# Maple integration test file: "1 Algebraic functions\1.2 Trinomial products\1.2.1 Quadratic\1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.txt"

lst:=[

# Integrands of the form (d x)^m (a+b x+c x^2)^p

# Integrands of the form (d x)^m (a+b x+c x^2)^p when a=0

# Integrands of the form x^m (b x+c x^2)^(p/2)

# p>0
[x^3*(b*x+c*x^2)^(1/2),x,6,7/48*b^2*(b*x+c*x^2)^(3/2)/c^3-7/40*b*x*(b*x+c*x^2)^(3/2)/c^2+1/5*x^2*(b*x+c*x^2)^(3/2)/c+7/128*b^5*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(9/2)-7/128*b^3*(b+2*c*x)*sqrt(b*x+c*x^2)/c^4],
[x^2*(b*x+c*x^2)^(1/2),x,5,-5/24*b*(b*x+c*x^2)^(3/2)/c^2+1/4*x*(b*x+c*x^2)^(3/2)/c-5/64*b^4*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(7/2)+5/64*b^2*(b+2*c*x)*sqrt(b*x+c*x^2)/c^3],
[x*(b*x+c*x^2)^(1/2),x,4,1/3*(b*x+c*x^2)^(3/2)/c+1/8*b^3*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(5/2)-1/8*b*(b+2*c*x)*sqrt(b*x+c*x^2)/c^2],
[(b*x+c*x^2)^(1/2),x,3,-1/4*b^2*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(3/2)+1/4*(b+2*c*x)*sqrt(b*x+c*x^2)/c],
[(b*x+c*x^2)^(1/2)/x,x,3,b*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/sqrt(c)+sqrt(b*x+c*x^2)],
[(b*x+c*x^2)^(1/2)/x^2,x,3,2*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))*sqrt(c)-2*sqrt(b*x+c*x^2)/x],
[(b*x+c*x^2)^(1/2)/x^3,x,1,-2/3*(b*x+c*x^2)^(3/2)/(b*x^3)],
[(b*x+c*x^2)^(1/2)/x^4,x,2,-2/5*(b*x+c*x^2)^(3/2)/(b*x^4)+4/15*c*(b*x+c*x^2)^(3/2)/(b^2*x^3)],
[(b*x+c*x^2)^(1/2)/x^5,x,3,-2/7*(b*x+c*x^2)^(3/2)/(b*x^5)+8/35*c*(b*x+c*x^2)^(3/2)/(b^2*x^4)-16/105*c^2*(b*x+c*x^2)^(3/2)/(b^3*x^3)],
[(b*x+c*x^2)^(1/2)/x^6,x,4,-2/9*(b*x+c*x^2)^(3/2)/(b*x^6)+4/21*c*(b*x+c*x^2)^(3/2)/(b^2*x^5)-16/105*c^2*(b*x+c*x^2)^(3/2)/(b^3*x^4)+32/315*c^3*(b*x+c*x^2)^(3/2)/(b^4*x^3)],
[(b*x+c*x^2)^(1/2)/x^7,x,5,-2/11*(b*x+c*x^2)^(3/2)/(b*x^7)+16/99*c*(b*x+c*x^2)^(3/2)/(b^2*x^6)-32/231*c^2*(b*x+c*x^2)^(3/2)/(b^3*x^5)+128/1155*c^3*(b*x+c*x^2)^(3/2)/(b^4*x^4)-256/3465*c^4*(b*x+c*x^2)^(3/2)/(b^5*x^3)],
[x^2*(b*x+c*x^2)^(3/2),x,6,7/192*b^2*(b+2*c*x)*(b*x+c*x^2)^(3/2)/c^3-7/60*b*(b*x+c*x^2)^(5/2)/c^2+1/6*x*(b*x+c*x^2)^(5/2)/c+7/512*b^6*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(9/2)-7/512*b^4*(b+2*c*x)*sqrt(b*x+c*x^2)/c^4],
[x*(b*x+c*x^2)^(3/2),x,5,-1/16*b*(b+2*c*x)*(b*x+c*x^2)^(3/2)/c^2+1/5*(b*x+c*x^2)^(5/2)/c-3/128*b^5*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(7/2)+3/128*b^3*(b+2*c*x)*sqrt(b*x+c*x^2)/c^3],
[(b*x+c*x^2)^(3/2),x,4,1/8*(b+2*c*x)*(b*x+c*x^2)^(3/2)/c+3/64*b^4*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(5/2)-3/64*b^2*(b+2*c*x)*sqrt(b*x+c*x^2)/c^2],
[(b*x+c*x^2)^(3/2)/x,x,4,1/3*(b*x+c*x^2)^(3/2)-1/8*b^3*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(3/2)+1/8*b*(b+2*c*x)*sqrt(b*x+c*x^2)/c],
[(b*x+c*x^2)^(3/2)/x^2,x,4,1/2*(b*x+c*x^2)^(3/2)/x+3/4*b^2*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/sqrt(c)+3/4*b*sqrt(b*x+c*x^2)],
[(b*x+c*x^2)^(3/2)/x^3,x,4,-2*(b*x+c*x^2)^(3/2)/x^2+3*b*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))*sqrt(c)+3*c*sqrt(b*x+c*x^2)],
[(b*x+c*x^2)^(3/2)/x^4,x,4,-2/3*(b*x+c*x^2)^(3/2)/x^3+2*c^(3/2)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))-2*c*sqrt(b*x+c*x^2)/x],
[(b*x+c*x^2)^(3/2)/x^5,x,1,-2/5*(b*x+c*x^2)^(5/2)/(b*x^5)],
[(b*x+c*x^2)^(3/2)/x^6,x,2,-2/7*(b*x+c*x^2)^(5/2)/(b*x^6)+4/35*c*(b*x+c*x^2)^(5/2)/(b^2*x^5)],
[(b*x+c*x^2)^(3/2)/x^7,x,3,-2/9*(b*x+c*x^2)^(5/2)/(b*x^7)+8/63*c*(b*x+c*x^2)^(5/2)/(b^2*x^6)-16/315*c^2*(b*x+c*x^2)^(5/2)/(b^3*x^5)],
[(b*x+c*x^2)^(3/2)/x^8,x,4,-2/11*(b*x+c*x^2)^(5/2)/(b*x^8)+4/33*c*(b*x+c*x^2)^(5/2)/(b^2*x^7)-16/231*c^2*(b*x+c*x^2)^(5/2)/(b^3*x^6)+32/1155*c^3*(b*x+c*x^2)^(5/2)/(b^4*x^5)],
[(b*x+c*x^2)^(3/2)/x^9,x,5,-2/13*(b*x+c*x^2)^(5/2)/(b*x^9)+16/143*c*(b*x+c*x^2)^(5/2)/(b^2*x^8)-32/429*c^2*(b*x+c*x^2)^(5/2)/(b^3*x^7)+128/3003*c^3*(b*x+c*x^2)^(5/2)/(b^4*x^6)-256/15015*c^4*(b*x+c*x^2)^(5/2)/(b^5*x^5)],
[x^2*(a*x+b*x^2)^(5/2),x,7,-15/2048*a^4*(a+2*b*x)*(a*x+b*x^2)^(3/2)/b^4+3/128*a^2*(a+2*b*x)*(a*x+b*x^2)^(5/2)/b^3-9/112*a*(a*x+b*x^2)^(7/2)/b^2+1/8*x*(a*x+b*x^2)^(7/2)/b-45/16384*a^8*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))/b^(11/2)+45/16384*a^6*(a+2*b*x)*sqrt(a*x+b*x^2)/b^5],
[x*(a*x+b*x^2)^(5/2),x,6,5/384*a^3*(a+2*b*x)*(a*x+b*x^2)^(3/2)/b^3-1/24*a*(a+2*b*x)*(a*x+b*x^2)^(5/2)/b^2+1/7*(a*x+b*x^2)^(7/2)/b+5/1024*a^7*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))/b^(9/2)-5/1024*a^5*(a+2*b*x)*sqrt(a*x+b*x^2)/b^4],
[(a*x+b*x^2)^(5/2),x,5,-5/192*a^2*(a+2*b*x)*(a*x+b*x^2)^(3/2)/b^2+1/12*(a+2*b*x)*(a*x+b*x^2)^(5/2)/b-5/512*a^6*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))/b^(7/2)+5/512*a^4*(a+2*b*x)*sqrt(a*x+b*x^2)/b^3],
[(a*x+b*x^2)^(5/2)/x,x,5,1/16*a*(a+2*b*x)*(a*x+b*x^2)^(3/2)/b+1/5*(a*x+b*x^2)^(5/2)+3/128*a^5*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))/b^(5/2)-3/128*a^3*(a+2*b*x)*sqrt(a*x+b*x^2)/b^2],
[(a*x+b*x^2)^(5/2)/x^2,x,5,5/24*a*(a*x+b*x^2)^(3/2)+1/4*(a*x+b*x^2)^(5/2)/x-5/64*a^4*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))/b^(3/2)+5/64*a^2*(a+2*b*x)*sqrt(a*x+b*x^2)/b],
[(a*x+b*x^2)^(5/2)/x^3,x,5,-5/3*b*(a*x+b*x^2)^(3/2)+2*(a*x+b*x^2)^(5/2)/x^2+5/8*a^3*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))/sqrt(b)-5/8*a*(a+2*b*x)*sqrt(a*x+b*x^2)],
[(a*x+b*x^2)^(5/2)/x^4,x,5,5/2*b*(a*x+b*x^2)^(3/2)/x-2*(a*x+b*x^2)^(5/2)/x^3+15/4*a^2*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))*sqrt(b)+15/4*a*b*sqrt(a*x+b*x^2)],
[(a*x+b*x^2)^(5/2)/x^5,x,5,-10/3*b*(a*x+b*x^2)^(3/2)/x^2-2/3*(a*x+b*x^2)^(5/2)/x^4+5*a*b^(3/2)*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))+5*b^2*sqrt(a*x+b*x^2)],
[(a*x+b*x^2)^(5/2)/x^6,x,5,-2/3*b*(a*x+b*x^2)^(3/2)/x^3-2/5*(a*x+b*x^2)^(5/2)/x^5+2*b^(5/2)*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))-2*b^2*sqrt(a*x+b*x^2)/x],
[(a*x+b*x^2)^(5/2)/x^7,x,1,-2/7*(a*x+b*x^2)^(7/2)/(a*x^7)],
[(a*x+b*x^2)^(5/2)/x^8,x,2,-2/9*(a*x+b*x^2)^(7/2)/(a*x^8)+4/63*b*(a*x+b*x^2)^(7/2)/(a^2*x^7)],
[(a*x+b*x^2)^(5/2)/x^9,x,3,-2/11*(a*x+b*x^2)^(7/2)/(a*x^9)+8/99*b*(a*x+b*x^2)^(7/2)/(a^2*x^8)-16/693*b^2*(a*x+b*x^2)^(7/2)/(a^3*x^7)],
[(a*x+b*x^2)^(5/2)/x^10,x,4,-2/13*(a*x+b*x^2)^(7/2)/(a*x^10)+12/143*b*(a*x+b*x^2)^(7/2)/(a^2*x^9)-16/429*b^2*(a*x+b*x^2)^(7/2)/(a^3*x^8)+32/3003*b^3*(a*x+b*x^2)^(7/2)/(a^4*x^7)],
[(a*x+b*x^2)^(5/2)/x^11,x,5,-2/15*(a*x+b*x^2)^(7/2)/(a*x^11)+16/195*b*(a*x+b*x^2)^(7/2)/(a^2*x^10)-32/715*b^2*(a*x+b*x^2)^(7/2)/(a^3*x^9)+128/6435*b^3*(a*x+b*x^2)^(7/2)/(a^4*x^8)-256/45045*b^4*(a*x+b*x^2)^(7/2)/(a^5*x^7)],
[(a*x+b*x^2)^(5/2)/x^12,x,6,-2/17*(a*x+b*x^2)^(7/2)/(a*x^12)+4/51*b*(a*x+b*x^2)^(7/2)/(a^2*x^11)-32/663*b^2*(a*x+b*x^2)^(7/2)/(a^3*x^10)+64/2431*b^3*(a*x+b*x^2)^(7/2)/(a^4*x^9)-256/21879*b^4*(a*x+b*x^2)^(7/2)/(a^5*x^8)+512/153153*b^5*(a*x+b*x^2)^(7/2)/(a^6*x^7)],
[x*sqrt(2*x-x^2),x,4,-1/3*(2*x-x^2)^(3/2)-1/2*arcsin(1-x)-1/2*(1-x)*sqrt(2*x-x^2)],
[x*sqrt(3*x-4*x^2),x,4,-1/12*(3*x-4*x^2)^(3/2)-27/512*arcsin(1-8/3*x)-3/128*(3-8*x)*sqrt(3*x-4*x^2)],
[x*sqrt(x+x^2),x,4,1/3*(x+x^2)^(3/2)+1/8*arctanh(x/sqrt(x+x^2))-1/8*(1+2*x)*sqrt(x+x^2)],

# p<0
[x^4/(b*x+c*x^2)^(1/2),x,6,35/64*b^4*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(9/2)-35/64*b^3*sqrt(b*x+c*x^2)/c^4+35/96*b^2*x*sqrt(b*x+c*x^2)/c^3-7/24*b*x^2*sqrt(b*x+c*x^2)/c^2+1/4*x^3*sqrt(b*x+c*x^2)/c],
[x^3/(b*x+c*x^2)^(1/2),x,5,-5/8*b^3*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(7/2)+5/8*b^2*sqrt(b*x+c*x^2)/c^3-5/12*b*x*sqrt(b*x+c*x^2)/c^2+1/3*x^2*sqrt(b*x+c*x^2)/c],
[x^2/(b*x+c*x^2)^(1/2),x,4,3/4*b^2*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(5/2)-3/4*b*sqrt(b*x+c*x^2)/c^2+1/2*x*sqrt(b*x+c*x^2)/c],
[x/(b*x+c*x^2)^(1/2),x,3,-b*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(3/2)+sqrt(b*x+c*x^2)/c],
[1/(b*x+c*x^2)^(1/2),x,2,2*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/sqrt(c)],
[1/(x*(b*x+c*x^2)^(1/2)),x,1,-2*sqrt(b*x+c*x^2)/(b*x)],
[1/(x^2*(b*x+c*x^2)^(1/2)),x,2,-2/3*sqrt(b*x+c*x^2)/(b*x^2)+4/3*c*sqrt(b*x+c*x^2)/(b^2*x)],
[1/(x^3*(b*x+c*x^2)^(1/2)),x,3,-2/5*sqrt(b*x+c*x^2)/(b*x^3)+8/15*c*sqrt(b*x+c*x^2)/(b^2*x^2)-16/15*c^2*sqrt(b*x+c*x^2)/(b^3*x)],
[1/(x^4*(b*x+c*x^2)^(1/2)),x,4,-2/7*sqrt(b*x+c*x^2)/(b*x^4)+12/35*c*sqrt(b*x+c*x^2)/(b^2*x^3)-16/35*c^2*sqrt(b*x+c*x^2)/(b^3*x^2)+32/35*c^3*sqrt(b*x+c*x^2)/(b^4*x)],
[1/(x^5*(b*x+c*x^2)^(1/2)),x,5,-2/9*sqrt(b*x+c*x^2)/(b*x^5)+16/63*c*sqrt(b*x+c*x^2)/(b^2*x^4)-32/105*c^2*sqrt(b*x+c*x^2)/(b^3*x^3)+128/315*c^3*sqrt(b*x+c*x^2)/(b^4*x^2)-256/315*c^4*sqrt(b*x+c*x^2)/(b^5*x)],
[x^4/(b*x+c*x^2)^(3/2),x,5,15/4*b^2*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(7/2)-2*x^3/(c*sqrt(b*x+c*x^2))-15/4*b*sqrt(b*x+c*x^2)/c^3+5/2*x*sqrt(b*x+c*x^2)/c^2],
[x^3/(b*x+c*x^2)^(3/2),x,4,-3*b*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(5/2)-2*x^2/(c*sqrt(b*x+c*x^2))+3*sqrt(b*x+c*x^2)/c^2],
[x^2/(b*x+c*x^2)^(3/2),x,3,2*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(3/2)-2*x/(c*sqrt(b*x+c*x^2))],
[x/(b*x+c*x^2)^(3/2),x,1,2*x/(b*sqrt(b*x+c*x^2))],
[1/(b*x+c*x^2)^(3/2),x,1,-2*(b+2*c*x)/(b^2*sqrt(b*x+c*x^2))],
[1/(x*(b*x+c*x^2)^(3/2)),x,2,(-2/3)/(b*x*sqrt(b*x+c*x^2))+8/3*c*(b+2*c*x)/(b^3*sqrt(b*x+c*x^2))],
[1/(x^2*(b*x+c*x^2)^(3/2)),x,3,(-2/5)/(b*x^2*sqrt(b*x+c*x^2))+4/5*c/(b^2*x*sqrt(b*x+c*x^2))-16/5*c^2*(b+2*c*x)/(b^4*sqrt(b*x+c*x^2))],
[1/(x^3*(b*x+c*x^2)^(3/2)),x,4,(-2/7)/(b*x^3*sqrt(b*x+c*x^2))+16/35*c/(b^2*x^2*sqrt(b*x+c*x^2))-32/35*c^2/(b^3*x*sqrt(b*x+c*x^2))+128/35*c^3*(b+2*c*x)/(b^5*sqrt(b*x+c*x^2))],
[x^6/(a*x+b*x^2)^(5/2),x,6,-2/3*x^5/(b*(a*x+b*x^2)^(3/2))+35/4*a^2*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))/b^(9/2)-14/3*x^3/(b^2*sqrt(a*x+b*x^2))-35/4*a*sqrt(a*x+b*x^2)/b^4+35/6*x*sqrt(a*x+b*x^2)/b^3],
[x^5/(a*x+b*x^2)^(5/2),x,5,-2/3*x^4/(b*(a*x+b*x^2)^(3/2))-5*a*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))/b^(7/2)-10/3*x^2/(b^2*sqrt(a*x+b*x^2))+5*sqrt(a*x+b*x^2)/b^3],
[x^4/(a*x+b*x^2)^(5/2),x,4,-2/3*x^3/(b*(a*x+b*x^2)^(3/2))+2*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))/b^(5/2)-2*x/(b^2*sqrt(a*x+b*x^2))],
[x^3/(a*x+b*x^2)^(5/2),x,1,2/3*x^3/(a*(a*x+b*x^2)^(3/2))],
[x^2/(a*x+b*x^2)^(5/2),x,2,-2/3*x/(b*(a*x+b*x^2)^(3/2))+2/3*(a+2*b*x)/(a^2*b*sqrt(a*x+b*x^2))],
[x/(a*x+b*x^2)^(5/2),x,2,2/3*x/(a*(a*x+b*x^2)^(3/2))-8/3*(a+2*b*x)/(a^3*sqrt(a*x+b*x^2))],
[1/(a*x+b*x^2)^(5/2),x,2,-2/3*(a+2*b*x)/(a^2*(a*x+b*x^2)^(3/2))+16/3*b*(a+2*b*x)/(a^4*sqrt(a*x+b*x^2))],
[1/(x*(a*x+b*x^2)^(5/2)),x,3,(-2/5)/(a*x*(a*x+b*x^2)^(3/2))+16/15*b*(a+2*b*x)/(a^3*(a*x+b*x^2)^(3/2))-128/15*b^2*(a+2*b*x)/(a^5*sqrt(a*x+b*x^2))],
[1/(x^2*(a*x+b*x^2)^(5/2)),x,4,(-2/7)/(a*x^2*(a*x+b*x^2)^(3/2))+4/7*b/(a^2*x*(a*x+b*x^2)^(3/2))-32/21*b^2*(a+2*b*x)/(a^4*(a*x+b*x^2)^(3/2))+256/21*b^3*(a+2*b*x)/(a^6*sqrt(a*x+b*x^2))],
[x/sqrt(4*x-x^2),x,3,-2*arcsin(1-1/2*x)-sqrt(4*x-x^2)],
[x/sqrt(-4*x+x^2),x,3,4*arctanh(x/sqrt(-4*x+x^2))+sqrt(-4*x+x^2)],
[x^2/sqrt(2*x-x^2),x,4,-3/2*arcsin(1-x)-3/2*sqrt(2*x-x^2)-1/2*x*sqrt(2*x-x^2)],

# Integrands of the form x^(m/2) (b x+c x^2)^(p/2)

# p>0
[x^(7/2)*(b*x+c*x^2)^(1/2),x,5,256/3465*b^4*(b*x+c*x^2)^(3/2)/(c^5*x^(3/2))-16/99*b*x^(3/2)*(b*x+c*x^2)^(3/2)/c^2+2/11*x^(5/2)*(b*x+c*x^2)^(3/2)/c-128/1155*b^3*(b*x+c*x^2)^(3/2)/(c^4*sqrt(x))+32/231*b^2*(b*x+c*x^2)^(3/2)*sqrt(x)/c^3],
[x^(5/2)*(b*x+c*x^2)^(1/2),x,4,-32/315*b^3*(b*x+c*x^2)^(3/2)/(c^4*x^(3/2))+2/9*x^(3/2)*(b*x+c*x^2)^(3/2)/c+16/105*b^2*(b*x+c*x^2)^(3/2)/(c^3*sqrt(x))-4/21*b*(b*x+c*x^2)^(3/2)*sqrt(x)/c^2],
[x^(3/2)*(b*x+c*x^2)^(1/2),x,3,16/105*b^2*(b*x+c*x^2)^(3/2)/(c^3*x^(3/2))-8/35*b*(b*x+c*x^2)^(3/2)/(c^2*sqrt(x))+2/7*(b*x+c*x^2)^(3/2)*sqrt(x)/c],
[x^(1/2)*(b*x+c*x^2)^(1/2),x,2,-4/15*b*(b*x+c*x^2)^(3/2)/(c^2*x^(3/2))+2/5*(b*x+c*x^2)^(3/2)/(c*sqrt(x))],
[(b*x+c*x^2)^(1/2)/x^(1/2),x,1,2/3*(b*x+c*x^2)^(3/2)/(c*x^(3/2))],
[(b*x+c*x^2)^(1/2)/x^(3/2),x,3,-2*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))*sqrt(b)+2*sqrt(b*x+c*x^2)/sqrt(x)],
[(b*x+c*x^2)^(1/2)/x^(5/2),x,3,-c*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))/sqrt(b)-sqrt(b*x+c*x^2)/x^(3/2)],
[(b*x+c*x^2)^(1/2)/x^(7/2),x,4,1/4*c^2*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))/b^(3/2)-1/2*sqrt(b*x+c*x^2)/x^(5/2)-1/4*c*sqrt(b*x+c*x^2)/(b*x^(3/2))],
[(b*x+c*x^2)^(1/2)/x^(9/2),x,5,-1/8*c^3*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))/b^(5/2)-1/3*sqrt(b*x+c*x^2)/x^(7/2)-1/12*c*sqrt(b*x+c*x^2)/(b*x^(5/2))+1/8*c^2*sqrt(b*x+c*x^2)/(b^2*x^(3/2))],
[(b*x+c*x^2)^(1/2)/x^(11/2),x,6,5/64*c^4*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))/b^(7/2)-1/4*sqrt(b*x+c*x^2)/x^(9/2)-1/24*c*sqrt(b*x+c*x^2)/(b*x^(7/2))+5/96*c^2*sqrt(b*x+c*x^2)/(b^2*x^(5/2))-5/64*c^3*sqrt(b*x+c*x^2)/(b^3*x^(3/2))],
[x^(7/2)*(b*x+c*x^2)^(3/2),x,6,-512/45045*b^5*(b*x+c*x^2)^(5/2)/(c^6*x^(5/2))+256/9009*b^4*(b*x+c*x^2)^(5/2)/(c^5*x^(3/2))-4/39*b*x^(3/2)*(b*x+c*x^2)^(5/2)/c^2+2/15*x^(5/2)*(b*x+c*x^2)^(5/2)/c-64/1287*b^3*(b*x+c*x^2)^(5/2)/(c^4*sqrt(x))+32/429*b^2*(b*x+c*x^2)^(5/2)*sqrt(x)/c^3],
[x^(5/2)*(b*x+c*x^2)^(3/2),x,5,256/15015*b^4*(b*x+c*x^2)^(5/2)/(c^5*x^(5/2))-128/3003*b^3*(b*x+c*x^2)^(5/2)/(c^4*x^(3/2))+2/13*x^(3/2)*(b*x+c*x^2)^(5/2)/c+32/429*b^2*(b*x+c*x^2)^(5/2)/(c^3*sqrt(x))-16/143*b*(b*x+c*x^2)^(5/2)*sqrt(x)/c^2],
[x^(3/2)*(b*x+c*x^2)^(3/2),x,4,-32/1155*b^3*(b*x+c*x^2)^(5/2)/(c^4*x^(5/2))+16/231*b^2*(b*x+c*x^2)^(5/2)/(c^3*x^(3/2))-4/33*b*(b*x+c*x^2)^(5/2)/(c^2*sqrt(x))+2/11*(b*x+c*x^2)^(5/2)*sqrt(x)/c],
[x^(1/2)*(b*x+c*x^2)^(3/2),x,3,16/315*b^2*(b*x+c*x^2)^(5/2)/(c^3*x^(5/2))-8/63*b*(b*x+c*x^2)^(5/2)/(c^2*x^(3/2))+2/9*(b*x+c*x^2)^(5/2)/(c*sqrt(x))],
[(b*x+c*x^2)^(3/2)/x^(1/2),x,2,-4/35*b*(b*x+c*x^2)^(5/2)/(c^2*x^(5/2))+2/7*(b*x+c*x^2)^(5/2)/(c*x^(3/2))],
[(b*x+c*x^2)^(3/2)/x^(3/2),x,1,2/5*(b*x+c*x^2)^(5/2)/(c*x^(5/2))],
[(b*x+c*x^2)^(3/2)/x^(5/2),x,4,2/3*(b*x+c*x^2)^(3/2)/x^(3/2)-2*b^(3/2)*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))+2*b*sqrt(b*x+c*x^2)/sqrt(x)],
[(b*x+c*x^2)^(3/2)/x^(7/2),x,4,-(b*x+c*x^2)^(3/2)/x^(5/2)-3*c*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))*sqrt(b)+3*c*sqrt(b*x+c*x^2)/sqrt(x)],
[(b*x+c*x^2)^(3/2)/x^(9/2),x,4,-1/2*(b*x+c*x^2)^(3/2)/x^(7/2)-3/4*c^2*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))/sqrt(b)-3/4*c*sqrt(b*x+c*x^2)/x^(3/2)],
[(b*x+c*x^2)^(3/2)/x^(11/2),x,5,-1/3*(b*x+c*x^2)^(3/2)/x^(9/2)+1/8*c^3*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))/b^(3/2)-1/4*c*sqrt(b*x+c*x^2)/x^(5/2)-1/8*c^2*sqrt(b*x+c*x^2)/(b*x^(3/2))],
[(b*x+c*x^2)^(3/2)/x^(13/2),x,6,-1/4*(b*x+c*x^2)^(3/2)/x^(11/2)-3/64*c^4*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))/b^(5/2)-1/8*c*sqrt(b*x+c*x^2)/x^(7/2)-1/32*c^2*sqrt(b*x+c*x^2)/(b*x^(5/2))+3/64*c^3*sqrt(b*x+c*x^2)/(b^2*x^(3/2))],
[(b*x+c*x^2)^(3/2)/x^(15/2),x,7,-1/5*(b*x+c*x^2)^(3/2)/x^(13/2)+3/128*c^5*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))/b^(7/2)-3/40*c*sqrt(b*x+c*x^2)/x^(9/2)-1/80*c^2*sqrt(b*x+c*x^2)/(b*x^(7/2))+1/64*c^3*sqrt(b*x+c*x^2)/(b^2*x^(5/2))-3/128*c^4*sqrt(b*x+c*x^2)/(b^3*x^(3/2))],

# p<0
[x^(7/2)/(b*x+c*x^2)^(1/2),x,4,-12/35*b*x^(3/2)*sqrt(b*x+c*x^2)/c^2+2/7*x^(5/2)*sqrt(b*x+c*x^2)/c-32/35*b^3*sqrt(b*x+c*x^2)/(c^4*sqrt(x))+16/35*b^2*sqrt(x)*sqrt(b*x+c*x^2)/c^3],
[x^(5/2)/(b*x+c*x^2)^(1/2),x,3,2/5*x^(3/2)*sqrt(b*x+c*x^2)/c+16/15*b^2*sqrt(b*x+c*x^2)/(c^3*sqrt(x))-8/15*b*sqrt(x)*sqrt(b*x+c*x^2)/c^2],
[x^(3/2)/(b*x+c*x^2)^(1/2),x,2,-4/3*b*sqrt(b*x+c*x^2)/(c^2*sqrt(x))+2/3*sqrt(x)*sqrt(b*x+c*x^2)/c],
[x^(1/2)/(b*x+c*x^2)^(1/2),x,1,2*sqrt(b*x+c*x^2)/(c*sqrt(x))],
[1/(x^(1/2)*(b*x+c*x^2)^(1/2)),x,2,-2*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))/sqrt(b)],
[1/(x^(3/2)*(b*x+c*x^2)^(1/2)),x,3,c*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))/b^(3/2)-sqrt(b*x+c*x^2)/(b*x^(3/2))],
[1/(x^(5/2)*(b*x+c*x^2)^(1/2)),x,4,-3/4*c^2*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))/b^(5/2)-1/2*sqrt(b*x+c*x^2)/(b*x^(5/2))+3/4*c*sqrt(b*x+c*x^2)/(b^2*x^(3/2))],
[1/(x^(7/2)*(b*x+c*x^2)^(1/2)),x,5,5/8*c^3*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))/b^(7/2)-1/3*sqrt(b*x+c*x^2)/(b*x^(7/2))+5/12*c*sqrt(b*x+c*x^2)/(b^2*x^(5/2))-5/8*c^2*sqrt(b*x+c*x^2)/(b^3*x^(3/2))],
[x^(13/2)/(b*x+c*x^2)^(3/2),x,6,256/63*b^4*x^(3/2)/(c^5*sqrt(b*x+c*x^2))-64/63*b^3*x^(5/2)/(c^4*sqrt(b*x+c*x^2))+32/63*b^2*x^(7/2)/(c^3*sqrt(b*x+c*x^2))-20/63*b*x^(9/2)/(c^2*sqrt(b*x+c*x^2))+2/9*x^(11/2)/(c*sqrt(b*x+c*x^2))+512/63*b^5*sqrt(x)/(c^6*sqrt(b*x+c*x^2))],
[x^(11/2)/(b*x+c*x^2)^(3/2),x,5,-128/35*b^3*x^(3/2)/(c^4*sqrt(b*x+c*x^2))+32/35*b^2*x^(5/2)/(c^3*sqrt(b*x+c*x^2))-16/35*b*x^(7/2)/(c^2*sqrt(b*x+c*x^2))+2/7*x^(9/2)/(c*sqrt(b*x+c*x^2))-256/35*b^4*sqrt(x)/(c^5*sqrt(b*x+c*x^2))],
[x^(9/2)/(b*x+c*x^2)^(3/2),x,4,16/5*b^2*x^(3/2)/(c^3*sqrt(b*x+c*x^2))-4/5*b*x^(5/2)/(c^2*sqrt(b*x+c*x^2))+2/5*x^(7/2)/(c*sqrt(b*x+c*x^2))+32/5*b^3*sqrt(x)/(c^4*sqrt(b*x+c*x^2))],
[x^(7/2)/(b*x+c*x^2)^(3/2),x,3,-8/3*b*x^(3/2)/(c^2*sqrt(b*x+c*x^2))+2/3*x^(5/2)/(c*sqrt(b*x+c*x^2))-16/3*b^2*sqrt(x)/(c^3*sqrt(b*x+c*x^2))],
[x^(5/2)/(b*x+c*x^2)^(3/2),x,2,2*x^(3/2)/(c*sqrt(b*x+c*x^2))+4*b*sqrt(x)/(c^2*sqrt(b*x+c*x^2))],
[x^(3/2)/(b*x+c*x^2)^(3/2),x,1,-2*sqrt(x)/(c*sqrt(b*x+c*x^2))],
[x^(1/2)/(b*x+c*x^2)^(3/2),x,3,-2*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))/b^(3/2)+2*sqrt(x)/(b*sqrt(b*x+c*x^2))],
[1/(x^(1/2)*(b*x+c*x^2)^(3/2)),x,4,3*c*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))/b^(5/2)+(-1)/(b*sqrt(x)*sqrt(b*x+c*x^2))-3*c*sqrt(x)/(b^2*sqrt(b*x+c*x^2))],
[1/(x^(3/2)*(b*x+c*x^2)^(3/2)),x,5,-15/4*c^2*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))/b^(7/2)+(-1/2)/(b*x^(3/2)*sqrt(b*x+c*x^2))+5/4*c/(b^2*sqrt(x)*sqrt(b*x+c*x^2))+15/4*c^2*sqrt(x)/(b^3*sqrt(b*x+c*x^2))],
[1/(x^(5/2)*(b*x+c*x^2)^(3/2)),x,6,35/8*c^3*arctanh(sqrt(b*x+c*x^2)/(sqrt(b)*sqrt(x)))/b^(9/2)+(-1/3)/(b*x^(5/2)*sqrt(b*x+c*x^2))+7/12*c/(b^2*x^(3/2)*sqrt(b*x+c*x^2))-35/24*c^2/(b^3*sqrt(x)*sqrt(b*x+c*x^2))-35/8*c^3*sqrt(x)/(b^4*sqrt(b*x+c*x^2))],

# Integrands of the form (d x)^m (b x+c x^2)^p when m symbolic
[(d*x)^m*(b*x+c*x^2)^3,x,3,b^3*(d*x)^(4+m)/(d^4*(4+m))+3*b^2*c*(d*x)^(5+m)/(d^5*(5+m))+3*b*c^2*(d*x)^(6+m)/(d^6*(6+m))+c^3*(d*x)^(7+m)/(d^7*(7+m))],
[(d*x)^m*(b*x+c*x^2)^2,x,3,b^2*(d*x)^(3+m)/(d^3*(3+m))+2*b*c*(d*x)^(4+m)/(d^4*(4+m))+c^2*(d*x)^(5+m)/(d^5*(5+m))],
[(d*x)^m*(b*x+c*x^2),x,2,b*(d*x)^(2+m)/(d^2*(2+m))+c*(d*x)^(3+m)/(d^3*(3+m))],
[(d*x)^m/(b*x+c*x^2),x,2,(d*x)^m*hypergeom([1,m],[1+m],-c*x/b)/(b*m)],
[(d*x)^m/(b*x+c*x^2)^2,x,2,-d*(d*x)^(-1+m)*hypergeom([2,-1+m],[m],-c*x/b)/(b^2*(1-m))],
[(d*x)^m/(b*x+c*x^2)^3,x,2,-d^2*(d*x)^(-2+m)*hypergeom([3,-2+m],[-1+m],-c*x/b)/(b^3*(2-m))],
[(d*x)^m*(b*x+c*x^2)^(5/2),x,3,2/7*b^2*(-c*x/b)^(-1/2-m)*(d*x)^m*(b+c*x)*(b*x+c*x^2)^(5/2)*hypergeom([7/2,-5/2-m],[9/2],1+c*x/b)/(c^3*x^2)],
[(d*x)^m*(b*x+c*x^2)^(3/2),x,3,-2/5*b*(-c*x/b)^(-1/2-m)*(d*x)^m*(b+c*x)*(b*x+c*x^2)^(3/2)*hypergeom([5/2,-3/2-m],[7/2],1+c*x/b)/(c^2*x)],
[(d*x)^m*(b*x+c*x^2)^(1/2),x,3,2/3*(-c*x/b)^(-1/2-m)*(d*x)^m*(b+c*x)*hypergeom([3/2,-1/2-m],[5/2],1+c*x/b)*sqrt(b*x+c*x^2)/c],
[(d*x)^m/(b*x+c*x^2)^(1/2),x,3,2*(-c*x/b)^(1/2-m)*(d*x)^m*(b+c*x)*hypergeom([1/2,1/2-m],[3/2],1+c*x/b)/(c*sqrt(b*x+c*x^2))],
[(d*x)^m/(b*x+c*x^2)^(3/2),x,3,2*x*(-c*x/b)^(1/2-m)*(d*x)^m*(b+c*x)*hypergeom([-1/2,3/2-m],[1/2],1+c*x/b)/(b*(b*x+c*x^2)^(3/2))],
[(d*x)^m/(b*x+c*x^2)^(5/2),x,3,-2/3*c*x^2*(-c*x/b)^(1/2-m)*(d*x)^m*(b+c*x)*hypergeom([-3/2,5/2-m],[-1/2],1+c*x/b)/(b^2*(b*x+c*x^2)^(5/2))],

# Integrands of the form (d x)^m (b x+c x^2)^p when p symbolic
[(d*x)^m*(b*x+c*x^2)^p,x,3,x*(d*x)^m*(b*x+c*x^2)^p*hypergeom([-p,1+m+p],[2+m+p],-c*x/b)/((1+m+p)*(1+c*x/b)^p)],
[x^3*(b*x+c*x^2)^p,x,3,x^4*(b*x+c*x^2)^p*hypergeom([-p,4+p],[5+p],-c*x/b)/((4+p)*(1+c*x/b)^p)],
[x^2*(b*x+c*x^2)^p,x,3,x^3*(b*x+c*x^2)^p*hypergeom([-p,3+p],[4+p],-c*x/b)/((3+p)*(1+c*x/b)^p)],
[x*(b*x+c*x^2)^p,x,2,x^2*(b*x+c*x^2)^p*hypergeom([-p,2+p],[3+p],-c*x/b)/((2+p)*(1+c*x/b)^p),1/2*(b*x+c*x^2)^(1+p)/(c*(1+p))+1/2*(-c*x/b)^(-1-p)*(b*x+c*x^2)^(1+p)*hypergeom([-p,1+p],[2+p],(b+c*x)/b)/(c*(1+p))],
[(b*x+c*x^2)^p/x,x,3,(b*x+c*x^2)^p*hypergeom([-p,p],[1+p],-c*x/b)/(p*(1+c*x/b)^p)],
[(b*x+c*x^2)^p/x^2,x,3,-(b*x+c*x^2)^p*hypergeom([-1+p,-p],[p],-c*x/b)/((1-p)*x*(1+c*x/b)^p)],
[(b*x+c*x^2)^p/x^3,x,3,-(b*x+c*x^2)^p*hypergeom([-2+p,-p],[-1+p],-c*x/b)/((2-p)*x^2*(1+c*x/b)^p)],
[(d*x)^(5/2)*(b*x+c*x^2)^p,x,3,2*x*(d*x)^(5/2)*(b*x+c*x^2)^p*hypergeom([-p,7/2+p],[9/2+p],-c*x/b)/((7+2*p)*(1+c*x/b)^p)],
[(d*x)^(3/2)*(b*x+c*x^2)^p,x,3,2*x*(d*x)^(3/2)*(b*x+c*x^2)^p*hypergeom([-p,5/2+p],[7/2+p],-c*x/b)/((5+2*p)*(1+c*x/b)^p)],
[(d*x)^(1/2)*(b*x+c*x^2)^p,x,3,2*x*(b*x+c*x^2)^p*hypergeom([-p,3/2+p],[5/2+p],-c*x/b)*sqrt(d*x)/((3+2*p)*(1+c*x/b)^p)],
[(b*x+c*x^2)^p/(d*x)^(1/2),x,3,2*x*(b*x+c*x^2)^p*hypergeom([-p,1/2+p],[3/2+p],-c*x/b)/((1+2*p)*(1+c*x/b)^p*sqrt(d*x))],
[(b*x+c*x^2)^p/(d*x)^(3/2),x,3,-2*x*(b*x+c*x^2)^p*hypergeom([-1/2+p,-p],[1/2+p],-c*x/b)/((1-2*p)*(d*x)^(3/2)*(1+c*x/b)^p)],
[(b*x+c*x^2)^p/(d*x)^(5/2),x,3,-2*x*(b*x+c*x^2)^p*hypergeom([-3/2+p,-p],[-1/2+p],-c*x/b)/((3-2*p)*(d*x)^(5/2)*(1+c*x/b)^p)],

# Integrands of the form (d x)^m (a+b x+c x^2)^p when b^2-4 a c=0

# Integrands of the form x^m (a^2+2 a b x+b^2 x^2)^p

# Integrands of the form x^m (a^2+2 a b x+b^2 x^2)^(p/2)

# p>0
[x^4*sqrt(a^2+2*a*b*x+b^2*x^2),x,3,1/5*a*x^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/6*b*x^6*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[x^3*sqrt(a^2+2*a*b*x+b^2*x^2),x,3,1/4*a*x^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/5*b*x^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[x^2*sqrt(a^2+2*a*b*x+b^2*x^2),x,3,1/3*a*x^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/4*b*x^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[x*sqrt(a^2+2*a*b*x+b^2*x^2),x,2,1/3*(a^2+2*a*b*x+b^2*x^2)^(3/2)/b^2-1/2*a*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/b^2],
[sqrt(a^2+2*a*b*x+b^2*x^2),x,1,1/2*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/b],
[sqrt(a^2+2*a*b*x+b^2*x^2)/x,x,3,b*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+a*log(x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[sqrt(a^2+2*a*b*x+b^2*x^2)/x^2,x,3,-a*sqrt(a^2+2*a*b*x+b^2*x^2)/(x*(a+b*x))+b*log(x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[sqrt(a^2+2*a*b*x+b^2*x^2)/x^3,x,2,-1/2*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a*x^2)],
[sqrt(a^2+2*a*b*x+b^2*x^2)/x^4,x,3,-1/3*a*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^3*(a+b*x))-1/2*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^2*(a+b*x))],
[sqrt(a^2+2*a*b*x+b^2*x^2)/x^5,x,3,-1/4*a*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^4*(a+b*x))-1/3*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^3*(a+b*x))],
[sqrt(a^2+2*a*b*x+b^2*x^2)/x^6,x,3,-1/5*a*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^5*(a+b*x))-1/4*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^4*(a+b*x))],
[x^5*(a^2+2*a*b*x+b^2*x^2)^(3/2),x,3,1/6*a^3*x^6*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+3/7*a^2*b*x^7*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+3/8*a*b^2*x^8*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/9*b^3*x^9*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[x^4*(a^2+2*a*b*x+b^2*x^2)^(3/2),x,3,1/5*a^3*x^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/2*a^2*b*x^6*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+3/7*a*b^2*x^7*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/8*b^3*x^8*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[x^3*(a^2+2*a*b*x+b^2*x^2)^(3/2),x,3,1/4*a^3*x^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+3/5*a^2*b*x^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/2*a*b^2*x^6*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/7*b^3*x^7*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[x^2*(a^2+2*a*b*x+b^2*x^2)^(3/2),x,2,1/4*a^2*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/b^3-2/5*a*(a^2+2*a*b*x+b^2*x^2)^(5/2)/b^3+1/6*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/b^3,1/4*a^2*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/b^3-2/5*a*(a+b*x)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/b^3+1/6*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/b^3],
[x*(a^2+2*a*b*x+b^2*x^2)^(3/2),x,2,-1/4*a*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/b^2+1/5*(a^2+2*a*b*x+b^2*x^2)^(5/2)/b^2],
[(a^2+2*a*b*x+b^2*x^2)^(3/2),x,1,1/4*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/b],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/x,x,3,3*a^2*b*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+3/2*a*b^2*x^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/3*b^3*x^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+a^3*log(x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/x^2,x,3,-a^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(x*(a+b*x))+3*a*b^2*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/2*b^3*x^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+3*a^2*b*log(x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/x^3,x,3,-1/2*a^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^2*(a+b*x))-3*a^2*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(x*(a+b*x))+b^3*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+3*a*b^2*log(x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/x^4,x,3,-1/3*a^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^3*(a+b*x))-3/2*a^2*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^2*(a+b*x))-3*a*b^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(x*(a+b*x))+b^3*log(x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/x^5,x,2,-1/4*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(a*x^4)],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/x^6,x,3,-1/5*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(a*x^5)+1/20*b*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(a^2*x^4)],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/x^7,x,3,-1/6*a^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^6*(a+b*x))-3/5*a^2*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^5*(a+b*x))-3/4*a*b^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^4*(a+b*x))-1/3*b^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^3*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/x^8,x,3,-1/7*a^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^7*(a+b*x))-1/2*a^2*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^6*(a+b*x))-3/5*a*b^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^5*(a+b*x))-1/4*b^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^4*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/x^9,x,3,-1/8*a^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^8*(a+b*x))-3/7*a^2*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^7*(a+b*x))-1/2*a*b^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^6*(a+b*x))-1/5*b^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^5*(a+b*x))],
[x^5*(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,1/6*a^5*x^6*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+5/7*a^4*b*x^7*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+5/4*a^3*b^2*x^8*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+10/9*a^2*b^3*x^9*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/2*a*b^4*x^10*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/11*b^5*x^11*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[x^4*(a^2+2*a*b*x+b^2*x^2)^(5/2),x,2,1/6*a^4*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/b^5-4/7*a^3*(a+b*x)^6*sqrt(a^2+2*a*b*x+b^2*x^2)/b^5+3/4*a^2*(a+b*x)^7*sqrt(a^2+2*a*b*x+b^2*x^2)/b^5-4/9*a*(a+b*x)^8*sqrt(a^2+2*a*b*x+b^2*x^2)/b^5+1/10*(a+b*x)^9*sqrt(a^2+2*a*b*x+b^2*x^2)/b^5],
[x^3*(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,-1/6*a^3*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/b^4+3/7*a^2*(a+b*x)^6*sqrt(a^2+2*a*b*x+b^2*x^2)/b^4-3/8*a*(a+b*x)^7*sqrt(a^2+2*a*b*x+b^2*x^2)/b^4+1/9*(a+b*x)^8*sqrt(a^2+2*a*b*x+b^2*x^2)/b^4],
[x^2*(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,1/6*a^2*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/b^3-2/7*a*(a+b*x)^6*sqrt(a^2+2*a*b*x+b^2*x^2)/b^3+1/8*(a+b*x)^7*sqrt(a^2+2*a*b*x+b^2*x^2)/b^3],
[x*(a^2+2*a*b*x+b^2*x^2)^(5/2),x,2,-1/6*a*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/b^2+1/7*(a^2+2*a*b*x+b^2*x^2)^(7/2)/b^2],
[(a^2+2*a*b*x+b^2*x^2)^(5/2),x,1,1/6*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/b],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/x,x,3,5*a^4*b*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+5*a^3*b^2*x^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+10/3*a^2*b^3*x^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+5/4*a*b^4*x^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/5*b^5*x^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+a^5*log(x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/x^2,x,3,-a^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(x*(a+b*x))+10*a^3*b^2*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+5*a^2*b^3*x^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+5/3*a*b^4*x^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/4*b^5*x^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+5*a^4*b*log(x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/x^3,x,3,-1/2*a^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^2*(a+b*x))-5*a^4*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(x*(a+b*x))+10*a^2*b^3*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+5/2*a*b^4*x^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/3*b^5*x^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+10*a^3*b^2*log(x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/x^4,x,3,-1/3*a^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^3*(a+b*x))-5/2*a^4*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^2*(a+b*x))-10*a^3*b^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(x*(a+b*x))+5*a*b^4*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+1/2*b^5*x^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+10*a^2*b^3*log(x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/x^5,x,3,-1/4*a^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^4*(a+b*x))-5/3*a^4*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^3*(a+b*x))-5*a^3*b^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^2*(a+b*x))-10*a^2*b^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(x*(a+b*x))+b^5*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)+5*a*b^4*log(x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/x^6,x,3,-1/5*a^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^5*(a+b*x))-5/4*a^4*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^4*(a+b*x))-10/3*a^3*b^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^3*(a+b*x))-5*a^2*b^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^2*(a+b*x))-5*a*b^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(x*(a+b*x))+b^5*log(x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(a+b*x)],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/x^7,x,2,-1/6*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(a*x^6)],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/x^8,x,3,-1/7*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(a*x^7)+1/42*b*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(a^2*x^6)],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/x^9,x,4,-1/8*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(a*x^8)+1/28*b*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(a^2*x^7)-1/168*b^2*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(a^3*x^6)],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/x^10,x,3,-1/9*a^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^9*(a+b*x))-5/8*a^4*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^8*(a+b*x))-10/7*a^3*b^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^7*(a+b*x))-5/3*a^2*b^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^6*(a+b*x))-a*b^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^5*(a+b*x))-1/4*b^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^4*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/x^11,x,3,-1/10*a^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^10*(a+b*x))-5/9*a^4*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^9*(a+b*x))-5/4*a^3*b^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^8*(a+b*x))-10/7*a^2*b^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^7*(a+b*x))-5/6*a*b^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^6*(a+b*x))-1/5*b^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^5*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/x^12,x,3,-1/11*a^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^11*(a+b*x))-1/2*a^4*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^10*(a+b*x))-10/9*a^3*b^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^9*(a+b*x))-5/4*a^2*b^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^8*(a+b*x))-5/7*a*b^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^7*(a+b*x))-1/6*b^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(x^6*(a+b*x))],

# p<0
[x^4/sqrt(a^2+2*a*b*x+b^2*x^2),x,3,-a^3*x*(a+b*x)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*a^2*x^2*(a+b*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))-1/3*a*x^3*(a+b*x)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))+1/4*x^4*(a+b*x)/(b*sqrt(a^2+2*a*b*x+b^2*x^2))+a^4*(a+b*x)*log(a+b*x)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))],
[x^3/sqrt(a^2+2*a*b*x+b^2*x^2),x,3,a^2*x*(a+b*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))-1/2*a*x^2*(a+b*x)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))+1/3*x^3*(a+b*x)/(b*sqrt(a^2+2*a*b*x+b^2*x^2))-a^3*(a+b*x)*log(a+b*x)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))],
[x^2/sqrt(a^2+2*a*b*x+b^2*x^2),x,3,-a*x*(a+b*x)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*x^2*(a+b*x)/(b*sqrt(a^2+2*a*b*x+b^2*x^2))+a^2*(a+b*x)*log(a+b*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))],
[x/sqrt(a^2+2*a*b*x+b^2*x^2),x,3,-a*(a+b*x)*log(a+b*x)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))+sqrt(a^2+2*a*b*x+b^2*x^2)/b^2],
[1/sqrt(a^2+2*a*b*x+b^2*x^2),x,2,(a+b*x)*log(a+b*x)/(b*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/(x*sqrt(a^2+2*a*b*x+b^2*x^2)),x,4,(a+b*x)*log(x)/(a*sqrt(a^2+2*a*b*x+b^2*x^2))-(a+b*x)*log(a+b*x)/(a*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/(x^2*sqrt(a^2+2*a*b*x+b^2*x^2)),x,3,(-a-b*x)/(a*x*sqrt(a^2+2*a*b*x+b^2*x^2))-b*(a+b*x)*log(x)/(a^2*sqrt(a^2+2*a*b*x+b^2*x^2))+b*(a+b*x)*log(a+b*x)/(a^2*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/(x^3*sqrt(a^2+2*a*b*x+b^2*x^2)),x,3,1/2*(-a-b*x)/(a*x^2*sqrt(a^2+2*a*b*x+b^2*x^2))+b*(a+b*x)/(a^2*x*sqrt(a^2+2*a*b*x+b^2*x^2))+b^2*(a+b*x)*log(x)/(a^3*sqrt(a^2+2*a*b*x+b^2*x^2))-b^2*(a+b*x)*log(a+b*x)/(a^3*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/(x^4*sqrt(a^2+2*a*b*x+b^2*x^2)),x,3,1/3*(-a-b*x)/(a*x^3*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*b*(a+b*x)/(a^2*x^2*sqrt(a^2+2*a*b*x+b^2*x^2))-b^2*(a+b*x)/(a^3*x*sqrt(a^2+2*a*b*x+b^2*x^2))-b^3*(a+b*x)*log(x)/(a^4*sqrt(a^2+2*a*b*x+b^2*x^2))+b^3*(a+b*x)*log(a+b*x)/(a^4*sqrt(a^2+2*a*b*x+b^2*x^2))],
[x^4/(a^2+2*a*b*x+b^2*x^2)^(3/2),x,3,4*a^3/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))-1/2*a^4/(b^5*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))-3*a*x*(a+b*x)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*x^2*(a+b*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))+6*a^2*(a+b*x)*log(a+b*x)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))],
[x^3/(a^2+2*a*b*x+b^2*x^2)^(3/2),x,3,-3*a^2/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*a^3/(b^4*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+x*(a+b*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))-3*a*(a+b*x)*log(a+b*x)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))],
[x^2/(a^2+2*a*b*x+b^2*x^2)^(3/2),x,3,2*a/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))-1/2*a^2/(b^3*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+(a+b*x)*log(a+b*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))],
[x/(a^2+2*a*b*x+b^2*x^2)^(3/2),x,2,(-1)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*a/(b^2*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/(a^2+2*a*b*x+b^2*x^2)^(3/2),x,1,(-1/2)/(b*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/(x*(a^2+2*a*b*x+b^2*x^2)^(3/2)),x,3,1/(a^2*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2/(a*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+(a+b*x)*log(x)/(a^3*sqrt(a^2+2*a*b*x+b^2*x^2))-(a+b*x)*log(a+b*x)/(a^3*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/(x^2*(a^2+2*a*b*x+b^2*x^2)^(3/2)),x,3,-2*b/(a^3*sqrt(a^2+2*a*b*x+b^2*x^2))-1/2*b/(a^2*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+(-a-b*x)/(a^3*x*sqrt(a^2+2*a*b*x+b^2*x^2))-3*b*(a+b*x)*log(x)/(a^4*sqrt(a^2+2*a*b*x+b^2*x^2))+3*b*(a+b*x)*log(a+b*x)/(a^4*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/(x^3*(a^2+2*a*b*x+b^2*x^2)^(3/2)),x,3,3*b^2/(a^4*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*b^2/(a^3*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*(-a-b*x)/(a^3*x^2*sqrt(a^2+2*a*b*x+b^2*x^2))+3*b*(a+b*x)/(a^4*x*sqrt(a^2+2*a*b*x+b^2*x^2))+6*b^2*(a+b*x)*log(x)/(a^5*sqrt(a^2+2*a*b*x+b^2*x^2))-6*b^2*(a+b*x)*log(a+b*x)/(a^5*sqrt(a^2+2*a*b*x+b^2*x^2))],
[x^6/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,20*a^3/(b^7*sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*a^6/(b^7*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))+2*a^5/(b^7*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-15/2*a^4/(b^7*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))-5*a*x*(a+b*x)/(b^6*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*x^2*(a+b*x)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))+15*a^2*(a+b*x)*log(a+b*x)/(b^7*sqrt(a^2+2*a*b*x+b^2*x^2))],
[x^5/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,-10*a^2/(b^6*sqrt(a^2+2*a*b*x+b^2*x^2))+1/4*a^5/(b^6*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-5/3*a^4/(b^6*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))+5*a^3/(b^6*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+x*(a+b*x)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))-5*a*(a+b*x)*log(a+b*x)/(b^6*sqrt(a^2+2*a*b*x+b^2*x^2))],
[x^4/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,4*a/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*a^4/(b^5*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))+4/3*a^3/(b^5*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-3*a^2/(b^5*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+(a+b*x)*log(a+b*x)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))],
[x^3/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,2,1/4*x^4/(a*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))],
[x^2/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,-1/4*a^2/(b^3*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))+2/3*a/(b^3*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))+(-1/2)/(b^3*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[x/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,2,(-1/3)/(b^2*(a^2+2*a*b*x+b^2*x^2)^(3/2))+1/4*a/(b^2*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2))],
[1/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,1,(-1/4)/(b*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2))],
[1/(x*(a^2+2*a*b*x+b^2*x^2)^(5/2)),x,3,1/(a^4*sqrt(a^2+2*a*b*x+b^2*x^2))+1/4/(a*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))+1/3/(a^2*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2/(a^3*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+(a+b*x)*log(x)/(a^5*sqrt(a^2+2*a*b*x+b^2*x^2))-(a+b*x)*log(a+b*x)/(a^5*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/(x^2*(a^2+2*a*b*x+b^2*x^2)^(5/2)),x,3,-4*b/(a^5*sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*b/(a^2*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-2/3*b/(a^3*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-3/2*b/(a^4*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+(-a-b*x)/(a^5*x*sqrt(a^2+2*a*b*x+b^2*x^2))-5*b*(a+b*x)*log(x)/(a^6*sqrt(a^2+2*a*b*x+b^2*x^2))+5*b*(a+b*x)*log(a+b*x)/(a^6*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/(x^3*(a^2+2*a*b*x+b^2*x^2)^(5/2)),x,3,10*b^2/(a^6*sqrt(a^2+2*a*b*x+b^2*x^2))+1/4*b^2/(a^3*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))+b^2/(a^4*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))+3*b^2/(a^5*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*(-a-b*x)/(a^5*x^2*sqrt(a^2+2*a*b*x+b^2*x^2))+5*b*(a+b*x)/(a^6*x*sqrt(a^2+2*a*b*x+b^2*x^2))+15*b^2*(a+b*x)*log(x)/(a^7*sqrt(a^2+2*a*b*x+b^2*x^2))-15*b^2*(a+b*x)*log(a+b*x)/(a^7*sqrt(a^2+2*a*b*x+b^2*x^2))],
[x*(9+12*x+4*x^2)^(5/2),x,2,-1/8*(3+2*x)*(9+12*x+4*x^2)^(5/2)+1/28*(9+12*x+4*x^2)^(7/2)],
[x*(9+12*x+4*x^2)^(3/2),x,2,-3/16*(3+2*x)*(9+12*x+4*x^2)^(3/2)+1/20*(9+12*x+4*x^2)^(5/2)],
[x*(9+12*x+4*x^2)^(1/2),x,2,1/12*(9+12*x+4*x^2)^(3/2)-3/8*(3+2*x)*sqrt(9+12*x+4*x^2)],
[x/(9+12*x+4*x^2)^(1/2),x,3,-3/4*(3+2*x)*log(3+2*x)/sqrt(9+12*x+4*x^2)+1/4*sqrt(9+12*x+4*x^2)],
[x/(9+12*x+4*x^2)^(3/2),x,2,(-1/4)/sqrt(9+12*x+4*x^2)+3/8/((3+2*x)*sqrt(9+12*x+4*x^2))],
[x/(9+12*x+4*x^2)^(5/2),x,2,(-1/12)/(9+12*x+4*x^2)^(3/2)+3/16/((3+2*x)*(9+12*x+4*x^2)^(3/2))],
[x/(9+12*x+4*x^2)^(7/2),x,2,(-1/20)/(9+12*x+4*x^2)^(5/2)+1/8/((3+2*x)*(9+12*x+4*x^2)^(5/2))],
[x/sqrt(4+12*x+9*x^2),x,3,-2/9*(2+3*x)*log(2+3*x)/sqrt(4+12*x+9*x^2)+1/9*sqrt(4+12*x+9*x^2)],
[x/sqrt(4-12*x+9*x^2),x,3,-2/9*(2-3*x)*log(2-3*x)/sqrt(4-12*x+9*x^2)+1/9*sqrt(4-12*x+9*x^2)],
[x/sqrt(-4+12*x-9*x^2),x,3,-2/9*(2-3*x)*log(2-3*x)/sqrt(-4+12*x-9*x^2)-1/9*sqrt(-4+12*x-9*x^2)],
[x/sqrt(-4-12*x-9*x^2),x,3,-2/9*(2+3*x)*log(2+3*x)/sqrt(-4-12*x-9*x^2)-1/9*sqrt(-4-12*x-9*x^2)],

# Integrands of the form (d x)^m (a+b x+c x^2)^p

# Integrands of the form (d+e x)^m (a+b x+c x^2)^p

# Integrands of the form (d+e x)^m (a+b x+c x^2)^p when a=0

# Integrands of the form (b d+2 c d x)^m (b x+c x^2)^p

# p>0

# p<0
[(1+x)/(2*x+x^2),x,1,1/2*log(2*x+x^2)],
[(a+2*b*x)/(a*x+b*x^2),x,1,log(a*x+b*x^2)],

# Integrands of the form (d+e x)^m (b x+c x^2)^p

# p>0
[(d+e*x)^4*(b*x+c*x^2),x,2,1/5*d*(c*d-b*e)*(d+e*x)^5/e^3-1/6*(2*c*d-b*e)*(d+e*x)^6/e^3+1/7*c*(d+e*x)^7/e^3],
[(d+e*x)^3*(b*x+c*x^2),x,2,1/4*d*(c*d-b*e)*(d+e*x)^4/e^3-1/5*(2*c*d-b*e)*(d+e*x)^5/e^3+1/6*c*(d+e*x)^6/e^3],
[(d+e*x)^2*(b*x+c*x^2),x,2,1/2*b*d^2*x^2+1/3*d*(c*d+2*b*e)*x^3+1/4*e*(2*c*d+b*e)*x^4+1/5*c*e^2*x^5],
[(d+e*x)*(b*x+c*x^2),x,2,1/2*b*d*x^2+1/3*(c*d+b*e)*x^3+1/4*c*e*x^4],
[b*x+c*x^2,x,1,1/2*b*x^2+1/3*c*x^3],
[(b*x+c*x^2)/(d+e*x),x,2,-(c*d-b*e)*x/e^2+1/2*c*x^2/e+d*(c*d-b*e)*log(d+e*x)/e^3],
[(b*x+c*x^2)/(d+e*x)^2,x,2,c*x/e^2-d*(c*d-b*e)/(e^3*(d+e*x))-(2*c*d-b*e)*log(d+e*x)/e^3],
[(b*x+c*x^2)/(d+e*x)^3,x,2,-1/2*d*(c*d-b*e)/(e^3*(d+e*x)^2)+(2*c*d-b*e)/(e^3*(d+e*x))+c*log(d+e*x)/e^3],
[(b*x+c*x^2)/(d+e*x)^4,x,2,-1/3*d*(c*d-b*e)/(e^3*(d+e*x)^3)+1/2*(2*c*d-b*e)/(e^3*(d+e*x)^2)-c/(e^3*(d+e*x))],
[(b*x+c*x^2)/(d+e*x)^5,x,2,-1/4*d*(c*d-b*e)/(e^3*(d+e*x)^4)+1/3*(2*c*d-b*e)/(e^3*(d+e*x)^3)-1/2*c/(e^3*(d+e*x)^2)],
[(d+e*x)^4*(b*x+c*x^2)^2,x,2,1/5*d^2*(c*d-b*e)^2*(d+e*x)^5/e^5-1/3*d*(c*d-b*e)*(2*c*d-b*e)*(d+e*x)^6/e^5+1/7*(6*c^2*d^2-6*b*c*d*e+b^2*e^2)*(d+e*x)^7/e^5-1/4*c*(2*c*d-b*e)*(d+e*x)^8/e^5+1/9*c^2*(d+e*x)^9/e^5],
[(d+e*x)^3*(b*x+c*x^2)^2,x,2,1/3*b^2*d^3*x^3+1/4*b*d^2*(2*c*d+3*b*e)*x^4+1/5*d*(c^2*d^2+6*b*c*d*e+3*b^2*e^2)*x^5+1/6*e*(3*c^2*d^2+6*b*c*d*e+b^2*e^2)*x^6+1/7*c*e^2*(3*c*d+2*b*e)*x^7+1/8*c^2*e^3*x^8],
[(d+e*x)^2*(b*x+c*x^2)^2,x,2,1/3*b^2*d^2*x^3+1/2*b*d*(c*d+b*e)*x^4+1/5*(c^2*d^2+4*b*c*d*e+b^2*e^2)*x^5+1/3*c*e*(c*d+b*e)*x^6+1/7*c^2*e^2*x^7],
[(d+e*x)*(b*x+c*x^2)^2,x,2,1/3*b^2*d*x^3+1/4*b*(2*c*d+b*e)*x^4+1/5*c*(c*d+2*b*e)*x^5+1/6*c^2*e*x^6],
[(b*x+c*x^2)^2,x,2,1/3*b^2*x^3+1/2*b*c*x^4+1/5*c^2*x^5],
[(b*x+c*x^2)^2/(d+e*x),x,2,-d*(c*d-b*e)^2*x/e^4+1/2*(c*d-b*e)^2*x^2/e^3-1/3*c*(c*d-2*b*e)*x^3/e^2+1/4*c^2*x^4/e+d^2*(c*d-b*e)^2*log(d+e*x)/e^5],
[(b*x+c*x^2)^2/(d+e*x)^2,x,2,(c*d-b*e)*(3*c*d-b*e)*x/e^4-c*(c*d-b*e)*x^2/e^3+1/3*c^2*x^3/e^2-d^2*(c*d-b*e)^2/(e^5*(d+e*x))-2*d*(c*d-b*e)*(2*c*d-b*e)*log(d+e*x)/e^5],
[(b*x+c*x^2)^2/(d+e*x)^3,x,2,-c*(3*c*d-2*b*e)*x/e^4+1/2*c^2*x^2/e^3-1/2*d^2*(c*d-b*e)^2/(e^5*(d+e*x)^2)+2*d*(c*d-b*e)*(2*c*d-b*e)/(e^5*(d+e*x))+(6*c^2*d^2-6*b*c*d*e+b^2*e^2)*log(d+e*x)/e^5],
[(b*x+c*x^2)^2/(d+e*x)^4,x,2,c^2*x/e^4-1/3*d^2*(c*d-b*e)^2/(e^5*(d+e*x)^3)+d*(c*d-b*e)*(2*c*d-b*e)/(e^5*(d+e*x)^2)+(-6*c^2*d^2+6*b*c*d*e-b^2*e^2)/(e^5*(d+e*x))-2*c*(2*c*d-b*e)*log(d+e*x)/e^5],
[(b*x+c*x^2)^2/(d+e*x)^5,x,2,-1/4*d^2*(c*d-b*e)^2/(e^5*(d+e*x)^4)+2/3*d*(c*d-b*e)*(2*c*d-b*e)/(e^5*(d+e*x)^3)+1/2*(-6*c^2*d^2+6*b*c*d*e-b^2*e^2)/(e^5*(d+e*x)^2)+2*c*(2*c*d-b*e)/(e^5*(d+e*x))+c^2*log(d+e*x)/e^5],
[(b*x+c*x^2)^2/(d+e*x)^6,x,2,-1/5*d^2*(c*d-b*e)^2/(e^5*(d+e*x)^5)+1/2*d*(c*d-b*e)*(2*c*d-b*e)/(e^5*(d+e*x)^4)+1/3*(-6*c^2*d^2+6*b*c*d*e-b^2*e^2)/(e^5*(d+e*x)^3)+c*(2*c*d-b*e)/(e^5*(d+e*x)^2)-c^2/(e^5*(d+e*x))],
[(b*x+c*x^2)^2/(d+e*x)^7,x,2,-1/6*d^2*(c*d-b*e)^2/(e^5*(d+e*x)^6)+2/5*d*(c*d-b*e)*(2*c*d-b*e)/(e^5*(d+e*x)^5)+1/4*(-6*c^2*d^2+6*b*c*d*e-b^2*e^2)/(e^5*(d+e*x)^4)+2/3*c*(2*c*d-b*e)/(e^5*(d+e*x)^3)-1/2*c^2/(e^5*(d+e*x)^2)],
[(b*x+c*x^2)^2/(d+e*x)^8,x,2,-1/7*d^2*(c*d-b*e)^2/(e^5*(d+e*x)^7)+1/3*d*(c*d-b*e)*(2*c*d-b*e)/(e^5*(d+e*x)^6)+1/5*(-6*c^2*d^2+6*b*c*d*e-b^2*e^2)/(e^5*(d+e*x)^5)+1/2*c*(2*c*d-b*e)/(e^5*(d+e*x)^4)-1/3*c^2/(e^5*(d+e*x)^3)],
[(d+e*x)^4*(b*x+c*x^2)^3,x,2,1/4*b^3*d^4*x^4+1/5*b^2*d^3*(3*c*d+4*b*e)*x^5+1/2*b*d^2*(c^2*d^2+4*b*c*d*e+2*b^2*e^2)*x^6+1/7*d*(c^3*d^3+12*b*c^2*d^2*e+18*b^2*c*d*e^2+4*b^3*e^3)*x^7+1/8*e*(4*c^3*d^3+18*b*c^2*d^2*e+12*b^2*c*d*e^2+b^3*e^3)*x^8+1/3*c*e^2*(2*c^2*d^2+4*b*c*d*e+b^2*e^2)*x^9+1/10*c^2*e^3*(4*c*d+3*b*e)*x^10+1/11*c^3*e^4*x^11],
[(d+e*x)^3*(b*x+c*x^2)^3,x,2,1/4*b^3*d^3*x^4+3/5*b^2*d^2*(c*d+b*e)*x^5+1/2*b*d*(c^2*d^2+3*b*c*d*e+b^2*e^2)*x^6+1/7*(c*d+b*e)*(c^2*d^2+8*b*c*d*e+b^2*e^2)*x^7+3/8*c*e*(c^2*d^2+3*b*c*d*e+b^2*e^2)*x^8+1/3*c^2*e^2*(c*d+b*e)*x^9+1/10*c^3*e^3*x^10],
[(d+e*x)^2*(b*x+c*x^2)^3,x,2,1/4*b^3*d^2*x^4+1/5*b^2*d*(3*c*d+2*b*e)*x^5+1/6*b*(3*c^2*d^2+6*b*c*d*e+b^2*e^2)*x^6+1/7*c*(c^2*d^2+6*b*c*d*e+3*b^2*e^2)*x^7+1/8*c^2*e*(2*c*d+3*b*e)*x^8+1/9*c^3*e^2*x^9],
[(d+e*x)*(b*x+c*x^2)^3,x,2,1/4*b^3*d*x^4+1/5*b^2*(3*c*d+b*e)*x^5+1/2*b*c*(c*d+b*e)*x^6+1/7*c^2*(c*d+3*b*e)*x^7+1/8*c^3*e*x^8],
[(b*x+c*x^2)^3,x,2,1/4*b^3*x^4+3/5*b^2*c*x^5+1/2*b*c^2*x^6+1/7*c^3*x^7],
[(b*x+c*x^2)^3/(d+e*x),x,2,-d^2*(c*d-b*e)^3*x/e^6+1/2*d*(c*d-b*e)^3*x^2/e^5-1/3*(c*d-b*e)^3*x^3/e^4+1/4*c*(c^2*d^2-3*b*c*d*e+3*b^2*e^2)*x^4/e^3-1/5*c^2*(c*d-3*b*e)*x^5/e^2+1/6*c^3*x^6/e+d^3*(c*d-b*e)^3*log(d+e*x)/e^7],
[(b*x+c*x^2)^3/(d+e*x)^2,x,2,d*(5*c*d-2*b*e)*(c*d-b*e)^2*x/e^6-1/2*(c*d-b*e)^2*(4*c*d-b*e)*x^2/e^5+c*(c*d-b*e)^2*x^3/e^4-1/4*c^2*(2*c*d-3*b*e)*x^4/e^3+1/5*c^3*x^5/e^2-d^3*(c*d-b*e)^3/(e^7*(d+e*x))-3*d^2*(c*d-b*e)^2*(2*c*d-b*e)*log(d+e*x)/e^7],
[(b*x+c*x^2)^3/(d+e*x)^3,x,2,-(c*d-b*e)*(10*c^2*d^2-8*b*c*d*e+b^2*e^2)*x/e^6+3/2*c*(c*d-b*e)*(2*c*d-b*e)*x^2/e^5-c^2*(c*d-b*e)*x^3/e^4+1/4*c^3*x^4/e^3-1/2*d^3*(c*d-b*e)^3/(e^7*(d+e*x)^2)+3*d^2*(c*d-b*e)^2*(2*c*d-b*e)/(e^7*(d+e*x))+3*d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*log(d+e*x)/e^7],
[(b*x+c*x^2)^3/(d+e*x)^4,x,2,c*(10*c^2*d^2-12*b*c*d*e+3*b^2*e^2)*x/e^6-1/2*c^2*(4*c*d-3*b*e)*x^2/e^5+1/3*c^3*x^3/e^4-1/3*d^3*(c*d-b*e)^3/(e^7*(d+e*x)^3)+3/2*d^2*(c*d-b*e)^2*(2*c*d-b*e)/(e^7*(d+e*x)^2)-3*d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)/(e^7*(d+e*x))-(2*c*d-b*e)*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)*log(d+e*x)/e^7],
[(b*x+c*x^2)^3/(d+e*x)^5,x,2,-c^2*(5*c*d-3*b*e)*x/e^6+1/2*c^3*x^2/e^5-1/4*d^3*(c*d-b*e)^3/(e^7*(d+e*x)^4)+d^2*(c*d-b*e)^2*(2*c*d-b*e)/(e^7*(d+e*x)^3)-3/2*d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)/(e^7*(d+e*x)^2)+(2*c*d-b*e)*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)/(e^7*(d+e*x))+3*c*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*log(d+e*x)/e^7],
[(b*x+c*x^2)^3/(d+e*x)^6,x,2,c^3*x/e^6-1/5*d^3*(c*d-b*e)^3/(e^7*(d+e*x)^5)+3/4*d^2*(c*d-b*e)^2*(2*c*d-b*e)/(e^7*(d+e*x)^4)-d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)/(e^7*(d+e*x)^3)+1/2*(2*c*d-b*e)*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)/(e^7*(d+e*x)^2)-3*c*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)/(e^7*(d+e*x))-3*c^2*(2*c*d-b*e)*log(d+e*x)/e^7],
[(b*x+c*x^2)^3/(d+e*x)^7,x,2,-1/6*d^3*(c*d-b*e)^3/(e^7*(d+e*x)^6)+3/5*d^2*(c*d-b*e)^2*(2*c*d-b*e)/(e^7*(d+e*x)^5)-3/4*d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)/(e^7*(d+e*x)^4)+1/3*(2*c*d-b*e)*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)/(e^7*(d+e*x)^3)-3/2*c*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)/(e^7*(d+e*x)^2)+3*c^2*(2*c*d-b*e)/(e^7*(d+e*x))+c^3*log(d+e*x)/e^7],
[(b*x+c*x^2)^3/(d+e*x)^8,x,2,-1/7*d^3*(c*d-b*e)^3/(e^7*(d+e*x)^7)+1/2*d^2*(c*d-b*e)^2*(2*c*d-b*e)/(e^7*(d+e*x)^6)-3/5*d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)/(e^7*(d+e*x)^5)+1/4*(2*c*d-b*e)*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)/(e^7*(d+e*x)^4)-c*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)/(e^7*(d+e*x)^3)+3/2*c^2*(2*c*d-b*e)/(e^7*(d+e*x)^2)-c^3/(e^7*(d+e*x))],
[(b*x+c*x^2)^3/(d+e*x)^9,x,2,-1/8*d^3*(c*d-b*e)^3/(e^7*(d+e*x)^8)+3/7*d^2*(c*d-b*e)^2*(2*c*d-b*e)/(e^7*(d+e*x)^7)-1/2*d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)/(e^7*(d+e*x)^6)+1/5*(2*c*d-b*e)*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)/(e^7*(d+e*x)^5)-3/4*c*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)/(e^7*(d+e*x)^4)+c^2*(2*c*d-b*e)/(e^7*(d+e*x)^3)-1/2*c^3/(e^7*(d+e*x)^2)],
[(b*x+c*x^2)^3/(d+e*x)^10,x,2,-1/9*d^3*(c*d-b*e)^3/(e^7*(d+e*x)^9)+3/8*d^2*(c*d-b*e)^2*(2*c*d-b*e)/(e^7*(d+e*x)^8)-3/7*d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)/(e^7*(d+e*x)^7)+1/6*(2*c*d-b*e)*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)/(e^7*(d+e*x)^6)-3/5*c*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)/(e^7*(d+e*x)^5)+3/4*c^2*(2*c*d-b*e)/(e^7*(d+e*x)^4)-1/3*c^3/(e^7*(d+e*x)^3)],

# p<0
[(d+e*x)^4/(b*x+c*x^2),x,2,e^2*(6*c^2*d^2-4*b*c*d*e+b^2*e^2)*x/c^3+1/2*e^3*(4*c*d-b*e)*x^2/c^2+1/3*e^4*x^3/c+d^4*log(x)/b-(c*d-b*e)^4*log(b+c*x)/(b*c^4)],
[(d+e*x)^3/(b*x+c*x^2),x,2,e^2*(3*c*d-b*e)*x/c^2+1/2*e^3*x^2/c+d^3*log(x)/b-(c*d-b*e)^3*log(b+c*x)/(b*c^3)],
[(d+e*x)^2/(b*x+c*x^2),x,2,e^2*x/c+d^2*log(x)/b-(c*d-b*e)^2*log(b+c*x)/(b*c^2)],
[(d+e*x)/(b*x+c*x^2),x,2,d*log(x)/b-(c*d-b*e)*log(b+c*x)/(b*c)],
[1/(b*x+c*x^2),x,1,log(x)/b-log(b+c*x)/b],
[1/((d+e*x)*(b*x+c*x^2)),x,2,log(x)/(b*d)-c*log(b+c*x)/(b*(c*d-b*e))+e*log(d+e*x)/(d*(c*d-b*e))],
[1/((d+e*x)^2*(b*x+c*x^2)),x,2,-e/(d*(c*d-b*e)*(d+e*x))+log(x)/(b*d^2)-c^2*log(b+c*x)/(b*(c*d-b*e)^2)+e*(2*c*d-b*e)*log(d+e*x)/(d^2*(c*d-b*e)^2)],
[1/((d+e*x)^3*(b*x+c*x^2)),x,2,-1/2*e/(d*(c*d-b*e)*(d+e*x)^2)-e*(2*c*d-b*e)/(d^2*(c*d-b*e)^2*(d+e*x))+log(x)/(b*d^3)-c^3*log(b+c*x)/(b*(c*d-b*e)^3)+e*(3*c^2*d^2-3*b*c*d*e+b^2*e^2)*log(d+e*x)/(d^3*(c*d-b*e)^3)],
[(d+e*x)^5/(b*x+c*x^2)^2,x,2,-d^5/(b^2*x)+e^4*(5*c*d-2*b*e)*x/c^3+1/2*e^5*x^2/c^2-(c*d-b*e)^5/(b^2*c^4*(b+c*x))-d^4*(2*c*d-5*b*e)*log(x)/b^3+(c*d-b*e)^4*(2*c*d+3*b*e)*log(b+c*x)/(b^3*c^4)],
[(d+e*x)^4/(b*x+c*x^2)^2,x,2,-d^4/(b^2*x)+e^4*x/c^2-(c*d-b*e)^4/(b^2*c^3*(b+c*x))-2*d^3*(c*d-2*b*e)*log(x)/b^3+2*(c*d-b*e)^3*(c*d+b*e)*log(b+c*x)/(b^3*c^3)],
[(d+e*x)^3/(b*x+c*x^2)^2,x,2,-d^3/(b^2*x)-(c*d-b*e)^3/(b^2*c^2*(b+c*x))-d^2*(2*c*d-3*b*e)*log(x)/b^3+(c*d-b*e)^2*(2*c*d+b*e)*log(b+c*x)/(b^3*c^2)],
[(d+e*x)^2/(b*x+c*x^2)^2,x,2,-d^2/(b^2*x)-(c*d-b*e)^2/(b^2*c*(b+c*x))-2*d*(c*d-b*e)*log(x)/b^3+2*d*(c*d-b*e)*log(b+c*x)/b^3],
[(d+e*x)/(b*x+c*x^2)^2,x,2,-d/(b^2*x)+(-c*d+b*e)/(b^2*(b+c*x))-(2*c*d-b*e)*log(x)/b^3+(2*c*d-b*e)*log(b+c*x)/b^3],
[1/(b*x+c*x^2)^2,x,2,(-b-2*c*x)/(b^2*(b*x+c*x^2))-2*c*log(x)/b^3+2*c*log(b+c*x)/b^3],
[1/((d+e*x)*(b*x+c*x^2)^2),x,2,(-1)/(b^2*d*x)-c^2/(b^2*(c*d-b*e)*(b+c*x))-(2*c*d+b*e)*log(x)/(b^3*d^2)+c^2*(2*c*d-3*b*e)*log(b+c*x)/(b^3*(c*d-b*e)^2)+e^3*log(d+e*x)/(d^2*(c*d-b*e)^2)],
[1/((d+e*x)^2*(b*x+c*x^2)^2),x,2,(-1)/(b^2*d^2*x)-c^3/(b^2*(c*d-b*e)^2*(b+c*x))-e^3/(d^2*(c*d-b*e)^2*(d+e*x))-2*(c*d+b*e)*log(x)/(b^3*d^3)+2*c^3*(c*d-2*b*e)*log(b+c*x)/(b^3*(c*d-b*e)^3)+2*e^3*(2*c*d-b*e)*log(d+e*x)/(d^3*(c*d-b*e)^3)],
[(d+e*x)^7/(b*x+c*x^2)^3,x,2,-1/2*d^7/(b^3*x^2)+d^6*(3*c*d-7*b*e)/(b^4*x)+e^6*(7*c*d-3*b*e)*x/c^4+1/2*e^7*x^2/c^3+1/2*(c*d-b*e)^7/(b^3*c^5*(b+c*x)^2)+(c*d-b*e)^6*(3*c*d+4*b*e)/(b^4*c^5*(b+c*x))+3*d^5*(2*c^2*d^2-7*b*c*d*e+7*b^2*e^2)*log(x)/b^5-3*(c*d-b*e)^5*(2*c^2*d^2+3*b*c*d*e+2*b^2*e^2)*log(b+c*x)/(b^5*c^5)],
[(d+e*x)^6/(b*x+c*x^2)^3,x,2,-1/2*d^6/(b^3*x^2)+3*d^5*(c*d-2*b*e)/(b^4*x)+e^6*x/c^3+1/2*(c*d-b*e)^6/(b^3*c^4*(b+c*x)^2)+3*(c*d-b*e)^5*(c*d+b*e)/(b^4*c^4*(b+c*x))+3*d^4*(2*c^2*d^2-6*b*c*d*e+5*b^2*e^2)*log(x)/b^5-3*(c*d-b*e)^4*(2*c^2*d^2+2*b*c*d*e+b^2*e^2)*log(b+c*x)/(b^5*c^4)],
[(d+e*x)^5/(b*x+c*x^2)^3,x,2,-1/2*d^5/(b^3*x^2)+d^4*(3*c*d-5*b*e)/(b^4*x)+1/2*(c*d-b*e)^5/(b^3*c^3*(b+c*x)^2)+(c*d-b*e)^4*(3*c*d+2*b*e)/(b^4*c^3*(b+c*x))+d^3*(6*c^2*d^2-15*b*c*d*e+10*b^2*e^2)*log(x)/b^5-(c*d-b*e)^3*(6*c^2*d^2+3*b*c*d*e+b^2*e^2)*log(b+c*x)/(b^5*c^3)],
[(d+e*x)^4/(b*x+c*x^2)^3,x,2,-1/2*d^4/(b^3*x^2)+d^3*(3*c*d-4*b*e)/(b^4*x)+1/2*(c*d-b*e)^4/(b^3*c^2*(b+c*x)^2)+(c*d-b*e)^3*(3*c*d+b*e)/(b^4*c^2*(b+c*x))+6*d^2*(c*d-b*e)^2*log(x)/b^5-6*d^2*(c*d-b*e)^2*log(b+c*x)/b^5],
[(d+e*x)^3/(b*x+c*x^2)^3,x,2,-1/2*d^3/(b^3*x^2)+3*d^2*(c*d-b*e)/(b^4*x)+1/2*(c*d-b*e)^3/(b^3*c*(b+c*x)^2)+3*d*(c*d-b*e)^2/(b^4*(b+c*x))+3*d*(c*d-b*e)*(2*c*d-b*e)*log(x)/b^5-3*d*(c*d-b*e)*(2*c*d-b*e)*log(b+c*x)/b^5],
[(d+e*x)^2/(b*x+c*x^2)^3,x,2,-1/2*d^2/(b^3*x^2)+d*(3*c*d-2*b*e)/(b^4*x)+1/2*(c*d-b*e)^2/(b^3*(b+c*x)^2)+(c*d-b*e)*(3*c*d-b*e)/(b^4*(b+c*x))+(6*c^2*d^2-6*b*c*d*e+b^2*e^2)*log(x)/b^5-(6*c^2*d^2-6*b*c*d*e+b^2*e^2)*log(b+c*x)/b^5],
[(d+e*x)/(b*x+c*x^2)^3,x,2,-1/2*d/(b^3*x^2)+(3*c*d-b*e)/(b^4*x)+1/2*c*(c*d-b*e)/(b^3*(b+c*x)^2)+c*(3*c*d-2*b*e)/(b^4*(b+c*x))+3*c*(2*c*d-b*e)*log(x)/b^5-3*c*(2*c*d-b*e)*log(b+c*x)/b^5],
[1/(b*x+c*x^2)^3,x,3,1/2*(-b-2*c*x)/(b^2*(b*x+c*x^2)^2)+3*c*(b+2*c*x)/(b^4*(b*x+c*x^2))+6*c^2*log(x)/b^5-6*c^2*log(b+c*x)/b^5],
[1/((d+e*x)*(b*x+c*x^2)^3),x,2,(-1/2)/(b^3*d*x^2)+(3*c*d+b*e)/(b^4*d^2*x)+1/2*c^3/(b^3*(c*d-b*e)*(b+c*x)^2)+c^3*(3*c*d-4*b*e)/(b^4*(c*d-b*e)^2*(b+c*x))+(6*c^2*d^2+3*b*c*d*e+b^2*e^2)*log(x)/(b^5*d^3)-c^3*(6*c^2*d^2-15*b*c*d*e+10*b^2*e^2)*log(b+c*x)/(b^5*(c*d-b*e)^3)+e^5*log(d+e*x)/(d^3*(c*d-b*e)^3)],
[1/((d+e*x)^2*(b*x+c*x^2)^3),x,2,(-1/2)/(b^3*d^2*x^2)+(3*c*d+2*b*e)/(b^4*d^3*x)+1/2*c^4/(b^3*(c*d-b*e)^2*(b+c*x)^2)+c^4*(3*c*d-5*b*e)/(b^4*(c*d-b*e)^3*(b+c*x))-e^5/(d^3*(c*d-b*e)^3*(d+e*x))+3*(2*c^2*d^2+2*b*c*d*e+b^2*e^2)*log(x)/(b^5*d^4)-3*c^4*(2*c^2*d^2-6*b*c*d*e+5*b^2*e^2)*log(b+c*x)/(b^5*(c*d-b*e)^4)+3*e^5*(2*c*d-b*e)*log(d+e*x)/(d^4*(c*d-b*e)^4)],

