R8alphabeta[x_]=1/11520 (-4194304 s^6+3932160 s^5 (\[Alpha]+\[Beta])-81920 s^4 (-8+9 \[Alpha]^2+50 \[Alpha] \[Beta]+9 \[Beta]^2)-20480 s^3 (15 \[Alpha]+13 \[Alpha]^3+15 \[Beta]-58 \[Alpha]^2 \[Beta]-58 \[Alpha] \[Beta]^2+13 \[Beta]^3)+64 s^2 (-544+1545 \[Alpha]^4+3272 \[Alpha] \[Beta]-252 \[Beta]^2+1545 \[Beta]^4-2 \[Alpha]^2 (126+3737 \[Beta]^2))-5 (\[Alpha]-\[Beta])^2 (-144+35 \[Alpha]^4-444 \[Alpha]^3 \[Beta]+216 \[Beta]^2+35 \[Beta]^4+\[Alpha]^2 (216-1102 \[Beta]^2)+\[Alpha] (720 \[Beta]-444 \[Beta]^3))-4 s (1737 \[Alpha]^5+8965 \[Alpha]^4 \[Beta]-2 \[Alpha]^3 (2168+6311 \[Beta]^2)+\[Alpha]^2 (6256 \[Beta]-12622 \[Beta]^3)+\[Beta] (-720-4336 \[Beta]^2+1737 \[Beta]^4)+\[Alpha] (-720+6256 \[Beta]^2+8965 \[Beta]^4)))+x*(-(16384/45) s^6 (-1+\[Beta])+1/3 (\[Alpha]-\[Beta])^2 (\[Alpha]+\[Beta])+1024/15 s^5 (4+5 \[Alpha] (-1+\[Beta])+9 (-1+\[Beta]) \[Beta])+1/120 (\[Alpha]-\[Beta])^2 (\[Alpha]+\[Beta]) (-68+\[Alpha]^2+46 \[Alpha] \[Beta]+\[Beta]^2)-1/11520 (-1+\[Beta]) (\[Alpha]+\[Beta]) (5 \[Alpha] (-144+216 \[Alpha]^2+35 \[Alpha]^4)+3 (720+1016 \[Alpha]^2-947 \[Alpha]^4) \[Beta]-2 \[Alpha] (3228+1817 \[Alpha]^2) \[Beta]^2+2 (-1716+3595 \[Alpha]^2) \[Beta]^3+4131 \[Alpha] \[Beta]^4+739 \[Beta]^5)+64/9 s^4 (-24 (\[Alpha]+\[Beta])-(-1+\[Beta]) (-8+9 \[Alpha]^2+74 \[Alpha] \[Beta]+57 \[Beta]^2))+16/9 s^3 (48+3 (-24+\[Alpha]^2+26 \[Alpha] \[Beta]+\[Beta]^2)+(-1+\[Beta]) (-15 \[Alpha]-13 \[Alpha]^3-39 \[Beta]+61 \[Alpha]^2 \[Beta]+172 \[Alpha] \[Beta]^2+74 \[Beta]^3))+s^2 (-16 (\[Alpha]+\[Beta])+1/3 (\[Alpha]+\[Beta]) (76+35 \[Alpha]^2-110 \[Alpha] \[Beta]+35 \[Beta]^2)+1/180 (-1+\[Beta]) (-544+1545 \[Alpha]^4+2100 \[Alpha]^3 \[Beta]+5268 \[Beta]^2-3915 \[Beta]^4+4 \[Alpha] \[Beta] (1238-3705 \[Beta]^2)-2 \[Alpha]^2 (126+5447 \[Beta]^2)))+1/2880 s (3648-1737 \[Alpha]^5 (-1+\[Beta])-55 \[Alpha]^4 (84+247 (-1+\[Beta]) \[Beta])-2 \[Alpha]^3 (2168+\[Beta] (112+349 (-1+\[Beta]) \[Beta]))+2 \[Alpha]^2 (2352+\[Beta] (776+\[Beta] (10708+19295 (-1+\[Beta]) \[Beta])))+\[Beta] (-4368+\[Beta] (9072+\[Beta] (14480+\[Beta] (-19100+5043 (-1+\[Beta]) \[Beta]))))+\[Alpha] (-720+\[Beta] (-14064+5 \[Beta] (5072+\[Beta] (-5984+5767 (-1+\[Beta]) \[Beta]))))))+x^2*(1/11520 (4194304 s^6-3932160 s^5 (\[Alpha]+\[Beta])+81920 s^4 (-32+9 \[Alpha]^2+50 \[Alpha] \[Beta]+9 \[Beta]^2)+20480 s^3 (51 \[Alpha]+13 \[Alpha]^3+51 \[Beta]-58 \[Alpha]^2 \[Beta]-58 \[Alpha] \[Beta]^2+13 \[Beta]^3)-64 s^2 (-4384+1545 \[Alpha]^4+10712 \[Alpha] \[Beta]-1332 \[Beta]^2+1545 \[Beta]^4-74 \[Alpha]^2 (18+101 \[Beta]^2))+4 s (1737 \[Alpha]^5+8965 \[Alpha]^4 \[Beta]-2 \[Alpha]^2 \[Beta] (-8948+6311 \[Beta]^2)-2 \[Alpha]^3 (6548+6311 \[Beta]^2)+\[Beta] (-5040-13096 \[Beta]^2+1737 \[Beta]^4)+\[Alpha] (-5040+17896 \[Beta]^2+8965 \[Beta]^4))+5 (-576+35 \[Alpha]^6-514 \[Alpha]^5 \[Beta]-1008 \[Beta]^2+468 \[Beta]^4+35 \[Beta]^6+\[Alpha]^4 (468-179 \[Beta]^2)+28 \[Alpha]^3 \[Beta] (36+47 \[Beta]^2)-\[Alpha]^2 (1008+2952 \[Beta]^2+179 \[Beta]^4)+\[Alpha] (2016 \[Beta]+1008 \[Beta]^3-514 \[Beta]^5))))+x^3*1/240 (-65536 s^5+40960 s^4 (\[Alpha]+\[Beta])-1280 s^3 (-24+\[Alpha]^2+26 \[Alpha] \[Beta]+\[Beta]^2)-2 (\[Alpha]-\[Beta])^2 (-68 \[Alpha]+\[Alpha]^3-68 \[Beta]+47 \[Alpha]^2 \[Beta]+47 \[Alpha] \[Beta]^2+\[Beta]^3)-80 s^2 (76 \[Alpha]+35 \[Alpha]^3+76 \[Beta]-75 \[Alpha]^2 \[Beta]-75 \[Alpha] \[Beta]^2+35 \[Beta]^3)+s (-1904+385 \[Alpha]^4+380 \[Alpha]^3 \[Beta]-1112 \[Beta]^2+385 \[Beta]^4+4 \[Alpha] \[Beta] (828+95 \[Beta]^2)-2 \[Alpha]^2 (556+957 \[Beta]^2)))+x^4*((512 s^4)/3-64 s^3 (\[Alpha]+\[Beta])-2/3 s^2 (80+9 \[Alpha]^2-62 \[Alpha] \[Beta]+9 \[Beta]^2)+1/24 s (84 \[Alpha]+73 \[Alpha]^3+84 \[Beta]-97 \[Alpha]^2 \[Beta]-97 \[Alpha] \[Beta]^2+73 \[Beta]^3)+1/64 (80-7 \[Alpha]^4-20 \[Alpha]^3 \[Beta]+56 \[Beta]^2-7 \[Beta]^4-4 \[Alpha] \[Beta] (28+5 \[Beta]^2)+\[Alpha]^2 (56+54 \[Beta]^2)))+x^5*(1/3 (-256 s^3+48 s^2 (\[Alpha]+\[Beta])-(\[Alpha]-\[Beta])^2 (\[Alpha]+\[Beta])+s (44+9 \[Alpha]^2-26 \[Alpha] \[Beta]+9 \[Beta]^2)))+x^6*(1/2 (-4+64 s^2-4 s \[Alpha]-\[Alpha]^2-4 s \[Beta]+2 \[Alpha] \[Beta]-\[Beta]^2))+(-8*s)*x^7+x^8;
Coefficient[R8alphabeta[x],x,8]
1
Coefficient[R8alphabeta[x],x,7]//Simplify
-8 s
Coefficient[R8alphabeta[x],x,6]//Simplify
