a4=1;
a3=-4 s;
a2=(5 \[Alpha]^6+2 \[Alpha]^5 (4+7 \[Beta])+\[Alpha]^4 (-8+28 \[Beta]+15 \[Beta]^2)-4 \[Alpha]^3 (-8+3 \[Beta]^3)+\[Alpha]^2 (-48+64 \[Beta]+112 \[Beta]^2-56 \[Beta]^3-77 \[Beta]^4)+2 \[Alpha] \[Beta] (16+16 \[Beta]-32 \[Beta]^2-20 \[Beta]^3+15 \[Beta]^4)+\[Beta] (64+16 \[Beta]-128 \[Beta]^2-40 \[Beta]^3+60 \[Beta]^4+25 \[Beta]^5))/(4 (4+\[Alpha]^2+2 \[Alpha] \[Beta]-3 \[Beta]^2)^2);
a1=-(((\[Alpha]-\[Beta]) (-4+\[Alpha]^2+2 \[Alpha] \[Beta]+5 \[Beta]^2) (\[Alpha]^4+4 \[Beta]-5 \[Beta]^3+\[Alpha]^3 (5+\[Beta])+\[Alpha]^2 (-4+9 \[Beta]+3 \[Beta]^2)+\[Alpha] (12+4 \[Beta]-9 \[Beta]^2-5 \[Beta]^3)))/(4 (4+\[Alpha]^2+2 \[Alpha] \[Beta]-3 \[Beta]^2)^2));
a0=((\[Alpha]-\[Beta]) (-2 \[Alpha]^5 (-4+\[Beta]) \[Beta]+\[Alpha]^6 (1+\[Beta])+(-1+\[Beta]) \[Beta]^2 (-4+\[Beta]^2)^2-\[Alpha]^4 (8+8 \[Beta]-21 \[Beta]^2+\[Beta]^3)+4 \[Alpha]^3 \[Beta] (-4+2 \[Beta]^2+\[Beta]^3)+\[Alpha]^2 (16+16 \[Beta]-64 \[Beta]^2+16 \[Beta]^3+11 \[Beta]^4-\[Beta]^5)-2 \[Alpha] \[Beta] (32-16 \[Beta]-40 \[Beta]^2+24 \[Beta]^4+\[Beta]^5)))/(8 (4+\[Alpha]^2+2 \[Alpha] \[Beta]-3 \[Beta]^2)^2);
S4\[Alpha]\[Beta][x_]=a0+a1 x+a2 x^2+a3 x^3+a4 x^4;