# Integrands of the form (d+e x)^m (b x+c x^2)^(p/2)

# p>0
[(d+e*x)^3*(b*x+c*x^2)^(1/2),x,5,1/5*e*(d+e*x)^2*(b*x+c*x^2)^(3/2)/c+1/240*e*(192*c^2*d^2-150*b*c*d*e+35*b^2*e^2+42*c*e*(2*c*d-b*e)*x)*(b*x+c*x^2)^(3/2)/c^3-1/128*b^2*(2*c*d-b*e)*(16*c^2*d^2-16*b*c*d*e+7*b^2*e^2)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(9/2)+1/128*(2*c*d-b*e)*(16*c^2*d^2-16*b*c*d*e+7*b^2*e^2)*(b+2*c*x)*sqrt(b*x+c*x^2)/c^4],
[(d+e*x)^2*(b*x+c*x^2)^(1/2),x,5,5/24*e*(2*c*d-b*e)*(b*x+c*x^2)^(3/2)/c^2+1/4*e*(d+e*x)*(b*x+c*x^2)^(3/2)/c-1/64*b^2*(16*c^2*d^2-16*b*c*d*e+5*b^2*e^2)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(7/2)+1/64*(16*c^2*d^2-16*b*c*d*e+5*b^2*e^2)*(b+2*c*x)*sqrt(b*x+c*x^2)/c^3],
[(d+e*x)*(b*x+c*x^2)^(1/2),x,4,1/3*e*(b*x+c*x^2)^(3/2)/c-1/8*b^2*(2*c*d-b*e)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(5/2)+1/8*(2*c*d-b*e)*(b+2*c*x)*sqrt(b*x+c*x^2)/c^2],
[(b*x+c*x^2)^(1/2),x,3,-1/4*b^2*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(3/2)+1/4*(b+2*c*x)*sqrt(b*x+c*x^2)/c],
[(b*x+c*x^2)^(1/2)/(d+e*x),x,6,-(2*c*d-b*e)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/(e^2*sqrt(c))+arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))*sqrt(d)*sqrt(c*d-b*e)/e^2+sqrt(b*x+c*x^2)/e],
[(b*x+c*x^2)^(1/2)/(d+e*x)^2,x,6,2*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))*sqrt(c)/e^2-1/2*(2*c*d-b*e)*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/(e^2*sqrt(d)*sqrt(c*d-b*e))-sqrt(b*x+c*x^2)/(e*(d+e*x))],
[(b*x+c*x^2)^(1/2)/(d+e*x)^3,x,3,-1/8*b^2*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/(d^(3/2)*(c*d-b*e)^(3/2))+1/4*(b*d+(2*c*d-b*e)*x)*sqrt(b*x+c*x^2)/(d*(c*d-b*e)*(d+e*x)^2)],
[(b*x+c*x^2)^(1/2)/(d+e*x)^4,x,4,-1/3*e*(b*x+c*x^2)^(3/2)/(d*(c*d-b*e)*(d+e*x)^3)-1/16*b^2*(2*c*d-b*e)*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/(d^(5/2)*(c*d-b*e)^(5/2))+1/8*(2*c*d-b*e)*(b*d+(2*c*d-b*e)*x)*sqrt(b*x+c*x^2)/(d^2*(c*d-b*e)^2*(d+e*x)^2)],
[(b*x+c*x^2)^(1/2)/(d+e*x)^5,x,5,-1/4*e*(b*x+c*x^2)^(3/2)/(d*(c*d-b*e)*(d+e*x)^4)-5/24*e*(2*c*d-b*e)*(b*x+c*x^2)^(3/2)/(d^2*(c*d-b*e)^2*(d+e*x)^3)-1/128*b^2*(16*c^2*d^2-16*b*c*d*e+5*b^2*e^2)*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/(d^(7/2)*(c*d-b*e)^(7/2))+1/64*(16*c^2*d^2-16*b*c*d*e+5*b^2*e^2)*(b*d+(2*c*d-b*e)*x)*sqrt(b*x+c*x^2)/(d^3*(c*d-b*e)^3*(d+e*x)^2)],
[(b*x+c*x^2)^(1/2)/(d+e*x)^6,x,6,-1/5*e*(b*x+c*x^2)^(3/2)/(d*(c*d-b*e)*(d+e*x)^5)-7/40*e*(2*c*d-b*e)*(b*x+c*x^2)^(3/2)/(d^2*(c*d-b*e)^2*(d+e*x)^4)-1/240*e*(108*c^2*d^2-108*b*c*d*e+35*b^2*e^2)*(b*x+c*x^2)^(3/2)/(d^3*(c*d-b*e)^3*(d+e*x)^3)-1/256*b^2*(2*c*d-b*e)*(16*c^2*d^2-16*b*c*d*e+7*b^2*e^2)*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/(d^(9/2)*(c*d-b*e)^(9/2))+1/128*(2*c*d-b*e)*(16*c^2*d^2-16*b*c*d*e+7*b^2*e^2)*(b*d+(2*c*d-b*e)*x)*sqrt(b*x+c*x^2)/(d^4*(c*d-b*e)^4*(d+e*x)^2)],
[(d+e*x)^3*(b*x+c*x^2)^(3/2),x,6,1/128*(2*c*d-b*e)*(8*c^2*d^2-8*b*c*d*e+3*b^2*e^2)*(b+2*c*x)*(b*x+c*x^2)^(3/2)/c^4+1/7*e*(d+e*x)^2*(b*x+c*x^2)^(5/2)/c+1/280*e*(128*c^2*d^2-98*b*c*d*e+21*b^2*e^2+30*c*e*(2*c*d-b*e)*x)*(b*x+c*x^2)^(5/2)/c^3+3/1024*b^4*(2*c*d-b*e)*(8*c^2*d^2-8*b*c*d*e+3*b^2*e^2)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(11/2)-3/1024*b^2*(2*c*d-b*e)*(8*c^2*d^2-8*b*c*d*e+3*b^2*e^2)*(b+2*c*x)*sqrt(b*x+c*x^2)/c^5],
[(d+e*x)^2*(b*x+c*x^2)^(3/2),x,6,1/192*(24*c^2*d^2-24*b*c*d*e+7*b^2*e^2)*(b+2*c*x)*(b*x+c*x^2)^(3/2)/c^3+7/60*e*(2*c*d-b*e)*(b*x+c*x^2)^(5/2)/c^2+1/6*e*(d+e*x)*(b*x+c*x^2)^(5/2)/c+1/512*b^4*(24*c^2*d^2-24*b*c*d*e+7*b^2*e^2)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(9/2)-1/512*b^2*(24*c^2*d^2-24*b*c*d*e+7*b^2*e^2)*(b+2*c*x)*sqrt(b*x+c*x^2)/c^4],
[(d+e*x)*(b*x+c*x^2)^(3/2),x,5,1/16*(2*c*d-b*e)*(b+2*c*x)*(b*x+c*x^2)^(3/2)/c^2+1/5*e*(b*x+c*x^2)^(5/2)/c+3/128*b^4*(2*c*d-b*e)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(7/2)-3/128*b^2*(2*c*d-b*e)*(b+2*c*x)*sqrt(b*x+c*x^2)/c^3],
[(b*x+c*x^2)^(3/2),x,4,1/8*(b+2*c*x)*(b*x+c*x^2)^(3/2)/c+3/64*b^4*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(5/2)-3/64*b^2*(b+2*c*x)*sqrt(b*x+c*x^2)/c^2],
[(b*x+c*x^2)^(3/2)/(d+e*x),x,7,1/3*(b*x+c*x^2)^(3/2)/e-1/8*(2*c*d-b*e)*(8*c^2*d^2-8*b*c*d*e-b^2*e^2)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/(c^(3/2)*e^4)+d^(3/2)*(c*d-b*e)^(3/2)*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/e^4+1/8*(8*c^2*d^2-10*b*c*d*e+b^2*e^2-2*c*e*(2*c*d-b*e)*x)*sqrt(b*x+c*x^2)/(c*e^3)],
[(b*x+c*x^2)^(3/2)/(d+e*x)^2,x,7,-(b*x+c*x^2)^(3/2)/(e*(d+e*x))+3/4*(8*c^2*d^2-8*b*c*d*e+b^2*e^2)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/(e^4*sqrt(c))-3/2*(2*c*d-b*e)*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))*sqrt(d)*sqrt(c*d-b*e)/e^4-3/4*(4*c*d-3*b*e-2*c*e*x)*sqrt(b*x+c*x^2)/e^3],
[(b*x+c*x^2)^(3/2)/(d+e*x)^3,x,7,-1/2*(b*x+c*x^2)^(3/2)/(e*(d+e*x)^2)-3*(2*c*d-b*e)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))*sqrt(c)/e^4+3/8*(8*c^2*d^2-8*b*c*d*e+b^2*e^2)*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/(e^4*sqrt(d)*sqrt(c*d-b*e))+3/4*(4*c*d-b*e+2*c*e*x)*sqrt(b*x+c*x^2)/(e^3*(d+e*x))],
[(d+e*x)^3*(b*x+c*x^2)^(5/2),x,7,-5/12288*b^2*(2*c*d-b*e)*(32*c^2*d^2-32*b*c*d*e+11*b^2*e^2)*(b+2*c*x)*(b*x+c*x^2)^(3/2)/c^5+1/768*(2*c*d-b*e)*(32*c^2*d^2-32*b*c*d*e+11*b^2*e^2)*(b+2*c*x)*(b*x+c*x^2)^(5/2)/c^4+1/9*e*(d+e*x)^2*(b*x+c*x^2)^(7/2)/c+1/2016*e*(640*c^2*d^2-486*b*c*d*e+99*b^2*e^2+154*c*e*(2*c*d-b*e)*x)*(b*x+c*x^2)^(7/2)/c^3-5/32768*b^6*(2*c*d-b*e)*(32*c^2*d^2-32*b*c*d*e+11*b^2*e^2)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(13/2)+5/32768*b^4*(2*c*d-b*e)*(32*c^2*d^2-32*b*c*d*e+11*b^2*e^2)*(b+2*c*x)*sqrt(b*x+c*x^2)/c^6],
[(d+e*x)^2*(b*x+c*x^2)^(5/2),x,7,-5/6144*b^2*(32*c^2*d^2-32*b*c*d*e+9*b^2*e^2)*(b+2*c*x)*(b*x+c*x^2)^(3/2)/c^4+1/384*(32*c^2*d^2-32*b*c*d*e+9*b^2*e^2)*(b+2*c*x)*(b*x+c*x^2)^(5/2)/c^3+9/112*e*(2*c*d-b*e)*(b*x+c*x^2)^(7/2)/c^2+1/8*e*(d+e*x)*(b*x+c*x^2)^(7/2)/c-5/16384*b^6*(32*c^2*d^2-32*b*c*d*e+9*b^2*e^2)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(11/2)+5/16384*b^4*(32*c^2*d^2-32*b*c*d*e+9*b^2*e^2)*(b+2*c*x)*sqrt(b*x+c*x^2)/c^5],
[(d+e*x)*(b*x+c*x^2)^(5/2),x,6,-5/384*b^2*(2*c*d-b*e)*(b+2*c*x)*(b*x+c*x^2)^(3/2)/c^3+1/24*(2*c*d-b*e)*(b+2*c*x)*(b*x+c*x^2)^(5/2)/c^2+1/7*e*(b*x+c*x^2)^(7/2)/c-5/1024*b^6*(2*c*d-b*e)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(9/2)+5/1024*b^4*(2*c*d-b*e)*(b+2*c*x)*sqrt(b*x+c*x^2)/c^4],
[(b*x+c*x^2)^(5/2),x,5,-5/192*b^2*(b+2*c*x)*(b*x+c*x^2)^(3/2)/c^2+1/12*(b+2*c*x)*(b*x+c*x^2)^(5/2)/c-5/512*b^6*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(7/2)+5/512*b^4*(b+2*c*x)*sqrt(b*x+c*x^2)/c^3],
[(b*x+c*x^2)^(5/2)/(d+e*x),x,8,1/48*(16*c^2*d^2-22*b*c*d*e+3*b^2*e^2-6*c*e*(2*c*d-b*e)*x)*(b*x+c*x^2)^(3/2)/(c*e^3)+1/5*(b*x+c*x^2)^(5/2)/e-1/128*(2*c*d-b*e)*(128*c^4*d^4-256*b*c^3*d^3*e+112*b^2*c^2*d^2*e^2+16*b^3*c*d*e^3+3*b^4*e^4)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/(c^(5/2)*e^6)+d^(5/2)*(c*d-b*e)^(5/2)*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/e^6+1/128*(128*c^4*d^4-288*b*c^3*d^3*e+176*b^2*c^2*d^2*e^2-10*b^3*c*d*e^3-3*b^4*e^4-2*c*e*(2*c*d-b*e)*(16*c^2*d^2-16*b*c*d*e-3*b^2*e^2)*x)*sqrt(b*x+c*x^2)/(c^2*e^5)],
[(b*x+c*x^2)^(5/2)/(d+e*x)^2,x,8,-5/24*(8*c*d-7*b*e-6*c*e*x)*(b*x+c*x^2)^(3/2)/e^3-(b*x+c*x^2)^(5/2)/(e*(d+e*x))+5/64*(128*c^4*d^4-256*b*c^3*d^3*e+144*b^2*c^2*d^2*e^2-16*b^3*c*d*e^3-b^4*e^4)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/(c^(3/2)*e^6)-5/2*d^(3/2)*(c*d-b*e)^(3/2)*(2*c*d-b*e)*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/e^6-5/64*(64*c^3*d^3-112*b*c^2*d^2*e+48*b^2*c*d*e^2-b^3*e^3-2*c*e*(16*c^2*d^2-16*b*c*d*e+b^2*e^2)*x)*sqrt(b*x+c*x^2)/(c*e^5)],
[(b*x+c*x^2)^(5/2)/(d+e*x)^3,x,8,5/12*(8*c*d-3*b*e+2*c*e*x)*(b*x+c*x^2)^(3/2)/(e^3*(d+e*x))-1/2*(b*x+c*x^2)^(5/2)/(e*(d+e*x)^2)-5/8*(2*c*d-b*e)*(16*c^2*d^2-16*b*c*d*e+b^2*e^2)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/(e^6*sqrt(c))+5/8*(4*c*d-3*b*e)*(4*c*d-b*e)*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))*sqrt(d)*sqrt(c*d-b*e)/e^6+5/8*(16*c^2*d^2-20*b*c*d*e+5*b^2*e^2-4*c*e*(2*c*d-b*e)*x)*sqrt(b*x+c*x^2)/e^5],

#  Integrands of the form (d+e*x)^m*(b*x+c*x^2)^p where 2*c*d-b*e=0 
[sqrt(2*x+x^2)/(1+x),x,3,-arctan(sqrt(2*x+x^2))+sqrt(2*x+x^2)],
[(2*x-x^2)^(3/2)/(2-2*x),x,4,-1/6*(2*x-x^2)^(3/2)+1/2*arctanh(sqrt(2*x-x^2))-1/2*sqrt(2*x-x^2)],
[(2*x-x^2)^(1/2)/(2-2*x),x,3,1/2*arctanh(sqrt(2*x-x^2))-1/2*sqrt(2*x-x^2)],
[1/((2-2*x)*(2*x-x^2)^(1/2)),x,2,1/2*arctanh(sqrt(2*x-x^2))],
[1/((2-2*x)*(2*x-x^2)^(3/2)),x,3,1/2*arctanh(sqrt(2*x-x^2))+(-1/2)/sqrt(2*x-x^2)],
[1/((2-2*x)*(2*x-x^2)^(5/2)),x,4,(-1/6)/(2*x-x^2)^(3/2)+1/2*arctanh(sqrt(2*x-x^2))+(-1/2)/sqrt(2*x-x^2)],

# p<0
[(d+e*x)^3/(b*x+c*x^2)^(1/2),x,4,1/8*(2*c*d-b*e)*(8*c^2*d^2-8*b*c*d*e+5*b^2*e^2)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(7/2)+1/3*e*(d+e*x)^2*sqrt(b*x+c*x^2)/c+1/24*e*(64*c^2*d^2-54*b*c*d*e+15*b^2*e^2+10*c*e*(2*c*d-b*e)*x)*sqrt(b*x+c*x^2)/c^3],
[(d+e*x)^2/(b*x+c*x^2)^(1/2),x,4,1/4*(8*c^2*d^2-8*b*c*d*e+3*b^2*e^2)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(5/2)+3/4*e*(2*c*d-b*e)*sqrt(b*x+c*x^2)/c^2+1/2*e*(d+e*x)*sqrt(b*x+c*x^2)/c],
[(d+e*x)/(b*x+c*x^2)^(1/2),x,3,(2*c*d-b*e)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(3/2)+e*sqrt(b*x+c*x^2)/c],
[1/(b*x+c*x^2)^(1/2),x,2,2*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/sqrt(c)],
[1/((d+e*x)*(b*x+c*x^2)^(1/2)),x,2,arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/(sqrt(d)*sqrt(c*d-b*e))],
[1/((d+e*x)^2*(b*x+c*x^2)^(1/2)),x,3,1/2*(2*c*d-b*e)*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/(d^(3/2)*(c*d-b*e)^(3/2))-e*sqrt(b*x+c*x^2)/(d*(c*d-b*e)*(d+e*x))],
[1/((d+e*x)^3*(b*x+c*x^2)^(1/2)),x,4,1/8*(8*c^2*d^2-8*b*c*d*e+3*b^2*e^2)*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/(d^(5/2)*(c*d-b*e)^(5/2))-1/2*e*sqrt(b*x+c*x^2)/(d*(c*d-b*e)*(d+e*x)^2)-3/4*e*(2*c*d-b*e)*sqrt(b*x+c*x^2)/(d^2*(c*d-b*e)^2*(d+e*x))],
[(d+e*x)^3/(b*x+c*x^2)^(3/2),x,4,3*e^2*(2*c*d-b*e)*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(5/2)-2*(d+e*x)^2*(b*d+(2*c*d-b*e)*x)/(b^2*sqrt(b*x+c*x^2))+e*(8*c^2*d^2-6*b*c*d*e+3*b^2*e^2+2*c*e*(2*c*d-b*e)*x)*sqrt(b*x+c*x^2)/(b^2*c^2)],
[(d+e*x)^2/(b*x+c*x^2)^(3/2),x,4,2*e^2*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(3/2)-2*(d+e*x)*(b*d+(2*c*d-b*e)*x)/(b^2*sqrt(b*x+c*x^2))+2*e*(2*c*d-b*e)*sqrt(b*x+c*x^2)/(b^2*c)],
[(d+e*x)/(b*x+c*x^2)^(3/2),x,1,-2*(b*d+(2*c*d-b*e)*x)/(b^2*sqrt(b*x+c*x^2))],
[1/(b*x+c*x^2)^(3/2),x,1,-2*(b+2*c*x)/(b^2*sqrt(b*x+c*x^2))],
[1/((d+e*x)*(b*x+c*x^2)^(3/2)),x,4,e^2*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/(d^(3/2)*(c*d-b*e)^(3/2))-2*(b*(c*d-b*e)+c*(2*c*d-b*e)*x)/(b^2*d*(c*d-b*e)*sqrt(b*x+c*x^2))],
[1/((d+e*x)^2*(b*x+c*x^2)^(3/2)),x,4,3/2*e^2*(2*c*d-b*e)*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/(d^(5/2)*(c*d-b*e)^(5/2))-2*(b*(c*d-b*e)+c*(2*c*d-b*e)*x)/(b^2*d*(c*d-b*e)*(d+e*x)*sqrt(b*x+c*x^2))-e*(4*c^2*d^2-4*b*c*d*e+3*b^2*e^2)*sqrt(b*x+c*x^2)/(b^2*d^2*(c*d-b*e)^2*(d+e*x))],
[1/((d+e*x)^3*(b*x+c*x^2)^(3/2)),x,5,3/8*e^2*(16*c^2*d^2-16*b*c*d*e+5*b^2*e^2)*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/(d^(7/2)*(c*d-b*e)^(7/2))-2*(b*(c*d-b*e)+c*(2*c*d-b*e)*x)/(b^2*d*(c*d-b*e)*(d+e*x)^2*sqrt(b*x+c*x^2))-1/2*e*(8*c^2*d^2-8*b*c*d*e+5*b^2*e^2)*sqrt(b*x+c*x^2)/(b^2*d^2*(c*d-b*e)^2*(d+e*x)^2)-1/4*e*(2*c*d-b*e)*(8*c^2*d^2-8*b*c*d*e+15*b^2*e^2)*sqrt(b*x+c*x^2)/(b^2*d^3*(c*d-b*e)^3*(d+e*x))],
[(d+e*x)^4/(b*x+c*x^2)^(5/2),x,5,-2/3*(d+e*x)^3*(b*d+(2*c*d-b*e)*x)/(b^2*(b*x+c*x^2)^(3/2))+2*e^4*arctanh(x*sqrt(c)/sqrt(b*x+c*x^2))/c^(5/2)+4/3*(d+e*x)*(b*c*d^2*(4*c*d-5*b*e)+(2*c*d-b*e)*(4*c^2*d^2-4*b*c*d*e-b^2*e^2)*x)/(b^4*c*sqrt(b*x+c*x^2))-2/3*e*(2*c*d-b*e)*(8*c^2*d^2-8*b*c*d*e-3*b^2*e^2)*sqrt(b*x+c*x^2)/(b^4*c^2)],
[(d+e*x)^3/(b*x+c*x^2)^(5/2),x,2,-2/3*(d+e*x)^2*(b*d+(2*c*d-b*e)*x)/(b^2*(b*x+c*x^2)^(3/2))+16/3*d*(c*d-b*e)*(b*d+(2*c*d-b*e)*x)/(b^4*sqrt(b*x+c*x^2))],
[(d+e*x)^2/(b*x+c*x^2)^(5/2),x,2,-2/3*(b+2*c*x)*(d+e*x)^2/(b^2*(b*x+c*x^2)^(3/2))+8/3*(2*c*d-b*e)*(b*d+(2*c*d-b*e)*x)/(b^4*sqrt(b*x+c*x^2))],
[(d+e*x)/(b*x+c*x^2)^(5/2),x,2,-2/3*(b*d+(2*c*d-b*e)*x)/(b^2*(b*x+c*x^2)^(3/2))+8/3*(2*c*d-b*e)*(b+2*c*x)/(b^4*sqrt(b*x+c*x^2))],
[1/(b*x+c*x^2)^(5/2),x,2,-2/3*(b+2*c*x)/(b^2*(b*x+c*x^2)^(3/2))+16/3*c*(b+2*c*x)/(b^4*sqrt(b*x+c*x^2))],
[1/((d+e*x)*(b*x+c*x^2)^(5/2)),x,5,-2/3*(b*(c*d-b*e)+c*(2*c*d-b*e)*x)/(b^2*d*(c*d-b*e)*(b*x+c*x^2)^(3/2))+e^4*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/(d^(5/2)*(c*d-b*e)^(5/2))+2/3*(b*(c*d-b*e)*(8*c^2*d^2-4*b*c*d*e-3*b^2*e^2)+c*(2*c*d-b*e)*(8*c^2*d^2-8*b*c*d*e-3*b^2*e^2)*x)/(b^4*d^2*(c*d-b*e)^2*sqrt(b*x+c*x^2))],
[1/((d+e*x)^2*(b*x+c*x^2)^(5/2)),x,5,-2/3*(b*(c*d-b*e)+c*(2*c*d-b*e)*x)/(b^2*d*(c*d-b*e)*(d+e*x)*(b*x+c*x^2)^(3/2))+5/2*e^4*(2*c*d-b*e)*arctanh(1/2*(b*d+(2*c*d-b*e)*x)/(sqrt(d)*sqrt(c*d-b*e)*sqrt(b*x+c*x^2)))/(d^(7/2)*(c*d-b*e)^(7/2))+2/3*(b*(c*d-b*e)*(8*c^2*d^2-2*b*c*d*e-5*b^2*e^2)+c*(2*c*d-b*e)*(8*c^2*d^2-8*b*c*d*e-5*b^2*e^2)*x)/(b^4*d^2*(c*d-b*e)^2*(d+e*x)*sqrt(b*x+c*x^2))+1/3*e*(32*c^4*d^4-64*b*c^3*d^3*e+12*b^2*c^2*d^2*e^2+20*b^3*c*d*e^3-15*b^4*e^4)*sqrt(b*x+c*x^2)/(b^4*d^3*(c*d-b*e)^3*(d+e*x))],
[1/((2+x)*sqrt(2*x+x^2)),x,1,sqrt(2*x+x^2)/(2+x)],

# Integrands of the form (d+e x)^(m/2) (b x+c x^2)^p

# p>0
[(d+e*x)^(7/2)*(b*x+c*x^2),x,2,2/9*d*(c*d-b*e)*(d+e*x)^(9/2)/e^3-2/11*(2*c*d-b*e)*(d+e*x)^(11/2)/e^3+2/13*c*(d+e*x)^(13/2)/e^3],
[(d+e*x)^(5/2)*(b*x+c*x^2),x,2,2/7*d*(c*d-b*e)*(d+e*x)^(7/2)/e^3-2/9*(2*c*d-b*e)*(d+e*x)^(9/2)/e^3+2/11*c*(d+e*x)^(11/2)/e^3],
[(d+e*x)^(3/2)*(b*x+c*x^2),x,2,2/5*d*(c*d-b*e)*(d+e*x)^(5/2)/e^3-2/7*(2*c*d-b*e)*(d+e*x)^(7/2)/e^3+2/9*c*(d+e*x)^(9/2)/e^3],
[(d+e*x)^(1/2)*(b*x+c*x^2),x,2,2/3*d*(c*d-b*e)*(d+e*x)^(3/2)/e^3-2/5*(2*c*d-b*e)*(d+e*x)^(5/2)/e^3+2/7*c*(d+e*x)^(7/2)/e^3],
[(b*x+c*x^2)/(d+e*x)^(1/2),x,2,-2/3*(2*c*d-b*e)*(d+e*x)^(3/2)/e^3+2/5*c*(d+e*x)^(5/2)/e^3+2*d*(c*d-b*e)*sqrt(d+e*x)/e^3],
[(b*x+c*x^2)/(d+e*x)^(3/2),x,2,2/3*c*(d+e*x)^(3/2)/e^3-2*d*(c*d-b*e)/(e^3*sqrt(d+e*x))-2*(2*c*d-b*e)*sqrt(d+e*x)/e^3],
[(b*x+c*x^2)/(d+e*x)^(5/2),x,2,-2/3*d*(c*d-b*e)/(e^3*(d+e*x)^(3/2))+2*(2*c*d-b*e)/(e^3*sqrt(d+e*x))+2*c*sqrt(d+e*x)/e^3],
[(b*x+c*x^2)/(d+e*x)^(7/2),x,2,-2/5*d*(c*d-b*e)/(e^3*(d+e*x)^(5/2))+2/3*(2*c*d-b*e)/(e^3*(d+e*x)^(3/2))-2*c/(e^3*sqrt(d+e*x))],
[(d+e*x)^(7/2)*(b*x+c*x^2)^2,x,2,2/9*d^2*(c*d-b*e)^2*(d+e*x)^(9/2)/e^5-4/11*d*(c*d-b*e)*(2*c*d-b*e)*(d+e*x)^(11/2)/e^5+2/13*(6*c^2*d^2-6*b*c*d*e+b^2*e^2)*(d+e*x)^(13/2)/e^5-4/15*c*(2*c*d-b*e)*(d+e*x)^(15/2)/e^5+2/17*c^2*(d+e*x)^(17/2)/e^5],
[(d+e*x)^(5/2)*(b*x+c*x^2)^2,x,2,2/7*d^2*(c*d-b*e)^2*(d+e*x)^(7/2)/e^5-4/9*d*(c*d-b*e)*(2*c*d-b*e)*(d+e*x)^(9/2)/e^5+2/11*(6*c^2*d^2-6*b*c*d*e+b^2*e^2)*(d+e*x)^(11/2)/e^5-4/13*c*(2*c*d-b*e)*(d+e*x)^(13/2)/e^5+2/15*c^2*(d+e*x)^(15/2)/e^5],
[(d+e*x)^(3/2)*(b*x+c*x^2)^2,x,2,2/5*d^2*(c*d-b*e)^2*(d+e*x)^(5/2)/e^5-4/7*d*(c*d-b*e)*(2*c*d-b*e)*(d+e*x)^(7/2)/e^5+2/9*(6*c^2*d^2-6*b*c*d*e+b^2*e^2)*(d+e*x)^(9/2)/e^5-4/11*c*(2*c*d-b*e)*(d+e*x)^(11/2)/e^5+2/13*c^2*(d+e*x)^(13/2)/e^5],
[(d+e*x)^(1/2)*(b*x+c*x^2)^2,x,2,2/3*d^2*(c*d-b*e)^2*(d+e*x)^(3/2)/e^5-4/5*d*(c*d-b*e)*(2*c*d-b*e)*(d+e*x)^(5/2)/e^5+2/7*(6*c^2*d^2-6*b*c*d*e+b^2*e^2)*(d+e*x)^(7/2)/e^5-4/9*c*(2*c*d-b*e)*(d+e*x)^(9/2)/e^5+2/11*c^2*(d+e*x)^(11/2)/e^5],
[(b*x+c*x^2)^2/(d+e*x)^(1/2),x,2,-4/3*d*(c*d-b*e)*(2*c*d-b*e)*(d+e*x)^(3/2)/e^5+2/5*(6*c^2*d^2-6*b*c*d*e+b^2*e^2)*(d+e*x)^(5/2)/e^5-4/7*c*(2*c*d-b*e)*(d+e*x)^(7/2)/e^5+2/9*c^2*(d+e*x)^(9/2)/e^5+2*d^2*(c*d-b*e)^2*sqrt(d+e*x)/e^5],
[(b*x+c*x^2)^2/(d+e*x)^(3/2),x,2,2/3*(6*c^2*d^2-6*b*c*d*e+b^2*e^2)*(d+e*x)^(3/2)/e^5-4/5*c*(2*c*d-b*e)*(d+e*x)^(5/2)/e^5+2/7*c^2*(d+e*x)^(7/2)/e^5-2*d^2*(c*d-b*e)^2/(e^5*sqrt(d+e*x))-4*d*(c*d-b*e)*(2*c*d-b*e)*sqrt(d+e*x)/e^5],
[(b*x+c*x^2)^2/(d+e*x)^(5/2),x,2,-2/3*d^2*(c*d-b*e)^2/(e^5*(d+e*x)^(3/2))-4/3*c*(2*c*d-b*e)*(d+e*x)^(3/2)/e^5+2/5*c^2*(d+e*x)^(5/2)/e^5+4*d*(c*d-b*e)*(2*c*d-b*e)/(e^5*sqrt(d+e*x))+2*(6*c^2*d^2-6*b*c*d*e+b^2*e^2)*sqrt(d+e*x)/e^5],
[(b*x+c*x^2)^2/(d+e*x)^(7/2),x,2,-2/5*d^2*(c*d-b*e)^2/(e^5*(d+e*x)^(5/2))+4/3*d*(c*d-b*e)*(2*c*d-b*e)/(e^5*(d+e*x)^(3/2))+2/3*c^2*(d+e*x)^(3/2)/e^5-2*(6*c^2*d^2-6*b*c*d*e+b^2*e^2)/(e^5*sqrt(d+e*x))-4*c*(2*c*d-b*e)*sqrt(d+e*x)/e^5],
[(d+e*x)^(7/2)*(b*x+c*x^2)^3,x,2,2/9*d^3*(c*d-b*e)^3*(d+e*x)^(9/2)/e^7-6/11*d^2*(c*d-b*e)^2*(2*c*d-b*e)*(d+e*x)^(11/2)/e^7+6/13*d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*(d+e*x)^(13/2)/e^7-2/15*(2*c*d-b*e)*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)*(d+e*x)^(15/2)/e^7+6/17*c*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*(d+e*x)^(17/2)/e^7-6/19*c^2*(2*c*d-b*e)*(d+e*x)^(19/2)/e^7+2/21*c^3*(d+e*x)^(21/2)/e^7],
[(d+e*x)^(5/2)*(b*x+c*x^2)^3,x,2,2/7*d^3*(c*d-b*e)^3*(d+e*x)^(7/2)/e^7-2/3*d^2*(c*d-b*e)^2*(2*c*d-b*e)*(d+e*x)^(9/2)/e^7+6/11*d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*(d+e*x)^(11/2)/e^7-2/13*(2*c*d-b*e)*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)*(d+e*x)^(13/2)/e^7+2/5*c*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*(d+e*x)^(15/2)/e^7-6/17*c^2*(2*c*d-b*e)*(d+e*x)^(17/2)/e^7+2/19*c^3*(d+e*x)^(19/2)/e^7],
[(d+e*x)^(3/2)*(b*x+c*x^2)^3,x,2,2/5*d^3*(c*d-b*e)^3*(d+e*x)^(5/2)/e^7-6/7*d^2*(c*d-b*e)^2*(2*c*d-b*e)*(d+e*x)^(7/2)/e^7+2/3*d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*(d+e*x)^(9/2)/e^7-2/11*(2*c*d-b*e)*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)*(d+e*x)^(11/2)/e^7+6/13*c*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*(d+e*x)^(13/2)/e^7-2/5*c^2*(2*c*d-b*e)*(d+e*x)^(15/2)/e^7+2/17*c^3*(d+e*x)^(17/2)/e^7],
[(d+e*x)^(1/2)*(b*x+c*x^2)^3,x,2,2/3*d^3*(c*d-b*e)^3*(d+e*x)^(3/2)/e^7-6/5*d^2*(c*d-b*e)^2*(2*c*d-b*e)*(d+e*x)^(5/2)/e^7+6/7*d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*(d+e*x)^(7/2)/e^7-2/9*(2*c*d-b*e)*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)*(d+e*x)^(9/2)/e^7+6/11*c*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*(d+e*x)^(11/2)/e^7-6/13*c^2*(2*c*d-b*e)*(d+e*x)^(13/2)/e^7+2/15*c^3*(d+e*x)^(15/2)/e^7],
[(b*x+c*x^2)^3/(d+e*x)^(1/2),x,2,-2*d^2*(c*d-b*e)^2*(2*c*d-b*e)*(d+e*x)^(3/2)/e^7+6/5*d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*(d+e*x)^(5/2)/e^7-2/7*(2*c*d-b*e)*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)*(d+e*x)^(7/2)/e^7+2/3*c*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*(d+e*x)^(9/2)/e^7-6/11*c^2*(2*c*d-b*e)*(d+e*x)^(11/2)/e^7+2/13*c^3*(d+e*x)^(13/2)/e^7+2*d^3*(c*d-b*e)^3*sqrt(d+e*x)/e^7],
[(b*x+c*x^2)^3/(d+e*x)^(3/2),x,2,2*d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*(d+e*x)^(3/2)/e^7-2/5*(2*c*d-b*e)*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)*(d+e*x)^(5/2)/e^7+6/7*c*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*(d+e*x)^(7/2)/e^7-2/3*c^2*(2*c*d-b*e)*(d+e*x)^(9/2)/e^7+2/11*c^3*(d+e*x)^(11/2)/e^7-2*d^3*(c*d-b*e)^3/(e^7*sqrt(d+e*x))-6*d^2*(c*d-b*e)^2*(2*c*d-b*e)*sqrt(d+e*x)/e^7],
[(b*x+c*x^2)^3/(d+e*x)^(5/2),x,2,-2/3*d^3*(c*d-b*e)^3/(e^7*(d+e*x)^(3/2))-2/3*(2*c*d-b*e)*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)*(d+e*x)^(3/2)/e^7+6/5*c*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*(d+e*x)^(5/2)/e^7-6/7*c^2*(2*c*d-b*e)*(d+e*x)^(7/2)/e^7+2/9*c^3*(d+e*x)^(9/2)/e^7+6*d^2*(c*d-b*e)^2*(2*c*d-b*e)/(e^7*sqrt(d+e*x))+6*d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*sqrt(d+e*x)/e^7],
[(b*x+c*x^2)^3/(d+e*x)^(7/2),x,2,-2/5*d^3*(c*d-b*e)^3/(e^7*(d+e*x)^(5/2))+2*d^2*(c*d-b*e)^2*(2*c*d-b*e)/(e^7*(d+e*x)^(3/2))+2*c*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*(d+e*x)^(3/2)/e^7-6/5*c^2*(2*c*d-b*e)*(d+e*x)^(5/2)/e^7+2/7*c^3*(d+e*x)^(7/2)/e^7-6*d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)/(e^7*sqrt(d+e*x))-2*(2*c*d-b*e)*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)*sqrt(d+e*x)/e^7],

# p<0
[(d+e*x)^(7/2)/(b*x+c*x^2),x,7,2/3*e*(2*c*d-b*e)*(d+e*x)^(3/2)/c^2+2/5*e*(d+e*x)^(5/2)/c-2*d^(7/2)*arctanh(sqrt(d+e*x)/sqrt(d))/b+2*(c*d-b*e)^(7/2)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b*c^(7/2))+2*e*(3*c^2*d^2-3*b*c*d*e+b^2*e^2)*sqrt(d+e*x)/c^3],
[(d+e*x)^(5/2)/(b*x+c*x^2),x,6,2/3*e*(d+e*x)^(3/2)/c-2*d^(5/2)*arctanh(sqrt(d+e*x)/sqrt(d))/b+2*(c*d-b*e)^(5/2)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b*c^(5/2))+2*e*(2*c*d-b*e)*sqrt(d+e*x)/c^2],
[(d+e*x)^(3/2)/(b*x+c*x^2),x,5,-2*d^(3/2)*arctanh(sqrt(d+e*x)/sqrt(d))/b+2*(c*d-b*e)^(3/2)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b*c^(3/2))+2*e*sqrt(d+e*x)/c],
[(d+e*x)^(1/2)/(b*x+c*x^2),x,4,-2*arctanh(sqrt(d+e*x)/sqrt(d))*sqrt(d)/b+2*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))*sqrt(c*d-b*e)/(b*sqrt(c))],
[1/((d+e*x)^(1/2)*(b*x+c*x^2)),x,4,-2*arctanh(sqrt(d+e*x)/sqrt(d))/(b*sqrt(d))+2*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))*sqrt(c)/(b*sqrt(c*d-b*e))],
[1/((d+e*x)^(3/2)*(b*x+c*x^2)),x,5,-2*arctanh(sqrt(d+e*x)/sqrt(d))/(b*d^(3/2))+2*c^(3/2)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b*(c*d-b*e)^(3/2))-2*e/(d*(c*d-b*e)*sqrt(d+e*x))],
[1/((d+e*x)^(5/2)*(b*x+c*x^2)),x,6,-2/3*e/(d*(c*d-b*e)*(d+e*x)^(3/2))-2*arctanh(sqrt(d+e*x)/sqrt(d))/(b*d^(5/2))+2*c^(5/2)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b*(c*d-b*e)^(5/2))-2*e*(2*c*d-b*e)/(d^2*(c*d-b*e)^2*sqrt(d+e*x))],
[1/((d+e*x)^(7/2)*(b*x+c*x^2)),x,7,-2/5*e/(d*(c*d-b*e)*(d+e*x)^(5/2))-2/3*e*(2*c*d-b*e)/(d^2*(c*d-b*e)^2*(d+e*x)^(3/2))-2*arctanh(sqrt(d+e*x)/sqrt(d))/(b*d^(7/2))+2*c^(7/2)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b*(c*d-b*e)^(7/2))-2*e*(3*c^2*d^2-3*b*c*d*e+b^2*e^2)/(d^3*(c*d-b*e)^3*sqrt(d+e*x))],
[(d+e*x)^(9/2)/(b*x+c*x^2)^2,x,8,1/3*e*(6*c^2*d^2-6*b*c*d*e+5*b^2*e^2)*(d+e*x)^(3/2)/(b^2*c^2)+e*(2*c*d-b*e)*(d+e*x)^(5/2)/(b^2*c)-(d+e*x)^(7/2)*(b*d+(2*c*d-b*e)*x)/(b^2*(b*x+c*x^2))+d^(7/2)*(4*c*d-9*b*e)*arctanh(sqrt(d+e*x)/sqrt(d))/b^3-(c*d-b*e)^(7/2)*(4*c*d+5*b*e)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b^3*c^(7/2))+e*(2*c*d-b*e)*(c^2*d^2-b*c*d*e+5*b^2*e^2)*sqrt(d+e*x)/(b^2*c^3)],
[(d+e*x)^(7/2)/(b*x+c*x^2)^2,x,7,e*(2*c*d-b*e)*(d+e*x)^(3/2)/(b^2*c)-(d+e*x)^(5/2)*(b*d+(2*c*d-b*e)*x)/(b^2*(b*x+c*x^2))+d^(5/2)*(4*c*d-7*b*e)*arctanh(sqrt(d+e*x)/sqrt(d))/b^3-(c*d-b*e)^(5/2)*(4*c*d+3*b*e)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b^3*c^(5/2))+e*(2*c^2*d^2-2*b*c*d*e+3*b^2*e^2)*sqrt(d+e*x)/(b^2*c^2)],
[(d+e*x)^(5/2)/(b*x+c*x^2)^2,x,6,-(d+e*x)^(3/2)*(b*d+(2*c*d-b*e)*x)/(b^2*(b*x+c*x^2))+d^(3/2)*(4*c*d-5*b*e)*arctanh(sqrt(d+e*x)/sqrt(d))/b^3-(c*d-b*e)^(3/2)*(4*c*d+b*e)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b^3*c^(3/2))+e*(2*c*d-b*e)*sqrt(d+e*x)/(b^2*c)],
[(d+e*x)^(3/2)/(b*x+c*x^2)^2,x,5,(4*c*d-3*b*e)*arctanh(sqrt(d+e*x)/sqrt(d))*sqrt(d)/b^3-(4*c*d-b*e)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))*sqrt(c*d-b*e)/(b^3*sqrt(c))-(b*d+(2*c*d-b*e)*x)*sqrt(d+e*x)/(b^2*(b*x+c*x^2))],
[(d+e*x)^(1/2)/(b*x+c*x^2)^2,x,5,(4*c*d-b*e)*arctanh(sqrt(d+e*x)/sqrt(d))/(b^3*sqrt(d))-(4*c*d-3*b*e)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))*sqrt(c)/(b^3*sqrt(c*d-b*e))-(b+2*c*x)*sqrt(d+e*x)/(b^2*(b*x+c*x^2))],
[1/((d+e*x)^(1/2)*(b*x+c*x^2)^2),x,5,(4*c*d+b*e)*arctanh(sqrt(d+e*x)/sqrt(d))/(b^3*d^(3/2))-c^(3/2)*(4*c*d-5*b*e)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b^3*(c*d-b*e)^(3/2))-(b*(c*d-b*e)+c*(2*c*d-b*e)*x)*sqrt(d+e*x)/(b^2*d*(c*d-b*e)*(b*x+c*x^2))],
[1/((d+e*x)^(3/2)*(b*x+c*x^2)^2),x,6,(4*c*d+3*b*e)*arctanh(sqrt(d+e*x)/sqrt(d))/(b^3*d^(5/2))-c^(5/2)*(4*c*d-7*b*e)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b^3*(c*d-b*e)^(5/2))-e*(2*c^2*d^2-2*b*c*d*e+3*b^2*e^2)/(b^2*d^2*(c*d-b*e)^2*sqrt(d+e*x))+(-b*(c*d-b*e)-c*(2*c*d-b*e)*x)/(b^2*d*(c*d-b*e)*(b*x+c*x^2)*sqrt(d+e*x))],
[1/((d+e*x)^(5/2)*(b*x+c*x^2)^2),x,7,-1/3*e*(6*c^2*d^2-6*b*c*d*e+5*b^2*e^2)/(b^2*d^2*(c*d-b*e)^2*(d+e*x)^(3/2))+(-b*(c*d-b*e)-c*(2*c*d-b*e)*x)/(b^2*d*(c*d-b*e)*(d+e*x)^(3/2)*(b*x+c*x^2))+(4*c*d+5*b*e)*arctanh(sqrt(d+e*x)/sqrt(d))/(b^3*d^(7/2))-c^(7/2)*(4*c*d-9*b*e)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b^3*(c*d-b*e)^(7/2))-e*(2*c*d-b*e)*(c^2*d^2-b*c*d*e+5*b^2*e^2)/(b^2*d^3*(c*d-b*e)^3*sqrt(d+e*x))],
[1/((d+e*x)^(7/2)*(b*x+c*x^2)^2),x,8,-1/5*e*(10*c^2*d^2-10*b*c*d*e+7*b^2*e^2)/(b^2*d^2*(c*d-b*e)^2*(d+e*x)^(5/2))-1/3*e*(2*c*d-b*e)*(3*c^2*d^2-3*b*c*d*e+7*b^2*e^2)/(b^2*d^3*(c*d-b*e)^3*(d+e*x)^(3/2))+(-b*(c*d-b*e)-c*(2*c*d-b*e)*x)/(b^2*d*(c*d-b*e)*(d+e*x)^(5/2)*(b*x+c*x^2))+(4*c*d+7*b*e)*arctanh(sqrt(d+e*x)/sqrt(d))/(b^3*d^(9/2))-c^(9/2)*(4*c*d-11*b*e)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b^3*(c*d-b*e)^(9/2))-e*(2*c^4*d^4-4*b*c^3*d^3*e+26*b^2*c^2*d^2*e^2-24*b^3*c*d*e^3+7*b^4*e^4)/(b^2*d^4*(c*d-b*e)^4*sqrt(d+e*x))],
[(d+e*x)^(9/2)/(b*x+c*x^2)^3,x,7,-1/2*(d+e*x)^(7/2)*(b*d+(2*c*d-b*e)*x)/(b^2*(b*x+c*x^2)^2)+1/4*(d+e*x)^(3/2)*(b*c*d^2*(12*c*d-13*b*e)+(2*c*d-b*e)*(12*c^2*d^2-12*b*c*d*e-b^2*e^2)*x)/(b^4*c*(b*x+c*x^2))-3/4*d^(5/2)*(16*c^2*d^2-36*b*c*d*e+21*b^2*e^2)*arctanh(sqrt(d+e*x)/sqrt(d))/b^5+3/4*(c*d-b*e)^(5/2)*(16*c^2*d^2+4*b*c*d*e+b^2*e^2)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b^5*c^(5/2))-3/4*e*(2*c*d-b*e)*(4*c^2*d^2-4*b*c*d*e-b^2*e^2)*sqrt(d+e*x)/(b^4*c^2)],
[(d+e*x)^(7/2)/(b*x+c*x^2)^3,x,6,-1/2*(d+e*x)^(5/2)*(b*d+(2*c*d-b*e)*x)/(b^2*(b*x+c*x^2)^2)-1/4*d^(3/2)*(48*c^2*d^2-84*b*c*d*e+35*b^2*e^2)*arctanh(sqrt(d+e*x)/sqrt(d))/b^5+1/4*(c*d-b*e)^(3/2)*(48*c^2*d^2-12*b*c*d*e-b^2*e^2)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b^5*c^(3/2))+1/4*(b*c*d^2*(12*c*d-11*b*e)+(2*c*d-b*e)*(12*c^2*d^2-12*b*c*d*e+b^2*e^2)*x)*sqrt(d+e*x)/(b^4*c*(b*x+c*x^2))],
[(d+e*x)^(5/2)/(b*x+c*x^2)^3,x,6,-1/2*(d+e*x)^(3/2)*(b*d+(2*c*d-b*e)*x)/(b^2*(b*x+c*x^2)^2)-3/4*(16*c^2*d^2-20*b*c*d*e+5*b^2*e^2)*arctanh(sqrt(d+e*x)/sqrt(d))*sqrt(d)/b^5+3/4*(16*c^2*d^2-12*b*c*d*e+b^2*e^2)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))*sqrt(c*d-b*e)/(b^5*sqrt(c))+3/4*(b*d*(4*c*d-3*b*e)+(8*c^2*d^2-8*b*c*d*e+b^2*e^2)*x)*sqrt(d+e*x)/(b^4*(b*x+c*x^2))],
[(d+e*x)^(3/2)/(b*x+c*x^2)^3,x,6,-3/4*(16*c^2*d^2-12*b*c*d*e+b^2*e^2)*arctanh(sqrt(d+e*x)/sqrt(d))/(b^5*sqrt(d))+3/4*(16*c^2*d^2-20*b*c*d*e+5*b^2*e^2)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))*sqrt(c)/(b^5*sqrt(c*d-b*e))-1/2*(b*d+(2*c*d-b*e)*x)*sqrt(d+e*x)/(b^2*(b*x+c*x^2)^2)+1/4*(b*(12*c*d-7*b*e)*(c*d-b*e)+12*c*(c*d-b*e)*(2*c*d-b*e)*x)*sqrt(d+e*x)/(b^4*(c*d-b*e)*(b*x+c*x^2))],
[(d+e*x)^(1/2)/(b*x+c*x^2)^3,x,6,-1/4*(48*c^2*d^2-12*b*c*d*e-b^2*e^2)*arctanh(sqrt(d+e*x)/sqrt(d))/(b^5*d^(3/2))+1/4*c^(3/2)*(48*c^2*d^2-84*b*c*d*e+35*b^2*e^2)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b^5*(c*d-b*e)^(3/2))-1/2*(b+2*c*x)*sqrt(d+e*x)/(b^2*(b*x+c*x^2)^2)+1/4*(b*(c*d-b*e)*(12*c*d-b*e)+c*(24*c^2*d^2-24*b*c*d*e+b^2*e^2)*x)*sqrt(d+e*x)/(b^4*d*(c*d-b*e)*(b*x+c*x^2))],
[1/((d+e*x)^(1/2)*(b*x+c*x^2)^3),x,6,-3/4*(16*c^2*d^2+4*b*c*d*e+b^2*e^2)*arctanh(sqrt(d+e*x)/sqrt(d))/(b^5*d^(5/2))+3/4*c^(5/2)*(16*c^2*d^2-36*b*c*d*e+21*b^2*e^2)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b^5*(c*d-b*e)^(5/2))-1/2*(b*(c*d-b*e)+c*(2*c*d-b*e)*x)*sqrt(d+e*x)/(b^2*d*(c*d-b*e)*(b*x+c*x^2)^2)+1/4*(b*(c*d-b*e)*(12*c^2*d^2-7*b*c*d*e-3*b^2*e^2)+3*c*(2*c*d-b*e)*(4*c^2*d^2-4*b*c*d*e-b^2*e^2)*x)*sqrt(d+e*x)/(b^4*d^2*(c*d-b*e)^2*(b*x+c*x^2))],
[1/((d+e*x)^(3/2)*(b*x+c*x^2)^3),x,7,-3/4*(16*c^2*d^2+12*b*c*d*e+5*b^2*e^2)*arctanh(sqrt(d+e*x)/sqrt(d))/(b^5*d^(7/2))+3/4*c^(7/2)*(16*c^2*d^2-44*b*c*d*e+33*b^2*e^2)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b^5*(c*d-b*e)^(7/2))+3/4*e*(c^2*d^2-b*c*d*e-b^2*e^2)*(8*c^2*d^2-8*b*c*d*e+5*b^2*e^2)/(b^4*d^3*(c*d-b*e)^3*sqrt(d+e*x))+1/2*(-b*(c*d-b*e)-c*(2*c*d-b*e)*x)/(b^2*d*(c*d-b*e)*(b*x+c*x^2)^2*sqrt(d+e*x))+1/4*(b*(12*c^3*d^3-17*b*c^2*d^2*e+5*b^3*e^3)+c*(2*c*d-b*e)*(12*c^2*d^2-12*b*c*d*e-5*b^2*e^2)*x)/(b^4*d^2*(c*d-b*e)^2*(b*x+c*x^2)*sqrt(d+e*x))],
[1/((d+e*x)^(5/2)*(b*x+c*x^2)^3),x,8,1/12*e*(72*c^4*d^4-144*b*c^3*d^3*e+27*b^2*c^2*d^2*e^2+45*b^3*c*d*e^3-35*b^4*e^4)/(b^4*d^3*(c*d-b*e)^3*(d+e*x)^(3/2))+1/2*(-b*(c*d-b*e)-c*(2*c*d-b*e)*x)/(b^2*d*(c*d-b*e)*(d+e*x)^(3/2)*(b*x+c*x^2)^2)+1/4*(b*(c*d-b*e)*(12*c^2*d^2-3*b*c*d*e-7*b^2*e^2)+c*(2*c*d-b*e)*(12*c^2*d^2-12*b*c*d*e-7*b^2*e^2)*x)/(b^4*d^2*(c*d-b*e)^2*(d+e*x)^(3/2)*(b*x+c*x^2))-1/4*(48*c^2*d^2+60*b*c*d*e+35*b^2*e^2)*arctanh(sqrt(d+e*x)/sqrt(d))/(b^5*d^(9/2))+1/4*c^(9/2)*(48*c^2*d^2-156*b*c*d*e+143*b^2*e^2)*arctanh(sqrt(c)*sqrt(d+e*x)/sqrt(c*d-b*e))/(b^5*(c*d-b*e)^(9/2))+1/4*e*(2*c*d-b*e)*(12*c^4*d^4-24*b*c^3*d^3*e+2*b^2*c^2*d^2*e^2+10*b^3*c*d*e^3-35*b^4*e^4)/(b^4*d^4*(c*d-b*e)^4*sqrt(d+e*x))],

# Integrands of the form (d+e x)^(m/2) (b x+c x^2)^(p/2)

# p>0
[(d+e*x)^(3/2)*sqrt(b*x+c*x^2),x,9,2/7*e*(b*x+c*x^2)^(3/2)*sqrt(d+e*x)/c-2/105*(2*c*d-b*e)*(3*c^2*d^2-3*b*c*d*e+8*b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(c^(5/2)*e^2*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+4/105*d*(c*d-b*e)*(3*c^2*d^2-3*b*c*d*e+2*b^2*e^2)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(c^(5/2)*e^2*sqrt(d+e*x)*sqrt(b*x+c*x^2))+2/105*(3*c^2*d^2+9*b*c*d*e-4*b^2*e^2+12*c*e*(2*c*d-b*e)*x)*sqrt(d+e*x)*sqrt(b*x+c*x^2)/(c^2*e)],
[sqrt(d+e*x)*sqrt(b*x+c*x^2),x,9,-4/15*(c^2*d^2-b*c*d*e+b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(c^(3/2)*e^2*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+2/15*d*(c*d-b*e)*(2*c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(c^(3/2)*e^2*sqrt(d+e*x)*sqrt(b*x+c*x^2))+2/5*(d+e*x)^(3/2)*sqrt(b*x+c*x^2)/e-2/15*(2*c*d-b*e)*sqrt(d+e*x)*sqrt(b*x+c*x^2)/(c*e)],
[sqrt(b*x+c*x^2)/sqrt(d+e*x),x,8,-2/3*(2*c*d-b*e)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(e^2*sqrt(c)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+4/3*d*(c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(e^2*sqrt(c)*sqrt(d+e*x)*sqrt(b*x+c*x^2))+2/3*sqrt(d+e*x)*sqrt(b*x+c*x^2)/e],
[sqrt(b*x+c*x^2)/(d+e*x)^(3/2),x,8,4*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(e^2*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-2*(2*c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(e^2*sqrt(c)*sqrt(d+e*x)*sqrt(b*x+c*x^2))-2*sqrt(b*x+c*x^2)/(e*sqrt(d+e*x))],
[sqrt(b*x+c*x^2)/(d+e*x)^(5/2),x,9,-2/3*(2*c*d-b*e)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(d*e^2*(c*d-b*e)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+4/3*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(e^2*sqrt(d+e*x)*sqrt(b*x+c*x^2))-2/3*sqrt(b*x+c*x^2)/(e*(d+e*x)^(3/2))+2/3*(2*c*d-b*e)*sqrt(b*x+c*x^2)/(d*e*(c*d-b*e)*sqrt(d+e*x))],
[sqrt(b*x+c*x^2)/(d+e*x)^(7/2),x,10,-4/15*(c^2*d^2-b*c*d*e+b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(d^2*e^2*(c*d-b*e)^2*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+2/15*(2*c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(d*e^2*(c*d-b*e)*sqrt(d+e*x)*sqrt(b*x+c*x^2))-2/5*sqrt(b*x+c*x^2)/(e*(d+e*x)^(5/2))+2/15*(2*c*d-b*e)*sqrt(b*x+c*x^2)/(d*e*(c*d-b*e)*(d+e*x)^(3/2))+4/15*(c^2*d^2-b*c*d*e+b^2*e^2)*sqrt(b*x+c*x^2)/(d^2*e*(c*d-b*e)^2*sqrt(d+e*x))],
[(d+e*x)^(3/2)*(b*x+c*x^2)^(3/2),x,10,2/231*(c^2*d^2+13*b*c*d*e-6*b^2*e^2+14*c*e*(2*c*d-b*e)*x)*(b*x+c*x^2)^(3/2)*sqrt(d+e*x)/(c^2*e)+2/11*e*(b*x+c*x^2)^(5/2)*sqrt(d+e*x)/c-16/1155*(c*d-2*b*e)*(2*c*d-b*e)*(c*d+b*e)*(c^2*d^2-b*c*d*e+b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(c^(7/2)*e^4*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+2/1155*d*(c*d-b*e)*(16*c^4*d^4-32*b*c^3*d^3*e+3*b^2*c^2*d^2*e^2+13*b^3*c*d*e^3-8*b^4*e^4)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(c^(7/2)*e^4*sqrt(d+e*x)*sqrt(b*x+c*x^2))+2/1155*(8*c^4*d^4-19*b*c^3*d^3*e+6*b^2*c^2*d^2*e^2-19*b^3*c*d*e^3+8*b^4*e^4-3*c*e*(2*c*d-b*e)*(c^2*d^2-b*c*d*e+8*b^2*e^2)*x)*sqrt(d+e*x)*sqrt(b*x+c*x^2)/(c^3*e^3)],
[(b*x+c*x^2)^(3/2)*sqrt(d+e*x),x,10,2/9*(d+e*x)^(3/2)*(b*x+c*x^2)^(3/2)/e-2/21*(2*c*d-b*e)*(b*x+c*x^2)^(3/2)*sqrt(d+e*x)/(c*e)-2/315*(16*c^4*d^4-32*b*c^3*d^3*e+9*b^2*c^2*d^2*e^2+7*b^3*c*d*e^3-8*b^4*e^4)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(c^(5/2)*e^4*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+8/315*d*(c*d-b*e)*(2*c*d-b*e)*(2*c^2*d^2-2*b*c*d*e-b^2*e^2)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(c^(5/2)*e^4*sqrt(d+e*x)*sqrt(b*x+c*x^2))+2/315*(8*c^3*d^3-15*b*c^2*d^2*e+3*b^2*c*d*e^2-4*b^3*e^3-6*c*e*(c^2*d^2-b*c*d*e+2*b^2*e^2)*x)*sqrt(d+e*x)*sqrt(b*x+c*x^2)/(c^2*e^3)],
[(b*x+c*x^2)^(3/2)/sqrt(d+e*x),x,9,2/7*(b*x+c*x^2)^(3/2)*sqrt(d+e*x)/e-4/35*(2*c*d-b*e)*(4*c^2*d^2-4*b*c*d*e-b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(c^(3/2)*e^4*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+2/35*d*(c*d-b*e)*(16*c^2*d^2-16*b*c*d*e-b^2*e^2)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(c^(3/2)*e^4*sqrt(d+e*x)*sqrt(b*x+c*x^2))+2/35*(8*c^2*d^2-11*b*c*d*e+b^2*e^2-3*c*e*(2*c*d-b*e)*x)*sqrt(d+e*x)*sqrt(b*x+c*x^2)/(c*e^3)],
[(b*x+c*x^2)^(3/2)/(d+e*x)^(3/2),x,9,-2*(b*x+c*x^2)^(3/2)/(e*sqrt(d+e*x))+2/5*(16*c^2*d^2-16*b*c*d*e+b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(e^4*sqrt(c)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-16/5*d*(c*d-b*e)*(2*c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(e^4*sqrt(c)*sqrt(d+e*x)*sqrt(b*x+c*x^2))-2/5*(8*c*d-7*b*e-6*c*e*x)*sqrt(d+e*x)*sqrt(b*x+c*x^2)/e^3],
[(b*x+c*x^2)^(3/2)/(d+e*x)^(5/2),x,9,-2/3*(b*x+c*x^2)^(3/2)/(e*(d+e*x)^(3/2))-16/3*(2*c*d-b*e)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(e^4*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+2/3*(4*c*d-3*b*e)*(4*c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(e^4*sqrt(c)*sqrt(d+e*x)*sqrt(b*x+c*x^2))+2/3*(8*c*d-3*b*e+2*c*e*x)*sqrt(b*x+c*x^2)/(e^3*sqrt(d+e*x))],
[(b*x+c*x^2)^(3/2)/(d+e*x)^(7/2),x,9,-2/5*(b*x+c*x^2)^(3/2)/(e*(d+e*x)^(5/2))+2/5*(16*c^2*d^2-16*b*c*d*e+b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(d*e^4*(c*d-b*e)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-16/5*(2*c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(e^4*sqrt(d+e*x)*sqrt(b*x+c*x^2))-2/5*(c*d^2*(8*c*d-7*b*e)+e*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)*x)*sqrt(b*x+c*x^2)/(d*e^3*(c*d-b*e)*(d+e*x)^(3/2))],
[(b*x+c*x^2)^(3/2)/(d+e*x)^(9/2),x,10,-2/7*(b*x+c*x^2)^(3/2)/(e*(d+e*x)^(7/2))-4/35*(2*c*d-b*e)*(4*c^2*d^2-4*b*c*d*e-b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(d^2*e^4*(c*d-b*e)^2*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+2/35*(16*c^2*d^2-16*b*c*d*e-b^2*e^2)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(d*e^4*(c*d-b*e)*sqrt(d+e*x)*sqrt(b*x+c*x^2))-2/35*(d*(8*c^2*d^2-5*b*c*d*e-2*b^2*e^2)+e*(14*c^2*d^2-14*b*c*d*e+b^2*e^2)*x)*sqrt(b*x+c*x^2)/(d*e^3*(c*d-b*e)*(d+e*x)^(5/2))+4/35*(2*c*d-b*e)*(4*c^2*d^2-4*b*c*d*e-b^2*e^2)*sqrt(b*x+c*x^2)/(d^2*e^3*(c*d-b*e)^2*sqrt(d+e*x))],
[(b*x+c*x^2)^(5/2)*sqrt(d+e*x),x,11,2/13*(d+e*x)^(3/2)*(b*x+c*x^2)^(5/2)/e+10/9009*(16*c^3*d^3-31*b*c^2*d^2*e+9*b^2*c*d*e^2-18*b^3*e^3-14*c*e*(c^2*d^2-b*c*d*e+3*b^2*e^2)*x)*(b*x+c*x^2)^(3/2)*sqrt(d+e*x)/(c^2*e^3)-10/143*(2*c*d-b*e)*(b*x+c*x^2)^(5/2)*sqrt(d+e*x)/(c*e)-4/9009*(128*c^6*d^6-384*b*c^5*d^5*e+343*b^2*c^4*d^4*e^2-46*b^3*c^3*d^3*e^3-21*b^4*c^2*d^2*e^4-20*b^5*c*d*e^5+24*b^6*e^6)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(c^(7/2)*e^6*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+2/9009*d*(c*d-b*e)*(2*c*d-b*e)*(128*c^4*d^4-256*b*c^3*d^3*e+79*b^2*c^2*d^2*e^2+49*b^3*c*d*e^3+24*b^4*e^4)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(c^(7/2)*e^6*sqrt(d+e*x)*sqrt(b*x+c*x^2))+2/9009*(128*c^5*d^5-368*b*c^4*d^4*e+303*b^2*c^3*d^3*e^2-22*b^3*c^2*d^2*e^3-17*b^4*c*d*e^4+24*b^5*e^5-3*c*e*(32*c^4*d^4-64*b*c^3*d^3*e+21*b^2*c^2*d^2*e^2+11*b^3*c*d*e^3-24*b^4*e^4)*x)*sqrt(d+e*x)*sqrt(b*x+c*x^2)/(c^3*e^5)],
[(b*x+c*x^2)^(5/2)/sqrt(d+e*x),x,10,10/693*(16*c^2*d^2-23*b*c*d*e+3*b^2*e^2-7*c*e*(2*c*d-b*e)*x)*(b*x+c*x^2)^(3/2)*sqrt(d+e*x)/(c*e^3)+2/11*(b*x+c*x^2)^(5/2)*sqrt(d+e*x)/e-2/693*(2*c*d-b*e)*(128*c^4*d^4-256*b*c^3*d^3*e+99*b^2*c^2*d^2*e^2+29*b^3*c*d*e^3+8*b^4*e^4)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(c^(5/2)*e^6*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+4/693*d*(c*d-b*e)*(128*c^4*d^4-256*b*c^3*d^3*e+123*b^2*c^2*d^2*e^2+5*b^3*c*d*e^3+2*b^4*e^4)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(c^(5/2)*e^6*sqrt(d+e*x)*sqrt(b*x+c*x^2))+2/693*(128*c^4*d^4-304*b*c^3*d^3*e+195*b^2*c^2*d^2*e^2-7*b^3*c*d*e^3-4*b^4*e^4-12*c*e*(2*c*d-b*e)*(4*c^2*d^2-4*b*c*d*e-b^2*e^2)*x)*sqrt(d+e*x)*sqrt(b*x+c*x^2)/(c^2*e^5)],
[(b*x+c*x^2)^(5/2)/(d+e*x)^(3/2),x,10,-2*(b*x+c*x^2)^(5/2)/(e*sqrt(d+e*x))-10/63*(16*c*d-15*b*e-14*c*e*x)*(b*x+c*x^2)^(3/2)*sqrt(d+e*x)/e^3+4/63*(128*c^4*d^4-256*b*c^3*d^3*e+135*b^2*c^2*d^2*e^2-7*b^3*c*d*e^3-b^4*e^4)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(c^(3/2)*e^6*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-2/63*d*(c*d-b*e)*(2*c*d-b*e)*(128*c^2*d^2-128*b*c*d*e-b^2*e^2)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(c^(3/2)*e^6*sqrt(d+e*x)*sqrt(b*x+c*x^2))-2/63*(128*c^3*d^3-240*b*c^2*d^2*e+111*b^2*c*d*e^2-b^3*e^3-3*c*e*(32*c^2*d^2-32*b*c*d*e+b^2*e^2)*x)*sqrt(d+e*x)*sqrt(b*x+c*x^2)/(c*e^5)],
[(b*x+c*x^2)^(5/2)/(d+e*x)^(5/2),x,10,-2/3*(b*x+c*x^2)^(5/2)/(e*(d+e*x)^(3/2))+10/21*(16*c*d-7*b*e+2*c*e*x)*(b*x+c*x^2)^(3/2)/(e^3*sqrt(d+e*x))-2/21*(2*c*d-b*e)*(128*c^2*d^2-128*b*c*d*e+3*b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(e^6*sqrt(c)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+4/21*d*(c*d-b*e)*(128*c^2*d^2-128*b*c*d*e+27*b^2*e^2)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(e^6*sqrt(c)*sqrt(d+e*x)*sqrt(b*x+c*x^2))+2/21*(128*c^2*d^2-176*b*c*d*e+51*b^2*e^2-48*c*e*(2*c*d-b*e)*x)*sqrt(d+e*x)*sqrt(b*x+c*x^2)/e^5],
[(b*x+c*x^2)^(5/2)/(d+e*x)^(7/2),x,10,2/15*(16*c*d-5*b*e+6*c*e*x)*(b*x+c*x^2)^(3/2)/(e^3*(d+e*x)^(3/2))-2/5*(b*x+c*x^2)^(5/2)/(e*(d+e*x)^(5/2))+4/15*(128*c^2*d^2-128*b*c*d*e+23*b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(e^6*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-2/15*(2*c*d-b*e)*(128*c^2*d^2-128*b*c*d*e+15*b^2*e^2)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(e^6*sqrt(c)*sqrt(d+e*x)*sqrt(b*x+c*x^2))-2/15*(128*c^2*d^2-112*b*c*d*e+15*b^2*e^2+16*c*e*(2*c*d-b*e)*x)*sqrt(b*x+c*x^2)/(e^5*sqrt(d+e*x))],
[(b*x+c*x^2)^(5/2)/(d+e*x)^(9/2),x,10,-2/21*(c*d^2*(16*c*d-13*b*e)+e*(22*c^2*d^2-22*b*c*d*e+3*b^2*e^2)*x)*(b*x+c*x^2)^(3/2)/(d*e^3*(c*d-b*e)*(d+e*x)^(5/2))-2/7*(b*x+c*x^2)^(5/2)/(e*(d+e*x)^(7/2))-2/21*(2*c*d-b*e)*(128*c^2*d^2-128*b*c*d*e+3*b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(d*e^6*(c*d-b*e)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+4/21*(128*c^2*d^2-128*b*c*d*e+27*b^2*e^2)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(e^6*sqrt(d+e*x)*sqrt(b*x+c*x^2))+2/21*c*(d*(128*c^2*d^2-176*b*c*d*e+51*b^2*e^2)+e*(32*c^2*d^2-32*b*c*d*e+3*b^2*e^2)*x)*sqrt(b*x+c*x^2)/(d*e^5*(c*d-b*e)*sqrt(d+e*x))],
[(b*x+c*x^2)^(5/2)/(d+e*x)^(11/2),x,10,-2/63*(d*(16*c^2*d^2-11*b*c*d*e-2*b^2*e^2)+e*(26*c^2*d^2-26*b*c*d*e+3*b^2*e^2)*x)*(b*x+c*x^2)^(3/2)/(d*e^3*(c*d-b*e)*(d+e*x)^(7/2))-2/9*(b*x+c*x^2)^(5/2)/(e*(d+e*x)^(9/2))+4/63*(128*c^4*d^4-256*b*c^3*d^3*e+135*b^2*c^2*d^2*e^2-7*b^3*c*d*e^3-b^4*e^4)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(d^2*e^6*(c*d-b*e)^2*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-2/63*(2*c*d-b*e)*(128*c^2*d^2-128*b*c*d*e-b^2*e^2)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(d*e^6*(c*d-b*e)*sqrt(d+e*x)*sqrt(b*x+c*x^2))-2/63*(c*d^2*(128*c^3*d^3-240*b*c^2*d^2*e+111*b^2*c*d*e^2-b^3*e^3)+e*(160*c^4*d^4-320*b*c^3*d^3*e+171*b^2*c^2*d^2*e^2-11*b^3*c*d*e^3-2*b^4*e^4)*x)*sqrt(b*x+c*x^2)/(d^2*e^5*(c*d-b*e)^2*(d+e*x)^(3/2))],

# p<0
[(d+e*x)^(7/2)/sqrt(b*x+c*x^2),x,10,16/105*(2*c*d-b*e)*(11*c^2*d^2-11*b*c*d*e+6*b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(c^(7/2)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-2/105*d*(c*d-b*e)*(71*c^2*d^2-71*b*c*d*e+24*b^2*e^2)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(c^(7/2)*sqrt(d+e*x)*sqrt(b*x+c*x^2))+12/35*e*(2*c*d-b*e)*(d+e*x)^(3/2)*sqrt(b*x+c*x^2)/c^2+2/7*e*(d+e*x)^(5/2)*sqrt(b*x+c*x^2)/c+2/105*e*(71*c^2*d^2-71*b*c*d*e+24*b^2*e^2)*sqrt(d+e*x)*sqrt(b*x+c*x^2)/c^3],
[(d+e*x)^(5/2)/sqrt(b*x+c*x^2),x,9,2/15*(23*c^2*d^2-23*b*c*d*e+8*b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(c^(5/2)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-8/15*d*(c*d-b*e)*(2*c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(c^(5/2)*sqrt(d+e*x)*sqrt(b*x+c*x^2))+2/5*e*(d+e*x)^(3/2)*sqrt(b*x+c*x^2)/c+8/15*e*(2*c*d-b*e)*sqrt(d+e*x)*sqrt(b*x+c*x^2)/c^2],
[(d+e*x)^(3/2)/sqrt(b*x+c*x^2),x,8,4/3*(2*c*d-b*e)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(c^(3/2)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-2/3*d*(c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(c^(3/2)*sqrt(d+e*x)*sqrt(b*x+c*x^2))+2/3*e*sqrt(d+e*x)*sqrt(b*x+c*x^2)/c],
[sqrt(d+e*x)/sqrt(b*x+c*x^2),x,3,2*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(sqrt(c)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))],
[1/(sqrt(d+e*x)*sqrt(b*x+c*x^2)),x,3,2*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(sqrt(c)*sqrt(d+e*x)*sqrt(b*x+c*x^2))],
[1/((d+e*x)^(3/2)*sqrt(b*x+c*x^2)),x,5,2*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(d*(c*d-b*e)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-2*e*sqrt(b*x+c*x^2)/(d*(c*d-b*e)*sqrt(d+e*x))],
[1/((d+e*x)^(5/2)*sqrt(b*x+c*x^2)),x,9,4/3*(2*c*d-b*e)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(d^2*(c*d-b*e)^2*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-2/3*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(d*(c*d-b*e)*sqrt(d+e*x)*sqrt(b*x+c*x^2))-2/3*e*sqrt(b*x+c*x^2)/(d*(c*d-b*e)*(d+e*x)^(3/2))-4/3*e*(2*c*d-b*e)*sqrt(b*x+c*x^2)/(d^2*(c*d-b*e)^2*sqrt(d+e*x))],
[1/((d+e*x)^(7/2)*sqrt(b*x+c*x^2)),x,10,2/15*(23*c^2*d^2-23*b*c*d*e+8*b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/(d^3*(c*d-b*e)^3*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-8/15*(2*c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(-b)*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/(d^2*(c*d-b*e)^2*sqrt(d+e*x)*sqrt(b*x+c*x^2))-2/5*e*sqrt(b*x+c*x^2)/(d*(c*d-b*e)*(d+e*x)^(5/2))-8/15*e*(2*c*d-b*e)*sqrt(b*x+c*x^2)/(d^2*(c*d-b*e)^2*(d+e*x)^(3/2))-2/15*e*(23*c^2*d^2-23*b*c*d*e+8*b^2*e^2)*sqrt(b*x+c*x^2)/(d^3*(c*d-b*e)^3*sqrt(d+e*x))],
[(d+e*x)^(7/2)/(b*x+c*x^2)^(3/2),x,10,-2*(d+e*x)^(5/2)*(b*d+(2*c*d-b*e)*x)/(b^2*sqrt(b*x+c*x^2))+2/3*(2*c*d-b*e)*(3*c^2*d^2-3*b*c*d*e+8*b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/((-b)^(3/2)*c^(5/2)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-4/3*d*(c*d-b*e)*(3*c^2*d^2-3*b*c*d*e+2*b^2*e^2)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/((-b)^(3/2)*c^(5/2)*sqrt(d+e*x)*sqrt(b*x+c*x^2))+2*e*(2*c*d-b*e)*(d+e*x)^(3/2)*sqrt(b*x+c*x^2)/(b^2*c)+4/3*e*(3*c^2*d^2-3*b*c*d*e+2*b^2*e^2)*sqrt(d+e*x)*sqrt(b*x+c*x^2)/(b^2*c^2)],
[(d+e*x)^(5/2)/(b*x+c*x^2)^(3/2),x,9,-2*(d+e*x)^(3/2)*(b*d+(2*c*d-b*e)*x)/(b^2*sqrt(b*x+c*x^2))+4*(c^2*d^2-b*c*d*e+b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/((-b)^(3/2)*c^(3/2)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-2*d*(c*d-b*e)*(2*c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/((-b)^(3/2)*c^(3/2)*sqrt(d+e*x)*sqrt(b*x+c*x^2))+2*e*(2*c*d-b*e)*sqrt(d+e*x)*sqrt(b*x+c*x^2)/(b^2*c)],
[(d+e*x)^(3/2)/(b*x+c*x^2)^(3/2),x,8,-2*(b*d+(2*c*d-b*e)*x)*sqrt(d+e*x)/(b^2*sqrt(b*x+c*x^2))+2*(2*c*d-b*e)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/((-b)^(3/2)*sqrt(c)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-4*d*(c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/((-b)^(3/2)*sqrt(c)*sqrt(d+e*x)*sqrt(b*x+c*x^2))],
[sqrt(d+e*x)/(b*x+c*x^2)^(3/2),x,8,-2*(b+2*c*x)*sqrt(d+e*x)/(b^2*sqrt(b*x+c*x^2))+4*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/((-b)^(3/2)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-2*(2*c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/((-b)^(3/2)*sqrt(c)*sqrt(d+e*x)*sqrt(b*x+c*x^2))],
[1/((b*x+c*x^2)^(3/2)*sqrt(d+e*x)),x,8,-2*(b*(c*d-b*e)+c*(2*c*d-b*e)*x)*sqrt(d+e*x)/(b^2*d*(c*d-b*e)*sqrt(b*x+c*x^2))+2*(2*c*d-b*e)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/((-b)^(3/2)*d*(c*d-b*e)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-4*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/((-b)^(3/2)*sqrt(d+e*x)*sqrt(b*x+c*x^2))],
[1/((d+e*x)^(3/2)*(b*x+c*x^2)^(3/2)),x,9,-2*(b*(c*d-b*e)+c*(2*c*d-b*e)*x)/(b^2*d*(c*d-b*e)*sqrt(d+e*x)*sqrt(b*x+c*x^2))+4*(c^2*d^2-b*c*d*e+b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/((-b)^(3/2)*d^2*(c*d-b*e)^2*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-2*(2*c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/((-b)^(3/2)*d*(c*d-b*e)*sqrt(d+e*x)*sqrt(b*x+c*x^2))-4*e*(c^2*d^2-b*c*d*e+b^2*e^2)*sqrt(b*x+c*x^2)/(b^2*d^2*(c*d-b*e)^2*sqrt(d+e*x))],
[1/((d+e*x)^(5/2)*(b*x+c*x^2)^(3/2)),x,10,-2*(b*(c*d-b*e)+c*(2*c*d-b*e)*x)/(b^2*d*(c*d-b*e)*(d+e*x)^(3/2)*sqrt(b*x+c*x^2))+2/3*(2*c*d-b*e)*(3*c^2*d^2-3*b*c*d*e+8*b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/((-b)^(3/2)*d^3*(c*d-b*e)^3*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))-4/3*(3*c^2*d^2-3*b*c*d*e+2*b^2*e^2)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/((-b)^(3/2)*d^2*(c*d-b*e)^2*sqrt(d+e*x)*sqrt(b*x+c*x^2))-4/3*e*(3*c^2*d^2-3*b*c*d*e+2*b^2*e^2)*sqrt(b*x+c*x^2)/(b^2*d^2*(c*d-b*e)^2*(d+e*x)^(3/2))-2/3*e*(2*c*d-b*e)*(3*c^2*d^2-3*b*c*d*e+8*b^2*e^2)*sqrt(b*x+c*x^2)/(b^2*d^3*(c*d-b*e)^3*sqrt(d+e*x))],
[(d+e*x)^(9/2)/(b*x+c*x^2)^(5/2),x,10,-2/3*(d+e*x)^(7/2)*(b*d+(2*c*d-b*e)*x)/(b^2*(b*x+c*x^2)^(3/2))+2/3*(d+e*x)^(3/2)*(b*c*d^2*(8*c*d-11*b*e)+(2*c*d-b*e)*(8*c^2*d^2-8*b*c*d*e-3*b^2*e^2)*x)/(b^4*c*sqrt(b*x+c*x^2))-2/3*(16*c^4*d^4-32*b*c^3*d^3*e+9*b^2*c^2*d^2*e^2+7*b^3*c*d*e^3-8*b^4*e^4)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/((-b)^(7/2)*c^(5/2)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+8/3*d*(c*d-b*e)*(2*c*d-b*e)*(2*c^2*d^2-2*b*c*d*e-b^2*e^2)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/((-b)^(7/2)*c^(5/2)*sqrt(d+e*x)*sqrt(b*x+c*x^2))-8/3*e*(4*c^3*d^3-6*b*c^2*d^2*e+b^3*e^3)*sqrt(d+e*x)*sqrt(b*x+c*x^2)/(b^4*c^2)],
[(d+e*x)^(7/2)/(b*x+c*x^2)^(5/2),x,9,-2/3*(d+e*x)^(5/2)*(b*d+(2*c*d-b*e)*x)/(b^2*(b*x+c*x^2)^(3/2))+2/3*(b*c*d^2*(8*c*d-9*b*e)+(2*c*d-b*e)*(8*c^2*d^2-8*b*c*d*e-b^2*e^2)*x)*sqrt(d+e*x)/(b^4*c*sqrt(b*x+c*x^2))-4/3*(2*c*d-b*e)*(4*c^2*d^2-4*b*c*d*e-b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/((-b)^(7/2)*c^(3/2)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+2/3*d*(c*d-b*e)*(16*c^2*d^2-16*b*c*d*e-b^2*e^2)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/((-b)^(7/2)*c^(3/2)*sqrt(d+e*x)*sqrt(b*x+c*x^2))],
[(d+e*x)^(5/2)/(b*x+c*x^2)^(5/2),x,9,-2/3*(d+e*x)^(3/2)*(b*d+(2*c*d-b*e)*x)/(b^2*(b*x+c*x^2)^(3/2))+2/3*(b*d*(8*c*d-7*b*e)+(16*c^2*d^2-16*b*c*d*e+b^2*e^2)*x)*sqrt(d+e*x)/(b^4*sqrt(b*x+c*x^2))-2/3*(16*c^2*d^2-16*b*c*d*e+b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/((-b)^(7/2)*sqrt(c)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+16/3*d*(c*d-b*e)*(2*c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/((-b)^(7/2)*sqrt(c)*sqrt(d+e*x)*sqrt(b*x+c*x^2))],
[(d+e*x)^(3/2)/(b*x+c*x^2)^(5/2),x,9,-2/3*(b*d+(2*c*d-b*e)*x)*sqrt(d+e*x)/(b^2*(b*x+c*x^2)^(3/2))+2/3*(b*(8*c*d-5*b*e)*(c*d-b*e)+8*c*(c*d-b*e)*(2*c*d-b*e)*x)*sqrt(d+e*x)/(b^4*(c*d-b*e)*sqrt(b*x+c*x^2))-16/3*(2*c*d-b*e)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/((-b)^(7/2)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+2/3*(4*c*d-3*b*e)*(4*c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/((-b)^(7/2)*sqrt(c)*sqrt(d+e*x)*sqrt(b*x+c*x^2))],
[sqrt(d+e*x)/(b*x+c*x^2)^(5/2),x,9,-2/3*(b+2*c*x)*sqrt(d+e*x)/(b^2*(b*x+c*x^2)^(3/2))+2/3*(b*(c*d-b*e)*(8*c*d-b*e)+c*(16*c^2*d^2-16*b*c*d*e+b^2*e^2)*x)*sqrt(d+e*x)/(b^4*d*(c*d-b*e)*sqrt(b*x+c*x^2))-2/3*(16*c^2*d^2-16*b*c*d*e+b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/((-b)^(7/2)*d*(c*d-b*e)*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+16/3*(2*c*d-b*e)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/((-b)^(7/2)*sqrt(d+e*x)*sqrt(b*x+c*x^2))],
[1/((b*x+c*x^2)^(5/2)*sqrt(d+e*x)),x,9,-2/3*(b*(c*d-b*e)+c*(2*c*d-b*e)*x)*sqrt(d+e*x)/(b^2*d*(c*d-b*e)*(b*x+c*x^2)^(3/2))+2/3*(b*(c*d-b*e)*(8*c^2*d^2-5*b*c*d*e-2*b^2*e^2)+2*c*(2*c*d-b*e)*(4*c^2*d^2-4*b*c*d*e-b^2*e^2)*x)*sqrt(d+e*x)/(b^4*d^2*(c*d-b*e)^2*sqrt(b*x+c*x^2))-4/3*(2*c*d-b*e)*(4*c^2*d^2-4*b*c*d*e-b^2*e^2)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/((-b)^(7/2)*d^2*(c*d-b*e)^2*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+2/3*(16*c^2*d^2-16*b*c*d*e-b^2*e^2)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/((-b)^(7/2)*d*(c*d-b*e)*sqrt(d+e*x)*sqrt(b*x+c*x^2))],
[1/((d+e*x)^(3/2)*(b*x+c*x^2)^(5/2)),x,10,-2/3*(b*(c*d-b*e)+c*(2*c*d-b*e)*x)/(b^2*d*(c*d-b*e)*(b*x+c*x^2)^(3/2)*sqrt(d+e*x))+2/3*(b*(c*d-b*e)*(8*c^2*d^2-3*b*c*d*e-4*b^2*e^2)+4*c*(4*c^3*d^3-6*b*c^2*d^2*e+b^3*e^3)*x)/(b^4*d^2*(c*d-b*e)^2*sqrt(d+e*x)*sqrt(b*x+c*x^2))-2/3*(16*c^4*d^4-32*b*c^3*d^3*e+9*b^2*c^2*d^2*e^2+7*b^3*c*d*e^3-8*b^4*e^4)*EllipticE(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(d+e*x)/((-b)^(7/2)*d^3*(c*d-b*e)^3*sqrt(1+e*x/d)*sqrt(b*x+c*x^2))+8/3*(2*c*d-b*e)*(2*c^2*d^2-2*b*c*d*e-b^2*e^2)*EllipticF(sqrt(c)*sqrt(x)/sqrt(-b),sqrt(b*e/(c*d)))*sqrt(c)*sqrt(x)*sqrt(1+c*x/b)*sqrt(1+e*x/d)/((-b)^(7/2)*d^2*(c*d-b*e)^2*sqrt(d+e*x)*sqrt(b*x+c*x^2))+2/3*e*(16*c^4*d^4-32*b*c^3*d^3*e+9*b^2*c^2*d^2*e^2+7*b^3*c*d*e^3-8*b^4*e^4)*sqrt(b*x+c*x^2)/(b^4*d^3*(c*d-b*e)^3*sqrt(d+e*x))],
[sqrt(d+e*x)/sqrt(2*x-3*x^2),x,4,2*EllipticE(sqrt(3/2)*sqrt(x),sqrt(-2/3*e/d))*sqrt(d+e*x)/(sqrt(3)*sqrt(1+e*x/d))],
[1/(sqrt(d+e*x)*sqrt(2*x-3*x^2)),x,4,2*EllipticF(sqrt(3/2)*sqrt(x),sqrt(-2/3*e/d))*sqrt(1+e*x/d)/(sqrt(3)*sqrt(d+e*x))],
[sqrt(d+e*x)/sqrt(-2*x-3*x^2),x,4,-2*EllipticE(sqrt(3/2)*sqrt(-x),sqrt(2/3*e/d))*sqrt(d+e*x)/(sqrt(3)*sqrt(1+e*x/d))],
[1/(sqrt(d+e*x)*sqrt(-2*x-3*x^2)),x,4,-2*EllipticF(sqrt(3/2)*sqrt(-x),sqrt(2/3*e/d))*sqrt(1+e*x/d)/(sqrt(3)*sqrt(d+e*x))],

#  Note: Integrands are equal. 
[sqrt(1-x)/(sqrt(-x)*sqrt(1+x)),x,1,-2*EllipticE(sqrt(-x),I)],
[sqrt(1-x)/sqrt(-x-x^2),x,2,-2*EllipticE(sqrt(-x),I)],

# Integrands of the form (d+e x)^m (b x+c x^2)^p when c d-b e=0
[(d+e*x)^m*(c*d*x+c*e*x^2)^3,x,4,-c^3*d^3*(d+e*x)^(4+m)/(e^4*(4+m))+3*c^3*d^2*(d+e*x)^(5+m)/(e^4*(5+m))-3*c^3*d*(d+e*x)^(6+m)/(e^4*(6+m))+c^3*(d+e*x)^(7+m)/(e^4*(7+m))],
[(d+e*x)^m*(c*d*x+c*e*x^2)^2,x,4,c^2*d^2*(d+e*x)^(3+m)/(e^3*(3+m))-2*c^2*d*(d+e*x)^(4+m)/(e^3*(4+m))+c^2*(d+e*x)^(5+m)/(e^3*(5+m))],
[(d+e*x)^m*(c*d*x+c*e*x^2),x,4,-c*d*(d+e*x)^(2+m)/(e^2*(2+m))+c*(d+e*x)^(3+m)/(e^2*(3+m))],
[(d+e*x)^m,x,1,(d+e*x)^(1+m)/(e*(1+m))],
[(d+e*x)^m/(c*d*x+c*e*x^2),x,3,-(d+e*x)^m*hypergeom([1,m],[1+m],1+e*x/d)/(c*d*m)],
[(d+e*x)^m/(c*d*x+c*e*x^2)^2,x,3,-e*(d+e*x)^(-1+m)*hypergeom([2,-1+m],[m],1+e*x/d)/(c^2*d^2*(1-m))],

# Integrands of the form (d+e x)^m (b x+c x^2)^p when m symbolic
[(d+e*x)^m*(b*x+c*x^2)^3,x,2,d^3*(c*d-b*e)^3*(d+e*x)^(1+m)/(e^7*(1+m))-3*d^2*(c*d-b*e)^2*(2*c*d-b*e)*(d+e*x)^(2+m)/(e^7*(2+m))+3*d*(c*d-b*e)*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*(d+e*x)^(3+m)/(e^7*(3+m))-(2*c*d-b*e)*(10*c^2*d^2-10*b*c*d*e+b^2*e^2)*(d+e*x)^(4+m)/(e^7*(4+m))+3*c*(5*c^2*d^2-5*b*c*d*e+b^2*e^2)*(d+e*x)^(5+m)/(e^7*(5+m))-3*c^2*(2*c*d-b*e)*(d+e*x)^(6+m)/(e^7*(6+m))+c^3*(d+e*x)^(7+m)/(e^7*(7+m))],
[(d+e*x)^m*(b*x+c*x^2)^2,x,2,d^2*(c*d-b*e)^2*(d+e*x)^(1+m)/(e^5*(1+m))-2*d*(c*d-b*e)*(2*c*d-b*e)*(d+e*x)^(2+m)/(e^5*(2+m))+(6*c^2*d^2-6*b*c*d*e+b^2*e^2)*(d+e*x)^(3+m)/(e^5*(3+m))-2*c*(2*c*d-b*e)*(d+e*x)^(4+m)/(e^5*(4+m))+c^2*(d+e*x)^(5+m)/(e^5*(5+m))],
[(d+e*x)^m*(b*x+c*x^2),x,2,d*(c*d-b*e)*(d+e*x)^(1+m)/(e^3*(1+m))-(2*c*d-b*e)*(d+e*x)^(2+m)/(e^3*(2+m))+c*(d+e*x)^(3+m)/(e^3*(3+m))],
[(d+e*x)^m/(b*x+c*x^2),x,4,c*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],c*(d+e*x)/(c*d-b*e))/(b*(c*d-b*e)*(1+m))-(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],1+e*x/d)/(b*d*(1+m))],
[(d+e*x)^m/(b*x+c*x^2)^2,x,5,-(d+e*x)^(1+m)*(b*(c*d-b*e)+c*(2*c*d-b*e)*x)/(b^2*d*(c*d-b*e)*(b*x+c*x^2))-c^2*(2*c*d-b*e*(2-m))*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],c*(d+e*x)/(c*d-b*e))/(b^3*(c*d-b*e)^2*(1+m))+(2*c*d-b*e*m)*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],1+e*x/d)/(b^3*d^2*(1+m))],
[(d+e*x)^m/(b*x+c*x^2)^3,x,6,-1/2*(d+e*x)^(1+m)*(b*(c*d-b*e)+c*(2*c*d-b*e)*x)/(b^2*d*(c*d-b*e)*(b*x+c*x^2)^2)+1/2*(d+e*x)^(1+m)*(b*(c*d-b*e)*(6*c^2*d^2-b^2*e^2*(1-m)-b*c*d*e*(4+m))+c*(2*c*d-b*e)*(6*c^2*d^2-6*b*c*d*e-b^2*e^2*(1-m))*x)/(b^4*d^2*(c*d-b*e)^2*(b*x+c*x^2))+1/2*c^3*(12*c^2*d^2-6*b*c*d*e*(4-m)+b^2*e^2*(12-7*m+m^2))*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],c*(d+e*x)/(c*d-b*e))/(b^5*(c*d-b*e)^3*(1+m))-1/2*(12*c^2*d^2-6*b*c*d*e*m-b^2*e^2*(1-m)*m)*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],1+e*x/d)/(b^5*d^3*(1+m))],
[(d+e*x)^m*(b*x+c*x^2)^(3/2),x,2,(d+e*x)^(1+m)*(b*x+c*x^2)^(3/2)*AppellF1(1+m,-3/2,-3/2,2+m,(d+e*x)/d,c*(d+e*x)/(c*d-b*e))/(e*(1+m)*(-e*x/d)^(3/2)*(1-c*(d+e*x)/(c*d-b*e))^(3/2))],
[(d+e*x)^m*(b*x+c*x^2)^(1/2),x,2,(d+e*x)^(1+m)*AppellF1(1+m,-1/2,-1/2,2+m,(d+e*x)/d,c*(d+e*x)/(c*d-b*e))*sqrt(b*x+c*x^2)/(e*(1+m)*sqrt(-e*x/d)*sqrt(1-c*(d+e*x)/(c*d-b*e)))],
[(d+e*x)^m/(b*x+c*x^2)^(1/2),x,2,(d+e*x)^(1+m)*AppellF1(1+m,1/2,1/2,2+m,(d+e*x)/d,c*(d+e*x)/(c*d-b*e))*sqrt(-e*x/d)*sqrt(1-c*(d+e*x)/(c*d-b*e))/(e*(1+m)*sqrt(b*x+c*x^2))],
[(d+e*x)^m/(b*x+c*x^2)^(3/2),x,2,(-e*x/d)^(3/2)*(d+e*x)^(1+m)*(1-c*(d+e*x)/(c*d-b*e))^(3/2)*AppellF1(1+m,3/2,3/2,2+m,(d+e*x)/d,c*(d+e*x)/(c*d-b*e))/(e*(1+m)*(b*x+c*x^2)^(3/2))],

# Integrands of the form (d+e x)^m (b x+c x^2)^p when p symbolic
[(d+e*x)^m*(b*x+c*x^2)^p,x,2,(d+e*x)^(1+m)*(b*x+c*x^2)^p*AppellF1(1+m,-p,-p,2+m,(d+e*x)/d,c*(d+e*x)/(c*d-b*e))/(e*(1+m)*(-e*x/d)^p*(1-c*(d+e*x)/(c*d-b*e))^p)],

