]^3-19 \[Alpha]^2 \[Beta]+\[Alpha] (18-19 \[Beta]^2)+2 \[Beta] (9+8 \[Beta]^2)))
a10=Coefficient[R13alphabeta[x],x,10]//Simplify
-(13/48) (1352 s^3-156 s^2 (\[Alpha]+\[Beta])+2 (\[Alpha]-\[Beta])^2 (\[Alpha]+\[Beta])-s (148+33 \[Alpha]^2-82 \[Alpha] \[Beta]+33 \[Beta]^2))
a11=Coefficient[R13alphabeta[x],x,11]//Simplify
13/16 (-4+104 s^2-\[Alpha]^2+2 \[Alpha] \[Beta]-\[Beta]^2-4 s (\[Alpha]+\[Beta]))
a12=Coefficient[R13alphabeta[x],x,12]//Simplify
-13 s
a13=Coefficient[R13alphabeta[x],x,13]//Simplify
1