-2+32 s^2-\[Alpha]^2/2+\[Alpha] \[Beta]-\[Beta]^2/2-2 s (\[Alpha]+\[Beta])
Coefficient[R8alphabeta[x],x,5]//Simplify
1/3 (-256 s^3+48 s^2 (\[Alpha]+\[Beta])-(\[Alpha]-\[Beta])^2 (\[Alpha]+\[Beta])+s (44+9 \[Alpha]^2-26 \[Alpha] \[Beta]+9 \[Beta]^2))
Coefficient[R8alphabeta[x],x,4]//Simplify
(512 s^4)/3-64 s^3 (\[Alpha]+\[Beta])-2/3 s^2 (80+9 \[Alpha]^2-62 \[Alpha] \[Beta]+9 \[Beta]^2)+1/24 s (84 \[Alpha]+73 \[Alpha]^3+84 \[Beta]-97 \[Alpha]^2 \[Beta]-97 \[Alpha] \[Beta]^2+73 \[Beta]^3)+1/64 (80-7 \[Alpha]^4-20 \[Alpha]^3 \[Beta]+56 \[Beta]^2-7 \[Beta]^4-4 \[Alpha] \[Beta] (28+5 \[Beta]^2)+\[Alpha]^2 (56+54 \[Beta]^2))
Coefficient[R8alphabeta[x],x,3]//Simplify
1/240 (-65536 s^5+40960 s^4 (\[Alpha]+\[Beta])-1280 s^3 (-24+\[Alpha]^2+26 \[Alpha] \[Beta]+\[Beta]^2)-2 (\[Alpha]-\[Beta])^2 (-68 \[Alpha]+\[Alpha]^3-68 \[Beta]+47 \[Alpha]^2 \[Beta]+47 \[Alpha] \[Beta]^2+\[Beta]^3)-80 s^2 (76 \[Alpha]+35 \[Alpha]^3+76 \[Beta]-75 \[Alpha]^2 \[Beta]-75 \[Alpha] \[Beta]^2+35 \[Beta]^3)+s (-1904+385 \[Alpha]^4+380 \[Alpha]^3 \[Beta]-1112 \[Beta]^2+385 \[Beta]^4+4 \[Alpha] \[Beta] (828+95 \[Beta]^2)-2 \[Alpha]^2 (556+957 \[Beta]^2)))
Coefficient[R8alphabeta[x],x,2]//Simplify
1/11520 (4194304 s^6-3932160 s^5 (\[Alpha]+\[Beta])+81920 s^4 (-32+9 \[Alpha]^2+50 \[Alpha] \[Beta]+9 \[Beta]^2)+20480 s^3 (51 \[Alpha]+13 \[Alpha]^3+51 \[Beta]-58 \[Alpha]^2 \[Beta]-58 \[Alpha] \[Beta]^2+13 \[Beta]^3)-64 s^2 (-4384+1545 \[Alpha]^4+10712 \[Alpha] \[Beta]-1332 \[Beta]^2+1545 \[Beta]^4-74 \[Alpha]^2 (18+101 \[Beta]^2))+4 s (1737 \[Alpha]^5+8965 \[Alpha]^4 \[Beta]-2 \[Alpha]^2 \[Beta] (-8948+6311 \[Beta]^2)-2 \[Alpha]^3 (6548+6311 \[Beta]^2)+\[Beta] (-5040-13096 \[Beta]^2+1737 \[Beta]^4)+\[Alpha] (-5040+17896 \[Beta]^2+8965 \[Beta]^4))+5 (-576+35 \[Alpha]^6-514 \[Alpha]^5 \[Beta]-1008 \[Beta]^2+468 \[Beta]^4+35 \[Beta]^6+\[Alpha]^4 (468-179 \[Beta]^2)+28 \[Alpha]^3 \[Beta] (36+47 \[Beta]^2)-\[Alpha]^2 (1008+2952 \[Beta]^2+179 \[Beta]^4)+\[Alpha] (2016 \[Beta]+1008 \[Beta]^3-514 \[Beta]^5)))
Coefficient[R8alphabeta[x],x,1]//Simplify