# Integrands of the form (d+e x)^m (a+b x+c x^2)^p when b=0

# Integrands of the form (d+e x)^m (a+c x^2)^p

# p>0
[(d+e*x)^4*(a+c*x^2),x,2,1/5*(c*d^2+a*e^2)*(d+e*x)^5/e^3-1/3*c*d*(d+e*x)^6/e^3+1/7*c*(d+e*x)^7/e^3],
[(d+e*x)^3*(a+c*x^2),x,2,1/4*(c*d^2+a*e^2)*(d+e*x)^4/e^3-2/5*c*d*(d+e*x)^5/e^3+1/6*c*(d+e*x)^6/e^3],
[(d+e*x)^2*(a+c*x^2),x,2,1/3*(c*d^2+a*e^2)*(d+e*x)^3/e^3-1/2*c*d*(d+e*x)^4/e^3+1/5*c*(d+e*x)^5/e^3],
[(d+e*x)*(a+c*x^2),x,2,a*d*x+1/3*c*d*x^3+1/4*e*(a+c*x^2)^2/c],
[(a+c*x^2)/(d+e*x),x,2,-c*d*x/e^2+1/2*c*x^2/e+(c*d^2+a*e^2)*log(d+e*x)/e^3],
[(a+c*x^2)/(d+e*x)^2,x,2,c*x/e^2+(-c*d^2-a*e^2)/(e^3*(d+e*x))-2*c*d*log(d+e*x)/e^3],
[(a+c*x^2)/(d+e*x)^3,x,2,1/2*(-c*d^2-a*e^2)/(e^3*(d+e*x)^2)+2*c*d/(e^3*(d+e*x))+c*log(d+e*x)/e^3],
[(a+c*x^2)/(d+e*x)^4,x,2,1/3*(-c*d^2-a*e^2)/(e^3*(d+e*x)^3)+c*d/(e^3*(d+e*x)^2)-c/(e^3*(d+e*x))],
[(a+c*x^2)/(d+e*x)^5,x,2,1/4*(-c*d^2-a*e^2)/(e^3*(d+e*x)^4)+2/3*c*d/(e^3*(d+e*x)^3)-1/2*c/(e^3*(d+e*x)^2)],
[(d+e*x)^4*(a+c*x^2)^2,x,2,1/5*(c*d^2+a*e^2)^2*(d+e*x)^5/e^5-2/3*c*d*(c*d^2+a*e^2)*(d+e*x)^6/e^5+2/7*c*(3*c*d^2+a*e^2)*(d+e*x)^7/e^5-1/2*c^2*d*(d+e*x)^8/e^5+1/9*c^2*(d+e*x)^9/e^5],
[(d+e*x)^3*(a+c*x^2)^2,x,2,1/4*(c*d^2+a*e^2)^2*(d+e*x)^4/e^5-4/5*c*d*(c*d^2+a*e^2)*(d+e*x)^5/e^5+1/3*c*(3*c*d^2+a*e^2)*(d+e*x)^6/e^5-4/7*c^2*d*(d+e*x)^7/e^5+1/8*c^2*(d+e*x)^8/e^5],
[(d+e*x)^2*(a+c*x^2)^2,x,3,a^2*d^2*x+1/3*a*(2*c*d^2+a*e^2)*x^3+1/5*c*(c*d^2+2*a*e^2)*x^5+1/7*c^2*e^2*x^7+1/3*d*e*(a+c*x^2)^3/c],
[(d+e*x)*(a+c*x^2)^2,x,3,a^2*d*x+2/3*a*c*d*x^3+1/5*c^2*d*x^5+1/6*e*(a+c*x^2)^3/c],
[(a+c*x^2)^2/(d+e*x),x,2,-c*d*(c*d^2+2*a*e^2)*x/e^4+1/2*c*(c*d^2+2*a*e^2)*x^2/e^3-1/3*c^2*d*x^3/e^2+1/4*c^2*x^4/e+(c*d^2+a*e^2)^2*log(d+e*x)/e^5],
[(a+c*x^2)^2/(d+e*x)^2,x,2,c*(3*c*d^2+2*a*e^2)*x/e^4-c^2*d*x^2/e^3+1/3*c^2*x^3/e^2-(c*d^2+a*e^2)^2/(e^5*(d+e*x))-4*c*d*(c*d^2+a*e^2)*log(d+e*x)/e^5],
[(a+c*x^2)^2/(d+e*x)^3,x,2,-3*c^2*d*x/e^4+1/2*c^2*x^2/e^3-1/2*(c*d^2+a*e^2)^2/(e^5*(d+e*x)^2)+4*c*d*(c*d^2+a*e^2)/(e^5*(d+e*x))+2*c*(3*c*d^2+a*e^2)*log(d+e*x)/e^5],
[(a+c*x^2)^2/(d+e*x)^4,x,2,c^2*x/e^4-1/3*(c*d^2+a*e^2)^2/(e^5*(d+e*x)^3)+2*c*d*(c*d^2+a*e^2)/(e^5*(d+e*x)^2)-2*c*(3*c*d^2+a*e^2)/(e^5*(d+e*x))-4*c^2*d*log(d+e*x)/e^5],
[(a+c*x^2)^2/(d+e*x)^5,x,2,-1/4*(c*d^2+a*e^2)^2/(e^5*(d+e*x)^4)+4/3*c*d*(c*d^2+a*e^2)/(e^5*(d+e*x)^3)-c*(3*c*d^2+a*e^2)/(e^5*(d+e*x)^2)+4*c^2*d/(e^5*(d+e*x))+c^2*log(d+e*x)/e^5],
[(a+c*x^2)^2/(d+e*x)^6,x,2,-1/5*(c*d^2+a*e^2)^2/(e^5*(d+e*x)^5)+c*d*(c*d^2+a*e^2)/(e^5*(d+e*x)^4)-2/3*c*(3*c*d^2+a*e^2)/(e^5*(d+e*x)^3)+2*c^2*d/(e^5*(d+e*x)^2)-c^2/(e^5*(d+e*x))],
[(a+c*x^2)^2/(d+e*x)^7,x,2,-1/6*(c*d^2+a*e^2)^2/(e^5*(d+e*x)^6)+4/5*c*d*(c*d^2+a*e^2)/(e^5*(d+e*x)^5)-1/2*c*(3*c*d^2+a*e^2)/(e^5*(d+e*x)^4)+4/3*c^2*d/(e^5*(d+e*x)^3)-1/2*c^2/(e^5*(d+e*x)^2)],
[(a+c*x^2)^2/(d+e*x)^8,x,2,-1/7*(c*d^2+a*e^2)^2/(e^5*(d+e*x)^7)+2/3*c*d*(c*d^2+a*e^2)/(e^5*(d+e*x)^6)-2/5*c*(3*c*d^2+a*e^2)/(e^5*(d+e*x)^5)+c^2*d/(e^5*(d+e*x)^4)-1/3*c^2/(e^5*(d+e*x)^3)],
[(d+e*x)^6*(a+c*x^2)^3,x,2,1/7*(c*d^2+a*e^2)^3*(d+e*x)^7/e^7-3/4*c*d*(c*d^2+a*e^2)^2*(d+e*x)^8/e^7+1/3*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)*(d+e*x)^9/e^7-2/5*c^2*d*(5*c*d^2+3*a*e^2)*(d+e*x)^10/e^7+3/11*c^2*(5*c*d^2+a*e^2)*(d+e*x)^11/e^7-1/2*c^3*d*(d+e*x)^12/e^7+1/13*c^3*(d+e*x)^13/e^7],
[(d+e*x)^5*(a+c*x^2)^3,x,2,1/6*(c*d^2+a*e^2)^3*(d+e*x)^6/e^7-6/7*c*d*(c*d^2+a*e^2)^2*(d+e*x)^7/e^7+3/8*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)*(d+e*x)^8/e^7-4/9*c^2*d*(5*c*d^2+3*a*e^2)*(d+e*x)^9/e^7+3/10*c^2*(5*c*d^2+a*e^2)*(d+e*x)^10/e^7-6/11*c^3*d*(d+e*x)^11/e^7+1/12*c^3*(d+e*x)^12/e^7],
[(d+e*x)^4*(a+c*x^2)^3,x,2,1/5*(c*d^2+a*e^2)^3*(d+e*x)^5/e^7-c*d*(c*d^2+a*e^2)^2*(d+e*x)^6/e^7+3/7*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)*(d+e*x)^7/e^7-1/2*c^2*d*(5*c*d^2+3*a*e^2)*(d+e*x)^8/e^7+1/3*c^2*(5*c*d^2+a*e^2)*(d+e*x)^9/e^7-3/5*c^3*d*(d+e*x)^10/e^7+1/11*c^3*(d+e*x)^11/e^7],
[(d+e*x)^3*(a+c*x^2)^3,x,3,a^3*d^3*x+a^2*d*(c*d^2+a*e^2)*x^3+1/4*a^3*e^3*x^4+3/5*a*c*d*(c*d^2+3*a*e^2)*x^5+1/2*a^2*c*e^3*x^6+1/7*c^2*d*(c*d^2+9*a*e^2)*x^7+3/8*a*c^2*e^3*x^8+1/3*c^3*d*e^2*x^9+1/10*c^3*e^3*x^10+3/8*d^2*e*(a+c*x^2)^4/c],
[(d+e*x)^2*(a+c*x^2)^3,x,3,a^3*d^2*x+1/3*a^2*(3*c*d^2+a*e^2)*x^3+3/5*a*c*(c*d^2+a*e^2)*x^5+1/7*c^2*(c*d^2+3*a*e^2)*x^7+1/9*c^3*e^2*x^9+1/4*d*e*(a+c*x^2)^4/c],
[(d+e*x)*(a+c*x^2)^3,x,3,a^3*d*x+a^2*c*d*x^3+3/5*a*c^2*d*x^5+1/7*c^3*d*x^7+1/8*e*(a+c*x^2)^4/c],
[(a+c*x^2)^3/(d+e*x),x,2,-c*d*(c^2*d^4+3*a*c*d^2*e^2+3*a^2*e^4)*x/e^6+1/2*c*(c^2*d^4+3*a*c*d^2*e^2+3*a^2*e^4)*x^2/e^5-1/3*c^2*d*(c*d^2+3*a*e^2)*x^3/e^4+1/4*c^2*(c*d^2+3*a*e^2)*x^4/e^3-1/5*c^3*d*x^5/e^2+1/6*c^3*x^6/e+(c*d^2+a*e^2)^3*log(d+e*x)/e^7],
[(a+c*x^2)^3/(d+e*x)^2,x,2,c*(5*c^2*d^4+9*a*c*d^2*e^2+3*a^2*e^4)*x/e^6-c^2*d*(2*c*d^2+3*a*e^2)*x^2/e^5+c^2*(c*d^2+a*e^2)*x^3/e^4-1/2*c^3*d*x^4/e^3+1/5*c^3*x^5/e^2-(c*d^2+a*e^2)^3/(e^7*(d+e*x))-6*c*d*(c*d^2+a*e^2)^2*log(d+e*x)/e^7],
[(a+c*x^2)^3/(d+e*x)^3,x,2,-c^2*d*(10*c*d^2+9*a*e^2)*x/e^6+3/2*c^2*(2*c*d^2+a*e^2)*x^2/e^5-c^3*d*x^3/e^4+1/4*c^3*x^4/e^3-1/2*(c*d^2+a*e^2)^3/(e^7*(d+e*x)^2)+6*c*d*(c*d^2+a*e^2)^2/(e^7*(d+e*x))+3*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)*log(d+e*x)/e^7],
[(a+c*x^2)^3/(d+e*x)^4,x,2,c^2*(10*c*d^2+3*a*e^2)*x/e^6-2*c^3*d*x^2/e^5+1/3*c^3*x^3/e^4-1/3*(c*d^2+a*e^2)^3/(e^7*(d+e*x)^3)+3*c*d*(c*d^2+a*e^2)^2/(e^7*(d+e*x)^2)-3*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)/(e^7*(d+e*x))-4*c^2*d*(5*c*d^2+3*a*e^2)*log(d+e*x)/e^7],
[(a+c*x^2)^3/(d+e*x)^5,x,2,-5*c^3*d*x/e^6+1/2*c^3*x^2/e^5-1/4*(c*d^2+a*e^2)^3/(e^7*(d+e*x)^4)+2*c*d*(c*d^2+a*e^2)^2/(e^7*(d+e*x)^3)-3/2*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)/(e^7*(d+e*x)^2)+4*c^2*d*(5*c*d^2+3*a*e^2)/(e^7*(d+e*x))+3*c^2*(5*c*d^2+a*e^2)*log(d+e*x)/e^7],
[(a+c*x^2)^3/(d+e*x)^6,x,2,c^3*x/e^6-1/5*(c*d^2+a*e^2)^3/(e^7*(d+e*x)^5)+3/2*c*d*(c*d^2+a*e^2)^2/(e^7*(d+e*x)^4)-c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)/(e^7*(d+e*x)^3)+2*c^2*d*(5*c*d^2+3*a*e^2)/(e^7*(d+e*x)^2)-3*c^2*(5*c*d^2+a*e^2)/(e^7*(d+e*x))-6*c^3*d*log(d+e*x)/e^7],
[(a+c*x^2)^3/(d+e*x)^7,x,2,-1/6*(c*d^2+a*e^2)^3/(e^7*(d+e*x)^6)+6/5*c*d*(c*d^2+a*e^2)^2/(e^7*(d+e*x)^5)-3/4*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)/(e^7*(d+e*x)^4)+4/3*c^2*d*(5*c*d^2+3*a*e^2)/(e^7*(d+e*x)^3)-3/2*c^2*(5*c*d^2+a*e^2)/(e^7*(d+e*x)^2)+6*c^3*d/(e^7*(d+e*x))+c^3*log(d+e*x)/e^7],
[(a+c*x^2)^3/(d+e*x)^8,x,2,-1/7*(c*d^2+a*e^2)^3/(e^7*(d+e*x)^7)+c*d*(c*d^2+a*e^2)^2/(e^7*(d+e*x)^6)-3/5*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)/(e^7*(d+e*x)^5)+c^2*d*(5*c*d^2+3*a*e^2)/(e^7*(d+e*x)^4)-c^2*(5*c*d^2+a*e^2)/(e^7*(d+e*x)^3)+3*c^3*d/(e^7*(d+e*x)^2)-c^3/(e^7*(d+e*x))],
[(a+c*x^2)^3/(d+e*x)^9,x,2,-1/8*(c*d^2+a*e^2)^3/(e^7*(d+e*x)^8)+6/7*c*d*(c*d^2+a*e^2)^2/(e^7*(d+e*x)^7)-1/2*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)/(e^7*(d+e*x)^6)+4/5*c^2*d*(5*c*d^2+3*a*e^2)/(e^7*(d+e*x)^5)-3/4*c^2*(5*c*d^2+a*e^2)/(e^7*(d+e*x)^4)+2*c^3*d/(e^7*(d+e*x)^3)-1/2*c^3/(e^7*(d+e*x)^2)],
[(a+c*x^2)^3/(d+e*x)^10,x,2,-1/9*(c*d^2+a*e^2)^3/(e^7*(d+e*x)^9)+3/4*c*d*(c*d^2+a*e^2)^2/(e^7*(d+e*x)^8)-3/7*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)/(e^7*(d+e*x)^7)+2/3*c^2*d*(5*c*d^2+3*a*e^2)/(e^7*(d+e*x)^6)-3/5*c^2*(5*c*d^2+a*e^2)/(e^7*(d+e*x)^5)+3/2*c^3*d/(e^7*(d+e*x)^4)-1/3*c^3/(e^7*(d+e*x)^3)],
[(d+e*x)^7*(a+c*x^2)^4,x,2,1/8*(c*d^2+a*e^2)^4*(d+e*x)^8/e^9-8/9*c*d*(c*d^2+a*e^2)^3*(d+e*x)^9/e^9+2/5*c*(c*d^2+a*e^2)^2*(7*c*d^2+a*e^2)*(d+e*x)^10/e^9-8/11*c^2*d*(c*d^2+a*e^2)*(7*c*d^2+3*a*e^2)*(d+e*x)^11/e^9+1/6*c^2*(35*c^2*d^4+30*a*c*d^2*e^2+3*a^2*e^4)*(d+e*x)^12/e^9-8/13*c^3*d*(7*c*d^2+3*a*e^2)*(d+e*x)^13/e^9+2/7*c^3*(7*c*d^2+a*e^2)*(d+e*x)^14/e^9-8/15*c^4*d*(d+e*x)^15/e^9+1/16*c^4*(d+e*x)^16/e^9],
[(d+e*x)^6*(a+c*x^2)^4,x,2,1/7*(c*d^2+a*e^2)^4*(d+e*x)^7/e^9-c*d*(c*d^2+a*e^2)^3*(d+e*x)^8/e^9+4/9*c*(c*d^2+a*e^2)^2*(7*c*d^2+a*e^2)*(d+e*x)^9/e^9-4/5*c^2*d*(c*d^2+a*e^2)*(7*c*d^2+3*a*e^2)*(d+e*x)^10/e^9+2/11*c^2*(35*c^2*d^4+30*a*c*d^2*e^2+3*a^2*e^4)*(d+e*x)^11/e^9-2/3*c^3*d*(7*c*d^2+3*a*e^2)*(d+e*x)^12/e^9+4/13*c^3*(7*c*d^2+a*e^2)*(d+e*x)^13/e^9-4/7*c^4*d*(d+e*x)^14/e^9+1/15*c^4*(d+e*x)^15/e^9],
[(d+e*x)^5*(a+c*x^2)^4,x,2,1/6*(c*d^2+a*e^2)^4*(d+e*x)^6/e^9-8/7*c*d*(c*d^2+a*e^2)^3*(d+e*x)^7/e^9+1/2*c*(c*d^2+a*e^2)^2*(7*c*d^2+a*e^2)*(d+e*x)^8/e^9-8/9*c^2*d*(c*d^2+a*e^2)*(7*c*d^2+3*a*e^2)*(d+e*x)^9/e^9+1/5*c^2*(35*c^2*d^4+30*a*c*d^2*e^2+3*a^2*e^4)*(d+e*x)^10/e^9-8/11*c^3*d*(7*c*d^2+3*a*e^2)*(d+e*x)^11/e^9+1/3*c^3*(7*c*d^2+a*e^2)*(d+e*x)^12/e^9-8/13*c^4*d*(d+e*x)^13/e^9+1/14*c^4*(d+e*x)^14/e^9],
[(d+e*x)^4*(a+c*x^2)^4,x,3,a^4*d^4*x+2/3*a^3*d^2*(2*c*d^2+3*a*e^2)*x^3+a^4*d*e^3*x^4+1/5*a^2*(6*c^2*d^4+24*a*c*d^2*e^2+a^2*e^4)*x^5+8/3*a^3*c*d*e^3*x^6+4/7*a*c*(c^2*d^4+9*a*c*d^2*e^2+a^2*e^4)*x^7+3*a^2*c^2*d*e^3*x^8+1/9*c^2*(c^2*d^4+24*a*c*d^2*e^2+6*a^2*e^4)*x^9+8/5*a*c^3*d*e^3*x^10+2/11*c^3*e^2*(3*c*d^2+2*a*e^2)*x^11+1/3*c^4*d*e^3*x^12+1/13*c^4*e^4*x^13+2/5*d^3*e*(a+c*x^2)^5/c],
[(d+e*x)^3*(a+c*x^2)^4,x,3,a^4*d^3*x+1/3*a^3*d*(4*c*d^2+3*a*e^2)*x^3+1/4*a^4*e^3*x^4+6/5*a^2*c*d*(c*d^2+2*a*e^2)*x^5+2/3*a^3*c*e^3*x^6+2/7*a*c^2*d*(2*c*d^2+9*a*e^2)*x^7+3/4*a^2*c^2*e^3*x^8+1/9*c^3*d*(c*d^2+12*a*e^2)*x^9+2/5*a*c^3*e^3*x^10+3/11*c^4*d*e^2*x^11+1/12*c^4*e^3*x^12+3/10*d^2*e*(a+c*x^2)^5/c],
[(d+e*x)^2*(a+c*x^2)^4,x,3,a^4*d^2*x+1/3*a^3*(4*c*d^2+a*e^2)*x^3+2/5*a^2*c*(3*c*d^2+2*a*e^2)*x^5+2/7*a*c^2*(2*c*d^2+3*a*e^2)*x^7+1/9*c^3*(c*d^2+4*a*e^2)*x^9+1/11*c^4*e^2*x^11+1/5*d*e*(a+c*x^2)^5/c],
[(d+e*x)*(a+c*x^2)^4,x,3,a^4*d*x+4/3*a^3*c*d*x^3+6/5*a^2*c^2*d*x^5+4/7*a*c^3*d*x^7+1/9*c^4*d*x^9+1/10*e*(a+c*x^2)^5/c],
[(a+c*x^2)^4/(d+e*x),x,2,-8*c*d*(c*d^2+a*e^2)^3*x/e^8+2*c*(c*d^2+a*e^2)^2*(7*c*d^2+a*e^2)*(d+e*x)^2/e^9-8/3*c^2*d*(c*d^2+a*e^2)*(7*c*d^2+3*a*e^2)*(d+e*x)^3/e^9+1/2*c^2*(35*c^2*d^4+30*a*c*d^2*e^2+3*a^2*e^4)*(d+e*x)^4/e^9-8/5*c^3*d*(7*c*d^2+3*a*e^2)*(d+e*x)^5/e^9+2/3*c^3*(7*c*d^2+a*e^2)*(d+e*x)^6/e^9-8/7*c^4*d*(d+e*x)^7/e^9+1/8*c^4*(d+e*x)^8/e^9+(c*d^2+a*e^2)^4*log(d+e*x)/e^9],
[(a+c*x^2)^4/(d+e*x)^2,x,2,c*(7*c^3*d^6+20*a*c^2*d^4*e^2+18*a^2*c*d^2*e^4+4*a^3*e^6)*x/e^8-c^2*d*(3*c^2*d^4+8*a*c*d^2*e^2+6*a^2*e^4)*x^2/e^7+1/3*c^2*(5*c^2*d^4+12*a*c*d^2*e^2+6*a^2*e^4)*x^3/e^6-c^3*d*(c*d^2+2*a*e^2)*x^4/e^5+1/5*c^3*(3*c*d^2+4*a*e^2)*x^5/e^4-1/3*c^4*d*x^6/e^3+1/7*c^4*x^7/e^2-(c*d^2+a*e^2)^4/(e^9*(d+e*x))-8*c*d*(c*d^2+a*e^2)^3*log(d+e*x)/e^9],
[(c+d*x)*(a+b*x^2)^4,x,3,a^4*c*x+4/3*a^3*b*c*x^3+6/5*a^2*b^2*c*x^5+4/7*a*b^3*c*x^7+1/9*b^4*c*x^9+1/10*d*(a+b*x^2)^5/b],

# p<0
[(d+e*x)^4/(a+c*x^2),x,5,e^2*(6*c*d^2-a*e^2)*x/c^2+2*d*e^3*x^2/c+1/3*e^4*x^3/c+2*d*e*(c*d^2-a*e^2)*log(a+c*x^2)/c^2+(c^2*d^4-6*a*c*d^2*e^2+a^2*e^4)*arctan(x*sqrt(c)/sqrt(a))/(c^(5/2)*sqrt(a))],
[(d+e*x)^3/(a+c*x^2),x,5,3*d*e^2*x/c+1/2*e^3*x^2/c+1/2*e*(3*c*d^2-a*e^2)*log(a+c*x^2)/c^2+d*(c*d^2-3*a*e^2)*arctan(x*sqrt(c)/sqrt(a))/(c^(3/2)*sqrt(a))],
[(d+e*x)^2/(a+c*x^2),x,5,e^2*x/c+d*e*log(a+c*x^2)/c+(c*d^2-a*e^2)*arctan(x*sqrt(c)/sqrt(a))/(c^(3/2)*sqrt(a))],
[(d+e*x)/(a+c*x^2),x,3,1/2*e*log(a+c*x^2)/c+d*arctan(x*sqrt(c)/sqrt(a))/(sqrt(a)*sqrt(c))],
[1/((d+e*x)*(a+c*x^2)),x,5,e*log(d+e*x)/(c*d^2+a*e^2)-1/2*e*log(a+c*x^2)/(c*d^2+a*e^2)+d*arctan(x*sqrt(c)/sqrt(a))*sqrt(c)/((c*d^2+a*e^2)*sqrt(a))],
[1/((d+e*x)^2*(a+c*x^2)),x,6,-e/((c*d^2+a*e^2)*(d+e*x))+2*c*d*e*log(d+e*x)/(c*d^2+a*e^2)^2-c*d*e*log(a+c*x^2)/(c*d^2+a*e^2)^2+(c*d^2-a*e^2)*arctan(x*sqrt(c)/sqrt(a))*sqrt(c)/((c*d^2+a*e^2)^2*sqrt(a))],
[1/((d+e*x)^3*(a+c*x^2)),x,6,-1/2*e/((c*d^2+a*e^2)*(d+e*x)^2)-2*c*d*e/((c*d^2+a*e^2)^2*(d+e*x))+c*e*(3*c*d^2-a*e^2)*log(d+e*x)/(c*d^2+a*e^2)^3-1/2*c*e*(3*c*d^2-a*e^2)*log(a+c*x^2)/(c*d^2+a*e^2)^3+c^(3/2)*d*(c*d^2-3*a*e^2)*arctan(x*sqrt(c)/sqrt(a))/((c*d^2+a*e^2)^3*sqrt(a))],
[(d+e*x)^5/(a+c*x^2)^2,x,6,-3/2*d*e^2*(2*c*d^2-5*a*e^2)*x/(a*c^2)-e^3*(2*c*d^2-a*e^2)*x^2/(a*c^2)-1/2*d*e^4*x^3/(a*c)-1/2*(a*e-c*d*x)*(d+e*x)^4/(a*c*(a+c*x^2))+1/2*d*(c^2*d^4+10*a*c*d^2*e^2-15*a^2*e^4)*arctan(x*sqrt(c)/sqrt(a))/(a^(3/2)*c^(5/2))+e^3*(5*c*d^2-a*e^2)*log(a+c*x^2)/c^3],
[(d+e*x)^4/(a+c*x^2)^2,x,6,-3/2*e^2*(c*d^2-a*e^2)*x/(a*c^2)-1/2*d*e^3*x^2/(a*c)-1/2*(a*e-c*d*x)*(d+e*x)^3/(a*c*(a+c*x^2))+1/2*(c^2*d^4+6*a*c*d^2*e^2-3*a^2*e^4)*arctan(x*sqrt(c)/sqrt(a))/(a^(3/2)*c^(5/2))+2*d*e^3*log(a+c*x^2)/c^2],
[(d+e*x)^3/(a+c*x^2)^2,x,5,-1/2*d*e^2*x/(a*c)-1/2*(a*e-c*d*x)*(d+e*x)^2/(a*c*(a+c*x^2))+1/2*d*(c*d^2+3*a*e^2)*arctan(x*sqrt(c)/sqrt(a))/(a^(3/2)*c^(3/2))+1/2*e^3*log(a+c*x^2)/c^2],
[(d+e*x)^2/(a+c*x^2)^2,x,2,-1/2*(a*e-c*d*x)*(d+e*x)/(a*c*(a+c*x^2))+1/2*(c*d^2+a*e^2)*arctan(x*sqrt(c)/sqrt(a))/(a^(3/2)*c^(3/2))],
[(d+e*x)/(a+c*x^2)^2,x,2,1/2*(-a*e+c*d*x)/(a*c*(a+c*x^2))+1/2*d*arctan(x*sqrt(c)/sqrt(a))/(a^(3/2)*sqrt(c))],
[1/((d+e*x)*(a+c*x^2)^2),x,6,1/2*(a*e+c*d*x)/(a*(c*d^2+a*e^2)*(a+c*x^2))+e^3*log(d+e*x)/(c*d^2+a*e^2)^2-1/2*e^3*log(a+c*x^2)/(c*d^2+a*e^2)^2+1/2*d*(c*d^2+3*a*e^2)*arctan(x*sqrt(c)/sqrt(a))*sqrt(c)/(a^(3/2)*(c*d^2+a*e^2)^2)],
[1/((d+e*x)^2*(a+c*x^2)^2),x,6,1/2*e*(c*d^2-3*a*e^2)/(a*(c*d^2+a*e^2)^2*(d+e*x))+1/2*(a*e+c*d*x)/(a*(c*d^2+a*e^2)*(d+e*x)*(a+c*x^2))+4*c*d*e^3*log(d+e*x)/(c*d^2+a*e^2)^3-2*c*d*e^3*log(a+c*x^2)/(c*d^2+a*e^2)^3+1/2*(c^2*d^4+6*a*c*d^2*e^2-3*a^2*e^4)*arctan(x*sqrt(c)/sqrt(a))*sqrt(c)/(a^(3/2)*(c*d^2+a*e^2)^3)],
[(d+e*x)^5/(a+c*x^2)^3,x,6,-1/8*d*e^2*(3*c*d^2+7*a*e^2)*x/(a^2*c^2)-1/4*(a*e-c*d*x)*(d+e*x)^4/(a*c*(a+c*x^2)^2)-1/8*(d+e*x)^2*(2*a*e*(c*d^2+2*a*e^2)-c*d*(3*c*d^2+5*a*e^2)*x)/(a^2*c^2*(a+c*x^2))+1/8*d*(3*c^2*d^4+10*a*c*d^2*e^2+15*a^2*e^4)*arctan(x*sqrt(c)/sqrt(a))/(a^(5/2)*c^(5/2))+1/2*e^5*log(a+c*x^2)/c^3],
[(d+e*x)^4/(a+c*x^2)^3,x,3,-1/4*(a*e-c*d*x)*(d+e*x)^3/(a*c*(a+c*x^2)^2)-3/8*(c*d^2+a*e^2)*(a*e-c*d*x)*(d+e*x)/(a^2*c^2*(a+c*x^2))+3/8*(c*d^2+a*e^2)^2*arctan(x*sqrt(c)/sqrt(a))/(a^(5/2)*c^(5/2))],
[(d+e*x)^3/(a+c*x^2)^3,x,3,1/4*x*(d+e*x)^3/(a*(a+c*x^2)^2)-3/8*d*(a*e-c*d*x)*(d+e*x)/(a^2*c*(a+c*x^2))+3/8*d*(c*d^2+a*e^2)*arctan(x*sqrt(c)/sqrt(a))/(a^(5/2)*c^(3/2))],
[(d+e*x)^2/(a+c*x^2)^3,x,3,-1/4*(a*e-c*d*x)*(d+e*x)/(a*c*(a+c*x^2)^2)+1/8*(-2*a*d*e+(3*c*d^2+a*e^2)*x)/(a^2*c*(a+c*x^2))+1/8*(3*c*d^2+a*e^2)*arctan(x*sqrt(c)/sqrt(a))/(a^(5/2)*c^(3/2))],
[(d+e*x)/(a+c*x^2)^3,x,3,1/4*(-a*e+c*d*x)/(a*c*(a+c*x^2)^2)+3/8*d*x/(a^2*(a+c*x^2))+3/8*d*arctan(x*sqrt(c)/sqrt(a))/(a^(5/2)*sqrt(c))],
[1/((d+e*x)*(a+c*x^2)^3),x,7,1/4*(a*e+c*d*x)/(a*(c*d^2+a*e^2)*(a+c*x^2)^2)+1/8*(4*a^2*e^3+c*d*(3*c*d^2+7*a*e^2)*x)/(a^2*(c*d^2+a*e^2)^2*(a+c*x^2))+e^5*log(d+e*x)/(c*d^2+a*e^2)^3-1/2*e^5*log(a+c*x^2)/(c*d^2+a*e^2)^3+1/8*d*(3*c^2*d^4+10*a*c*d^2*e^2+15*a^2*e^4)*arctan(x*sqrt(c)/sqrt(a))*sqrt(c)/(a^(5/2)*(c*d^2+a*e^2)^3)],
[1/((d+e*x)^2*(a+c*x^2)^3),x,7,3/8*e*(c*d^2-a*e^2)*(c*d^2+5*a*e^2)/(a^2*(c*d^2+a*e^2)^3*(d+e*x))+1/4*(a*e+c*d*x)/(a*(c*d^2+a*e^2)*(d+e*x)*(a+c*x^2)^2)+1/8*(-a*e*(c*d^2-5*a*e^2)+3*c*d*(c*d^2+3*a*e^2)*x)/(a^2*(c*d^2+a*e^2)^2*(d+e*x)*(a+c*x^2))+6*c*d*e^5*log(d+e*x)/(c*d^2+a*e^2)^4-3*c*d*e^5*log(a+c*x^2)/(c*d^2+a*e^2)^4+3/8*(c^3*d^6+5*a*c^2*d^4*e^2+15*a^2*c*d^2*e^4-5*a^3*e^6)*arctan(x*sqrt(c)/sqrt(a))*sqrt(c)/(a^(5/2)*(c*d^2+a*e^2)^4)],
[(d+e*x)^4/(a+c*x^2)^4,x,4,1/6*x*(d+e*x)^4/(a*(a+c*x^2)^3)-1/24*(a*e-5*c*d*x)*(d+e*x)^3/(a^2*c*(a+c*x^2)^2)-1/16*(5*c*d^2+a*e^2)*(a*e-c*d*x)*(d+e*x)/(a^3*c^2*(a+c*x^2))+1/16*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)*arctan(x*sqrt(c)/sqrt(a))/(a^(7/2)*c^(5/2))],
[(d+e*x)^3/(a+c*x^2)^4,x,4,1/6*x*(d+e*x)^3/(a*(a+c*x^2)^3)-1/24*(2*a*e-5*c*d*x)*(d+e*x)^2/(a^2*c*(a+c*x^2)^2)+1/48*(-4*a*e*(5*c*d^2+a*e^2)+c*d*(15*c*d^2-a*e^2)*x)/(a^3*c^2*(a+c*x^2))+1/16*d*(5*c*d^2+3*a*e^2)*arctan(x*sqrt(c)/sqrt(a))/(a^(7/2)*c^(3/2))],
[(d+e*x)^2/(a+c*x^2)^4,x,4,-1/6*(a*e-c*d*x)*(d+e*x)/(a*c*(a+c*x^2)^3)+1/24*(-4*a*d*e+(5*c*d^2+a*e^2)*x)/(a^2*c*(a+c*x^2)^2)+1/16*(5*c*d^2+a*e^2)*x/(a^3*c*(a+c*x^2))+1/16*(5*c*d^2+a*e^2)*arctan(x*sqrt(c)/sqrt(a))/(a^(7/2)*c^(3/2))],
[(d+e*x)/(a+c*x^2)^4,x,4,1/6*(-a*e+c*d*x)/(a*c*(a+c*x^2)^3)+5/24*d*x/(a^2*(a+c*x^2)^2)+5/16*d*x/(a^3*(a+c*x^2))+5/16*d*arctan(x*sqrt(c)/sqrt(a))/(a^(7/2)*sqrt(c))],
[1/((d+e*x)*(a+c*x^2)^4),x,8,1/6*(a*e+c*d*x)/(a*(c*d^2+a*e^2)*(a+c*x^2)^3)+1/24*(6*a^2*e^3+c*d*(5*c*d^2+11*a*e^2)*x)/(a^2*(c*d^2+a*e^2)^2*(a+c*x^2)^2)+1/16*(8*a^3*e^5+c*d*(5*c^2*d^4+16*a*c*d^2*e^2+19*a^2*e^4)*x)/(a^3*(c*d^2+a*e^2)^3*(a+c*x^2))+e^7*log(d+e*x)/(c*d^2+a*e^2)^4-1/2*e^7*log(a+c*x^2)/(c*d^2+a*e^2)^4+1/16*d*(5*c^3*d^6+21*a*c^2*d^4*e^2+35*a^2*c*d^2*e^4+35*a^3*e^6)*arctan(x*sqrt(c)/sqrt(a))*sqrt(c)/(a^(7/2)*(c*d^2+a*e^2)^4)],
[1/((d+e*x)^2*(a+c*x^2)^4),x,8,1/16*e*(5*c^3*d^6+23*a*c^2*d^4*e^2+47*a^2*c*d^2*e^4-35*a^3*e^6)/(a^3*(c*d^2+a*e^2)^4*(d+e*x))+1/6*(a*e+c*d*x)/(a*(c*d^2+a*e^2)*(d+e*x)*(a+c*x^2)^3)+1/24*(-a*e*(c*d^2-7*a*e^2)+c*d*(5*c*d^2+13*a*e^2)*x)/(a^2*(c*d^2+a*e^2)^2*(d+e*x)*(a+c*x^2)^2)+1/48*(-a*e*(5*c*d^2-7*a*e^2)*(c*d^2+5*a*e^2)+3*c*d*(5*c^2*d^4+18*a*c*d^2*e^2+29*a^2*e^4)*x)/(a^3*(c*d^2+a*e^2)^3*(d+e*x)*(a+c*x^2))+8*c*d*e^7*log(d+e*x)/(c*d^2+a*e^2)^5-4*c*d*e^7*log(a+c*x^2)/(c*d^2+a*e^2)^5+1/16*(5*c^4*d^8+28*a*c^3*d^6*e^2+70*a^2*c^2*d^4*e^4+140*a^3*c*d^2*e^6-35*a^4*e^8)*arctan(x*sqrt(c)/sqrt(a))*sqrt(c)/(a^(7/2)*(c*d^2+a*e^2)^5)],

# Integrands of the form (d+e x)^m (a+c x^2)^(p/2)

# p>0
[(d+e*x)^4*sqrt(a+c*x^2),x,6,3/10*d*e*(d+e*x)^2*(a+c*x^2)^(3/2)/c+1/6*e*(d+e*x)^3*(a+c*x^2)^(3/2)/c+1/120*e*(8*d*(13*c*d^2-8*a*e^2)+3*e*(16*c*d^2-5*a*e^2)*x)*(a+c*x^2)^(3/2)/c^2+1/16*a*(8*c^2*d^4-12*a*c*d^2*e^2+a^2*e^4)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(5/2)+1/16*(8*c^2*d^4-12*a*c*d^2*e^2+a^2*e^4)*x*sqrt(a+c*x^2)/c^2],
[(d+e*x)^3*sqrt(a+c*x^2),x,5,1/5*e*(d+e*x)^2*(a+c*x^2)^(3/2)/c+1/60*e*(8*(6*c*d^2-a*e^2)+21*c*d*e*x)*(a+c*x^2)^(3/2)/c^2+1/8*a*d*(4*c*d^2-3*a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(3/2)+1/8*d*(4*c*d^2-3*a*e^2)*x*sqrt(a+c*x^2)/c],
[(d+e*x)^2*sqrt(a+c*x^2),x,5,5/12*d*e*(a+c*x^2)^(3/2)/c+1/4*e*(d+e*x)*(a+c*x^2)^(3/2)/c+1/8*a*(4*c*d^2-a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(3/2)+1/8*(4*c*d^2-a*e^2)*x*sqrt(a+c*x^2)/c],
[(d+e*x)*sqrt(a+c*x^2),x,4,1/3*e*(a+c*x^2)^(3/2)/c+1/2*a*d*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/sqrt(c)+1/2*d*x*sqrt(a+c*x^2)],
[sqrt(a+c*x^2)/(d+e*x),x,6,-d*arctanh(x*sqrt(c)/sqrt(a+c*x^2))*sqrt(c)/e^2-arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))*sqrt(c*d^2+a*e^2)/e^2+sqrt(a+c*x^2)/e],
[sqrt(a+c*x^2)/(d+e*x)^2,x,6,arctanh(x*sqrt(c)/sqrt(a+c*x^2))*sqrt(c)/e^2+c*d*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(e^2*sqrt(c*d^2+a*e^2))-sqrt(a+c*x^2)/(e*(d+e*x))],
[sqrt(a+c*x^2)/(d+e*x)^3,x,3,-1/2*a*c*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(3/2)-1/2*(a*e-c*d*x)*sqrt(a+c*x^2)/((c*d^2+a*e^2)*(d+e*x)^2)],
[sqrt(a+c*x^2)/(d+e*x)^4,x,4,-1/3*e*(a+c*x^2)^(3/2)/((c*d^2+a*e^2)*(d+e*x)^3)-1/2*a*c^2*d*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(5/2)-1/2*c*d*(a*e-c*d*x)*sqrt(a+c*x^2)/((c*d^2+a*e^2)^2*(d+e*x)^2)],
[sqrt(a+c*x^2)/(d+e*x)^5,x,5,-1/4*e*(a+c*x^2)^(3/2)/((c*d^2+a*e^2)*(d+e*x)^4)-5/12*c*d*e*(a+c*x^2)^(3/2)/((c*d^2+a*e^2)^2*(d+e*x)^3)-1/8*a*c^2*(4*c*d^2-a*e^2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(7/2)-1/8*c*(4*c*d^2-a*e^2)*(a*e-c*d*x)*sqrt(a+c*x^2)/((c*d^2+a*e^2)^3*(d+e*x)^2)],
[(d+e*x)^4*(a+c*x^2)^(3/2),x,7,1/64*(16*c^2*d^4-16*a*c*d^2*e^2+a^2*e^4)*x*(a+c*x^2)^(3/2)/c^2+11/56*d*e*(d+e*x)^2*(a+c*x^2)^(5/2)/c+1/8*e*(d+e*x)^3*(a+c*x^2)^(5/2)/c+1/560*e*(4*d*(67*c*d^2-32*a*e^2)+5*e*(26*c*d^2-7*a*e^2)*x)*(a+c*x^2)^(5/2)/c^2+3/128*a^2*(16*c^2*d^4-16*a*c*d^2*e^2+a^2*e^4)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(5/2)+3/128*a*(16*c^2*d^4-16*a*c*d^2*e^2+a^2*e^4)*x*sqrt(a+c*x^2)/c^2],
[(d+e*x)^3*(a+c*x^2)^(3/2),x,6,1/8*d*(2*c*d^2-a*e^2)*x*(a+c*x^2)^(3/2)/c+1/7*e*(d+e*x)^2*(a+c*x^2)^(5/2)/c+1/70*e*(4*(8*c*d^2-a*e^2)+15*c*d*e*x)*(a+c*x^2)^(5/2)/c^2+3/16*a^2*d*(2*c*d^2-a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(3/2)+3/16*a*d*(2*c*d^2-a*e^2)*x*sqrt(a+c*x^2)/c],
[(d+e*x)^2*(a+c*x^2)^(3/2),x,6,1/24*(6*c*d^2-a*e^2)*x*(a+c*x^2)^(3/2)/c+7/30*d*e*(a+c*x^2)^(5/2)/c+1/6*e*(d+e*x)*(a+c*x^2)^(5/2)/c+1/16*a^2*(6*c*d^2-a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(3/2)+1/16*a*(6*c*d^2-a*e^2)*x*sqrt(a+c*x^2)/c],
[(d+e*x)*(a+c*x^2)^(3/2),x,5,1/4*d*x*(a+c*x^2)^(3/2)+1/5*e*(a+c*x^2)^(5/2)/c+3/8*a^2*d*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/sqrt(c)+3/8*a*d*x*sqrt(a+c*x^2)],
[(a+c*x^2)^(3/2)/(d+e*x),x,7,1/3*(a+c*x^2)^(3/2)/e-(c*d^2+a*e^2)^(3/2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/e^4-1/2*d*(2*c*d^2+3*a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))*sqrt(c)/e^4+1/2*(2*(c*d^2+a*e^2)-c*d*e*x)*sqrt(a+c*x^2)/e^3],
[(a+c*x^2)^(3/2)/(d+e*x)^2,x,7,-(a+c*x^2)^(3/2)/(e*(d+e*x))+3/2*(2*c*d^2+a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))*sqrt(c)/e^4+3*c*d*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))*sqrt(c*d^2+a*e^2)/e^4-3/2*c*(2*d-e*x)*sqrt(a+c*x^2)/e^3],
[(a+c*x^2)^(3/2)/(d+e*x)^3,x,7,-1/2*(a+c*x^2)^(3/2)/(e*(d+e*x)^2)-3*c^(3/2)*d*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/e^4-3/2*c*(2*c*d^2+a*e^2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(e^4*sqrt(c*d^2+a*e^2))+3/2*c*(2*d+e*x)*sqrt(a+c*x^2)/(e^3*(d+e*x))],
[(a+c*x^2)^(3/2)/(d+e*x)^4,x,7,-1/3*(a+c*x^2)^(3/2)/(e*(d+e*x)^3)+c^(3/2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/e^4+1/2*c^2*d*(2*c*d^2+3*a*e^2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(e^4*(c*d^2+a*e^2)^(3/2))-1/2*c*(d*(2*c*d^2+a*e^2)+e*(3*c*d^2+2*a*e^2)*x)*sqrt(a+c*x^2)/(e^3*(c*d^2+a*e^2)*(d+e*x)^2)],
[(a+c*x^2)^(3/2)/(d+e*x)^5,x,4,-1/4*(a*e-c*d*x)*(a+c*x^2)^(3/2)/((c*d^2+a*e^2)*(d+e*x)^4)-3/8*a^2*c^2*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(5/2)-3/8*a*c*(a*e-c*d*x)*sqrt(a+c*x^2)/((c*d^2+a*e^2)^2*(d+e*x)^2)],
[(a+c*x^2)^(3/2)/(d+e*x)^6,x,5,-1/4*c*d*(a*e-c*d*x)*(a+c*x^2)^(3/2)/((c*d^2+a*e^2)^2*(d+e*x)^4)-1/5*e*(a+c*x^2)^(5/2)/((c*d^2+a*e^2)*(d+e*x)^5)-3/8*a^2*c^3*d*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(7/2)-3/8*a*c^2*d*(a*e-c*d*x)*sqrt(a+c*x^2)/((c*d^2+a*e^2)^3*(d+e*x)^2)],
[(a+c*x^2)^(3/2)/(d+e*x)^7,x,6,-1/24*c*(6*c*d^2-a*e^2)*(a*e-c*d*x)*(a+c*x^2)^(3/2)/((c*d^2+a*e^2)^3*(d+e*x)^4)-1/6*e*(a+c*x^2)^(5/2)/((c*d^2+a*e^2)*(d+e*x)^6)-7/30*c*d*e*(a+c*x^2)^(5/2)/((c*d^2+a*e^2)^2*(d+e*x)^5)-1/16*a^2*c^3*(6*c*d^2-a*e^2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(9/2)-1/16*a*c^2*(6*c*d^2-a*e^2)*(a*e-c*d*x)*sqrt(a+c*x^2)/((c*d^2+a*e^2)^4*(d+e*x)^2)],
[(d+e*x)^4*(a+c*x^2)^(5/2),x,8,1/384*a*(80*c^2*d^4-60*a*c*d^2*e^2+3*a^2*e^4)*x*(a+c*x^2)^(3/2)/c^2+1/480*(80*c^2*d^4-60*a*c*d^2*e^2+3*a^2*e^4)*x*(a+c*x^2)^(5/2)/c^2+13/90*d*e*(d+e*x)^2*(a+c*x^2)^(7/2)/c+1/10*e*(d+e*x)^3*(a+c*x^2)^(7/2)/c+1/5040*e*(16*d*(103*c*d^2-40*a*e^2)+7*e*(116*c*d^2-27*a*e^2)*x)*(a+c*x^2)^(7/2)/c^2+1/256*a^3*(80*c^2*d^4-60*a*c*d^2*e^2+3*a^2*e^4)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(5/2)+1/256*a^2*(80*c^2*d^4-60*a*c*d^2*e^2+3*a^2*e^4)*x*sqrt(a+c*x^2)/c^2],
[(d+e*x)^3*(a+c*x^2)^(5/2),x,7,5/192*a*d*(8*c*d^2-3*a*e^2)*x*(a+c*x^2)^(3/2)/c+1/48*d*(8*c*d^2-3*a*e^2)*x*(a+c*x^2)^(5/2)/c+1/9*e*(d+e*x)^2*(a+c*x^2)^(7/2)/c+1/504*e*(16*(10*c*d^2-a*e^2)+77*c*d*e*x)*(a+c*x^2)^(7/2)/c^2+5/128*a^3*d*(8*c*d^2-3*a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(3/2)+5/128*a^2*d*(8*c*d^2-3*a*e^2)*x*sqrt(a+c*x^2)/c],
[(d+e*x)^2*(a+c*x^2)^(5/2),x,7,5/192*a*(8*c*d^2-a*e^2)*x*(a+c*x^2)^(3/2)/c+1/48*(8*c*d^2-a*e^2)*x*(a+c*x^2)^(5/2)/c+9/56*d*e*(a+c*x^2)^(7/2)/c+1/8*e*(d+e*x)*(a+c*x^2)^(7/2)/c+5/128*a^3*(8*c*d^2-a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(3/2)+5/128*a^2*(8*c*d^2-a*e^2)*x*sqrt(a+c*x^2)/c],
[(d+e*x)*(a+c*x^2)^(5/2),x,6,5/24*a*d*x*(a+c*x^2)^(3/2)+1/6*d*x*(a+c*x^2)^(5/2)+1/7*e*(a+c*x^2)^(7/2)/c+5/16*a^3*d*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/sqrt(c)+5/16*a^2*d*x*sqrt(a+c*x^2)],
[(a+c*x^2)^(5/2)/(d+e*x),x,8,1/12*(4*(c*d^2+a*e^2)-3*c*d*e*x)*(a+c*x^2)^(3/2)/e^3+1/5*(a+c*x^2)^(5/2)/e-(c*d^2+a*e^2)^(5/2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/e^6-1/8*d*(8*c^2*d^4+20*a*c*d^2*e^2+15*a^2*e^4)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))*sqrt(c)/e^6+1/8*(8*(c*d^2+a*e^2)^2-c*d*e*(4*c*d^2+7*a*e^2)*x)*sqrt(a+c*x^2)/e^5],
[(a+c*x^2)^(5/2)/(d+e*x)^2,x,8,-5/12*c*(4*d-3*e*x)*(a+c*x^2)^(3/2)/e^3-(a+c*x^2)^(5/2)/(e*(d+e*x))+5*c*d*(c*d^2+a*e^2)^(3/2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/e^6+5/8*(8*c^2*d^4+12*a*c*d^2*e^2+3*a^2*e^4)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))*sqrt(c)/e^6-5/8*c*(8*d*(c*d^2+a*e^2)-e*(4*c*d^2+3*a*e^2)*x)*sqrt(a+c*x^2)/e^5],
[(a+c*x^2)^(5/2)/(d+e*x)^3,x,8,5/6*c*(4*d+e*x)*(a+c*x^2)^(3/2)/(e^3*(d+e*x))-1/2*(a+c*x^2)^(5/2)/(e*(d+e*x)^2)-5/2*c^(3/2)*d*(4*c*d^2+3*a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/e^6-5/2*c*(4*c*d^2+a*e^2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))*sqrt(c*d^2+a*e^2)/e^6+5/2*c*(4*c*d^2+a*e^2-2*c*d*e*x)*sqrt(a+c*x^2)/e^5],
[(a+c*x^2)^(5/2)/(d+e*x)^4,x,8,5/6*c*(2*d+e*x)*(a+c*x^2)^(3/2)/(e^3*(d+e*x)^2)-1/3*(a+c*x^2)^(5/2)/(e*(d+e*x)^3)+5/2*c^(3/2)*(4*c*d^2+a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/e^6+5/2*c^2*d*(4*c*d^2+3*a*e^2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(e^6*sqrt(c*d^2+a*e^2))-5/2*c*(4*c*d^2+a*e^2+2*c*d*e*x)*sqrt(a+c*x^2)/(e^5*(d+e*x))],
[(a+c*x^2)^(5/2)/(d+e*x)^5,x,8,-5/24*c*(d*(4*c*d^2+a*e^2)+3*e*(2*c*d^2+a*e^2)*x)*(a+c*x^2)^(3/2)/(e^3*(c*d^2+a*e^2)*(d+e*x)^3)-1/4*(a+c*x^2)^(5/2)/(e*(d+e*x)^4)-5*c^(5/2)*d*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/e^6-5/8*c^2*(8*c^2*d^4+12*a*c*d^2*e^2+3*a^2*e^4)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(e^6*(c*d^2+a*e^2)^(3/2))+5/8*c^2*(8*d*(c*d^2+a*e^2)+e*(4*c*d^2+3*a*e^2)*x)*sqrt(a+c*x^2)/(e^5*(c*d^2+a*e^2)*(d+e*x))],
[(a+c*x^2)^(5/2)/(d+e*x)^6,x,8,-1/12*c*(d*(4*c*d^2+a*e^2)+e*(7*c*d^2+4*a*e^2)*x)*(a+c*x^2)^(3/2)/(e^3*(c*d^2+a*e^2)*(d+e*x)^4)-1/5*(a+c*x^2)^(5/2)/(e*(d+e*x)^5)+c^(5/2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/e^6+1/8*c^3*d*(8*c^2*d^4+20*a*c*d^2*e^2+15*a^2*e^4)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(e^6*(c*d^2+a*e^2)^(5/2))-1/8*c^2*(d*(8*c^2*d^4+12*a*c*d^2*e^2+a^2*e^4)+e*(12*c^2*d^4+23*a*c*d^2*e^2+8*a^2*e^4)*x)*sqrt(a+c*x^2)/(e^5*(c*d^2+a*e^2)^2*(d+e*x)^2)],
[(a+c*x^2)^(5/2)/(d+e*x)^7,x,5,-5/24*a*c*(a*e-c*d*x)*(a+c*x^2)^(3/2)/((c*d^2+a*e^2)^2*(d+e*x)^4)-1/6*(a*e-c*d*x)*(a+c*x^2)^(5/2)/((c*d^2+a*e^2)*(d+e*x)^6)-5/16*a^3*c^3*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(7/2)-5/16*a^2*c^2*(a*e-c*d*x)*sqrt(a+c*x^2)/((c*d^2+a*e^2)^3*(d+e*x)^2)],
[(a+c*x^2)^(5/2)/(d+e*x)^8,x,6,-5/24*a*c^2*d*(a*e-c*d*x)*(a+c*x^2)^(3/2)/((c*d^2+a*e^2)^3*(d+e*x)^4)-1/6*c*d*(a*e-c*d*x)*(a+c*x^2)^(5/2)/((c*d^2+a*e^2)^2*(d+e*x)^6)-1/7*e*(a+c*x^2)^(7/2)/((c*d^2+a*e^2)*(d+e*x)^7)-5/16*a^3*c^4*d*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(9/2)-5/16*a^2*c^3*d*(a*e-c*d*x)*sqrt(a+c*x^2)/((c*d^2+a*e^2)^4*(d+e*x)^2)],
[(a+c*x^2)^(5/2)/(d+e*x)^9,x,7,-5/192*a*c^2*(8*c*d^2-a*e^2)*(a*e-c*d*x)*(a+c*x^2)^(3/2)/((c*d^2+a*e^2)^4*(d+e*x)^4)-1/48*c*(8*c*d^2-a*e^2)*(a*e-c*d*x)*(a+c*x^2)^(5/2)/((c*d^2+a*e^2)^3*(d+e*x)^6)-1/8*e*(a+c*x^2)^(7/2)/((c*d^2+a*e^2)*(d+e*x)^8)-9/56*c*d*e*(a+c*x^2)^(7/2)/((c*d^2+a*e^2)^2*(d+e*x)^7)-5/128*a^3*c^4*(8*c*d^2-a*e^2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(11/2)-5/128*a^2*c^3*(8*c*d^2-a*e^2)*(a*e-c*d*x)*sqrt(a+c*x^2)/((c*d^2+a*e^2)^5*(d+e*x)^2)],
[sqrt(2+x^2)/(1+4*x),x,5,-1/16*arcsinh(x/sqrt(2))-1/16*arctanh((8-x)/(sqrt(33)*sqrt(2+x^2)))*sqrt(33)+1/4*sqrt(2+x^2)],
[sqrt(2+4*x^2)/(5+4*x),x,5,-5/8*arcsinh(x*sqrt(2))-1/8*arctanh((2-5*x)*sqrt(2/33)/sqrt(1+2*x^2))*sqrt(33)+1/2*sqrt(1+2*x^2)/sqrt(2)],
[(2+3*x)*sqrt(-5+7*x^2),x,4,1/7*(-5+7*x^2)^(3/2)-5*arctanh(x*sqrt(7)/sqrt(-5+7*x^2))/sqrt(7)+x*sqrt(-5+7*x^2)],

# p<0
[(d+e*x)^4/sqrt(a+c*x^2),x,5,1/8*(8*c^2*d^4-24*a*c*d^2*e^2+3*a^2*e^4)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(5/2)+7/12*d*e*(d+e*x)^2*sqrt(a+c*x^2)/c+1/4*e*(d+e*x)^3*sqrt(a+c*x^2)/c+1/24*e*(4*d*(19*c*d^2-16*a*e^2)+e*(26*c*d^2-9*a*e^2)*x)*sqrt(a+c*x^2)/c^2],
[(d+e*x)^3/sqrt(a+c*x^2),x,4,1/2*d*(2*c*d^2-3*a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(3/2)+1/3*e*(d+e*x)^2*sqrt(a+c*x^2)/c+1/6*e*(4*(4*c*d^2-a*e^2)+5*c*d*e*x)*sqrt(a+c*x^2)/c^2],
[(d+e*x)^2/sqrt(a+c*x^2),x,4,1/2*(2*c*d^2-a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(3/2)+3/2*d*e*sqrt(a+c*x^2)/c+1/2*e*(d+e*x)*sqrt(a+c*x^2)/c],
[(d+e*x)/sqrt(a+c*x^2),x,3,d*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/sqrt(c)+e*sqrt(a+c*x^2)/c],
[1/((d+e*x)*sqrt(a+c*x^2)),x,2,-arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/sqrt(c*d^2+a*e^2)],
[1/((d+e*x)^2*sqrt(a+c*x^2)),x,3,-c*d*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(3/2)-e*sqrt(a+c*x^2)/((c*d^2+a*e^2)*(d+e*x))],
[1/((d+e*x)^3*sqrt(a+c*x^2)),x,4,-1/2*c*(2*c*d^2-a*e^2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(5/2)-1/2*e*sqrt(a+c*x^2)/((c*d^2+a*e^2)*(d+e*x)^2)-3/2*c*d*e*sqrt(a+c*x^2)/((c*d^2+a*e^2)^2*(d+e*x))],
[1/((d+e*x)^4*sqrt(a+c*x^2)),x,5,-1/2*c^2*d*(2*c*d^2-3*a*e^2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(7/2)-1/3*e*sqrt(a+c*x^2)/((c*d^2+a*e^2)*(d+e*x)^3)-5/6*c*d*e*sqrt(a+c*x^2)/((c*d^2+a*e^2)^2*(d+e*x)^2)-1/6*c*e*(11*c*d^2-4*a*e^2)*sqrt(a+c*x^2)/((c*d^2+a*e^2)^3*(d+e*x))],
[(d+e*x)^4/(a+c*x^2)^(3/2),x,5,3/2*e^2*(4*c*d^2-a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(5/2)-(a*e-c*d*x)*(d+e*x)^3/(a*c*sqrt(a+c*x^2))-d*e*(d+e*x)^2*sqrt(a+c*x^2)/(a*c)-1/2*e*(4*d*(c*d^2-4*a*e^2)+e*(2*c*d^2-3*a*e^2)*x)*sqrt(a+c*x^2)/(a*c^2)],
[(d+e*x)^3/(a+c*x^2)^(3/2),x,4,3*d*e^2*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(3/2)-(a*e-c*d*x)*(d+e*x)^2/(a*c*sqrt(a+c*x^2))-e*(2*(c*d^2-a*e^2)+c*d*e*x)*sqrt(a+c*x^2)/(a*c^2)],
[(d+e*x)^2/(a+c*x^2)^(3/2),x,4,e^2*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(3/2)-(a*e-c*d*x)*(d+e*x)/(a*c*sqrt(a+c*x^2))-d*e*sqrt(a+c*x^2)/(a*c)],
[(d+e*x)/(a+c*x^2)^(3/2),x,1,(-a*e+c*d*x)/(a*c*sqrt(a+c*x^2))],
[1/((d+e*x)*(a+c*x^2)^(3/2)),x,4,-e^2*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(3/2)+(a*e+c*d*x)/(a*(c*d^2+a*e^2)*sqrt(a+c*x^2))],
[1/((d+e*x)^2*(a+c*x^2)^(3/2)),x,4,-3*c*d*e^2*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(5/2)+(a*e+c*d*x)/(a*(c*d^2+a*e^2)*(d+e*x)*sqrt(a+c*x^2))+e*(c*d^2-2*a*e^2)*sqrt(a+c*x^2)/(a*(c*d^2+a*e^2)^2*(d+e*x))],
[1/((d+e*x)^3*(a+c*x^2)^(3/2)),x,5,-3/2*c*e^2*(4*c*d^2-a*e^2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(7/2)+(a*e+c*d*x)/(a*(c*d^2+a*e^2)*(d+e*x)^2*sqrt(a+c*x^2))+1/2*e*(2*c*d^2-3*a*e^2)*sqrt(a+c*x^2)/(a*(c*d^2+a*e^2)^2*(d+e*x)^2)+1/2*c*d*e*(2*c*d^2-13*a*e^2)*sqrt(a+c*x^2)/(a*(c*d^2+a*e^2)^3*(d+e*x))],
[1/((d+e*x)^4*(a+c*x^2)^(3/2)),x,6,-5/2*c^2*d*e^2*(4*c*d^2-3*a*e^2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(9/2)+(a*e+c*d*x)/(a*(c*d^2+a*e^2)*(d+e*x)^3*sqrt(a+c*x^2))+1/3*e*(3*c*d^2-4*a*e^2)*sqrt(a+c*x^2)/(a*(c*d^2+a*e^2)^2*(d+e*x)^3)+1/6*c*d*e*(6*c*d^2-29*a*e^2)*sqrt(a+c*x^2)/(a*(c*d^2+a*e^2)^3*(d+e*x)^2)+1/6*c*e*(6*c^2*d^4-83*a*c*d^2*e^2+16*a^2*e^4)*sqrt(a+c*x^2)/(a*(c*d^2+a*e^2)^4*(d+e*x))],
[(d+e*x)^5/(a+c*x^2)^(5/2),x,5,-1/3*(a*e-c*d*x)*(d+e*x)^4/(a*c*(a+c*x^2)^(3/2))+5*d*e^4*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(5/2)-2/3*(d+e*x)^2*(2*a^2*e^3-c*d*(c*d^2+3*a*e^2)*x)/(a^2*c^2*sqrt(a+c*x^2))-1/3*e*(4*(c^2*d^4+4*a*c*d^2*e^2-2*a^2*e^4)+c*d*e*(2*c*d^2+7*a*e^2)*x)*sqrt(a+c*x^2)/(a^2*c^3)],
[(d+e*x)^4/(a+c*x^2)^(5/2),x,5,-1/3*(a*e-c*d*x)*(d+e*x)^3/(a*c*(a+c*x^2)^(3/2))+e^4*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/c^(5/2)-1/3*(d+e*x)*(a*e*(c*d^2+3*a*e^2)-2*c*d*(c*d^2+2*a*e^2)*x)/(a^2*c^2*sqrt(a+c*x^2))-1/3*d*e*(2*c*d^2+5*a*e^2)*sqrt(a+c*x^2)/(a^2*c^2)],
[(d+e*x)^3/(a+c*x^2)^(5/2),x,2,-1/3*(a*e-c*d*x)*(d+e*x)^2/(a*c*(a+c*x^2)^(3/2))-2/3*(c*d^2+a*e^2)*(a*e-c*d*x)/(a^2*c^2*sqrt(a+c*x^2))],
[(d+e*x)^2/(a+c*x^2)^(5/2),x,2,1/3*x*(d+e*x)^2/(a*(a+c*x^2)^(3/2))-2/3*d*(a*e-c*d*x)/(a^2*c*sqrt(a+c*x^2))],
[(d+e*x)/(a+c*x^2)^(5/2),x,2,1/3*(-a*e+c*d*x)/(a*c*(a+c*x^2)^(3/2))+2/3*d*x/(a^2*sqrt(a+c*x^2))],
[1/((d+e*x)*(a+c*x^2)^(5/2)),x,5,1/3*(a*e+c*d*x)/(a*(c*d^2+a*e^2)*(a+c*x^2)^(3/2))-e^4*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(5/2)+1/3*(3*a^2*e^3+c*d*(2*c*d^2+5*a*e^2)*x)/(a^2*(c*d^2+a*e^2)^2*sqrt(a+c*x^2))],
[1/((d+e*x)^2*(a+c*x^2)^(5/2)),x,5,1/3*(a*e+c*d*x)/(a*(c*d^2+a*e^2)*(d+e*x)*(a+c*x^2)^(3/2))-5*c*d*e^4*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(7/2)+1/3*(-a*e*(c*d^2-4*a*e^2)+c*d*(2*c*d^2+7*a*e^2)*x)/(a^2*(c*d^2+a*e^2)^2*(d+e*x)*sqrt(a+c*x^2))+1/3*e*(2*c^2*d^4+9*a*c*d^2*e^2-8*a^2*e^4)*sqrt(a+c*x^2)/(a^2*(c*d^2+a*e^2)^3*(d+e*x))],
[1/((d+e*x)^3*(a+c*x^2)^(5/2)),x,6,1/3*(a*e+c*d*x)/(a*(c*d^2+a*e^2)*(d+e*x)^2*(a+c*x^2)^(3/2))-5/2*c*e^4*(6*c*d^2-a*e^2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(9/2)+1/3*(-a*e*(2*c*d^2-5*a*e^2)+c*d*(2*c*d^2+9*a*e^2)*x)/(a^2*(c*d^2+a*e^2)^2*(d+e*x)^2*sqrt(a+c*x^2))+1/6*e*(4*c^2*d^4+24*a*c*d^2*e^2-15*a^2*e^4)*sqrt(a+c*x^2)/(a^2*(c*d^2+a*e^2)^3*(d+e*x)^2)+1/6*c*d*e*(4*c^2*d^4+28*a*c*d^2*e^2-81*a^2*e^4)*sqrt(a+c*x^2)/(a^2*(c*d^2+a*e^2)^4*(d+e*x))],
[(3+x)/sqrt(1-x^2),x,2,3*arcsin(x)-sqrt(1-x^2)],
[(1+x)/sqrt(4-x^2),x,2,arcsin(1/2*x)-sqrt(4-x^2)],
[(2+x)/sqrt(9+x^2),x,2,2*arcsinh(1/3*x)+sqrt(9+x^2)],
[(a+b*x)^2/sqrt(1-x^2),x,3,1/2*(2*a^2+b^2)*arcsin(x)-3/2*a*b*sqrt(1-x^2)-1/2*b*(a+b*x)*sqrt(1-x^2)],
[(a+b*x)^2/sqrt(1+x^2),x,3,1/2*(2*a^2-b^2)*arcsinh(x)+3/2*a*b*sqrt(1+x^2)+1/2*b*(a+b*x)*sqrt(1+x^2)],
[(2+3*x)/(4+x^2)^(3/2),x,1,1/2*(-6+x)/sqrt(4+x^2)],

# Integrands of the form (d+e x)^(m/2) (a+c x^2)^p

# p>0
[(d+e*x)^(5/2)*(a+c*x^2),x,2,2/7*(c*d^2+a*e^2)*(d+e*x)^(7/2)/e^3-4/9*c*d*(d+e*x)^(9/2)/e^3+2/11*c*(d+e*x)^(11/2)/e^3],
[(d+e*x)^(3/2)*(a+c*x^2),x,2,2/5*(c*d^2+a*e^2)*(d+e*x)^(5/2)/e^3-4/7*c*d*(d+e*x)^(7/2)/e^3+2/9*c*(d+e*x)^(9/2)/e^3],
[(a+c*x^2)*sqrt(d+e*x),x,2,2/3*(c*d^2+a*e^2)*(d+e*x)^(3/2)/e^3-4/5*c*d*(d+e*x)^(5/2)/e^3+2/7*c*(d+e*x)^(7/2)/e^3],
[(a+c*x^2)/sqrt(d+e*x),x,2,-4/3*c*d*(d+e*x)^(3/2)/e^3+2/5*c*(d+e*x)^(5/2)/e^3+2*(c*d^2+a*e^2)*sqrt(d+e*x)/e^3],
[(a+c*x^2)/(d+e*x)^(3/2),x,2,2/3*c*(d+e*x)^(3/2)/e^3-2*(c*d^2+a*e^2)/(e^3*sqrt(d+e*x))-4*c*d*sqrt(d+e*x)/e^3],
[(a+c*x^2)/(d+e*x)^(5/2),x,2,-2/3*(c*d^2+a*e^2)/(e^3*(d+e*x)^(3/2))+4*c*d/(e^3*sqrt(d+e*x))+2*c*sqrt(d+e*x)/e^3],
[(a+c*x^2)/(d+e*x)^(7/2),x,2,-2/5*(c*d^2+a*e^2)/(e^3*(d+e*x)^(5/2))+4/3*c*d/(e^3*(d+e*x)^(3/2))-2*c/(e^3*sqrt(d+e*x))],
[(d+e*x)^(5/2)*(a+c*x^2)^2,x,2,2/7*(c*d^2+a*e^2)^2*(d+e*x)^(7/2)/e^5-8/9*c*d*(c*d^2+a*e^2)*(d+e*x)^(9/2)/e^5+4/11*c*(3*c*d^2+a*e^2)*(d+e*x)^(11/2)/e^5-8/13*c^2*d*(d+e*x)^(13/2)/e^5+2/15*c^2*(d+e*x)^(15/2)/e^5],
[(d+e*x)^(3/2)*(a+c*x^2)^2,x,2,2/5*(c*d^2+a*e^2)^2*(d+e*x)^(5/2)/e^5-8/7*c*d*(c*d^2+a*e^2)*(d+e*x)^(7/2)/e^5+4/9*c*(3*c*d^2+a*e^2)*(d+e*x)^(9/2)/e^5-8/11*c^2*d*(d+e*x)^(11/2)/e^5+2/13*c^2*(d+e*x)^(13/2)/e^5],
[(a+c*x^2)^2*sqrt(d+e*x),x,2,2/3*(c*d^2+a*e^2)^2*(d+e*x)^(3/2)/e^5-8/5*c*d*(c*d^2+a*e^2)*(d+e*x)^(5/2)/e^5+4/7*c*(3*c*d^2+a*e^2)*(d+e*x)^(7/2)/e^5-8/9*c^2*d*(d+e*x)^(9/2)/e^5+2/11*c^2*(d+e*x)^(11/2)/e^5],
[(a+c*x^2)^2/sqrt(d+e*x),x,2,-8/3*c*d*(c*d^2+a*e^2)*(d+e*x)^(3/2)/e^5+4/5*c*(3*c*d^2+a*e^2)*(d+e*x)^(5/2)/e^5-8/7*c^2*d*(d+e*x)^(7/2)/e^5+2/9*c^2*(d+e*x)^(9/2)/e^5+2*(c*d^2+a*e^2)^2*sqrt(d+e*x)/e^5],
[(a+c*x^2)^2/(d+e*x)^(3/2),x,2,4/3*c*(3*c*d^2+a*e^2)*(d+e*x)^(3/2)/e^5-8/5*c^2*d*(d+e*x)^(5/2)/e^5+2/7*c^2*(d+e*x)^(7/2)/e^5-2*(c*d^2+a*e^2)^2/(e^5*sqrt(d+e*x))-8*c*d*(c*d^2+a*e^2)*sqrt(d+e*x)/e^5],
[(a+c*x^2)^2/(d+e*x)^(5/2),x,2,-2/3*(c*d^2+a*e^2)^2/(e^5*(d+e*x)^(3/2))-8/3*c^2*d*(d+e*x)^(3/2)/e^5+2/5*c^2*(d+e*x)^(5/2)/e^5+8*c*d*(c*d^2+a*e^2)/(e^5*sqrt(d+e*x))+4*c*(3*c*d^2+a*e^2)*sqrt(d+e*x)/e^5],
[(a+c*x^2)^2/(d+e*x)^(7/2),x,2,-2/5*(c*d^2+a*e^2)^2/(e^5*(d+e*x)^(5/2))+8/3*c*d*(c*d^2+a*e^2)/(e^5*(d+e*x)^(3/2))+2/3*c^2*(d+e*x)^(3/2)/e^5-4*c*(3*c*d^2+a*e^2)/(e^5*sqrt(d+e*x))-8*c^2*d*sqrt(d+e*x)/e^5],
[(d+e*x)^(5/2)*(a+c*x^2)^3,x,2,2/7*(c*d^2+a*e^2)^3*(d+e*x)^(7/2)/e^7-4/3*c*d*(c*d^2+a*e^2)^2*(d+e*x)^(9/2)/e^7+6/11*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)*(d+e*x)^(11/2)/e^7-8/13*c^2*d*(5*c*d^2+3*a*e^2)*(d+e*x)^(13/2)/e^7+2/5*c^2*(5*c*d^2+a*e^2)*(d+e*x)^(15/2)/e^7-12/17*c^3*d*(d+e*x)^(17/2)/e^7+2/19*c^3*(d+e*x)^(19/2)/e^7],
[(d+e*x)^(3/2)*(a+c*x^2)^3,x,2,2/5*(c*d^2+a*e^2)^3*(d+e*x)^(5/2)/e^7-12/7*c*d*(c*d^2+a*e^2)^2*(d+e*x)^(7/2)/e^7+2/3*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)*(d+e*x)^(9/2)/e^7-8/11*c^2*d*(5*c*d^2+3*a*e^2)*(d+e*x)^(11/2)/e^7+6/13*c^2*(5*c*d^2+a*e^2)*(d+e*x)^(13/2)/e^7-4/5*c^3*d*(d+e*x)^(15/2)/e^7+2/17*c^3*(d+e*x)^(17/2)/e^7],
[(a+c*x^2)^3*sqrt(d+e*x),x,2,2/3*(c*d^2+a*e^2)^3*(d+e*x)^(3/2)/e^7-12/5*c*d*(c*d^2+a*e^2)^2*(d+e*x)^(5/2)/e^7+6/7*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)*(d+e*x)^(7/2)/e^7-8/9*c^2*d*(5*c*d^2+3*a*e^2)*(d+e*x)^(9/2)/e^7+6/11*c^2*(5*c*d^2+a*e^2)*(d+e*x)^(11/2)/e^7-12/13*c^3*d*(d+e*x)^(13/2)/e^7+2/15*c^3*(d+e*x)^(15/2)/e^7],
[(a+c*x^2)^3/sqrt(d+e*x),x,2,-4*c*d*(c*d^2+a*e^2)^2*(d+e*x)^(3/2)/e^7+6/5*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)*(d+e*x)^(5/2)/e^7-8/7*c^2*d*(5*c*d^2+3*a*e^2)*(d+e*x)^(7/2)/e^7+2/3*c^2*(5*c*d^2+a*e^2)*(d+e*x)^(9/2)/e^7-12/11*c^3*d*(d+e*x)^(11/2)/e^7+2/13*c^3*(d+e*x)^(13/2)/e^7+2*(c*d^2+a*e^2)^3*sqrt(d+e*x)/e^7],
[(a+c*x^2)^3/(d+e*x)^(3/2),x,2,2*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)*(d+e*x)^(3/2)/e^7-8/5*c^2*d*(5*c*d^2+3*a*e^2)*(d+e*x)^(5/2)/e^7+6/7*c^2*(5*c*d^2+a*e^2)*(d+e*x)^(7/2)/e^7-4/3*c^3*d*(d+e*x)^(9/2)/e^7+2/11*c^3*(d+e*x)^(11/2)/e^7-2*(c*d^2+a*e^2)^3/(e^7*sqrt(d+e*x))-12*c*d*(c*d^2+a*e^2)^2*sqrt(d+e*x)/e^7],
[(a+c*x^2)^3/(d+e*x)^(5/2),x,2,-2/3*(c*d^2+a*e^2)^3/(e^7*(d+e*x)^(3/2))-8/3*c^2*d*(5*c*d^2+3*a*e^2)*(d+e*x)^(3/2)/e^7+6/5*c^2*(5*c*d^2+a*e^2)*(d+e*x)^(5/2)/e^7-12/7*c^3*d*(d+e*x)^(7/2)/e^7+2/9*c^3*(d+e*x)^(9/2)/e^7+12*c*d*(c*d^2+a*e^2)^2/(e^7*sqrt(d+e*x))+6*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)*sqrt(d+e*x)/e^7],
[(a+c*x^2)^3/(d+e*x)^(7/2),x,2,-2/5*(c*d^2+a*e^2)^3/(e^7*(d+e*x)^(5/2))+4*c*d*(c*d^2+a*e^2)^2/(e^7*(d+e*x)^(3/2))+2*c^2*(5*c*d^2+a*e^2)*(d+e*x)^(3/2)/e^7-12/5*c^3*d*(d+e*x)^(5/2)/e^7+2/7*c^3*(d+e*x)^(7/2)/e^7-6*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)/(e^7*sqrt(d+e*x))-8*c^2*d*(5*c*d^2+3*a*e^2)*sqrt(d+e*x)/e^7],

# p<0
[(d+e*x)^(5/2)/(a-c*x^2),x,6,-2/3*e*(d+e*x)^(3/2)/c-arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))*(-e*sqrt(a)+d*sqrt(c))^(5/2)/(c^(7/4)*sqrt(a))+arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))*(e*sqrt(a)+d*sqrt(c))^(5/2)/(c^(7/4)*sqrt(a))-4*d*e*sqrt(d+e*x)/c],
[(d+e*x)^(3/2)/(a-c*x^2),x,5,-arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))*(-e*sqrt(a)+d*sqrt(c))^(3/2)/(c^(5/4)*sqrt(a))+arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))*(e*sqrt(a)+d*sqrt(c))^(3/2)/(c^(5/4)*sqrt(a))-2*e*sqrt(d+e*x)/c],
[sqrt(d+e*x)/(a-c*x^2),x,4,-arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))*sqrt(-e*sqrt(a)+d*sqrt(c))/(c^(3/4)*sqrt(a))+arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))*sqrt(e*sqrt(a)+d*sqrt(c))/(c^(3/4)*sqrt(a))],
[1/((a-c*x^2)*sqrt(d+e*x)),x,4,-arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))/(c^(1/4)*sqrt(a)*sqrt(-e*sqrt(a)+d*sqrt(c)))+arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))/(c^(1/4)*sqrt(a)*sqrt(e*sqrt(a)+d*sqrt(c)))],
[1/((d+e*x)^(3/2)*(a-c*x^2)),x,5,-c^(1/4)*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))/(sqrt(a)*(-e*sqrt(a)+d*sqrt(c))^(3/2))+c^(1/4)*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))/(sqrt(a)*(e*sqrt(a)+d*sqrt(c))^(3/2))+2*e/((c*d^2-a*e^2)*sqrt(d+e*x))],
[1/((d+e*x)^(5/2)*(a-c*x^2)),x,6,2/3*e/((c*d^2-a*e^2)*(d+e*x)^(3/2))-c^(3/4)*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))/(sqrt(a)*(-e*sqrt(a)+d*sqrt(c))^(5/2))+c^(3/4)*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))/(sqrt(a)*(e*sqrt(a)+d*sqrt(c))^(5/2))+4*c*d*e/((c*d^2-a*e^2)^2*sqrt(d+e*x))],
[(d+e*x)^(5/2)/(a+c*x^2),x,12,2/3*e*(d+e*x)^(3/2)/c+4*d*e*sqrt(d+e*x)/c-e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(2*c^(3/2)*d^3+2*a*d*e^2*sqrt(c)-(3*c*d^2-a*e^2)*sqrt(c*d^2+a*e^2))/(c^(7/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))+e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(2*c^(3/2)*d^3+2*a*d*e^2*sqrt(c)-(3*c*d^2-a*e^2)*sqrt(c*d^2+a*e^2))/(c^(7/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))+1/2*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(2*c^(3/2)*d^3+2*a*d*e^2*sqrt(c)+(3*c*d^2-a*e^2)*sqrt(c*d^2+a*e^2))/(c^(7/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))-1/2*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(2*c^(3/2)*d^3+2*a*d*e^2*sqrt(c)+(3*c*d^2-a*e^2)*sqrt(c*d^2+a*e^2))/(c^(7/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[(d+e*x)^(3/2)/(a+c*x^2),x,11,2*e*sqrt(d+e*x)/c-e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(c*d^2+a*e^2-2*d*sqrt(c)*sqrt(c*d^2+a*e^2))/(c^(5/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))+e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(c*d^2+a*e^2-2*d*sqrt(c)*sqrt(c*d^2+a*e^2))/(c^(5/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))+1/2*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(c*d^2+a*e^2+2*d*sqrt(c)*sqrt(c*d^2+a*e^2))/(c^(5/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))-1/2*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(c*d^2+a*e^2+2*d*sqrt(c)*sqrt(c*d^2+a*e^2))/(c^(5/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[sqrt(d+e*x)/(a+c*x^2),x,10,e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))/(c^(3/4)*sqrt(2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))/(c^(3/4)*sqrt(2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))+1/2*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/(c^(3/4)*sqrt(2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))-1/2*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/(c^(3/4)*sqrt(2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[1/((a+c*x^2)*sqrt(d+e*x)),x,10,e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))/(c^(1/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))/(c^(1/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/2*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/(c^(1/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))+1/2*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/(c^(1/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[1/((d+e*x)^(3/2)*(a+c*x^2)),x,11,-2*e/((c*d^2+a*e^2)*sqrt(d+e*x))+c^(1/4)*e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(2*d*sqrt(c)-sqrt(c*d^2+a*e^2))/((c*d^2+a*e^2)^(3/2)*sqrt(2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-c^(1/4)*e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(2*d*sqrt(c)-sqrt(c*d^2+a*e^2))/((c*d^2+a*e^2)^(3/2)*sqrt(2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/2*c^(1/4)*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(2*d*sqrt(c)+sqrt(c*d^2+a*e^2))/((c*d^2+a*e^2)^(3/2)*sqrt(2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))+1/2*c^(1/4)*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(2*d*sqrt(c)+sqrt(c*d^2+a*e^2))/((c*d^2+a*e^2)^(3/2)*sqrt(2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[1/((d+e*x)^(5/2)*(a+c*x^2)),x,12,-2/3*e/((c*d^2+a*e^2)*(d+e*x)^(3/2))-4*c*d*e/((c*d^2+a*e^2)^2*sqrt(d+e*x))+c^(3/4)*e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(3*c*d^2-a*e^2-2*d*sqrt(c)*sqrt(c*d^2+a*e^2))/((c*d^2+a*e^2)^(5/2)*sqrt(2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-c^(3/4)*e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(3*c*d^2-a*e^2-2*d*sqrt(c)*sqrt(c*d^2+a*e^2))/((c*d^2+a*e^2)^(5/2)*sqrt(2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/2*c^(3/4)*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(3*c*d^2-a*e^2+2*d*sqrt(c)*sqrt(c*d^2+a*e^2))/((c*d^2+a*e^2)^(5/2)*sqrt(2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))+1/2*c^(3/4)*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(3*c*d^2-a*e^2+2*d*sqrt(c)*sqrt(c*d^2+a*e^2))/((c*d^2+a*e^2)^(5/2)*sqrt(2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[(d+e*x)^(7/2)/(a-c*x^2)^2,x,7,1/2*d*e*(d+e*x)^(3/2)/(a*c)+1/2*(a*e+c*d*x)*(d+e*x)^(5/2)/(a*c*(a-c*x^2))+1/4*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))*(e*sqrt(a)+d*sqrt(c))^(5/2)*(-5*e*sqrt(a)+2*d*sqrt(c))/(a^(3/2)*c^(9/4))-1/4*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))*(-e*sqrt(a)+d*sqrt(c))^(5/2)*(5*e*sqrt(a)+2*d*sqrt(c))/(a^(3/2)*c^(9/4))+1/2*e*(c*d^2+5*a*e^2)*sqrt(d+e*x)/(a*c^2)],
[(d+e*x)^(5/2)/(a-c*x^2)^2,x,6,1/2*(a*e+c*d*x)*(d+e*x)^(3/2)/(a*c*(a-c*x^2))+1/4*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))*(e*sqrt(a)+d*sqrt(c))^(3/2)*(-3*e*sqrt(a)+2*d*sqrt(c))/(a^(3/2)*c^(7/4))-1/4*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))*(-e*sqrt(a)+d*sqrt(c))^(3/2)*(3*e*sqrt(a)+2*d*sqrt(c))/(a^(3/2)*c^(7/4))+1/2*d*e*sqrt(d+e*x)/(a*c)],
[(d+e*x)^(3/2)/(a-c*x^2)^2,x,5,1/2*(a*e+c*d*x)*sqrt(d+e*x)/(a*c*(a-c*x^2))-1/4*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))*(e*sqrt(a)+2*d*sqrt(c))*sqrt(-e*sqrt(a)+d*sqrt(c))/(a^(3/2)*c^(5/4))+1/4*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))*(-e*sqrt(a)+2*d*sqrt(c))*sqrt(e*sqrt(a)+d*sqrt(c))/(a^(3/2)*c^(5/4))],
[sqrt(d+e*x)/(a-c*x^2)^2,x,5,1/2*x*sqrt(d+e*x)/(a*(a-c*x^2))-1/4*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))*(-e+2*d*sqrt(c)/sqrt(a))/(a*c^(3/4)*sqrt(-e*sqrt(a)+d*sqrt(c)))+1/4*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))*(e+2*d*sqrt(c)/sqrt(a))/(a*c^(3/4)*sqrt(e*sqrt(a)+d*sqrt(c)))],
[1/((a-c*x^2)^2*sqrt(d+e*x)),x,5,-1/4*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))*(-3*e*sqrt(a)+2*d*sqrt(c))/(a^(3/2)*c^(1/4)*(-e*sqrt(a)+d*sqrt(c))^(3/2))+1/4*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))*(3*e*sqrt(a)+2*d*sqrt(c))/(a^(3/2)*c^(1/4)*(e*sqrt(a)+d*sqrt(c))^(3/2))-1/2*(a*e-c*d*x)*sqrt(d+e*x)/(a*(c*d^2-a*e^2)*(a-c*x^2))],
[1/((d+e*x)^(3/2)*(a-c*x^2)^2),x,6,-1/4*c^(1/4)*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))*(-5*e*sqrt(a)+2*d*sqrt(c))/(a^(3/2)*(-e*sqrt(a)+d*sqrt(c))^(5/2))+1/4*c^(1/4)*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))*(5*e*sqrt(a)+2*d*sqrt(c))/(a^(3/2)*(e*sqrt(a)+d*sqrt(c))^(5/2))-1/2*e*(c*d^2+5*a*e^2)/(a*(c*d^2-a*e^2)^2*sqrt(d+e*x))+1/2*(-a*e+c*d*x)/(a*(c*d^2-a*e^2)*(a-c*x^2)*sqrt(d+e*x))],
[1/((d+e*x)^(5/2)*(a-c*x^2)^2),x,7,-1/6*e*(3*c*d^2+7*a*e^2)/(a*(c*d^2-a*e^2)^2*(d+e*x)^(3/2))+1/2*(-a*e+c*d*x)/(a*(c*d^2-a*e^2)*(d+e*x)^(3/2)*(a-c*x^2))-1/4*c^(3/4)*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))*(-7*e*sqrt(a)+2*d*sqrt(c))/(a^(3/2)*(-e*sqrt(a)+d*sqrt(c))^(7/2))+1/4*c^(3/4)*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))*(7*e*sqrt(a)+2*d*sqrt(c))/(a^(3/2)*(e*sqrt(a)+d*sqrt(c))^(7/2))-1/2*c*d*e*(c*d^2+19*a*e^2)/(a*(c*d^2-a*e^2)^3*sqrt(d+e*x))],
[(d+e*x)^(7/2)/(a+c*x^2)^2,x,13,-1/2*d*e*(d+e*x)^(3/2)/(a*c)-1/2*(a*e-c*d*x)*(d+e*x)^(5/2)/(a*c*(a+c*x^2))-1/2*e*(c*d^2-5*a*e^2)*sqrt(d+e*x)/(a*c^2)+1/4*e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(c^2*d^4-4*a*c*d^2*e^2-5*a^2*e^4+d*(c*d^2+13*a*e^2)*sqrt(c)*sqrt(c*d^2+a*e^2))/(a*c^(9/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/4*e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(c^2*d^4-4*a*c*d^2*e^2-5*a^2*e^4+d*(c*d^2+13*a*e^2)*sqrt(c)*sqrt(c*d^2+a*e^2))/(a*c^(9/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/8*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(c^2*d^4-4*a*c*d^2*e^2-5*a^2*e^4-d*(c*d^2+13*a*e^2)*sqrt(c)*sqrt(c*d^2+a*e^2))/(a*c^(9/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))+1/8*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(c^2*d^4-4*a*c*d^2*e^2-5*a^2*e^4-d*(c*d^2+13*a*e^2)*sqrt(c)*sqrt(c*d^2+a*e^2))/(a*c^(9/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[(d+e*x)^(5/2)/(a+c*x^2)^2,x,12,-1/2*(a*e-c*d*x)*(d+e*x)^(3/2)/(a*c*(a+c*x^2))-1/2*d*e*sqrt(d+e*x)/(a*c)+1/4*e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(c^(3/2)*d^3+a*d*e^2*sqrt(c)+(c*d^2+3*a*e^2)*sqrt(c*d^2+a*e^2))/(a*c^(7/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/4*e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(c^(3/2)*d^3+a*d*e^2*sqrt(c)+(c*d^2+3*a*e^2)*sqrt(c*d^2+a*e^2))/(a*c^(7/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/8*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(c^(3/2)*d^3+a*d*e^2*sqrt(c)-(c*d^2+3*a*e^2)*sqrt(c*d^2+a*e^2))/(a*c^(7/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))+1/8*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(c^(3/2)*d^3+a*d*e^2*sqrt(c)-(c*d^2+3*a*e^2)*sqrt(c*d^2+a*e^2))/(a*c^(7/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[(d+e*x)^(3/2)/(a+c*x^2)^2,x,11,-1/2*(a*e-c*d*x)*sqrt(d+e*x)/(a*c*(a+c*x^2))+1/4*e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(c*d^2+a*e^2+d*sqrt(c)*sqrt(c*d^2+a*e^2))/(a*c^(5/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/4*e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(c*d^2+a*e^2+d*sqrt(c)*sqrt(c*d^2+a*e^2))/(a*c^(5/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/8*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(c*d^2+a*e^2-d*sqrt(c)*sqrt(c*d^2+a*e^2))/(a*c^(5/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))+1/8*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(c*d^2+a*e^2-d*sqrt(c)*sqrt(c*d^2+a*e^2))/(a*c^(5/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[sqrt(d+e*x)/(a+c*x^2)^2,x,11,1/2*x*sqrt(d+e*x)/(a*(a+c*x^2))+1/4*e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(d*sqrt(c)+sqrt(c*d^2+a*e^2))/(a*c^(3/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/4*e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(d*sqrt(c)+sqrt(c*d^2+a*e^2))/(a*c^(3/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/8*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(d-sqrt(c*d^2+a*e^2)/sqrt(c))/(a*c^(1/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))+1/8*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(d-sqrt(c*d^2+a*e^2)/sqrt(c))/(a*c^(1/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[1/((a+c*x^2)^2*sqrt(d+e*x)),x,11,1/2*(a*e+c*d*x)*sqrt(d+e*x)/(a*(c*d^2+a*e^2)*(a+c*x^2))+1/4*e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(c*d^2+3*a*e^2+d*sqrt(c)*sqrt(c*d^2+a*e^2))/(a*c^(1/4)*(c*d^2+a*e^2)^(3/2)*sqrt(2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/4*e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(c*d^2+3*a*e^2+d*sqrt(c)*sqrt(c*d^2+a*e^2))/(a*c^(1/4)*(c*d^2+a*e^2)^(3/2)*sqrt(2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/8*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(c*d^2+3*a*e^2-d*sqrt(c)*sqrt(c*d^2+a*e^2))/(a*c^(1/4)*(c*d^2+a*e^2)^(3/2)*sqrt(2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))+1/8*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(c*d^2+3*a*e^2-d*sqrt(c)*sqrt(c*d^2+a*e^2))/(a*c^(1/4)*(c*d^2+a*e^2)^(3/2)*sqrt(2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[1/((d+e*x)^(3/2)*(a+c*x^2)^2),x,12,1/2*e*(c*d^2-5*a*e^2)/(a*(c*d^2+a*e^2)^2*sqrt(d+e*x))+1/2*(a*e+c*d*x)/(a*(c*d^2+a*e^2)*(a+c*x^2)*sqrt(d+e*x))+1/4*c^(1/4)*e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(c^(3/2)*d^3+13*a*d*e^2*sqrt(c)+(c*d^2-5*a*e^2)*sqrt(c*d^2+a*e^2))/(a*(c*d^2+a*e^2)^(5/2)*sqrt(2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/4*c^(1/4)*e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(c^(3/2)*d^3+13*a*d*e^2*sqrt(c)+(c*d^2-5*a*e^2)*sqrt(c*d^2+a*e^2))/(a*(c*d^2+a*e^2)^(5/2)*sqrt(2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/8*c^(1/4)*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(c^(3/2)*d^3+13*a*d*e^2*sqrt(c)-(c*d^2-5*a*e^2)*sqrt(c*d^2+a*e^2))/(a*(c*d^2+a*e^2)^(5/2)*sqrt(2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))+1/8*c^(1/4)*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(c^(3/2)*d^3+13*a*d*e^2*sqrt(c)-(c*d^2-5*a*e^2)*sqrt(c*d^2+a*e^2))/(a*(c*d^2+a*e^2)^(5/2)*sqrt(2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[1/((d+e*x)^(5/2)*(a+c*x^2)^2),x,13,1/6*e*(3*c*d^2-7*a*e^2)/(a*(c*d^2+a*e^2)^2*(d+e*x)^(3/2))+1/2*(a*e+c*d*x)/(a*(c*d^2+a*e^2)*(d+e*x)^(3/2)*(a+c*x^2))+1/2*c*d*e*(c*d^2-19*a*e^2)/(a*(c*d^2+a*e^2)^3*sqrt(d+e*x))+1/4*c^(3/4)*e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(c^2*d^4+34*a*c*d^2*e^2-7*a^2*e^4+d*(c*d^2-19*a*e^2)*sqrt(c)*sqrt(c*d^2+a*e^2))/(a*(c*d^2+a*e^2)^(7/2)*sqrt(2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/4*c^(3/4)*e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(c^2*d^4+34*a*c*d^2*e^2-7*a^2*e^4+d*(c*d^2-19*a*e^2)*sqrt(c)*sqrt(c*d^2+a*e^2))/(a*(c*d^2+a*e^2)^(7/2)*sqrt(2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/8*c^(3/4)*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(c^2*d^4+34*a*c*d^2*e^2-7*a^2*e^4-d*(c*d^2-19*a*e^2)*sqrt(c)*sqrt(c*d^2+a*e^2))/(a*(c*d^2+a*e^2)^(7/2)*sqrt(2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))+1/8*c^(3/4)*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(c^2*d^4+34*a*c*d^2*e^2-7*a^2*e^4-d*(c*d^2-19*a*e^2)*sqrt(c)*sqrt(c*d^2+a*e^2))/(a*(c*d^2+a*e^2)^(7/2)*sqrt(2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[(d+e*x)^(7/2)/(a-c*x^2)^3,x,6,1/4*(a*e+c*d*x)*(d+e*x)^(5/2)/(a*c*(a-c*x^2)^2)+1/32*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))*(e*sqrt(a)+d*sqrt(c))^(3/2)*(12*c*d^2+5*a*e^2-18*d*e*sqrt(a)*sqrt(c))/(a^(5/2)*c^(9/4))-1/32*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))*(-e*sqrt(a)+d*sqrt(c))^(3/2)*(12*c*d^2+5*a*e^2+18*d*e*sqrt(a)*sqrt(c))/(a^(5/2)*c^(9/4))+1/16*(a*e*(7*c*d^2-5*a*e^2)+2*c*d*(3*c*d^2-2*a*e^2)*x)*sqrt(d+e*x)/(a^2*c^2*(a-c*x^2))],
[(d+e*x)^(5/2)/(a-c*x^2)^3,x,6,1/4*(a*e+c*d*x)*(d+e*x)^(3/2)/(a*c*(a-c*x^2)^2)+3/16*(a*d*e+(2*c*d^2-a*e^2)*x)*sqrt(d+e*x)/(a^2*c*(a-c*x^2))-3/32*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))*(4*c*d^2-a*e^2+2*d*e*sqrt(a)*sqrt(c))*sqrt(-e*sqrt(a)+d*sqrt(c))/(a^(5/2)*c^(7/4))+3/32*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))*(4*c*d^2-a*e^2-2*d*e*sqrt(a)*sqrt(c))*sqrt(e*sqrt(a)+d*sqrt(c))/(a^(5/2)*c^(7/4))],
[(d+e*x)^(3/2)/(a-c*x^2)^3,x,6,1/4*(a*e+c*d*x)*sqrt(d+e*x)/(a*c*(a-c*x^2)^2)-1/16*(a*e-6*c*d*x)*sqrt(d+e*x)/(a^2*c*(a-c*x^2))-3/32*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))*(4*c*d^2-a*e^2-2*d*e*sqrt(a)*sqrt(c))/(a^(5/2)*c^(5/4)*sqrt(-e*sqrt(a)+d*sqrt(c)))+3/32*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))*(4*c*d^2-a*e^2+2*d*e*sqrt(a)*sqrt(c))/(a^(5/2)*c^(5/4)*sqrt(e*sqrt(a)+d*sqrt(c)))],
[sqrt(d+e*x)/(a-c*x^2)^3,x,6,-1/32*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))*(12*c*d^2+5*a*e^2-18*d*e*sqrt(a)*sqrt(c))/(a^(5/2)*c^(3/4)*(-e*sqrt(a)+d*sqrt(c))^(3/2))+1/32*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))*(12*c*d^2+5*a*e^2+18*d*e*sqrt(a)*sqrt(c))/(a^(5/2)*c^(3/4)*(e*sqrt(a)+d*sqrt(c))^(3/2))+1/4*x*sqrt(d+e*x)/(a*(a-c*x^2)^2)-1/16*(a*d*e-(6*c*d^2-5*a*e^2)*x)*sqrt(d+e*x)/(a^2*(c*d^2-a*e^2)*(a-c*x^2))],
[1/((a-c*x^2)^3*sqrt(d+e*x)),x,6,-3/32*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(-e*sqrt(a)+d*sqrt(c)))*(4*c*d^2+7*a*e^2-10*d*e*sqrt(a)*sqrt(c))/(a^(5/2)*c^(1/4)*(-e*sqrt(a)+d*sqrt(c))^(5/2))+3/32*arctanh(c^(1/4)*sqrt(d+e*x)/sqrt(e*sqrt(a)+d*sqrt(c)))*(4*c*d^2+7*a*e^2+10*d*e*sqrt(a)*sqrt(c))/(a^(5/2)*c^(1/4)*(e*sqrt(a)+d*sqrt(c))^(5/2))-1/4*(a*e-c*d*x)*sqrt(d+e*x)/(a*(c*d^2-a*e^2)*(a-c*x^2)^2)-1/16*(a*e*(c*d^2-7*a*e^2)-6*c*d*(c*d^2-2*a*e^2)*x)*sqrt(d+e*x)/(a^2*(c*d^2-a*e^2)^2*(a-c*x^2))],
[(d+e*x)^(7/2)/(a+c*x^2)^3,x,12,-1/4*(a*e-c*d*x)*(d+e*x)^(5/2)/(a*c*(a+c*x^2)^2)-1/16*(a*e*(7*c*d^2+5*a*e^2)-2*c*d*(3*c*d^2+2*a*e^2)*x)*sqrt(d+e*x)/(a^2*c^2*(a+c*x^2))+1/32*e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(6*c^2*d^4+11*a*c*d^2*e^2+5*a^2*e^4+d*(6*c*d^2+8*a*e^2)*sqrt(c)*sqrt(c*d^2+a*e^2))/(a^2*c^(9/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/32*e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(6*c^2*d^4+11*a*c*d^2*e^2+5*a^2*e^4+d*(6*c*d^2+8*a*e^2)*sqrt(c)*sqrt(c*d^2+a*e^2))/(a^2*c^(9/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/64*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(6*c^2*d^4+11*a*c*d^2*e^2+5*a^2*e^4-2*d*(3*c*d^2+4*a*e^2)*sqrt(c)*sqrt(c*d^2+a*e^2))/(a^2*c^(9/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))+1/64*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(6*c^2*d^4+11*a*c*d^2*e^2+5*a^2*e^4-2*d*(3*c*d^2+4*a*e^2)*sqrt(c)*sqrt(c*d^2+a*e^2))/(a^2*c^(9/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[(d+e*x)^(5/2)/(a+c*x^2)^3,x,12,-1/4*(a*e-c*d*x)*(d+e*x)^(3/2)/(a*c*(a+c*x^2)^2)-3/16*(a*d*e-(2*c*d^2+a*e^2)*x)*sqrt(d+e*x)/(a^2*c*(a+c*x^2))+3/32*e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(2*c^(3/2)*d^3+2*a*d*e^2*sqrt(c)+(2*c*d^2+a*e^2)*sqrt(c*d^2+a*e^2))/(a^2*c^(7/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-3/32*e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(2*c^(3/2)*d^3+2*a*d*e^2*sqrt(c)+(2*c*d^2+a*e^2)*sqrt(c*d^2+a*e^2))/(a^2*c^(7/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-3/64*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(2*c^(3/2)*d^3+2*a*d*e^2*sqrt(c)-(2*c*d^2+a*e^2)*sqrt(c*d^2+a*e^2))/(a^2*c^(7/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))+3/64*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(2*c^(3/2)*d^3+2*a*d*e^2*sqrt(c)-(2*c*d^2+a*e^2)*sqrt(c*d^2+a*e^2))/(a^2*c^(7/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[(d+e*x)^(3/2)/(a+c*x^2)^3,x,12,-1/4*(a*e-c*d*x)*sqrt(d+e*x)/(a*c*(a+c*x^2)^2)+1/16*(a*e+6*c*d*x)*sqrt(d+e*x)/(a^2*c*(a+c*x^2))+3/32*e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(2*c*d^2+a*e^2+2*d*sqrt(c)*sqrt(c*d^2+a*e^2))/(a^2*c^(5/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-3/32*e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(2*c*d^2+a*e^2+2*d*sqrt(c)*sqrt(c*d^2+a*e^2))/(a^2*c^(5/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-3/64*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(2*c*d^2+a*e^2-2*d*sqrt(c)*sqrt(c*d^2+a*e^2))/(a^2*c^(5/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))+3/64*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(2*c*d^2+a*e^2-2*d*sqrt(c)*sqrt(c*d^2+a*e^2))/(a^2*c^(5/4)*sqrt(2)*sqrt(c*d^2+a*e^2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[sqrt(d+e*x)/(a+c*x^2)^3,x,12,1/4*x*sqrt(d+e*x)/(a*(a+c*x^2)^2)+1/16*(a*d*e+(6*c*d^2+5*a*e^2)*x)*sqrt(d+e*x)/(a^2*(c*d^2+a*e^2)*(a+c*x^2))+1/32*e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(6*c^(3/2)*d^3+8*a*d*e^2*sqrt(c)+(6*c*d^2+5*a*e^2)*sqrt(c*d^2+a*e^2))/(a^2*c^(3/4)*(c*d^2+a*e^2)^(3/2)*sqrt(2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/32*e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(6*c^(3/2)*d^3+8*a*d*e^2*sqrt(c)+(6*c*d^2+5*a*e^2)*sqrt(c*d^2+a*e^2))/(a^2*c^(3/4)*(c*d^2+a*e^2)^(3/2)*sqrt(2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-1/64*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(6*c^(3/2)*d^3+8*a*d*e^2*sqrt(c)-(6*c*d^2+5*a*e^2)*sqrt(c*d^2+a*e^2))/(a^2*c^(3/4)*(c*d^2+a*e^2)^(3/2)*sqrt(2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))+1/64*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(6*c^(3/2)*d^3+8*a*d*e^2*sqrt(c)-(6*c*d^2+5*a*e^2)*sqrt(c*d^2+a*e^2))/(a^2*c^(3/4)*(c*d^2+a*e^2)^(3/2)*sqrt(2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[1/((a+c*x^2)^3*sqrt(d+e*x)),x,12,1/4*(a*e+c*d*x)*sqrt(d+e*x)/(a*(c*d^2+a*e^2)*(a+c*x^2)^2)+1/16*(a*e*(c*d^2+7*a*e^2)+6*c*d*(c*d^2+2*a*e^2)*x)*sqrt(d+e*x)/(a^2*(c*d^2+a*e^2)^2*(a+c*x^2))+3/32*e*arctanh((-c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(2*c^2*d^4+5*a*c*d^2*e^2+7*a^2*e^4+2*d*(c*d^2+2*a*e^2)*sqrt(c)*sqrt(c*d^2+a*e^2))/(a^2*c^(1/4)*(c*d^2+a*e^2)^(5/2)*sqrt(2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-3/32*e*arctanh((c^(1/4)*sqrt(2)*sqrt(d+e*x)+sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))/sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))*(2*c^2*d^4+5*a*c*d^2*e^2+7*a^2*e^4+2*d*(c*d^2+2*a*e^2)*sqrt(c)*sqrt(c*d^2+a*e^2))/(a^2*c^(1/4)*(c*d^2+a*e^2)^(5/2)*sqrt(2)*sqrt(d*sqrt(c)-sqrt(c*d^2+a*e^2)))-3/64*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)-c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(2*c^2*d^4+5*a*c*d^2*e^2+7*a^2*e^4-2*d*(c*d^2+2*a*e^2)*sqrt(c)*sqrt(c*d^2+a*e^2))/(a^2*c^(1/4)*(c*d^2+a*e^2)^(5/2)*sqrt(2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))+3/64*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2+a*e^2)+c^(1/4)*sqrt(2)*sqrt(d+e*x)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))*(2*c^2*d^4+5*a*c*d^2*e^2+7*a^2*e^4-2*d*(c*d^2+2*a*e^2)*sqrt(c)*sqrt(c*d^2+a*e^2))/(a^2*c^(1/4)*(c*d^2+a*e^2)^(5/2)*sqrt(2)*sqrt(d*sqrt(c)+sqrt(c*d^2+a*e^2)))],
[sqrt(2+3*x)/(1+x^2),x,10,-3*arctan((-2*sqrt(2+3*x)+sqrt(2*(2+sqrt(13))))/sqrt(2*(-2+sqrt(13))))/sqrt(2*(-2+sqrt(13)))+3*arctan((2*sqrt(2+3*x)+sqrt(2*(2+sqrt(13))))/sqrt(2*(-2+sqrt(13))))/sqrt(2*(-2+sqrt(13)))+3/2*log(2+3*x+sqrt(13)-sqrt(2+3*x)*sqrt(2*(2+sqrt(13))))/sqrt(2*(2+sqrt(13)))-3/2*log(2+3*x+sqrt(13)+sqrt(2+3*x)*sqrt(2*(2+sqrt(13))))/sqrt(2*(2+sqrt(13)))],
[sqrt(c+d*x)/(1+x^2),x,10,d*arctanh((-sqrt(2)*sqrt(c+d*x)+sqrt(c+sqrt(c^2+d^2)))/sqrt(c-sqrt(c^2+d^2)))/(sqrt(2)*sqrt(c-sqrt(c^2+d^2)))-d*arctanh((sqrt(2)*sqrt(c+d*x)+sqrt(c+sqrt(c^2+d^2)))/sqrt(c-sqrt(c^2+d^2)))/(sqrt(2)*sqrt(c-sqrt(c^2+d^2)))+1/2*d*log(c+d*x+sqrt(c^2+d^2)-sqrt(2)*sqrt(c+d*x)*sqrt(c+sqrt(c^2+d^2)))/(sqrt(2)*sqrt(c+sqrt(c^2+d^2)))-1/2*d*log(c+d*x+sqrt(c^2+d^2)+sqrt(2)*sqrt(c+d*x)*sqrt(c+sqrt(c^2+d^2)))/(sqrt(2)*sqrt(c+sqrt(c^2+d^2)))],
[sqrt(2+3*x)/(1-x^2),x,4,-arctan(sqrt(2+3*x))+arctanh(sqrt(2+3*x)/sqrt(5))*sqrt(5)],
[sqrt(c+d*x)/(1-x^2),x,4,-arctanh(sqrt(c+d*x)/sqrt(c-d))*sqrt(c-d)+arctanh(sqrt(c+d*x)/sqrt(c+d))*sqrt(c+d)],
[sqrt(2+3*x)/(a+b*x^2),x,10,3*arctanh((-b^(1/4)*sqrt(2)*sqrt(2+3*x)+sqrt(2*sqrt(b)+sqrt(9*a+4*b)))/sqrt(2*sqrt(b)-sqrt(9*a+4*b)))/(b^(3/4)*sqrt(2)*sqrt(2*sqrt(b)-sqrt(9*a+4*b)))-3*arctanh((b^(1/4)*sqrt(2)*sqrt(2+3*x)+sqrt(2*sqrt(b)+sqrt(9*a+4*b)))/sqrt(2*sqrt(b)-sqrt(9*a+4*b)))/(b^(3/4)*sqrt(2)*sqrt(2*sqrt(b)-sqrt(9*a+4*b)))+3/2*log((2+3*x)*sqrt(b)+sqrt(9*a+4*b)-b^(1/4)*sqrt(2)*sqrt(2+3*x)*sqrt(2*sqrt(b)+sqrt(9*a+4*b)))/(b^(3/4)*sqrt(2)*sqrt(2*sqrt(b)+sqrt(9*a+4*b)))-3/2*log((2+3*x)*sqrt(b)+sqrt(9*a+4*b)+b^(1/4)*sqrt(2)*sqrt(2+3*x)*sqrt(2*sqrt(b)+sqrt(9*a+4*b)))/(b^(3/4)*sqrt(2)*sqrt(2*sqrt(b)+sqrt(9*a+4*b)))],
[sqrt(2+3*x)/(a-b*x^2),x,4,-arctan(b^(1/4)*sqrt(2+3*x)/sqrt(3*sqrt(a)-2*sqrt(b)))*sqrt(3*sqrt(a)-2*sqrt(b))/(b^(3/4)*sqrt(a))+arctanh(b^(1/4)*sqrt(2+3*x)/sqrt(3*sqrt(a)+2*sqrt(b)))*sqrt(3*sqrt(a)+2*sqrt(b))/(b^(3/4)*sqrt(a))],
[sqrt(1+x)/(1+x^2),x,10,-arctan((-2*sqrt(1+x)+sqrt(2*(1+sqrt(2))))/sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))+arctan((2*sqrt(1+x)+sqrt(2*(1+sqrt(2))))/sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))+1/2*log(1+x+sqrt(2)-sqrt(1+x)*sqrt(2*(1+sqrt(2))))/sqrt(2*(1+sqrt(2)))-1/2*log(1+x+sqrt(2)+sqrt(1+x)*sqrt(2*(1+sqrt(2))))/sqrt(2*(1+sqrt(2)))],
[1/((1+x^2)*sqrt(1+x)),x,10,-1/4*log(1+x+sqrt(2)-sqrt(1+x)*sqrt(2*(1+sqrt(2))))/sqrt(1+sqrt(2))+1/4*log(1+x+sqrt(2)+sqrt(1+x)*sqrt(2*(1+sqrt(2))))/sqrt(1+sqrt(2))-1/2*arctan((-2*sqrt(1+x)+sqrt(2*(1+sqrt(2))))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))+1/2*arctan((2*sqrt(1+x)+sqrt(2*(1+sqrt(2))))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))],
[sqrt(-1+x)/(1+x^2)^3,x,12,1/4*x*sqrt(-1+x)/(1+x^2)^2-1/32*(1-11*x)*sqrt(-1+x)/(1+x^2)-1/64*arctan((-2*sqrt(-1+x)+sqrt(2*(-1+sqrt(2))))/sqrt(2*(1+sqrt(2))))*sqrt(1/2*(-527+373*sqrt(2)))+1/64*arctan((2*sqrt(-1+x)+sqrt(2*(-1+sqrt(2))))/sqrt(2*(1+sqrt(2))))*sqrt(1/2*(-527+373*sqrt(2)))-1/128*log(1-x-sqrt(2)-sqrt(-1+x)*sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(527+373*sqrt(2)))+1/128*log(1-x-sqrt(2)+sqrt(-1+x)*sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(527+373*sqrt(2)))],

# Integrands of the form (d+e x)^(m/2) (a+c x^2)^(p/2)

# p>0
[(d+e*x)^(3/2)*sqrt(a+c*x^2),x,7,2/7*e*(a+c*x^2)^(3/2)*sqrt(d+e*x)/c+2/105*(3*c*d^2-5*a*e^2+24*c*d*e*x)*sqrt(d+e*x)*sqrt(a+c*x^2)/(c*e)+4/105*d*(3*c*d^2-29*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^2*sqrt(c)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-4/105*(3*c*d^2-5*a*e^2)*(c*d^2+a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(c^(3/2)*e^2*sqrt(d+e*x)*sqrt(a+c*x^2))],
[sqrt(d+e*x)*sqrt(a+c*x^2),x,7,2/5*(d+e*x)^(3/2)*sqrt(a+c*x^2)/e-4/15*d*sqrt(d+e*x)*sqrt(a+c*x^2)/e+4/15*(c*d^2-3*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^2*sqrt(c)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-4/15*d*(c*d^2+a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^2*sqrt(c)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[sqrt(a+c*x^2)/sqrt(d+e*x),x,6,2/3*sqrt(d+e*x)*sqrt(a+c*x^2)/e+4/3*d*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^2*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-4/3*(c*d^2+a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^2*sqrt(c)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[sqrt(a+c*x^2)/(d+e*x)^(3/2),x,6,-2*sqrt(a+c*x^2)/(e*sqrt(d+e*x))-4*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^2*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+4*d*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^2*sqrt(d+e*x)*sqrt(a+c*x^2))],
[sqrt(a+c*x^2)/(d+e*x)^(5/2),x,7,-2/3*sqrt(a+c*x^2)/(e*(d+e*x)^(3/2))+4/3*c*d*sqrt(a+c*x^2)/(e*(c*d^2+a*e^2)*sqrt(d+e*x))+4/3*c^(3/2)*d*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^2*(c*d^2+a*e^2)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-4/3*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^2*sqrt(d+e*x)*sqrt(a+c*x^2))],
[sqrt(a+c*x^2)/(d+e*x)^(7/2),x,8,-2/5*sqrt(a+c*x^2)/(e*(d+e*x)^(5/2))+4/15*c*d*sqrt(a+c*x^2)/(e*(c*d^2+a*e^2)*(d+e*x)^(3/2))+4/15*c*(c*d^2-3*a*e^2)*sqrt(a+c*x^2)/(e*(c*d^2+a*e^2)^2*sqrt(d+e*x))+4/15*c^(3/2)*(c*d^2-3*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^2*(c*d^2+a*e^2)^2*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-4/15*c^(3/2)*d*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^2*(c*d^2+a*e^2)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(d+e*x)^(3/2)*(a+c*x^2)^(3/2),x,8,2/231*(c*d^2-3*a*e^2+28*c*d*e*x)*(a+c*x^2)^(3/2)*sqrt(d+e*x)/(c*e)+2/11*e*(a+c*x^2)^(5/2)*sqrt(d+e*x)/c+4/1155*(4*c^2*d^4+21*a*c*d^2*e^2-15*a^2*e^4-3*c*d*e*(c*d^2-31*a*e^2)*x)*sqrt(d+e*x)*sqrt(a+c*x^2)/(c*e^3)+32/1155*d*(c*d^2-3*a*e^2)*(c*d^2+9*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^4*sqrt(c)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-8/1155*(c*d^2+a*e^2)*(4*c^2*d^4+21*a*c*d^2*e^2-15*a^2*e^4)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(c^(3/2)*e^4*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(a+c*x^2)^(3/2)*sqrt(d+e*x),x,8,2/9*(d+e*x)^(3/2)*(a+c*x^2)^(3/2)/e-4/21*d*(a+c*x^2)^(3/2)*sqrt(d+e*x)/e+4/315*(4*d*(c*d^2+3*a*e^2)-3*e*(c*d^2-7*a*e^2)*x)*sqrt(d+e*x)*sqrt(a+c*x^2)/e^3+8/315*(4*c^2*d^4+15*a*c*d^2*e^2-21*a^2*e^4)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^4*sqrt(c)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-32/315*d*(c*d^2+a*e^2)*(c*d^2+3*a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^4*sqrt(c)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(a+c*x^2)^(3/2)/sqrt(d+e*x),x,7,2/7*(a+c*x^2)^(3/2)*sqrt(d+e*x)/e+4/35*(4*c*d^2+5*a*e^2-3*c*d*e*x)*sqrt(d+e*x)*sqrt(a+c*x^2)/e^3+32/35*d*(c*d^2+2*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^4*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-8/35*(c*d^2+a*e^2)*(4*c*d^2+5*a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^4*sqrt(c)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(a+c*x^2)^(3/2)/(d+e*x)^(3/2),x,7,-2*(a+c*x^2)^(3/2)/(e*sqrt(d+e*x))-4/5*c*(4*d-3*e*x)*sqrt(d+e*x)*sqrt(a+c*x^2)/e^3-8/5*(4*c*d^2+3*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^4*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+32/5*d*(c*d^2+a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^4*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(a+c*x^2)^(3/2)/(d+e*x)^(5/2),x,7,-2/3*(a+c*x^2)^(3/2)/(e*(d+e*x)^(3/2))+4/3*c*(4*d+e*x)*sqrt(a+c*x^2)/(e^3*sqrt(d+e*x))+32/3*c^(3/2)*d*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^4*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-8/3*(4*c*d^2+a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^4*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(a+c*x^2)^(3/2)/(d+e*x)^(7/2),x,7,-2/5*(a+c*x^2)^(3/2)/(e*(d+e*x)^(5/2))-4/5*c*(2*d*(2*c*d^2+a*e^2)+e*(5*c*d^2+3*a*e^2)*x)*sqrt(a+c*x^2)/(e^3*(c*d^2+a*e^2)*(d+e*x)^(3/2))-8/5*c^(3/2)*(4*c*d^2+3*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^4*(c*d^2+a*e^2)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+32/5*c^(3/2)*d*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^4*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(a+c*x^2)^(3/2)/(d+e*x)^(9/2),x,8,-2/7*(a+c*x^2)^(3/2)/(e*(d+e*x)^(7/2))-4/35*c*(2*d*(2*c*d^2+a*e^2)+e*(7*c*d^2+5*a*e^2)*x)*sqrt(a+c*x^2)/(e^3*(c*d^2+a*e^2)*(d+e*x)^(5/2))+32/35*c^2*d*(c*d^2+2*a*e^2)*sqrt(a+c*x^2)/(e^3*(c*d^2+a*e^2)^2*sqrt(d+e*x))+32/35*c^(5/2)*d*(c*d^2+2*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^4*(c*d^2+a*e^2)^2*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-8/35*c^(3/2)*(4*c*d^2+5*a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^4*(c*d^2+a*e^2)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(a+c*x^2)^(5/2)*sqrt(d+e*x),x,9,2/13*(d+e*x)^(3/2)*(a+c*x^2)^(5/2)/e+20/9009*(4*d*(2*c*d^2+5*a*e^2)-7*e*(c*d^2-11*a*e^2)*x)*(a+c*x^2)^(3/2)*sqrt(d+e*x)/e^3-20/143*d*(a+c*x^2)^(5/2)*sqrt(d+e*x)/e+8/9009*(d*(32*c^2*d^4+113*a*c*d^2*e^2+177*a^2*e^4)-3*e*(8*c^2*d^4+27*a*c*d^2*e^2-77*a^2*e^4)*x)*sqrt(d+e*x)*sqrt(a+c*x^2)/e^5+16/9009*(32*c^3*d^6+137*a*c^2*d^4*e^2+258*a^2*c*d^2*e^4-231*a^3*e^6)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^6*sqrt(c)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-16/9009*d*(c*d^2+a*e^2)*(32*c^2*d^4+113*a*c*d^2*e^2+177*a^2*e^4)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^6*sqrt(c)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(a+c*x^2)^(5/2)/sqrt(d+e*x),x,8,20/693*(8*c*d^2+9*a*e^2-7*c*d*e*x)*(a+c*x^2)^(3/2)*sqrt(d+e*x)/e^3+2/11*(a+c*x^2)^(5/2)*sqrt(d+e*x)/e+8/693*(32*c^2*d^4+69*a*c*d^2*e^2+45*a^2*e^4-24*c*d*e*(c*d^2+2*a*e^2)*x)*sqrt(d+e*x)*sqrt(a+c*x^2)/e^5+16/693*d*(32*c^2*d^4+93*a*c*d^2*e^2+93*a^2*e^4)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^6*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-16/693*(c*d^2+a*e^2)*(32*c^2*d^4+69*a*c*d^2*e^2+45*a^2*e^4)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^6*sqrt(c)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(a+c*x^2)^(5/2)/(d+e*x)^(3/2),x,8,-2*(a+c*x^2)^(5/2)/(e*sqrt(d+e*x))-20/63*c*(8*d-7*e*x)*(a+c*x^2)^(3/2)*sqrt(d+e*x)/e^3-8/63*c*(d*(32*c*d^2+33*a*e^2)-3*e*(8*c*d^2+7*a*e^2)*x)*sqrt(d+e*x)*sqrt(a+c*x^2)/e^5-16/63*(32*c^2*d^4+57*a*c*d^2*e^2+21*a^2*e^4)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^6*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+16/63*d*(c*d^2+a*e^2)*(32*c*d^2+33*a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^6*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(a+c*x^2)^(5/2)/(d+e*x)^(5/2),x,8,-2/3*(a+c*x^2)^(5/2)/(e*(d+e*x)^(3/2))+20/21*c*(8*d+e*x)*(a+c*x^2)^(3/2)/(e^3*sqrt(d+e*x))+8/21*c*(32*c*d^2+5*a*e^2-24*c*d*e*x)*sqrt(d+e*x)*sqrt(a+c*x^2)/e^5+16/21*c^(3/2)*d*(32*c*d^2+29*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^6*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-16/21*(c*d^2+a*e^2)*(32*c*d^2+5*a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^6*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(a+c*x^2)^(5/2)/(d+e*x)^(7/2),x,8,4/15*c*(8*d+3*e*x)*(a+c*x^2)^(3/2)/(e^3*(d+e*x)^(3/2))-2/5*(a+c*x^2)^(5/2)/(e*(d+e*x)^(5/2))-8/15*c*(32*c*d^2+9*a*e^2+8*c*d*e*x)*sqrt(a+c*x^2)/(e^5*sqrt(d+e*x))-16/15*c^(3/2)*(32*c*d^2+9*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^6*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+16/15*c^(3/2)*d*(32*c*d^2+17*a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^6*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(a+c*x^2)^(5/2)/(d+e*x)^(9/2),x,8,-4/21*c*(2*d*(4*c*d^2+a*e^2)+e*(11*c*d^2+5*a*e^2)*x)*(a+c*x^2)^(3/2)/(e^3*(c*d^2+a*e^2)*(d+e*x)^(5/2))-2/7*(a+c*x^2)^(5/2)/(e*(d+e*x)^(7/2))+8/21*c^2*(d*(32*c*d^2+29*a*e^2)+e*(8*c*d^2+5*a*e^2)*x)*sqrt(a+c*x^2)/(e^5*(c*d^2+a*e^2)*sqrt(d+e*x))+16/21*c^(5/2)*d*(32*c*d^2+29*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^6*(c*d^2+a*e^2)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-16/21*c^(3/2)*(32*c*d^2+5*a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^6*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(a+c*x^2)^(5/2)/(d+e*x)^(11/2),x,8,-4/63*c*(2*d*(4*c*d^2+a*e^2)+e*(13*c*d^2+7*a*e^2)*x)*(a+c*x^2)^(3/2)/(e^3*(c*d^2+a*e^2)*(d+e*x)^(7/2))-2/9*(a+c*x^2)^(5/2)/(e*(d+e*x)^(9/2))-8/63*c^2*(d*(32*c^2*d^4+49*a*c*d^2*e^2+9*a^2*e^4)+e*(40*c^2*d^4+69*a*c*d^2*e^2+21*a^2*e^4)*x)*sqrt(a+c*x^2)/(e^5*(c*d^2+a*e^2)^2*(d+e*x)^(3/2))-16/63*c^(5/2)*(32*c^2*d^4+57*a*c*d^2*e^2+21*a^2*e^4)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(e^6*(c*d^2+a*e^2)^2*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+16/63*c^(5/2)*d*(32*c*d^2+33*a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(e^6*(c*d^2+a*e^2)*sqrt(d+e*x)*sqrt(a+c*x^2))],

# p<0
[(d+e*x)^(7/2)/sqrt(a+c*x^2),x,8,24/35*d*e*(d+e*x)^(3/2)*sqrt(a+c*x^2)/c+2/7*e*(d+e*x)^(5/2)*sqrt(a+c*x^2)/c+2/105*e*(71*c*d^2-25*a*e^2)*sqrt(d+e*x)*sqrt(a+c*x^2)/c^2-32/105*d*(11*c*d^2-13*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(c^(3/2)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+2/105*(71*c*d^2-25*a*e^2)*(c*d^2+a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(c^(5/2)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(d+e*x)^(5/2)/sqrt(a+c*x^2),x,7,2/5*e*(d+e*x)^(3/2)*sqrt(a+c*x^2)/c+16/15*d*e*sqrt(d+e*x)*sqrt(a+c*x^2)/c-2/15*(23*c*d^2-9*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(c^(3/2)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+16/15*d*(c*d^2+a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(c^(3/2)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(d+e*x)^(3/2)/sqrt(a+c*x^2),x,6,2/3*e*sqrt(d+e*x)*sqrt(a+c*x^2)/c-8/3*d*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(sqrt(c)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+2/3*(c*d^2+a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(c^(3/2)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[sqrt(d+e*x)/sqrt(a+c*x^2),x,2,-2*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(sqrt(c)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))],
[1/(sqrt(d+e*x)*sqrt(a+c*x^2)),x,2,-2*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(sqrt(c)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[1/((d+e*x)^(3/2)*sqrt(a+c*x^2)),x,4,-2*e*sqrt(a+c*x^2)/((c*d^2+a*e^2)*sqrt(d+e*x))-2*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/((c*d^2+a*e^2)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))],
[1/((d+e*x)^(5/2)*sqrt(a+c*x^2)),x,7,-2/3*e*sqrt(a+c*x^2)/((c*d^2+a*e^2)*(d+e*x)^(3/2))-8/3*c*d*e*sqrt(a+c*x^2)/((c*d^2+a*e^2)^2*sqrt(d+e*x))-8/3*c^(3/2)*d*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/((c*d^2+a*e^2)^2*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+2/3*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/((c*d^2+a*e^2)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[1/((d+e*x)^(7/2)*sqrt(a+c*x^2)),x,8,-2/5*e*sqrt(a+c*x^2)/((c*d^2+a*e^2)*(d+e*x)^(5/2))-16/15*c*d*e*sqrt(a+c*x^2)/((c*d^2+a*e^2)^2*(d+e*x)^(3/2))-2/15*c*e*(23*c*d^2-9*a*e^2)*sqrt(a+c*x^2)/((c*d^2+a*e^2)^3*sqrt(d+e*x))-2/15*c^(3/2)*(23*c*d^2-9*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/((c*d^2+a*e^2)^3*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+16/15*c^(3/2)*d*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/((c*d^2+a*e^2)^2*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(d+e*x)^(7/2)/(a+c*x^2)^(3/2),x,8,-(a*e-c*d*x)*(d+e*x)^(5/2)/(a*c*sqrt(a+c*x^2))-d*e*(d+e*x)^(3/2)*sqrt(a+c*x^2)/(a*c)-1/3*e*(3*c*d^2-5*a*e^2)*sqrt(d+e*x)*sqrt(a+c*x^2)/(a*c^2)-1/3*d*(3*c*d^2-29*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(c^(3/2)*sqrt(-a)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+1/3*(3*c*d^2-5*a*e^2)*(c*d^2+a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(c^(5/2)*sqrt(-a)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(d+e*x)^(5/2)/(a+c*x^2)^(3/2),x,7,-(a*e-c*d*x)*(d+e*x)^(3/2)/(a*c*sqrt(a+c*x^2))-d*e*sqrt(d+e*x)*sqrt(a+c*x^2)/(a*c)-(c*d^2-3*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(c^(3/2)*sqrt(-a)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+d*(c*d^2+a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(c^(3/2)*sqrt(-a)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(d+e*x)^(3/2)/(a+c*x^2)^(3/2),x,6,-(a*e-c*d*x)*sqrt(d+e*x)/(a*c*sqrt(a+c*x^2))-d*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(sqrt(-a)*sqrt(c)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+(c*d^2+a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(c^(3/2)*sqrt(-a)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[sqrt(d+e*x)/(a+c*x^2)^(3/2),x,7,x*sqrt(d+e*x)/(a*sqrt(a+c*x^2))-EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(d+e*x)*sqrt(1+c*x^2/a)/(sqrt(-a)*sqrt(c)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+d*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(sqrt(-a)*sqrt(c)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[1/((a+c*x^2)^(3/2)*sqrt(d+e*x)),x,6,(a*e+c*d*x)*sqrt(d+e*x)/(a*(c*d^2+a*e^2)*sqrt(a+c*x^2))-d*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(c)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/((c*d^2+a*e^2)*sqrt(-a)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(sqrt(-a)*sqrt(c)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[1/((d+e*x)^(3/2)*(a+c*x^2)^(3/2)),x,7,(a*e+c*d*x)/(a*(c*d^2+a*e^2)*sqrt(d+e*x)*sqrt(a+c*x^2))+e*(c*d^2-3*a*e^2)*sqrt(a+c*x^2)/(a*(c*d^2+a*e^2)^2*sqrt(d+e*x))-(c*d^2-3*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(c)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/((c*d^2+a*e^2)^2*sqrt(-a)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+d*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/((c*d^2+a*e^2)*sqrt(-a)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[1/((d+e*x)^(5/2)*(a+c*x^2)^(3/2)),x,8,(a*e+c*d*x)/(a*(c*d^2+a*e^2)*(d+e*x)^(3/2)*sqrt(a+c*x^2))+1/3*e*(3*c*d^2-5*a*e^2)*sqrt(a+c*x^2)/(a*(c*d^2+a*e^2)^2*(d+e*x)^(3/2))+1/3*c*d*e*(3*c*d^2-29*a*e^2)*sqrt(a+c*x^2)/(a*(c*d^2+a*e^2)^3*sqrt(d+e*x))-1/3*c^(3/2)*d*(3*c*d^2-29*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(d+e*x)*sqrt(1+c*x^2/a)/((c*d^2+a*e^2)^3*sqrt(-a)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))+1/3*(3*c*d^2-5*a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/((c*d^2+a*e^2)^2*sqrt(-a)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(d+e*x)^(9/2)/(a+c*x^2)^(5/2),x,8,-1/3*(a*e-c*d*x)*(d+e*x)^(7/2)/(a*c*(a+c*x^2)^(3/2))-1/6*(d+e*x)^(3/2)*(a*e*(c*d^2+7*a*e^2)-2*c*d*(2*c*d^2+5*a*e^2)*x)/(a^2*c^2*sqrt(a+c*x^2))-2/3*d*e*(c*d^2+3*a*e^2)*sqrt(d+e*x)*sqrt(a+c*x^2)/(a^2*c^2)+1/6*(4*c^2*d^4+15*a*c*d^2*e^2-21*a^2*e^4)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(d+e*x)*sqrt(1+c*x^2/a)/((-a)^(3/2)*c^(5/2)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-2/3*d*(c*d^2+a*e^2)*(c*d^2+3*a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/((-a)^(3/2)*c^(5/2)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(d+e*x)^(7/2)/(a+c*x^2)^(5/2),x,7,-1/3*(a*e-c*d*x)*(d+e*x)^(5/2)/(a*c*(a+c*x^2)^(3/2))-1/6*(a*e*(3*c*d^2+5*a*e^2)-2*c*d*(2*c*d^2+3*a*e^2)*x)*sqrt(d+e*x)/(a^2*c^2*sqrt(a+c*x^2))+2/3*d*(c*d^2+2*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(d+e*x)*sqrt(1+c*x^2/a)/((-a)^(3/2)*c^(3/2)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-1/6*(c*d^2+a*e^2)*(4*c*d^2+5*a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/((-a)^(3/2)*c^(5/2)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(d+e*x)^(5/2)/(a+c*x^2)^(5/2),x,7,-1/3*(a*e-c*d*x)*(d+e*x)^(3/2)/(a*c*(a+c*x^2)^(3/2))-1/6*(a*d*e-(4*c*d^2+3*a*e^2)*x)*sqrt(d+e*x)/(a^2*c*sqrt(a+c*x^2))+1/6*(4*c*d^2+3*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(d+e*x)*sqrt(1+c*x^2/a)/((-a)^(3/2)*c^(3/2)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-2/3*d*(c*d^2+a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/((-a)^(3/2)*c^(3/2)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[(d+e*x)^(3/2)/(a+c*x^2)^(5/2),x,7,-1/3*(a*e-c*d*x)*sqrt(d+e*x)/(a*c*(a+c*x^2)^(3/2))+1/6*(a*e+4*c*d*x)*sqrt(d+e*x)/(a^2*c*sqrt(a+c*x^2))+2/3*d*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(d+e*x)*sqrt(1+c*x^2/a)/((-a)^(3/2)*sqrt(c)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-1/6*(4*c*d^2+a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/((-a)^(3/2)*c^(3/2)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[sqrt(d+e*x)/(a+c*x^2)^(5/2),x,7,1/3*x*sqrt(d+e*x)/(a*(a+c*x^2)^(3/2))+1/6*(a*d*e+(4*c*d^2+3*a*e^2)*x)*sqrt(d+e*x)/(a^2*(c*d^2+a*e^2)*sqrt(a+c*x^2))+1/6*(4*c*d^2+3*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(d+e*x)*sqrt(1+c*x^2/a)/((-a)^(3/2)*(c*d^2+a*e^2)*sqrt(c)*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-2/3*d*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/((-a)^(3/2)*sqrt(c)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[1/((a+c*x^2)^(5/2)*sqrt(d+e*x)),x,7,1/3*(a*e+c*d*x)*sqrt(d+e*x)/(a*(c*d^2+a*e^2)*(a+c*x^2)^(3/2))+1/6*(a*e*(c*d^2+5*a*e^2)+4*c*d*(c*d^2+2*a*e^2)*x)*sqrt(d+e*x)/(a^2*(c*d^2+a*e^2)^2*sqrt(a+c*x^2))+2/3*d*(c*d^2+2*a*e^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(c)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/((-a)^(3/2)*(c*d^2+a*e^2)^2*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-1/6*(4*c*d^2+5*a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/((-a)^(3/2)*(c*d^2+a*e^2)*sqrt(c)*sqrt(d+e*x)*sqrt(a+c*x^2))],
[1/((d+e*x)^(3/2)*(a+c*x^2)^(5/2)),x,8,1/3*(a*e+c*d*x)/(a*(c*d^2+a*e^2)*(a+c*x^2)^(3/2)*sqrt(d+e*x))+1/6*(-a*e*(c*d^2-7*a*e^2)+4*c*d*(c*d^2+3*a*e^2)*x)/(a^2*(c*d^2+a*e^2)^2*sqrt(d+e*x)*sqrt(a+c*x^2))+1/6*e*(4*c^2*d^4+15*a*c*d^2*e^2-21*a^2*e^4)*sqrt(a+c*x^2)/(a^2*(c*d^2+a*e^2)^3*sqrt(d+e*x))+1/6*(4*c^2*d^4+15*a*c*d^2*e^2-21*a^2*e^4)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(c)*sqrt(d+e*x)*sqrt(1+c*x^2/a)/((-a)^(3/2)*(c*d^2+a*e^2)^3*sqrt(a+c*x^2)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c))))-2/3*d*(c*d^2+3*a*e^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*e/(-a*e+d*sqrt(-a)*sqrt(c))))*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/((-a)^(3/2)*(c*d^2+a*e^2)^2*sqrt(d+e*x)*sqrt(a+c*x^2))],

# Integrands of the form (d+e x)^m (a+c x^2)^(p/3) with c d^2-3 a e^2=0
[1/((d+e*x)*(d^2+3*e^2*x^2)^(1/3)),x,1,-1/2*log(d+e*x)/(2^(2/3)*d^(2/3)*e)+1/2*log(3*d*e^2-3*e^3*x-3*2^(1/3)*d^(1/3)*e^2*(d^2+3*e^2*x^2)^(1/3))/(2^(2/3)*d^(2/3)*e)-arctan(1/sqrt(3)+2^(2/3)*(d-e*x)/(d^(1/3)*(d^2+3*e^2*x^2)^(1/3)*sqrt(3)))/(2^(2/3)*d^(2/3)*e*sqrt(3))],
[(2+3*x)^3/(4+27*x^2)^(1/3),x,6,1/30*(2+3*x)^2*(4+27*x^2)^(2/3)+4/35*(7+4*x)*(4+27*x^2)^(2/3)-96/7*x/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))-32/63*2^(5/6)*(2^(2/3)-(4+27*x^2)^(1/3))*EllipticF((-(4+27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4+27*x^2)^(1/3)+(4+27*x^2)^(2/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)/(3^(1/4)*x*sqrt((-2^(2/3)+(4+27*x^2)^(1/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))+16/21*2^(1/3)*(2^(2/3)-(4+27*x^2)^(1/3))*EllipticE((-(4+27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4+27*x^2)^(1/3)+(4+27*x^2)^(2/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(3/4)*x*sqrt((-2^(2/3)+(4+27*x^2)^(1/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))],
[(2+3*x)^2/(4+27*x^2)^(1/3),x,6,5/21*(4+27*x^2)^(2/3)+1/21*(2+3*x)*(4+27*x^2)^(2/3)-72/7*x/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))-8/21*2^(5/6)*(2^(2/3)-(4+27*x^2)^(1/3))*EllipticF((-(4+27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4+27*x^2)^(1/3)+(4+27*x^2)^(2/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)/(3^(1/4)*x*sqrt((-2^(2/3)+(4+27*x^2)^(1/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))+4/7*2^(1/3)*(2^(2/3)-(4+27*x^2)^(1/3))*EllipticE((-(4+27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4+27*x^2)^(1/3)+(4+27*x^2)^(2/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(3/4)*x*sqrt((-2^(2/3)+(4+27*x^2)^(1/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))],
[(2+3*x)/(4+27*x^2)^(1/3),x,5,1/12*(4+27*x^2)^(2/3)-6*x/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))-2/9*2^(5/6)*(2^(2/3)-(4+27*x^2)^(1/3))*EllipticF((-(4+27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4+27*x^2)^(1/3)+(4+27*x^2)^(2/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)/(3^(1/4)*x*sqrt((-2^(2/3)+(4+27*x^2)^(1/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))+1/3*2^(1/3)*(2^(2/3)-(4+27*x^2)^(1/3))*EllipticE((-(4+27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4+27*x^2)^(1/3)+(4+27*x^2)^(2/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(3/4)*x*sqrt((-2^(2/3)+(4+27*x^2)^(1/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))],
[1/((2+3*x)*(4+27*x^2)^(1/3)),x,1,-1/12*log(2+3*x)/2^(1/3)+1/12*log(54-81*x-27*2^(2/3)*(4+27*x^2)^(1/3))/2^(1/3)-1/6*arctan(1/sqrt(3)+2^(1/3)*(2-3*x)/((4+27*x^2)^(1/3)*sqrt(3)))/(2^(1/3)*sqrt(3))],
[1/((2+3*x)^2*(4+27*x^2)^(1/3)),x,7,-1/48*(4+27*x^2)^(2/3)/(2+3*x)-1/48*log(2+3*x)/2^(1/3)+1/48*log(54-81*x-27*2^(2/3)*(4+27*x^2)^(1/3))/2^(1/3)-3/16*x/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))-1/24*arctan(1/sqrt(3)+2^(1/3)*(2-3*x)/((4+27*x^2)^(1/3)*sqrt(3)))/(2^(1/3)*sqrt(3))-1/72*(2^(2/3)-(4+27*x^2)^(1/3))*EllipticF((-(4+27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4+27*x^2)^(1/3)+(4+27*x^2)^(2/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)/(2^(1/6)*3^(1/4)*x*sqrt((-2^(2/3)+(4+27*x^2)^(1/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))+1/48*(2^(2/3)-(4+27*x^2)^(1/3))*EllipticE((-(4+27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4+27*x^2)^(1/3)+(4+27*x^2)^(2/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(2^(2/3)*3^(3/4)*x*sqrt((-2^(2/3)+(4+27*x^2)^(1/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))],
[1/((2+3*x)^3*(4+27*x^2)^(1/3)),x,8,-1/96*(4+27*x^2)^(2/3)/(2+3*x)^2-1/96*(4+27*x^2)^(2/3)/(2+3*x)-1/192*log(2+3*x)/2^(1/3)+1/192*log(54-81*x-27*2^(2/3)*(4+27*x^2)^(1/3))/2^(1/3)-3/32*x/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))-1/96*arctan(1/sqrt(3)+2^(1/3)*(2-3*x)/((4+27*x^2)^(1/3)*sqrt(3)))/(2^(1/3)*sqrt(3))-1/144*(2^(2/3)-(4+27*x^2)^(1/3))*EllipticF((-(4+27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4+27*x^2)^(1/3)+(4+27*x^2)^(2/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)/(2^(1/6)*3^(1/4)*x*sqrt((-2^(2/3)+(4+27*x^2)^(1/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))+1/96*(2^(2/3)-(4+27*x^2)^(1/3))*EllipticE((-(4+27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4+27*x^2)^(1/3)+(4+27*x^2)^(2/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(2^(2/3)*3^(3/4)*x*sqrt((-2^(2/3)+(4+27*x^2)^(1/3))/(-(4+27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))],
[(2+3*I*x)^3/(4-27*x^2)^(1/3),x,6,-4/35*(7*I-4*x)*(4-27*x^2)^(2/3)-1/30*I*(2+3*I*x)^2*(4-27*x^2)^(2/3)-96/7*x/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))+32/63*2^(5/6)*(2^(2/3)-(4-27*x^2)^(1/3))*EllipticF((-(4-27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4-27*x^2)^(1/3)+(4-27*x^2)^(2/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)/(3^(1/4)*x*sqrt((-2^(2/3)+(4-27*x^2)^(1/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))-16/21*2^(1/3)*(2^(2/3)-(4-27*x^2)^(1/3))*EllipticE((-(4-27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4-27*x^2)^(1/3)+(4-27*x^2)^(2/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(3/4)*x*sqrt((-2^(2/3)+(4-27*x^2)^(1/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))],
[(2+3*I*x)^2/(4-27*x^2)^(1/3),x,6,-5/21*I*(4-27*x^2)^(2/3)-1/21*I*(2+3*I*x)*(4-27*x^2)^(2/3)-72/7*x/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))+8/21*2^(5/6)*(2^(2/3)-(4-27*x^2)^(1/3))*EllipticF((-(4-27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4-27*x^2)^(1/3)+(4-27*x^2)^(2/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)/(3^(1/4)*x*sqrt((-2^(2/3)+(4-27*x^2)^(1/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))-4/7*2^(1/3)*(2^(2/3)-(4-27*x^2)^(1/3))*EllipticE((-(4-27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4-27*x^2)^(1/3)+(4-27*x^2)^(2/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(3/4)*x*sqrt((-2^(2/3)+(4-27*x^2)^(1/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))],
[(2+3*I*x)/(4-27*x^2)^(1/3),x,5,-1/12*I*(4-27*x^2)^(2/3)-6*x/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))+2/9*2^(5/6)*(2^(2/3)-(4-27*x^2)^(1/3))*EllipticF((-(4-27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4-27*x^2)^(1/3)+(4-27*x^2)^(2/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)/(3^(1/4)*x*sqrt((-2^(2/3)+(4-27*x^2)^(1/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))-1/3*2^(1/3)*(2^(2/3)-(4-27*x^2)^(1/3))*EllipticE((-(4-27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4-27*x^2)^(1/3)+(4-27*x^2)^(2/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(3/4)*x*sqrt((-2^(2/3)+(4-27*x^2)^(1/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))],
[1/((2+3*I*x)*(4-27*x^2)^(1/3)),x,1,1/12*I*log(2+3*I*x)/2^(1/3)-1/12*I*log(-54+81*I*x+27*2^(2/3)*(4-27*x^2)^(1/3))/2^(1/3)+1/6*I*arctan(1/sqrt(3)+2^(1/3)*(2-3*I*x)/((4-27*x^2)^(1/3)*sqrt(3)))/(2^(1/3)*sqrt(3))],
[1/((2+3*I*x)^2*(4-27*x^2)^(1/3)),x,7,1/48*I*(4-27*x^2)^(2/3)/(2+3*I*x)+1/48*I*log(2+3*I*x)/2^(1/3)-1/48*I*log(-54+81*I*x+27*2^(2/3)*(4-27*x^2)^(1/3))/2^(1/3)-3/16*x/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))+1/24*I*arctan(1/sqrt(3)+2^(1/3)*(2-3*I*x)/((4-27*x^2)^(1/3)*sqrt(3)))/(2^(1/3)*sqrt(3))+1/72*(2^(2/3)-(4-27*x^2)^(1/3))*EllipticF((-(4-27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4-27*x^2)^(1/3)+(4-27*x^2)^(2/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)/(2^(1/6)*3^(1/4)*x*sqrt((-2^(2/3)+(4-27*x^2)^(1/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))-1/48*(2^(2/3)-(4-27*x^2)^(1/3))*EllipticE((-(4-27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4-27*x^2)^(1/3)+(4-27*x^2)^(2/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(2^(2/3)*3^(3/4)*x*sqrt((-2^(2/3)+(4-27*x^2)^(1/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))],
[1/((2+3*I*x)^3*(4-27*x^2)^(1/3)),x,8,1/96*I*(4-27*x^2)^(2/3)/(2+3*I*x)^2+1/96*I*(4-27*x^2)^(2/3)/(2+3*I*x)+1/192*I*log(2+3*I*x)/2^(1/3)-1/192*I*log(-54+81*I*x+27*2^(2/3)*(4-27*x^2)^(1/3))/2^(1/3)-3/32*x/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))+1/96*I*arctan(1/sqrt(3)+2^(1/3)*(2-3*I*x)/((4-27*x^2)^(1/3)*sqrt(3)))/(2^(1/3)*sqrt(3))+1/144*(2^(2/3)-(4-27*x^2)^(1/3))*EllipticF((-(4-27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4-27*x^2)^(1/3)+(4-27*x^2)^(2/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)/(2^(1/6)*3^(1/4)*x*sqrt((-2^(2/3)+(4-27*x^2)^(1/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))-1/96*(2^(2/3)-(4-27*x^2)^(1/3))*EllipticE((-(4-27*x^2)^(1/3)+2^(2/3)*(1+sqrt(3)))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((2*2^(1/3)+2^(2/3)*(4-27*x^2)^(1/3)+(4-27*x^2)^(2/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(2^(2/3)*3^(3/4)*x*sqrt((-2^(2/3)+(4-27*x^2)^(1/3))/(-(4-27*x^2)^(1/3)+2^(2/3)*(1-sqrt(3)))^2))],
[1/((1+x^2)^(1/3)*(x+sqrt(3))),x,1,-1/2*log(x+sqrt(3))/2^(2/3)+1/2*log(-x+sqrt(3)-2^(1/3)*(1+x^2)^(1/3)*sqrt(3))/2^(2/3)-arctan(1/sqrt(3)+1/3*2^(2/3)*(-x+sqrt(3))/(1+x^2)^(1/3))/(2^(2/3)*sqrt(3))],
[1/((1+x^2)^(1/3)*(-x+sqrt(3))),x,1,1/2*log(-x+sqrt(3))/2^(2/3)-1/2*log(x+sqrt(3)-2^(1/3)*(1+x^2)^(1/3)*sqrt(3))/2^(2/3)+arctan(1/sqrt(3)+1/3*2^(2/3)*(x+sqrt(3))/(1+x^2)^(1/3))/(2^(2/3)*sqrt(3))],

# Integrands of the form (d+e x)^m (a+c x^2)^(p/3) with c d^2+9 a e^2=0
[1/((3-x)*(1-x^2)^(1/3)),x,2,-1/4*log(3-x)+3/8*log(-(1-x)^(1/3)-1/2*(1+x)^(2/3))-1/4*arctan(1/sqrt(3)-(1+x)^(2/3)/((1-x)^(1/3)*sqrt(3)))*sqrt(3)],
[1/((3+x)*(1-x^2)^(1/3)),x,2,1/4*log(3+x)-3/8*log(-1/2*(1-x)^(2/3)-(1+x)^(1/3))+1/4*arctan(1/sqrt(3)-(1-x)^(2/3)/((1+x)^(1/3)*sqrt(3)))*sqrt(3)],
[1/((d+e*x)*(d^2-9*e^2*x^2)^(1/3)),x,3,1/4*(1-9*e^2*x^2/d^2)^(1/3)*log(d+e*x)/(e*(d^2-9*e^2*x^2)^(1/3))-3/8*(1-9*e^2*x^2/d^2)^(1/3)*log(-1/2*(1-3*e*x/d)^(2/3)-(1+3*e*x/d)^(1/3))/(e*(d^2-9*e^2*x^2)^(1/3))+1/4*(1-9*e^2*x^2/d^2)^(1/3)*arctan(1/sqrt(3)-(1-3*e*x/d)^(2/3)/((1+3*e*x/d)^(1/3)*sqrt(3)))*sqrt(3)/(e*(d^2-9*e^2*x^2)^(1/3))],

# Integrands of the form (d+e x)^m (a+c x^2)^(p/4)
[1/((a+b*x)*(c+d*x^2)^(1/4)),x,10,arctan((c+d*x^2)^(1/4)*sqrt(b)/(b^2*c+a^2*d)^(1/4))/((b^2*c+a^2*d)^(1/4)*sqrt(b))-arctanh((c+d*x^2)^(1/4)*sqrt(b)/(b^2*c+a^2*d)^(1/4))/((b^2*c+a^2*d)^(1/4)*sqrt(b))-a*c^(1/4)*EllipticPi((c+d*x^2)^(1/4)/c^(1/4),-b*sqrt(c)/sqrt(b^2*c+a^2*d),I)*sqrt(-d*x^2/c)/(b*x*sqrt(b^2*c+a^2*d))+a*c^(1/4)*EllipticPi((c+d*x^2)^(1/4)/c^(1/4),b*sqrt(c)/sqrt(b^2*c+a^2*d),I)*sqrt(-d*x^2/c)/(b*x*sqrt(b^2*c+a^2*d))],
[1/((a+b*x)*(c+d*x^2)^(3/4)),x,11,-arctan((c+d*x^2)^(1/4)*sqrt(b)/(b^2*c+a^2*d)^(1/4))*sqrt(b)/(b^2*c+a^2*d)^(3/4)-arctanh((c+d*x^2)^(1/4)*sqrt(b)/(b^2*c+a^2*d)^(1/4))*sqrt(b)/(b^2*c+a^2*d)^(3/4)+a*c^(1/4)*EllipticPi((c+d*x^2)^(1/4)/c^(1/4),-b*sqrt(c)/sqrt(b^2*c+a^2*d),I)*sqrt(-d*x^2/c)/((b^2*c+a^2*d)*x)+a*c^(1/4)*EllipticPi((c+d*x^2)^(1/4)/c^(1/4),b*sqrt(c)/sqrt(b^2*c+a^2*d),I)*sqrt(-d*x^2/c)/((b^2*c+a^2*d)*x)],
[1/((d+e*x)^(3/2)*(a+c*x^2)^(1/4)),x,1,-2*hypergeom([-1/2,1/4],[1/2],2*(d+e*x)*sqrt(-a)*sqrt(c)/((-e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)-x*sqrt(c))))*(sqrt(-a)-x*sqrt(c))*(-(e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)+x*sqrt(c))/((-e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)-x*sqrt(c))))^(1/4)/((a+c*x^2)^(1/4)*(e*sqrt(-a)+d*sqrt(c))*sqrt(d+e*x))],

# Integrands of the form (d+e x)^m (a+c x^2)^(p/6)
[1/((1+x)*(1+x^2)^(1/6)),x,15,x*AppellF1(1/2,1,1/6,3/2,x^2,-x^2)-arctanh((1+x^2)^(1/6)/2^(1/6))/2^(1/6)+1/4*log(2^(1/3)-2^(1/6)*(1+x^2)^(1/6)+(1+x^2)^(1/3))/2^(1/6)-1/4*log(2^(1/3)+2^(1/6)*(1+x^2)^(1/6)+(1+x^2)^(1/3))/2^(1/6)-1/2*arctan((1-2^(5/6)*(1+x^2)^(1/6))/sqrt(3))*sqrt(3)/2^(1/6)+1/2*arctan((1+2^(5/6)*(1+x^2)^(1/6))/sqrt(3))*sqrt(3)/2^(1/6)],

# Integrands of the form (d+e x)^m (a+c x^2)^p when m symbolic
[(d+e*x)^m*(a+c*x^2)^3,x,2,(c*d^2+a*e^2)^3*(d+e*x)^(1+m)/(e^7*(1+m))-6*c*d*(c*d^2+a*e^2)^2*(d+e*x)^(2+m)/(e^7*(2+m))+3*c*(c*d^2+a*e^2)*(5*c*d^2+a*e^2)*(d+e*x)^(3+m)/(e^7*(3+m))-4*c^2*d*(5*c*d^2+3*a*e^2)*(d+e*x)^(4+m)/(e^7*(4+m))+3*c^2*(5*c*d^2+a*e^2)*(d+e*x)^(5+m)/(e^7*(5+m))-6*c^3*d*(d+e*x)^(6+m)/(e^7*(6+m))+c^3*(d+e*x)^(7+m)/(e^7*(7+m))],
[(d+e*x)^m*(a+c*x^2)^2,x,2,(c*d^2+a*e^2)^2*(d+e*x)^(1+m)/(e^5*(1+m))-4*c*d*(c*d^2+a*e^2)*(d+e*x)^(2+m)/(e^5*(2+m))+2*c*(3*c*d^2+a*e^2)*(d+e*x)^(3+m)/(e^5*(3+m))-4*c^2*d*(d+e*x)^(4+m)/(e^5*(4+m))+c^2*(d+e*x)^(5+m)/(e^5*(5+m))],
[(d+e*x)^m*(a+c*x^2),x,2,(c*d^2+a*e^2)*(d+e*x)^(1+m)/(e^3*(1+m))-2*c*d*(d+e*x)^(2+m)/(e^3*(2+m))+c*(d+e*x)^(3+m)/(e^3*(3+m))],
[(d+e*x)^m/(a+c*x^2),x,4,1/2*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))/((1+m)*sqrt(-a)*(-e*sqrt(-a)+d*sqrt(c)))-1/2*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/((1+m)*sqrt(-a)*(e*sqrt(-a)+d*sqrt(c)))],
[(d+e*x)^m/(a+c*x^2)^2,x,5,1/2*(a*e+c*d*x)*(d+e*x)^(1+m)/(a*(c*d^2+a*e^2)*(a+c*x^2))+1/4*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))*(c*d^2+a*e^2*(1-m)-d*e*m*sqrt(-a)*sqrt(c))/((-a)^(3/2)*(c*d^2+a*e^2)*(1+m)*(e*sqrt(-a)+d*sqrt(c)))-1/4*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))*(c*d^2+a*e^2*(1-m)+d*e*m*sqrt(-a)*sqrt(c))/((-a)^(3/2)*(c*d^2+a*e^2)*(1+m)*(-e*sqrt(-a)+d*sqrt(c)))],
[(d+e*x)^m/(a+c*x^2)^3,x,6,1/4*(a*e+c*d*x)*(d+e*x)^(1+m)/(a*(c*d^2+a*e^2)*(a+c*x^2)^2)+1/8*(d+e*x)^(1+m)*(a*e*(a*e^2*(3-m)+c*d^2*(1+m))+c*d*(3*c*d^2+a*e^2*(5-2*m))*x)/(a^2*(c*d^2+a*e^2)^2*(a+c*x^2))+1/16*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))*(-(3*c^2*d^4+a*c*d^2*e^2*(6-2*m-m^2)+a^2*e^4*(3-4*m+m^2))*sqrt(-a)+a*d*e*(3*c*d^2+a*e^2*(5-2*m))*m*sqrt(c))/(a^3*(c*d^2+a*e^2)^2*(1+m)*(-e*sqrt(-a)+d*sqrt(c)))+1/16*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))*((3*c^2*d^4+a*c*d^2*e^2*(6-2*m-m^2)+a^2*e^4*(3-4*m+m^2))*sqrt(-a)+a*d*e*(3*c*d^2+a*e^2*(5-2*m))*m*sqrt(c))/(a^3*(c*d^2+a*e^2)^2*(1+m)*(e*sqrt(-a)+d*sqrt(c)))],
[(d+e*x)^m*(a+c*x^2)^(3/2),x,2,(d+e*x)^(1+m)*(a+c*x^2)^(3/2)*AppellF1(1+m,-3/2,-3/2,2+m,(d+e*x)/(d-e*sqrt(-a)/sqrt(c)),(d+e*x)/(d+e*sqrt(-a)/sqrt(c)))/(e*(1+m)*(1+(-d-e*x)/(d-e*sqrt(-a)/sqrt(c)))^(3/2)*(1+(-d-e*x)/(d+e*sqrt(-a)/sqrt(c)))^(3/2))],
[(d+e*x)^m*(a+c*x^2)^(1/2),x,2,(d+e*x)^(1+m)*AppellF1(1+m,-1/2,-1/2,2+m,(d+e*x)/(d-e*sqrt(-a)/sqrt(c)),(d+e*x)/(d+e*sqrt(-a)/sqrt(c)))*sqrt(a+c*x^2)/(e*(1+m)*sqrt(1+(-d-e*x)/(d-e*sqrt(-a)/sqrt(c)))*sqrt(1+(-d-e*x)/(d+e*sqrt(-a)/sqrt(c))))],
[(d+e*x)^m/(a+c*x^2)^(1/2),x,2,(d+e*x)^(1+m)*AppellF1(1+m,1/2,1/2,2+m,(d+e*x)/(d-e*sqrt(-a)/sqrt(c)),(d+e*x)/(d+e*sqrt(-a)/sqrt(c)))*sqrt(1+(-d-e*x)/(d-e*sqrt(-a)/sqrt(c)))*sqrt(1+(-d-e*x)/(d+e*sqrt(-a)/sqrt(c)))/(e*(1+m)*sqrt(a+c*x^2))],
[(d+e*x)^m/(a+c*x^2)^(3/2),x,2,(d+e*x)^(1+m)*AppellF1(1+m,3/2,3/2,2+m,(d+e*x)/(d-e*sqrt(-a)/sqrt(c)),(d+e*x)/(d+e*sqrt(-a)/sqrt(c)))*(1+(-d-e*x)/(d-e*sqrt(-a)/sqrt(c)))^(3/2)*(1+(-d-e*x)/(d+e*sqrt(-a)/sqrt(c)))^(3/2)/(e*(1+m)*(a+c*x^2)^(3/2))],

# Integrands of the form (d+e x)^m (a+c x^2)^p when p symbolic
[(d+e*x)^m*(a+c*x^2)^p,x,2,(d+e*x)^(1+m)*(a+c*x^2)^p*AppellF1(1+m,-p,-p,2+m,(d+e*x)/(d-e*sqrt(-a)/sqrt(c)),(d+e*x)/(d+e*sqrt(-a)/sqrt(c)))/(e*(1+m)*(1+(-d-e*x)/(d-e*sqrt(-a)/sqrt(c)))^p*(1+(-d-e*x)/(d+e*sqrt(-a)/sqrt(c)))^p)],
[(d+e*x)^3*(a+c*x^2)^p,x,4,1/2*e*(d+e*x)^2*(a+c*x^2)^(1+p)/(c*(2+p))-1/2*e*((3+2*p)*(a*e^2-c*d^2*(5+2*p))-2*c*d*e*(1+p)*(3+p)*x)*(a+c*x^2)^(1+p)/(c^2*(2+p)*(3+5*p+2*p^2))-d*(3*a*e^2-c*d^2*(3+2*p))*x*(a+c*x^2)^p*hypergeom([1/2,-p],[3/2],-c*x^2/a)/(c*(3+2*p)*(1+c*x^2/a)^p)],
[(d+e*x)^2*(a+c*x^2)^p,x,4,d*e*(2+p)*(a+c*x^2)^(1+p)/(c*(3+5*p+2*p^2))+e*(d+e*x)*(a+c*x^2)^(1+p)/(c*(3+2*p))-(a*e^2-c*d^2*(3+2*p))*x*(a+c*x^2)^p*hypergeom([1/2,-p],[3/2],-c*x^2/a)/(c*(3+2*p)*(1+c*x^2/a)^p)],
[(d+e*x)*(a+c*x^2)^p,x,3,1/2*e*(a+c*x^2)^(1+p)/(c*(1+p))+d*x*(a+c*x^2)^(1+p)*hypergeom([1,3/2+p],[3/2],-c*x^2/a)/a,1/2*e*(a+c*x^2)^(1+p)/(c*(1+p))+d*x*(a+c*x^2)^p*hypergeom([1/2,-p],[3/2],-c*x^2/a)/(1+c*x^2/a)^p],
[(a+c*x^2)^p,x,2,x*(a+c*x^2)^(1+p)*hypergeom([1,3/2+p],[3/2],-c*x^2/a)/a,x*(a+c*x^2)^p*hypergeom([1/2,-p],[3/2],-c*x^2/a)/(1+c*x^2/a)^p],
[(a+c*x^2)^p/(d+e*x),x,6,x*(a+c*x^2)^p*AppellF1(1/2,-p,1,3/2,-c*x^2/a,e^2*x^2/d^2)/(d*(1+c*x^2/a)^p)-1/2*e*(a+c*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+c*x^2)/(c*d^2+a*e^2))/((c*d^2+a*e^2)*(1+p))],
[(a+c*x^2)^p/(d+e*x)^2,x,8,x*(a+c*x^2)^p*AppellF1(1/2,-p,2,3/2,-c*x^2/a,e^2*x^2/d^2)/(d^2*(1+c*x^2/a)^p)+1/3*e^2*x^3*(a+c*x^2)^p*AppellF1(3/2,-p,2,5/2,-c*x^2/a,e^2*x^2/d^2)/(d^4*(1+c*x^2/a)^p)-c*d*e*(a+c*x^2)^(1+p)*hypergeom([2,1+p],[2+p],e^2*(a+c*x^2)/(c*d^2+a*e^2))/((c*d^2+a*e^2)^2*(1+p))],
[(a+c*x^2)^p/(d+e*x)^3,x,11,-1/4*d^2*e*(a+c*x^2)^(1+p)/((c*d^2+a*e^2)*(d^2-e^2*x^2)^2)+x*(a+c*x^2)^p*AppellF1(1/2,-p,3,3/2,-c*x^2/a,e^2*x^2/d^2)/(d^3*(1+c*x^2/a)^p)+e^2*x^3*(a+c*x^2)^p*AppellF1(3/2,-p,3,5/2,-c*x^2/a,e^2*x^2/d^2)/(d^5*(1+c*x^2/a)^p)+1/4*c*e*(2*a*e^2+c*d^2*(1+p))*(a+c*x^2)^(1+p)*hypergeom([2,1+p],[2+p],e^2*(a+c*x^2)/(c*d^2+a*e^2))/((c*d^2+a*e^2)^3*(1+p))-3/2*c^2*d^2*e*(a+c*x^2)^(1+p)*hypergeom([3,1+p],[2+p],e^2*(a+c*x^2)/(c*d^2+a*e^2))/((c*d^2+a*e^2)^3*(1+p))],
[(a+c*x^2)^p/(d+e*x)^(2*p),x,2,(d+e*x)^(1-2*p)*(a+c*x^2)^p*AppellF1(1-2*p,-p,-p,2-2*p,(d+e*x)/(d-e*sqrt(-a)/sqrt(c)),(d+e*x)/(d+e*sqrt(-a)/sqrt(c)))/(e*(1-2*p)*(1+(-d-e*x)/(d-e*sqrt(-a)/sqrt(c)))^p*(1+(-d-e*x)/(d+e*sqrt(-a)/sqrt(c)))^p)],
[(d+e*x)^(-1-2*p)*(a+c*x^2)^p,x,2,-1/2*(a+c*x^2)^p*AppellF1(-2*p,-p,-p,1-2*p,(d+e*x)/(d-e*sqrt(-a)/sqrt(c)),(d+e*x)/(d+e*sqrt(-a)/sqrt(c)))/(e*p*(d+e*x)^(2*p)*(1+(-d-e*x)/(d-e*sqrt(-a)/sqrt(c)))^p*(1+(-d-e*x)/(d+e*sqrt(-a)/sqrt(c)))^p)],
[(d+e*x)^(-2-2*p)*(a+c*x^2)^p,x,1,-(d+e*x)^(-1-2*p)*(a+c*x^2)^p*hypergeom([-1-2*p,-p],[-2*p],2*(d+e*x)*sqrt(-a)*sqrt(c)/((-e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)-x*sqrt(c))))*(sqrt(-a)-x*sqrt(c))/((1+2*p)*(e*sqrt(-a)+d*sqrt(c))*(-(e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)+x*sqrt(c))/((-e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)-x*sqrt(c))))^p)],
[(d+e*x)^(-3-2*p)*(a+c*x^2)^p,x,2,-1/2*e*(a+c*x^2)^(1+p)/((c*d^2+a*e^2)*(1+p)*(d+e*x)^(2*(1+p)))-c*d*(d+e*x)^(-1-2*p)*(a+c*x^2)^p*hypergeom([-1-2*p,-p],[-2*p],2*(d+e*x)*sqrt(-a)*sqrt(c)/((-e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)-x*sqrt(c))))*(sqrt(-a)-x*sqrt(c))/((c*d^2+a*e^2)*(1+2*p)*(e*sqrt(-a)+d*sqrt(c))*(-(e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)+x*sqrt(c))/((-e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)-x*sqrt(c))))^p)],
[(d+e*x)^(-4-2*p)*(a+c*x^2)^p,x,3,-e*(d+e*x)^(-3-2*p)*(a+c*x^2)^(1+p)/((c*d^2+a*e^2)*(3+2*p))-c*d*e*(2+p)*(a+c*x^2)^(1+p)/((c*d^2+a*e^2)^2*(1+p)*(3+2*p)*(d+e*x)^(2*(1+p)))+c*(a*e^2-c*d^2*(3+2*p))*(d+e*x)^(-1-2*p)*(a+c*x^2)^p*hypergeom([-1-2*p,-p],[-2*p],2*(d+e*x)*sqrt(-a)*sqrt(c)/((-e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)-x*sqrt(c))))*(sqrt(-a)-x*sqrt(c))/((c*d^2+a*e^2)^2*(1+2*p)*(3+2*p)*(e*sqrt(-a)+d*sqrt(c))*(-(e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)+x*sqrt(c))/((-e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)-x*sqrt(c))))^p)],
[(d+e*x)^(-5-2*p)*(a+c*x^2)^p,x,4,-c*d*e*(3+p)*(d+e*x)^(-3-2*p)*(a+c*x^2)^(1+p)/((c*d^2+a*e^2)^2*(2+p)*(3+2*p))+1/2*c*e*(a*e^2*(3+2*p)-c*d^2*(9+8*p+2*p^2))*(a+c*x^2)^(1+p)/((c*d^2+a*e^2)^3*(1+p)*(2+p)*(3+2*p)*(d+e*x)^(2*(1+p)))-1/2*e*(a+c*x^2)^(1+p)/((c*d^2+a*e^2)*(2+p)*(d+e*x)^(2*(2+p)))+c^2*d*(3*a*e^2-c*d^2*(3+2*p))*(d+e*x)^(-1-2*p)*(a+c*x^2)^p*hypergeom([-1-2*p,-p],[-2*p],2*(d+e*x)*sqrt(-a)*sqrt(c)/((-e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)-x*sqrt(c))))*(sqrt(-a)-x*sqrt(c))/((c*d^2+a*e^2)^3*(1+2*p)*(3+2*p)*(e*sqrt(-a)+d*sqrt(c))*(-(e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)+x*sqrt(c))/((-e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)-x*sqrt(c))))^p)],
[(d+e*x)^(-6-2*p)*(a+c*x^2)^p,x,5,-e*(d+e*x)^(-5-2*p)*(a+c*x^2)^(1+p)/((c*d^2+a*e^2)*(5+2*p))+c*e*(3*a*e^2*(2+p)-c*d^2*(18+11*p+2*p^2))*(d+e*x)^(-3-2*p)*(a+c*x^2)^(1+p)/((c*d^2+a*e^2)^3*(2+p)*(3+2*p)*(5+2*p))+c^2*d*e*(3+p)*(a*e^2*(8+5*p)-c*d^2*(8+7*p+2*p^2))*(a+c*x^2)^(1+p)/((c*d^2+a*e^2)^4*(1+p)*(2+p)*(3+2*p)*(5+2*p)*(d+e*x)^(2*(1+p)))-c*d*e*(4+p)*(a+c*x^2)^(1+p)/((c*d^2+a*e^2)^2*(2+p)*(5+2*p)*(d+e*x)^(2*(2+p)))-c^2*(3*a^2*e^4-6*a*c*d^2*e^2*(5+2*p)+c^2*d^4*(15+16*p+4*p^2))*(d+e*x)^(-1-2*p)*(a+c*x^2)^p*hypergeom([-1-2*p,-p],[-2*p],2*(d+e*x)*sqrt(-a)*sqrt(c)/((-e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)-x*sqrt(c))))*(sqrt(-a)-x*sqrt(c))/((c*d^2+a*e^2)^4*(1+2*p)*(3+2*p)*(5+2*p)*(e*sqrt(-a)+d*sqrt(c))*(-(e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)+x*sqrt(c))/((-e*sqrt(-a)+d*sqrt(c))*(sqrt(-a)-x*sqrt(c))))^p)],
[(3-4*x)^n/sqrt(1-x^2),x,2,7^n*AppellF1(1/2,-n,1/2,3/2,4/7*(1+x),1/2*(1+x))*sqrt(2)*sqrt(1+x)],

# Integrands of the form (d+e x)^m (a+b x+c x^2)^p when b=0 and c d^2+a e^2=0

# Integrands of the form (d+e x)^m (d^2 - e^2 x^2)^p

# p>0

# p<0
[(a+b*x)^6/(a^2-b^2*x^2),x,3,-16*a^4*x-4*a^3*(a+b*x)^2/b-4/3*a^2*(a+b*x)^3/b-1/2*a*(a+b*x)^4/b-1/5*(a+b*x)^5/b-32*a^5*log(a-b*x)/b],
[(a+b*x)^5/(a^2-b^2*x^2),x,3,-8*a^3*x-2*a^2*(a+b*x)^2/b-2/3*a*(a+b*x)^3/b-1/4*(a+b*x)^4/b-16*a^4*log(a-b*x)/b],
[(a+b*x)^4/(a^2-b^2*x^2),x,3,-4*a^2*x-a*(a+b*x)^2/b-1/3*(a+b*x)^3/b-8*a^3*log(a-b*x)/b],
[(a+b*x)^3/(a^2-b^2*x^2),x,3,-3*a*x-1/2*b*x^2-4*a^2*log(a-b*x)/b],
[(a+b*x)^2/(a^2-b^2*x^2),x,3,-x-2*a*log(a-b*x)/b],
[(a+b*x)/(a^2-b^2*x^2),x,2,-log(a-b*x)/b],
[1/((a+b*x)*(a^2-b^2*x^2)),x,4,(-1/2)/(a*b*(a+b*x))+1/2*arctanh(b*x/a)/(a^2*b)],
[1/((a+b*x)^2*(a^2-b^2*x^2)),x,4,(-1/4)/(a*b*(a+b*x)^2)+(-1/4)/(a^2*b*(a+b*x))+1/4*arctanh(b*x/a)/(a^3*b)],
[1/((a+b*x)^3*(a^2-b^2*x^2)),x,4,(-1/6)/(a*b*(a+b*x)^3)+(-1/8)/(a^2*b*(a+b*x)^2)+(-1/8)/(a^3*b*(a+b*x))+1/8*arctanh(b*x/a)/(a^4*b)],
[1/((a+b*x)^4*(a^2-b^2*x^2)),x,4,(-1/8)/(a*b*(a+b*x)^4)+(-1/12)/(a^2*b*(a+b*x)^3)+(-1/16)/(a^3*b*(a+b*x)^2)+(-1/16)/(a^4*b*(a+b*x))+1/16*arctanh(b*x/a)/(a^5*b)],
[(a+b*x)^7/(a^2-b^2*x^2)^2,x,3,49*a^3*x+23/2*a^2*b*x^2+7/3*a*b^2*x^3+1/4*b^3*x^4+32*a^5/(b*(a-b*x))+80*a^4*log(a-b*x)/b],
[(a+b*x)^6/(a^2-b^2*x^2)^2,x,3,17*a^2*x+3*a*b*x^2+1/3*b^2*x^3+16*a^4/(b*(a-b*x))+32*a^3*log(a-b*x)/b],
[(a+b*x)^5/(a^2-b^2*x^2)^2,x,3,5*a*x+1/2*b*x^2+8*a^3/(b*(a-b*x))+12*a^2*log(a-b*x)/b],
[(a+b*x)^4/(a^2-b^2*x^2)^2,x,3,x+4*a^2/(b*(a-b*x))+4*a*log(a-b*x)/b],
[(a+b*x)^3/(a^2-b^2*x^2)^2,x,3,2*a/(b*(a-b*x))+log(a-b*x)/b],
[(a+b*x)^2/(a^2-b^2*x^2)^2,x,2,1/(b*(a-b*x))],
[(a+b*x)/(a^2-b^2*x^2)^2,x,4,1/2/(a*b*(a-b*x))+1/2*arctanh(b*x/a)/(a^2*b)],
[1/((a+b*x)*(a^2-b^2*x^2)^2),x,4,1/8/(a^3*b*(a-b*x))+(-1/8)/(a^2*b*(a+b*x)^2)+(-1/4)/(a^3*b*(a+b*x))+3/8*arctanh(b*x/a)/(a^4*b)],
[1/((a+b*x)^2*(a^2-b^2*x^2)^2),x,4,1/16/(a^4*b*(a-b*x))+(-1/12)/(a^2*b*(a+b*x)^3)+(-1/8)/(a^3*b*(a+b*x)^2)+(-3/16)/(a^4*b*(a+b*x))+1/4*arctanh(b*x/a)/(a^5*b)],
[1/((a+b*x)^3*(a^2-b^2*x^2)^2),x,4,1/32/(a^5*b*(a-b*x))+(-1/16)/(a^2*b*(a+b*x)^4)+(-1/12)/(a^3*b*(a+b*x)^3)+(-3/32)/(a^4*b*(a+b*x)^2)+(-1/8)/(a^5*b*(a+b*x))+5/32*arctanh(b*x/a)/(a^6*b)],
[(a+b*x)^8/(a^2-b^2*x^2)^3,x,3,-31*a^2*x-4*a*b*x^2-1/3*b^2*x^3+16*a^5/(b*(a-b*x)^2)-80*a^4/(b*(a-b*x))-80*a^3*log(a-b*x)/b],
[(a+b*x)^7/(a^2-b^2*x^2)^3,x,3,-7*a*x-1/2*b*x^2+8*a^4/(b*(a-b*x)^2)-32*a^3/(b*(a-b*x))-24*a^2*log(a-b*x)/b],
[(a+b*x)^6/(a^2-b^2*x^2)^3,x,3,-x+4*a^3/(b*(a-b*x)^2)-12*a^2/(b*(a-b*x))-6*a*log(a-b*x)/b],
[(a+b*x)^5/(a^2-b^2*x^2)^3,x,3,2*a^2/(b*(a-b*x)^2)-4*a/(b*(a-b*x))-log(a-b*x)/b],
[(a+b*x)^4/(a^2-b^2*x^2)^3,x,2,x/(a-b*x)^2],
[(a+b*x)^3/(a^2-b^2*x^2)^3,x,2,1/2/(b*(a-b*x)^2)],
[(a+b*x)^2/(a^2-b^2*x^2)^3,x,4,1/4/(a*b*(a-b*x)^2)+1/4/(a^2*b*(a-b*x))+1/4*arctanh(b*x/a)/(a^3*b)],
[(a+b*x)/(a^2-b^2*x^2)^3,x,4,1/8/(a^2*b*(a-b*x)^2)+1/4/(a^3*b*(a-b*x))+(-1/8)/(a^3*b*(a+b*x))+3/8*arctanh(b*x/a)/(a^4*b)],
[1/((a+b*x)*(a^2-b^2*x^2)^3),x,4,1/32/(a^4*b*(a-b*x)^2)+1/8/(a^5*b*(a-b*x))+(-1/24)/(a^3*b*(a+b*x)^3)+(-3/32)/(a^4*b*(a+b*x)^2)+(-3/16)/(a^5*b*(a+b*x))+5/16*arctanh(b*x/a)/(a^6*b)],
[1/((a+b*x)^2*(a^2-b^2*x^2)^3),x,4,1/64/(a^5*b*(a-b*x)^2)+5/64/(a^6*b*(a-b*x))+(-1/32)/(a^3*b*(a+b*x)^4)+(-1/16)/(a^4*b*(a+b*x)^3)+(-3/32)/(a^5*b*(a+b*x)^2)+(-5/32)/(a^6*b*(a+b*x))+15/64*arctanh(b*x/a)/(a^7*b)],