-(16384/45) s^6 (-1+\[Beta])+1/3 (\[Alpha]-\[Beta])^2 (\[Alpha]+\[Beta])+1024/15 s^5 (4+5 \[Alpha] (-1+\[Beta])+9 (-1+\[Beta]) \[Beta])+1/120 (\[Alpha]-\[Beta])^2 (\[Alpha]+\[Beta]) (-68+\[Alpha]^2+46 \[Alpha] \[Beta]+\[Beta]^2)-1/11520 (-1+\[Beta]) (\[Alpha]+\[Beta]) (5 \[Alpha] (-144+216 \[Alpha]^2+35 \[Alpha]^4)+3 (720+1016 \[Alpha]^2-947 \[Alpha]^4) \[Beta]-2 \[Alpha] (3228+1817 \[Alpha]^2) \[Beta]^2+2 (-1716+3595 \[Alpha]^2) \[Beta]^3+4131 \[Alpha] \[Beta]^4+739 \[Beta]^5)+64/9 s^4 (-24 (\[Alpha]+\[Beta])-(-1+\[Beta]) (-8+9 \[Alpha]^2+74 \[Alpha] \[Beta]+57 \[Beta]^2))+16/9 s^3 (48+3 (-24+\[Alpha]^2+26 \[Alpha] \[Beta]+\[Beta]^2)+(-1+\[Beta]) (-15 \[Alpha]-13 \[Alpha]^3-39 \[Beta]+61 \[Alpha]^2 \[Beta]+172 \[Alpha] \[Beta]^2+74 \[Beta]^3))+s^2 (-16 (\[Alpha]+\[Beta])+1/3 (\[Alpha]+\[Beta]) (76+35 \[Alpha]^2-110 \[Alpha] \[Beta]+35 \[Beta]^2)+1/180 (-1+\[Beta]) (-544+1545 \[Alpha]^4+2100 \[Alpha]^3 \[Beta]+5268 \[Beta]^2-3915 \[Beta]^4+4 \[Alpha] \[Beta] (1238-3705 \[Beta]^2)-2 \[Alpha]^2 (126+5447 \[Beta]^2)))+1/2880 s (3648-1737 \[Alpha]^5 (-1+\[Beta])-55 \[Alpha]^4 (84+247 (-1+\[Beta]) \[Beta])-2 \[Alpha]^3 (2168+\[Beta] (112+349 (-1+\[Beta]) \[Beta]))+2 \[Alpha]^2 (2352+\[Beta] (776+\[Beta] (10708+19295 (-1+\[Beta]) \[Beta])))+\[Beta] (-4368+\[Beta] (9072+\[Beta] (14480+\[Beta] (-19100+5043 (-1+\[Beta]) \[Beta]))))+\[Alpha] (-720+\[Beta] (-14064+5 \[Beta] (5072+\[Beta] (-5984+5767 (-1+\[Beta]) \[Beta])))))
Coefficient[R8alphabeta[x],x,0]//Simplify
1/11520 (-4194304 s^6+3932160 s^5 (\[Alpha]+\[Beta])-81920 s^4 (-8+9 \[Alpha]^2+50 \[Alpha] \[Beta]+9 \[Beta]^2)-20480 s^3 (15 \[Alpha]+13 \[Alpha]^3+15 \[Beta]-58 \[Alpha]^2 \[Beta]-58 \[Alpha] \[Beta]^2+13 \[Beta]^3)+64 s^2 (-544+1545 \[Alpha]^4+3272 \[Alpha] \[Beta]-252 \[Beta]^2+1545 \[Beta]^4-2 \[Alpha]^2 (126+3737 \[Beta]^2))-5 (\[Alpha]-\[Beta])^2 (-144+35 \[Alpha]^4-444 \[Alpha]^3 \[Beta]+216 \[Beta]^2+35 \[Beta]^4+\[Alpha]^2 (216-1102 \[Beta]^2)+\[Alpha] (720 \[Beta]-444 \[Beta]^3))-4 s (1737 \[Alpha]^5+8965 \[Alpha]^4 \[Beta]-2 \[Alpha]^3 (2168+6311 \[Beta]^2)+\[Alpha]^2 (6256 \[Beta]-12622 \[Beta]^3)+\[Beta] (-720-4336 \[Beta]^2+1737 \[Beta]^4)+\[Alpha] (-720+6256 \[Beta]^2+8965 \[Beta]^4)))