# Integrands of the form (d+e x)^m (d^2 - e^2 x^2)^(p/2)

# p>0
[(a+b*x)^4*sqrt(a^2-b^2*x^2),x,7,-7/8*a^3*(a^2-b^2*x^2)^(3/2)/b-21/40*a^2*(a+b*x)*(a^2-b^2*x^2)^(3/2)/b-3/10*a*(a+b*x)^2*(a^2-b^2*x^2)^(3/2)/b-1/6*(a+b*x)^3*(a^2-b^2*x^2)^(3/2)/b+21/16*a^6*arctan(b*x/sqrt(a^2-b^2*x^2))/b+21/16*a^4*x*sqrt(a^2-b^2*x^2)],
[(a+b*x)^3*sqrt(a^2-b^2*x^2),x,6,-7/12*a^2*(a^2-b^2*x^2)^(3/2)/b-7/20*a*(a+b*x)*(a^2-b^2*x^2)^(3/2)/b-1/5*(a+b*x)^2*(a^2-b^2*x^2)^(3/2)/b+7/8*a^5*arctan(b*x/sqrt(a^2-b^2*x^2))/b+7/8*a^3*x*sqrt(a^2-b^2*x^2)],
[(a+b*x)^2*sqrt(a^2-b^2*x^2),x,5,-5/12*a*(a^2-b^2*x^2)^(3/2)/b-1/4*(a+b*x)*(a^2-b^2*x^2)^(3/2)/b+5/8*a^4*arctan(b*x/sqrt(a^2-b^2*x^2))/b+5/8*a^2*x*sqrt(a^2-b^2*x^2)],
[(a+b*x)*sqrt(a^2-b^2*x^2),x,4,-1/3*(a^2-b^2*x^2)^(3/2)/b+1/2*a^3*arctan(b*x/sqrt(a^2-b^2*x^2))/b+1/2*a*x*sqrt(a^2-b^2*x^2)],
[sqrt(a^2-b^2*x^2)/(a+b*x),x,3,a*arctan(b*x/sqrt(a^2-b^2*x^2))/b+sqrt(a^2-b^2*x^2)/b],
[sqrt(a^2-b^2*x^2)/(a+b*x)^2,x,3,-arctan(b*x/sqrt(a^2-b^2*x^2))/b-2*sqrt(a^2-b^2*x^2)/(b*(a+b*x))],
[sqrt(a^2-b^2*x^2)/(a+b*x)^3,x,1,-1/3*(a^2-b^2*x^2)^(3/2)/(a*b*(a+b*x)^3)],
[sqrt(a^2-b^2*x^2)/(a+b*x)^4,x,2,-1/5*(a^2-b^2*x^2)^(3/2)/(a*b*(a+b*x)^4)-1/15*(a^2-b^2*x^2)^(3/2)/(a^2*b*(a+b*x)^3)],
[sqrt(a^2-b^2*x^2)/(a+b*x)^5,x,3,-1/7*(a^2-b^2*x^2)^(3/2)/(a*b*(a+b*x)^5)-2/35*(a^2-b^2*x^2)^(3/2)/(a^2*b*(a+b*x)^4)-2/105*(a^2-b^2*x^2)^(3/2)/(a^3*b*(a+b*x)^3)],
[sqrt(a^2-b^2*x^2)/(a+b*x)^6,x,4,-1/9*(a^2-b^2*x^2)^(3/2)/(a*b*(a+b*x)^6)-1/21*(a^2-b^2*x^2)^(3/2)/(a^2*b*(a+b*x)^5)-2/105*(a^2-b^2*x^2)^(3/2)/(a^3*b*(a+b*x)^4)-2/315*(a^2-b^2*x^2)^(3/2)/(a^4*b*(a+b*x)^3)],
[sqrt(a^2-b^2*x^2)/(a+b*x)^7,x,5,-1/11*(a^2-b^2*x^2)^(3/2)/(a*b*(a+b*x)^7)-4/99*(a^2-b^2*x^2)^(3/2)/(a^2*b*(a+b*x)^6)-4/231*(a^2-b^2*x^2)^(3/2)/(a^3*b*(a+b*x)^5)-8/1155*(a^2-b^2*x^2)^(3/2)/(a^4*b*(a+b*x)^4)-8/3465*(a^2-b^2*x^2)^(3/2)/(a^5*b*(a+b*x)^3)],
[(a+b*x)^3*(a^2-b^2*x^2)^(3/2),x,7,3/8*a^3*x*(a^2-b^2*x^2)^(3/2)-3/10*a^2*(a^2-b^2*x^2)^(5/2)/b-3/14*a*(a+b*x)*(a^2-b^2*x^2)^(5/2)/b-1/7*(a+b*x)^2*(a^2-b^2*x^2)^(5/2)/b+9/16*a^7*arctan(b*x/sqrt(a^2-b^2*x^2))/b+9/16*a^5*x*sqrt(a^2-b^2*x^2)],
[(a+b*x)^2*(a^2-b^2*x^2)^(3/2),x,6,7/24*a^2*x*(a^2-b^2*x^2)^(3/2)-7/30*a*(a^2-b^2*x^2)^(5/2)/b-1/6*(a+b*x)*(a^2-b^2*x^2)^(5/2)/b+7/16*a^6*arctan(b*x/sqrt(a^2-b^2*x^2))/b+7/16*a^4*x*sqrt(a^2-b^2*x^2)],
[(a+b*x)*(a^2-b^2*x^2)^(3/2),x,5,1/4*a*x*(a^2-b^2*x^2)^(3/2)-1/5*(a^2-b^2*x^2)^(5/2)/b+3/8*a^5*arctan(b*x/sqrt(a^2-b^2*x^2))/b+3/8*a^3*x*sqrt(a^2-b^2*x^2)],
[(a^2-b^2*x^2)^(3/2)/(a+b*x),x,4,1/3*(a^2-b^2*x^2)^(3/2)/b+1/2*a^3*arctan(b*x/sqrt(a^2-b^2*x^2))/b+1/2*a*x*sqrt(a^2-b^2*x^2)],
[(a^2-b^2*x^2)^(3/2)/(a+b*x)^2,x,4,1/2*(a^2-b^2*x^2)^(3/2)/(b*(a+b*x))+3/2*a^2*arctan(b*x/sqrt(a^2-b^2*x^2))/b+3/2*a*sqrt(a^2-b^2*x^2)/b],
[(a^2-b^2*x^2)^(3/2)/(a+b*x)^3,x,4,-2*(a^2-b^2*x^2)^(3/2)/(b*(a+b*x)^2)-3*a*arctan(b*x/sqrt(a^2-b^2*x^2))/b-3*sqrt(a^2-b^2*x^2)/b],
[(a^2-b^2*x^2)^(3/2)/(a+b*x)^4,x,4,-2/3*(a^2-b^2*x^2)^(3/2)/(b*(a+b*x)^3)+arctan(b*x/sqrt(a^2-b^2*x^2))/b+2*sqrt(a^2-b^2*x^2)/(b*(a+b*x))],
[(a^2-b^2*x^2)^(3/2)/(a+b*x)^5,x,1,-1/5*(a^2-b^2*x^2)^(5/2)/(a*b*(a+b*x)^5)],
[(a^2-b^2*x^2)^(3/2)/(a+b*x)^6,x,2,-1/7*(a^2-b^2*x^2)^(5/2)/(a*b*(a+b*x)^6)-1/35*(a^2-b^2*x^2)^(5/2)/(a^2*b*(a+b*x)^5)],
[(a^2-b^2*x^2)^(3/2)/(a+b*x)^7,x,3,-1/9*(a^2-b^2*x^2)^(5/2)/(a*b*(a+b*x)^7)-2/63*(a^2-b^2*x^2)^(5/2)/(a^2*b*(a+b*x)^6)-2/315*(a^2-b^2*x^2)^(5/2)/(a^3*b*(a+b*x)^5)],
[(a^2-b^2*x^2)^(3/2)/(a+b*x)^8,x,4,-1/11*(a^2-b^2*x^2)^(5/2)/(a*b*(a+b*x)^8)-1/33*(a^2-b^2*x^2)^(5/2)/(a^2*b*(a+b*x)^7)-2/231*(a^2-b^2*x^2)^(5/2)/(a^3*b*(a+b*x)^6)-2/1155*(a^2-b^2*x^2)^(5/2)/(a^4*b*(a+b*x)^5)],
[(a^2-b^2*x^2)^(3/2)/(a+b*x)^9,x,5,-1/13*(a^2-b^2*x^2)^(5/2)/(a*b*(a+b*x)^9)-4/143*(a^2-b^2*x^2)^(5/2)/(a^2*b*(a+b*x)^8)-4/429*(a^2-b^2*x^2)^(5/2)/(a^3*b*(a+b*x)^7)-8/3003*(a^2-b^2*x^2)^(5/2)/(a^4*b*(a+b*x)^6)-8/15015*(a^2-b^2*x^2)^(5/2)/(a^5*b*(a+b*x)^5)],
[(d+e*x)^3*(d^2-e^2*x^2)^(7/2),x,9,91/384*d^7*x*(d^2-e^2*x^2)^(3/2)+91/480*d^5*x*(d^2-e^2*x^2)^(5/2)+13/80*d^3*x*(d^2-e^2*x^2)^(7/2)-13/90*d^2*(d^2-e^2*x^2)^(9/2)/e-13/110*d*(d+e*x)*(d^2-e^2*x^2)^(9/2)/e-1/11*(d+e*x)^2*(d^2-e^2*x^2)^(9/2)/e+91/256*d^11*arctan(e*x/sqrt(d^2-e^2*x^2))/e+91/256*d^9*x*sqrt(d^2-e^2*x^2)],
[(d+e*x)^2*(d^2-e^2*x^2)^(7/2),x,8,77/384*d^6*x*(d^2-e^2*x^2)^(3/2)+77/480*d^4*x*(d^2-e^2*x^2)^(5/2)+11/80*d^2*x*(d^2-e^2*x^2)^(7/2)-11/90*d*(d^2-e^2*x^2)^(9/2)/e-1/10*(d+e*x)*(d^2-e^2*x^2)^(9/2)/e+77/256*d^10*arctan(e*x/sqrt(d^2-e^2*x^2))/e+77/256*d^8*x*sqrt(d^2-e^2*x^2)],
[(d+e*x)*(d^2-e^2*x^2)^(7/2),x,7,35/192*d^5*x*(d^2-e^2*x^2)^(3/2)+7/48*d^3*x*(d^2-e^2*x^2)^(5/2)+1/8*d*x*(d^2-e^2*x^2)^(7/2)-1/9*(d^2-e^2*x^2)^(9/2)/e+35/128*d^9*arctan(e*x/sqrt(d^2-e^2*x^2))/e+35/128*d^7*x*sqrt(d^2-e^2*x^2)],
[(d^2-e^2*x^2)^(7/2)/(d+e*x),x,6,5/24*d^3*x*(d^2-e^2*x^2)^(3/2)+1/6*d*x*(d^2-e^2*x^2)^(5/2)+1/7*(d^2-e^2*x^2)^(7/2)/e+5/16*d^7*arctan(e*x/sqrt(d^2-e^2*x^2))/e+5/16*d^5*x*sqrt(d^2-e^2*x^2)],
[(d^2-e^2*x^2)^(7/2)/(d+e*x)^2,x,7,7/24*d^2*x*(d^2-e^2*x^2)^(3/2)+7/30*d*(d^2-e^2*x^2)^(5/2)/e+1/6*(d-e*x)*(d^2-e^2*x^2)^(5/2)/e+7/16*d^6*arctan(e*x/sqrt(d^2-e^2*x^2))/e+7/16*d^4*x*sqrt(d^2-e^2*x^2)],
[(d^2-e^2*x^2)^(7/2)/(d+e*x)^3,x,7,7/12*d^2*(d^2-e^2*x^2)^(3/2)/e+7/20*d*(d-e*x)*(d^2-e^2*x^2)^(3/2)/e+1/5*(d-e*x)^2*(d^2-e^2*x^2)^(3/2)/e+7/8*d^5*arctan(e*x/sqrt(d^2-e^2*x^2))/e+7/8*d^3*x*sqrt(d^2-e^2*x^2)],
[(d^2-e^2*x^2)^(7/2)/(d+e*x)^4,x,7,35/12*d*(d^2-e^2*x^2)^(3/2)/e+7/4*(d-e*x)*(d^2-e^2*x^2)^(3/2)/e+2*(d^2-e^2*x^2)^(7/2)/(e*(d+e*x)^3)+35/8*d^4*arctan(e*x/sqrt(d^2-e^2*x^2))/e+35/8*d^2*x*sqrt(d^2-e^2*x^2)],
[(d^2-e^2*x^2)^(7/2)/(d+e*x)^5,x,6,-35/3*(d^2-e^2*x^2)^(3/2)/e-14*(d^2-e^2*x^2)^(5/2)/(e*(d+e*x)^2)-2*(d^2-e^2*x^2)^(7/2)/(e*(d+e*x)^4)-35/2*d^3*arctan(e*x/sqrt(d^2-e^2*x^2))/e-35/2*d*x*sqrt(d^2-e^2*x^2)],
[(d^2-e^2*x^2)^(7/2)/(d+e*x)^6,x,6,35/6*(d^2-e^2*x^2)^(3/2)/(e*(d+e*x))+14/3*(d^2-e^2*x^2)^(5/2)/(e*(d+e*x)^3)-2/3*(d^2-e^2*x^2)^(7/2)/(e*(d+e*x)^5)+35/2*d^2*arctan(e*x/sqrt(d^2-e^2*x^2))/e+35/2*d*sqrt(d^2-e^2*x^2)/e],
[(d^2-e^2*x^2)^(7/2)/(d+e*x)^7,x,6,-14/3*(d^2-e^2*x^2)^(3/2)/(e*(d+e*x)^2)+14/15*(d^2-e^2*x^2)^(5/2)/(e*(d+e*x)^4)-2/5*(d^2-e^2*x^2)^(7/2)/(e*(d+e*x)^6)-7*d*arctan(e*x/sqrt(d^2-e^2*x^2))/e-7*sqrt(d^2-e^2*x^2)/e],
[(d^2-e^2*x^2)^(7/2)/(d+e*x)^8,x,6,-2/3*(d^2-e^2*x^2)^(3/2)/(e*(d+e*x)^3)+2/5*(d^2-e^2*x^2)^(5/2)/(e*(d+e*x)^5)-2/7*(d^2-e^2*x^2)^(7/2)/(e*(d+e*x)^7)+arctan(e*x/sqrt(d^2-e^2*x^2))/e+2*sqrt(d^2-e^2*x^2)/(e*(d+e*x))],
[(d^2-e^2*x^2)^(7/2)/(d+e*x)^9,x,1,-1/9*(d^2-e^2*x^2)^(9/2)/(d*e*(d+e*x)^9)],
[(d^2-e^2*x^2)^(7/2)/(d+e*x)^10,x,2,-1/11*(d^2-e^2*x^2)^(9/2)/(d*e*(d+e*x)^10)-1/99*(d^2-e^2*x^2)^(9/2)/(d^2*e*(d+e*x)^9)],
[(d^2-e^2*x^2)^(7/2)/(d+e*x)^11,x,3,-1/13*(d^2-e^2*x^2)^(9/2)/(d*e*(d+e*x)^11)-2/143*(d^2-e^2*x^2)^(9/2)/(d^2*e*(d+e*x)^10)-2/1287*(d^2-e^2*x^2)^(9/2)/(d^3*e*(d+e*x)^9)],
[(d^2-e^2*x^2)^(7/2)/(d+e*x)^12,x,4,-1/15*(d^2-e^2*x^2)^(9/2)/(d*e*(d+e*x)^12)-1/65*(d^2-e^2*x^2)^(9/2)/(d^2*e*(d+e*x)^11)-2/715*(d^2-e^2*x^2)^(9/2)/(d^3*e*(d+e*x)^10)-2/6435*(d^2-e^2*x^2)^(9/2)/(d^4*e*(d+e*x)^9)],
[(d^2-e^2*x^2)^(7/2)/(d+e*x)^13,x,5,-1/17*(d^2-e^2*x^2)^(9/2)/(d*e*(d+e*x)^13)-4/255*(d^2-e^2*x^2)^(9/2)/(d^2*e*(d+e*x)^12)-4/1105*(d^2-e^2*x^2)^(9/2)/(d^3*e*(d+e*x)^11)-8/12155*(d^2-e^2*x^2)^(9/2)/(d^4*e*(d+e*x)^10)-8/109395*(d^2-e^2*x^2)^(9/2)/(d^5*e*(d+e*x)^9)],
[sqrt(a^2-b^2*x^2)/(a-b*x),x,3,a*arctan(b*x/sqrt(a^2-b^2*x^2))/b-sqrt(a^2-b^2*x^2)/b],
[(a+b*x)^2*sqrt(-a^2*c/b^2+c*x^2),x,5,5/12*a*b*(-a^2*c/b^2+c*x^2)^(3/2)/c+1/4*b*(a+b*x)*(-a^2*c/b^2+c*x^2)^(3/2)/c-5/8*a^4*arctanh(x*sqrt(c)/sqrt(-a^2*c/b^2+c*x^2))*sqrt(c)/b^2+5/8*a^2*x*sqrt(-a^2*c/b^2+c*x^2)],
[(a+b*x)^3*sqrt(-a^2*c/b^2+c*x^2),x,6,7/12*a^2*b*(-a^2*c/b^2+c*x^2)^(3/2)/c+7/20*a*b*(a+b*x)*(-a^2*c/b^2+c*x^2)^(3/2)/c+1/5*b*(a+b*x)^2*(-a^2*c/b^2+c*x^2)^(3/2)/c-7/8*a^5*arctanh(x*sqrt(c)/sqrt(-a^2*c/b^2+c*x^2))*sqrt(c)/b^2+7/8*a^3*x*sqrt(-a^2*c/b^2+c*x^2)],
[(1+x)*sqrt(-1+x^2),x,4,1/3*(-1+x^2)^(3/2)-1/2*arctanh(x/sqrt(-1+x^2))+1/2*x*sqrt(-1+x^2)],
[(1+x)*sqrt(1-x^2),x,3,-1/3*(1-x^2)^(3/2)+1/2*arcsin(x)+1/2*x*sqrt(1-x^2)],
[sqrt(1-x^2)/(1+x),x,2,arcsin(x)+sqrt(1-x^2)],
[(1-x)*sqrt(1-x^2),x,3,1/3*(1-x^2)^(3/2)+1/2*arcsin(x)+1/2*x*sqrt(1-x^2)],
[sqrt(1-x^2)/(1-x),x,2,arcsin(x)-sqrt(1-x^2)],
[sqrt(1-x^2)/(1-x)^2,x,2,-arcsin(x)+2*sqrt(1-x^2)/(1-x)],
[sqrt(1-x^2)/(1-x)^3,x,1,1/3*(1-x^2)^(3/2)/(1-x)^3],

# p<0
[(d+e*x)^5/sqrt(d^2-e^2*x^2),x,7,63/8*d^5*arctan(e*x/sqrt(d^2-e^2*x^2))/e-63/8*d^4*sqrt(d^2-e^2*x^2)/e-21/8*d^3*(d+e*x)*sqrt(d^2-e^2*x^2)/e-21/20*d^2*(d+e*x)^2*sqrt(d^2-e^2*x^2)/e-9/20*d*(d+e*x)^3*sqrt(d^2-e^2*x^2)/e-1/5*(d+e*x)^4*sqrt(d^2-e^2*x^2)/e],
[(d+e*x)^4/sqrt(d^2-e^2*x^2),x,6,35/8*d^4*arctan(e*x/sqrt(d^2-e^2*x^2))/e-35/8*d^3*sqrt(d^2-e^2*x^2)/e-35/24*d^2*(d+e*x)*sqrt(d^2-e^2*x^2)/e-7/12*d*(d+e*x)^2*sqrt(d^2-e^2*x^2)/e-1/4*(d+e*x)^3*sqrt(d^2-e^2*x^2)/e],
[(d+e*x)^3/sqrt(d^2-e^2*x^2),x,5,5/2*d^3*arctan(e*x/sqrt(d^2-e^2*x^2))/e-5/2*d^2*sqrt(d^2-e^2*x^2)/e-5/6*d*(d+e*x)*sqrt(d^2-e^2*x^2)/e-1/3*(d+e*x)^2*sqrt(d^2-e^2*x^2)/e],
[(d+e*x)^2/sqrt(d^2-e^2*x^2),x,4,3/2*d^2*arctan(e*x/sqrt(d^2-e^2*x^2))/e-2*d*sqrt(d^2-e^2*x^2)/e-1/2*x*sqrt(d^2-e^2*x^2),3/2*d^2*arctan(e*x/sqrt(d^2-e^2*x^2))/e-3/2*d*sqrt(d^2-e^2*x^2)/e-1/2*(d+e*x)*sqrt(d^2-e^2*x^2)/e],
[(d+e*x)/sqrt(d^2-e^2*x^2),x,3,d*arctan(e*x/sqrt(d^2-e^2*x^2))/e-sqrt(d^2-e^2*x^2)/e],
[1/((d+e*x)*sqrt(d^2-e^2*x^2)),x,1,-sqrt(d^2-e^2*x^2)/(d*e*(d+e*x))],
[1/((d+e*x)^2*sqrt(d^2-e^2*x^2)),x,2,-1/3*sqrt(d^2-e^2*x^2)/(d*e*(d+e*x)^2)-1/3*sqrt(d^2-e^2*x^2)/(d^2*e*(d+e*x))],
[1/((d+e*x)^3*sqrt(d^2-e^2*x^2)),x,3,-1/5*sqrt(d^2-e^2*x^2)/(d*e*(d+e*x)^3)-2/15*sqrt(d^2-e^2*x^2)/(d^2*e*(d+e*x)^2)-2/15*sqrt(d^2-e^2*x^2)/(d^3*e*(d+e*x))],
[1/((d+e*x)^4*sqrt(d^2-e^2*x^2)),x,4,-1/7*sqrt(d^2-e^2*x^2)/(d*e*(d+e*x)^4)-3/35*sqrt(d^2-e^2*x^2)/(d^2*e*(d+e*x)^3)-2/35*sqrt(d^2-e^2*x^2)/(d^3*e*(d+e*x)^2)-2/35*sqrt(d^2-e^2*x^2)/(d^4*e*(d+e*x))],
[1/((d+e*x)^5*sqrt(d^2-e^2*x^2)),x,5,-1/9*sqrt(d^2-e^2*x^2)/(d*e*(d+e*x)^5)-4/63*sqrt(d^2-e^2*x^2)/(d^2*e*(d+e*x)^4)-4/105*sqrt(d^2-e^2*x^2)/(d^3*e*(d+e*x)^3)-8/315*sqrt(d^2-e^2*x^2)/(d^4*e*(d+e*x)^2)-8/315*sqrt(d^2-e^2*x^2)/(d^5*e*(d+e*x))],
[(d+e*x)^6/(d^2-e^2*x^2)^(5/2),x,6,2/3*(d+e*x)^5/(e*(d^2-e^2*x^2)^(3/2))+35/2*d^2*arctan(e*x/sqrt(d^2-e^2*x^2))/e-14/3*(d+e*x)^3/(e*sqrt(d^2-e^2*x^2))-35/2*d*sqrt(d^2-e^2*x^2)/e-35/6*(d+e*x)*sqrt(d^2-e^2*x^2)/e],
[(d+e*x)^5/(d^2-e^2*x^2)^(5/2),x,5,2/3*(d+e*x)^4/(e*(d^2-e^2*x^2)^(3/2))+5*d*arctan(e*x/sqrt(d^2-e^2*x^2))/e-10/3*(d+e*x)^2/(e*sqrt(d^2-e^2*x^2))-5*sqrt(d^2-e^2*x^2)/e],
[(d+e*x)^4/(d^2-e^2*x^2)^(5/2),x,4,2/3*(d+e*x)^3/(e*(d^2-e^2*x^2)^(3/2))+arctan(e*x/sqrt(d^2-e^2*x^2))/e-2*(d+e*x)/(e*sqrt(d^2-e^2*x^2))],
[(d+e*x)^3/(d^2-e^2*x^2)^(5/2),x,1,1/3*(d+e*x)^3/(d*e*(d^2-e^2*x^2)^(3/2))],
[(d+e*x)^2/(d^2-e^2*x^2)^(5/2),x,2,2/3*(d+e*x)/(e*(d^2-e^2*x^2)^(3/2))+1/3*x/(d^2*sqrt(d^2-e^2*x^2))],
[(d+e*x)/(d^2-e^2*x^2)^(5/2),x,2,1/3*(d+e*x)/(d*e*(d^2-e^2*x^2)^(3/2))+2/3*x/(d^3*sqrt(d^2-e^2*x^2))],
[1/((d+e*x)*(d^2-e^2*x^2)^(5/2)),x,3,4/15*x/(d^3*(d^2-e^2*x^2)^(3/2))+(-1/5)/(d*e*(d+e*x)*(d^2-e^2*x^2)^(3/2))+8/15*x/(d^5*sqrt(d^2-e^2*x^2))],
[1/((d+e*x)^2*(d^2-e^2*x^2)^(5/2)),x,4,4/21*x/(d^4*(d^2-e^2*x^2)^(3/2))+(-1/7)/(d*e*(d+e*x)^2*(d^2-e^2*x^2)^(3/2))+(-1/7)/(d^2*e*(d+e*x)*(d^2-e^2*x^2)^(3/2))+8/21*x/(d^6*sqrt(d^2-e^2*x^2))],
[1/((d+e*x)^3*(d^2-e^2*x^2)^(5/2)),x,5,8/63*x/(d^5*(d^2-e^2*x^2)^(3/2))+(-1/9)/(d*e*(d+e*x)^3*(d^2-e^2*x^2)^(3/2))+(-2/21)/(d^2*e*(d+e*x)^2*(d^2-e^2*x^2)^(3/2))+(-2/21)/(d^3*e*(d+e*x)*(d^2-e^2*x^2)^(3/2))+16/63*x/(d^7*sqrt(d^2-e^2*x^2))],
[1/((d+e*x)^4*(d^2-e^2*x^2)^(5/2)),x,6,8/99*x/(d^6*(d^2-e^2*x^2)^(3/2))+(-1/11)/(d*e*(d+e*x)^4*(d^2-e^2*x^2)^(3/2))+(-7/99)/(d^2*e*(d+e*x)^3*(d^2-e^2*x^2)^(3/2))+(-2/33)/(d^3*e*(d+e*x)^2*(d^2-e^2*x^2)^(3/2))+(-2/33)/(d^4*e*(d+e*x)*(d^2-e^2*x^2)^(3/2))+16/99*x/(d^8*sqrt(d^2-e^2*x^2))],
[(d+e*x)^9/(d^2-e^2*x^2)^(7/2),x,8,2/5*(d+e*x)^8/(e*(d^2-e^2*x^2)^(5/2))-22/15*(d+e*x)^6/(e*(d^2-e^2*x^2)^(3/2))-231/2*d^3*arctan(e*x/sqrt(d^2-e^2*x^2))/e+66/5*(d+e*x)^4/(e*sqrt(d^2-e^2*x^2))+231/2*d^2*sqrt(d^2-e^2*x^2)/e+77/2*d*(d+e*x)*sqrt(d^2-e^2*x^2)/e+77/5*(d+e*x)^2*sqrt(d^2-e^2*x^2)/e],
[(d+e*x)^8/(d^2-e^2*x^2)^(7/2),x,7,2/5*(d+e*x)^7/(e*(d^2-e^2*x^2)^(5/2))-6/5*(d+e*x)^5/(e*(d^2-e^2*x^2)^(3/2))-63/2*d^2*arctan(e*x/sqrt(d^2-e^2*x^2))/e+42/5*(d+e*x)^3/(e*sqrt(d^2-e^2*x^2))+63/2*d*sqrt(d^2-e^2*x^2)/e+21/2*(d+e*x)*sqrt(d^2-e^2*x^2)/e],
[(d+e*x)^7/(d^2-e^2*x^2)^(7/2),x,6,2/5*(d+e*x)^6/(e*(d^2-e^2*x^2)^(5/2))-14/15*(d+e*x)^4/(e*(d^2-e^2*x^2)^(3/2))-7*d*arctan(e*x/sqrt(d^2-e^2*x^2))/e+14/3*(d+e*x)^2/(e*sqrt(d^2-e^2*x^2))+7*sqrt(d^2-e^2*x^2)/e],
[(d+e*x)^6/(d^2-e^2*x^2)^(7/2),x,5,2/5*(d+e*x)^5/(e*(d^2-e^2*x^2)^(5/2))-2/3*(d+e*x)^3/(e*(d^2-e^2*x^2)^(3/2))-arctan(e*x/sqrt(d^2-e^2*x^2))/e+2*(d+e*x)/(e*sqrt(d^2-e^2*x^2))],
[(d+e*x)^5/(d^2-e^2*x^2)^(7/2),x,1,1/5*(d+e*x)^5/(d*e*(d^2-e^2*x^2)^(5/2))],
[(d+e*x)^4/(d^2-e^2*x^2)^(7/2),x,2,1/3*(d+e*x)^4/(d*e*(d^2-e^2*x^2)^(5/2))-1/15*(d+e*x)^5/(d^2*e*(d^2-e^2*x^2)^(5/2))],
[(d+e*x)^3/(d^2-e^2*x^2)^(7/2),x,4,1/5*sqrt(d^2-e^2*x^2)/(d*e*(d-e*x)^3)+2/15*sqrt(d^2-e^2*x^2)/(d^2*e*(d-e*x)^2)+2/15*sqrt(d^2-e^2*x^2)/(d^3*e*(d-e*x))],
[(d+e*x)^2/(d^2-e^2*x^2)^(7/2),x,3,2/5*(d+e*x)/(e*(d^2-e^2*x^2)^(5/2))+1/5*x/(d^2*(d^2-e^2*x^2)^(3/2))+2/5*x/(d^4*sqrt(d^2-e^2*x^2))],
[(d+e*x)/(d^2-e^2*x^2)^(7/2),x,3,1/5*(d+e*x)/(d*e*(d^2-e^2*x^2)^(5/2))+4/15*x/(d^3*(d^2-e^2*x^2)^(3/2))+8/15*x/(d^5*sqrt(d^2-e^2*x^2))],
[1/((d+e*x)*(d^2-e^2*x^2)^(7/2)),x,4,6/35*x/(d^3*(d^2-e^2*x^2)^(5/2))+(-1/7)/(d*e*(d+e*x)*(d^2-e^2*x^2)^(5/2))+8/35*x/(d^5*(d^2-e^2*x^2)^(3/2))+16/35*x/(d^7*sqrt(d^2-e^2*x^2))],
[1/((d+e*x)^2*(d^2-e^2*x^2)^(7/2)),x,5,2/15*x/(d^4*(d^2-e^2*x^2)^(5/2))+(-1/9)/(d*e*(d+e*x)^2*(d^2-e^2*x^2)^(5/2))+(-1/9)/(d^2*e*(d+e*x)*(d^2-e^2*x^2)^(5/2))+8/45*x/(d^6*(d^2-e^2*x^2)^(3/2))+16/45*x/(d^8*sqrt(d^2-e^2*x^2))],
[1/((d+e*x)^3*(d^2-e^2*x^2)^(7/2)),x,6,16/165*x/(d^5*(d^2-e^2*x^2)^(5/2))+(-1/11)/(d*e*(d+e*x)^3*(d^2-e^2*x^2)^(5/2))+(-8/99)/(d^2*e*(d+e*x)^2*(d^2-e^2*x^2)^(5/2))+(-8/99)/(d^3*e*(d+e*x)*(d^2-e^2*x^2)^(5/2))+64/495*x/(d^7*(d^2-e^2*x^2)^(3/2))+128/495*x/(d^9*sqrt(d^2-e^2*x^2))],
[1/((d+e*x)^4*(d^2-e^2*x^2)^(7/2)),x,7,48/715*x/(d^6*(d^2-e^2*x^2)^(5/2))+(-1/13)/(d*e*(d+e*x)^4*(d^2-e^2*x^2)^(5/2))+(-9/143)/(d^2*e*(d+e*x)^3*(d^2-e^2*x^2)^(5/2))+(-8/143)/(d^3*e*(d+e*x)^2*(d^2-e^2*x^2)^(5/2))+(-8/143)/(d^4*e*(d+e*x)*(d^2-e^2*x^2)^(5/2))+64/715*x/(d^8*(d^2-e^2*x^2)^(3/2))+128/715*x/(d^10*sqrt(d^2-e^2*x^2))],
[1/((d+e*x)^5*(d^2-e^2*x^2)^(7/2)),x,8,32/715*x/(d^7*(d^2-e^2*x^2)^(5/2))+(-1/15)/(d*e*(d+e*x)^5*(d^2-e^2*x^2)^(5/2))+(-2/39)/(d^2*e*(d+e*x)^4*(d^2-e^2*x^2)^(5/2))+(-6/143)/(d^3*e*(d+e*x)^3*(d^2-e^2*x^2)^(5/2))+(-16/429)/(d^4*e*(d+e*x)^2*(d^2-e^2*x^2)^(5/2))+(-16/429)/(d^5*e*(d+e*x)*(d^2-e^2*x^2)^(5/2))+128/2145*x/(d^9*(d^2-e^2*x^2)^(3/2))+256/2145*x/(d^11*sqrt(d^2-e^2*x^2))],
[(1+x)/sqrt(1-x^2),x,2,arcsin(x)-sqrt(1-x^2)],
[(1-x)/sqrt(1-x^2),x,2,arcsin(x)+sqrt(1-x^2)],

# Integrands of the form (d+e x)^(m/2) (c d^2 - c e^2 x^2)^(p/2)

# p>0
[(d+e*x)^(5/2)*(c*d^2-c*e^2*x^2)^(1/2),x,4,-256/315*d^3*(c*d^2-c*e^2*x^2)^(3/2)/(c*e*(d+e*x)^(3/2))-2/9*(d+e*x)^(3/2)*(c*d^2-c*e^2*x^2)^(3/2)/(c*e)-64/105*d^2*(c*d^2-c*e^2*x^2)^(3/2)/(c*e*sqrt(d+e*x))-8/21*d*(c*d^2-c*e^2*x^2)^(3/2)*sqrt(d+e*x)/(c*e)],
[(d+e*x)^(3/2)*(c*d^2-c*e^2*x^2)^(1/2),x,3,-64/105*d^2*(c*d^2-c*e^2*x^2)^(3/2)/(c*e*(d+e*x)^(3/2))-16/35*d*(c*d^2-c*e^2*x^2)^(3/2)/(c*e*sqrt(d+e*x))-2/7*(c*d^2-c*e^2*x^2)^(3/2)*sqrt(d+e*x)/(c*e)],
[(d+e*x)^(1/2)*(c*d^2-c*e^2*x^2)^(1/2),x,2,-8/15*d*(c*d^2-c*e^2*x^2)^(3/2)/(c*e*(d+e*x)^(3/2))-2/5*(c*d^2-c*e^2*x^2)^(3/2)/(c*e*sqrt(d+e*x))],
[(c*d^2-c*e^2*x^2)^(1/2)/(d+e*x)^(1/2),x,1,-2/3*(c*d^2-c*e^2*x^2)^(3/2)/(c*e*(d+e*x)^(3/2))],
[(c*d^2-c*e^2*x^2)^(1/2)/(d+e*x)^(3/2),x,3,-2*arctanh(sqrt(c*d^2-c*e^2*x^2)/(sqrt(2)*sqrt(c)*sqrt(d)*sqrt(d+e*x)))*sqrt(2)*sqrt(c)*sqrt(d)/e+2*sqrt(c*d^2-c*e^2*x^2)/(e*sqrt(d+e*x))],
[(c*d^2-c*e^2*x^2)^(1/2)/(d+e*x)^(5/2),x,3,arctanh(sqrt(c*d^2-c*e^2*x^2)/(sqrt(2)*sqrt(c)*sqrt(d)*sqrt(d+e*x)))*sqrt(c)/(e*sqrt(2)*sqrt(d))-sqrt(c*d^2-c*e^2*x^2)/(e*(d+e*x)^(3/2))],
[(c*d^2-c*e^2*x^2)^(1/2)/(d+e*x)^(7/2),x,4,1/8*arctanh(sqrt(c*d^2-c*e^2*x^2)/(sqrt(2)*sqrt(c)*sqrt(d)*sqrt(d+e*x)))*sqrt(c)/(d^(3/2)*e*sqrt(2))-1/2*sqrt(c*d^2-c*e^2*x^2)/(e*(d+e*x)^(5/2))+1/8*sqrt(c*d^2-c*e^2*x^2)/(d*e*(d+e*x)^(3/2))],
[(d+e*x)^(5/2)*(c*d^2-c*e^2*x^2)^(3/2),x,5,-4096/15015*d^4*(c*d^2-c*e^2*x^2)^(5/2)/(c*e*(d+e*x)^(5/2))-1024/3003*d^3*(c*d^2-c*e^2*x^2)^(5/2)/(c*e*(d+e*x)^(3/2))-2/13*(d+e*x)^(3/2)*(c*d^2-c*e^2*x^2)^(5/2)/(c*e)-128/429*d^2*(c*d^2-c*e^2*x^2)^(5/2)/(c*e*sqrt(d+e*x))-32/143*d*(c*d^2-c*e^2*x^2)^(5/2)*sqrt(d+e*x)/(c*e)],
[(d+e*x)^(3/2)*(c*d^2-c*e^2*x^2)^(3/2),x,4,-256/1155*d^3*(c*d^2-c*e^2*x^2)^(5/2)/(c*e*(d+e*x)^(5/2))-64/231*d^2*(c*d^2-c*e^2*x^2)^(5/2)/(c*e*(d+e*x)^(3/2))-8/33*d*(c*d^2-c*e^2*x^2)^(5/2)/(c*e*sqrt(d+e*x))-2/11*(c*d^2-c*e^2*x^2)^(5/2)*sqrt(d+e*x)/(c*e)],
[(d+e*x)^(1/2)*(c*d^2-c*e^2*x^2)^(3/2),x,3,-64/315*d^2*(c*d^2-c*e^2*x^2)^(5/2)/(c*e*(d+e*x)^(5/2))-16/63*d*(c*d^2-c*e^2*x^2)^(5/2)/(c*e*(d+e*x)^(3/2))-2/9*(c*d^2-c*e^2*x^2)^(5/2)/(c*e*sqrt(d+e*x))],
[(c*d^2-c*e^2*x^2)^(3/2)/(d+e*x)^(1/2),x,2,-8/35*d*(c*d^2-c*e^2*x^2)^(5/2)/(c*e*(d+e*x)^(5/2))-2/7*(c*d^2-c*e^2*x^2)^(5/2)/(c*e*(d+e*x)^(3/2))],
[(c*d^2-c*e^2*x^2)^(3/2)/(d+e*x)^(3/2),x,1,-2/5*(c*d^2-c*e^2*x^2)^(5/2)/(c*e*(d+e*x)^(5/2))],
[(c*d^2-c*e^2*x^2)^(3/2)/(d+e*x)^(5/2),x,4,2/3*(c*d^2-c*e^2*x^2)^(3/2)/(e*(d+e*x)^(3/2))-4*c^(3/2)*d^(3/2)*arctanh(sqrt(c*d^2-c*e^2*x^2)/(sqrt(2)*sqrt(c)*sqrt(d)*sqrt(d+e*x)))*sqrt(2)/e+4*c*d*sqrt(c*d^2-c*e^2*x^2)/(e*sqrt(d+e*x))],
[(c*d^2-c*e^2*x^2)^(3/2)/(d+e*x)^(7/2),x,4,-(c*d^2-c*e^2*x^2)^(3/2)/(e*(d+e*x)^(5/2))+3*c^(3/2)*arctanh(sqrt(c*d^2-c*e^2*x^2)/(sqrt(2)*sqrt(c)*sqrt(d)*sqrt(d+e*x)))*sqrt(2)*sqrt(d)/e-3*c*sqrt(c*d^2-c*e^2*x^2)/(e*sqrt(d+e*x))],
[(c*d^2-c*e^2*x^2)^(3/2)/(d+e*x)^(9/2),x,4,-1/2*(c*d^2-c*e^2*x^2)^(3/2)/(e*(d+e*x)^(7/2))-3/4*c^(3/2)*arctanh(sqrt(c*d^2-c*e^2*x^2)/(sqrt(2)*sqrt(c)*sqrt(d)*sqrt(d+e*x)))/(e*sqrt(2)*sqrt(d))+3/4*c*sqrt(c*d^2-c*e^2*x^2)/(e*(d+e*x)^(3/2))],
[(c*d^2-c*e^2*x^2)^(3/2)/(d+e*x)^(11/2),x,5,-1/3*(c*d^2-c*e^2*x^2)^(3/2)/(e*(d+e*x)^(9/2))-1/16*c^(3/2)*arctanh(sqrt(c*d^2-c*e^2*x^2)/(sqrt(2)*sqrt(c)*sqrt(d)*sqrt(d+e*x)))/(d^(3/2)*e*sqrt(2))+1/4*c*sqrt(c*d^2-c*e^2*x^2)/(e*(d+e*x)^(5/2))-1/16*c*sqrt(c*d^2-c*e^2*x^2)/(d*e*(d+e*x)^(3/2))],
[(c*d^2-c*e^2*x^2)^(3/2)/(d+e*x)^(13/2),x,6,-1/4*(c*d^2-c*e^2*x^2)^(3/2)/(e*(d+e*x)^(11/2))-3/256*c^(3/2)*arctanh(sqrt(c*d^2-c*e^2*x^2)/(sqrt(2)*sqrt(c)*sqrt(d)*sqrt(d+e*x)))/(d^(5/2)*e*sqrt(2))+1/8*c*sqrt(c*d^2-c*e^2*x^2)/(e*(d+e*x)^(7/2))-1/64*c*sqrt(c*d^2-c*e^2*x^2)/(d*e*(d+e*x)^(5/2))-3/256*c*sqrt(c*d^2-c*e^2*x^2)/(d^2*e*(d+e*x)^(3/2))],

# p<0
[(d+e*x)^(7/2)/(c*d^2-c*e^2*x^2)^(1/2),x,4,-24/35*d*(d+e*x)^(3/2)*sqrt(c*d^2-c*e^2*x^2)/(c*e)-2/7*(d+e*x)^(5/2)*sqrt(c*d^2-c*e^2*x^2)/(c*e)-256/35*d^3*sqrt(c*d^2-c*e^2*x^2)/(c*e*sqrt(d+e*x))-64/35*d^2*sqrt(d+e*x)*sqrt(c*d^2-c*e^2*x^2)/(c*e)],
[(d+e*x)^(5/2)/(c*d^2-c*e^2*x^2)^(1/2),x,3,-2/5*(d+e*x)^(3/2)*sqrt(c*d^2-c*e^2*x^2)/(c*e)-64/15*d^2*sqrt(c*d^2-c*e^2*x^2)/(c*e*sqrt(d+e*x))-16/15*d*sqrt(d+e*x)*sqrt(c*d^2-c*e^2*x^2)/(c*e)],
[(d+e*x)^(3/2)/(c*d^2-c*e^2*x^2)^(1/2),x,2,-8/3*d*sqrt(c*d^2-c*e^2*x^2)/(c*e*sqrt(d+e*x))-2/3*sqrt(d+e*x)*sqrt(c*d^2-c*e^2*x^2)/(c*e)],
[(d+e*x)^(1/2)/(c*d^2-c*e^2*x^2)^(1/2),x,1,-2*sqrt(c*d^2-c*e^2*x^2)/(c*e*sqrt(d+e*x))],
[1/((d+e*x)^(1/2)*(c*d^2-c*e^2*x^2)^(1/2)),x,2,-arctanh(sqrt(c*d^2-c*e^2*x^2)/(sqrt(2)*sqrt(c)*sqrt(d)*sqrt(d+e*x)))*sqrt(2)/(e*sqrt(c)*sqrt(d))],
[1/((d+e*x)^(3/2)*(c*d^2-c*e^2*x^2)^(1/2)),x,3,-1/2*arctanh(sqrt(c*d^2-c*e^2*x^2)/(sqrt(2)*sqrt(c)*sqrt(d)*sqrt(d+e*x)))/(d^(3/2)*e*sqrt(2)*sqrt(c))-1/2*sqrt(c*d^2-c*e^2*x^2)/(c*d*e*(d+e*x)^(3/2))],
[1/((d+e*x)^(5/2)*(c*d^2-c*e^2*x^2)^(1/2)),x,4,-3/16*arctanh(sqrt(c*d^2-c*e^2*x^2)/(sqrt(2)*sqrt(c)*sqrt(d)*sqrt(d+e*x)))/(d^(5/2)*e*sqrt(2)*sqrt(c))-1/4*sqrt(c*d^2-c*e^2*x^2)/(c*d*e*(d+e*x)^(5/2))-3/16*sqrt(c*d^2-c*e^2*x^2)/(c*d^2*e*(d+e*x)^(3/2))],
[(d+e*x)^(9/2)/(c*d^2-c*e^2*x^2)^(3/2),x,4,-64/5*d^2*(d+e*x)^(3/2)/(c*e*sqrt(c*d^2-c*e^2*x^2))-8/5*d*(d+e*x)^(5/2)/(c*e*sqrt(c*d^2-c*e^2*x^2))-2/5*(d+e*x)^(7/2)/(c*e*sqrt(c*d^2-c*e^2*x^2))+256/5*d^3*sqrt(d+e*x)/(c*e*sqrt(c*d^2-c*e^2*x^2))],
[(d+e*x)^(7/2)/(c*d^2-c*e^2*x^2)^(3/2),x,3,-16/3*d*(d+e*x)^(3/2)/(c*e*sqrt(c*d^2-c*e^2*x^2))-2/3*(d+e*x)^(5/2)/(c*e*sqrt(c*d^2-c*e^2*x^2))+64/3*d^2*sqrt(d+e*x)/(c*e*sqrt(c*d^2-c*e^2*x^2))],
[(d+e*x)^(5/2)/(c*d^2-c*e^2*x^2)^(3/2),x,2,-2*(d+e*x)^(3/2)/(c*e*sqrt(c*d^2-c*e^2*x^2))+8*d*sqrt(d+e*x)/(c*e*sqrt(c*d^2-c*e^2*x^2))],
[(d+e*x)^(3/2)/(c*d^2-c*e^2*x^2)^(3/2),x,1,2*sqrt(d+e*x)/(c*e*sqrt(c*d^2-c*e^2*x^2))],
[(d+e*x)^(1/2)/(c*d^2-c*e^2*x^2)^(3/2),x,3,-arctanh(sqrt(c*d^2-c*e^2*x^2)/(sqrt(2)*sqrt(c)*sqrt(d)*sqrt(d+e*x)))/(c^(3/2)*d^(3/2)*e*sqrt(2))+sqrt(d+e*x)/(c*d*e*sqrt(c*d^2-c*e^2*x^2))],
[1/((d+e*x)^(1/2)*(c*d^2-c*e^2*x^2)^(3/2)),x,4,-3/4*arctanh(sqrt(c*d^2-c*e^2*x^2)/(sqrt(2)*sqrt(c)*sqrt(d)*sqrt(d+e*x)))/(c^(3/2)*d^(5/2)*e*sqrt(2))+(-1/2)/(c*d*e*sqrt(d+e*x)*sqrt(c*d^2-c*e^2*x^2))+3/4*sqrt(d+e*x)/(c*d^2*e*sqrt(c*d^2-c*e^2*x^2))],
[1/((d+e*x)^(3/2)*(c*d^2-c*e^2*x^2)^(3/2)),x,5,-15/32*arctanh(sqrt(c*d^2-c*e^2*x^2)/(sqrt(2)*sqrt(c)*sqrt(d)*sqrt(d+e*x)))/(c^(3/2)*d^(7/2)*e*sqrt(2))+(-1/4)/(c*d*e*(d+e*x)^(3/2)*sqrt(c*d^2-c*e^2*x^2))+(-5/16)/(c*d^2*e*sqrt(d+e*x)*sqrt(c*d^2-c*e^2*x^2))+15/32*sqrt(d+e*x)/(c*d^3*e*sqrt(c*d^2-c*e^2*x^2))],
[1/(sqrt(-1+x)*sqrt(1-x^2)),x,2,arctan(sqrt(1-x^2)/(sqrt(2)*sqrt(-1+x)))*sqrt(2)],

# Integrands of the form (d+e x)^(m/2) (c d^2 - c e^2 x^2)^(p/2) when d>0 and c>0

# p>0
[(2+e*x)^(5/2)*(12-3*e^2*x^2)^(1/2),x,3,-128*(2-e*x)^(3/2)/(e*sqrt(3))+2/3*(2-e*x)^(9/2)/(e*sqrt(3))+96/5*(2-e*x)^(5/2)*sqrt(3)/e-24/7*(2-e*x)^(7/2)*sqrt(3)/e],
[(2+e*x)^(3/2)*(12-3*e^2*x^2)^(1/2),x,3,-32*(2-e*x)^(3/2)/(e*sqrt(3))+16/5*(2-e*x)^(5/2)*sqrt(3)/e-2/7*(2-e*x)^(7/2)*sqrt(3)/e],
[(2+e*x)^(1/2)*(12-3*e^2*x^2)^(1/2),x,3,-8*(2-e*x)^(3/2)/(e*sqrt(3))+2/5*(2-e*x)^(5/2)*sqrt(3)/e],
[(12-3*e^2*x^2)^(1/2)/(2+e*x)^(1/2),x,2,-2*(2-e*x)^(3/2)/(e*sqrt(3))],
[(12-3*e^2*x^2)^(1/2)/(2+e*x)^(3/2),x,4,-4*arctanh(1/2*sqrt(2-e*x))*sqrt(3)/e+2*sqrt(3)*sqrt(2-e*x)/e],
[(12-3*e^2*x^2)^(1/2)/(2+e*x)^(5/2),x,4,1/2*arctanh(1/2*sqrt(2-e*x))*sqrt(3)/e-sqrt(3)*sqrt(2-e*x)/(e*(2+e*x))],
[(12-3*e^2*x^2)^(1/2)/(2+e*x)^(7/2),x,5,1/32*arctanh(1/2*sqrt(2-e*x))*sqrt(3)/e-1/2*sqrt(3)*sqrt(2-e*x)/(e*(2+e*x)^2)+1/16*sqrt(3)*sqrt(2-e*x)/(e*(2+e*x))],
[(2+e*x)^(5/2)*(12-3*e^2*x^2)^(3/2),x,3,-1536/5*(2-e*x)^(5/2)*sqrt(3)/e+1536/7*(2-e*x)^(7/2)*sqrt(3)/e-64*(2-e*x)^(9/2)*sqrt(3)/e+96/11*(2-e*x)^(11/2)*sqrt(3)/e-6/13*(2-e*x)^(13/2)*sqrt(3)/e],
[(2+e*x)^(3/2)*(12-3*e^2*x^2)^(3/2),x,3,-384/5*(2-e*x)^(5/2)*sqrt(3)/e+288/7*(2-e*x)^(7/2)*sqrt(3)/e-8*(2-e*x)^(9/2)*sqrt(3)/e+6/11*(2-e*x)^(11/2)*sqrt(3)/e],
[(2+e*x)^(1/2)*(12-3*e^2*x^2)^(3/2),x,3,-2*(2-e*x)^(9/2)/(e*sqrt(3))-96/5*(2-e*x)^(5/2)*sqrt(3)/e+48/7*(2-e*x)^(7/2)*sqrt(3)/e],
[(12-3*e^2*x^2)^(3/2)/(2+e*x)^(1/2),x,3,-24/5*(2-e*x)^(5/2)*sqrt(3)/e+6/7*(2-e*x)^(7/2)*sqrt(3)/e],
[(12-3*e^2*x^2)^(3/2)/(2+e*x)^(3/2),x,2,-6/5*(2-e*x)^(5/2)*sqrt(3)/e],
[(12-3*e^2*x^2)^(3/2)/(2+e*x)^(5/2),x,5,2*(2-e*x)^(3/2)*sqrt(3)/e-48*arctanh(1/2*sqrt(2-e*x))*sqrt(3)/e+24*sqrt(3)*sqrt(2-e*x)/e],
[(12-3*e^2*x^2)^(3/2)/(2+e*x)^(7/2),x,5,-3*(2-e*x)^(3/2)*sqrt(3)/(e*(2+e*x))+18*arctanh(1/2*sqrt(2-e*x))*sqrt(3)/e-9*sqrt(3)*sqrt(2-e*x)/e],
[(12-3*e^2*x^2)^(3/2)/(2+e*x)^(9/2),x,5,-3/2*(2-e*x)^(3/2)*sqrt(3)/(e*(2+e*x)^2)-9/8*arctanh(1/2*sqrt(2-e*x))*sqrt(3)/e+9/4*sqrt(3)*sqrt(2-e*x)/(e*(2+e*x))],
[(12-3*e^2*x^2)^(3/2)/(2+e*x)^(11/2),x,6,-(2-e*x)^(3/2)*sqrt(3)/(e*(2+e*x)^3)-3/64*arctanh(1/2*sqrt(2-e*x))*sqrt(3)/e+3/4*sqrt(3)*sqrt(2-e*x)/(e*(2+e*x)^2)-3/32*sqrt(3)*sqrt(2-e*x)/(e*(2+e*x))],
[(12-3*e^2*x^2)^(3/2)/(2+e*x)^(13/2),x,7,-3/4*(2-e*x)^(3/2)*sqrt(3)/(e*(2+e*x)^4)-9/2048*arctanh(1/2*sqrt(2-e*x))*sqrt(3)/e+3/8*sqrt(3)*sqrt(2-e*x)/(e*(2+e*x)^3)-3/128*sqrt(3)*sqrt(2-e*x)/(e*(2+e*x)^2)-9/1024*sqrt(3)*sqrt(2-e*x)/(e*(2+e*x))],

# p<0
[(2+e*x)^(7/2)/(12-3*e^2*x^2)^(1/2),x,3,32*(2-e*x)^(3/2)/(e*sqrt(3))+2/7*(2-e*x)^(7/2)/(e*sqrt(3))-8/5*(2-e*x)^(5/2)*sqrt(3)/e-128*sqrt(2-e*x)/(e*sqrt(3))],
[(2+e*x)^(5/2)/(12-3*e^2*x^2)^(1/2),x,3,16/3*(2-e*x)^(3/2)/(e*sqrt(3))-2/5*(2-e*x)^(5/2)/(e*sqrt(3))-32*sqrt(2-e*x)/(e*sqrt(3))],
[(2+e*x)^(3/2)/(12-3*e^2*x^2)^(1/2),x,3,2/3*(2-e*x)^(3/2)/(e*sqrt(3))-8*sqrt(2-e*x)/(e*sqrt(3))],
[(2+e*x)^(1/2)/(12-3*e^2*x^2)^(1/2),x,2,-2*sqrt(2-e*x)/(e*sqrt(3))],
[1/((2+e*x)^(1/2)*(12-3*e^2*x^2)^(1/2)),x,3,-arctanh(1/2*sqrt(2-e*x))/(e*sqrt(3))],
[1/((2+e*x)^(3/2)*(12-3*e^2*x^2)^(1/2)),x,4,-1/8*arctanh(1/2*sqrt(2-e*x))/(e*sqrt(3))-1/4*sqrt(2-e*x)/(e*(2+e*x)*sqrt(3))],
[1/((2+e*x)^(5/2)*(12-3*e^2*x^2)^(1/2)),x,5,-1/128*arctanh(1/2*sqrt(2-e*x))*sqrt(3)/e-1/8*sqrt(2-e*x)/(e*(2+e*x)^2*sqrt(3))-1/64*sqrt(3)*sqrt(2-e*x)/(e*(2+e*x))],
[(2+e*x)^(11/2)/(12-3*e^2*x^2)^(3/2),x,3,-64/3*(2-e*x)^(3/2)/(e*sqrt(3))+32/15*(2-e*x)^(5/2)/(e*sqrt(3))-2/21*(2-e*x)^(7/2)/(e*sqrt(3))+512/3/(e*sqrt(3)*sqrt(2-e*x))+512/3*sqrt(2-e*x)/(e*sqrt(3))],
[(2+e*x)^(9/2)/(12-3*e^2*x^2)^(3/2),x,3,-8/3*(2-e*x)^(3/2)/(e*sqrt(3))+2/15*(2-e*x)^(5/2)/(e*sqrt(3))+128/3/(e*sqrt(3)*sqrt(2-e*x))+32*sqrt(2-e*x)/(e*sqrt(3))],
[(2+e*x)^(7/2)/(12-3*e^2*x^2)^(3/2),x,3,-2/9*(2-e*x)^(3/2)/(e*sqrt(3))+32/3/(e*sqrt(3)*sqrt(2-e*x))+16/3*sqrt(2-e*x)/(e*sqrt(3))],
[(2+e*x)^(5/2)/(12-3*e^2*x^2)^(3/2),x,3,8/3/(e*sqrt(3)*sqrt(2-e*x))+2/3*sqrt(2-e*x)/(e*sqrt(3))],
[(2+e*x)^(3/2)/(12-3*e^2*x^2)^(3/2),x,2,2/3/(e*sqrt(3)*sqrt(2-e*x))],
[(2+e*x)^(1/2)/(12-3*e^2*x^2)^(3/2),x,4,-1/12*arctanh(1/2*sqrt(2-e*x))/(e*sqrt(3))+1/6/(e*sqrt(3)*sqrt(2-e*x))],
[1/((2+e*x)^(1/2)*(12-3*e^2*x^2)^(3/2)),x,5,-1/32*arctanh(1/2*sqrt(2-e*x))/(e*sqrt(3))+1/16/(e*sqrt(3)*sqrt(2-e*x))+(-1/12)/(e*(2+e*x)*sqrt(3)*sqrt(2-e*x))],
[1/((2+e*x)^(3/2)*(12-3*e^2*x^2)^(3/2)),x,6,-5/512*arctanh(1/2*sqrt(2-e*x))/(e*sqrt(3))+5/256/(e*sqrt(3)*sqrt(2-e*x))+(-1/24)/(e*(2+e*x)^2*sqrt(3)*sqrt(2-e*x))+(-5/192)/(e*(2+e*x)*sqrt(3)*sqrt(2-e*x))],

#  The following pairs of integrands are equal: 
[1/((1+x)*sqrt(1-x)),x,2,-arctanh(sqrt(1-x)/sqrt(2))*sqrt(2)],
[1/(sqrt(1+x)*sqrt(1-x^2)),x,3,-arctanh(sqrt(1-x)/sqrt(2))*sqrt(2)],
[1/((1+a*x)*sqrt(1-a*x)),x,2,-arctanh(sqrt(1-a*x)/sqrt(2))*sqrt(2)/a],
[1/(sqrt(1+a*x)*sqrt(1-a^2*x^2)),x,3,-arctanh(sqrt(1-a*x)/sqrt(2))*sqrt(2)/a],

# Integrands of the form (d+e x)^(m/2) (c d^2 - c e^2 x^2)^(p/4) when d>0 and c>0

# p>0
[(2+e*x)^(1/2)*(12-3*e^2*x^2)^(1/4),x,14,3/2*3^(1/4)*(2-e*x)^(1/4)*(2+e*x)^(3/4)/e-1/2*3^(1/4)*(2-e*x)^(5/4)*(2+e*x)^(3/4)/e+3*3^(1/4)*arctan(1-(2-e*x)^(1/4)*sqrt(2)/(2+e*x)^(1/4))/(e*sqrt(2))-3*3^(1/4)*arctan(1+(2-e*x)^(1/4)*sqrt(2)/(2+e*x)^(1/4))/(e*sqrt(2))+3/2*3^(1/4)*log(sqrt(3)-(2-e*x)^(1/4)*sqrt(6)/(2+e*x)^(1/4)+sqrt(3)*sqrt(2-e*x)/sqrt(2+e*x))/(e*sqrt(2))-3/2*3^(1/4)*log(sqrt(3)+(2-e*x)^(1/4)*sqrt(6)/(2+e*x)^(1/4)+sqrt(3)*sqrt(2-e*x)/sqrt(2+e*x))/(e*sqrt(2))],
[(12-3*e^2*x^2)^(1/4)/(2+e*x)^(1/2),x,13,3^(1/4)*(2-e*x)^(1/4)*(2+e*x)^(3/4)/e+3^(1/4)*log(sqrt(3)-(2-e*x)^(1/4)*sqrt(6)/(2+e*x)^(1/4)+sqrt(3)*sqrt(2-e*x)/sqrt(2+e*x))/(e*sqrt(2))-3^(1/4)*log(sqrt(3)+(2-e*x)^(1/4)*sqrt(6)/(2+e*x)^(1/4)+sqrt(3)*sqrt(2-e*x)/sqrt(2+e*x))/(e*sqrt(2))+3^(1/4)*arctan(1-(2-e*x)^(1/4)*sqrt(2)/(2+e*x)^(1/4))*sqrt(2)/e-3^(1/4)*arctan(1+(2-e*x)^(1/4)*sqrt(2)/(2+e*x)^(1/4))*sqrt(2)/e],
[(12-3*e^2*x^2)^(1/4)/(2+e*x)^(3/2),x,13,-4*3^(1/4)*(2-e*x)^(1/4)/(e*(2+e*x)^(1/4))-3^(1/4)*log(sqrt(3)-(2-e*x)^(1/4)*sqrt(6)/(2+e*x)^(1/4)+sqrt(3)*sqrt(2-e*x)/sqrt(2+e*x))/(e*sqrt(2))+3^(1/4)*log(sqrt(3)+(2-e*x)^(1/4)*sqrt(6)/(2+e*x)^(1/4)+sqrt(3)*sqrt(2-e*x)/sqrt(2+e*x))/(e*sqrt(2))-3^(1/4)*arctan(1-(2-e*x)^(1/4)*sqrt(2)/(2+e*x)^(1/4))*sqrt(2)/e+3^(1/4)*arctan(1+(2-e*x)^(1/4)*sqrt(2)/(2+e*x)^(1/4))*sqrt(2)/e],
[(12-3*e^2*x^2)^(1/4)/(2+e*x)^(5/2),x,1,-1/5*3^(1/4)*(4-e^2*x^2)^(5/4)/(e*(2+e*x)^(5/2))],
[(12-3*e^2*x^2)^(1/4)/(2+e*x)^(7/2),x,2,-1/3*(4-e^2*x^2)^(5/4)/(3^(3/4)*e*(2+e*x)^(7/2))-1/15*(4-e^2*x^2)^(5/4)/(3^(3/4)*e*(2+e*x)^(5/2))],
[(12-3*e^2*x^2)^(1/4)/(2+e*x)^(9/2),x,3,-1/13*3^(1/4)*(4-e^2*x^2)^(5/4)/(e*(2+e*x)^(9/2))-2/39*(4-e^2*x^2)^(5/4)/(3^(3/4)*e*(2+e*x)^(7/2))-2/195*(4-e^2*x^2)^(5/4)/(3^(3/4)*e*(2+e*x)^(5/2))],
[(12-3*e^2*x^2)^(1/4)/(2+e*x)^(11/2),x,4,-1/17*3^(1/4)*(4-e^2*x^2)^(5/4)/(e*(2+e*x)^(11/2))-3/221*3^(1/4)*(4-e^2*x^2)^(5/4)/(e*(2+e*x)^(9/2))-2/221*(4-e^2*x^2)^(5/4)/(3^(3/4)*e*(2+e*x)^(7/2))-2/1105*(4-e^2*x^2)^(5/4)/(3^(3/4)*e*(2+e*x)^(5/2))],

# p<0
[(2+e*x)^(5/2)/(12-3*e^2*x^2)^(1/4),x,15,-5/2*3^(3/4)*(2-e*x)^(3/4)*(2+e*x)^(1/4)/e-1/2*3^(3/4)*(2-e*x)^(3/4)*(2+e*x)^(5/4)/e-1/3*(2-e*x)^(3/4)*(2+e*x)^(9/4)/(3^(1/4)*e)+5*3^(3/4)*arctan(1-(2-e*x)^(1/4)*sqrt(2)/(2+e*x)^(1/4))/(e*sqrt(2))-5*3^(3/4)*arctan(1+(2-e*x)^(1/4)*sqrt(2)/(2+e*x)^(1/4))/(e*sqrt(2))-5/2*3^(3/4)*log(sqrt(3)-(2-e*x)^(1/4)*sqrt(6)/(2+e*x)^(1/4)+sqrt(3)*sqrt(2-e*x)/sqrt(2+e*x))/(e*sqrt(2))+5/2*3^(3/4)*log(sqrt(3)+(2-e*x)^(1/4)*sqrt(6)/(2+e*x)^(1/4)+sqrt(3)*sqrt(2-e*x)/sqrt(2+e*x))/(e*sqrt(2))],
[(2+e*x)^(3/2)/(12-3*e^2*x^2)^(1/4),x,14,-5/2*(2-e*x)^(3/4)*(2+e*x)^(1/4)/(3^(1/4)*e)-1/2*(2-e*x)^(3/4)*(2+e*x)^(5/4)/(3^(1/4)*e)+5*arctan(1-(2-e*x)^(1/4)*sqrt(2)/(2+e*x)^(1/4))/(3^(1/4)*e*sqrt(2))-5*arctan(1+(2-e*x)^(1/4)*sqrt(2)/(2+e*x)^(1/4))/(3^(1/4)*e*sqrt(2))-5/2*log(sqrt(3)-(2-e*x)^(1/4)*sqrt(6)/(2+e*x)^(1/4)+sqrt(3)*sqrt(2-e*x)/sqrt(2+e*x))/(3^(1/4)*e*sqrt(2))+5/2*log(sqrt(3)+(2-e*x)^(1/4)*sqrt(6)/(2+e*x)^(1/4)+sqrt(3)*sqrt(2-e*x)/sqrt(2+e*x))/(3^(1/4)*e*sqrt(2))],
[(2+e*x)^(1/2)/(12-3*e^2*x^2)^(1/4),x,13,-(2-e*x)^(3/4)*(2+e*x)^(1/4)/(3^(1/4)*e)-log(sqrt(3)-(2-e*x)^(1/4)*sqrt(6)/(2+e*x)^(1/4)+sqrt(3)*sqrt(2-e*x)/sqrt(2+e*x))/(3^(1/4)*e*sqrt(2))+log(sqrt(3)+(2-e*x)^(1/4)*sqrt(6)/(2+e*x)^(1/4)+sqrt(3)*sqrt(2-e*x)/sqrt(2+e*x))/(3^(1/4)*e*sqrt(2))+arctan(1-(2-e*x)^(1/4)*sqrt(2)/(2+e*x)^(1/4))*sqrt(2)/(3^(1/4)*e)-arctan(1+(2-e*x)^(1/4)*sqrt(2)/(2+e*x)^(1/4))*sqrt(2)/(3^(1/4)*e)],
[1/((2+e*x)^(1/2)*(12-3*e^2*x^2)^(1/4)),x,12,-log(sqrt(3)-(2-e*x)^(1/4)*sqrt(6)/(2+e*x)^(1/4)+sqrt(3)*sqrt(2-e*x)/sqrt(2+e*x))/(3^(1/4)*e*sqrt(2))+log(sqrt(3)+(2-e*x)^(1/4)*sqrt(6)/(2+e*x)^(1/4)+sqrt(3)*sqrt(2-e*x)/sqrt(2+e*x))/(3^(1/4)*e*sqrt(2))+arctan(1-(2-e*x)^(1/4)*sqrt(2)/(2+e*x)^(1/4))*sqrt(2)/(3^(1/4)*e)-arctan(1+(2-e*x)^(1/4)*sqrt(2)/(2+e*x)^(1/4))*sqrt(2)/(3^(1/4)*e)],
[1/((2+e*x)^(3/2)*(12-3*e^2*x^2)^(1/4)),x,1,-1/3*(4-e^2*x^2)^(3/4)/(3^(1/4)*e*(2+e*x)^(3/2))],
[1/((2+e*x)^(5/2)*(12-3*e^2*x^2)^(1/4)),x,2,-1/7*(4-e^2*x^2)^(3/4)/(3^(1/4)*e*(2+e*x)^(5/2))-1/21*(4-e^2*x^2)^(3/4)/(3^(1/4)*e*(2+e*x)^(3/2))],
[1/((2+e*x)^(7/2)*(12-3*e^2*x^2)^(1/4)),x,3,-1/11*(4-e^2*x^2)^(3/4)/(3^(1/4)*e*(2+e*x)^(7/2))-2/77*(4-e^2*x^2)^(3/4)/(3^(1/4)*e*(2+e*x)^(5/2))-2/231*(4-e^2*x^2)^(3/4)/(3^(1/4)*e*(2+e*x)^(3/2))],
[1/((2+e*x)^(9/2)*(12-3*e^2*x^2)^(1/4)),x,4,-1/15*(4-e^2*x^2)^(3/4)/(3^(1/4)*e*(2+e*x)^(9/2))-1/55*(4-e^2*x^2)^(3/4)/(3^(1/4)*e*(2+e*x)^(7/2))-2/385*(4-e^2*x^2)^(3/4)/(3^(1/4)*e*(2+e*x)^(5/2))-2/1155*(4-e^2*x^2)^(3/4)/(3^(1/4)*e*(2+e*x)^(3/2))],

# Integrands of the form (d+e x)^m (d^2 - e^2 x^2)^p when m symbolic
[(a+b*x)^m*(a^2-b^2*x^2)^3,x,3,8*a^3*(a+b*x)^(4+m)/(b*(4+m))-12*a^2*(a+b*x)^(5+m)/(b*(5+m))+6*a*(a+b*x)^(6+m)/(b*(6+m))-(a+b*x)^(7+m)/(b*(7+m))],
[(a+b*x)^m*(a^2-b^2*x^2)^2,x,3,4*a^2*(a+b*x)^(3+m)/(b*(3+m))-4*a*(a+b*x)^(4+m)/(b*(4+m))+(a+b*x)^(5+m)/(b*(5+m))],
[(a+b*x)^m*(a^2-b^2*x^2),x,3,2*a*(a+b*x)^(2+m)/(b*(2+m))-(a+b*x)^(3+m)/(b*(3+m))],
[(a+b*x)^m/(a^2-b^2*x^2),x,2,1/2*(a+b*x)^m*hypergeom([1,m],[1+m],1/2*(a+b*x)/a)/(a*b*m)],
[(a+b*x)^m/(a^2-b^2*x^2)^2,x,2,-1/4*(a+b*x)^(-1+m)*hypergeom([2,-1+m],[m],1/2*(a+b*x)/a)/(a^2*b*(1-m))],
[(a+b*x)^m/(a^2-b^2*x^2)^3,x,2,-1/8*(a+b*x)^(-2+m)*hypergeom([3,-2+m],[-1+m],1/2*(a+b*x)/a)/(a^3*b*(2-m))],
[(d+e*x)^m*(d^2-e^2*x^2)^(7/2),x,3,(d+e*x)^m*(d^2-e^2*x^2)^(9/2)*hypergeom([1,9+m],[11/2+m],1/2*(d+e*x)/d)/(d*e*(9+2*m)),-1/9*2^(9/2+m)*(d+e*x)^m*(1+e*x/d)^(-9/2-m)*(d^2-e^2*x^2)^(9/2)*hypergeom([9/2,-7/2-m],[11/2],1/2*(d-e*x)/d)/(d*e)],
[(d+e*x)^m*(d^2-e^2*x^2)^(5/2),x,3,-1/7*2^(7/2+m)*(d+e*x)^m*(1+e*x/d)^(-7/2-m)*(d^2-e^2*x^2)^(7/2)*hypergeom([7/2,-5/2-m],[9/2],1/2*(d-e*x)/d)/(d*e)],
[(d+e*x)^m*(d^2-e^2*x^2)^(3/2),x,3,(d+e*x)^m*(d^2-e^2*x^2)^(5/2)*hypergeom([1,5+m],[7/2+m],1/2*(d+e*x)/d)/(d*e*(5+2*m)),-1/5*2^(5/2+m)*(d+e*x)^m*(1+e*x/d)^(-5/2-m)*(d^2-e^2*x^2)^(5/2)*hypergeom([5/2,-3/2-m],[7/2],1/2*(d-e*x)/d)/(d*e)],
[(d+e*x)^m*(d^2-e^2*x^2)^(1/2),x,3,(d-e*x)*(d+e*x)^(1+m)*hypergeom([1,3+m],[5/2+m],1/2*(d+e*x)/d)*sqrt(d^2-e^2*x^2)/(d*e*(3+2*m)),-1/3*2^(3/2+m)*(d+e*x)^m*(1+e*x/d)^(-3/2-m)*(d^2-e^2*x^2)^(3/2)*hypergeom([3/2,-1/2-m],[5/2],1/2*(d-e*x)/d)/(d*e)],
[(d+e*x)^m/(d^2-e^2*x^2)^(1/2),x,3,(d-e*x)*(d+e*x)^(1+m)*hypergeom([1,1+m],[3/2+m],1/2*(d+e*x)/d)/(d*e*(1+2*m)*sqrt(d^2-e^2*x^2)),-2^(1/2+m)*(d+e*x)^m*(1+e*x/d)^(-1/2-m)*hypergeom([1/2,1/2-m],[3/2],1/2*(d-e*x)/d)*sqrt(d^2-e^2*x^2)/(d*e)],
[(d+e*x)^m/(d^2-e^2*x^2)^(3/2),x,3,2^(-1/2+m)*(d+e*x)^m*(1+e*x/d)^(1/2-m)*hypergeom([-1/2,3/2-m],[1/2],1/2*(d-e*x)/d)/(d*e*sqrt(d^2-e^2*x^2))],
[(d+e*x)^m/(d^2-e^2*x^2)^(5/2),x,3,-(d+e*x)^m*hypergeom([1,-3+m],[-1/2+m],1/2*(d+e*x)/d)/(d*e*(3-2*m)*(d^2-e^2*x^2)^(3/2)),1/3*2^(-3/2+m)*(d+e*x)^m*(1+e*x/d)^(3/2-m)*hypergeom([-3/2,5/2-m],[-1/2],1/2*(d-e*x)/d)/(d*e*(d^2-e^2*x^2)^(3/2))],
[(d+e*x)^m/(d^2-e^2*x^2)^(7/2),x,3,-(d-e*x)*(d+e*x)^(1+m)*hypergeom([1,-5+m],[-3/2+m],1/2*(d+e*x)/d)/(d*e*(5-2*m)*(d^2-e^2*x^2)^(7/2)),1/5*2^(-5/2+m)*(d+e*x)^m*(1+e*x/d)^(5/2-m)*hypergeom([-5/2,7/2-m],[-3/2],1/2*(d-e*x)/d)/(d*e*(d^2-e^2*x^2)^(5/2))],

# Integrands of the form (d+e x)^m (d^2 - e^2 x^2)^p when p symbolic
[(a+b*x)^m*(a^2-b^2*x^2)^p,x,3,1/2*(a+b*x)^m*(a^2-b^2*x^2)^(1+p)*hypergeom([1,2+m+2*p],[2+m+p],1/2*(a+b*x)/a)/(a*b*(1+m+p)),-2^(m+p)*(a+b*x)^m*(1+b*x/a)^(-1-m-p)*(a^2-b^2*x^2)^(1+p)*hypergeom([-m-p,1+p],[2+p],1/2*(a-b*x)/a)/(a*b*(1+p))],
[(d+e*x)^3*(1-e^2*x^2/d^2)^p,x,2,-2^(3+p)*d^4*((d-e*x)/d)^(1+p)*hypergeom([-3-p,1+p],[2+p],1/2*(d-e*x)/d)/(e*(1+p))],
[(d+e*x)^2*(1-e^2*x^2/d^2)^p,x,2,-2^(2+p)*d^3*((d-e*x)/d)^(1+p)*hypergeom([-2-p,1+p],[2+p],1/2*(d-e*x)/d)/(e*(1+p))],
[(d+e*x)*(1-e^2*x^2/d^2)^p,x,2,-2^(1+p)*d^2*((d-e*x)/d)^(1+p)*hypergeom([-1-p,1+p],[2+p],1/2*(d-e*x)/d)/(e*(1+p)),-1/2*d^2*(1-e^2*x^2/d^2)^(1+p)/(e*(1+p))+d*x*hypergeom([1/2,-p],[3/2],e^2*x^2/d^2)],
[(1-e^2*x^2/d^2)^p,x,1,x*hypergeom([1/2,-p],[3/2],e^2*x^2/d^2)],
[(1-e^2*x^2/d^2)^p/(d+e*x),x,2,2^p*((d+e*x)/d)^p*hypergeom([-p,p],[1+p],1/2*(d+e*x)/d)/(e*p),-2^(-1+p)*((d-e*x)/d)^(1+p)*hypergeom([1-p,1+p],[2+p],1/2*(d-e*x)/d)/(e*(1+p))],
[(1-e^2*x^2/d^2)^p/(d+e*x)^2,x,2,-2^(-2+p)*((d-e*x)/d)^(1+p)*hypergeom([2-p,1+p],[2+p],1/2*(d-e*x)/d)/(d*e*(1+p))],
[(1-e^2*x^2/d^2)^p/(d+e*x)^3,x,2,-2^(-3+p)*((d-e*x)/d)^(1+p)*hypergeom([3-p,1+p],[2+p],1/2*(d-e*x)/d)/(d^2*e*(1+p))],
[(a+b*x)^3*(a^2-b^2*x^2)^p,x,2,1/2*(a+b*x)^3*(a^2-b^2*x^2)^(1+p)*hypergeom([1,5+2*p],[5+p],1/2*(a+b*x)/a)/(a*b*(4+p)),-2^(3+p)*a^2*(1+b*x/a)^(-1-p)*(a^2-b^2*x^2)^(1+p)*hypergeom([-3-p,1+p],[2+p],1/2*(a-b*x)/a)/(b*(1+p))],
[(a+b*x)^2*(a^2-b^2*x^2)^p,x,2,1/2*(a+b*x)^2*(a^2-b^2*x^2)^(1+p)*hypergeom([1,2*(2+p)],[4+p],1/2*(a+b*x)/a)/(a*b*(3+p)),-2^(2+p)*a*(1+b*x/a)^(-1-p)*(a^2-b^2*x^2)^(1+p)*hypergeom([-2-p,1+p],[2+p],1/2*(a-b*x)/a)/(b*(1+p))],
[(a+b*x)*(a^2-b^2*x^2)^p,x,3,-1/2*(a^2-b^2*x^2)^(1+p)/(b*(1+p))+a*x*(a^2-b^2*x^2)^p*hypergeom([1/2,-p],[3/2],b^2*x^2/a^2)/(1-b^2*x^2/a^2)^p],
[(a^2-b^2*x^2)^p/(a+b*x),x,2,1/2*(a-b*x)*(a^2-b^2*x^2)^p*hypergeom([1,1+2*p],[1+p],1/2*(a+b*x)/a)/(a*b*p),-2^(-1+p)*(1+b*x/a)^(-1-p)*(a^2-b^2*x^2)^(1+p)*hypergeom([1-p,1+p],[2+p],1/2*(a-b*x)/a)/(a^2*b*(1+p))],
[(a^2-b^2*x^2)^p/(a+b*x)^2,x,2,-1/2*(a^2-b^2*x^2)^(1+p)*hypergeom([1,2*p],[p],1/2*(a+b*x)/a)/(a*b*(1-p)*(a+b*x)^2),-2^(-2+p)*(1+b*x/a)^(-1-p)*(a^2-b^2*x^2)^(1+p)*hypergeom([2-p,1+p],[2+p],1/2*(a-b*x)/a)/(a^3*b*(1+p))],
[(a^2-b^2*x^2)^p/(a+b*x)^3,x,2,-1/2*(a^2-b^2*x^2)^(1+p)*hypergeom([1,-1+2*p],[-1+p],1/2*(a+b*x)/a)/(a*b*(2-p)*(a+b*x)^3),-2^(-3+p)*(1+b*x/a)^(-1-p)*(a^2-b^2*x^2)^(1+p)*hypergeom([3-p,1+p],[2+p],1/2*(a-b*x)/a)/(a^4*b*(1+p))],
[(a+b*x)^(3/2)*(a^2-b^2*x^2)^p,x,3,-2^(3/2+p)*(1+b*x/a)^(-3/2-p)*(a^2-b^2*x^2)^(1+p)*hypergeom([-3/2-p,1+p],[2+p],1/2*(a-b*x)/a)*sqrt(a+b*x)/(b*(1+p))],
[(a+b*x)^(1/2)*(a^2-b^2*x^2)^p,x,3,-2^(1/2+p)*(1+b*x/a)^(-3/2-p)*(a^2-b^2*x^2)^(1+p)*hypergeom([-1/2-p,1+p],[2+p],1/2*(a-b*x)/a)*sqrt(a+b*x)/(a*b*(1+p))],
[(a^2-b^2*x^2)^p/(a+b*x)^(1/2),x,3,-2^(-1/2+p)*(1+b*x/a)^(-1/2-p)*(a^2-b^2*x^2)^(1+p)*hypergeom([1/2-p,1+p],[2+p],1/2*(a-b*x)/a)/(a*b*(1+p)*sqrt(a+b*x))],
[(a^2-b^2*x^2)^p/(a+b*x)^(3/2),x,3,-2^(-3/2+p)*(1+b*x/a)^(-1/2-p)*(a^2-b^2*x^2)^(1+p)*hypergeom([3/2-p,1+p],[2+p],1/2*(a-b*x)/a)/(a^2*b*(1+p)*sqrt(a+b*x))],
[-a*(a^2-b^2*x^2)^p+(a+b*x)*(a^2-b^2*x^2)^p,x,6,-1/2*(a^2-b^2*x^2)^(1+p)/(b*(1+p))],

# Integrands of the form (d+e x)^m (a+b x+c x^2)^p when b^2-4 a c=0 and 2 c d-b e=0

# Integrands of the form (d+e x)^m (c d^2+2 c d e x+c e^2 x^2)^p

# p>0
[(d+e*x)^2*(c*d^2+2*c*d*e*x+c*e^2*x^2),x,3,1/5*c*(d+e*x)^5/e],
[(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2),x,3,1/4*c*(d+e*x)^4/e],
[c*d^2+2*c*d*e*x+c*e^2*x^2,x,1,c*d^2*x+c*d*e*x^2+1/3*c*e^2*x^3],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)/(d+e*x),x,2,c*d*x+1/2*c*e*x^2],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)/(d+e*x)^2,x,3,c*x],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)/(d+e*x)^3,x,3,c*log(d+e*x)/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)/(d+e*x)^4,x,3,-c/(e*(d+e*x))],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)/(d+e*x)^5,x,3,-1/2*c/(e*(d+e*x)^2)],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)/(d+e*x)^6,x,3,-1/3*c/(e*(d+e*x)^3)],
[(d+e*x)^2*(c*d^2+2*c*d*e*x+c*e^2*x^2)^2,x,3,1/7*c^2*(d+e*x)^7/e],
[(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^2,x,3,1/6*c^2*(d+e*x)^6/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^2,x,3,1/5*c^2*(d+e*x)^5/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^2/(d+e*x),x,3,1/4*c^2*(d+e*x)^4/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^2/(d+e*x)^2,x,3,1/3*c^2*(d+e*x)^3/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^2/(d+e*x)^3,x,2,1/2*c^2*(d+e*x)^2/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^2/(d+e*x)^4,x,2,c^2*x],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^2/(d+e*x)^5,x,3,c^2*log(d+e*x)/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^2/(d+e*x)^6,x,3,-c^2/(e*(d+e*x))],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^2/(d+e*x)^7,x,3,-1/2*c^2/(e*(d+e*x)^2)],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^2/(d+e*x)^8,x,3,-1/3*c^2/(e*(d+e*x)^3)],

# p<0
[(d+e*x)^5/(c*d^2+2*c*d*e*x+c*e^2*x^2),x,3,1/4*(d+e*x)^4/(c*e)],
[(d+e*x)^4/(c*d^2+2*c*d*e*x+c*e^2*x^2),x,3,1/3*(d+e*x)^3/(c*e)],
[(d+e*x)^3/(c*d^2+2*c*d*e*x+c*e^2*x^2),x,2,1/2*(d+e*x)^2/(c*e)],
[(d+e*x)^2/(c*d^2+2*c*d*e*x+c*e^2*x^2),x,2,x/c],
[(d+e*x)/(c*d^2+2*c*d*e*x+c*e^2*x^2),x,3,log(d+e*x)/(c*e)],
[1/(c*d^2+2*c*d*e*x+c*e^2*x^2),x,3,(-1)/(c*e*(d+e*x))],
[1/((d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)),x,3,(-1/2)/(c*e*(d+e*x)^2)],
[1/((d+e*x)^2*(c*d^2+2*c*d*e*x+c*e^2*x^2)),x,3,(-1/3)/(c*e*(d+e*x)^3)],
[1/((d+e*x)^3*(c*d^2+2*c*d*e*x+c*e^2*x^2)),x,3,(-1/4)/(c*e*(d+e*x)^4)],
[(d+e*x)^7/(c*d^2+2*c*d*e*x+c*e^2*x^2)^2,x,3,1/4*(d+e*x)^4/(c^2*e)],
[(d+e*x)^6/(c*d^2+2*c*d*e*x+c*e^2*x^2)^2,x,3,1/3*(d+e*x)^3/(c^2*e)],
[(d+e*x)^5/(c*d^2+2*c*d*e*x+c*e^2*x^2)^2,x,2,1/2*(d+e*x)^2/(c^2*e)],
[(d+e*x)^4/(c*d^2+2*c*d*e*x+c*e^2*x^2)^2,x,2,x/c^2],
[(d+e*x)^3/(c*d^2+2*c*d*e*x+c*e^2*x^2)^2,x,3,log(d+e*x)/(c^2*e)],
[(d+e*x)^2/(c*d^2+2*c*d*e*x+c*e^2*x^2)^2,x,3,(-1)/(c^2*e*(d+e*x))],
[(d+e*x)/(c*d^2+2*c*d*e*x+c*e^2*x^2)^2,x,3,(-1/2)/(c^2*e*(d+e*x)^2)],
[1/(c*d^2+2*c*d*e*x+c*e^2*x^2)^2,x,3,(-1/3)/(c^2*e*(d+e*x)^3)],
[1/((d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^2),x,3,(-1/4)/(c^2*e*(d+e*x)^4)],
[1/((d+e*x)^2*(c*d^2+2*c*d*e*x+c*e^2*x^2)^2),x,3,(-1/5)/(c^2*e*(d+e*x)^5)],
[(d+e*x)^9/(c*d^2+2*c*d*e*x+c*e^2*x^2)^3,x,3,1/4*(d+e*x)^4/(c^3*e)],
[(d+e*x)^8/(c*d^2+2*c*d*e*x+c*e^2*x^2)^3,x,3,1/3*(d+e*x)^3/(c^3*e)],
[(d+e*x)^7/(c*d^2+2*c*d*e*x+c*e^2*x^2)^3,x,2,1/2*(d+e*x)^2/(c^3*e)],
[(d+e*x)^6/(c*d^2+2*c*d*e*x+c*e^2*x^2)^3,x,2,x/c^3],
[(d+e*x)^5/(c*d^2+2*c*d*e*x+c*e^2*x^2)^3,x,3,log(d+e*x)/(c^3*e)],
[(d+e*x)^4/(c*d^2+2*c*d*e*x+c*e^2*x^2)^3,x,3,(-1)/(c^3*e*(d+e*x))],
[(d+e*x)^3/(c*d^2+2*c*d*e*x+c*e^2*x^2)^3,x,3,(-1/2)/(c^3*e*(d+e*x)^2)],
[(d+e*x)^2/(c*d^2+2*c*d*e*x+c*e^2*x^2)^3,x,3,(-1/3)/(c^3*e*(d+e*x)^3)],
[(d+e*x)/(c*d^2+2*c*d*e*x+c*e^2*x^2)^3,x,3,(-1/4)/(c^3*e*(d+e*x)^4)],
[1/(c*d^2+2*c*d*e*x+c*e^2*x^2)^3,x,3,(-1/5)/(c^3*e*(d+e*x)^5)],
[1/((d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^3),x,3,(-1/6)/(c^3*e*(d+e*x)^6)],
[1/((d+e*x)^2*(c*d^2+2*c*d*e*x+c*e^2*x^2)^3),x,3,(-1/7)/(c^3*e*(d+e*x)^7)],

# Integrands of the form (d+e x)^m (c d^2+2 c d e x+c e^2 x^2)^(p/2)

# p>0
[(d+e*x)^3*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2),x,2,1/5*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2)/(c^2*e)],
[(d+e*x)^2*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2),x,2,1/4*(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)/(c*e)],
[(d+e*x)*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2),x,1,1/3*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)/(c*e)],
[sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2),x,1,1/2*(d+e*x)*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/e],
[sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/(d+e*x),x,2,sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/e],
[sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/(d+e*x)^2,x,3,c*(d+e*x)*log(d+e*x)/(e*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/(d+e*x)^3,x,2,-c/(e*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/(d+e*x)^4,x,2,-1/2*c/(e*(d+e*x)*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/(d+e*x)^5,x,2,-1/3*c^2/(e*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2))],
[sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/(d+e*x)^6,x,2,-1/4*c^2/(e*(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2))],
[(d+e*x)^3*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2),x,2,1/7*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(7/2)/(c^2*e)],
[(d+e*x)^2*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2),x,2,1/6*(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2)/(c*e)],
[(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2),x,1,1/5*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2)/(c*e)],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2),x,1,1/4*(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)/(d+e*x),x,2,1/3*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)/(d+e*x)^2,x,2,1/2*c*(d+e*x)*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)/(d+e*x)^3,x,2,c*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)/(d+e*x)^4,x,3,c^2*(d+e*x)*log(d+e*x)/(e*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)/(d+e*x)^5,x,2,-c^2/(e*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)/(d+e*x)^6,x,2,-1/2*c^2/(e*(d+e*x)*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)/(d+e*x)^7,x,2,-1/3*c^3/(e*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2))],
[(d+e*x)^3*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2),x,2,1/9*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(9/2)/(c^2*e)],
[(d+e*x)^2*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2),x,2,1/8*(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(7/2)/(c*e)],
[(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2),x,1,1/7*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(7/2)/(c*e)],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2),x,1,1/6*(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2)/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2)/(d+e*x),x,2,1/5*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2)/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2)/(d+e*x)^2,x,2,1/4*c*(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2)/(d+e*x)^3,x,2,1/3*c*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2)/(d+e*x)^4,x,2,1/2*c^2*(d+e*x)*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2)/(d+e*x)^5,x,2,c^2*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/e],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2)/(d+e*x)^6,x,3,c^3*(d+e*x)*log(d+e*x)/(e*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2)/(d+e*x)^7,x,2,-c^3/(e*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2)/(d+e*x)^8,x,2,-1/2*c^3/(e*(d+e*x)*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],

# p<0
[(d+e*x)^4/sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2),x,2,1/4*(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)/(c^2*e)],
[(d+e*x)^3/sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2),x,2,1/3*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)/(c^2*e)],
[(d+e*x)^2/sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2),x,2,1/2*(d+e*x)*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/(c*e)],
[(d+e*x)/sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2),x,1,sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/(c*e)],
[1/sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2),x,2,(d+e*x)*log(d+e*x)/(e*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[1/((d+e*x)*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)),x,2,(-1)/(e*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[1/((d+e*x)^2*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)),x,2,(-1/2)/(e*(d+e*x)*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[1/((d+e*x)^3*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)),x,2,-1/3*c/(e*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2))],
[1/((d+e*x)^4*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)),x,2,-1/4*c/(e*(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2))],
[(d+e*x)^4/(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2),x,2,1/2*(d+e*x)*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/(c^2*e)],
[(d+e*x)^3/(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2),x,2,sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/(c^2*e)],
[(d+e*x)^2/(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2),x,3,(d+e*x)*log(d+e*x)/(c*e*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[(d+e*x)/(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2),x,1,(-1)/(c*e*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[1/(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2),x,1,(-1/2)/(c*e*(d+e*x)*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[1/((d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)),x,2,(-1/3)/(e*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2))],
[1/((d+e*x)^2*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)),x,2,(-1/4)/(e*(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2))],
[1/((d+e*x)^3*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)),x,2,-1/5*c/(e*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2))],
[(d+e*x)^6/(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2),x,2,1/2*(d+e*x)*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/(c^3*e)],
[(d+e*x)^5/(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2),x,2,sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/(c^3*e)],
[(d+e*x)^4/(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2),x,3,(d+e*x)*log(d+e*x)/(c^2*e*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[(d+e*x)^3/(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2),x,2,(-1)/(c^2*e*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[(d+e*x)^2/(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2),x,2,(-1/2)/(c^2*e*(d+e*x)*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[(d+e*x)/(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2),x,1,(-1/3)/(c*e*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2))],
[1/(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2),x,1,(-1/4)/(c*e*(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2))],
[1/((d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2)),x,2,(-1/5)/(e*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2))],
[1/((d+e*x)^2*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2)),x,2,(-1/6)/(e*(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2))],
[1/((d+e*x)^3*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(5/2)),x,2,-1/7*c/(e*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(7/2))],

# Integrands of the form (d+e x)^m (c d^2+2 c d e x+c e^2 x^2)^p when m symbolic
[(d+e*x)^m*(c*d^2+2*c*d*e*x+c*e^2*x^2)^2,x,3,c^2*(d+e*x)^(5+m)/(e*(5+m))],
[(d+e*x)^m*(c*d^2+2*c*d*e*x+c*e^2*x^2),x,3,c*(d+e*x)^(3+m)/(e*(3+m))],
[(d+e*x)^m/(c*d^2+2*c*d*e*x+c*e^2*x^2),x,3,-(d+e*x)^(-1+m)/(c*e*(1-m))],
[(d+e*x)^m/(c*d^2+2*c*d*e*x+c*e^2*x^2)^2,x,3,-(d+e*x)^(-3+m)/(c^2*e*(3-m))],
[(d+e*x)^m/(c*d^2+2*c*d*e*x+c*e^2*x^2)^3,x,3,-(d+e*x)^(-5+m)/(c^3*e*(5-m))],
[(d+e*x)^m*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2),x,2,(d+e*x)^(1+m)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2)/(e*(4+m))],
[(d+e*x)^m*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(1/2),x,2,(d+e*x)^(1+m)*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2)/(e*(2+m))],
[(d+e*x)^m/(c*d^2+2*c*d*e*x+c*e^2*x^2)^(1/2),x,2,(d+e*x)^(1+m)/(e*m*sqrt(c*d^2+2*c*d*e*x+c*e^2*x^2))],
[(d+e*x)^m/(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2),x,2,-(d+e*x)^(1+m)/(e*(2-m)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(3/2))],

# Integrands of the form (d+e x)^m (c d^2+2 c d e x+c e^2 x^2)^p when p symbolic
[(d+e*x)^m*(c*d^2+2*c*d*e*x+c*e^2*x^2)^p,x,2,(d+e*x)^(1+m)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^p/(e*(1+m+2*p))],
[(d+e*x)^p/(c*d^2+2*c*d*e*x+c*e^2*x^2)^p,x,2,(d+e*x)^(1+p)/(e*(1-p)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^p)],
[(d+e*x)^3*(c*d^2+2*c*d*e*x+c*e^2*x^2)^p,x,2,1/2*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(2+p)/(c^2*e*(2+p))],
[(d+e*x)^2*(c*d^2+2*c*d*e*x+c*e^2*x^2)^p,x,2,(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(1+p)/(c*e*(3+2*p))],
[(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^p,x,1,1/2*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(1+p)/(c*e*(1+p))],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^p,x,1,(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^p/(e*(1+2*p))],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^p/(d+e*x),x,2,1/2*(c*d^2+2*c*d*e*x+c*e^2*x^2)^p/(e*p)],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^p/(d+e*x)^2,x,2,-c*(d+e*x)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(-1+p)/(e*(1-2*p))],
[(c*d^2+2*c*d*e*x+c*e^2*x^2)^p/(d+e*x)^3,x,2,-1/2*c*(c*d^2+2*c*d*e*x+c*e^2*x^2)^(-1+p)/(e*(1-p))],
[(d+e*x)^(-1-2*p)*(c*d^2+2*c*d*e*x+c*e^2*x^2)^p,x,2,(c*d^2+2*c*d*e*x+c*e^2*x^2)^p*log(d+e*x)/(e*(d+e*x)^(2*p))],
[(d+e*x)^(-1+2*p)/(c*d^2+2*c*d*e*x+c*e^2*x^2)^p,x,2,(d+e*x)^(2*p)*log(d+e*x)/(e*(c*d^2+2*c*d*e*x+c*e^2*x^2)^p)],

# Integrands of the form (d+e x)^m (a+b x+c x^2)^p when 2 c d-b e=0

# Integrands of the form (b d+2 c d x)^m (a+b x+c x^2)^p

# p>0
[(b*d+2*c*d*x)^4*(a+b*x+c*x^2),x,2,-1/40*(b^2-4*a*c)*d^4*(b+2*c*x)^5/c^2+1/56*d^4*(b+2*c*x)^7/c^2],
[(b*d+2*c*d*x)^3*(a+b*x+c*x^2),x,2,-1/32*(b^2-4*a*c)*d^3*(b+2*c*x)^4/c^2+1/48*d^3*(b+2*c*x)^6/c^2],
[(b*d+2*c*d*x)^2*(a+b*x+c*x^2),x,2,-1/24*(b^2-4*a*c)*d^2*(b+2*c*x)^3/c^2+1/40*d^2*(b+2*c*x)^5/c^2],
[(b*d+2*c*d*x)*(a+b*x+c*x^2),x,1,1/2*d*(a+b*x+c*x^2)^2],
[(a+b*x+c*x^2)/(b*d+2*c*d*x),x,2,1/4*b*x/(c*d)+1/4*x^2/d-1/8*(b^2-4*a*c)*log(b+2*c*x)/(c^2*d)],
[(a+b*x+c*x^2)/(b*d+2*c*d*x)^2,x,2,1/4*x/(c*d^2)+1/8*(b^2-4*a*c)/(c^2*d^2*(b+2*c*x))],
[(a+b*x+c*x^2)/(b*d+2*c*d*x)^3,x,2,1/16*(b^2-4*a*c)/(c^2*d^3*(b+2*c*x)^2)+1/8*log(b+2*c*x)/(c^2*d^3)],
[(a+b*x+c*x^2)/(b*d+2*c*d*x)^4,x,2,1/24*(b^2-4*a*c)/(c^2*d^4*(b+2*c*x)^3)+(-1/8)/(c^2*d^4*(b+2*c*x))],
[(a+b*x+c*x^2)/(b*d+2*c*d*x)^5,x,1,1/2*(a+b*x+c*x^2)^2/((b^2-4*a*c)*d^5*(b+2*c*x)^4)],
[(a+b*x+c*x^2)/(b*d+2*c*d*x)^6,x,2,1/40*(b^2-4*a*c)/(c^2*d^6*(b+2*c*x)^5)+(-1/24)/(c^2*d^6*(b+2*c*x)^3)],
[(a+b*x+c*x^2)/(b*d+2*c*d*x)^7,x,2,1/48*(b^2-4*a*c)/(c^2*d^7*(b+2*c*x)^6)+(-1/32)/(c^2*d^7*(b+2*c*x)^4)],
[(a+b*x+c*x^2)/(b*d+2*c*d*x)^8,x,2,1/56*(b^2-4*a*c)/(c^2*d^8*(b+2*c*x)^7)+(-1/40)/(c^2*d^8*(b+2*c*x)^5)],
[(b*d+2*c*d*x)^5*(a+b*x+c*x^2)^2,x,2,1/192*(b^2-4*a*c)^2*d^5*(b+2*c*x)^6/c^3-1/128*(b^2-4*a*c)*d^5*(b+2*c*x)^8/c^3+1/320*d^5*(b+2*c*x)^10/c^3],
[(b*d+2*c*d*x)^4*(a+b*x+c*x^2)^2,x,2,1/160*(b^2-4*a*c)^2*d^4*(b+2*c*x)^5/c^3-1/112*(b^2-4*a*c)*d^4*(b+2*c*x)^7/c^3+1/288*d^4*(b+2*c*x)^9/c^3],
[(b*d+2*c*d*x)^3*(a+b*x+c*x^2)^2,x,2,1/12*(b^2-4*a*c)*d^3*(a+b*x+c*x^2)^3+1/4*d^3*(b+2*c*x)^2*(a+b*x+c*x^2)^3],
[(b*d+2*c*d*x)^2*(a+b*x+c*x^2)^2,x,2,1/96*(b^2-4*a*c)^2*d^2*(b+2*c*x)^3/c^3-1/80*(b^2-4*a*c)*d^2*(b+2*c*x)^5/c^3+1/224*d^2*(b+2*c*x)^7/c^3],
[(b*d+2*c*d*x)*(a+b*x+c*x^2)^2,x,1,1/3*d*(a+b*x+c*x^2)^3],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x),x,2,-1/32*(b^2-4*a*c)*(b+2*c*x)^2/(c^3*d)+1/128*(b+2*c*x)^4/(c^3*d)+1/32*(b^2-4*a*c)^2*log(b+2*c*x)/(c^3*d)],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x)^2,x,2,-1/16*(b^2-8*a*c)*x/(c^2*d^2)+1/8*b*x^2/(c*d^2)+1/12*x^3/d^2-1/32*(b^2-4*a*c)^2/(c^3*d^2*(b+2*c*x))],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x)^3,x,2,1/16*b*x/(c^2*d^3)+1/16*x^2/(c*d^3)-1/64*(b^2-4*a*c)^2/(c^3*d^3*(b+2*c*x)^2)-1/16*(b^2-4*a*c)*log(b+2*c*x)/(c^3*d^3)],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x)^4,x,2,1/16*x/(c^2*d^4)-1/96*(b^2-4*a*c)^2/(c^3*d^4*(b+2*c*x)^3)+1/16*(b^2-4*a*c)/(c^3*d^4*(b+2*c*x))],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x)^5,x,2,-1/128*(b^2-4*a*c)^2/(c^3*d^5*(b+2*c*x)^4)+1/32*(b^2-4*a*c)/(c^3*d^5*(b+2*c*x)^2)+1/32*log(b+2*c*x)/(c^3*d^5)],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x)^6,x,2,-1/160*(b^2-4*a*c)^2/(c^3*d^6*(b+2*c*x)^5)+1/48*(b^2-4*a*c)/(c^3*d^6*(b+2*c*x)^3)+(-1/32)/(c^3*d^6*(b+2*c*x))],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x)^7,x,1,1/3*(a+b*x+c*x^2)^3/((b^2-4*a*c)*d^7*(b+2*c*x)^6)],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x)^8,x,2,-1/224*(b^2-4*a*c)^2/(c^3*d^8*(b+2*c*x)^7)+1/80*(b^2-4*a*c)/(c^3*d^8*(b+2*c*x)^5)+(-1/96)/(c^3*d^8*(b+2*c*x)^3)],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x)^9,x,2,-1/256*(b^2-4*a*c)^2/(c^3*d^9*(b+2*c*x)^8)+1/96*(b^2-4*a*c)/(c^3*d^9*(b+2*c*x)^6)+(-1/128)/(c^3*d^9*(b+2*c*x)^4)],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x)^10,x,2,-1/288*(b^2-4*a*c)^2/(c^3*d^10*(b+2*c*x)^9)+1/112*(b^2-4*a*c)/(c^3*d^10*(b+2*c*x)^7)+(-1/160)/(c^3*d^10*(b+2*c*x)^5)],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x)^11,x,2,-1/320*(b^2-4*a*c)^2/(c^3*d^11*(b+2*c*x)^10)+1/128*(b^2-4*a*c)/(c^3*d^11*(b+2*c*x)^8)+(-1/192)/(c^3*d^11*(b+2*c*x)^6)],
[(b*d+2*c*d*x)^5*(a+b*x+c*x^2)^3,x,2,-1/768*(b^2-4*a*c)^3*d^5*(b+2*c*x)^6/c^4+3/1024*(b^2-4*a*c)^2*d^5*(b+2*c*x)^8/c^4-3/1280*(b^2-4*a*c)*d^5*(b+2*c*x)^10/c^4+1/1536*d^5*(b+2*c*x)^12/c^4],
[(b*d+2*c*d*x)^4*(a+b*x+c*x^2)^3,x,2,-1/640*(b^2-4*a*c)^3*d^4*(b+2*c*x)^5/c^4+3/896*(b^2-4*a*c)^2*d^4*(b+2*c*x)^7/c^4-1/384*(b^2-4*a*c)*d^4*(b+2*c*x)^9/c^4+1/1408*d^4*(b+2*c*x)^11/c^4],
[(b*d+2*c*d*x)^3*(a+b*x+c*x^2)^3,x,2,1/20*(b^2-4*a*c)*d^3*(a+b*x+c*x^2)^4+1/5*d^3*(b+2*c*x)^2*(a+b*x+c*x^2)^4],
[(b*d+2*c*d*x)^2*(a+b*x+c*x^2)^3,x,2,-1/384*(b^2-4*a*c)^3*d^2*(b+2*c*x)^3/c^4+3/640*(b^2-4*a*c)^2*d^2*(b+2*c*x)^5/c^4-3/896*(b^2-4*a*c)*d^2*(b+2*c*x)^7/c^4+1/1152*d^2*(b+2*c*x)^9/c^4],
[(b*d+2*c*d*x)*(a+b*x+c*x^2)^3,x,1,1/4*d*(a+b*x+c*x^2)^4],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x),x,2,3/256*(b^2-4*a*c)^2*(b+2*c*x)^2/(c^4*d)-3/512*(b^2-4*a*c)*(b+2*c*x)^4/(c^4*d)+1/768*(b+2*c*x)^6/(c^4*d)-1/128*(b^2-4*a*c)^3*log(b+2*c*x)/(c^4*d)],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^2,x,2,3/64*(b^2-4*a*c)^2*x/(c^3*d^2)+1/128*(b^2-4*a*c)^3/(c^4*d^2*(b+2*c*x))-1/128*(b^2-4*a*c)*(b+2*c*x)^3/(c^4*d^2)+1/640*(b+2*c*x)^5/(c^4*d^2)],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^3,x,2,1/256*(b^2-4*a*c)^3/(c^4*d^3*(b+2*c*x)^2)-3/256*(b^2-4*a*c)*(b+2*c*x)^2/(c^4*d^3)+1/512*(b+2*c*x)^4/(c^4*d^3)+3/128*(b^2-4*a*c)^2*log(b+2*c*x)/(c^4*d^3)],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^4,x,2,-1/32*(b^2-6*a*c)*x/(c^3*d^4)+1/32*b*x^2/(c^2*d^4)+1/48*x^3/(c*d^4)+1/384*(b^2-4*a*c)^3/(c^4*d^4*(b+2*c*x)^3)-3/128*(b^2-4*a*c)^2/(c^4*d^4*(b+2*c*x))],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^5,x,2,1/64*b*x/(c^3*d^5)+1/64*x^2/(c^2*d^5)+1/512*(b^2-4*a*c)^3/(c^4*d^5*(b+2*c*x)^4)-3/256*(b^2-4*a*c)^2/(c^4*d^5*(b+2*c*x)^2)-3/128*(b^2-4*a*c)*log(b+2*c*x)/(c^4*d^5)],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^6,x,2,1/64*x/(c^3*d^6)+1/640*(b^2-4*a*c)^3/(c^4*d^6*(b+2*c*x)^5)-1/128*(b^2-4*a*c)^2/(c^4*d^6*(b+2*c*x)^3)+3/128*(b^2-4*a*c)/(c^4*d^6*(b+2*c*x))],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^7,x,2,1/768*(b^2-4*a*c)^3/(c^4*d^7*(b+2*c*x)^6)-3/512*(b^2-4*a*c)^2/(c^4*d^7*(b+2*c*x)^4)+3/256*(b^2-4*a*c)/(c^4*d^7*(b+2*c*x)^2)+1/128*log(b+2*c*x)/(c^4*d^7)],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^8,x,2,1/896*(b^2-4*a*c)^3/(c^4*d^8*(b+2*c*x)^7)-3/640*(b^2-4*a*c)^2/(c^4*d^8*(b+2*c*x)^5)+1/128*(b^2-4*a*c)/(c^4*d^8*(b+2*c*x)^3)+(-1/128)/(c^4*d^8*(b+2*c*x))],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^9,x,1,1/4*(a+b*x+c*x^2)^4/((b^2-4*a*c)*d^9*(b+2*c*x)^8)],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^10,x,2,1/1152*(b^2-4*a*c)^3/(c^4*d^10*(b+2*c*x)^9)-3/896*(b^2-4*a*c)^2/(c^4*d^10*(b+2*c*x)^7)+3/640*(b^2-4*a*c)/(c^4*d^10*(b+2*c*x)^5)+(-1/384)/(c^4*d^10*(b+2*c*x)^3)],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^11,x,2,1/1280*(b^2-4*a*c)^3/(c^4*d^11*(b+2*c*x)^10)-3/1024*(b^2-4*a*c)^2/(c^4*d^11*(b+2*c*x)^8)+1/256*(b^2-4*a*c)/(c^4*d^11*(b+2*c*x)^6)+(-1/512)/(c^4*d^11*(b+2*c*x)^4)],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^12,x,2,1/1408*(b^2-4*a*c)^3/(c^4*d^12*(b+2*c*x)^11)-1/384*(b^2-4*a*c)^2/(c^4*d^12*(b+2*c*x)^9)+3/896*(b^2-4*a*c)/(c^4*d^12*(b+2*c*x)^7)+(-1/640)/(c^4*d^12*(b+2*c*x)^5)],

# p<0
[(b*d+2*c*d*x)^8/(a+b*x+c*x^2),x,6,2*(b^2-4*a*c)^3*d^8*(b+2*c*x)+2/3*(b^2-4*a*c)^2*d^8*(b+2*c*x)^3+2/5*(b^2-4*a*c)*d^8*(b+2*c*x)^5+2/7*d^8*(b+2*c*x)^7-2*(b^2-4*a*c)^(7/2)*d^8*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))],
[(b*d+2*c*d*x)^7/(a+b*x+c*x^2),x,4,(b^2-4*a*c)^2*d^7*(b+2*c*x)^2+1/2*(b^2-4*a*c)*d^7*(b+2*c*x)^4+1/3*d^7*(b+2*c*x)^6+(b^2-4*a*c)^3*d^7*log(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^6/(a+b*x+c*x^2),x,5,2*(b^2-4*a*c)^2*d^6*(b+2*c*x)+2/3*(b^2-4*a*c)*d^6*(b+2*c*x)^3+2/5*d^6*(b+2*c*x)^5-2*(b^2-4*a*c)^(5/2)*d^6*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))],
[(b*d+2*c*d*x)^5/(a+b*x+c*x^2),x,3,(b^2-4*a*c)*d^5*(b+2*c*x)^2+1/2*d^5*(b+2*c*x)^4+(b^2-4*a*c)^2*d^5*log(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^4/(a+b*x+c*x^2),x,4,2*(b^2-4*a*c)*d^4*(b+2*c*x)+2/3*d^4*(b+2*c*x)^3-2*(b^2-4*a*c)^(3/2)*d^4*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))],
[(b*d+2*c*d*x)^3/(a+b*x+c*x^2),x,2,d^3*(b+2*c*x)^2+(b^2-4*a*c)*d^3*log(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^2/(a+b*x+c*x^2),x,3,2*d^2*(b+2*c*x)-2*d^2*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))*sqrt(b^2-4*a*c)],
[(b*d+2*c*d*x)/(a+b*x+c*x^2),x,1,d*log(a+b*x+c*x^2)],
[1/((b*d+2*c*d*x)*(a+b*x+c*x^2)),x,3,-2*log(b+2*c*x)/((b^2-4*a*c)*d)+log(a+b*x+c*x^2)/((b^2-4*a*c)*d)],
[1/((b*d+2*c*d*x)^2*(a+b*x+c*x^2)),x,3,2/((b^2-4*a*c)*d^2*(b+2*c*x))-2*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*d^2)],
[1/((b*d+2*c*d*x)^3*(a+b*x+c*x^2)),x,4,1/((b^2-4*a*c)*d^3*(b+2*c*x)^2)-2*log(b+2*c*x)/((b^2-4*a*c)^2*d^3)+log(a+b*x+c*x^2)/((b^2-4*a*c)^2*d^3)],
[1/((b*d+2*c*d*x)^4*(a+b*x+c*x^2)),x,4,2/3/((b^2-4*a*c)*d^4*(b+2*c*x)^3)+2/((b^2-4*a*c)^2*d^4*(b+2*c*x))-2*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(5/2)*d^4)],
[(b*d+2*c*d*x)^8/(a+b*x+c*x^2)^2,x,6,28*c*(b^2-4*a*c)^2*d^8*(b+2*c*x)+28/3*c*(b^2-4*a*c)*d^8*(b+2*c*x)^3+28/5*c*d^8*(b+2*c*x)^5-d^8*(b+2*c*x)^7/(a+b*x+c*x^2)-28*c*(b^2-4*a*c)^(5/2)*d^8*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))],
[(b*d+2*c*d*x)^7/(a+b*x+c*x^2)^2,x,4,12*c*(b^2-4*a*c)*d^7*(b+2*c*x)^2+6*c*d^7*(b+2*c*x)^4-d^7*(b+2*c*x)^6/(a+b*x+c*x^2)+12*c*(b^2-4*a*c)^2*d^7*log(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^6/(a+b*x+c*x^2)^2,x,5,20*c*(b^2-4*a*c)*d^6*(b+2*c*x)+20/3*c*d^6*(b+2*c*x)^3-d^6*(b+2*c*x)^5/(a+b*x+c*x^2)-20*c*(b^2-4*a*c)^(3/2)*d^6*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))],
[(b*d+2*c*d*x)^5/(a+b*x+c*x^2)^2,x,3,8*c*d^5*(b+2*c*x)^2-d^5*(b+2*c*x)^4/(a+b*x+c*x^2)+8*c*(b^2-4*a*c)*d^5*log(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^4/(a+b*x+c*x^2)^2,x,4,12*c*d^4*(b+2*c*x)-d^4*(b+2*c*x)^3/(a+b*x+c*x^2)-12*c*d^4*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))*sqrt(b^2-4*a*c)],
[(b*d+2*c*d*x)^3/(a+b*x+c*x^2)^2,x,2,-d^3*(b+2*c*x)^2/(a+b*x+c*x^2)+4*c*d^3*log(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^2/(a+b*x+c*x^2)^2,x,3,-d^2*(b+2*c*x)/(a+b*x+c*x^2)-4*c*d^2*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c)],
[(b*d+2*c*d*x)/(a+b*x+c*x^2)^2,x,1,-d/(a+b*x+c*x^2)],
[1/((b*d+2*c*d*x)*(a+b*x+c*x^2)^2),x,4,(-1)/((b^2-4*a*c)*d*(a+b*x+c*x^2))+8*c*log(b+2*c*x)/((b^2-4*a*c)^2*d)-4*c*log(a+b*x+c*x^2)/((b^2-4*a*c)^2*d)],
[1/((b*d+2*c*d*x)^2*(a+b*x+c*x^2)^2),x,4,-12*c/((b^2-4*a*c)^2*d^2*(b+2*c*x))+(-1)/((b^2-4*a*c)*d^2*(b+2*c*x)*(a+b*x+c*x^2))+12*c*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(5/2)*d^2)],
[1/((b*d+2*c*d*x)^3*(a+b*x+c*x^2)^2),x,5,-8*c/((b^2-4*a*c)^2*d^3*(b+2*c*x)^2)+(-1)/((b^2-4*a*c)*d^3*(b+2*c*x)^2*(a+b*x+c*x^2))+16*c*log(b+2*c*x)/((b^2-4*a*c)^3*d^3)-8*c*log(a+b*x+c*x^2)/((b^2-4*a*c)^3*d^3)],
[(b*d+2*c*d*x)^10/(a+b*x+c*x^2)^3,x,7,252*c^2*(b^2-4*a*c)^2*d^10*(b+2*c*x)+84*c^2*(b^2-4*a*c)*d^10*(b+2*c*x)^3+252/5*c^2*d^10*(b+2*c*x)^5-1/2*d^10*(b+2*c*x)^9/(a+b*x+c*x^2)^2-9*c*d^10*(b+2*c*x)^7/(a+b*x+c*x^2)-252*c^2*(b^2-4*a*c)^(5/2)*d^10*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))],
[(b*d+2*c*d*x)^9/(a+b*x+c*x^2)^3,x,5,96*c^2*(b^2-4*a*c)*d^9*(b+2*c*x)^2+48*c^2*d^9*(b+2*c*x)^4-1/2*d^9*(b+2*c*x)^8/(a+b*x+c*x^2)^2-8*c*d^9*(b+2*c*x)^6/(a+b*x+c*x^2)+96*c^2*(b^2-4*a*c)^2*d^9*log(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^8/(a+b*x+c*x^2)^3,x,6,140*c^2*(b^2-4*a*c)*d^8*(b+2*c*x)+140/3*c^2*d^8*(b+2*c*x)^3-1/2*d^8*(b+2*c*x)^7/(a+b*x+c*x^2)^2-7*c*d^8*(b+2*c*x)^5/(a+b*x+c*x^2)-140*c^2*(b^2-4*a*c)^(3/2)*d^8*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))],
[(b*d+2*c*d*x)^7/(a+b*x+c*x^2)^3,x,4,48*c^2*d^7*(b+2*c*x)^2-1/2*d^7*(b+2*c*x)^6/(a+b*x+c*x^2)^2-6*c*d^7*(b+2*c*x)^4/(a+b*x+c*x^2)+48*c^2*(b^2-4*a*c)*d^7*log(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^6/(a+b*x+c*x^2)^3,x,5,60*c^2*d^6*(b+2*c*x)-1/2*d^6*(b+2*c*x)^5/(a+b*x+c*x^2)^2-5*c*d^6*(b+2*c*x)^3/(a+b*x+c*x^2)-60*c^2*d^6*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))*sqrt(b^2-4*a*c)],
[(b*d+2*c*d*x)^5/(a+b*x+c*x^2)^3,x,3,-1/2*d^5*(b+2*c*x)^4/(a+b*x+c*x^2)^2-4*c*d^5*(b+2*c*x)^2/(a+b*x+c*x^2)+16*c^2*d^5*log(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^4/(a+b*x+c*x^2)^3,x,4,-1/2*d^4*(b+2*c*x)^3/(a+b*x+c*x^2)^2-3*c*d^4*(b+2*c*x)/(a+b*x+c*x^2)-12*c^2*d^4*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c)],
[(b*d+2*c*d*x)^3/(a+b*x+c*x^2)^3,x,1,-1/2*d^3*(b+2*c*x)^4/((b^2-4*a*c)*(a+b*x+c*x^2)^2)],
[(b*d+2*c*d*x)^2/(a+b*x+c*x^2)^3,x,4,-1/2*d^2*(b+2*c*x)/(a+b*x+c*x^2)^2-c*d^2*(b+2*c*x)/((b^2-4*a*c)*(a+b*x+c*x^2))+4*c^2*d^2*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)],
[(b*d+2*c*d*x)/(a+b*x+c*x^2)^3,x,1,-1/2*d/(a+b*x+c*x^2)^2],
[1/((b*d+2*c*d*x)*(a+b*x+c*x^2)^3),x,5,(-1/2)/((b^2-4*a*c)*d*(a+b*x+c*x^2)^2)+4*c/((b^2-4*a*c)^2*d*(a+b*x+c*x^2))-32*c^2*log(b+2*c*x)/((b^2-4*a*c)^3*d)+16*c^2*log(a+b*x+c*x^2)/((b^2-4*a*c)^3*d)],
[1/((b*d+2*c*d*x)^2*(a+b*x+c*x^2)^3),x,5,60*c^2/((b^2-4*a*c)^3*d^2*(b+2*c*x))+(-1/2)/((b^2-4*a*c)*d^2*(b+2*c*x)*(a+b*x+c*x^2)^2)+5*c/((b^2-4*a*c)^2*d^2*(b+2*c*x)*(a+b*x+c*x^2))-60*c^2*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(7/2)*d^2)],
[1/((b*d+2*c*d*x)^3*(a+b*x+c*x^2)^3),x,6,48*c^2/((b^2-4*a*c)^3*d^3*(b+2*c*x)^2)+(-1/2)/((b^2-4*a*c)*d^3*(b+2*c*x)^2*(a+b*x+c*x^2)^2)+6*c/((b^2-4*a*c)^2*d^3*(b+2*c*x)^2*(a+b*x+c*x^2))-96*c^2*log(b+2*c*x)/((b^2-4*a*c)^4*d^3)+48*c^2*log(a+b*x+c*x^2)/((b^2-4*a*c)^4*d^3)],
[1/((b*d+2*c*d*x)^4*(a+b*x+c*x^2)^3),x,6,140/3*c^2/((b^2-4*a*c)^3*d^4*(b+2*c*x)^3)+140*c^2/((b^2-4*a*c)^4*d^4*(b+2*c*x))+(-1/2)/((b^2-4*a*c)*d^4*(b+2*c*x)^3*(a+b*x+c*x^2)^2)+7*c/((b^2-4*a*c)^2*d^4*(b+2*c*x)^3*(a+b*x+c*x^2))-140*c^2*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(9/2)*d^4)],

# Integrands of the form (b d+2 c d x)^m (a+b x+c x^2)^(p/2)

# p>0
[(b*d+2*c*d*x)^4*(a+b*x+c*x^2)^(1/2),x,5,-1/64*(b^2-4*a*c)^3*d^4*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(3/2)-1/32*(b^2-4*a*c)^2*d^4*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c-1/48*(b^2-4*a*c)*d^4*(b+2*c*x)^3*sqrt(a+b*x+c*x^2)/c+1/12*d^4*(b+2*c*x)^5*sqrt(a+b*x+c*x^2)/c],
[(b*d+2*c*d*x)^3*(a+b*x+c*x^2)^(1/2),x,2,4/15*(b^2-4*a*c)*d^3*(a+b*x+c*x^2)^(3/2)+2/5*d^3*(b+2*c*x)^2*(a+b*x+c*x^2)^(3/2)],
[(b*d+2*c*d*x)^2*(a+b*x+c*x^2)^(1/2),x,4,-1/32*(b^2-4*a*c)^2*d^2*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(3/2)-1/16*(b^2-4*a*c)*d^2*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c+1/8*d^2*(b+2*c*x)^3*sqrt(a+b*x+c*x^2)/c],
[(b*d+2*c*d*x)*(a+b*x+c*x^2)^(1/2),x,1,2/3*d*(a+b*x+c*x^2)^(3/2)],
[(a+b*x+c*x^2)^(1/2)/(b*d+2*c*d*x),x,3,-1/4*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt(b^2-4*a*c)/(c^(3/2)*d)+1/2*sqrt(a+b*x+c*x^2)/(c*d)],
[(a+b*x+c*x^2)^(1/2)/(b*d+2*c*d*x)^2,x,3,1/4*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*d^2)-1/2*sqrt(a+b*x+c*x^2)/(c*d^2*(b+2*c*x))],
[(a+b*x+c*x^2)^(1/2)/(b*d+2*c*d*x)^3,x,3,1/8*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))/(c^(3/2)*d^3*sqrt(b^2-4*a*c))-1/4*sqrt(a+b*x+c*x^2)/(c*d^3*(b+2*c*x)^2)],
[(a+b*x+c*x^2)^(1/2)/(b*d+2*c*d*x)^4,x,1,2/3*(a+b*x+c*x^2)^(3/2)/((b^2-4*a*c)*d^4*(b+2*c*x)^3)],
[(a+b*x+c*x^2)^(1/2)/(b*d+2*c*d*x)^5,x,4,1/32*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))/(c^(3/2)*(b^2-4*a*c)^(3/2)*d^5)-1/8*sqrt(a+b*x+c*x^2)/(c*d^5*(b+2*c*x)^4)+1/16*sqrt(a+b*x+c*x^2)/(c*(b^2-4*a*c)*d^5*(b+2*c*x)^2)],
[(a+b*x+c*x^2)^(1/2)/(b*d+2*c*d*x)^6,x,2,2/5*(a+b*x+c*x^2)^(3/2)/((b^2-4*a*c)*d^6*(b+2*c*x)^5)+4/15*(a+b*x+c*x^2)^(3/2)/((b^2-4*a*c)^2*d^6*(b+2*c*x)^3)],
[(a+b*x+c*x^2)^(1/2)/(b*d+2*c*d*x)^7,x,5,1/64*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))/(c^(3/2)*(b^2-4*a*c)^(5/2)*d^7)-1/12*sqrt(a+b*x+c*x^2)/(c*d^7*(b+2*c*x)^6)+1/48*sqrt(a+b*x+c*x^2)/(c*(b^2-4*a*c)*d^7*(b+2*c*x)^4)+1/32*sqrt(a+b*x+c*x^2)/(c*(b^2-4*a*c)^2*d^7*(b+2*c*x)^2)],
[(b*d+2*c*d*x)^5*(a+b*x+c*x^2)^(3/2),x,3,16/315*(b^2-4*a*c)^2*d^5*(a+b*x+c*x^2)^(5/2)+8/63*(b^2-4*a*c)*d^5*(b+2*c*x)^2*(a+b*x+c*x^2)^(5/2)+2/9*d^5*(b+2*c*x)^4*(a+b*x+c*x^2)^(5/2)],
[(b*d+2*c*d*x)^4*(a+b*x+c*x^2)^(3/2),x,6,1/16*d^4*(b+2*c*x)^5*(a+b*x+c*x^2)^(3/2)/c+3/2048*(b^2-4*a*c)^4*d^4*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(5/2)+3/1024*(b^2-4*a*c)^3*d^4*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c^2+1/512*(b^2-4*a*c)^2*d^4*(b+2*c*x)^3*sqrt(a+b*x+c*x^2)/c^2-1/128*(b^2-4*a*c)*d^4*(b+2*c*x)^5*sqrt(a+b*x+c*x^2)/c^2],
[(b*d+2*c*d*x)^3*(a+b*x+c*x^2)^(3/2),x,2,4/35*(b^2-4*a*c)*d^3*(a+b*x+c*x^2)^(5/2)+2/7*d^3*(b+2*c*x)^2*(a+b*x+c*x^2)^(5/2)],
[(b*d+2*c*d*x)^2*(a+b*x+c*x^2)^(3/2),x,5,1/12*d^2*(b+2*c*x)^3*(a+b*x+c*x^2)^(3/2)/c+1/256*(b^2-4*a*c)^3*d^2*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(5/2)+1/128*(b^2-4*a*c)^2*d^2*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c^2-1/64*(b^2-4*a*c)*d^2*(b+2*c*x)^3*sqrt(a+b*x+c*x^2)/c^2],
[(b*d+2*c*d*x)*(a+b*x+c*x^2)^(3/2),x,1,2/5*d*(a+b*x+c*x^2)^(5/2)],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x),x,4,1/6*(a+b*x+c*x^2)^(3/2)/(c*d)+1/16*(b^2-4*a*c)^(3/2)*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))/(c^(5/2)*d)-1/8*(b^2-4*a*c)*sqrt(a+b*x+c*x^2)/(c^2*d)],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^2,x,4,-1/2*(a+b*x+c*x^2)^(3/2)/(c*d^2*(b+2*c*x))-3/32*(b^2-4*a*c)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(5/2)*d^2)+3/16*(b+2*c*x)*sqrt(a+b*x+c*x^2)/(c^2*d^2)],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^3,x,4,-1/4*(a+b*x+c*x^2)^(3/2)/(c*d^3*(b+2*c*x)^2)-3/32*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt(b^2-4*a*c)/(c^(5/2)*d^3)+3/16*sqrt(a+b*x+c*x^2)/(c^2*d^3)],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^4,x,4,-1/6*(a+b*x+c*x^2)^(3/2)/(c*d^4*(b+2*c*x)^3)+1/16*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(5/2)*d^4)-1/8*sqrt(a+b*x+c*x^2)/(c^2*d^4*(b+2*c*x))],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^5,x,4,-1/8*(a+b*x+c*x^2)^(3/2)/(c*d^5*(b+2*c*x)^4)+3/128*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))/(c^(5/2)*d^5*sqrt(b^2-4*a*c))-3/64*sqrt(a+b*x+c*x^2)/(c^2*d^5*(b+2*c*x)^2)],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^6,x,1,2/5*(a+b*x+c*x^2)^(5/2)/((b^2-4*a*c)*d^6*(b+2*c*x)^5)],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^7,x,5,-1/12*(a+b*x+c*x^2)^(3/2)/(c*d^7*(b+2*c*x)^6)+1/256*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))/(c^(5/2)*(b^2-4*a*c)^(3/2)*d^7)-1/64*sqrt(a+b*x+c*x^2)/(c^2*d^7*(b+2*c*x)^4)+1/128*sqrt(a+b*x+c*x^2)/(c^2*(b^2-4*a*c)*d^7*(b+2*c*x)^2)],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^8,x,2,2/7*(a+b*x+c*x^2)^(5/2)/((b^2-4*a*c)*d^8*(b+2*c*x)^7)+4/35*(a+b*x+c*x^2)^(5/2)/((b^2-4*a*c)^2*d^8*(b+2*c*x)^5)],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^9,x,6,-1/16*(a+b*x+c*x^2)^(3/2)/(c*d^9*(b+2*c*x)^8)+3/2048*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))/(c^(5/2)*(b^2-4*a*c)^(5/2)*d^9)-1/128*sqrt(a+b*x+c*x^2)/(c^2*d^9*(b+2*c*x)^6)+1/512*sqrt(a+b*x+c*x^2)/(c^2*(b^2-4*a*c)*d^9*(b+2*c*x)^4)+3/1024*sqrt(a+b*x+c*x^2)/(c^2*(b^2-4*a*c)^2*d^9*(b+2*c*x)^2)],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^10,x,3,2/9*(a+b*x+c*x^2)^(5/2)/((b^2-4*a*c)*d^10*(b+2*c*x)^9)+8/63*(a+b*x+c*x^2)^(5/2)/((b^2-4*a*c)^2*d^10*(b+2*c*x)^7)+16/315*(a+b*x+c*x^2)^(5/2)/((b^2-4*a*c)^3*d^10*(b+2*c*x)^5)],
[(b*d+2*c*d*x)^5*(a+b*x+c*x^2)^(5/2),x,3,16/693*(b^2-4*a*c)^2*d^5*(a+b*x+c*x^2)^(7/2)+8/99*(b^2-4*a*c)*d^5*(b+2*c*x)^2*(a+b*x+c*x^2)^(7/2)+2/11*d^5*(b+2*c*x)^4*(a+b*x+c*x^2)^(7/2)],
[(b*d+2*c*d*x)^4*(a+b*x+c*x^2)^(5/2),x,7,-1/128*(b^2-4*a*c)*d^4*(b+2*c*x)^5*(a+b*x+c*x^2)^(3/2)/c^2+1/20*d^4*(b+2*c*x)^5*(a+b*x+c*x^2)^(5/2)/c-3/16384*(b^2-4*a*c)^5*d^4*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(7/2)-3/8192*(b^2-4*a*c)^4*d^4*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c^3-1/4096*(b^2-4*a*c)^3*d^4*(b+2*c*x)^3*sqrt(a+b*x+c*x^2)/c^3+1/1024*(b^2-4*a*c)^2*d^4*(b+2*c*x)^5*sqrt(a+b*x+c*x^2)/c^3],
[(b*d+2*c*d*x)^3*(a+b*x+c*x^2)^(5/2),x,2,4/63*(b^2-4*a*c)*d^3*(a+b*x+c*x^2)^(7/2)+2/9*d^3*(b+2*c*x)^2*(a+b*x+c*x^2)^(7/2)],
[(b*d+2*c*d*x)^2*(a+b*x+c*x^2)^(5/2),x,6,-5/384*(b^2-4*a*c)*d^2*(b+2*c*x)^3*(a+b*x+c*x^2)^(3/2)/c^2+1/16*d^2*(b+2*c*x)^3*(a+b*x+c*x^2)^(5/2)/c-5/8192*(b^2-4*a*c)^4*d^2*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(7/2)-5/4096*(b^2-4*a*c)^3*d^2*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c^3+5/2048*(b^2-4*a*c)^2*d^2*(b+2*c*x)^3*sqrt(a+b*x+c*x^2)/c^3],
[(b*d+2*c*d*x)*(a+b*x+c*x^2)^(5/2),x,1,2/7*d*(a+b*x+c*x^2)^(7/2)],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x),x,5,-1/24*(b^2-4*a*c)*(a+b*x+c*x^2)^(3/2)/(c^2*d)+1/10*(a+b*x+c*x^2)^(5/2)/(c*d)-1/64*(b^2-4*a*c)^(5/2)*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))/(c^(7/2)*d)+1/32*(b^2-4*a*c)^2*sqrt(a+b*x+c*x^2)/(c^3*d)],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^2,x,5,5/32*(b+2*c*x)*(a+b*x+c*x^2)^(3/2)/(c^2*d^2)-1/2*(a+b*x+c*x^2)^(5/2)/(c*d^2*(b+2*c*x))+15/512*(b^2-4*a*c)^2*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(7/2)*d^2)-15/256*(b^2-4*a*c)*(b+2*c*x)*sqrt(a+b*x+c*x^2)/(c^3*d^2)],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^3,x,5,5/48*(a+b*x+c*x^2)^(3/2)/(c^2*d^3)-1/4*(a+b*x+c*x^2)^(5/2)/(c*d^3*(b+2*c*x)^2)+5/128*(b^2-4*a*c)^(3/2)*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))/(c^(7/2)*d^3)-5/64*(b^2-4*a*c)*sqrt(a+b*x+c*x^2)/(c^3*d^3)],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^4,x,5,-5/24*(a+b*x+c*x^2)^(3/2)/(c^2*d^4*(b+2*c*x))-1/6*(a+b*x+c*x^2)^(5/2)/(c*d^4*(b+2*c*x)^3)-5/128*(b^2-4*a*c)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(7/2)*d^4)+5/64*(b+2*c*x)*sqrt(a+b*x+c*x^2)/(c^3*d^4)],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^5,x,5,-5/64*(a+b*x+c*x^2)^(3/2)/(c^2*d^5*(b+2*c*x)^2)-1/8*(a+b*x+c*x^2)^(5/2)/(c*d^5*(b+2*c*x)^4)-15/512*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt(b^2-4*a*c)/(c^(7/2)*d^5)+15/256*sqrt(a+b*x+c*x^2)/(c^3*d^5)],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^6,x,5,-1/24*(a+b*x+c*x^2)^(3/2)/(c^2*d^6*(b+2*c*x)^3)-1/10*(a+b*x+c*x^2)^(5/2)/(c*d^6*(b+2*c*x)^5)+1/64*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(7/2)*d^6)-1/32*sqrt(a+b*x+c*x^2)/(c^3*d^6*(b+2*c*x))],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^7,x,5,-5/192*(a+b*x+c*x^2)^(3/2)/(c^2*d^7*(b+2*c*x)^4)-1/12*(a+b*x+c*x^2)^(5/2)/(c*d^7*(b+2*c*x)^6)+5/1024*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))/(c^(7/2)*d^7*sqrt(b^2-4*a*c))-5/512*sqrt(a+b*x+c*x^2)/(c^3*d^7*(b+2*c*x)^2)],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^8,x,1,2/7*(a+b*x+c*x^2)^(7/2)/((b^2-4*a*c)*d^8*(b+2*c*x)^7)],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^9,x,6,-5/384*(a+b*x+c*x^2)^(3/2)/(c^2*d^9*(b+2*c*x)^6)-1/16*(a+b*x+c*x^2)^(5/2)/(c*d^9*(b+2*c*x)^8)+5/8192*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))/(c^(7/2)*(b^2-4*a*c)^(3/2)*d^9)-5/2048*sqrt(a+b*x+c*x^2)/(c^3*d^9*(b+2*c*x)^4)+5/4096*sqrt(a+b*x+c*x^2)/(c^3*(b^2-4*a*c)*d^9*(b+2*c*x)^2)],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^10,x,2,2/9*(a+b*x+c*x^2)^(7/2)/((b^2-4*a*c)*d^10*(b+2*c*x)^9)+4/63*(a+b*x+c*x^2)^(7/2)/((b^2-4*a*c)^2*d^10*(b+2*c*x)^7)],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^11,x,7,-1/128*(a+b*x+c*x^2)^(3/2)/(c^2*d^11*(b+2*c*x)^8)-1/20*(a+b*x+c*x^2)^(5/2)/(c*d^11*(b+2*c*x)^10)+3/16384*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))/(c^(7/2)*(b^2-4*a*c)^(5/2)*d^11)-1/1024*sqrt(a+b*x+c*x^2)/(c^3*d^11*(b+2*c*x)^6)+1/4096*sqrt(a+b*x+c*x^2)/(c^3*(b^2-4*a*c)*d^11*(b+2*c*x)^4)+3/8192*sqrt(a+b*x+c*x^2)/(c^3*(b^2-4*a*c)^2*d^11*(b+2*c*x)^2)],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^12,x,3,2/11*(a+b*x+c*x^2)^(7/2)/((b^2-4*a*c)*d^12*(b+2*c*x)^11)+8/99*(a+b*x+c*x^2)^(7/2)/((b^2-4*a*c)^2*d^12*(b+2*c*x)^9)+16/693*(a+b*x+c*x^2)^(7/2)/((b^2-4*a*c)^3*d^12*(b+2*c*x)^7)],

# p<0
[(b*d+2*c*d*x)^4/(a+b*x+c*x^2)^(1/2),x,4,3/8*(b^2-4*a*c)^2*d^4*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/sqrt(c)+3/4*(b^2-4*a*c)*d^4*(b+2*c*x)*sqrt(a+b*x+c*x^2)+1/2*d^4*(b+2*c*x)^3*sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^3/(a+b*x+c*x^2)^(1/2),x,2,4/3*(b^2-4*a*c)*d^3*sqrt(a+b*x+c*x^2)+2/3*d^3*(b+2*c*x)^2*sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^2/(a+b*x+c*x^2)^(1/2),x,3,1/2*(b^2-4*a*c)*d^2*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/sqrt(c)+d^2*(b+2*c*x)*sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)/(a+b*x+c*x^2)^(1/2),x,1,2*d*sqrt(a+b*x+c*x^2)],
[1/((b*d+2*c*d*x)*(a+b*x+c*x^2)^(1/2)),x,2,arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))/(d*sqrt(c)*sqrt(b^2-4*a*c))],
[1/((b*d+2*c*d*x)^2*(a+b*x+c*x^2)^(1/2)),x,1,2*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*d^2*(b+2*c*x))],
[1/((b*d+2*c*d*x)^3*(a+b*x+c*x^2)^(1/2)),x,3,1/2*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*d^3*sqrt(c))+sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*d^3*(b+2*c*x)^2)],
[1/((b*d+2*c*d*x)^4*(a+b*x+c*x^2)^(1/2)),x,2,2/3*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*d^4*(b+2*c*x)^3)+4/3*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^2*d^4*(b+2*c*x))],
[(b*d+2*c*d*x)^4/(a+b*x+c*x^2)^(3/2),x,4,6*(b^2-4*a*c)*d^4*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))*sqrt(c)-2*d^4*(b+2*c*x)^3/sqrt(a+b*x+c*x^2)+12*c*d^4*(b+2*c*x)*sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^3/(a+b*x+c*x^2)^(3/2),x,2,-2*d^3*(b+2*c*x)^2/sqrt(a+b*x+c*x^2)+16*c*d^3*sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^2/(a+b*x+c*x^2)^(3/2),x,3,4*d^2*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))*sqrt(c)-2*d^2*(b+2*c*x)/sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)/(a+b*x+c*x^2)^(3/2),x,1,-2*d/sqrt(a+b*x+c*x^2)],
[1/((b*d+2*c*d*x)*(a+b*x+c*x^2)^(3/2)),x,3,-4*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt(c)/((b^2-4*a*c)^(3/2)*d)+(-2)/((b^2-4*a*c)*d*sqrt(a+b*x+c*x^2))],
[1/((b*d+2*c*d*x)^2*(a+b*x+c*x^2)^(3/2)),x,2,(-2)/((b^2-4*a*c)*d^2*(b+2*c*x)*sqrt(a+b*x+c*x^2))-16*c*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^2*d^2*(b+2*c*x))],
[1/((b*d+2*c*d*x)^3*(a+b*x+c*x^2)^(3/2)),x,4,-6*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt(c)/((b^2-4*a*c)^(5/2)*d^3)+(-2)/((b^2-4*a*c)*d^3*(b+2*c*x)^2*sqrt(a+b*x+c*x^2))-12*c*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^2*d^3*(b+2*c*x)^2)],
[1/((b*d+2*c*d*x)^4*(a+b*x+c*x^2)^(3/2)),x,3,(-2)/((b^2-4*a*c)*d^4*(b+2*c*x)^3*sqrt(a+b*x+c*x^2))-32/3*c*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^2*d^4*(b+2*c*x)^3)-64/3*c*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^3*d^4*(b+2*c*x))],
[(b*d+2*c*d*x)^6/(a+b*x+c*x^2)^(5/2),x,5,-2/3*d^6*(b+2*c*x)^5/(a+b*x+c*x^2)^(3/2)+40*c^(3/2)*(b^2-4*a*c)*d^6*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))-40/3*c*d^6*(b+2*c*x)^3/sqrt(a+b*x+c*x^2)+80*c^2*d^6*(b+2*c*x)*sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^5/(a+b*x+c*x^2)^(5/2),x,3,-2/3*d^5*(b+2*c*x)^4/(a+b*x+c*x^2)^(3/2)-32/3*c*d^5*(b+2*c*x)^2/sqrt(a+b*x+c*x^2)+256/3*c^2*d^5*sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^4/(a+b*x+c*x^2)^(5/2),x,4,-2/3*d^4*(b+2*c*x)^3/(a+b*x+c*x^2)^(3/2)+16*c^(3/2)*d^4*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))-8*c*d^4*(b+2*c*x)/sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^3/(a+b*x+c*x^2)^(5/2),x,2,-2/3*d^3*(b+2*c*x)^2/(a+b*x+c*x^2)^(3/2)-16/3*c*d^3/sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^2/(a+b*x+c*x^2)^(5/2),x,1,-2/3*d^2*(b+2*c*x)^3/((b^2-4*a*c)*(a+b*x+c*x^2)^(3/2))],
[(b*d+2*c*d*x)/(a+b*x+c*x^2)^(5/2),x,1,-2/3*d/(a+b*x+c*x^2)^(3/2)],
[1/((b*d+2*c*d*x)*(a+b*x+c*x^2)^(5/2)),x,4,(-2/3)/((b^2-4*a*c)*d*(a+b*x+c*x^2)^(3/2))+16*c^(3/2)*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(5/2)*d)+8*c/((b^2-4*a*c)^2*d*sqrt(a+b*x+c*x^2))],
[1/((b*d+2*c*d*x)^2*(a+b*x+c*x^2)^(5/2)),x,3,(-2/3)/((b^2-4*a*c)*d^2*(b+2*c*x)*(a+b*x+c*x^2)^(3/2))+32/3*c/((b^2-4*a*c)^2*d^2*(b+2*c*x)*sqrt(a+b*x+c*x^2))+256/3*c^2*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^3*d^2*(b+2*c*x))],
[1/((b*d+2*c*d*x)^3*(a+b*x+c*x^2)^(5/2)),x,5,(-2/3)/((b^2-4*a*c)*d^3*(b+2*c*x)^2*(a+b*x+c*x^2)^(3/2))+40*c^(3/2)*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(7/2)*d^3)+40/3*c/((b^2-4*a*c)^2*d^3*(b+2*c*x)^2*sqrt(a+b*x+c*x^2))+80*c^2*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^3*d^3*(b+2*c*x)^2)],
[1/((b*d+2*c*d*x)^4*(a+b*x+c*x^2)^(5/2)),x,4,(-2/3)/((b^2-4*a*c)*d^4*(b+2*c*x)^3*(a+b*x+c*x^2)^(3/2))+16*c/((b^2-4*a*c)^2*d^4*(b+2*c*x)^3*sqrt(a+b*x+c*x^2))+256/3*c^2*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^3*d^4*(b+2*c*x)^3)+512/3*c^2*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^4*d^4*(b+2*c*x))],
[1/((a+b*x)*sqrt(1+a^2+2*a*b*x+b^2*x^2)),x,2,-arctanh(sqrt(1+a^2+2*a*b*x+b^2*x^2))/b],

# Integrands of the form (b d+2 c d x)^(m/2) (a+b x+c x^2)^p

# p>0
[(b*d+2*c*d*x)^(5/2)*(a+b*x+c*x^2),x,2,-1/28*(b^2-4*a*c)*(b*d+2*c*d*x)^(7/2)/(c^2*d)+1/44*(b*d+2*c*d*x)^(11/2)/(c^2*d^3)],
[(b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2),x,2,-1/20*(b^2-4*a*c)*(b*d+2*c*d*x)^(5/2)/(c^2*d)+1/36*(b*d+2*c*d*x)^(9/2)/(c^2*d^3)],
[(b*d+2*c*d*x)^(1/2)*(a+b*x+c*x^2),x,2,-1/12*(b^2-4*a*c)*(b*d+2*c*d*x)^(3/2)/(c^2*d)+1/28*(b*d+2*c*d*x)^(7/2)/(c^2*d^3)],
[(a+b*x+c*x^2)/(b*d+2*c*d*x)^(1/2),x,2,1/20*(b*d+2*c*d*x)^(5/2)/(c^2*d^3)-1/4*(b^2-4*a*c)*sqrt(b*d+2*c*d*x)/(c^2*d)],
[(a+b*x+c*x^2)/(b*d+2*c*d*x)^(3/2),x,2,1/12*(b*d+2*c*d*x)^(3/2)/(c^2*d^3)+1/4*(b^2-4*a*c)/(c^2*d*sqrt(b*d+2*c*d*x))],
[(a+b*x+c*x^2)/(b*d+2*c*d*x)^(5/2),x,2,1/12*(b^2-4*a*c)/(c^2*d*(b*d+2*c*d*x)^(3/2))+1/4*sqrt(b*d+2*c*d*x)/(c^2*d^3)],
[(a+b*x+c*x^2)/(b*d+2*c*d*x)^(7/2),x,2,1/20*(b^2-4*a*c)/(c^2*d*(b*d+2*c*d*x)^(5/2))+(-1/4)/(c^2*d^3*sqrt(b*d+2*c*d*x))],
[(a+b*x+c*x^2)/(b*d+2*c*d*x)^(9/2),x,2,1/28*(b^2-4*a*c)/(c^2*d*(b*d+2*c*d*x)^(7/2))+(-1/12)/(c^2*d^3*(b*d+2*c*d*x)^(3/2))],
[(b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2)^2,x,2,1/80*(b^2-4*a*c)^2*(b*d+2*c*d*x)^(5/2)/(c^3*d)-1/72*(b^2-4*a*c)*(b*d+2*c*d*x)^(9/2)/(c^3*d^3)+1/208*(b*d+2*c*d*x)^(13/2)/(c^3*d^5)],
[(b*d+2*c*d*x)^(1/2)*(a+b*x+c*x^2)^2,x,2,1/48*(b^2-4*a*c)^2*(b*d+2*c*d*x)^(3/2)/(c^3*d)-1/56*(b^2-4*a*c)*(b*d+2*c*d*x)^(7/2)/(c^3*d^3)+1/176*(b*d+2*c*d*x)^(11/2)/(c^3*d^5)],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x)^(1/2),x,2,-1/40*(b^2-4*a*c)*(b*d+2*c*d*x)^(5/2)/(c^3*d^3)+1/144*(b*d+2*c*d*x)^(9/2)/(c^3*d^5)+1/16*(b^2-4*a*c)^2*sqrt(b*d+2*c*d*x)/(c^3*d)],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x)^(3/2),x,2,-1/24*(b^2-4*a*c)*(b*d+2*c*d*x)^(3/2)/(c^3*d^3)+1/112*(b*d+2*c*d*x)^(7/2)/(c^3*d^5)-1/16*(b^2-4*a*c)^2/(c^3*d*sqrt(b*d+2*c*d*x))],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x)^(5/2),x,2,-1/48*(b^2-4*a*c)^2/(c^3*d*(b*d+2*c*d*x)^(3/2))+1/80*(b*d+2*c*d*x)^(5/2)/(c^3*d^5)-1/8*(b^2-4*a*c)*sqrt(b*d+2*c*d*x)/(c^3*d^3)],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x)^(7/2),x,2,-1/80*(b^2-4*a*c)^2/(c^3*d*(b*d+2*c*d*x)^(5/2))+1/48*(b*d+2*c*d*x)^(3/2)/(c^3*d^5)+1/8*(b^2-4*a*c)/(c^3*d^3*sqrt(b*d+2*c*d*x))],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x)^(9/2),x,2,-1/112*(b^2-4*a*c)^2/(c^3*d*(b*d+2*c*d*x)^(7/2))+1/24*(b^2-4*a*c)/(c^3*d^3*(b*d+2*c*d*x)^(3/2))+1/16*sqrt(b*d+2*c*d*x)/(c^3*d^5)],
[(a+b*x+c*x^2)^2/(b*d+2*c*d*x)^(11/2),x,2,-1/144*(b^2-4*a*c)^2/(c^3*d*(b*d+2*c*d*x)^(9/2))+1/40*(b^2-4*a*c)/(c^3*d^3*(b*d+2*c*d*x)^(5/2))+(-1/16)/(c^3*d^5*sqrt(b*d+2*c*d*x))],
[(b*d+2*c*d*x)^(1/2)*(a+b*x+c*x^2)^3,x,2,-1/192*(b^2-4*a*c)^3*(b*d+2*c*d*x)^(3/2)/(c^4*d)+3/448*(b^2-4*a*c)^2*(b*d+2*c*d*x)^(7/2)/(c^4*d^3)-3/704*(b^2-4*a*c)*(b*d+2*c*d*x)^(11/2)/(c^4*d^5)+1/960*(b*d+2*c*d*x)^(15/2)/(c^4*d^7)],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^(1/2),x,2,3/320*(b^2-4*a*c)^2*(b*d+2*c*d*x)^(5/2)/(c^4*d^3)-1/192*(b^2-4*a*c)*(b*d+2*c*d*x)^(9/2)/(c^4*d^5)+1/832*(b*d+2*c*d*x)^(13/2)/(c^4*d^7)-1/64*(b^2-4*a*c)^3*sqrt(b*d+2*c*d*x)/(c^4*d)],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^(3/2),x,2,1/64*(b^2-4*a*c)^2*(b*d+2*c*d*x)^(3/2)/(c^4*d^3)-3/448*(b^2-4*a*c)*(b*d+2*c*d*x)^(7/2)/(c^4*d^5)+1/704*(b*d+2*c*d*x)^(11/2)/(c^4*d^7)+1/64*(b^2-4*a*c)^3/(c^4*d*sqrt(b*d+2*c*d*x))],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^(5/2),x,2,1/192*(b^2-4*a*c)^3/(c^4*d*(b*d+2*c*d*x)^(3/2))-3/320*(b^2-4*a*c)*(b*d+2*c*d*x)^(5/2)/(c^4*d^5)+1/576*(b*d+2*c*d*x)^(9/2)/(c^4*d^7)+3/64*(b^2-4*a*c)^2*sqrt(b*d+2*c*d*x)/(c^4*d^3)],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^(7/2),x,2,1/320*(b^2-4*a*c)^3/(c^4*d*(b*d+2*c*d*x)^(5/2))-1/64*(b^2-4*a*c)*(b*d+2*c*d*x)^(3/2)/(c^4*d^5)+1/448*(b*d+2*c*d*x)^(7/2)/(c^4*d^7)-3/64*(b^2-4*a*c)^2/(c^4*d^3*sqrt(b*d+2*c*d*x))],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^(9/2),x,2,1/448*(b^2-4*a*c)^3/(c^4*d*(b*d+2*c*d*x)^(7/2))-1/64*(b^2-4*a*c)^2/(c^4*d^3*(b*d+2*c*d*x)^(3/2))+1/320*(b*d+2*c*d*x)^(5/2)/(c^4*d^7)-3/64*(b^2-4*a*c)*sqrt(b*d+2*c*d*x)/(c^4*d^5)],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^(11/2),x,2,1/576*(b^2-4*a*c)^3/(c^4*d*(b*d+2*c*d*x)^(9/2))-3/320*(b^2-4*a*c)^2/(c^4*d^3*(b*d+2*c*d*x)^(5/2))+1/192*(b*d+2*c*d*x)^(3/2)/(c^4*d^7)+3/64*(b^2-4*a*c)/(c^4*d^5*sqrt(b*d+2*c*d*x))],
[(a+b*x+c*x^2)^3/(b*d+2*c*d*x)^(13/2),x,2,1/704*(b^2-4*a*c)^3/(c^4*d*(b*d+2*c*d*x)^(11/2))-3/448*(b^2-4*a*c)^2/(c^4*d^3*(b*d+2*c*d*x)^(7/2))+1/64*(b^2-4*a*c)/(c^4*d^5*(b*d+2*c*d*x)^(3/2))+1/64*sqrt(b*d+2*c*d*x)/(c^4*d^7)],

# p<0
[(b*d+2*c*d*x)^(11/2)/(a+b*x+c*x^2),x,8,4/5*(b^2-4*a*c)*d^3*(b*d+2*c*d*x)^(5/2)+4/9*d*(b*d+2*c*d*x)^(9/2)-2*(b^2-4*a*c)^(9/4)*d^(11/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))-2*(b^2-4*a*c)^(9/4)*d^(11/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))+4*(b^2-4*a*c)^2*d^5*sqrt(b*d+2*c*d*x)],
[(b*d+2*c*d*x)^(9/2)/(a+b*x+c*x^2),x,7,4/3*(b^2-4*a*c)*d^3*(b*d+2*c*d*x)^(3/2)+4/7*d*(b*d+2*c*d*x)^(7/2)+2*(b^2-4*a*c)^(7/4)*d^(9/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))-2*(b^2-4*a*c)^(7/4)*d^(9/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))],
[(b*d+2*c*d*x)^(7/2)/(a+b*x+c*x^2),x,7,4/5*d*(b*d+2*c*d*x)^(5/2)-2*(b^2-4*a*c)^(5/4)*d^(7/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))-2*(b^2-4*a*c)^(5/4)*d^(7/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))+4*(b^2-4*a*c)*d^3*sqrt(b*d+2*c*d*x)],
[(b*d+2*c*d*x)^(5/2)/(a+b*x+c*x^2),x,6,4/3*d*(b*d+2*c*d*x)^(3/2)+2*(b^2-4*a*c)^(3/4)*d^(5/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))-2*(b^2-4*a*c)^(3/4)*d^(5/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))],
[(b*d+2*c*d*x)^(3/2)/(a+b*x+c*x^2),x,6,-2*(b^2-4*a*c)^(1/4)*d^(3/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))-2*(b^2-4*a*c)^(1/4)*d^(3/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))+4*d*sqrt(b*d+2*c*d*x)],
[(b*d+2*c*d*x)^(1/2)/(a+b*x+c*x^2),x,5,2*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))*sqrt(d)/(b^2-4*a*c)^(1/4)-2*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))*sqrt(d)/(b^2-4*a*c)^(1/4)],
[1/((b*d+2*c*d*x)^(1/2)*(a+b*x+c*x^2)),x,5,-2*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(3/4)*sqrt(d))-2*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(3/4)*sqrt(d))],
[1/((b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2)),x,6,2*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(5/4)*d^(3/2))-2*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(5/4)*d^(3/2))+4/((b^2-4*a*c)*d*sqrt(b*d+2*c*d*x))],
[1/((b*d+2*c*d*x)^(5/2)*(a+b*x+c*x^2)),x,6,4/3/((b^2-4*a*c)*d*(b*d+2*c*d*x)^(3/2))-2*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(7/4)*d^(5/2))-2*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(7/4)*d^(5/2))],
[1/((b*d+2*c*d*x)^(7/2)*(a+b*x+c*x^2)),x,7,4/5/((b^2-4*a*c)*d*(b*d+2*c*d*x)^(5/2))+2*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(9/4)*d^(7/2))-2*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(9/4)*d^(7/2))+4/((b^2-4*a*c)^2*d^3*sqrt(b*d+2*c*d*x))],
[(b*d+2*c*d*x)^(15/2)/(a+b*x+c*x^2)^2,x,9,52/5*c*(b^2-4*a*c)*d^5*(b*d+2*c*d*x)^(5/2)+52/9*c*d^3*(b*d+2*c*d*x)^(9/2)-d*(b*d+2*c*d*x)^(13/2)/(a+b*x+c*x^2)-26*c*(b^2-4*a*c)^(9/4)*d^(15/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))-26*c*(b^2-4*a*c)^(9/4)*d^(15/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))+52*c*(b^2-4*a*c)^2*d^7*sqrt(b*d+2*c*d*x)],
[(b*d+2*c*d*x)^(13/2)/(a+b*x+c*x^2)^2,x,8,44/3*c*(b^2-4*a*c)*d^5*(b*d+2*c*d*x)^(3/2)+44/7*c*d^3*(b*d+2*c*d*x)^(7/2)-d*(b*d+2*c*d*x)^(11/2)/(a+b*x+c*x^2)+22*c*(b^2-4*a*c)^(7/4)*d^(13/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))-22*c*(b^2-4*a*c)^(7/4)*d^(13/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))],
[(b*d+2*c*d*x)^(11/2)/(a+b*x+c*x^2)^2,x,8,36/5*c*d^3*(b*d+2*c*d*x)^(5/2)-d*(b*d+2*c*d*x)^(9/2)/(a+b*x+c*x^2)-18*c*(b^2-4*a*c)^(5/4)*d^(11/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))-18*c*(b^2-4*a*c)^(5/4)*d^(11/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))+36*c*(b^2-4*a*c)*d^5*sqrt(b*d+2*c*d*x)],
[(b*d+2*c*d*x)^(9/2)/(a+b*x+c*x^2)^2,x,7,28/3*c*d^3*(b*d+2*c*d*x)^(3/2)-d*(b*d+2*c*d*x)^(7/2)/(a+b*x+c*x^2)+14*c*(b^2-4*a*c)^(3/4)*d^(9/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))-14*c*(b^2-4*a*c)^(3/4)*d^(9/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))],
[(b*d+2*c*d*x)^(7/2)/(a+b*x+c*x^2)^2,x,7,-d*(b*d+2*c*d*x)^(5/2)/(a+b*x+c*x^2)-10*c*(b^2-4*a*c)^(1/4)*d^(7/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))-10*c*(b^2-4*a*c)^(1/4)*d^(7/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))+20*c*d^3*sqrt(b*d+2*c*d*x)],
[(b*d+2*c*d*x)^(5/2)/(a+b*x+c*x^2)^2,x,6,-d*(b*d+2*c*d*x)^(3/2)/(a+b*x+c*x^2)+6*c*d^(5/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/(b^2-4*a*c)^(1/4)-6*c*d^(5/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/(b^2-4*a*c)^(1/4)],
[(b*d+2*c*d*x)^(3/2)/(a+b*x+c*x^2)^2,x,6,-2*c*d^(3/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/(b^2-4*a*c)^(3/4)-2*c*d^(3/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/(b^2-4*a*c)^(3/4)-d*sqrt(b*d+2*c*d*x)/(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^(1/2)/(a+b*x+c*x^2)^2,x,6,-(b*d+2*c*d*x)^(3/2)/((b^2-4*a*c)*d*(a+b*x+c*x^2))-2*c*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))*sqrt(d)/(b^2-4*a*c)^(5/4)+2*c*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))*sqrt(d)/(b^2-4*a*c)^(5/4)],
[1/((b*d+2*c*d*x)^(1/2)*(a+b*x+c*x^2)^2),x,6,6*c*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(7/4)*sqrt(d))+6*c*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(7/4)*sqrt(d))-sqrt(b*d+2*c*d*x)/((b^2-4*a*c)*d*(a+b*x+c*x^2))],
[1/((b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2)^2),x,7,-10*c*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(9/4)*d^(3/2))+10*c*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(9/4)*d^(3/2))-20*c/((b^2-4*a*c)^2*d*sqrt(b*d+2*c*d*x))+(-1)/((b^2-4*a*c)*d*(a+b*x+c*x^2)*sqrt(b*d+2*c*d*x))],
[1/((b*d+2*c*d*x)^(5/2)*(a+b*x+c*x^2)^2),x,7,-28/3*c/((b^2-4*a*c)^2*d*(b*d+2*c*d*x)^(3/2))+(-1)/((b^2-4*a*c)*d*(b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2))+14*c*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(11/4)*d^(5/2))+14*c*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(11/4)*d^(5/2))],
[1/((b*d+2*c*d*x)^(7/2)*(a+b*x+c*x^2)^2),x,8,-36/5*c/((b^2-4*a*c)^2*d*(b*d+2*c*d*x)^(5/2))+(-1)/((b^2-4*a*c)*d*(b*d+2*c*d*x)^(5/2)*(a+b*x+c*x^2))-18*c*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(13/4)*d^(7/2))+18*c*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(13/4)*d^(7/2))-36*c/((b^2-4*a*c)^3*d^3*sqrt(b*d+2*c*d*x))],
[(b*d+2*c*d*x)^(17/2)/(a+b*x+c*x^2)^3,x,9,110*c^2*(b^2-4*a*c)*d^7*(b*d+2*c*d*x)^(3/2)+330/7*c^2*d^5*(b*d+2*c*d*x)^(7/2)-1/2*d*(b*d+2*c*d*x)^(15/2)/(a+b*x+c*x^2)^2-15/2*c*d^3*(b*d+2*c*d*x)^(11/2)/(a+b*x+c*x^2)+165*c^2*(b^2-4*a*c)^(7/4)*d^(17/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))-165*c^2*(b^2-4*a*c)^(7/4)*d^(17/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))],
[(b*d+2*c*d*x)^(15/2)/(a+b*x+c*x^2)^3,x,9,234/5*c^2*d^5*(b*d+2*c*d*x)^(5/2)-1/2*d*(b*d+2*c*d*x)^(13/2)/(a+b*x+c*x^2)^2-13/2*c*d^3*(b*d+2*c*d*x)^(9/2)/(a+b*x+c*x^2)-117*c^2*(b^2-4*a*c)^(5/4)*d^(15/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))-117*c^2*(b^2-4*a*c)^(5/4)*d^(15/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))+234*c^2*(b^2-4*a*c)*d^7*sqrt(b*d+2*c*d*x)],
[(b*d+2*c*d*x)^(13/2)/(a+b*x+c*x^2)^3,x,8,154/3*c^2*d^5*(b*d+2*c*d*x)^(3/2)-1/2*d*(b*d+2*c*d*x)^(11/2)/(a+b*x+c*x^2)^2-11/2*c*d^3*(b*d+2*c*d*x)^(7/2)/(a+b*x+c*x^2)+77*c^2*(b^2-4*a*c)^(3/4)*d^(13/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))-77*c^2*(b^2-4*a*c)^(3/4)*d^(13/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))],
[(b*d+2*c*d*x)^(11/2)/(a+b*x+c*x^2)^3,x,8,-1/2*d*(b*d+2*c*d*x)^(9/2)/(a+b*x+c*x^2)^2-9/2*c*d^3*(b*d+2*c*d*x)^(5/2)/(a+b*x+c*x^2)-45*c^2*(b^2-4*a*c)^(1/4)*d^(11/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))-45*c^2*(b^2-4*a*c)^(1/4)*d^(11/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))+90*c^2*d^5*sqrt(b*d+2*c*d*x)],
[(b*d+2*c*d*x)^(9/2)/(a+b*x+c*x^2)^3,x,7,-1/2*d*(b*d+2*c*d*x)^(7/2)/(a+b*x+c*x^2)^2-7/2*c*d^3*(b*d+2*c*d*x)^(3/2)/(a+b*x+c*x^2)+21*c^2*d^(9/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/(b^2-4*a*c)^(1/4)-21*c^2*d^(9/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/(b^2-4*a*c)^(1/4)],
[(b*d+2*c*d*x)^(7/2)/(a+b*x+c*x^2)^3,x,7,-1/2*d*(b*d+2*c*d*x)^(5/2)/(a+b*x+c*x^2)^2-5*c^2*d^(7/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/(b^2-4*a*c)^(3/4)-5*c^2*d^(7/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/(b^2-4*a*c)^(3/4)-5/2*c*d^3*sqrt(b*d+2*c*d*x)/(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^(5/2)/(a+b*x+c*x^2)^3,x,7,-1/2*d*(b*d+2*c*d*x)^(3/2)/(a+b*x+c*x^2)^2-3/2*c*d*(b*d+2*c*d*x)^(3/2)/((b^2-4*a*c)*(a+b*x+c*x^2))-3*c^2*d^(5/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/(b^2-4*a*c)^(5/4)+3*c^2*d^(5/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/(b^2-4*a*c)^(5/4)],
[(b*d+2*c*d*x)^(3/2)/(a+b*x+c*x^2)^3,x,7,3*c^2*d^(3/2)*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/(b^2-4*a*c)^(7/4)+3*c^2*d^(3/2)*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/(b^2-4*a*c)^(7/4)-1/2*d*sqrt(b*d+2*c*d*x)/(a+b*x+c*x^2)^2-1/2*c*d*sqrt(b*d+2*c*d*x)/((b^2-4*a*c)*(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(1/2)/(a+b*x+c*x^2)^3,x,7,-1/2*(b*d+2*c*d*x)^(3/2)/((b^2-4*a*c)*d*(a+b*x+c*x^2)^2)+5/2*c*(b*d+2*c*d*x)^(3/2)/((b^2-4*a*c)^2*d*(a+b*x+c*x^2))+5*c^2*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))*sqrt(d)/(b^2-4*a*c)^(9/4)-5*c^2*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))*sqrt(d)/(b^2-4*a*c)^(9/4)],
[1/((b*d+2*c*d*x)^(1/2)*(a+b*x+c*x^2)^3),x,7,-21*c^2*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(11/4)*sqrt(d))-21*c^2*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(11/4)*sqrt(d))-1/2*sqrt(b*d+2*c*d*x)/((b^2-4*a*c)*d*(a+b*x+c*x^2)^2)+7/2*c*sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^2*d*(a+b*x+c*x^2))],
[1/((b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2)^3),x,8,45*c^2*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(13/4)*d^(3/2))-45*c^2*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(13/4)*d^(3/2))+90*c^2/((b^2-4*a*c)^3*d*sqrt(b*d+2*c*d*x))+(-1/2)/((b^2-4*a*c)*d*(a+b*x+c*x^2)^2*sqrt(b*d+2*c*d*x))+9/2*c/((b^2-4*a*c)^2*d*(a+b*x+c*x^2)*sqrt(b*d+2*c*d*x))],
[1/((b*d+2*c*d*x)^(5/2)*(a+b*x+c*x^2)^3),x,8,154/3*c^2/((b^2-4*a*c)^3*d*(b*d+2*c*d*x)^(3/2))+(-1/2)/((b^2-4*a*c)*d*(b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2)^2)+11/2*c/((b^2-4*a*c)^2*d*(b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2))-77*c^2*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(15/4)*d^(5/2))-77*c^2*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(15/4)*d^(5/2))],
[1/((b*d+2*c*d*x)^(7/2)*(a+b*x+c*x^2)^3),x,9,234/5*c^2/((b^2-4*a*c)^3*d*(b*d+2*c*d*x)^(5/2))+(-1/2)/((b^2-4*a*c)*d*(b*d+2*c*d*x)^(5/2)*(a+b*x+c*x^2)^2)+13/2*c/((b^2-4*a*c)^2*d*(b*d+2*c*d*x)^(5/2)*(a+b*x+c*x^2))+117*c^2*arctan(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(17/4)*d^(7/2))-117*c^2*arctanh(sqrt(d*(b+2*c*x))/((b^2-4*a*c)^(1/4)*sqrt(d)))/((b^2-4*a*c)^(17/4)*d^(7/2))+234*c^2/((b^2-4*a*c)^4*d^3*sqrt(b*d+2*c*d*x))],
[(1+2*x)^(7/2)/(1+x+x^2),x,13,4/5*(1+2*x)^(5/2)-3*3^(1/4)*log(1+2*x+sqrt(3)-3^(1/4)*sqrt(2)*sqrt(1+2*x))/sqrt(2)+3*3^(1/4)*log(1+2*x+sqrt(3)+3^(1/4)*sqrt(2)*sqrt(1+2*x))/sqrt(2)-3*3^(1/4)*arctan(1-sqrt(2)*sqrt(1+2*x)/3^(1/4))*sqrt(2)+3*3^(1/4)*arctan(1+sqrt(2)*sqrt(1+2*x)/3^(1/4))*sqrt(2)-12*sqrt(1+2*x)],
[(1+2*x)^(5/2)/(1+x+x^2),x,12,4/3*(1+2*x)^(3/2)-3^(3/4)*log(1+2*x+sqrt(3)-3^(1/4)*sqrt(2)*sqrt(1+2*x))/sqrt(2)+3^(3/4)*log(1+2*x+sqrt(3)+3^(1/4)*sqrt(2)*sqrt(1+2*x))/sqrt(2)+3^(3/4)*arctan(1-sqrt(2)*sqrt(1+2*x)/3^(1/4))*sqrt(2)-3^(3/4)*arctan(1+sqrt(2)*sqrt(1+2*x)/3^(1/4))*sqrt(2)],
[(1+2*x)^(3/2)/(1+x+x^2),x,12,3^(1/4)*log(1+2*x+sqrt(3)-3^(1/4)*sqrt(2)*sqrt(1+2*x))/sqrt(2)-3^(1/4)*log(1+2*x+sqrt(3)+3^(1/4)*sqrt(2)*sqrt(1+2*x))/sqrt(2)+3^(1/4)*arctan(1-sqrt(2)*sqrt(1+2*x)/3^(1/4))*sqrt(2)-3^(1/4)*arctan(1+sqrt(2)*sqrt(1+2*x)/3^(1/4))*sqrt(2)+4*sqrt(1+2*x)],
[(1+2*x)^(1/2)/(1+x+x^2),x,11,log(1+2*x+sqrt(3)-3^(1/4)*sqrt(2)*sqrt(1+2*x))/(3^(1/4)*sqrt(2))-log(1+2*x+sqrt(3)+3^(1/4)*sqrt(2)*sqrt(1+2*x))/(3^(1/4)*sqrt(2))-arctan(1-sqrt(2)*sqrt(1+2*x)/3^(1/4))*sqrt(2)/3^(1/4)+arctan(1+sqrt(2)*sqrt(1+2*x)/3^(1/4))*sqrt(2)/3^(1/4)],
[1/((1+2*x)^(1/2)*(1+x+x^2)),x,11,-log(1+2*x+sqrt(3)-3^(1/4)*sqrt(2)*sqrt(1+2*x))/(3^(3/4)*sqrt(2))+log(1+2*x+sqrt(3)+3^(1/4)*sqrt(2)*sqrt(1+2*x))/(3^(3/4)*sqrt(2))-arctan(1-sqrt(2)*sqrt(1+2*x)/3^(1/4))*sqrt(2)/3^(3/4)+arctan(1+sqrt(2)*sqrt(1+2*x)/3^(1/4))*sqrt(2)/3^(3/4)],
[1/((1+2*x)^(3/2)*(1+x+x^2)),x,12,-1/3*log(1+2*x+sqrt(3)-3^(1/4)*sqrt(2)*sqrt(1+2*x))/(3^(1/4)*sqrt(2))+1/3*log(1+2*x+sqrt(3)+3^(1/4)*sqrt(2)*sqrt(1+2*x))/(3^(1/4)*sqrt(2))+1/3*arctan(1-sqrt(2)*sqrt(1+2*x)/3^(1/4))*sqrt(2)/3^(1/4)-1/3*arctan(1+sqrt(2)*sqrt(1+2*x)/3^(1/4))*sqrt(2)/3^(1/4)+(-4/3)/sqrt(1+2*x)],
[1/((1+2*x)^(5/2)*(1+x+x^2)),x,12,(-4/9)/(1+2*x)^(3/2)+1/3*log(1+2*x+sqrt(3)-3^(1/4)*sqrt(2)*sqrt(1+2*x))/(3^(3/4)*sqrt(2))-1/3*log(1+2*x+sqrt(3)+3^(1/4)*sqrt(2)*sqrt(1+2*x))/(3^(3/4)*sqrt(2))+1/3*arctan(1-sqrt(2)*sqrt(1+2*x)/3^(1/4))*sqrt(2)/3^(3/4)-1/3*arctan(1+sqrt(2)*sqrt(1+2*x)/3^(1/4))*sqrt(2)/3^(3/4)],

# Integrands of the form (b d+2 c d x)^(m/2) (a+b x+c x^2)^(p/2)

# p>0
[(b*d+2*c*d*x)^(7/2)*(a+b*x+c*x^2)^(1/2),x,6,-2/77*(b^2-4*a*c)*d*(b*d+2*c*d*x)^(5/2)*sqrt(a+b*x+c*x^2)/c+1/11*(b*d+2*c*d*x)^(9/2)*sqrt(a+b*x+c*x^2)/(c*d)-10/231*(b^2-4*a*c)^2*d^3*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)/c-5/231*(b^2-4*a*c)^(13/4)*d^(7/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2)^(1/2),x,5,1/7*(b*d+2*c*d*x)^(5/2)*sqrt(a+b*x+c*x^2)/(c*d)-2/21*(b^2-4*a*c)*d*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)/c-1/21*(b^2-4*a*c)^(9/4)*d^(3/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(1/2)/(b*d+2*c*d*x)^(1/2),x,4,1/3*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)/(c*d)-1/3*(b^2-4*a*c)^(5/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*sqrt(d)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(1/2)/(b*d+2*c*d*x)^(5/2),x,4,-1/3*sqrt(a+b*x+c*x^2)/(c*d*(b*d+2*c*d*x)^(3/2))+1/3*(b^2-4*a*c)^(1/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*d^(5/2)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(1/2)/(b*d+2*c*d*x)^(9/2),x,5,-1/7*sqrt(a+b*x+c*x^2)/(c*d*(b*d+2*c*d*x)^(7/2))+2/21*sqrt(a+b*x+c*x^2)/(c*(b^2-4*a*c)*d^3*(b*d+2*c*d*x)^(3/2))+1/21*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*(b^2-4*a*c)^(3/4)*d^(9/2)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(1/2)/(b*d+2*c*d*x)^(13/2),x,6,-1/11*sqrt(a+b*x+c*x^2)/(c*d*(b*d+2*c*d*x)^(11/2))+2/77*sqrt(a+b*x+c*x^2)/(c*(b^2-4*a*c)*d^3*(b*d+2*c*d*x)^(7/2))+10/231*sqrt(a+b*x+c*x^2)/(c*(b^2-4*a*c)^2*d^5*(b*d+2*c*d*x)^(3/2))+5/231*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*(b^2-4*a*c)^(7/4)*d^(13/2)*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(5/2)*(a+b*x+c*x^2)^(1/2),x,8,-2/45*(b^2-4*a*c)*d*(b*d+2*c*d*x)^(3/2)*sqrt(a+b*x+c*x^2)/c+1/9*(b*d+2*c*d*x)^(7/2)*sqrt(a+b*x+c*x^2)/(c*d)-1/15*(b^2-4*a*c)^(11/4)*d^(5/2)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*sqrt(a+b*x+c*x^2))+1/15*(b^2-4*a*c)^(11/4)*d^(5/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(1/2)*(a+b*x+c*x^2)^(1/2),x,7,1/5*(b*d+2*c*d*x)^(3/2)*sqrt(a+b*x+c*x^2)/(c*d)-1/5*(b^2-4*a*c)^(7/4)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(d)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*sqrt(a+b*x+c*x^2))+1/5*(b^2-4*a*c)^(7/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(d)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(1/2)/(b*d+2*c*d*x)^(3/2),x,7,-sqrt(a+b*x+c*x^2)/(c*d*sqrt(b*d+2*c*d*x))+(b^2-4*a*c)^(3/4)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*d^(3/2)*sqrt(a+b*x+c*x^2))-(b^2-4*a*c)^(3/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*d^(3/2)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(1/2)/(b*d+2*c*d*x)^(7/2),x,8,-1/5*sqrt(a+b*x+c*x^2)/(c*d*(b*d+2*c*d*x)^(5/2))+2/5*sqrt(a+b*x+c*x^2)/(c*(b^2-4*a*c)*d^3*sqrt(b*d+2*c*d*x))-1/5*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*(b^2-4*a*c)^(1/4)*d^(7/2)*sqrt(a+b*x+c*x^2))+1/5*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*(b^2-4*a*c)^(1/4)*d^(7/2)*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(7/2)*(a+b*x+c*x^2)^(3/2),x,7,1/15*(b*d+2*c*d*x)^(9/2)*(a+b*x+c*x^2)^(3/2)/(c*d)+1/385*(b^2-4*a*c)^2*d*(b*d+2*c*d*x)^(5/2)*sqrt(a+b*x+c*x^2)/c^2-1/110*(b^2-4*a*c)*(b*d+2*c*d*x)^(9/2)*sqrt(a+b*x+c*x^2)/(c^2*d)+1/231*(b^2-4*a*c)^3*d^3*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)/c^2+1/462*(b^2-4*a*c)^(17/4)*d^(7/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2)^(3/2),x,6,1/11*(b*d+2*c*d*x)^(5/2)*(a+b*x+c*x^2)^(3/2)/(c*d)-3/154*(b^2-4*a*c)*(b*d+2*c*d*x)^(5/2)*sqrt(a+b*x+c*x^2)/(c^2*d)+1/77*(b^2-4*a*c)^2*d*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)/c^2+1/154*(b^2-4*a*c)^(13/4)*d^(3/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^(1/2),x,5,1/7*(a+b*x+c*x^2)^(3/2)*sqrt(b*d+2*c*d*x)/(c*d)-1/14*(b^2-4*a*c)*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)/(c^2*d)+1/14*(b^2-4*a*c)^(9/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*sqrt(d)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^(5/2),x,5,-1/3*(a+b*x+c*x^2)^(3/2)/(c*d*(b*d+2*c*d*x)^(3/2))+1/6*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)/(c^2*d^3)-1/6*(b^2-4*a*c)^(5/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*d^(5/2)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^(9/2),x,5,-1/7*(a+b*x+c*x^2)^(3/2)/(c*d*(b*d+2*c*d*x)^(7/2))-1/14*sqrt(a+b*x+c*x^2)/(c^2*d^3*(b*d+2*c*d*x)^(3/2))+1/14*(b^2-4*a*c)^(1/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*d^(9/2)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^(13/2),x,6,-1/11*(a+b*x+c*x^2)^(3/2)/(c*d*(b*d+2*c*d*x)^(11/2))-3/154*sqrt(a+b*x+c*x^2)/(c^2*d^3*(b*d+2*c*d*x)^(7/2))+1/77*sqrt(a+b*x+c*x^2)/(c^2*(b^2-4*a*c)*d^5*(b*d+2*c*d*x)^(3/2))+1/154*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*(b^2-4*a*c)^(3/4)*d^(13/2)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^(17/2),x,7,-1/15*(a+b*x+c*x^2)^(3/2)/(c*d*(b*d+2*c*d*x)^(15/2))-1/110*sqrt(a+b*x+c*x^2)/(c^2*d^3*(b*d+2*c*d*x)^(11/2))+1/385*sqrt(a+b*x+c*x^2)/(c^2*(b^2-4*a*c)*d^5*(b*d+2*c*d*x)^(7/2))+1/231*sqrt(a+b*x+c*x^2)/(c^2*(b^2-4*a*c)^2*d^7*(b*d+2*c*d*x)^(3/2))+1/462*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*(b^2-4*a*c)^(7/4)*d^(17/2)*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(5/2)*(a+b*x+c*x^2)^(3/2),x,9,1/13*(b*d+2*c*d*x)^(7/2)*(a+b*x+c*x^2)^(3/2)/(c*d)+1/195*(b^2-4*a*c)^2*d*(b*d+2*c*d*x)^(3/2)*sqrt(a+b*x+c*x^2)/c^2-1/78*(b^2-4*a*c)*(b*d+2*c*d*x)^(7/2)*sqrt(a+b*x+c*x^2)/(c^2*d)+1/130*(b^2-4*a*c)^(15/4)*d^(5/2)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*sqrt(a+b*x+c*x^2))-1/130*(b^2-4*a*c)^(15/4)*d^(5/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(1/2)*(a+b*x+c*x^2)^(3/2),x,8,1/9*(b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2)^(3/2)/(c*d)-1/30*(b^2-4*a*c)*(b*d+2*c*d*x)^(3/2)*sqrt(a+b*x+c*x^2)/(c^2*d)+1/30*(b^2-4*a*c)^(11/4)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(d)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*sqrt(a+b*x+c*x^2))-1/30*(b^2-4*a*c)^(11/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(d)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^(3/2),x,8,-(a+b*x+c*x^2)^(3/2)/(c*d*sqrt(b*d+2*c*d*x))+3/10*(b*d+2*c*d*x)^(3/2)*sqrt(a+b*x+c*x^2)/(c^2*d^3)-3/10*(b^2-4*a*c)^(7/4)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*d^(3/2)*sqrt(a+b*x+c*x^2))+3/10*(b^2-4*a*c)^(7/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*d^(3/2)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^(7/2),x,8,-1/5*(a+b*x+c*x^2)^(3/2)/(c*d*(b*d+2*c*d*x)^(5/2))-3/10*sqrt(a+b*x+c*x^2)/(c^2*d^3*sqrt(b*d+2*c*d*x))+3/10*(b^2-4*a*c)^(3/4)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*d^(7/2)*sqrt(a+b*x+c*x^2))-3/10*(b^2-4*a*c)^(3/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*d^(7/2)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(3/2)/(b*d+2*c*d*x)^(11/2),x,9,-1/9*(a+b*x+c*x^2)^(3/2)/(c*d*(b*d+2*c*d*x)^(9/2))-1/30*sqrt(a+b*x+c*x^2)/(c^2*d^3*(b*d+2*c*d*x)^(5/2))+1/15*sqrt(a+b*x+c*x^2)/(c^2*(b^2-4*a*c)*d^5*sqrt(b*d+2*c*d*x))-1/30*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*(b^2-4*a*c)^(1/4)*d^(11/2)*sqrt(a+b*x+c*x^2))+1/30*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*(b^2-4*a*c)^(1/4)*d^(11/2)*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(7/2)*(a+b*x+c*x^2)^(5/2),x,8,-1/114*(b^2-4*a*c)*(b*d+2*c*d*x)^(9/2)*(a+b*x+c*x^2)^(3/2)/(c^2*d)+1/19*(b*d+2*c*d*x)^(9/2)*(a+b*x+c*x^2)^(5/2)/(c*d)-1/2926*(b^2-4*a*c)^3*d*(b*d+2*c*d*x)^(5/2)*sqrt(a+b*x+c*x^2)/c^3+1/836*(b^2-4*a*c)^2*(b*d+2*c*d*x)^(9/2)*sqrt(a+b*x+c*x^2)/(c^3*d)-5/8778*(b^2-4*a*c)^4*d^3*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)/c^3-5/17556*(b^2-4*a*c)^(21/4)*d^(7/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2)^(5/2),x,7,-1/66*(b^2-4*a*c)*(b*d+2*c*d*x)^(5/2)*(a+b*x+c*x^2)^(3/2)/(c^2*d)+1/15*(b*d+2*c*d*x)^(5/2)*(a+b*x+c*x^2)^(5/2)/(c*d)+1/308*(b^2-4*a*c)^2*(b*d+2*c*d*x)^(5/2)*sqrt(a+b*x+c*x^2)/(c^3*d)-1/462*(b^2-4*a*c)^3*d*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)/c^3-1/924*(b^2-4*a*c)^(17/4)*d^(3/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^(1/2),x,6,-5/154*(b^2-4*a*c)*(a+b*x+c*x^2)^(3/2)*sqrt(b*d+2*c*d*x)/(c^2*d)+1/11*(a+b*x+c*x^2)^(5/2)*sqrt(b*d+2*c*d*x)/(c*d)+5/308*(b^2-4*a*c)^2*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)/(c^3*d)-5/308*(b^2-4*a*c)^(13/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*sqrt(d)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^(5/2),x,6,-1/3*(a+b*x+c*x^2)^(5/2)/(c*d*(b*d+2*c*d*x)^(3/2))+5/42*(a+b*x+c*x^2)^(3/2)*sqrt(b*d+2*c*d*x)/(c^2*d^3)-5/84*(b^2-4*a*c)*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)/(c^3*d^3)+5/84*(b^2-4*a*c)^(9/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*d^(5/2)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^(9/2),x,6,-5/42*(a+b*x+c*x^2)^(3/2)/(c^2*d^3*(b*d+2*c*d*x)^(3/2))-1/7*(a+b*x+c*x^2)^(5/2)/(c*d*(b*d+2*c*d*x)^(7/2))+5/84*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)/(c^3*d^5)-5/84*(b^2-4*a*c)^(5/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*d^(9/2)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^(13/2),x,6,-5/154*(a+b*x+c*x^2)^(3/2)/(c^2*d^3*(b*d+2*c*d*x)^(7/2))-1/11*(a+b*x+c*x^2)^(5/2)/(c*d*(b*d+2*c*d*x)^(11/2))-5/308*sqrt(a+b*x+c*x^2)/(c^3*d^5*(b*d+2*c*d*x)^(3/2))+5/308*(b^2-4*a*c)^(1/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*d^(13/2)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^(17/2),x,7,-1/66*(a+b*x+c*x^2)^(3/2)/(c^2*d^3*(b*d+2*c*d*x)^(11/2))-1/15*(a+b*x+c*x^2)^(5/2)/(c*d*(b*d+2*c*d*x)^(15/2))-1/308*sqrt(a+b*x+c*x^2)/(c^3*d^5*(b*d+2*c*d*x)^(7/2))+1/462*sqrt(a+b*x+c*x^2)/(c^3*(b^2-4*a*c)*d^7*(b*d+2*c*d*x)^(3/2))+1/924*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*(b^2-4*a*c)^(3/4)*d^(17/2)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^(21/2),x,8,-1/114*(a+b*x+c*x^2)^(3/2)/(c^2*d^3*(b*d+2*c*d*x)^(15/2))-1/19*(a+b*x+c*x^2)^(5/2)/(c*d*(b*d+2*c*d*x)^(19/2))-1/836*sqrt(a+b*x+c*x^2)/(c^3*d^5*(b*d+2*c*d*x)^(11/2))+1/2926*sqrt(a+b*x+c*x^2)/(c^3*(b^2-4*a*c)*d^7*(b*d+2*c*d*x)^(7/2))+5/8778*sqrt(a+b*x+c*x^2)/(c^3*(b^2-4*a*c)^2*d^9*(b*d+2*c*d*x)^(3/2))+5/17556*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*(b^2-4*a*c)^(7/4)*d^(21/2)*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(5/2)*(a+b*x+c*x^2)^(5/2),x,10,-5/442*(b^2-4*a*c)*(b*d+2*c*d*x)^(7/2)*(a+b*x+c*x^2)^(3/2)/(c^2*d)+1/17*(b*d+2*c*d*x)^(7/2)*(a+b*x+c*x^2)^(5/2)/(c*d)-1/1326*(b^2-4*a*c)^3*d*(b*d+2*c*d*x)^(3/2)*sqrt(a+b*x+c*x^2)/c^3+5/2652*(b^2-4*a*c)^2*(b*d+2*c*d*x)^(7/2)*sqrt(a+b*x+c*x^2)/(c^3*d)-1/884*(b^2-4*a*c)^(19/4)*d^(5/2)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*sqrt(a+b*x+c*x^2))+1/884*(b^2-4*a*c)^(19/4)*d^(5/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(1/2)*(a+b*x+c*x^2)^(5/2),x,9,-5/234*(b^2-4*a*c)*(b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2)^(3/2)/(c^2*d)+1/13*(b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2)^(5/2)/(c*d)+1/156*(b^2-4*a*c)^2*(b*d+2*c*d*x)^(3/2)*sqrt(a+b*x+c*x^2)/(c^3*d)-1/156*(b^2-4*a*c)^(15/4)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(d)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*sqrt(a+b*x+c*x^2))+1/156*(b^2-4*a*c)^(15/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(d)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^(3/2),x,9,5/18*(b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2)^(3/2)/(c^2*d^3)-(a+b*x+c*x^2)^(5/2)/(c*d*sqrt(b*d+2*c*d*x))-1/12*(b^2-4*a*c)*(b*d+2*c*d*x)^(3/2)*sqrt(a+b*x+c*x^2)/(c^3*d^3)+1/12*(b^2-4*a*c)^(11/4)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*d^(3/2)*sqrt(a+b*x+c*x^2))-1/12*(b^2-4*a*c)^(11/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*d^(3/2)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^(7/2),x,9,-1/5*(a+b*x+c*x^2)^(5/2)/(c*d*(b*d+2*c*d*x)^(5/2))-1/2*(a+b*x+c*x^2)^(3/2)/(c^2*d^3*sqrt(b*d+2*c*d*x))+3/20*(b*d+2*c*d*x)^(3/2)*sqrt(a+b*x+c*x^2)/(c^3*d^5)-3/20*(b^2-4*a*c)^(7/4)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*d^(7/2)*sqrt(a+b*x+c*x^2))+3/20*(b^2-4*a*c)^(7/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*d^(7/2)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^(11/2),x,9,-1/18*(a+b*x+c*x^2)^(3/2)/(c^2*d^3*(b*d+2*c*d*x)^(5/2))-1/9*(a+b*x+c*x^2)^(5/2)/(c*d*(b*d+2*c*d*x)^(9/2))-1/12*sqrt(a+b*x+c*x^2)/(c^3*d^5*sqrt(b*d+2*c*d*x))+1/12*(b^2-4*a*c)^(3/4)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*d^(11/2)*sqrt(a+b*x+c*x^2))-1/12*(b^2-4*a*c)^(3/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*d^(11/2)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(5/2)/(b*d+2*c*d*x)^(15/2),x,10,-5/234*(a+b*x+c*x^2)^(3/2)/(c^2*d^3*(b*d+2*c*d*x)^(9/2))-1/13*(a+b*x+c*x^2)^(5/2)/(c*d*(b*d+2*c*d*x)^(13/2))-1/156*sqrt(a+b*x+c*x^2)/(c^3*d^5*(b*d+2*c*d*x)^(5/2))+1/78*sqrt(a+b*x+c*x^2)/(c^3*(b^2-4*a*c)*d^7*sqrt(b*d+2*c*d*x))-1/156*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*(b^2-4*a*c)^(1/4)*d^(15/2)*sqrt(a+b*x+c*x^2))+1/156*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*(b^2-4*a*c)^(1/4)*d^(15/2)*sqrt(a+b*x+c*x^2))],

# p<0
[(b*d+2*c*d*x)^(7/2)/(a+b*x+c*x^2)^(1/2),x,5,4/7*d*(b*d+2*c*d*x)^(5/2)*sqrt(a+b*x+c*x^2)+20/21*(b^2-4*a*c)*d^3*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)+10/21*(b^2-4*a*c)^(9/4)*d^(7/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(3/2)/(a+b*x+c*x^2)^(1/2),x,4,4/3*d*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)+2/3*(b^2-4*a*c)^(5/4)*d^(3/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*sqrt(a+b*x+c*x^2))],
[1/((b*d+2*c*d*x)^(1/2)*(a+b*x+c*x^2)^(1/2)),x,3,2*(b^2-4*a*c)^(1/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*sqrt(d)*sqrt(a+b*x+c*x^2))],
[1/((b*d+2*c*d*x)^(5/2)*(a+b*x+c*x^2)^(1/2)),x,4,4/3*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*d*(b*d+2*c*d*x)^(3/2))+2/3*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*(b^2-4*a*c)^(3/4)*d^(5/2)*sqrt(a+b*x+c*x^2))],
[1/((b*d+2*c*d*x)^(9/2)*(a+b*x+c*x^2)^(1/2)),x,5,4/7*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*d*(b*d+2*c*d*x)^(7/2))+20/21*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^2*d^3*(b*d+2*c*d*x)^(3/2))+10/21*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*(b^2-4*a*c)^(7/4)*d^(9/2)*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(9/2)/(a+b*x+c*x^2)^(1/2),x,8,28/45*(b^2-4*a*c)*d^3*(b*d+2*c*d*x)^(3/2)*sqrt(a+b*x+c*x^2)+4/9*d*(b*d+2*c*d*x)^(7/2)*sqrt(a+b*x+c*x^2)+14/15*(b^2-4*a*c)^(11/4)*d^(9/2)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*sqrt(a+b*x+c*x^2))-14/15*(b^2-4*a*c)^(11/4)*d^(9/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(5/2)/(a+b*x+c*x^2)^(1/2),x,7,4/5*d*(b*d+2*c*d*x)^(3/2)*sqrt(a+b*x+c*x^2)+6/5*(b^2-4*a*c)^(7/4)*d^(5/2)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*sqrt(a+b*x+c*x^2))-6/5*(b^2-4*a*c)^(7/4)*d^(5/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(1/2)/(a+b*x+c*x^2)^(1/2),x,6,2*(b^2-4*a*c)^(3/4)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(d)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*sqrt(a+b*x+c*x^2))-2*(b^2-4*a*c)^(3/4)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(d)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*sqrt(a+b*x+c*x^2))],
[1/((b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2)^(1/2)),x,7,4*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*d*sqrt(b*d+2*c*d*x))-2*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*(b^2-4*a*c)^(1/4)*d^(3/2)*sqrt(a+b*x+c*x^2))+2*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*(b^2-4*a*c)^(1/4)*d^(3/2)*sqrt(a+b*x+c*x^2))],
[1/((b*d+2*c*d*x)^(7/2)*(a+b*x+c*x^2)^(1/2)),x,8,4/5*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*d*(b*d+2*c*d*x)^(5/2))+12/5*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^2*d^3*sqrt(b*d+2*c*d*x))-6/5*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*(b^2-4*a*c)^(5/4)*d^(7/2)*sqrt(a+b*x+c*x^2))+6/5*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*(b^2-4*a*c)^(5/4)*d^(7/2)*sqrt(a+b*x+c*x^2))],
[(3-2*x)^(3/2)/sqrt(1-3*x+x^2),x,4,-2/3*5^(3/4)*EllipticF(sqrt(3-2*x)/5^(1/4),I)*sqrt(-1+3*x-x^2)/sqrt(1-3*x+x^2)-4/3*sqrt(3-2*x)*sqrt(1-3*x+x^2)],
[1/((3-2*x)^(1/2)*sqrt(1-3*x+x^2)),x,3,-2*EllipticF(sqrt(3-2*x)/5^(1/4),I)*sqrt(-1+3*x-x^2)/(5^(1/4)*sqrt(1-3*x+x^2))],
[1/((3-2*x)^(5/2)*sqrt(1-3*x+x^2)),x,4,-2/15*EllipticF(sqrt(3-2*x)/5^(1/4),I)*sqrt(-1+3*x-x^2)/(5^(1/4)*sqrt(1-3*x+x^2))-4/15*sqrt(1-3*x+x^2)/(3-2*x)^(3/2)],
[(3-2*x)^(5/2)/sqrt(1-3*x+x^2),x,8,-6*5^(1/4)*EllipticE(sqrt(3-2*x)/5^(1/4),I)*sqrt(-1+3*x-x^2)/sqrt(1-3*x+x^2)+6*5^(1/4)*EllipticF(sqrt(3-2*x)/5^(1/4),I)*sqrt(-1+3*x-x^2)/sqrt(1-3*x+x^2)-4/5*(3-2*x)^(3/2)*sqrt(1-3*x+x^2)],
[(3-2*x)^(1/2)/sqrt(1-3*x+x^2),x,7,-2*5^(1/4)*EllipticE(sqrt(3-2*x)/5^(1/4),I)*sqrt(-1+3*x-x^2)/sqrt(1-3*x+x^2)+2*5^(1/4)*EllipticF(sqrt(3-2*x)/5^(1/4),I)*sqrt(-1+3*x-x^2)/sqrt(1-3*x+x^2)],
[1/((3-2*x)^(3/2)*sqrt(1-3*x+x^2)),x,8,2*EllipticE(sqrt(3-2*x)/5^(1/4),I)*sqrt(-1+3*x-x^2)/(5^(3/4)*sqrt(1-3*x+x^2))-2*EllipticF(sqrt(3-2*x)/5^(1/4),I)*sqrt(-1+3*x-x^2)/(5^(3/4)*sqrt(1-3*x+x^2))-4/5*sqrt(1-3*x+x^2)/sqrt(3-2*x)],
[(b*d+2*c*d*x)^(11/2)/(a+b*x+c*x^2)^(3/2),x,6,-2*d*(b*d+2*c*d*x)^(9/2)/sqrt(a+b*x+c*x^2)+72/7*c*d^3*(b*d+2*c*d*x)^(5/2)*sqrt(a+b*x+c*x^2)+120/7*c*(b^2-4*a*c)*d^5*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)+60/7*(b^2-4*a*c)^(9/4)*d^(11/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^(7/2)/(a+b*x+c*x^2)^(3/2),x,5,-2*d*(b*d+2*c*d*x)^(5/2)/sqrt(a+b*x+c*x^2)+40/3*c*d^3*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)+20/3*(b^2-4*a*c)^(5/4)*d^(7/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^(3/2)/(a+b*x+c*x^2)^(3/2),x,4,-2*d*sqrt(b*d+2*c*d*x)/sqrt(a+b*x+c*x^2)+4*(b^2-4*a*c)^(1/4)*d^(3/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/sqrt(a+b*x+c*x^2)],
[1/((b*d+2*c*d*x)^(1/2)*(a+b*x+c*x^2)^(3/2)),x,4,-2*sqrt(b*d+2*c*d*x)/((b^2-4*a*c)*d*sqrt(a+b*x+c*x^2))-4*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(3/4)*sqrt(d)*sqrt(a+b*x+c*x^2))],
[1/((b*d+2*c*d*x)^(5/2)*(a+b*x+c*x^2)^(3/2)),x,5,(-2)/((b^2-4*a*c)*d*(b*d+2*c*d*x)^(3/2)*sqrt(a+b*x+c*x^2))-40/3*c*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^2*d*(b*d+2*c*d*x)^(3/2))-20/3*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(7/4)*d^(5/2)*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(9/2)/(a+b*x+c*x^2)^(3/2),x,8,-2*d*(b*d+2*c*d*x)^(7/2)/sqrt(a+b*x+c*x^2)+56/5*c*d^3*(b*d+2*c*d*x)^(3/2)*sqrt(a+b*x+c*x^2)+84/5*(b^2-4*a*c)^(7/4)*d^(9/2)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/sqrt(a+b*x+c*x^2)-84/5*(b^2-4*a*c)^(7/4)*d^(9/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^(5/2)/(a+b*x+c*x^2)^(3/2),x,7,-2*d*(b*d+2*c*d*x)^(3/2)/sqrt(a+b*x+c*x^2)+12*(b^2-4*a*c)^(3/4)*d^(5/2)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/sqrt(a+b*x+c*x^2)-12*(b^2-4*a*c)^(3/4)*d^(5/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^(1/2)/(a+b*x+c*x^2)^(3/2),x,7,-2*(b*d+2*c*d*x)^(3/2)/((b^2-4*a*c)*d*sqrt(a+b*x+c*x^2))+4*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(d)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(1/4)*sqrt(a+b*x+c*x^2))-4*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(d)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(1/4)*sqrt(a+b*x+c*x^2))],
[1/((b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2)^(3/2)),x,8,(-2)/((b^2-4*a*c)*d*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2))-24*c*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^2*d*sqrt(b*d+2*c*d*x))+12*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(5/4)*d^(3/2)*sqrt(a+b*x+c*x^2))-12*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(5/4)*d^(3/2)*sqrt(a+b*x+c*x^2))],
[1/((b*d+2*c*d*x)^(7/2)*(a+b*x+c*x^2)^(3/2)),x,9,(-2)/((b^2-4*a*c)*d*(b*d+2*c*d*x)^(5/2)*sqrt(a+b*x+c*x^2))-56/5*c*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^2*d*(b*d+2*c*d*x)^(5/2))-168/5*c*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^3*d^3*sqrt(b*d+2*c*d*x))+84/5*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(9/4)*d^(7/2)*sqrt(a+b*x+c*x^2))-84/5*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(9/4)*d^(7/2)*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(15/2)/(a+b*x+c*x^2)^(5/2),x,7,-2/3*d*(b*d+2*c*d*x)^(13/2)/(a+b*x+c*x^2)^(3/2)-52/3*c*d^3*(b*d+2*c*d*x)^(9/2)/sqrt(a+b*x+c*x^2)+624/7*c^2*d^5*(b*d+2*c*d*x)^(5/2)*sqrt(a+b*x+c*x^2)+1040/7*c^2*(b^2-4*a*c)*d^7*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)+520/7*c*(b^2-4*a*c)^(9/4)*d^(15/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^(11/2)/(a+b*x+c*x^2)^(5/2),x,6,-2/3*d*(b*d+2*c*d*x)^(9/2)/(a+b*x+c*x^2)^(3/2)-12*c*d^3*(b*d+2*c*d*x)^(5/2)/sqrt(a+b*x+c*x^2)+80*c^2*d^5*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2)+40*c*(b^2-4*a*c)^(5/4)*d^(11/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^(7/2)/(a+b*x+c*x^2)^(5/2),x,5,-2/3*d*(b*d+2*c*d*x)^(5/2)/(a+b*x+c*x^2)^(3/2)-20/3*c*d^3*sqrt(b*d+2*c*d*x)/sqrt(a+b*x+c*x^2)+40/3*c*(b^2-4*a*c)^(1/4)*d^(7/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^(3/2)/(a+b*x+c*x^2)^(5/2),x,5,-2/3*d*sqrt(b*d+2*c*d*x)/(a+b*x+c*x^2)^(3/2)-4/3*c*d*sqrt(b*d+2*c*d*x)/((b^2-4*a*c)*sqrt(a+b*x+c*x^2))-8/3*c*d^(3/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(3/4)*sqrt(a+b*x+c*x^2))],
[1/((b*d+2*c*d*x)^(1/2)*(a+b*x+c*x^2)^(5/2)),x,5,-2/3*sqrt(b*d+2*c*d*x)/((b^2-4*a*c)*d*(a+b*x+c*x^2)^(3/2))+20/3*c*sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^2*d*sqrt(a+b*x+c*x^2))+40/3*c*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(7/4)*sqrt(d)*sqrt(a+b*x+c*x^2))],
[1/((b*d+2*c*d*x)^(5/2)*(a+b*x+c*x^2)^(5/2)),x,6,(-2/3)/((b^2-4*a*c)*d*(b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2)^(3/2))+12*c/((b^2-4*a*c)^2*d*(b*d+2*c*d*x)^(3/2)*sqrt(a+b*x+c*x^2))+80*c^2*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^3*d*(b*d+2*c*d*x)^(3/2))+40*c*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(11/4)*d^(5/2)*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(13/2)/(a+b*x+c*x^2)^(5/2),x,9,-2/3*d*(b*d+2*c*d*x)^(11/2)/(a+b*x+c*x^2)^(3/2)-44/3*c*d^3*(b*d+2*c*d*x)^(7/2)/sqrt(a+b*x+c*x^2)+1232/15*c^2*d^5*(b*d+2*c*d*x)^(3/2)*sqrt(a+b*x+c*x^2)+616/5*c*(b^2-4*a*c)^(7/4)*d^(13/2)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/sqrt(a+b*x+c*x^2)-616/5*c*(b^2-4*a*c)^(7/4)*d^(13/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^(9/2)/(a+b*x+c*x^2)^(5/2),x,8,-2/3*d*(b*d+2*c*d*x)^(7/2)/(a+b*x+c*x^2)^(3/2)-28/3*c*d^3*(b*d+2*c*d*x)^(3/2)/sqrt(a+b*x+c*x^2)+56*c*(b^2-4*a*c)^(3/4)*d^(9/2)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/sqrt(a+b*x+c*x^2)-56*c*(b^2-4*a*c)^(3/4)*d^(9/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/sqrt(a+b*x+c*x^2)],
[(b*d+2*c*d*x)^(5/2)/(a+b*x+c*x^2)^(5/2),x,8,-2/3*d*(b*d+2*c*d*x)^(3/2)/(a+b*x+c*x^2)^(3/2)-4*c*d*(b*d+2*c*d*x)^(3/2)/((b^2-4*a*c)*sqrt(a+b*x+c*x^2))+8*c*d^(5/2)*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(1/4)*sqrt(a+b*x+c*x^2))-8*c*d^(5/2)*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(1/4)*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^(1/2)/(a+b*x+c*x^2)^(5/2),x,8,-2/3*(b*d+2*c*d*x)^(3/2)/((b^2-4*a*c)*d*(a+b*x+c*x^2)^(3/2))+4*c*(b*d+2*c*d*x)^(3/2)/((b^2-4*a*c)^2*d*sqrt(a+b*x+c*x^2))-8*c*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(d)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(5/4)*sqrt(a+b*x+c*x^2))+8*c*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(d)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(5/4)*sqrt(a+b*x+c*x^2))],
[1/((b*d+2*c*d*x)^(3/2)*(a+b*x+c*x^2)^(5/2)),x,9,(-2/3)/((b^2-4*a*c)*d*(a+b*x+c*x^2)^(3/2)*sqrt(b*d+2*c*d*x))+28/3*c/((b^2-4*a*c)^2*d*sqrt(b*d+2*c*d*x)*sqrt(a+b*x+c*x^2))+112*c^2*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^3*d*sqrt(b*d+2*c*d*x))-56*c*EllipticE(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(9/4)*d^(3/2)*sqrt(a+b*x+c*x^2))+56*c*EllipticF(sqrt(b*d+2*c*d*x)/((b^2-4*a*c)^(1/4)*sqrt(d)),I)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(9/4)*d^(3/2)*sqrt(a+b*x+c*x^2))],
[(c*e+d*e*x)^(11/2)/sqrt(1-c^2-2*c*d*x-d^2*x^2),x,5,30/77*e^(11/2)*EllipticF(sqrt(c*e+d*e*x)/sqrt(e),I)/d-18/77*e^3*(c*e+d*e*x)^(5/2)*sqrt(1-c^2-2*c*d*x-d^2*x^2)/d-2/11*e*(c*e+d*e*x)^(9/2)*sqrt(1-c^2-2*c*d*x-d^2*x^2)/d-30/77*e^5*sqrt(c*e+d*e*x)*sqrt(1-c^2-2*c*d*x-d^2*x^2)/d],
[(c*e+d*e*x)^(7/2)/sqrt(1-c^2-2*c*d*x-d^2*x^2),x,4,10/21*e^(7/2)*EllipticF(sqrt(c*e+d*e*x)/sqrt(e),I)/d-2/7*e*(c*e+d*e*x)^(5/2)*sqrt(1-c^2-2*c*d*x-d^2*x^2)/d-10/21*e^3*sqrt(c*e+d*e*x)*sqrt(1-c^2-2*c*d*x-d^2*x^2)/d],
[(c*e+d*e*x)^(3/2)/sqrt(1-c^2-2*c*d*x-d^2*x^2),x,3,2/3*e^(3/2)*EllipticF(sqrt(c*e+d*e*x)/sqrt(e),I)/d-2/3*e*sqrt(c*e+d*e*x)*sqrt(1-c^2-2*c*d*x-d^2*x^2)/d],
[1/((c*e+d*e*x)^(1/2)*sqrt(1-c^2-2*c*d*x-d^2*x^2)),x,2,2*EllipticF(sqrt(c*e+d*e*x)/sqrt(e),I)/(d*sqrt(e))],
[1/((c*e+d*e*x)^(5/2)*sqrt(1-c^2-2*c*d*x-d^2*x^2)),x,3,2/3*EllipticF(sqrt(c*e+d*e*x)/sqrt(e),I)/(d*e^(5/2))-2/3*sqrt(1-c^2-2*c*d*x-d^2*x^2)/(d*e*(c*e+d*e*x)^(3/2))],
[1/((c*e+d*e*x)^(9/2)*sqrt(1-c^2-2*c*d*x-d^2*x^2)),x,4,10/21*EllipticF(sqrt(c*e+d*e*x)/sqrt(e),I)/(d*e^(9/2))-2/7*sqrt(1-c^2-2*c*d*x-d^2*x^2)/(d*e*(c*e+d*e*x)^(7/2))-10/21*sqrt(1-c^2-2*c*d*x-d^2*x^2)/(d*e^3*(c*e+d*e*x)^(3/2))],
[1/((c*e+d*e*x)^(13/2)*sqrt(1-c^2-2*c*d*x-d^2*x^2)),x,5,30/77*EllipticF(sqrt(c*e+d*e*x)/sqrt(e),I)/(d*e^(13/2))-2/11*sqrt(1-c^2-2*c*d*x-d^2*x^2)/(d*e*(c*e+d*e*x)^(11/2))-18/77*sqrt(1-c^2-2*c*d*x-d^2*x^2)/(d*e^3*(c*e+d*e*x)^(7/2))-30/77*sqrt(1-c^2-2*c*d*x-d^2*x^2)/(d*e^5*(c*e+d*e*x)^(3/2))],
[(c*e+d*e*x)^(9/2)/sqrt(1-c^2-2*c*d*x-d^2*x^2),x,7,14/15*e^(9/2)*EllipticE(sqrt(c*e+d*e*x)/sqrt(e),I)/d-14/15*e^(9/2)*EllipticF(sqrt(c*e+d*e*x)/sqrt(e),I)/d-14/45*e^3*(c*e+d*e*x)^(3/2)*sqrt(1-c^2-2*c*d*x-d^2*x^2)/d-2/9*e*(c*e+d*e*x)^(7/2)*sqrt(1-c^2-2*c*d*x-d^2*x^2)/d],
[(c*e+d*e*x)^(5/2)/sqrt(1-c^2-2*c*d*x-d^2*x^2),x,6,6/5*e^(5/2)*EllipticE(sqrt(c*e+d*e*x)/sqrt(e),I)/d-6/5*e^(5/2)*EllipticF(sqrt(c*e+d*e*x)/sqrt(e),I)/d-2/5*e*(c*e+d*e*x)^(3/2)*sqrt(1-c^2-2*c*d*x-d^2*x^2)/d],
[(c*e+d*e*x)^(1/2)/sqrt(1-c^2-2*c*d*x-d^2*x^2),x,5,2*EllipticE(sqrt(c*e+d*e*x)/sqrt(e),I)*sqrt(e)/d-2*EllipticF(sqrt(c*e+d*e*x)/sqrt(e),I)*sqrt(e)/d],
[1/((c*e+d*e*x)^(3/2)*sqrt(1-c^2-2*c*d*x-d^2*x^2)),x,6,-2*EllipticE(sqrt(c*e+d*e*x)/sqrt(e),I)/(d*e^(3/2))+2*EllipticF(sqrt(c*e+d*e*x)/sqrt(e),I)/(d*e^(3/2))-2*sqrt(1-c^2-2*c*d*x-d^2*x^2)/(d*e*sqrt(c*e+d*e*x))],
[1/((c*e+d*e*x)^(7/2)*sqrt(1-c^2-2*c*d*x-d^2*x^2)),x,7,-6/5*EllipticE(sqrt(c*e+d*e*x)/sqrt(e),I)/(d*e^(7/2))+6/5*EllipticF(sqrt(c*e+d*e*x)/sqrt(e),I)/(d*e^(7/2))-2/5*sqrt(1-c^2-2*c*d*x-d^2*x^2)/(d*e*(c*e+d*e*x)^(5/2))-6/5*sqrt(1-c^2-2*c*d*x-d^2*x^2)/(d*e^3*sqrt(c*e+d*e*x))],

# Integrands of the form (b d+2 c d x)^(m/3) (a+b x+c x^2)^(p/3)
[(a+b*x+c*x^2)^(4/3)/(b*d+2*c*d*x)^(11/3),x,8,-3/16*(d*(b+2*c*x))^(4/3)*(a+b*x+c*x^2)^(1/3)/(c^2*(b^2-4*a*c)*d^5)+9/16*(d*(b+2*c*x))^(4/3)*(a+b*x+c*x^2)^(4/3)/(c*(b^2-4*a*c)^2*d^5)+3/4*(a+b*x+c*x^2)^(7/3)/((b^2-4*a*c)*d*(b*d+2*c*d*x)^(8/3))-9/4*(a+b*x+c*x^2)^(7/3)/((b^2-4*a*c)^2*d^3*(b*d+2*c*d*x)^(2/3))-3/32*log((d*(b+2*c*x))^(2/3)-2^(2/3)*c^(1/3)*d^(2/3)*(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(7/3)*d^(11/3))-1/16*arctan((1+2^(1/3)*(d*(b+2*c*x))^(2/3)/(c^(1/3)*d^(2/3)*(a+b*x+c*x^2)^(1/3)))/sqrt(3))*sqrt(3)/(2^(2/3)*c^(7/3)*d^(11/3))],
[(a+b*x+c*x^2)^(4/3)/(b*d+2*c*d*x)^(17/3),x,1,3/7*(a+b*x+c*x^2)^(7/3)/((b^2-4*a*c)*d*(b*d+2*c*d*x)^(14/3))],
[(a+b*x+c*x^2)^(4/3)/(b*d+2*c*d*x)^(23/3),x,2,3/10*(a+b*x+c*x^2)^(7/3)/((b^2-4*a*c)*d*(b*d+2*c*d*x)^(20/3))+9/70*(a+b*x+c*x^2)^(7/3)/((b^2-4*a*c)^2*d^3*(b*d+2*c*d*x)^(14/3))],
[(a+b*x+c*x^2)^(4/3)/(b*d+2*c*d*x)^(29/3),x,3,3/13*(a+b*x+c*x^2)^(7/3)/((b^2-4*a*c)*d*(b*d+2*c*d*x)^(26/3))+9/65*(a+b*x+c*x^2)^(7/3)/((b^2-4*a*c)^2*d^3*(b*d+2*c*d*x)^(20/3))+27/455*(a+b*x+c*x^2)^(7/3)/((b^2-4*a*c)^3*d^5*(b*d+2*c*d*x)^(14/3))],
[(a+b*x+c*x^2)^(4/3)/(b*d+2*c*d*x)^(2/3),x,6,-1/9*(b^2-4*a*c)*(d*(b+2*c*x))^(1/3)*(a+b*x+c*x^2)^(1/3)/(c^2*d)+1/6*(d*(b+2*c*x))^(1/3)*(a+b*x+c*x^2)^(4/3)/(c*d)+1/72*(b^2-4*a*c)*(d*(b+2*c*x))^(1/3)*(b^2-4*a*c-(b+2*c*x)^2)*(2*c^(1/3)*d^(2/3)-2^(1/3)*(d*(b+2*c*x))^(2/3)/(a+b*x+c*x^2)^(1/3))*sqrt(cos(arccos((2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1-sqrt(3))/(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3))))^2)/cos(arccos((2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1-sqrt(3))/(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3))))*EllipticF(sin(arccos((2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1-sqrt(3))/(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3)))),sqrt(1/4*(2+sqrt(3))))*sqrt((2*2^(1/3)*c^(2/3)*d^(4/3)+(d*(b+2*c*x))^(4/3)/(a+b*x+c*x^2)^(2/3)+2^(2/3)*c^(1/3)*d^(2/3)*(d*(b+2*c*x))^(2/3)/(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3))^2)/(3^(1/4)*c^(10/3)*d^(5/3)*(a+b*x+c*x^2)^(2/3)*sqrt(-(d*(b+2*c*x))^(2/3)*(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)/(a+b*x+c*x^2)^(1/3))/((a+b*x+c*x^2)^(1/3)*(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3))^2)))],
[(a+b*x+c*x^2)^(4/3)/(b*d+2*c*d*x)^(8/3),x,6,1/5*(d*(b+2*c*x))^(1/3)*(a+b*x+c*x^2)^(1/3)/(c^2*d^3)-3/10*(a+b*x+c*x^2)^(4/3)/(c*d*(d*(b+2*c*x))^(5/3))-1/40*(d*(b+2*c*x))^(1/3)*(b^2-4*a*c-(b+2*c*x)^2)*(2*c^(1/3)*d^(2/3)-2^(1/3)*(d*(b+2*c*x))^(2/3)/(a+b*x+c*x^2)^(1/3))*sqrt(cos(arccos((2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1-sqrt(3))/(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3))))^2)/cos(arccos((2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1-sqrt(3))/(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3))))*EllipticF(sin(arccos((2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1-sqrt(3))/(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3)))),sqrt(1/4*(2+sqrt(3))))*sqrt((2*2^(1/3)*c^(2/3)*d^(4/3)+(d*(b+2*c*x))^(4/3)/(a+b*x+c*x^2)^(2/3)+2^(2/3)*c^(1/3)*d^(2/3)*(d*(b+2*c*x))^(2/3)/(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3))^2)/(3^(1/4)*c^(10/3)*d^(11/3)*(a+b*x+c*x^2)^(2/3)*sqrt(-(d*(b+2*c*x))^(2/3)*(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)/(a+b*x+c*x^2)^(1/3))/((a+b*x+c*x^2)^(1/3)*(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3))^2)))],
[(a+b*x+c*x^2)^(4/3)/(b*d+2*c*d*x)^(14/3),x,6,-3/55*(a+b*x+c*x^2)^(1/3)/(c^2*d^3*(d*(b+2*c*x))^(5/3))-3/22*(a+b*x+c*x^2)^(4/3)/(c*d*(d*(b+2*c*x))^(11/3))+1/440*3^(3/4)*(d*(b+2*c*x))^(1/3)*(b^2-4*a*c-(b+2*c*x)^2)*(2*c^(1/3)*d^(2/3)-2^(1/3)*(d*(b+2*c*x))^(2/3)/(a+b*x+c*x^2)^(1/3))*sqrt(cos(arccos((2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1-sqrt(3))/(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3))))^2)/cos(arccos((2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1-sqrt(3))/(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3))))*EllipticF(sin(arccos((2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1-sqrt(3))/(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3)))),sqrt(1/4*(2+sqrt(3))))*sqrt((2*2^(1/3)*c^(2/3)*d^(4/3)+(d*(b+2*c*x))^(4/3)/(a+b*x+c*x^2)^(2/3)+2^(2/3)*c^(1/3)*d^(2/3)*(d*(b+2*c*x))^(2/3)/(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3))^2)/(c^(10/3)*(b^2-4*a*c)*d^(17/3)*(a+b*x+c*x^2)^(2/3)*sqrt(-(d*(b+2*c*x))^(2/3)*(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)/(a+b*x+c*x^2)^(1/3))/((a+b*x+c*x^2)^(1/3)*(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3))^2)))],
[(a+b*x+c*x^2)^(4/3)/(b*d+2*c*d*x)^(20/3),x,7,-3/187*(a+b*x+c*x^2)^(1/3)/(c^2*d^3*(d*(b+2*c*x))^(11/3))+6/935*(a+b*x+c*x^2)^(1/3)/(c^2*(b^2-4*a*c)*d^5*(d*(b+2*c*x))^(5/3))-3/34*(a+b*x+c*x^2)^(4/3)/(c*d*(d*(b+2*c*x))^(17/3))+3/7480*3^(3/4)*(d*(b+2*c*x))^(1/3)*(b^2-4*a*c-(b+2*c*x)^2)*(2*c^(1/3)*d^(2/3)-2^(1/3)*(d*(b+2*c*x))^(2/3)/(a+b*x+c*x^2)^(1/3))*sqrt(cos(arccos((2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1-sqrt(3))/(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3))))^2)/cos(arccos((2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1-sqrt(3))/(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3))))*EllipticF(sin(arccos((2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1-sqrt(3))/(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3)))),sqrt(1/4*(2+sqrt(3))))*sqrt((2*2^(1/3)*c^(2/3)*d^(4/3)+(d*(b+2*c*x))^(4/3)/(a+b*x+c*x^2)^(2/3)+2^(2/3)*c^(1/3)*d^(2/3)*(d*(b+2*c*x))^(2/3)/(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3))^2)/(c^(10/3)*(b^2-4*a*c)^2*d^(23/3)*(a+b*x+c*x^2)^(2/3)*sqrt(-(d*(b+2*c*x))^(2/3)*(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)/(a+b*x+c*x^2)^(1/3))/((a+b*x+c*x^2)^(1/3)*(2^(2/3)*c^(1/3)*d^(2/3)-(d*(b+2*c*x))^(2/3)*(1+sqrt(3))/(a+b*x+c*x^2)^(1/3))^2)))],
[(a+b*x+c*x^2)^(4/3)/(b*d+2*c*d*x)^(4/3),x,3,3/8*(b^2-4*a*c)*(a+b*x+c*x^2)^(1/3)*hypergeom([-4/3,-1/6],[5/6],(b+2*c*x)^2/(b^2-4*a*c))/(c^2*d*(d*(b+2*c*x))^(1/3)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/3))],
[(a+b*x+c*x^2)^(4/3)/(b*d+2*c*d*x)^(10/3),x,3,3/56*(b^2-4*a*c)*(a+b*x+c*x^2)^(1/3)*hypergeom([-4/3,-7/6],[-1/6],(b+2*c*x)^2/(b^2-4*a*c))/(c^2*d*(d*(b+2*c*x))^(7/3)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/3))],
[(a+b*x+c*x^2)^(4/3)/(b*d+2*c*d*x)^(16/3),x,3,3/104*(b^2-4*a*c)*(a+b*x+c*x^2)^(1/3)*hypergeom([-13/6,-4/3],[-7/6],(b+2*c*x)^2/(b^2-4*a*c))/(c^2*d*(d*(b+2*c*x))^(13/3)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/3))],

# Integrands of the form (b d+2 c d x)^m (a+b x+c x^2)^p when m symbolic
[(b*d+2*c*d*x)^m*(a+b*x+c*x^2)^3,x,2,-1/128*(b^2-4*a*c)^3*(b*d+2*c*d*x)^(1+m)/(c^4*d*(1+m))+3/128*(b^2-4*a*c)^2*(b*d+2*c*d*x)^(3+m)/(c^4*d^3*(3+m))-3/128*(b^2-4*a*c)*(b*d+2*c*d*x)^(5+m)/(c^4*d^5*(5+m))+1/128*(b*d+2*c*d*x)^(7+m)/(c^4*d^7*(7+m))],
[(b*d+2*c*d*x)^m*(a+b*x+c*x^2)^2,x,2,1/32*(b^2-4*a*c)^2*(b*d+2*c*d*x)^(1+m)/(c^3*d*(1+m))-1/16*(b^2-4*a*c)*(b*d+2*c*d*x)^(3+m)/(c^3*d^3*(3+m))+1/32*(b*d+2*c*d*x)^(5+m)/(c^3*d^5*(5+m))],
[(b*d+2*c*d*x)^m*(a+b*x+c*x^2),x,2,-1/8*(b^2-4*a*c)*(b*d+2*c*d*x)^(1+m)/(c^2*d*(1+m))+1/8*(b*d+2*c*d*x)^(3+m)/(c^2*d^3*(3+m))],
[(b*d+2*c*d*x)^m/(a+b*x+c*x^2),x,2,-2*(d*(b+2*c*x))^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)*d*(1+m))],
[(b*d+2*c*d*x)^m/(a+b*x+c*x^2)^2,x,2,8*c*(d*(b+2*c*x))^(1+m)*hypergeom([2,1/2*(1+m)],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)^2*d*(1+m))],
[(b*d+2*c*d*x)^m/(a+b*x+c*x^2)^3,x,2,-32*c^2*(d*(b+2*c*x))^(1+m)*hypergeom([3,1/2*(1+m)],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)^3*d*(1+m))],
[(b*d+2*c*d*x)^m*(a+b*x+c*x^2)^(5/2),x,3,-2*(b*d+2*c*d*x)^(1+m)*(a+b*x+c*x^2)^(7/2)*hypergeom([1,1/2*(8+m)],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)*d*(1+m)),1/32*(b^2-4*a*c)^2*(d*(b+2*c*x))^(1+m)*hypergeom([-5/2,1/2*(1+m)],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))*sqrt(a+b*x+c*x^2)/(c^3*d*(1+m)*sqrt(1-(b+2*c*x)^2/(b^2-4*a*c)))],
[(b*d+2*c*d*x)^m*(a+b*x+c*x^2)^(3/2),x,3,-1/16*(b*d+2*c*d*x)^(1+m)*(4*a-b^2/c+(b+2*c*x)^2/c)^(5/2)*hypergeom([1,1/2*(6+m)],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)*d*(1+m)),-1/8*(b^2-4*a*c)*(d*(b+2*c*x))^(1+m)*hypergeom([-3/2,1/2*(1+m)],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))*sqrt(a+b*x+c*x^2)/(c^2*d*(1+m)*sqrt(1-(b+2*c*x)^2/(b^2-4*a*c)))],
[(b*d+2*c*d*x)^m*(a+b*x+c*x^2)^(1/2),x,3,-1/4*(b*d+2*c*d*x)^(1+m)*(4*a-b^2/c+(b+2*c*x)^2/c)^(3/2)*hypergeom([1,1/2*(4+m)],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)*d*(1+m)),1/2*(d*(b+2*c*x))^(1+m)*hypergeom([-1/2,1/2*(1+m)],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))*sqrt(a+b*x+c*x^2)/(c*d*(1+m)*sqrt(1-(b+2*c*x)^2/(b^2-4*a*c)))],
[(b*d+2*c*d*x)^m/(a+b*x+c*x^2)^(1/2),x,3,-(b*d+2*c*d*x)^(1+m)*hypergeom([1,1/2*(2+m)],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))*sqrt(4*a-b^2/c+(b+2*c*x)^2/c)/((b^2-4*a*c)*d*(1+m)),1/2*(d*(b+2*c*x))^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))*sqrt(1-(b+2*c*x)^2/(b^2-4*a*c))/(c*d*(1+m)*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^m/(a+b*x+c*x^2)^(3/2),x,3,-4*(b*d+2*c*d*x)^(1+m)*hypergeom([1,1/2*m],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)*d*(1+m)*sqrt(4*a-b^2/c+(b+2*c*x)^2/c)),-2*(d*(b+2*c*x))^(1+m)*hypergeom([3/2,1/2*(1+m)],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))*sqrt(1-(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)*d*(1+m)*sqrt(a+b*x+c*x^2))],
[(b*d+2*c*d*x)^m/(a+b*x+c*x^2)^(5/2),x,3,-16*(b*d+2*c*d*x)^(1+m)*hypergeom([1,1/2*(-2+m)],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)*d*(1+m)*(4*a-b^2/c+(b+2*c*x)^2/c)^(3/2)),8*c*(d*(b+2*c*x))^(1+m)*hypergeom([5/2,1/2*(1+m)],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))*sqrt(1-(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)^2*d*(1+m)*sqrt(a+b*x+c*x^2))],

# Integrands of the form (b d+2 c d x)^m (a+b x+c x^2)^p when p symbolic
[(b*d+2*c*d*x)^m*(a+b*x+c*x^2)^p,x,3,-2*(b*d+2*c*d*x)^(1+m)*(1/4*(4*a-b^2/c)+1/4*(b+2*c*x)^2/c)^(1+p)*hypergeom([1,1/2*(3+m+2*p)],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)*d*(1+m)),1/2*(d*(b+2*c*x))^(1+m)*(a+b*x+c*x^2)^p*hypergeom([1/2*(1+m),-p],[1/2*(3+m)],(b+2*c*x)^2/(b^2-4*a*c))/(c*d*(1+m)*(1-(b+2*c*x)^2/(b^2-4*a*c))^p)],
[(b*d+2*c*d*x)^5*(a+b*x+c*x^2)^p,x,3,2*(b^2-4*a*c)^2*d^5*(a+b*x+c*x^2)^(1+p)/((3+p)*(2+3*p+p^2))+2*(b^2-4*a*c)*d^5*(b+2*c*x)^2*(a+b*x+c*x^2)^(1+p)/(6+5*p+p^2)+d^5*(b+2*c*x)^4*(a+b*x+c*x^2)^(1+p)/(3+p)],
[(b*d+2*c*d*x)^4*(a+b*x+c*x^2)^p,x,3,-2/5*d^4*(b+2*c*x)^5*(1/4*(4*a-b^2/c)+1/4*(b+2*c*x)^2/c)^(1+p)*hypergeom([1,7/2+p],[7/2],(b+2*c*x)^2/(b^2-4*a*c))/(b^2-4*a*c),1/10*d^4*(b+2*c*x)^5*(a+b*x+c*x^2)^p*hypergeom([5/2,-p],[7/2],(b+2*c*x)^2/(b^2-4*a*c))/(c*(1-(b+2*c*x)^2/(b^2-4*a*c))^p)],
[(b*d+2*c*d*x)^3*(a+b*x+c*x^2)^p,x,2,(b^2-4*a*c)*d^3*(a+b*x+c*x^2)^(1+p)/(2+3*p+p^2)+d^3*(b+2*c*x)^2*(a+b*x+c*x^2)^(1+p)/(2+p)],
[(b*d+2*c*d*x)^2*(a+b*x+c*x^2)^p,x,3,-2/3*d^2*(b+2*c*x)^3*(1/4*(4*a-b^2/c)+1/4*(b+2*c*x)^2/c)^(1+p)*hypergeom([1,5/2+p],[5/2],(b+2*c*x)^2/(b^2-4*a*c))/(b^2-4*a*c),1/6*d^2*(b+2*c*x)^3*(a+b*x+c*x^2)^p*hypergeom([3/2,-p],[5/2],(b+2*c*x)^2/(b^2-4*a*c))/(c*(1-(b+2*c*x)^2/(b^2-4*a*c))^p)],
[(b*d+2*c*d*x)*(a+b*x+c*x^2)^p,x,1,d*(a+b*x+c*x^2)^(1+p)/(1+p)],
[(a+b*x+c*x^2)^p/(b*d+2*c*d*x),x,3,(a+b*x+c*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1-(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)*d*(1+p))],
[(a+b*x+c*x^2)^p/(b*d+2*c*d*x)^2,x,3,2*(1/4*(4*a-b^2/c)+1/4*(b+2*c*x)^2/c)^(1+p)*hypergeom([1,1/2+p],[1/2],(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)*d^2*(b+2*c*x)),-1/2*(a+b*x+c*x^2)^p*hypergeom([-1/2,-p],[1/2],(b+2*c*x)^2/(b^2-4*a*c))/(c*d^2*(b+2*c*x)*(1-(b+2*c*x)^2/(b^2-4*a*c))^p)],
[(a+b*x+c*x^2)^p/(b*d+2*c*d*x)^3,x,3,(a+b*x+c*x^2)^(1+p)*hypergeom([2,1+p],[2+p],1-(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)^2*d^3*(1+p))],
[(a+b*x+c*x^2)^p/(b*d+2*c*d*x)^4,x,3,2/3*(1/4*(4*a-b^2/c)+1/4*(b+2*c*x)^2/c)^(1+p)*hypergeom([1,-1/2+p],[-1/2],(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)*d^4*(b+2*c*x)^3),-1/6*(a+b*x+c*x^2)^p*hypergeom([-3/2,-p],[-1/2],(b+2*c*x)^2/(b^2-4*a*c))/(c*d^4*(b+2*c*x)^3*(1-(b+2*c*x)^2/(b^2-4*a*c))^p)],
[(a+b*x+c*x^2)^p/(b*d+2*c*d*x)^5,x,3,(a+b*x+c*x^2)^(1+p)*hypergeom([3,1+p],[2+p],1-(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)^3*d^5*(1+p))],
[(a+b*x+c*x^2)^p/(b*d+2*c*d*x)^6,x,3,2/5*(1/4*(4*a-b^2/c)+1/4*(b+2*c*x)^2/c)^(1+p)*hypergeom([1,-3/2+p],[-3/2],(b+2*c*x)^2/(b^2-4*a*c))/((b^2-4*a*c)*d^6*(b+2*c*x)^5),-1/10*(a+b*x+c*x^2)^p*hypergeom([-5/2,-p],[-3/2],(b+2*c*x)^2/(b^2-4*a*c))/(c*d^6*(b+2*c*x)^5*(1-(b+2*c*x)^2/(b^2-4*a*c))^p)],
[(1+x)/(-3+2*x+x^2)^(2/3),x,1,3/2*(-3+2*x+x^2)^(1/3)],
[(b+c*x)/(a+2*b*x+c*x^2)^(3/7),x,1,7/8*(a+2*b*x+c*x^2)^(4/7)],
[(1+x)^m*(1+2*x+x^2)^n,x,2,(1+x)^(1+m)*(1+2*x+x^2)^n/(1+m+2*n)],
[(1/2*b*e/c+e*x)^m*(1/4*b^2/c+b*x+c*x^2)^n,x,2,(1/2*b*e/c+e*x)^(1+m)*(1/4*b^2/c+b*x+c*x^2)^n/(e*(1+m+2*n))],

# Integrands of the form (d+e x)^m (a+b x+c x^2)^p when b^2-4 a c=0

# Integrands of the form (d+e x)^m (a^2+2 a b x+b^2 x^2)^p

# p>0
[(d+e*x)^4*(a^2+2*a*b*x+b^2*x^2),x,3,1/5*(b*d-a*e)^2*(d+e*x)^5/e^3-1/3*b*(b*d-a*e)*(d+e*x)^6/e^3+1/7*b^2*(d+e*x)^7/e^3],
[(d+e*x)^3*(a^2+2*a*b*x+b^2*x^2),x,3,1/4*(b*d-a*e)^2*(d+e*x)^4/e^3-2/5*b*(b*d-a*e)*(d+e*x)^5/e^3+1/6*b^2*(d+e*x)^6/e^3],
[(d+e*x)^2*(a^2+2*a*b*x+b^2*x^2),x,3,1/3*(b*d-a*e)^2*(a+b*x)^3/b^3+1/2*e*(b*d-a*e)*(a+b*x)^4/b^3+1/5*e^2*(a+b*x)^5/b^3],
[(d+e*x)*(a^2+2*a*b*x+b^2*x^2),x,3,1/3*(b*d-a*e)*(a+b*x)^3/b^2+1/4*e*(a+b*x)^4/b^2],
[a^2+2*a*b*x+b^2*x^2,x,1,a^2*x+a*b*x^2+1/3*b^2*x^3],
[(a^2+2*a*b*x+b^2*x^2)/(d+e*x),x,3,-b*(b*d-a*e)*x/e^2+1/2*(a+b*x)^2/e+(b*d-a*e)^2*log(d+e*x)/e^3],
[(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^2,x,3,b^2*x/e^2-(b*d-a*e)^2/(e^3*(d+e*x))-2*b*(b*d-a*e)*log(d+e*x)/e^3],
[(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^3,x,3,-1/2*(b*d-a*e)^2/(e^3*(d+e*x)^2)+2*b*(b*d-a*e)/(e^3*(d+e*x))+b^2*log(d+e*x)/e^3],
[(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^4,x,2,1/3*(a+b*x)^3/((b*d-a*e)*(d+e*x)^3)],
[(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^5,x,3,-1/4*(b*d-a*e)^2/(e^3*(d+e*x)^4)+2/3*b*(b*d-a*e)/(e^3*(d+e*x)^3)-1/2*b^2/(e^3*(d+e*x)^2)],
[(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^6,x,3,-1/5*(b*d-a*e)^2/(e^3*(d+e*x)^5)+1/2*b*(b*d-a*e)/(e^3*(d+e*x)^4)-1/3*b^2/(e^3*(d+e*x)^3)],
[(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^7,x,3,-1/6*(b*d-a*e)^2/(e^3*(d+e*x)^6)+2/5*b*(b*d-a*e)/(e^3*(d+e*x)^5)-1/4*b^2/(e^3*(d+e*x)^4)],
[(d+e*x)^6*(a^2+2*a*b*x+b^2*x^2)^2,x,3,1/7*(b*d-a*e)^4*(d+e*x)^7/e^5-1/2*b*(b*d-a*e)^3*(d+e*x)^8/e^5+2/3*b^2*(b*d-a*e)^2*(d+e*x)^9/e^5-2/5*b^3*(b*d-a*e)*(d+e*x)^10/e^5+1/11*b^4*(d+e*x)^11/e^5],
[(d+e*x)^5*(a^2+2*a*b*x+b^2*x^2)^2,x,3,1/6*(b*d-a*e)^4*(d+e*x)^6/e^5-4/7*b*(b*d-a*e)^3*(d+e*x)^7/e^5+3/4*b^2*(b*d-a*e)^2*(d+e*x)^8/e^5-4/9*b^3*(b*d-a*e)*(d+e*x)^9/e^5+1/10*b^4*(d+e*x)^10/e^5],
[(d+e*x)^4*(a^2+2*a*b*x+b^2*x^2)^2,x,3,1/5*(b*d-a*e)^4*(a+b*x)^5/b^5+2/3*e*(b*d-a*e)^3*(a+b*x)^6/b^5+6/7*e^2*(b*d-a*e)^2*(a+b*x)^7/b^5+1/2*e^3*(b*d-a*e)*(a+b*x)^8/b^5+1/9*e^4*(a+b*x)^9/b^5],
[(d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)^2,x,3,1/5*(b*d-a*e)^3*(a+b*x)^5/b^4+1/2*e*(b*d-a*e)^2*(a+b*x)^6/b^4+3/7*e^2*(b*d-a*e)*(a+b*x)^7/b^4+1/8*e^3*(a+b*x)^8/b^4],
[(d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^2,x,3,1/5*(b*d-a*e)^2*(a+b*x)^5/b^3+1/3*e*(b*d-a*e)*(a+b*x)^6/b^3+1/7*e^2*(a+b*x)^7/b^3],
[(d+e*x)*(a^2+2*a*b*x+b^2*x^2)^2,x,3,1/5*(b*d-a*e)*(a+b*x)^5/b^2+1/6*e*(a+b*x)^6/b^2],
[(a^2+2*a*b*x+b^2*x^2)^2,x,2,1/5*(a+b*x)^5/b],
[(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x),x,3,-b*(b*d-a*e)^3*x/e^4+1/2*(b*d-a*e)^2*(a+b*x)^2/e^3-1/3*(b*d-a*e)*(a+b*x)^3/e^2+1/4*(a+b*x)^4/e+(b*d-a*e)^4*log(d+e*x)/e^5],
[(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^2,x,3,6*b^2*(b*d-a*e)^2*x/e^4-(b*d-a*e)^4/(e^5*(d+e*x))-2*b^3*(b*d-a*e)*(d+e*x)^2/e^5+1/3*b^4*(d+e*x)^3/e^5-4*b*(b*d-a*e)^3*log(d+e*x)/e^5],
[(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^3,x,3,-b^3*(3*b*d-4*a*e)*x/e^4+1/2*b^4*x^2/e^3-1/2*(b*d-a*e)^4/(e^5*(d+e*x)^2)+4*b*(b*d-a*e)^3/(e^5*(d+e*x))+6*b^2*(b*d-a*e)^2*log(d+e*x)/e^5],
[(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^4,x,3,b^4*x/e^4-1/3*(b*d-a*e)^4/(e^5*(d+e*x)^3)+2*b*(b*d-a*e)^3/(e^5*(d+e*x)^2)-6*b^2*(b*d-a*e)^2/(e^5*(d+e*x))-4*b^3*(b*d-a*e)*log(d+e*x)/e^5],
[(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^5,x,3,-1/4*(b*d-a*e)^4/(e^5*(d+e*x)^4)+4/3*b*(b*d-a*e)^3/(e^5*(d+e*x)^3)-3*b^2*(b*d-a*e)^2/(e^5*(d+e*x)^2)+4*b^3*(b*d-a*e)/(e^5*(d+e*x))+b^4*log(d+e*x)/e^5],
[(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^6,x,2,1/5*(a+b*x)^5/((b*d-a*e)*(d+e*x)^5)],
[(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^7,x,3,1/6*(a+b*x)^5/((b*d-a*e)*(d+e*x)^6)+1/30*b*(a+b*x)^5/((b*d-a*e)^2*(d+e*x)^5)],
[(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^8,x,4,1/7*(a+b*x)^5/((b*d-a*e)*(d+e*x)^7)+1/21*b*(a+b*x)^5/((b*d-a*e)^2*(d+e*x)^6)+1/105*b^2*(a+b*x)^5/((b*d-a*e)^3*(d+e*x)^5)],
[(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^9,x,3,-1/8*(b*d-a*e)^4/(e^5*(d+e*x)^8)+4/7*b*(b*d-a*e)^3/(e^5*(d+e*x)^7)-b^2*(b*d-a*e)^2/(e^5*(d+e*x)^6)+4/5*b^3*(b*d-a*e)/(e^5*(d+e*x)^5)-1/4*b^4/(e^5*(d+e*x)^4)],
[(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^10,x,3,-1/9*(b*d-a*e)^4/(e^5*(d+e*x)^9)+1/2*b*(b*d-a*e)^3/(e^5*(d+e*x)^8)-6/7*b^2*(b*d-a*e)^2/(e^5*(d+e*x)^7)+2/3*b^3*(b*d-a*e)/(e^5*(d+e*x)^6)-1/5*b^4/(e^5*(d+e*x)^5)],
[(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^11,x,3,-1/10*(b*d-a*e)^4/(e^5*(d+e*x)^10)+4/9*b*(b*d-a*e)^3/(e^5*(d+e*x)^9)-3/4*b^2*(b*d-a*e)^2/(e^5*(d+e*x)^8)+4/7*b^3*(b*d-a*e)/(e^5*(d+e*x)^7)-1/6*b^4/(e^5*(d+e*x)^6)],
[(d+e*x)^8*(a^2+2*a*b*x+b^2*x^2)^3,x,3,1/9*(b*d-a*e)^6*(d+e*x)^9/e^7-3/5*b*(b*d-a*e)^5*(d+e*x)^10/e^7+15/11*b^2*(b*d-a*e)^4*(d+e*x)^11/e^7-5/3*b^3*(b*d-a*e)^3*(d+e*x)^12/e^7+15/13*b^4*(b*d-a*e)^2*(d+e*x)^13/e^7-3/7*b^5*(b*d-a*e)*(d+e*x)^14/e^7+1/15*b^6*(d+e*x)^15/e^7],
[(d+e*x)^7*(a^2+2*a*b*x+b^2*x^2)^3,x,3,1/8*(b*d-a*e)^6*(d+e*x)^8/e^7-2/3*b*(b*d-a*e)^5*(d+e*x)^9/e^7+3/2*b^2*(b*d-a*e)^4*(d+e*x)^10/e^7-20/11*b^3*(b*d-a*e)^3*(d+e*x)^11/e^7+5/4*b^4*(b*d-a*e)^2*(d+e*x)^12/e^7-6/13*b^5*(b*d-a*e)*(d+e*x)^13/e^7+1/14*b^6*(d+e*x)^14/e^7],
[(d+e*x)^6*(a^2+2*a*b*x+b^2*x^2)^3,x,3,1/7*(b*d-a*e)^6*(a+b*x)^7/b^7+3/4*e*(b*d-a*e)^5*(a+b*x)^8/b^7+5/3*e^2*(b*d-a*e)^4*(a+b*x)^9/b^7+2*e^3*(b*d-a*e)^3*(a+b*x)^10/b^7+15/11*e^4*(b*d-a*e)^2*(a+b*x)^11/b^7+1/2*e^5*(b*d-a*e)*(a+b*x)^12/b^7+1/13*e^6*(a+b*x)^13/b^7],
[(d+e*x)^5*(a^2+2*a*b*x+b^2*x^2)^3,x,3,1/7*(b*d-a*e)^5*(a+b*x)^7/b^6+5/8*e*(b*d-a*e)^4*(a+b*x)^8/b^6+10/9*e^2*(b*d-a*e)^3*(a+b*x)^9/b^6+e^3*(b*d-a*e)^2*(a+b*x)^10/b^6+5/11*e^4*(b*d-a*e)*(a+b*x)^11/b^6+1/12*e^5*(a+b*x)^12/b^6],
[(d+e*x)^4*(a^2+2*a*b*x+b^2*x^2)^3,x,3,1/7*(b*d-a*e)^4*(a+b*x)^7/b^5+1/2*e*(b*d-a*e)^3*(a+b*x)^8/b^5+2/3*e^2*(b*d-a*e)^2*(a+b*x)^9/b^5+2/5*e^3*(b*d-a*e)*(a+b*x)^10/b^5+1/11*e^4*(a+b*x)^11/b^5],
[(d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)^3,x,3,1/7*(b*d-a*e)^3*(a+b*x)^7/b^4+3/8*e*(b*d-a*e)^2*(a+b*x)^8/b^4+1/3*e^2*(b*d-a*e)*(a+b*x)^9/b^4+1/10*e^3*(a+b*x)^10/b^4],
[(d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^3,x,3,1/7*(b*d-a*e)^2*(a+b*x)^7/b^3+1/4*e*(b*d-a*e)*(a+b*x)^8/b^3+1/9*e^2*(a+b*x)^9/b^3],
[(d+e*x)*(a^2+2*a*b*x+b^2*x^2)^3,x,3,1/7*(b*d-a*e)*(a+b*x)^7/b^2+1/8*e*(a+b*x)^8/b^2],
[(a^2+2*a*b*x+b^2*x^2)^3,x,2,1/7*(a+b*x)^7/b],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x),x,3,-b*(b*d-a*e)^5*x/e^6+1/2*(b*d-a*e)^4*(a+b*x)^2/e^5-1/3*(b*d-a*e)^3*(a+b*x)^3/e^4+1/4*(b*d-a*e)^2*(a+b*x)^4/e^3-1/5*(b*d-a*e)*(a+b*x)^5/e^2+1/6*(a+b*x)^6/e+(b*d-a*e)^6*log(d+e*x)/e^7],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^2,x,3,15*b^2*(b*d-a*e)^4*x/e^6-(b*d-a*e)^6/(e^7*(d+e*x))-10*b^3*(b*d-a*e)^3*(d+e*x)^2/e^7+5*b^4*(b*d-a*e)^2*(d+e*x)^3/e^7-3/2*b^5*(b*d-a*e)*(d+e*x)^4/e^7+1/5*b^6*(d+e*x)^5/e^7-6*b*(b*d-a*e)^5*log(d+e*x)/e^7],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^3,x,3,-20*b^3*(b*d-a*e)^3*x/e^6-1/2*(b*d-a*e)^6/(e^7*(d+e*x)^2)+6*b*(b*d-a*e)^5/(e^7*(d+e*x))+15/2*b^4*(b*d-a*e)^2*(d+e*x)^2/e^7-2*b^5*(b*d-a*e)*(d+e*x)^3/e^7+1/4*b^6*(d+e*x)^4/e^7+15*b^2*(b*d-a*e)^4*log(d+e*x)/e^7],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^4,x,3,15*b^4*(b*d-a*e)^2*x/e^6-1/3*(b*d-a*e)^6/(e^7*(d+e*x)^3)+3*b*(b*d-a*e)^5/(e^7*(d+e*x)^2)-15*b^2*(b*d-a*e)^4/(e^7*(d+e*x))-3*b^5*(b*d-a*e)*(d+e*x)^2/e^7+1/3*b^6*(d+e*x)^3/e^7-20*b^3*(b*d-a*e)^3*log(d+e*x)/e^7],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^5,x,3,-b^5*(5*b*d-6*a*e)*x/e^6+1/2*b^6*x^2/e^5-1/4*(b*d-a*e)^6/(e^7*(d+e*x)^4)+2*b*(b*d-a*e)^5/(e^7*(d+e*x)^3)-15/2*b^2*(b*d-a*e)^4/(e^7*(d+e*x)^2)+20*b^3*(b*d-a*e)^3/(e^7*(d+e*x))+15*b^4*(b*d-a*e)^2*log(d+e*x)/e^7],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^6,x,3,b^6*x/e^6-1/5*(b*d-a*e)^6/(e^7*(d+e*x)^5)+3/2*b*(b*d-a*e)^5/(e^7*(d+e*x)^4)-5*b^2*(b*d-a*e)^4/(e^7*(d+e*x)^3)+10*b^3*(b*d-a*e)^3/(e^7*(d+e*x)^2)-15*b^4*(b*d-a*e)^2/(e^7*(d+e*x))-6*b^5*(b*d-a*e)*log(d+e*x)/e^7],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^7,x,3,-1/6*(b*d-a*e)^6/(e^7*(d+e*x)^6)+6/5*b*(b*d-a*e)^5/(e^7*(d+e*x)^5)-15/4*b^2*(b*d-a*e)^4/(e^7*(d+e*x)^4)+20/3*b^3*(b*d-a*e)^3/(e^7*(d+e*x)^3)-15/2*b^4*(b*d-a*e)^2/(e^7*(d+e*x)^2)+6*b^5*(b*d-a*e)/(e^7*(d+e*x))+b^6*log(d+e*x)/e^7],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^8,x,2,1/7*(a+b*x)^7/((b*d-a*e)*(d+e*x)^7)],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^9,x,3,1/8*(a+b*x)^7/((b*d-a*e)*(d+e*x)^8)+1/56*b*(a+b*x)^7/((b*d-a*e)^2*(d+e*x)^7)],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^10,x,4,1/9*(a+b*x)^7/((b*d-a*e)*(d+e*x)^9)+1/36*b*(a+b*x)^7/((b*d-a*e)^2*(d+e*x)^8)+1/252*b^2*(a+b*x)^7/((b*d-a*e)^3*(d+e*x)^7)],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^11,x,5,1/10*(a+b*x)^7/((b*d-a*e)*(d+e*x)^10)+1/30*b*(a+b*x)^7/((b*d-a*e)^2*(d+e*x)^9)+1/120*b^2*(a+b*x)^7/((b*d-a*e)^3*(d+e*x)^8)+1/840*b^3*(a+b*x)^7/((b*d-a*e)^4*(d+e*x)^7)],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^12,x,3,-1/11*(b*d-a*e)^6/(e^7*(d+e*x)^11)+3/5*b*(b*d-a*e)^5/(e^7*(d+e*x)^10)-5/3*b^2*(b*d-a*e)^4/(e^7*(d+e*x)^9)+5/2*b^3*(b*d-a*e)^3/(e^7*(d+e*x)^8)-15/7*b^4*(b*d-a*e)^2/(e^7*(d+e*x)^7)+b^5*(b*d-a*e)/(e^7*(d+e*x)^6)-1/5*b^6/(e^7*(d+e*x)^5)],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^13,x,3,-1/12*(b*d-a*e)^6/(e^7*(d+e*x)^12)+6/11*b*(b*d-a*e)^5/(e^7*(d+e*x)^11)-3/2*b^2*(b*d-a*e)^4/(e^7*(d+e*x)^10)+20/9*b^3*(b*d-a*e)^3/(e^7*(d+e*x)^9)-15/8*b^4*(b*d-a*e)^2/(e^7*(d+e*x)^8)+6/7*b^5*(b*d-a*e)/(e^7*(d+e*x)^7)-1/6*b^6/(e^7*(d+e*x)^6)],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^14,x,3,-1/13*(b*d-a*e)^6/(e^7*(d+e*x)^13)+1/2*b*(b*d-a*e)^5/(e^7*(d+e*x)^12)-15/11*b^2*(b*d-a*e)^4/(e^7*(d+e*x)^11)+2*b^3*(b*d-a*e)^3/(e^7*(d+e*x)^10)-5/3*b^4*(b*d-a*e)^2/(e^7*(d+e*x)^9)+3/4*b^5*(b*d-a*e)/(e^7*(d+e*x)^8)-1/7*b^6/(e^7*(d+e*x)^7)],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^15,x,3,-1/14*(b*d-a*e)^6/(e^7*(d+e*x)^14)+6/13*b*(b*d-a*e)^5/(e^7*(d+e*x)^13)-5/4*b^2*(b*d-a*e)^4/(e^7*(d+e*x)^12)+20/11*b^3*(b*d-a*e)^3/(e^7*(d+e*x)^11)-3/2*b^4*(b*d-a*e)^2/(e^7*(d+e*x)^10)+2/3*b^5*(b*d-a*e)/(e^7*(d+e*x)^9)-1/8*b^6/(e^7*(d+e*x)^8)],

# p<0
[(d+e*x)^5/(a^2+2*a*b*x+b^2*x^2),x,3,10*e^2*(b*d-a*e)^3*x/b^5-(b*d-a*e)^5/(b^6*(a+b*x))+5*e^3*(b*d-a*e)^2*(a+b*x)^2/b^6+5/3*e^4*(b*d-a*e)*(a+b*x)^3/b^6+1/4*e^5*(a+b*x)^4/b^6+5*e*(b*d-a*e)^4*log(a+b*x)/b^6],
[(d+e*x)^4/(a^2+2*a*b*x+b^2*x^2),x,3,6*e^2*(b*d-a*e)^2*x/b^4-(b*d-a*e)^4/(b^5*(a+b*x))+2*e^3*(b*d-a*e)*(a+b*x)^2/b^5+1/3*e^4*(a+b*x)^3/b^5+4*e*(b*d-a*e)^3*log(a+b*x)/b^5],
[(d+e*x)^3/(a^2+2*a*b*x+b^2*x^2),x,3,e^2*(3*b*d-2*a*e)*x/b^3+1/2*e^3*x^2/b^2-(b*d-a*e)^3/(b^4*(a+b*x))+3*e*(b*d-a*e)^2*log(a+b*x)/b^4],
[(d+e*x)^2/(a^2+2*a*b*x+b^2*x^2),x,3,e^2*x/b^2-(b*d-a*e)^2/(b^3*(a+b*x))+2*e*(b*d-a*e)*log(a+b*x)/b^3],
[(d+e*x)/(a^2+2*a*b*x+b^2*x^2),x,3,(-b*d+a*e)/(b^2*(a+b*x))+e*log(a+b*x)/b^2],
[1/(a^2+2*a*b*x+b^2*x^2),x,2,(-1)/(b*(a+b*x))],
[1/((d+e*x)*(a^2+2*a*b*x+b^2*x^2)),x,3,(-1)/((b*d-a*e)*(a+b*x))-e*log(a+b*x)/(b*d-a*e)^2+e*log(d+e*x)/(b*d-a*e)^2],
[1/((d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)),x,3,-b/((b*d-a*e)^2*(a+b*x))-e/((b*d-a*e)^2*(d+e*x))-2*b*e*log(a+b*x)/(b*d-a*e)^3+2*b*e*log(d+e*x)/(b*d-a*e)^3],
[1/((d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)),x,3,-b^2/((b*d-a*e)^3*(a+b*x))-1/2*e/((b*d-a*e)^2*(d+e*x)^2)-2*b*e/((b*d-a*e)^3*(d+e*x))-3*b^2*e*log(a+b*x)/(b*d-a*e)^4+3*b^2*e*log(d+e*x)/(b*d-a*e)^4],
[1/((d+e*x)^4*(a^2+2*a*b*x+b^2*x^2)),x,3,-b^3/((b*d-a*e)^4*(a+b*x))-1/3*e/((b*d-a*e)^2*(d+e*x)^3)-b*e/((b*d-a*e)^3*(d+e*x)^2)-3*b^2*e/((b*d-a*e)^4*(d+e*x))-4*b^3*e*log(a+b*x)/(b*d-a*e)^5+4*b^3*e*log(d+e*x)/(b*d-a*e)^5],
[(d+e*x)^6/(a^2+2*a*b*x+b^2*x^2)^2,x,3,15*e^4*(b*d-a*e)^2*x/b^6-1/3*(b*d-a*e)^6/(b^7*(a+b*x)^3)-3*e*(b*d-a*e)^5/(b^7*(a+b*x)^2)-15*e^2*(b*d-a*e)^4/(b^7*(a+b*x))+3*e^5*(b*d-a*e)*(a+b*x)^2/b^7+1/3*e^6*(a+b*x)^3/b^7+20*e^3*(b*d-a*e)^3*log(a+b*x)/b^7],
[(d+e*x)^5/(a^2+2*a*b*x+b^2*x^2)^2,x,3,e^4*(5*b*d-4*a*e)*x/b^5+1/2*e^5*x^2/b^4-1/3*(b*d-a*e)^5/(b^6*(a+b*x)^3)-5/2*e*(b*d-a*e)^4/(b^6*(a+b*x)^2)-10*e^2*(b*d-a*e)^3/(b^6*(a+b*x))+10*e^3*(b*d-a*e)^2*log(a+b*x)/b^6],
[(d+e*x)^4/(a^2+2*a*b*x+b^2*x^2)^2,x,3,e^4*x/b^4-1/3*(b*d-a*e)^4/(b^5*(a+b*x)^3)-2*e*(b*d-a*e)^3/(b^5*(a+b*x)^2)-6*e^2*(b*d-a*e)^2/(b^5*(a+b*x))+4*e^3*(b*d-a*e)*log(a+b*x)/b^5],
[(d+e*x)^3/(a^2+2*a*b*x+b^2*x^2)^2,x,3,-1/3*(b*d-a*e)^3/(b^4*(a+b*x)^3)-3/2*e*(b*d-a*e)^2/(b^4*(a+b*x)^2)-3*e^2*(b*d-a*e)/(b^4*(a+b*x))+e^3*log(a+b*x)/b^4],
[(d+e*x)^2/(a^2+2*a*b*x+b^2*x^2)^2,x,2,-1/3*(d+e*x)^3/((b*d-a*e)*(a+b*x)^3)],
[(d+e*x)/(a^2+2*a*b*x+b^2*x^2)^2,x,3,1/3*(-b*d+a*e)/(b^2*(a+b*x)^3)-1/2*e/(b^2*(a+b*x)^2)],
[1/(a^2+2*a*b*x+b^2*x^2)^2,x,2,(-1/3)/(b*(a+b*x)^3)],
[1/((d+e*x)*(a^2+2*a*b*x+b^2*x^2)^2),x,3,(-1/3)/((b*d-a*e)*(a+b*x)^3)+1/2*e/((b*d-a*e)^2*(a+b*x)^2)-e^2/((b*d-a*e)^3*(a+b*x))-e^3*log(a+b*x)/(b*d-a*e)^4+e^3*log(d+e*x)/(b*d-a*e)^4],
[1/((d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^2),x,3,-1/3*b/((b*d-a*e)^2*(a+b*x)^3)+b*e/((b*d-a*e)^3*(a+b*x)^2)-3*b*e^2/((b*d-a*e)^4*(a+b*x))-e^3/((b*d-a*e)^4*(d+e*x))-4*b*e^3*log(a+b*x)/(b*d-a*e)^5+4*b*e^3*log(d+e*x)/(b*d-a*e)^5],
[1/((d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)^2),x,3,-1/3*b^2/((b*d-a*e)^3*(a+b*x)^3)+3/2*b^2*e/((b*d-a*e)^4*(a+b*x)^2)-6*b^2*e^2/((b*d-a*e)^5*(a+b*x))-1/2*e^3/((b*d-a*e)^4*(d+e*x)^2)-4*b*e^3/((b*d-a*e)^5*(d+e*x))-10*b^2*e^3*log(a+b*x)/(b*d-a*e)^6+10*b^2*e^3*log(d+e*x)/(b*d-a*e)^6],
[(d+e*x)^8/(a^2+2*a*b*x+b^2*x^2)^3,x,3,28*e^6*(b*d-a*e)^2*x/b^8-1/5*(b*d-a*e)^8/(b^9*(a+b*x)^5)-2*e*(b*d-a*e)^7/(b^9*(a+b*x)^4)-28/3*e^2*(b*d-a*e)^6/(b^9*(a+b*x)^3)-28*e^3*(b*d-a*e)^5/(b^9*(a+b*x)^2)-70*e^4*(b*d-a*e)^4/(b^9*(a+b*x))+4*e^7*(b*d-a*e)*(a+b*x)^2/b^9+1/3*e^8*(a+b*x)^3/b^9+56*e^5*(b*d-a*e)^3*log(a+b*x)/b^9],
[(d+e*x)^7/(a^2+2*a*b*x+b^2*x^2)^3,x,3,e^6*(7*b*d-6*a*e)*x/b^7+1/2*e^7*x^2/b^6-1/5*(b*d-a*e)^7/(b^8*(a+b*x)^5)-7/4*e*(b*d-a*e)^6/(b^8*(a+b*x)^4)-7*e^2*(b*d-a*e)^5/(b^8*(a+b*x)^3)-35/2*e^3*(b*d-a*e)^4/(b^8*(a+b*x)^2)-35*e^4*(b*d-a*e)^3/(b^8*(a+b*x))+21*e^5*(b*d-a*e)^2*log(a+b*x)/b^8],
[(d+e*x)^6/(a^2+2*a*b*x+b^2*x^2)^3,x,3,e^6*x/b^6-1/5*(b*d-a*e)^6/(b^7*(a+b*x)^5)-3/2*e*(b*d-a*e)^5/(b^7*(a+b*x)^4)-5*e^2*(b*d-a*e)^4/(b^7*(a+b*x)^3)-10*e^3*(b*d-a*e)^3/(b^7*(a+b*x)^2)-15*e^4*(b*d-a*e)^2/(b^7*(a+b*x))+6*e^5*(b*d-a*e)*log(a+b*x)/b^7],
[(d+e*x)^5/(a^2+2*a*b*x+b^2*x^2)^3,x,3,-1/5*(b*d-a*e)^5/(b^6*(a+b*x)^5)-5/4*e*(b*d-a*e)^4/(b^6*(a+b*x)^4)-10/3*e^2*(b*d-a*e)^3/(b^6*(a+b*x)^3)-5*e^3*(b*d-a*e)^2/(b^6*(a+b*x)^2)-5*e^4*(b*d-a*e)/(b^6*(a+b*x))+e^5*log(a+b*x)/b^6],
[(d+e*x)^4/(a^2+2*a*b*x+b^2*x^2)^3,x,2,-1/5*(d+e*x)^5/((b*d-a*e)*(a+b*x)^5)],
[(d+e*x)^3/(a^2+2*a*b*x+b^2*x^2)^3,x,3,-1/5*(d+e*x)^4/((b*d-a*e)*(a+b*x)^5)+1/20*e*(d+e*x)^4/((b*d-a*e)^2*(a+b*x)^4)],
[(d+e*x)^2/(a^2+2*a*b*x+b^2*x^2)^3,x,3,-1/5*(b*d-a*e)^2/(b^3*(a+b*x)^5)-1/2*e*(b*d-a*e)/(b^3*(a+b*x)^4)-1/3*e^2/(b^3*(a+b*x)^3)],
[(d+e*x)/(a^2+2*a*b*x+b^2*x^2)^3,x,3,1/5*(-b*d+a*e)/(b^2*(a+b*x)^5)-1/4*e/(b^2*(a+b*x)^4)],
[1/(a^2+2*a*b*x+b^2*x^2)^3,x,2,(-1/5)/(b*(a+b*x)^5)],
[1/((d+e*x)*(a^2+2*a*b*x+b^2*x^2)^3),x,3,(-1/5)/((b*d-a*e)*(a+b*x)^5)+1/4*e/((b*d-a*e)^2*(a+b*x)^4)-1/3*e^2/((b*d-a*e)^3*(a+b*x)^3)+1/2*e^3/((b*d-a*e)^4*(a+b*x)^2)-e^4/((b*d-a*e)^5*(a+b*x))-e^5*log(a+b*x)/(b*d-a*e)^6+e^5*log(d+e*x)/(b*d-a*e)^6],
[1/((d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^3),x,3,-1/5*b/((b*d-a*e)^2*(a+b*x)^5)+1/2*b*e/((b*d-a*e)^3*(a+b*x)^4)-b*e^2/((b*d-a*e)^4*(a+b*x)^3)+2*b*e^3/((b*d-a*e)^5*(a+b*x)^2)-5*b*e^4/((b*d-a*e)^6*(a+b*x))-e^5/((b*d-a*e)^6*(d+e*x))-6*b*e^5*log(a+b*x)/(b*d-a*e)^7+6*b*e^5*log(d+e*x)/(b*d-a*e)^7],
[1/((d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)^3),x,3,-1/5*b^2/((b*d-a*e)^3*(a+b*x)^5)+3/4*b^2*e/((b*d-a*e)^4*(a+b*x)^4)-2*b^2*e^2/((b*d-a*e)^5*(a+b*x)^3)+5*b^2*e^3/((b*d-a*e)^6*(a+b*x)^2)-15*b^2*e^4/((b*d-a*e)^7*(a+b*x))-1/2*e^5/((b*d-a*e)^6*(d+e*x)^2)-6*b*e^5/((b*d-a*e)^7*(d+e*x))-21*b^2*e^5*log(a+b*x)/(b*d-a*e)^8+21*b^2*e^5*log(d+e*x)/(b*d-a*e)^8],
[(d+e*x)*(9+12*x+4*x^2)^3,x,3,1/28*(2*d-3*e)*(3+2*x)^7+1/32*e*(3+2*x)^8],
[(d+e*x)*(9+12*x+4*x^2)^2,x,3,1/20*(2*d-3*e)*(3+2*x)^5+1/24*e*(3+2*x)^6],
[(d+e*x)*(9+12*x+4*x^2),x,3,1/12*(2*d-3*e)*(3+2*x)^3+1/16*e*(3+2*x)^4],
[(d+e*x)/(9+12*x+4*x^2),x,3,1/4*(-2*d+3*e)/(3+2*x)+1/4*e*log(3+2*x)],
[(d+e*x)/(9+12*x+4*x^2)^2,x,3,1/12*(-2*d+3*e)/(3+2*x)^3-1/8*e/(3+2*x)^2],
[(d+e*x)/(9+12*x+4*x^2)^3,x,3,1/20*(-2*d+3*e)/(3+2*x)^5-1/16*e/(3+2*x)^4],

# Integrands of the form (d+e x)^m (a^2+2 a b x+b^2 x^2)^(p/2)

# p>0
[(d+e*x)^4*sqrt(a^2+2*a*b*x+b^2*x^2),x,3,-1/5*(b*d-a*e)*(d+e*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))+1/6*b*(d+e*x)^6*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))],
[(d+e*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2),x,3,-1/4*(b*d-a*e)*(d+e*x)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))+1/5*b*(d+e*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))],
[(d+e*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2),x,3,-1/3*(b*d-a*e)*(d+e*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))+1/4*b*(d+e*x)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))],
[(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2),x,2,1/3*e*(a^2+2*a*b*x+b^2*x^2)^(3/2)/b^2+1/2*(b*d-a*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/b^2],
[sqrt(a^2+2*a*b*x+b^2*x^2),x,1,1/2*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/b],
[sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x),x,3,b*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(e*(a+b*x))-(b*d-a*e)*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))],
[sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^2,x,3,(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x)*(d+e*x))+b*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))],
[sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^3,x,2,1/2*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)*(d+e*x)^2)],
[sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^4,x,3,1/3*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x)*(d+e*x)^3)-1/2*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x)*(d+e*x)^2)],
[sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^5,x,3,1/4*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x)*(d+e*x)^4)-1/3*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x)*(d+e*x)^3)],
[sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^6,x,3,1/5*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x)*(d+e*x)^5)-1/4*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x)*(d+e*x)^4)],
[(d+e*x)^5*(a^2+2*a*b*x+b^2*x^2)^(3/2),x,3,-1/6*(b*d-a*e)^3*(d+e*x)^6*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))+3/7*b*(b*d-a*e)^2*(d+e*x)^7*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))-3/8*b^2*(b*d-a*e)*(d+e*x)^8*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))+1/9*b^3*(d+e*x)^9*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))],
[(d+e*x)^4*(a^2+2*a*b*x+b^2*x^2)^(3/2),x,3,-1/5*(b*d-a*e)^3*(d+e*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))+1/2*b*(b*d-a*e)^2*(d+e*x)^6*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))-3/7*b^2*(b*d-a*e)*(d+e*x)^7*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))+1/8*b^3*(d+e*x)^8*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))],
[(d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)^(3/2),x,3,1/4*(b*d-a*e)^3*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/b^4+3/5*e*(b*d-a*e)^2*(a+b*x)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/b^4+1/2*e^2*(b*d-a*e)*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/b^4+1/7*e^3*(a+b*x)^6*sqrt(a^2+2*a*b*x+b^2*x^2)/b^4],
[(d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^(3/2),x,2,1/4*(b*d-a*e)^2*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/b^3+2/5*e*(b*d-a*e)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/b^3+1/6*e^2*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/b^3,1/4*(b*d-a*e)^2*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/b^3+2/5*e*(b*d-a*e)*(a+b*x)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/b^3+1/6*e^2*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/b^3],
[(d+e*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2),x,2,1/4*(b*d-a*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/b^2+1/5*e*(a^2+2*a*b*x+b^2*x^2)^(5/2)/b^2],
[(a^2+2*a*b*x+b^2*x^2)^(3/2),x,1,1/4*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/b],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x),x,3,b*(b*d-a*e)^2*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(a+b*x))-1/2*(b*d-a*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/e^2+1/3*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/e-(b*d-a*e)^3*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^2,x,3,-b^2*(2*b*d-3*a*e)*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(a+b*x))+1/2*b^3*x^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))+(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x))+3*b*(b*d-a*e)^2*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^3,x,3,b^3*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(a+b*x))+1/2*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^2)-3*b*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x))-3*b^2*(b*d-a*e)*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^4,x,3,1/3*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^3)-3/2*b*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^2)+3*b^2*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x))+b^3*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^5,x,2,1/4*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)*(d+e*x)^4)],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^6,x,3,1/5*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)*(d+e*x)^5)+1/20*b*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)^2*(d+e*x)^4)],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^7,x,3,1/6*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/((b*d-a*e)*(d+e*x)^6)+1/15*b*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/((b*d-a*e)^2*(d+e*x)^5)+1/60*b^2*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/((b*d-a*e)^3*(d+e*x)^4),1/6*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^6)-3/5*b*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^5)+3/4*b^2*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^4)-1/3*b^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^3)],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^8,x,3,1/7*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^7)-1/2*b*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^6)+3/5*b^2*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^5)-1/4*b^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^4)],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^9,x,3,1/8*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^8)-3/7*b*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^7)+1/2*b^2*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^6)-1/5*b^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^5)],
[(d+e*x)^5*(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,1/6*(b*d-a*e)^5*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/b^6+5/7*e*(b*d-a*e)^4*(a+b*x)^6*sqrt(a^2+2*a*b*x+b^2*x^2)/b^6+5/4*e^2*(b*d-a*e)^3*(a+b*x)^7*sqrt(a^2+2*a*b*x+b^2*x^2)/b^6+10/9*e^3*(b*d-a*e)^2*(a+b*x)^8*sqrt(a^2+2*a*b*x+b^2*x^2)/b^6+1/2*e^4*(b*d-a*e)*(a+b*x)^9*sqrt(a^2+2*a*b*x+b^2*x^2)/b^6+1/11*e^5*(a+b*x)^10*sqrt(a^2+2*a*b*x+b^2*x^2)/b^6],
[(d+e*x)^4*(a^2+2*a*b*x+b^2*x^2)^(5/2),x,2,1/6*(b*d-a*e)^4*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/b^5+4/7*e*(b*d-a*e)^3*(a+b*x)^6*sqrt(a^2+2*a*b*x+b^2*x^2)/b^5+3/4*e^2*(b*d-a*e)^2*(a+b*x)^7*sqrt(a^2+2*a*b*x+b^2*x^2)/b^5+4/9*e^3*(b*d-a*e)*(a+b*x)^8*sqrt(a^2+2*a*b*x+b^2*x^2)/b^5+1/10*e^4*(a+b*x)^9*sqrt(a^2+2*a*b*x+b^2*x^2)/b^5],
[(d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,1/6*(b*d-a*e)^3*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/b^4+3/7*e*(b*d-a*e)^2*(a+b*x)^6*sqrt(a^2+2*a*b*x+b^2*x^2)/b^4+3/8*e^2*(b*d-a*e)*(a+b*x)^7*sqrt(a^2+2*a*b*x+b^2*x^2)/b^4+1/9*e^3*(a+b*x)^8*sqrt(a^2+2*a*b*x+b^2*x^2)/b^4],
[(d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,1/6*(b*d-a*e)^2*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/b^3+2/7*e*(b*d-a*e)*(a+b*x)^6*sqrt(a^2+2*a*b*x+b^2*x^2)/b^3+1/8*e^2*(a+b*x)^7*sqrt(a^2+2*a*b*x+b^2*x^2)/b^3],
[(d+e*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2),x,2,1/6*(b*d-a*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/b^2+1/7*e*(a^2+2*a*b*x+b^2*x^2)^(7/2)/b^2],
[(a^2+2*a*b*x+b^2*x^2)^(5/2),x,1,1/6*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/b],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x),x,3,b*(b*d-a*e)^4*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(a+b*x))-1/2*(b*d-a*e)^3*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/e^4+1/3*(b*d-a*e)^2*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/e^3-1/4*(b*d-a*e)*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/e^2+1/5*(a+b*x)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/e-(b*d-a*e)^5*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^2,x,3,-10*b^2*(b*d-a*e)^3*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(a+b*x))+(b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x))+5*b^3*(b*d-a*e)^2*(d+e*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))-5/3*b^4*(b*d-a*e)*(d+e*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+1/4*b^5*(d+e*x)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+5*b*(b*d-a*e)^4*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^3,x,3,10*b^3*(b*d-a*e)^2*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(a+b*x))+1/2*(b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^2)-5*b*(b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x))-5/2*b^4*(b*d-a*e)*(d+e*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+1/3*b^5*(d+e*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))-10*b^2*(b*d-a*e)^3*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^4,x,3,-b^4*(4*b*d-5*a*e)*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(a+b*x))+1/2*b^5*x^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))+1/3*(b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^3)-5/2*b*(b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^2)+10*b^2*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x))+10*b^3*(b*d-a*e)^2*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^5,x,3,b^5*x*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(a+b*x))+1/4*(b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^4)-5/3*b*(b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^3)+5*b^2*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^2)-10*b^3*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x))-5*b^4*(b*d-a*e)*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^6,x,3,1/5*(b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^5)-5/4*b*(b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^4)+10/3*b^2*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^3)-5*b^3*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^2)+5*b^4*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x))+b^5*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^7,x,2,1/6*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)*(d+e*x)^6)],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^8,x,3,1/7*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)*(d+e*x)^7)+1/42*b*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)^2*(d+e*x)^6)],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^9,x,4,1/8*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)*(d+e*x)^8)+1/28*b*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)^2*(d+e*x)^7)+1/168*b^2*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)^3*(d+e*x)^6)],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^10,x,5,1/9*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)*(d+e*x)^9)+1/24*b*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)^2*(d+e*x)^8)+1/84*b^2*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)^3*(d+e*x)^7)+1/504*b^3*(a+b*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)^4*(d+e*x)^6)],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^11,x,3,1/10*(b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^10)-5/9*b*(b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^9)+5/4*b^2*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^8)-10/7*b^3*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^7)+5/6*b^4*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^6)-1/5*b^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^5)],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^12,x,3,1/11*(b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^11)-1/2*b*(b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^10)+10/9*b^2*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^9)-5/4*b^3*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^8)+5/7*b^4*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^7)-1/6*b^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^6)],

# p<0
[(d+e*x)^4/sqrt(a^2+2*a*b*x+b^2*x^2),x,3,e*(b*d-a*e)^3*x*(a+b*x)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*(b*d-a*e)^2*(a+b*x)*(d+e*x)^2/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))+1/3*(b*d-a*e)*(a+b*x)*(d+e*x)^3/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))+1/4*(a+b*x)*(d+e*x)^4/(b*sqrt(a^2+2*a*b*x+b^2*x^2))+(b*d-a*e)^4*(a+b*x)*log(a+b*x)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^3/sqrt(a^2+2*a*b*x+b^2*x^2),x,3,e*(b*d-a*e)^2*x*(a+b*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*(b*d-a*e)*(a+b*x)*(d+e*x)^2/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))+1/3*(a+b*x)*(d+e*x)^3/(b*sqrt(a^2+2*a*b*x+b^2*x^2))+(b*d-a*e)^3*(a+b*x)*log(a+b*x)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^2/sqrt(a^2+2*a*b*x+b^2*x^2),x,3,e*(b*d-a*e)*x*(a+b*x)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*(a+b*x)*(d+e*x)^2/(b*sqrt(a^2+2*a*b*x+b^2*x^2))+(b*d-a*e)^2*(a+b*x)*log(a+b*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)/sqrt(a^2+2*a*b*x+b^2*x^2),x,3,(b*d-a*e)*(a+b*x)*log(a+b*x)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))+e*sqrt(a^2+2*a*b*x+b^2*x^2)/b^2],
[1/sqrt(a^2+2*a*b*x+b^2*x^2),x,2,(a+b*x)*log(a+b*x)/(b*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)),x,4,(a+b*x)*log(a+b*x)/((b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2))-(a+b*x)*log(d+e*x)/((b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2)),x,3,(a+b*x)/((b*d-a*e)*(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+b*(a+b*x)*log(a+b*x)/((b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-b*(a+b*x)*log(d+e*x)/((b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2)),x,3,1/2*(a+b*x)/((b*d-a*e)*(d+e*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))+b*(a+b*x)/((b*d-a*e)^2*(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+b^2*(a+b*x)*log(a+b*x)/((b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-b^2*(a+b*x)*log(d+e*x)/((b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)^4*sqrt(a^2+2*a*b*x+b^2*x^2)),x,3,1/3*(a+b*x)/((b*d-a*e)*(d+e*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*b*(a+b*x)/((b*d-a*e)^2*(d+e*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))+b^2*(a+b*x)/((b*d-a*e)^3*(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+b^3*(a+b*x)*log(a+b*x)/((b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2))-b^3*(a+b*x)*log(d+e*x)/((b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^4/(a^2+2*a*b*x+b^2*x^2)^(3/2),x,3,-4*e*(b*d-a*e)^3/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))-1/2*(b*d-a*e)^4/(b^5*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+e^3*(4*b*d-3*a*e)*x*(a+b*x)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*e^4*x^2*(a+b*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))+6*e^2*(b*d-a*e)^2*(a+b*x)*log(a+b*x)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^3/(a^2+2*a*b*x+b^2*x^2)^(3/2),x,3,-3*e*(b*d-a*e)^2/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))-1/2*(b*d-a*e)^3/(b^4*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+e^3*x*(a+b*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))+3*e^2*(b*d-a*e)*(a+b*x)*log(a+b*x)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^2/(a^2+2*a*b*x+b^2*x^2)^(3/2),x,3,-2*e*(b*d-a*e)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))-1/2*(b*d-a*e)^2/(b^3*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+e^2*(a+b*x)*log(a+b*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)/(a^2+2*a*b*x+b^2*x^2)^(3/2),x,2,-e/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*(-b*d+a*e)/(b^2*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/(a^2+2*a*b*x+b^2*x^2)^(3/2),x,1,(-1/2)/(b*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)),x,3,e/((b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2))+(-1/2)/((b*d-a*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+e^2*(a+b*x)*log(a+b*x)/((b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-e^2*(a+b*x)*log(d+e*x)/((b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^(3/2)),x,3,2*b*e/((b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-1/2*b/((b*d-a*e)^2*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+e^2*(a+b*x)/((b*d-a*e)^3*(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+3*b*e^2*(a+b*x)*log(a+b*x)/((b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2))-3*b*e^2*(a+b*x)*log(d+e*x)/((b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)^(3/2)),x,3,3*b^2*e/((b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2))-1/2*b^2/((b*d-a*e)^3*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*e^2*(a+b*x)/((b*d-a*e)^3*(d+e*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))+3*b*e^2*(a+b*x)/((b*d-a*e)^4*(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+6*b^2*e^2*(a+b*x)*log(a+b*x)/((b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2))-6*b^2*e^2*(a+b*x)*log(d+e*x)/((b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^6/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,-20*e^3*(b*d-a*e)^3/(b^7*sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*(b*d-a*e)^6/(b^7*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-2*e*(b*d-a*e)^5/(b^7*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-15/2*e^2*(b*d-a*e)^4/(b^7*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+e^5*(6*b*d-5*a*e)*x*(a+b*x)/(b^6*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*e^6*x^2*(a+b*x)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))+15*e^4*(b*d-a*e)^2*(a+b*x)*log(a+b*x)/(b^7*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^5/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,-10*e^3*(b*d-a*e)^2/(b^6*sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*(b*d-a*e)^5/(b^6*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-5/3*e*(b*d-a*e)^4/(b^6*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-5*e^2*(b*d-a*e)^3/(b^6*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+e^5*x*(a+b*x)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))+5*e^4*(b*d-a*e)*(a+b*x)*log(a+b*x)/(b^6*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^4/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,-4*e^3*(b*d-a*e)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*(b*d-a*e)^4/(b^5*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-4/3*e*(b*d-a*e)^3/(b^5*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-3*e^2*(b*d-a*e)^2/(b^5*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+e^4*(a+b*x)*log(a+b*x)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^3/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,2,-1/4*(d+e*x)^4/((b*d-a*e)*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^2/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,-1/4*(b*d-a*e)^2/(b^3*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-2/3*e*(b*d-a*e)/(b^3*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-1/2*e^2/(b^3*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,2,-1/3*e/(b^2*(a^2+2*a*b*x+b^2*x^2)^(3/2))+1/4*(-b*d+a*e)/(b^2*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2))],
[1/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,1,(-1/4)/(b*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2))],
[1/((d+e*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)),x,3,e^3/((b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2))+(-1/4)/((b*d-a*e)*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))+1/3*e/((b*d-a*e)^2*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-1/2*e^2/((b*d-a*e)^3*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+e^4*(a+b*x)*log(a+b*x)/((b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2))-e^4*(a+b*x)*log(d+e*x)/((b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^(5/2)),x,3,4*b*e^3/((b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*b/((b*d-a*e)^2*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))+2/3*b*e/((b*d-a*e)^3*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-3/2*b*e^2/((b*d-a*e)^4*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+e^4*(a+b*x)/((b*d-a*e)^5*(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+5*b*e^4*(a+b*x)*log(a+b*x)/((b*d-a*e)^6*sqrt(a^2+2*a*b*x+b^2*x^2))-5*b*e^4*(a+b*x)*log(d+e*x)/((b*d-a*e)^6*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)^(5/2)),x,3,10*b^2*e^3/((b*d-a*e)^6*sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*b^2/((b*d-a*e)^3*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))+b^2*e/((b*d-a*e)^4*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-3*b^2*e^2/((b*d-a*e)^5*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+1/2*e^4*(a+b*x)/((b*d-a*e)^5*(d+e*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))+5*b*e^4*(a+b*x)/((b*d-a*e)^6*(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+15*b^2*e^4*(a+b*x)*log(a+b*x)/((b*d-a*e)^7*sqrt(a^2+2*a*b*x+b^2*x^2))-15*b^2*e^4*(a+b*x)*log(d+e*x)/((b*d-a*e)^7*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)*(9+12*x+4*x^2)^(5/2),x,2,1/24*(2*d-3*e)*(3+2*x)*(9+12*x+4*x^2)^(5/2)+1/28*e*(9+12*x+4*x^2)^(7/2)],
[(d+e*x)*(9+12*x+4*x^2)^(3/2),x,2,1/16*(2*d-3*e)*(3+2*x)*(9+12*x+4*x^2)^(3/2)+1/20*e*(9+12*x+4*x^2)^(5/2)],
[(d+e*x)*(9+12*x+4*x^2)^(1/2),x,2,1/12*e*(9+12*x+4*x^2)^(3/2)+1/8*(2*d-3*e)*(3+2*x)*sqrt(9+12*x+4*x^2)],
[(d+e*x)/(9+12*x+4*x^2)^(1/2),x,3,1/4*(2*d-3*e)*(3+2*x)*log(3+2*x)/sqrt(9+12*x+4*x^2)+1/4*e*sqrt(9+12*x+4*x^2)],
[(d+e*x)/(9+12*x+4*x^2)^(3/2),x,2,-1/4*e/sqrt(9+12*x+4*x^2)+1/8*(-2*d+3*e)/((3+2*x)*sqrt(9+12*x+4*x^2))],
[(d+e*x)/(9+12*x+4*x^2)^(5/2),x,2,-1/12*e/(9+12*x+4*x^2)^(3/2)+1/16*(-2*d+3*e)/((3+2*x)*(9+12*x+4*x^2)^(3/2))],
[(d+e*x)/(9+12*x+4*x^2)^(7/2),x,2,-1/20*e/(9+12*x+4*x^2)^(5/2)+1/24*(-2*d+3*e)/((3+2*x)*(9+12*x+4*x^2)^(5/2))],

# Integrands of the form (d+e x)^(m/2) (a^2+2 a b x+b^2 x^2)^p

# p>0
[(d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2),x,3,2/9*(b*d-a*e)^2*(d+e*x)^(9/2)/e^3-4/11*b*(b*d-a*e)*(d+e*x)^(11/2)/e^3+2/13*b^2*(d+e*x)^(13/2)/e^3],
[(d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2),x,3,2/7*(b*d-a*e)^2*(d+e*x)^(7/2)/e^3-4/9*b*(b*d-a*e)*(d+e*x)^(9/2)/e^3+2/11*b^2*(d+e*x)^(11/2)/e^3],
[(d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2),x,3,2/5*(b*d-a*e)^2*(d+e*x)^(5/2)/e^3-4/7*b*(b*d-a*e)*(d+e*x)^(7/2)/e^3+2/9*b^2*(d+e*x)^(9/2)/e^3],
[(a^2+2*a*b*x+b^2*x^2)*sqrt(d+e*x),x,3,2/3*(b*d-a*e)^2*(d+e*x)^(3/2)/e^3-4/5*b*(b*d-a*e)*(d+e*x)^(5/2)/e^3+2/7*b^2*(d+e*x)^(7/2)/e^3],
[(a^2+2*a*b*x+b^2*x^2)/sqrt(d+e*x),x,3,-4/3*b*(b*d-a*e)*(d+e*x)^(3/2)/e^3+2/5*b^2*(d+e*x)^(5/2)/e^3+2*(b*d-a*e)^2*sqrt(d+e*x)/e^3],
[(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^(3/2),x,3,2/3*b^2*(d+e*x)^(3/2)/e^3-2*(b*d-a*e)^2/(e^3*sqrt(d+e*x))-4*b*(b*d-a*e)*sqrt(d+e*x)/e^3],
[(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^(5/2),x,3,-2/3*(b*d-a*e)^2/(e^3*(d+e*x)^(3/2))+4*b*(b*d-a*e)/(e^3*sqrt(d+e*x))+2*b^2*sqrt(d+e*x)/e^3],
[(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^(7/2),x,3,-2/5*(b*d-a*e)^2/(e^3*(d+e*x)^(5/2))+4/3*b*(b*d-a*e)/(e^3*(d+e*x)^(3/2))-2*b^2/(e^3*sqrt(d+e*x))],
[(d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2)^2,x,3,2/9*(b*d-a*e)^4*(d+e*x)^(9/2)/e^5-8/11*b*(b*d-a*e)^3*(d+e*x)^(11/2)/e^5+12/13*b^2*(b*d-a*e)^2*(d+e*x)^(13/2)/e^5-8/15*b^3*(b*d-a*e)*(d+e*x)^(15/2)/e^5+2/17*b^4*(d+e*x)^(17/2)/e^5],
[(d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^2,x,3,2/7*(b*d-a*e)^4*(d+e*x)^(7/2)/e^5-8/9*b*(b*d-a*e)^3*(d+e*x)^(9/2)/e^5+12/11*b^2*(b*d-a*e)^2*(d+e*x)^(11/2)/e^5-8/13*b^3*(b*d-a*e)*(d+e*x)^(13/2)/e^5+2/15*b^4*(d+e*x)^(15/2)/e^5],
[(d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^2,x,3,2/5*(b*d-a*e)^4*(d+e*x)^(5/2)/e^5-8/7*b*(b*d-a*e)^3*(d+e*x)^(7/2)/e^5+4/3*b^2*(b*d-a*e)^2*(d+e*x)^(9/2)/e^5-8/11*b^3*(b*d-a*e)*(d+e*x)^(11/2)/e^5+2/13*b^4*(d+e*x)^(13/2)/e^5],
[(a^2+2*a*b*x+b^2*x^2)^2*sqrt(d+e*x),x,3,2/3*(b*d-a*e)^4*(d+e*x)^(3/2)/e^5-8/5*b*(b*d-a*e)^3*(d+e*x)^(5/2)/e^5+12/7*b^2*(b*d-a*e)^2*(d+e*x)^(7/2)/e^5-8/9*b^3*(b*d-a*e)*(d+e*x)^(9/2)/e^5+2/11*b^4*(d+e*x)^(11/2)/e^5],
[(a^2+2*a*b*x+b^2*x^2)^2/sqrt(d+e*x),x,3,-8/3*b*(b*d-a*e)^3*(d+e*x)^(3/2)/e^5+12/5*b^2*(b*d-a*e)^2*(d+e*x)^(5/2)/e^5-8/7*b^3*(b*d-a*e)*(d+e*x)^(7/2)/e^5+2/9*b^4*(d+e*x)^(9/2)/e^5+2*(b*d-a*e)^4*sqrt(d+e*x)/e^5],
[(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^(3/2),x,3,4*b^2*(b*d-a*e)^2*(d+e*x)^(3/2)/e^5-8/5*b^3*(b*d-a*e)*(d+e*x)^(5/2)/e^5+2/7*b^4*(d+e*x)^(7/2)/e^5-2*(b*d-a*e)^4/(e^5*sqrt(d+e*x))-8*b*(b*d-a*e)^3*sqrt(d+e*x)/e^5],
[(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^(5/2),x,3,-2/3*(b*d-a*e)^4/(e^5*(d+e*x)^(3/2))-8/3*b^3*(b*d-a*e)*(d+e*x)^(3/2)/e^5+2/5*b^4*(d+e*x)^(5/2)/e^5+8*b*(b*d-a*e)^3/(e^5*sqrt(d+e*x))+12*b^2*(b*d-a*e)^2*sqrt(d+e*x)/e^5],
[(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^(7/2),x,3,-2/5*(b*d-a*e)^4/(e^5*(d+e*x)^(5/2))+8/3*b*(b*d-a*e)^3/(e^5*(d+e*x)^(3/2))+2/3*b^4*(d+e*x)^(3/2)/e^5-12*b^2*(b*d-a*e)^2/(e^5*sqrt(d+e*x))-8*b^3*(b*d-a*e)*sqrt(d+e*x)/e^5],
[(d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2)^3,x,3,2/9*(b*d-a*e)^6*(d+e*x)^(9/2)/e^7-12/11*b*(b*d-a*e)^5*(d+e*x)^(11/2)/e^7+30/13*b^2*(b*d-a*e)^4*(d+e*x)^(13/2)/e^7-8/3*b^3*(b*d-a*e)^3*(d+e*x)^(15/2)/e^7+30/17*b^4*(b*d-a*e)^2*(d+e*x)^(17/2)/e^7-12/19*b^5*(b*d-a*e)*(d+e*x)^(19/2)/e^7+2/21*b^6*(d+e*x)^(21/2)/e^7],
[(d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^3,x,3,2/7*(b*d-a*e)^6*(d+e*x)^(7/2)/e^7-4/3*b*(b*d-a*e)^5*(d+e*x)^(9/2)/e^7+30/11*b^2*(b*d-a*e)^4*(d+e*x)^(11/2)/e^7-40/13*b^3*(b*d-a*e)^3*(d+e*x)^(13/2)/e^7+2*b^4*(b*d-a*e)^2*(d+e*x)^(15/2)/e^7-12/17*b^5*(b*d-a*e)*(d+e*x)^(17/2)/e^7+2/19*b^6*(d+e*x)^(19/2)/e^7],
[(d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^3,x,3,2/5*(b*d-a*e)^6*(d+e*x)^(5/2)/e^7-12/7*b*(b*d-a*e)^5*(d+e*x)^(7/2)/e^7+10/3*b^2*(b*d-a*e)^4*(d+e*x)^(9/2)/e^7-40/11*b^3*(b*d-a*e)^3*(d+e*x)^(11/2)/e^7+30/13*b^4*(b*d-a*e)^2*(d+e*x)^(13/2)/e^7-4/5*b^5*(b*d-a*e)*(d+e*x)^(15/2)/e^7+2/17*b^6*(d+e*x)^(17/2)/e^7],
[(a^2+2*a*b*x+b^2*x^2)^3*sqrt(d+e*x),x,3,2/3*(b*d-a*e)^6*(d+e*x)^(3/2)/e^7-12/5*b*(b*d-a*e)^5*(d+e*x)^(5/2)/e^7+30/7*b^2*(b*d-a*e)^4*(d+e*x)^(7/2)/e^7-40/9*b^3*(b*d-a*e)^3*(d+e*x)^(9/2)/e^7+30/11*b^4*(b*d-a*e)^2*(d+e*x)^(11/2)/e^7-12/13*b^5*(b*d-a*e)*(d+e*x)^(13/2)/e^7+2/15*b^6*(d+e*x)^(15/2)/e^7],
[(a^2+2*a*b*x+b^2*x^2)^3/sqrt(d+e*x),x,3,-4*b*(b*d-a*e)^5*(d+e*x)^(3/2)/e^7+6*b^2*(b*d-a*e)^4*(d+e*x)^(5/2)/e^7-40/7*b^3*(b*d-a*e)^3*(d+e*x)^(7/2)/e^7+10/3*b^4*(b*d-a*e)^2*(d+e*x)^(9/2)/e^7-12/11*b^5*(b*d-a*e)*(d+e*x)^(11/2)/e^7+2/13*b^6*(d+e*x)^(13/2)/e^7+2*(b*d-a*e)^6*sqrt(d+e*x)/e^7],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^(3/2),x,3,10*b^2*(b*d-a*e)^4*(d+e*x)^(3/2)/e^7-8*b^3*(b*d-a*e)^3*(d+e*x)^(5/2)/e^7+30/7*b^4*(b*d-a*e)^2*(d+e*x)^(7/2)/e^7-4/3*b^5*(b*d-a*e)*(d+e*x)^(9/2)/e^7+2/11*b^6*(d+e*x)^(11/2)/e^7-2*(b*d-a*e)^6/(e^7*sqrt(d+e*x))-12*b*(b*d-a*e)^5*sqrt(d+e*x)/e^7],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^(5/2),x,3,-2/3*(b*d-a*e)^6/(e^7*(d+e*x)^(3/2))-40/3*b^3*(b*d-a*e)^3*(d+e*x)^(3/2)/e^7+6*b^4*(b*d-a*e)^2*(d+e*x)^(5/2)/e^7-12/7*b^5*(b*d-a*e)*(d+e*x)^(7/2)/e^7+2/9*b^6*(d+e*x)^(9/2)/e^7+12*b*(b*d-a*e)^5/(e^7*sqrt(d+e*x))+30*b^2*(b*d-a*e)^4*sqrt(d+e*x)/e^7],
[(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^(7/2),x,3,-2/5*(b*d-a*e)^6/(e^7*(d+e*x)^(5/2))+4*b*(b*d-a*e)^5/(e^7*(d+e*x)^(3/2))+10*b^4*(b*d-a*e)^2*(d+e*x)^(3/2)/e^7-12/5*b^5*(b*d-a*e)*(d+e*x)^(5/2)/e^7+2/7*b^6*(d+e*x)^(7/2)/e^7-30*b^2*(b*d-a*e)^4/(e^7*sqrt(d+e*x))-40*b^3*(b*d-a*e)^3*sqrt(d+e*x)/e^7],

# p<0
[(d+e*x)^(9/2)/(a^2+2*a*b*x+b^2*x^2),x,8,3*e*(b*d-a*e)^2*(d+e*x)^(3/2)/b^4+9/5*e*(b*d-a*e)*(d+e*x)^(5/2)/b^3+9/7*e*(d+e*x)^(7/2)/b^2-(d+e*x)^(9/2)/(b*(a+b*x))-9*e*(b*d-a*e)^(7/2)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/b^(11/2)+9*e*(b*d-a*e)^3*sqrt(d+e*x)/b^5],
[(d+e*x)^(7/2)/(a^2+2*a*b*x+b^2*x^2),x,7,7/3*e*(b*d-a*e)*(d+e*x)^(3/2)/b^3+7/5*e*(d+e*x)^(5/2)/b^2-(d+e*x)^(7/2)/(b*(a+b*x))-7*e*(b*d-a*e)^(5/2)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/b^(9/2)+7*e*(b*d-a*e)^2*sqrt(d+e*x)/b^4],
[(d+e*x)^(5/2)/(a^2+2*a*b*x+b^2*x^2),x,6,5/3*e*(d+e*x)^(3/2)/b^2-(d+e*x)^(5/2)/(b*(a+b*x))-5*e*(b*d-a*e)^(3/2)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/b^(7/2)+5*e*(b*d-a*e)*sqrt(d+e*x)/b^3],
[(d+e*x)^(3/2)/(a^2+2*a*b*x+b^2*x^2),x,5,-(d+e*x)^(3/2)/(b*(a+b*x))-3*e*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b*d-a*e)/b^(5/2)+3*e*sqrt(d+e*x)/b^2],
[(d+e*x)^(1/2)/(a^2+2*a*b*x+b^2*x^2),x,4,-e*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(3/2)*sqrt(b*d-a*e))-sqrt(d+e*x)/(b*(a+b*x))],
[1/((d+e*x)^(1/2)*(a^2+2*a*b*x+b^2*x^2)),x,4,e*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/((b*d-a*e)^(3/2)*sqrt(b))-sqrt(d+e*x)/((b*d-a*e)*(a+b*x))],
[1/((d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)),x,5,3*e*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b)/(b*d-a*e)^(5/2)-3*e/((b*d-a*e)^2*sqrt(d+e*x))+(-1)/((b*d-a*e)*(a+b*x)*sqrt(d+e*x))],
[1/((d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)),x,6,-5/3*e/((b*d-a*e)^2*(d+e*x)^(3/2))+(-1)/((b*d-a*e)*(a+b*x)*(d+e*x)^(3/2))+5*b^(3/2)*e*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b*d-a*e)^(7/2)-5*b*e/((b*d-a*e)^3*sqrt(d+e*x))],
[1/((d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2)),x,7,-7/5*e/((b*d-a*e)^2*(d+e*x)^(5/2))+(-1)/((b*d-a*e)*(a+b*x)*(d+e*x)^(5/2))-7/3*b*e/((b*d-a*e)^3*(d+e*x)^(3/2))+7*b^(5/2)*e*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b*d-a*e)^(9/2)-7*b^2*e/((b*d-a*e)^4*sqrt(d+e*x))],
[(d+e*x)^(11/2)/(a^2+2*a*b*x+b^2*x^2)^2,x,9,77/8*e^3*(b*d-a*e)*(d+e*x)^(3/2)/b^5+231/40*e^3*(d+e*x)^(5/2)/b^4-33/8*e^2*(d+e*x)^(7/2)/(b^3*(a+b*x))-11/12*e*(d+e*x)^(9/2)/(b^2*(a+b*x)^2)-1/3*(d+e*x)^(11/2)/(b*(a+b*x)^3)-231/8*e^3*(b*d-a*e)^(5/2)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/b^(13/2)+231/8*e^3*(b*d-a*e)^2*sqrt(d+e*x)/b^6],
[(d+e*x)^(9/2)/(a^2+2*a*b*x+b^2*x^2)^2,x,8,35/8*e^3*(d+e*x)^(3/2)/b^4-21/8*e^2*(d+e*x)^(5/2)/(b^3*(a+b*x))-3/4*e*(d+e*x)^(7/2)/(b^2*(a+b*x)^2)-1/3*(d+e*x)^(9/2)/(b*(a+b*x)^3)-105/8*e^3*(b*d-a*e)^(3/2)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/b^(11/2)+105/8*e^3*(b*d-a*e)*sqrt(d+e*x)/b^5],
[(d+e*x)^(7/2)/(a^2+2*a*b*x+b^2*x^2)^2,x,7,-35/24*e^2*(d+e*x)^(3/2)/(b^3*(a+b*x))-7/12*e*(d+e*x)^(5/2)/(b^2*(a+b*x)^2)-1/3*(d+e*x)^(7/2)/(b*(a+b*x)^3)-35/8*e^3*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b*d-a*e)/b^(9/2)+35/8*e^3*sqrt(d+e*x)/b^4],
[(d+e*x)^(5/2)/(a^2+2*a*b*x+b^2*x^2)^2,x,6,-5/12*e*(d+e*x)^(3/2)/(b^2*(a+b*x)^2)-1/3*(d+e*x)^(5/2)/(b*(a+b*x)^3)-5/8*e^3*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(7/2)*sqrt(b*d-a*e))-5/8*e^2*sqrt(d+e*x)/(b^3*(a+b*x))],
[(d+e*x)^(3/2)/(a^2+2*a*b*x+b^2*x^2)^2,x,6,-1/3*(d+e*x)^(3/2)/(b*(a+b*x)^3)+1/8*e^3*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(5/2)*(b*d-a*e)^(3/2))-1/4*e*sqrt(d+e*x)/(b^2*(a+b*x)^2)-1/8*e^2*sqrt(d+e*x)/(b^2*(b*d-a*e)*(a+b*x))],
[(d+e*x)^(1/2)/(a^2+2*a*b*x+b^2*x^2)^2,x,6,-1/8*e^3*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(3/2)*(b*d-a*e)^(5/2))-1/3*sqrt(d+e*x)/(b*(a+b*x)^3)-1/12*e*sqrt(d+e*x)/(b*(b*d-a*e)*(a+b*x)^2)+1/8*e^2*sqrt(d+e*x)/(b*(b*d-a*e)^2*(a+b*x))],
[1/((d+e*x)^(1/2)*(a^2+2*a*b*x+b^2*x^2)^2),x,6,5/8*e^3*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/((b*d-a*e)^(7/2)*sqrt(b))-1/3*sqrt(d+e*x)/((b*d-a*e)*(a+b*x)^3)+5/12*e*sqrt(d+e*x)/((b*d-a*e)^2*(a+b*x)^2)-5/8*e^2*sqrt(d+e*x)/((b*d-a*e)^3*(a+b*x))],
[1/((d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^2),x,7,35/8*e^3*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b)/(b*d-a*e)^(9/2)-35/8*e^3/((b*d-a*e)^4*sqrt(d+e*x))+(-1/3)/((b*d-a*e)*(a+b*x)^3*sqrt(d+e*x))+7/12*e/((b*d-a*e)^2*(a+b*x)^2*sqrt(d+e*x))-35/24*e^2/((b*d-a*e)^3*(a+b*x)*sqrt(d+e*x))],
[1/((d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^2),x,8,-35/8*e^3/((b*d-a*e)^4*(d+e*x)^(3/2))+(-1/3)/((b*d-a*e)*(a+b*x)^3*(d+e*x)^(3/2))+3/4*e/((b*d-a*e)^2*(a+b*x)^2*(d+e*x)^(3/2))-21/8*e^2/((b*d-a*e)^3*(a+b*x)*(d+e*x)^(3/2))+105/8*b^(3/2)*e^3*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b*d-a*e)^(11/2)-105/8*b*e^3/((b*d-a*e)^5*sqrt(d+e*x))],
[1/((d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2)^2),x,9,-231/40*e^3/((b*d-a*e)^4*(d+e*x)^(5/2))+(-1/3)/((b*d-a*e)*(a+b*x)^3*(d+e*x)^(5/2))+11/12*e/((b*d-a*e)^2*(a+b*x)^2*(d+e*x)^(5/2))-33/8*e^2/((b*d-a*e)^3*(a+b*x)*(d+e*x)^(5/2))-77/8*b*e^3/((b*d-a*e)^5*(d+e*x)^(3/2))+231/8*b^(5/2)*e^3*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b*d-a*e)^(13/2)-231/8*b^2*e^3/((b*d-a*e)^6*sqrt(d+e*x))],
[(d+e*x)^(15/2)/(a^2+2*a*b*x+b^2*x^2)^3,x,11,3003/128*e^5*(b*d-a*e)*(d+e*x)^(3/2)/b^7+9009/640*e^5*(d+e*x)^(5/2)/b^6-1287/128*e^4*(d+e*x)^(7/2)/(b^5*(a+b*x))-143/64*e^3*(d+e*x)^(9/2)/(b^4*(a+b*x)^2)-13/16*e^2*(d+e*x)^(11/2)/(b^3*(a+b*x)^3)-3/8*e*(d+e*x)^(13/2)/(b^2*(a+b*x)^4)-1/5*(d+e*x)^(15/2)/(b*(a+b*x)^5)-9009/128*e^5*(b*d-a*e)^(5/2)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/b^(17/2)+9009/128*e^5*(b*d-a*e)^2*sqrt(d+e*x)/b^8],
[(d+e*x)^(13/2)/(a^2+2*a*b*x+b^2*x^2)^3,x,10,1001/128*e^5*(d+e*x)^(3/2)/b^6-3003/640*e^4*(d+e*x)^(5/2)/(b^5*(a+b*x))-429/320*e^3*(d+e*x)^(7/2)/(b^4*(a+b*x)^2)-143/240*e^2*(d+e*x)^(9/2)/(b^3*(a+b*x)^3)-13/40*e*(d+e*x)^(11/2)/(b^2*(a+b*x)^4)-1/5*(d+e*x)^(13/2)/(b*(a+b*x)^5)-3003/128*e^5*(b*d-a*e)^(3/2)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/b^(15/2)+3003/128*e^5*(b*d-a*e)*sqrt(d+e*x)/b^7],
[(d+e*x)^(11/2)/(a^2+2*a*b*x+b^2*x^2)^3,x,9,-231/128*e^4*(d+e*x)^(3/2)/(b^5*(a+b*x))-231/320*e^3*(d+e*x)^(5/2)/(b^4*(a+b*x)^2)-33/80*e^2*(d+e*x)^(7/2)/(b^3*(a+b*x)^3)-11/40*e*(d+e*x)^(9/2)/(b^2*(a+b*x)^4)-1/5*(d+e*x)^(11/2)/(b*(a+b*x)^5)-693/128*e^5*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b*d-a*e)/b^(13/2)+693/128*e^5*sqrt(d+e*x)/b^6],
[(d+e*x)^(9/2)/(a^2+2*a*b*x+b^2*x^2)^3,x,8,-21/64*e^3*(d+e*x)^(3/2)/(b^4*(a+b*x)^2)-21/80*e^2*(d+e*x)^(5/2)/(b^3*(a+b*x)^3)-9/40*e*(d+e*x)^(7/2)/(b^2*(a+b*x)^4)-1/5*(d+e*x)^(9/2)/(b*(a+b*x)^5)-63/128*e^5*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(11/2)*sqrt(b*d-a*e))-63/128*e^4*sqrt(d+e*x)/(b^5*(a+b*x))],
[(d+e*x)^(7/2)/(a^2+2*a*b*x+b^2*x^2)^3,x,8,-7/48*e^2*(d+e*x)^(3/2)/(b^3*(a+b*x)^3)-7/40*e*(d+e*x)^(5/2)/(b^2*(a+b*x)^4)-1/5*(d+e*x)^(7/2)/(b*(a+b*x)^5)+7/128*e^5*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(9/2)*(b*d-a*e)^(3/2))-7/64*e^3*sqrt(d+e*x)/(b^4*(a+b*x)^2)-7/128*e^4*sqrt(d+e*x)/(b^4*(b*d-a*e)*(a+b*x))],
[(d+e*x)^(5/2)/(a^2+2*a*b*x+b^2*x^2)^3,x,8,-1/8*e*(d+e*x)^(3/2)/(b^2*(a+b*x)^4)-1/5*(d+e*x)^(5/2)/(b*(a+b*x)^5)-3/128*e^5*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(7/2)*(b*d-a*e)^(5/2))-1/16*e^2*sqrt(d+e*x)/(b^3*(a+b*x)^3)-1/64*e^3*sqrt(d+e*x)/(b^3*(b*d-a*e)*(a+b*x)^2)+3/128*e^4*sqrt(d+e*x)/(b^3*(b*d-a*e)^2*(a+b*x))],
[(d+e*x)^(3/2)/(a^2+2*a*b*x+b^2*x^2)^3,x,8,-1/5*(d+e*x)^(3/2)/(b*(a+b*x)^5)+3/128*e^5*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(5/2)*(b*d-a*e)^(7/2))-3/40*e*sqrt(d+e*x)/(b^2*(a+b*x)^4)-1/80*e^2*sqrt(d+e*x)/(b^2*(b*d-a*e)*(a+b*x)^3)+1/64*e^3*sqrt(d+e*x)/(b^2*(b*d-a*e)^2*(a+b*x)^2)-3/128*e^4*sqrt(d+e*x)/(b^2*(b*d-a*e)^3*(a+b*x))],
[(d+e*x)^(1/2)/(a^2+2*a*b*x+b^2*x^2)^3,x,8,-7/128*e^5*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(3/2)*(b*d-a*e)^(9/2))-1/5*sqrt(d+e*x)/(b*(a+b*x)^5)-1/40*e*sqrt(d+e*x)/(b*(b*d-a*e)*(a+b*x)^4)+7/240*e^2*sqrt(d+e*x)/(b*(b*d-a*e)^2*(a+b*x)^3)-7/192*e^3*sqrt(d+e*x)/(b*(b*d-a*e)^3*(a+b*x)^2)+7/128*e^4*sqrt(d+e*x)/(b*(b*d-a*e)^4*(a+b*x))],
[1/((d+e*x)^(1/2)*(a^2+2*a*b*x+b^2*x^2)^3),x,8,63/128*e^5*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/((b*d-a*e)^(11/2)*sqrt(b))-1/5*sqrt(d+e*x)/((b*d-a*e)*(a+b*x)^5)+9/40*e*sqrt(d+e*x)/((b*d-a*e)^2*(a+b*x)^4)-21/80*e^2*sqrt(d+e*x)/((b*d-a*e)^3*(a+b*x)^3)+21/64*e^3*sqrt(d+e*x)/((b*d-a*e)^4*(a+b*x)^2)-63/128*e^4*sqrt(d+e*x)/((b*d-a*e)^5*(a+b*x))],
[1/((d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^3),x,9,693/128*e^5*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b)/(b*d-a*e)^(13/2)-693/128*e^5/((b*d-a*e)^6*sqrt(d+e*x))+(-1/5)/((b*d-a*e)*(a+b*x)^5*sqrt(d+e*x))+11/40*e/((b*d-a*e)^2*(a+b*x)^4*sqrt(d+e*x))-33/80*e^2/((b*d-a*e)^3*(a+b*x)^3*sqrt(d+e*x))+231/320*e^3/((b*d-a*e)^4*(a+b*x)^2*sqrt(d+e*x))-231/128*e^4/((b*d-a*e)^5*(a+b*x)*sqrt(d+e*x))],
[1/((d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^3),x,10,-1001/128*e^5/((b*d-a*e)^6*(d+e*x)^(3/2))+(-1/5)/((b*d-a*e)*(a+b*x)^5*(d+e*x)^(3/2))+13/40*e/((b*d-a*e)^2*(a+b*x)^4*(d+e*x)^(3/2))-143/240*e^2/((b*d-a*e)^3*(a+b*x)^3*(d+e*x)^(3/2))+429/320*e^3/((b*d-a*e)^4*(a+b*x)^2*(d+e*x)^(3/2))-3003/640*e^4/((b*d-a*e)^5*(a+b*x)*(d+e*x)^(3/2))+3003/128*b^(3/2)*e^5*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b*d-a*e)^(15/2)-3003/128*b*e^5/((b*d-a*e)^7*sqrt(d+e*x))],
[1/((d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2)^3),x,11,-9009/640*e^5/((b*d-a*e)^6*(d+e*x)^(5/2))+(-1/5)/((b*d-a*e)*(a+b*x)^5*(d+e*x)^(5/2))+3/8*e/((b*d-a*e)^2*(a+b*x)^4*(d+e*x)^(5/2))-13/16*e^2/((b*d-a*e)^3*(a+b*x)^3*(d+e*x)^(5/2))+143/64*e^3/((b*d-a*e)^4*(a+b*x)^2*(d+e*x)^(5/2))-1287/128*e^4/((b*d-a*e)^5*(a+b*x)*(d+e*x)^(5/2))-3003/128*b*e^5/((b*d-a*e)^7*(d+e*x)^(3/2))+9009/128*b^(5/2)*e^5*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b*d-a*e)^(17/2)-9009/128*b^2*e^5/((b*d-a*e)^8*sqrt(d+e*x))],

# Integrands of the form (d+e x)^(m/2) (a^2+2 a b x+b^2 x^2)^(p/2)

# p>0
[(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2),x,3,-2/7*(b*d-a*e)*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))+2/9*b*(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))],
[(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2),x,3,-2/5*(b*d-a*e)*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))+2/7*b*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))],
[sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2),x,3,-2/3*(b*d-a*e)*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))+2/5*b*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))],
[sqrt(a^2+2*a*b*x+b^2*x^2)/sqrt(d+e*x),x,3,2/3*b*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))-2*(b*d-a*e)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))],
[sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^(3/2),x,3,2*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x)*sqrt(d+e*x))+2*b*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x))],
[sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^(5/2),x,3,2/3*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x)*(d+e*x)^(3/2))-2*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x)*sqrt(d+e*x))],
[sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^(7/2),x,3,2/5*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x)*(d+e*x)^(5/2))-2/3*b*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(a+b*x)*(d+e*x)^(3/2))],
[(d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^(3/2),x,3,-2/7*(b*d-a*e)^3*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))+2/3*b*(b*d-a*e)^2*(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))-6/11*b^2*(b*d-a*e)*(d+e*x)^(11/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))+2/13*b^3*(d+e*x)^(13/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))],
[(d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^(3/2),x,3,-2/5*(b*d-a*e)^3*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))+6/7*b*(b*d-a*e)^2*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))-2/3*b^2*(b*d-a*e)*(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))+2/11*b^3*(d+e*x)^(11/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)*sqrt(d+e*x),x,3,-2/3*(b*d-a*e)^3*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))+6/5*b*(b*d-a*e)^2*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))-6/7*b^2*(b*d-a*e)*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))+2/9*b^3*(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/sqrt(d+e*x),x,3,2*b*(b*d-a*e)^2*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))-6/5*b^2*(b*d-a*e)*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))+2/7*b^3*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))-2*(b*d-a*e)^3*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^(3/2),x,3,-2*b^2*(b*d-a*e)*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))+2/5*b^3*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))+2*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*sqrt(d+e*x))+6*b*(b*d-a*e)^2*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^(5/2),x,3,2/3*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^(3/2))+2/3*b^3*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))-6*b*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*sqrt(d+e*x))-6*b^2*(b*d-a*e)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^(7/2),x,3,2/5*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^(5/2))-2*b*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^(3/2))+6*b^2*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*sqrt(d+e*x))+2*b^3*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^(9/2),x,3,2/7*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^(7/2))-6/5*b*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^(5/2))+2*b^2*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^(3/2))-2*b^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*sqrt(d+e*x))],
[(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^(11/2),x,3,2/9*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^(9/2))-6/7*b*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^(7/2))+6/5*b^2*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^(5/2))-2/3*b^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(a+b*x)*(d+e*x)^(3/2))],
[(d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,-2/7*(b*d-a*e)^5*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+10/9*b*(b*d-a*e)^4*(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))-20/11*b^2*(b*d-a*e)^3*(d+e*x)^(11/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+20/13*b^3*(b*d-a*e)^2*(d+e*x)^(13/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))-2/3*b^4*(b*d-a*e)*(d+e*x)^(15/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+2/17*b^5*(d+e*x)^(17/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))],
[(d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,-2/5*(b*d-a*e)^5*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+10/7*b*(b*d-a*e)^4*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))-20/9*b^2*(b*d-a*e)^3*(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+20/11*b^3*(b*d-a*e)^2*(d+e*x)^(11/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))-10/13*b^4*(b*d-a*e)*(d+e*x)^(13/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+2/15*b^5*(d+e*x)^(15/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)*sqrt(d+e*x),x,3,-2/3*(b*d-a*e)^5*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+2*b*(b*d-a*e)^4*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))-20/7*b^2*(b*d-a*e)^3*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+20/9*b^3*(b*d-a*e)^2*(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))-10/11*b^4*(b*d-a*e)*(d+e*x)^(11/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+2/13*b^5*(d+e*x)^(13/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/sqrt(d+e*x),x,3,10/3*b*(b*d-a*e)^4*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))-4*b^2*(b*d-a*e)^3*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+20/7*b^3*(b*d-a*e)^2*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))-10/9*b^4*(b*d-a*e)*(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+2/11*b^5*(d+e*x)^(11/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))-2*(b*d-a*e)^5*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^(3/2),x,3,-20/3*b^2*(b*d-a*e)^3*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+4*b^3*(b*d-a*e)^2*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))-10/7*b^4*(b*d-a*e)*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+2/9*b^5*(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+2*(b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*sqrt(d+e*x))+10*b*(b*d-a*e)^4*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^(5/2),x,3,2/3*(b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(3/2))+20/3*b^3*(b*d-a*e)^2*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))-2*b^4*(b*d-a*e)*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+2/7*b^5*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))-10*b*(b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*sqrt(d+e*x))-20*b^2*(b*d-a*e)^3*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^(7/2),x,3,2/5*(b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(5/2))-10/3*b*(b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(3/2))-10/3*b^4*(b*d-a*e)*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+2/5*b^5*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))+20*b^2*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*sqrt(d+e*x))+20*b^3*(b*d-a*e)^2*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^(9/2),x,3,2/7*(b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(7/2))-2*b*(b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(5/2))+20/3*b^2*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(3/2))+2/3*b^5*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))-20*b^3*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*sqrt(d+e*x))-10*b^4*(b*d-a*e)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^(11/2),x,3,2/9*(b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(9/2))-10/7*b*(b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(7/2))+4*b^2*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(5/2))-20/3*b^3*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(3/2))+10*b^4*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*sqrt(d+e*x))+2*b^5*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^(13/2),x,3,2/11*(b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(11/2))-10/9*b*(b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(9/2))+20/7*b^2*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(7/2))-4*b^3*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(5/2))+10/3*b^4*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(3/2))-2*b^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*sqrt(d+e*x))],
[(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^(15/2),x,3,2/13*(b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(13/2))-10/11*b*(b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(11/2))+20/9*b^2*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(9/2))-20/7*b^3*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(7/2))+2*b^4*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(5/2))-2/3*b^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(a+b*x)*(d+e*x)^(3/2))],

# p<0
[(d+e*x)^(7/2)/sqrt(a^2+2*a*b*x+b^2*x^2),x,7,2/3*(b*d-a*e)^2*(a+b*x)*(d+e*x)^(3/2)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))+2/5*(b*d-a*e)*(a+b*x)*(d+e*x)^(5/2)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))+2/7*(a+b*x)*(d+e*x)^(7/2)/(b*sqrt(a^2+2*a*b*x+b^2*x^2))-2*(b*d-a*e)^(7/2)*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+2*(b*d-a*e)^3*(a+b*x)*sqrt(d+e*x)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^(5/2)/sqrt(a^2+2*a*b*x+b^2*x^2),x,6,2/3*(b*d-a*e)*(a+b*x)*(d+e*x)^(3/2)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))+2/5*(a+b*x)*(d+e*x)^(5/2)/(b*sqrt(a^2+2*a*b*x+b^2*x^2))-2*(b*d-a*e)^(5/2)*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+2*(b*d-a*e)^2*(a+b*x)*sqrt(d+e*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^(3/2)/sqrt(a^2+2*a*b*x+b^2*x^2),x,5,2/3*(a+b*x)*(d+e*x)^(3/2)/(b*sqrt(a^2+2*a*b*x+b^2*x^2))-2*(b*d-a*e)^(3/2)*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+2*(b*d-a*e)*(a+b*x)*sqrt(d+e*x)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))],
[sqrt(d+e*x)/sqrt(a^2+2*a*b*x+b^2*x^2),x,4,-2*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b*d-a*e)/(b^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+2*(a+b*x)*sqrt(d+e*x)/(b*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/(sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)),x,3,-2*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(sqrt(b)*sqrt(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)),x,4,-2*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b)/((b*d-a*e)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+2*(a+b*x)/((b*d-a*e)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)),x,5,2/3*(a+b*x)/((b*d-a*e)*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))-2*b^(3/2)*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/((b*d-a*e)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+2*b*(a+b*x)/((b*d-a*e)^2*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)),x,6,2/5*(a+b*x)/((b*d-a*e)*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+2/3*b*(a+b*x)/((b*d-a*e)^2*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))-2*b^(5/2)*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/((b*d-a*e)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+2*b^2*(a+b*x)/((b*d-a*e)^3*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^(9/2)/(a^2+2*a*b*x+b^2*x^2)^(3/2),x,8,21/4*e^2*(b*d-a*e)*(a+b*x)*(d+e*x)^(3/2)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))+63/20*e^2*(a+b*x)*(d+e*x)^(5/2)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))-9/4*e*(d+e*x)^(7/2)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))-1/2*(d+e*x)^(9/2)/(b*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))-63/4*e^2*(b*d-a*e)^(5/2)*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(11/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+63/4*e^2*(b*d-a*e)^2*(a+b*x)*sqrt(d+e*x)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^(7/2)/(a^2+2*a*b*x+b^2*x^2)^(3/2),x,7,35/12*e^2*(a+b*x)*(d+e*x)^(3/2)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))-7/4*e*(d+e*x)^(5/2)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))-1/2*(d+e*x)^(7/2)/(b*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))-35/4*e^2*(b*d-a*e)^(3/2)*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+35/4*e^2*(b*d-a*e)*(a+b*x)*sqrt(d+e*x)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^(5/2)/(a^2+2*a*b*x+b^2*x^2)^(3/2),x,6,-5/4*e*(d+e*x)^(3/2)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))-1/2*(d+e*x)^(5/2)/(b*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))-15/4*e^2*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b*d-a*e)/(b^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+15/4*e^2*(a+b*x)*sqrt(d+e*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^(3/2)/(a^2+2*a*b*x+b^2*x^2)^(3/2),x,5,-1/2*(d+e*x)^(3/2)/(b*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))-3/4*e^2*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(5/2)*sqrt(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2))-3/4*e*sqrt(d+e*x)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))],
[sqrt(d+e*x)/(a^2+2*a*b*x+b^2*x^2)^(3/2),x,5,1/4*e^2*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(3/2)*(b*d-a*e)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*e*sqrt(d+e*x)/(b*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2))-1/2*sqrt(d+e*x)/(b*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((a^2+2*a*b*x+b^2*x^2)^(3/2)*sqrt(d+e*x)),x,5,-3/4*e^2*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/((b*d-a*e)^(5/2)*sqrt(b)*sqrt(a^2+2*a*b*x+b^2*x^2))+3/4*e*sqrt(d+e*x)/((b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-1/2*sqrt(d+e*x)/((b*d-a*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^(3/2)),x,6,-15/4*e^2*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b)/((b*d-a*e)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+5/4*e/((b*d-a*e)^2*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+(-1/2)/((b*d-a*e)*(a+b*x)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+15/4*e^2*(a+b*x)/((b*d-a*e)^3*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^(3/2)),x,7,7/4*e/((b*d-a*e)^2*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+(-1/2)/((b*d-a*e)*(a+b*x)*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+35/12*e^2*(a+b*x)/((b*d-a*e)^3*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))-35/4*b^(3/2)*e^2*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/((b*d-a*e)^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+35/4*b*e^2*(a+b*x)/((b*d-a*e)^4*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2)^(3/2)),x,8,9/4*e/((b*d-a*e)^2*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+(-1/2)/((b*d-a*e)*(a+b*x)*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+63/20*e^2*(a+b*x)/((b*d-a*e)^3*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+21/4*b*e^2*(a+b*x)/((b*d-a*e)^4*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))-63/4*b^(5/2)*e^2*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/((b*d-a*e)^(11/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+63/4*b^2*e^2*(a+b*x)/((b*d-a*e)^5*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^(13/2)/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,10,1001/64*e^4*(b*d-a*e)*(a+b*x)*(d+e*x)^(3/2)/(b^6*sqrt(a^2+2*a*b*x+b^2*x^2))+3003/320*e^4*(a+b*x)*(d+e*x)^(5/2)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))-429/64*e^3*(d+e*x)^(7/2)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))-143/96*e^2*(d+e*x)^(9/2)/(b^3*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))-13/24*e*(d+e*x)^(11/2)/(b^2*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*(d+e*x)^(13/2)/(b*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-3003/64*e^4*(b*d-a*e)^(5/2)*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(15/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+3003/64*e^4*(b*d-a*e)^2*(a+b*x)*sqrt(d+e*x)/(b^7*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^(11/2)/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,9,385/64*e^4*(a+b*x)*(d+e*x)^(3/2)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))-231/64*e^3*(d+e*x)^(5/2)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))-33/32*e^2*(d+e*x)^(7/2)/(b^3*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))-11/24*e*(d+e*x)^(9/2)/(b^2*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*(d+e*x)^(11/2)/(b*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-1155/64*e^4*(b*d-a*e)^(3/2)*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(13/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+1155/64*e^4*(b*d-a*e)*(a+b*x)*sqrt(d+e*x)/(b^6*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^(9/2)/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,8,-105/64*e^3*(d+e*x)^(3/2)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))-21/32*e^2*(d+e*x)^(5/2)/(b^3*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))-3/8*e*(d+e*x)^(7/2)/(b^2*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*(d+e*x)^(9/2)/(b*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-315/64*e^4*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b*d-a*e)/(b^(11/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+315/64*e^4*(a+b*x)*sqrt(d+e*x)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^(7/2)/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,7,-35/96*e^2*(d+e*x)^(3/2)/(b^3*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))-7/24*e*(d+e*x)^(5/2)/(b^2*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*(d+e*x)^(7/2)/(b*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-35/64*e^4*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(9/2)*sqrt(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2))-35/64*e^3*sqrt(d+e*x)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^(5/2)/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,7,-5/24*e*(d+e*x)^(3/2)/(b^2*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*(d+e*x)^(5/2)/(b*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))+5/64*e^4*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(7/2)*(b*d-a*e)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))-5/64*e^3*sqrt(d+e*x)/(b^3*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2))-5/32*e^2*sqrt(d+e*x)/(b^3*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^(3/2)/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,7,-1/4*(d+e*x)^(3/2)/(b*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-3/64*e^4*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(5/2)*(b*d-a*e)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+3/64*e^3*sqrt(d+e*x)/(b^2*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-1/8*e*sqrt(d+e*x)/(b^2*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-1/32*e^2*sqrt(d+e*x)/(b^2*(b*d-a*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[sqrt(d+e*x)/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,7,5/64*e^4*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(3/2)*(b*d-a*e)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2))-5/64*e^3*sqrt(d+e*x)/(b*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*sqrt(d+e*x)/(b*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-1/24*e*sqrt(d+e*x)/(b*(b*d-a*e)*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))+5/96*e^2*sqrt(d+e*x)/(b*(b*d-a*e)^2*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((a^2+2*a*b*x+b^2*x^2)^(5/2)*sqrt(d+e*x)),x,7,-35/64*e^4*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/((b*d-a*e)^(9/2)*sqrt(b)*sqrt(a^2+2*a*b*x+b^2*x^2))+35/64*e^3*sqrt(d+e*x)/((b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*sqrt(d+e*x)/((b*d-a*e)*(a+b*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))+7/24*e*sqrt(d+e*x)/((b*d-a*e)^2*(a+b*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-35/96*e^2*sqrt(d+e*x)/((b*d-a*e)^3*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^(5/2)),x,8,-315/64*e^4*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b)/((b*d-a*e)^(11/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+105/64*e^3/((b*d-a*e)^4*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+(-1/4)/((b*d-a*e)*(a+b*x)^3*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+3/8*e/((b*d-a*e)^2*(a+b*x)^2*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))-21/32*e^2/((b*d-a*e)^3*(a+b*x)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+315/64*e^4*(a+b*x)/((b*d-a*e)^5*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^(5/2)),x,9,231/64*e^3/((b*d-a*e)^4*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+(-1/4)/((b*d-a*e)*(a+b*x)^3*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+11/24*e/((b*d-a*e)^2*(a+b*x)^2*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))-33/32*e^2/((b*d-a*e)^3*(a+b*x)*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+385/64*e^4*(a+b*x)/((b*d-a*e)^5*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))-1155/64*b^(3/2)*e^4*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/((b*d-a*e)^(13/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+1155/64*b*e^4*(a+b*x)/((b*d-a*e)^6*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/((d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2)^(5/2)),x,10,429/64*e^3/((b*d-a*e)^4*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+(-1/4)/((b*d-a*e)*(a+b*x)^3*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+13/24*e/((b*d-a*e)^2*(a+b*x)^2*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2))-143/96*e^2/((b*d-a*e)^3*(a+b*x)*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+3003/320*e^4*(a+b*x)/((b*d-a*e)^5*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+1001/64*b*e^4*(a+b*x)/((b*d-a*e)^6*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))-3003/64*b^(5/2)*e^4*(a+b*x)*arctanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/((b*d-a*e)^(15/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+3003/64*b^2*e^4*(a+b*x)/((b*d-a*e)^7*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))],

# Integrands of the form (d+e x)^m (a^2+2 a b x+b^2 x^2)^p when m symbolic
[(d+e*x)^m*(a^2+2*a*b*x+b^2*x^2)^3,x,3,(b*d-a*e)^6*(d+e*x)^(1+m)/(e^7*(1+m))-6*b*(b*d-a*e)^5*(d+e*x)^(2+m)/(e^7*(2+m))+15*b^2*(b*d-a*e)^4*(d+e*x)^(3+m)/(e^7*(3+m))-20*b^3*(b*d-a*e)^3*(d+e*x)^(4+m)/(e^7*(4+m))+15*b^4*(b*d-a*e)^2*(d+e*x)^(5+m)/(e^7*(5+m))-6*b^5*(b*d-a*e)*(d+e*x)^(6+m)/(e^7*(6+m))+b^6*(d+e*x)^(7+m)/(e^7*(7+m))],
[(d+e*x)^m*(a^2+2*a*b*x+b^2*x^2)^2,x,3,(b*d-a*e)^4*(d+e*x)^(1+m)/(e^5*(1+m))-4*b*(b*d-a*e)^3*(d+e*x)^(2+m)/(e^5*(2+m))+6*b^2*(b*d-a*e)^2*(d+e*x)^(3+m)/(e^5*(3+m))-4*b^3*(b*d-a*e)*(d+e*x)^(4+m)/(e^5*(4+m))+b^4*(d+e*x)^(5+m)/(e^5*(5+m))],
[(d+e*x)^m*(a^2+2*a*b*x+b^2*x^2),x,3,(b*d-a*e)^2*(d+e*x)^(1+m)/(e^3*(1+m))-2*b*(b*d-a*e)*(d+e*x)^(2+m)/(e^3*(2+m))+b^2*(d+e*x)^(3+m)/(e^3*(3+m))],
[(d+e*x)^m/(a^2+2*a*b*x+b^2*x^2),x,2,e*(d+e*x)^(1+m)*hypergeom([2,1+m],[2+m],b*(d+e*x)/(b*d-a*e))/((b*d-a*e)^2*(1+m))],
[(d+e*x)^m/(a^2+2*a*b*x+b^2*x^2)^2,x,2,e^3*(d+e*x)^(1+m)*hypergeom([4,1+m],[2+m],b*(d+e*x)/(b*d-a*e))/((b*d-a*e)^4*(1+m))],
[(d+e*x)^m/(a^2+2*a*b*x+b^2*x^2)^3,x,2,e^5*(d+e*x)^(1+m)*hypergeom([6,1+m],[2+m],b*(d+e*x)/(b*d-a*e))/((b*d-a*e)^6*(1+m))],
[(d+e*x)^m*(a^2+2*a*b*x+b^2*x^2)^(5/2),x,3,-(b*d-a*e)^5*(d+e*x)^(1+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(1+m)*(a+b*x))+5*b*(b*d-a*e)^4*(d+e*x)^(2+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(2+m)*(a+b*x))-10*b^2*(b*d-a*e)^3*(d+e*x)^(3+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(3+m)*(a+b*x))+10*b^3*(b*d-a*e)^2*(d+e*x)^(4+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(4+m)*(a+b*x))-5*b^4*(b*d-a*e)*(d+e*x)^(5+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(5+m)*(a+b*x))+b^5*(d+e*x)^(6+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(6+m)*(a+b*x))],
[(d+e*x)^m*(a^2+2*a*b*x+b^2*x^2)^(3/2),x,3,-(b*d-a*e)^3*(d+e*x)^(1+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(1+m)*(a+b*x))+3*b*(b*d-a*e)^2*(d+e*x)^(2+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(2+m)*(a+b*x))-3*b^2*(b*d-a*e)*(d+e*x)^(3+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(3+m)*(a+b*x))+b^3*(d+e*x)^(4+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(4+m)*(a+b*x))],
[(d+e*x)^m*(a^2+2*a*b*x+b^2*x^2)^(1/2),x,3,-(b*d-a*e)*(d+e*x)^(1+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(1+m)*(a+b*x))+b*(d+e*x)^(2+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(2+m)*(a+b*x))],
[(d+e*x)^m/(a^2+2*a*b*x+b^2*x^2)^(1/2),x,2,-(a+b*x)*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],b*(d+e*x)/(b*d-a*e))/((b*d-a*e)*(1+m)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^m/(a^2+2*a*b*x+b^2*x^2)^(3/2),x,2,-e^2*(a+b*x)*(d+e*x)^(1+m)*hypergeom([3,1+m],[2+m],b*(d+e*x)/(b*d-a*e))/((b*d-a*e)^3*(1+m)*sqrt(a^2+2*a*b*x+b^2*x^2))],
[(d+e*x)^m/(a^2+2*a*b*x+b^2*x^2)^(5/2),x,2,-e^4*(a+b*x)*(d+e*x)^(1+m)*hypergeom([5,1+m],[2+m],b*(d+e*x)/(b*d-a*e))/((b*d-a*e)^5*(1+m)*sqrt(a^2+2*a*b*x+b^2*x^2))],

# Integrands of the form (d+e x)^m (a^2+2 a b x+b^2 x^2)^p when p symbolic
[(d+e*x)^m*(a^2+2*a*b*x+b^2*x^2)^p,x,3,(d+e*x)^(1+m)*(a^2+2*a*b*x+b^2*x^2)^p*hypergeom([1+m,-2*p],[2+m],b*(d+e*x)/(b*d-a*e))/(e*(1+m)*(-e*(a+b*x)/(b*d-a*e))^(2*p))],
[(d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)^p,x,3,(b*d-a*e)^3*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^p/(b^4*(1+2*p))+3/2*e*(b*d-a*e)^2*(a+b*x)^2*(a^2+2*a*b*x+b^2*x^2)^p/(b^4*(1+p))+3*e^2*(b*d-a*e)*(a+b*x)^3*(a^2+2*a*b*x+b^2*x^2)^p/(b^4*(3+2*p))+1/2*e^3*(a+b*x)^4*(a^2+2*a*b*x+b^2*x^2)^p/(b^4*(2+p))],
[(d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^p,x,3,(b*d-a*e)^2*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^p/(b^3*(1+2*p))+e*(b*d-a*e)*(a+b*x)^2*(a^2+2*a*b*x+b^2*x^2)^p/(b^3*(1+p))+e^2*(a+b*x)^3*(a^2+2*a*b*x+b^2*x^2)^p/(b^3*(3+2*p))],
[(d+e*x)*(a^2+2*a*b*x+b^2*x^2)^p,x,2,(b*d-a*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^p/(b^2*(1+2*p))+1/2*e*(a^2+2*a*b*x+b^2*x^2)^(1+p)/(b^2*(1+p))],
[(a^2+2*a*b*x+b^2*x^2)^p,x,1,(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^p/(b*(1+2*p))],
[(a^2+2*a*b*x+b^2*x^2)^p/(d+e*x),x,2,(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^p*hypergeom([1,1+2*p],[2*(1+p)],-e*(a+b*x)/(b*d-a*e))/((b*d-a*e)*(1+2*p))],
[(a^2+2*a*b*x+b^2*x^2)^p/(d+e*x)^2,x,2,b*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^p*hypergeom([2,1+2*p],[2*(1+p)],-e*(a+b*x)/(b*d-a*e))/((b*d-a*e)^2*(1+2*p))],
[(a^2+2*a*b*x+b^2*x^2)^p/(d+e*x)^3,x,2,b^2*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^p*hypergeom([3,1+2*p],[2*(1+p)],-e*(a+b*x)/(b*d-a*e))/((b*d-a*e)^3*(1+2*p))],
[(d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^p,x,3,2/5*(d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^p*hypergeom([5/2,-2*p],[7/2],b*(d+e*x)/(b*d-a*e))/(e*(-e*(a+b*x)/(b*d-a*e))^(2*p))],
[(d+e*x)^(1/2)*(a^2+2*a*b*x+b^2*x^2)^p,x,3,2/3*(d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^p*hypergeom([3/2,-2*p],[5/2],b*(d+e*x)/(b*d-a*e))/(e*(-e*(a+b*x)/(b*d-a*e))^(2*p))],
[(a^2+2*a*b*x+b^2*x^2)^p/(d+e*x)^(1/2),x,3,2*(a^2+2*a*b*x+b^2*x^2)^p*hypergeom([1/2,-2*p],[3/2],b*(d+e*x)/(b*d-a*e))*sqrt(d+e*x)/(e*(-e*(a+b*x)/(b*d-a*e))^(2*p))],
[(a^2+2*a*b*x+b^2*x^2)^p/(d+e*x)^(3/2),x,3,-2*(a^2+2*a*b*x+b^2*x^2)^p*hypergeom([-1/2,-2*p],[1/2],b*(d+e*x)/(b*d-a*e))/(e*(-e*(a+b*x)/(b*d-a*e))^(2*p)*sqrt(d+e*x))],
[(a^2+2*a*b*x+b^2*x^2)^p/(d+e*x)^(5/2),x,3,-2/3*(a^2+2*a*b*x+b^2*x^2)^p*hypergeom([-3/2,-2*p],[-1/2],b*(d+e*x)/(b*d-a*e))/(e*(-e*(a+b*x)/(b*d-a*e))^(2*p)*(d+e*x)^(3/2))],
[(d+e*x)^m*(a^2+2*a*b*x+b^2*x^2)^(5+p),x,3,(b*d-a*e)^10*(d+e*x)^(1+m)*(a^2+2*a*b*x+b^2*x^2)^p*hypergeom([1+m,-2*(5+p)],[2+m],b*(d+e*x)/(b*d-a*e))/(e^11*(1+m)*(-e*(a+b*x)/(b*d-a*e))^(2*p))],
[(d+e*x)^(-3-2*p)*(a^2+2*a*b*x+b^2*x^2)^p,x,3,1/2*b*(a+b*x)*(d+e*x)^(-1-2*p)*(a^2+2*a*b*x+b^2*x^2)^p/((b*d-a*e)^2*(1+p)*(1+2*p))+1/2*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^p/((b*d-a*e)*(1+p)*(d+e*x)^(2*(1+p)))],
[(d+e*x)*(9+12*x+4*x^2)^p,x,2,1/4*(2*d-3*e)*(3+2*x)*(9+12*x+4*x^2)^p/(1+2*p)+1/8*e*(9+12*x+4*x^2)^(1+p)/(1+p)],

# Integrands of the form (d+e x)^m (a+b x+c x^2)^p when c d^2-b d e+a e^2=0

# Integrands of the form (a+b x)^m (a c+(b c+a d) x+b d x^2)^p

# p>0
[(a+b*x)^3*(a*c+(b*c+a*d)*x+b*d*x^2),x,3,1/5*(b*c-a*d)*(a+b*x)^5/b^2+1/6*d*(a+b*x)^6/b^2],
[(a+b*x)^2*(a*c+(b*c+a*d)*x+b*d*x^2),x,3,1/4*(b*c-a*d)*(a+b*x)^4/b^2+1/5*d*(a+b*x)^5/b^2],
[(a+b*x)*(a*c+(b*c+a*d)*x+b*d*x^2),x,3,1/3*(b*c-a*d)*(a+b*x)^3/b^2+1/4*d*(a+b*x)^4/b^2],
[a*c+(b*c+a*d)*x+b*d*x^2,x,1,a*c*x+1/2*(b*c+a*d)*x^2+1/3*b*d*x^3],
[(a*c+(b*c+a*d)*x+b*d*x^2)/(a+b*x),x,2,c*x+1/2*d*x^2],
[(a*c+(b*c+a*d)*x+b*d*x^2)/(a+b*x)^2,x,3,d*x/b+(b*c-a*d)*log(a+b*x)/b^2],
[(a*c+(b*c+a*d)*x+b*d*x^2)/(a+b*x)^3,x,3,(-b*c+a*d)/(b^2*(a+b*x))+d*log(a+b*x)/b^2],
[(a*c+(b*c+a*d)*x+b*d*x^2)/(a+b*x)^4,x,2,-1/2*(c+d*x)^2/((b*c-a*d)*(a+b*x)^2)],
[(a*c+(b*c+a*d)*x+b*d*x^2)/(a+b*x)^5,x,3,1/3*(-b*c+a*d)/(b^2*(a+b*x)^3)-1/2*d/(b^2*(a+b*x)^2)],
[(a*c+(b*c+a*d)*x+b*d*x^2)/(a+b*x)^6,x,3,1/4*(-b*c+a*d)/(b^2*(a+b*x)^4)-1/3*d/(b^2*(a+b*x)^3)],
[(a+b*x)^3*(a*c+(b*c+a*d)*x+b*d*x^2)^2,x,3,1/6*(b*c-a*d)^2*(a+b*x)^6/b^3+2/7*d*(b*c-a*d)*(a+b*x)^7/b^3+1/8*d^2*(a+b*x)^8/b^3],
[(a+b*x)^2*(a*c+(b*c+a*d)*x+b*d*x^2)^2,x,3,1/5*(b*c-a*d)^2*(a+b*x)^5/b^3+1/3*d*(b*c-a*d)*(a+b*x)^6/b^3+1/7*d^2*(a+b*x)^7/b^3],
[(a+b*x)*(a*c+(b*c+a*d)*x+b*d*x^2)^2,x,3,1/4*(b*c-a*d)^2*(a+b*x)^4/b^3+2/5*d*(b*c-a*d)*(a+b*x)^5/b^3+1/6*d^2*(a+b*x)^6/b^3],
[(a*c+(b*c+a*d)*x+b*d*x^2)^2,x,3,1/3*(b*c-a*d)^2*(c+d*x)^3/d^3-1/2*b*(b*c-a*d)*(c+d*x)^4/d^3+1/5*b^2*(c+d*x)^5/d^3],
[(a*c+(b*c+a*d)*x+b*d*x^2)^2/(a+b*x),x,3,-1/3*(b*c-a*d)*(c+d*x)^3/d^2+1/4*b*(c+d*x)^4/d^2],
[(a*c+(b*c+a*d)*x+b*d*x^2)^2/(a+b*x)^2,x,2,1/3*(c+d*x)^3/d],
[(a*c+(b*c+a*d)*x+b*d*x^2)^2/(a+b*x)^3,x,3,d*(b*c-a*d)*x/b^2+1/2*(c+d*x)^2/b+(b*c-a*d)^2*log(a+b*x)/b^3],
[(a*c+(b*c+a*d)*x+b*d*x^2)^2/(a+b*x)^4,x,3,d^2*x/b^2-(b*c-a*d)^2/(b^3*(a+b*x))+2*d*(b*c-a*d)*log(a+b*x)/b^3],
[(a*c+(b*c+a*d)*x+b*d*x^2)^2/(a+b*x)^5,x,3,-1/2*(b*c-a*d)^2/(b^3*(a+b*x)^2)-2*d*(b*c-a*d)/(b^3*(a+b*x))+d^2*log(a+b*x)/b^3],
[(a*c+(b*c+a*d)*x+b*d*x^2)^2/(a+b*x)^6,x,2,-1/3*(c+d*x)^3/((b*c-a*d)*(a+b*x)^3)],
[(a*c+(b*c+a*d)*x+b*d*x^2)^2/(a+b*x)^7,x,3,-1/4*(b*c-a*d)^2/(b^3*(a+b*x)^4)-2/3*d*(b*c-a*d)/(b^3*(a+b*x)^3)-1/2*d^2/(b^3*(a+b*x)^2)],
[(a*c+(b*c+a*d)*x+b*d*x^2)^2/(a+b*x)^8,x,3,-1/5*(b*c-a*d)^2/(b^3*(a+b*x)^5)-1/2*d*(b*c-a*d)/(b^3*(a+b*x)^4)-1/3*d^2/(b^3*(a+b*x)^3)],
[(a*c+(b*c+a*d)*x+b*d*x^2)^2/(a+b*x)^9,x,3,-1/6*(b*c-a*d)^2/(b^3*(a+b*x)^6)-2/5*d*(b*c-a*d)/(b^3*(a+b*x)^5)-1/4*d^2/(b^3*(a+b*x)^4)],
[(a*c+(b*c+a*d)*x+b*d*x^2)^2/(a+b*x)^10,x,3,-1/7*(b*c-a*d)^2/(b^3*(a+b*x)^7)-1/3*d*(b*c-a*d)/(b^3*(a+b*x)^6)-1/5*d^2/(b^3*(a+b*x)^5)],
[(a+b*x)^3*(a*c+(b*c+a*d)*x+b*d*x^2)^3,x,3,1/7*(b*c-a*d)^3*(a+b*x)^7/b^4+3/8*d*(b*c-a*d)^2*(a+b*x)^8/b^4+1/3*d^2*(b*c-a*d)*(a+b*x)^9/b^4+1/10*d^3*(a+b*x)^10/b^4],
[(a+b*x)^2*(a*c+(b*c+a*d)*x+b*d*x^2)^3,x,3,1/6*(b*c-a*d)^3*(a+b*x)^6/b^4+3/7*d*(b*c-a*d)^2*(a+b*x)^7/b^4+3/8*d^2*(b*c-a*d)*(a+b*x)^8/b^4+1/9*d^3*(a+b*x)^9/b^4],
[(a+b*x)*(a*c+(b*c+a*d)*x+b*d*x^2)^3,x,3,1/5*(b*c-a*d)^3*(a+b*x)^5/b^4+1/2*d*(b*c-a*d)^2*(a+b*x)^6/b^4+3/7*d^2*(b*c-a*d)*(a+b*x)^7/b^4+1/8*d^3*(a+b*x)^8/b^4],
[(a*c+(b*c+a*d)*x+b*d*x^2)^3,x,3,-1/4*(b*c-a*d)^3*(c+d*x)^4/d^4+3/5*b*(b*c-a*d)^2*(c+d*x)^5/d^4-1/2*b^2*(b*c-a*d)*(c+d*x)^6/d^4+1/7*b^3*(c+d*x)^7/d^4],
[(a*c+(b*c+a*d)*x+b*d*x^2)^3/(a+b*x),x,3,1/4*(b*c-a*d)^2*(c+d*x)^4/d^3-2/5*b*(b*c-a*d)*(c+d*x)^5/d^3+1/6*b^2*(c+d*x)^6/d^3],
[(a*c+(b*c+a*d)*x+b*d*x^2)^3/(a+b*x)^2,x,3,-1/4*(b*c-a*d)*(c+d*x)^4/d^2+1/5*b*(c+d*x)^5/d^2],
[(a*c+(b*c+a*d)*x+b*d*x^2)^3/(a+b*x)^3,x,2,1/4*(c+d*x)^4/d],
[(a*c+(b*c+a*d)*x+b*d*x^2)^3/(a+b*x)^4,x,3,d*(b*c-a*d)^2*x/b^3+1/2*(b*c-a*d)*(c+d*x)^2/b^2+1/3*(c+d*x)^3/b+(b*c-a*d)^3*log(a+b*x)/b^4],
[(a*c+(b*c+a*d)*x+b*d*x^2)^3/(a+b*x)^5,x,3,d^2*(3*b*c-2*a*d)*x/b^3+1/2*d^3*x^2/b^2-(b*c-a*d)^3/(b^4*(a+b*x))+3*d*(b*c-a*d)^2*log(a+b*x)/b^4],
[(a*c+(b*c+a*d)*x+b*d*x^2)^3/(a+b*x)^6,x,3,d^3*x/b^3-1/2*(b*c-a*d)^3/(b^4*(a+b*x)^2)-3*d*(b*c-a*d)^2/(b^4*(a+b*x))+3*d^2*(b*c-a*d)*log(a+b*x)/b^4],
[(a*c+(b*c+a*d)*x+b*d*x^2)^3/(a+b*x)^7,x,3,-1/3*(b*c-a*d)^3/(b^4*(a+b*x)^3)-3/2*d*(b*c-a*d)^2/(b^4*(a+b*x)^2)-3*d^2*(b*c-a*d)/(b^4*(a+b*x))+d^3*log(a+b*x)/b^4],
[(a*c+(b*c+a*d)*x+b*d*x^2)^3/(a+b*x)^8,x,2,-1/4*(c+d*x)^4/((b*c-a*d)*(a+b*x)^4)],
[(a*c+(b*c+a*d)*x+b*d*x^2)^3/(a+b*x)^9,x,3,-1/5*(c+d*x)^4/((b*c-a*d)*(a+b*x)^5)+1/20*d*(c+d*x)^4/((b*c-a*d)^2*(a+b*x)^4)],
[(a*c+(b*c+a*d)*x+b*d*x^2)^3/(a+b*x)^10,x,3,-1/6*(b*c-a*d)^3/(b^4*(a+b*x)^6)-3/5*d*(b*c-a*d)^2/(b^4*(a+b*x)^5)-3/4*d^2*(b*c-a*d)/(b^4*(a+b*x)^4)-1/3*d^3/(b^4*(a+b*x)^3)],
[(a*c+(b*c+a*d)*x+b*d*x^2)^3/(a+b*x)^11,x,3,-1/7*(b*c-a*d)^3/(b^4*(a+b*x)^7)-1/2*d*(b*c-a*d)^2/(b^4*(a+b*x)^6)-3/5*d^2*(b*c-a*d)/(b^4*(a+b*x)^5)-1/4*d^3/(b^4*(a+b*x)^4)],
[(a*c+(b*c+a*d)*x+b*d*x^2)^3/(a+b*x)^12,x,3,-1/8*(b*c-a*d)^3/(b^4*(a+b*x)^8)-3/7*d*(b*c-a*d)^2/(b^4*(a+b*x)^7)-1/2*d^2*(b*c-a*d)/(b^4*(a+b*x)^6)-1/5*d^3/(b^4*(a+b*x)^5)],

# p<0
[(a+b*x)^6/(a*c+(b*c+a*d)*x+b*d*x^2),x,3,b*(b*c-a*d)^4*x/d^5-1/2*(b*c-a*d)^3*(a+b*x)^2/d^4+1/3*(b*c-a*d)^2*(a+b*x)^3/d^3-1/4*(b*c-a*d)*(a+b*x)^4/d^2+1/5*(a+b*x)^5/d-(b*c-a*d)^5*log(c+d*x)/d^6],
[(a+b*x)^5/(a*c+(b*c+a*d)*x+b*d*x^2),x,3,-b*(b*c-a*d)^3*x/d^4+1/2*(b*c-a*d)^2*(a+b*x)^2/d^3-1/3*(b*c-a*d)*(a+b*x)^3/d^2+1/4*(a+b*x)^4/d+(b*c-a*d)^4*log(c+d*x)/d^5],
[(a+b*x)^4/(a*c+(b*c+a*d)*x+b*d*x^2),x,3,b*(b*c-a*d)^2*x/d^3-1/2*(b*c-a*d)*(a+b*x)^2/d^2+1/3*(a+b*x)^3/d-(b*c-a*d)^3*log(c+d*x)/d^4],
[(a+b*x)^3/(a*c+(b*c+a*d)*x+b*d*x^2),x,3,-b*(b*c-a*d)*x/d^2+1/2*(a+b*x)^2/d+(b*c-a*d)^2*log(c+d*x)/d^3],
[(a+b*x)^2/(a*c+(b*c+a*d)*x+b*d*x^2),x,3,b*x/d-(b*c-a*d)*log(c+d*x)/d^2],
[(a+b*x)/(a*c+(b*c+a*d)*x+b*d*x^2),x,2,log(c+d*x)/d],
[1/(a*c+(b*c+a*d)*x+b*d*x^2),x,3,log(a+b*x)/(b*c-a*d)-log(c+d*x)/(b*c-a*d)],
[1/((a+b*x)*(a*c+(b*c+a*d)*x+b*d*x^2)),x,3,(-1)/((b*c-a*d)*(a+b*x))-d*log(a+b*x)/(b*c-a*d)^2+d*log(c+d*x)/(b*c-a*d)^2],
[1/((a+b*x)^2*(a*c+(b*c+a*d)*x+b*d*x^2)),x,3,(-1/2)/((b*c-a*d)*(a+b*x)^2)+d/((b*c-a*d)^2*(a+b*x))+d^2*log(a+b*x)/(b*c-a*d)^3-d^2*log(c+d*x)/(b*c-a*d)^3],
[1/((a+b*x)^3*(a*c+(b*c+a*d)*x+b*d*x^2)),x,3,(-1/3)/((b*c-a*d)*(a+b*x)^3)+1/2*d/((b*c-a*d)^2*(a+b*x)^2)-d^2/((b*c-a*d)^3*(a+b*x))-d^3*log(a+b*x)/(b*c-a*d)^4+d^3*log(c+d*x)/(b*c-a*d)^4],
[1/((a+b*x)^4*(a*c+(b*c+a*d)*x+b*d*x^2)),x,3,(-1/4)/((b*c-a*d)*(a+b*x)^4)+1/3*d/((b*c-a*d)^2*(a+b*x)^3)-1/2*d^2/((b*c-a*d)^3*(a+b*x)^2)+d^3/((b*c-a*d)^4*(a+b*x))+d^4*log(a+b*x)/(b*c-a*d)^5-d^4*log(c+d*x)/(b*c-a*d)^5],
[(a+b*x)^6/(a*c+(b*c+a*d)*x+b*d*x^2)^2,x,3,6*b^2*(b*c-a*d)^2*x/d^4-(b*c-a*d)^4/(d^5*(c+d*x))-2*b^3*(b*c-a*d)*(c+d*x)^2/d^5+1/3*b^4*(c+d*x)^3/d^5-4*b*(b*c-a*d)^3*log(c+d*x)/d^5],
[(a+b*x)^5/(a*c+(b*c+a*d)*x+b*d*x^2)^2,x,3,-b^2*(2*b*c-3*a*d)*x/d^3+1/2*b^3*x^2/d^2+(b*c-a*d)^3/(d^4*(c+d*x))+3*b*(b*c-a*d)^2*log(c+d*x)/d^4],
[(a+b*x)^4/(a*c+(b*c+a*d)*x+b*d*x^2)^2,x,3,b^2*x/d^2-(b*c-a*d)^2/(d^3*(c+d*x))-2*b*(b*c-a*d)*log(c+d*x)/d^3],
[(a+b*x)^3/(a*c+(b*c+a*d)*x+b*d*x^2)^2,x,3,(b*c-a*d)/(d^2*(c+d*x))+b*log(c+d*x)/d^2],
[(a+b*x)^2/(a*c+(b*c+a*d)*x+b*d*x^2)^2,x,2,(-1)/(d*(c+d*x))],
[(a+b*x)/(a*c+(b*c+a*d)*x+b*d*x^2)^2,x,3,1/((b*c-a*d)*(c+d*x))+b*log(a+b*x)/(b*c-a*d)^2-b*log(c+d*x)/(b*c-a*d)^2],
[1/(a*c+(b*c+a*d)*x+b*d*x^2)^2,x,4,(-b*c-a*d-2*b*d*x)/((b*c-a*d)^2*(a*c+(b*c+a*d)*x+b*d*x^2))-2*b*d*log(a+b*x)/(b*c-a*d)^3+2*b*d*log(c+d*x)/(b*c-a*d)^3],
[1/((a+b*x)*(a*c+(b*c+a*d)*x+b*d*x^2)^2),x,3,-1/2*b/((b*c-a*d)^2*(a+b*x)^2)+2*b*d/((b*c-a*d)^3*(a+b*x))+d^2/((b*c-a*d)^3*(c+d*x))+3*b*d^2*log(a+b*x)/(b*c-a*d)^4-3*b*d^2*log(c+d*x)/(b*c-a*d)^4],
[(a+b*x)^8/(a*c+(b*c+a*d)*x+b*d*x^2)^3,x,3,10*b^3*(b*c-a*d)^2*x/d^5+1/2*(b*c-a*d)^5/(d^6*(c+d*x)^2)-5*b*(b*c-a*d)^4/(d^6*(c+d*x))-5/2*b^4*(b*c-a*d)*(c+d*x)^2/d^6+1/3*b^5*(c+d*x)^3/d^6-10*b^2*(b*c-a*d)^3*log(c+d*x)/d^6],
[(a+b*x)^7/(a*c+(b*c+a*d)*x+b*d*x^2)^3,x,3,-b^3*(3*b*c-4*a*d)*x/d^4+1/2*b^4*x^2/d^3-1/2*(b*c-a*d)^4/(d^5*(c+d*x)^2)+4*b*(b*c-a*d)^3/(d^5*(c+d*x))+6*b^2*(b*c-a*d)^2*log(c+d*x)/d^5],
[(a+b*x)^6/(a*c+(b*c+a*d)*x+b*d*x^2)^3,x,3,b^3*x/d^3+1/2*(b*c-a*d)^3/(d^4*(c+d*x)^2)-3*b*(b*c-a*d)^2/(d^4*(c+d*x))-3*b^2*(b*c-a*d)*log(c+d*x)/d^4],
[(a+b*x)^5/(a*c+(b*c+a*d)*x+b*d*x^2)^3,x,3,-1/2*(b*c-a*d)^2/(d^3*(c+d*x)^2)+2*b*(b*c-a*d)/(d^3*(c+d*x))+b^2*log(c+d*x)/d^3],
[(a+b*x)^4/(a*c+(b*c+a*d)*x+b*d*x^2)^3,x,2,1/2*(a+b*x)^2/((b*c-a*d)*(c+d*x)^2)],
[(a+b*x)^3/(a*c+(b*c+a*d)*x+b*d*x^2)^3,x,2,(-1/2)/(d*(c+d*x)^2)],
[(a+b*x)^2/(a*c+(b*c+a*d)*x+b*d*x^2)^3,x,3,1/2/((b*c-a*d)*(c+d*x)^2)+b/((b*c-a*d)^2*(c+d*x))+b^2*log(a+b*x)/(b*c-a*d)^3-b^2*log(c+d*x)/(b*c-a*d)^3],
[(a+b*x)/(a*c+(b*c+a*d)*x+b*d*x^2)^3,x,3,-b^2/((b*c-a*d)^3*(a+b*x))-1/2*d/((b*c-a*d)^2*(c+d*x)^2)-2*b*d/((b*c-a*d)^3*(c+d*x))-3*b^2*d*log(a+b*x)/(b*c-a*d)^4+3*b^2*d*log(c+d*x)/(b*c-a*d)^4],
[1/(a*c+(b*c+a*d)*x+b*d*x^2)^3,x,5,1/2*(-b*c-a*d-2*b*d*x)/((b*c-a*d)^2*(a*c+(b*c+a*d)*x+b*d*x^2)^2)+3*b*d*(b*c+a*d+2*b*d*x)/((b*c-a*d)^4*(a*c+(b*c+a*d)*x+b*d*x^2))+6*b^2*d^2*log(a+b*x)/(b*c-a*d)^5-6*b^2*d^2*log(c+d*x)/(b*c-a*d)^5],
[1/((a+b*x)*(a*c+(b*c+a*d)*x+b*d*x^2)^3),x,3,-1/3*b^2/((b*c-a*d)^3*(a+b*x)^3)+3/2*b^2*d/((b*c-a*d)^4*(a+b*x)^2)-6*b^2*d^2/((b*c-a*d)^5*(a+b*x))-1/2*d^3/((b*c-a*d)^4*(c+d*x)^2)-4*b*d^3/((b*c-a*d)^5*(c+d*x))-10*b^2*d^3*log(a+b*x)/(b*c-a*d)^6+10*b^2*d^3*log(c+d*x)/(b*c-a*d)^6],

# Integrands of the form (d+e x)^m (a d e+(c d^2+a e^2)x+c d e x^2)^p

# p>0
[(d+e*x)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,3,1/6*(a-c*d^2/e^2)*(d+e*x)^6+1/7*c*d*(d+e*x)^7/e^2],
[(d+e*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,3,1/5*(a-c*d^2/e^2)*(d+e*x)^5+1/6*c*d*(d+e*x)^6/e^2],
[(d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,3,1/4*(a-c*d^2/e^2)*(d+e*x)^4+1/5*c*d*(d+e*x)^5/e^2],
[(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,3,1/3*(a-c*d^2/e^2)*(d+e*x)^3+1/4*c*d*(d+e*x)^4/e^2],
[a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2,x,1,a*d*e*x+1/2*(c*d^2+a*e^2)*x^2+1/3*c*d*e*x^3],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d+e*x),x,2,a*e*x+1/2*c*d*x^2],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d+e*x)^2,x,3,c*d*x/e+(a-c*d^2/e^2)*log(d+e*x)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d+e*x)^3,x,3,(-a+c*d^2/e^2)/(d+e*x)+c*d*log(d+e*x)/e^2],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d+e*x)^4,x,2,1/2*(a*e+c*d*x)^2/((c*d^2-a*e^2)*(d+e*x)^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d+e*x)^5,x,3,1/3*(-a+c*d^2/e^2)/(d+e*x)^3-1/2*c*d/(e^2*(d+e*x)^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d+e*x)^6,x,3,1/4*(-a+c*d^2/e^2)/(d+e*x)^4-1/3*c*d/(e^2*(d+e*x)^3)],
[(d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,3,1/5*(c*d^2-a*e^2)^2*(d+e*x)^5/e^3-1/3*c*d*(c*d^2-a*e^2)*(d+e*x)^6/e^3+1/7*c^2*d^2*(d+e*x)^7/e^3],
[(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,3,1/4*(c*d^2-a*e^2)^2*(d+e*x)^4/e^3-2/5*c*d*(c*d^2-a*e^2)*(d+e*x)^5/e^3+1/6*c^2*d^2*(d+e*x)^6/e^3],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,3,1/3*(c*d^2-a*e^2)^2*(d+e*x)^3/e^3-1/2*c*d*(c*d^2-a*e^2)*(d+e*x)^4/e^3+1/5*c^2*d^2*(d+e*x)^5/e^3],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2/(d+e*x),x,3,1/3*(c*d^2-a*e^2)*(a*e+c*d*x)^3/(c^2*d^2)+1/4*e*(a*e+c*d*x)^4/(c^2*d^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2/(d+e*x)^2,x,2,1/3*(a*e+c*d*x)^3/(c*d)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2/(d+e*x)^3,x,3,-c*d*(c*d^2-a*e^2)*x/e^2+1/2*(a*e+c*d*x)^2/e+(c*d^2-a*e^2)^2*log(d+e*x)/e^3],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2/(d+e*x)^4,x,3,c^2*d^2*x/e^2-(c*d^2-a*e^2)^2/(e^3*(d+e*x))-2*c*d*(c*d^2-a*e^2)*log(d+e*x)/e^3],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2/(d+e*x)^5,x,3,-1/2*(c*d^2-a*e^2)^2/(e^3*(d+e*x)^2)+2*c*d*(c*d^2-a*e^2)/(e^3*(d+e*x))+c^2*d^2*log(d+e*x)/e^3],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2/(d+e*x)^6,x,2,1/3*(a*e+c*d*x)^3/((c*d^2-a*e^2)*(d+e*x)^3)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2/(d+e*x)^7,x,3,-1/4*(c*d^2-a*e^2)^2/(e^3*(d+e*x)^4)+2/3*c*d*(c*d^2-a*e^2)/(e^3*(d+e*x)^3)-1/2*c^2*d^2/(e^3*(d+e*x)^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2/(d+e*x)^8,x,3,-1/5*(c*d^2-a*e^2)^2/(e^3*(d+e*x)^5)+1/2*c*d*(c*d^2-a*e^2)/(e^3*(d+e*x)^4)-1/3*c^2*d^2/(e^3*(d+e*x)^3)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2/(d+e*x)^9,x,3,-1/6*(c*d^2-a*e^2)^2/(e^3*(d+e*x)^6)+2/5*c*d*(c*d^2-a*e^2)/(e^3*(d+e*x)^5)-1/4*c^2*d^2/(e^3*(d+e*x)^4)],
[(d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,3,-1/6*(c*d^2-a*e^2)^3*(d+e*x)^6/e^4+3/7*c*d*(c*d^2-a*e^2)^2*(d+e*x)^7/e^4-3/8*c^2*d^2*(c*d^2-a*e^2)*(d+e*x)^8/e^4+1/9*c^3*d^3*(d+e*x)^9/e^4],
[(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,3,-1/5*(c*d^2-a*e^2)^3*(d+e*x)^5/e^4+1/2*c*d*(c*d^2-a*e^2)^2*(d+e*x)^6/e^4-3/7*c^2*d^2*(c*d^2-a*e^2)*(d+e*x)^7/e^4+1/8*c^3*d^3*(d+e*x)^8/e^4],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,3,-1/4*(c*d^2-a*e^2)^3*(d+e*x)^4/e^4+3/5*c*d*(c*d^2-a*e^2)^2*(d+e*x)^5/e^4-1/2*c^2*d^2*(c*d^2-a*e^2)*(d+e*x)^6/e^4+1/7*c^3*d^3*(d+e*x)^7/e^4],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x),x,3,1/4*(c*d^2-a*e^2)^2*(a*e+c*d*x)^4/(c^3*d^3)+2/5*e*(c*d^2-a*e^2)*(a*e+c*d*x)^5/(c^3*d^3)+1/6*e^2*(a*e+c*d*x)^6/(c^3*d^3)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x)^2,x,3,1/4*(c*d^2-a*e^2)*(a*e+c*d*x)^4/(c^2*d^2)+1/5*e*(a*e+c*d*x)^5/(c^2*d^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x)^3,x,2,1/4*(a*e+c*d*x)^4/(c*d)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x)^4,x,3,c*d*(c*d^2-a*e^2)^2*x/e^3+1/2*(a-c*d^2/e^2)*(a*e+c*d*x)^2+1/3*(a*e+c*d*x)^3/e-(c*d^2-a*e^2)^3*log(d+e*x)/e^4],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x)^5,x,3,-c^2*d^2*(2*c*d^2-3*a*e^2)*x/e^3+1/2*c^3*d^3*x^2/e^2+(c*d^2-a*e^2)^3/(e^4*(d+e*x))+3*c*d*(c*d^2-a*e^2)^2*log(d+e*x)/e^4],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x)^6,x,3,c^3*d^3*x/e^3+1/2*(c*d^2-a*e^2)^3/(e^4*(d+e*x)^2)-3*c*d*(c*d^2-a*e^2)^2/(e^4*(d+e*x))-3*c^2*d^2*(c*d^2-a*e^2)*log(d+e*x)/e^4],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x)^7,x,3,1/3*(c*d^2-a*e^2)^3/(e^4*(d+e*x)^3)-3/2*c*d*(c*d^2-a*e^2)^2/(e^4*(d+e*x)^2)+3*c^2*d^2*(c*d^2-a*e^2)/(e^4*(d+e*x))+c^3*d^3*log(d+e*x)/e^4],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x)^8,x,2,1/4*(a*e+c*d*x)^4/((c*d^2-a*e^2)*(d+e*x)^4)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x)^9,x,3,1/5*(a*e+c*d*x)^4/((c*d^2-a*e^2)*(d+e*x)^5)+1/20*c*d*(a*e+c*d*x)^4/((c*d^2-a*e^2)^2*(d+e*x)^4)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x)^10,x,3,1/6*(c*d^2-a*e^2)^3/(e^4*(d+e*x)^6)-3/5*c*d*(c*d^2-a*e^2)^2/(e^4*(d+e*x)^5)+3/4*c^2*d^2*(c*d^2-a*e^2)/(e^4*(d+e*x)^4)-1/3*c^3*d^3/(e^4*(d+e*x)^3)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x)^11,x,3,1/7*(c*d^2-a*e^2)^3/(e^4*(d+e*x)^7)-1/2*c*d*(c*d^2-a*e^2)^2/(e^4*(d+e*x)^6)+3/5*c^2*d^2*(c*d^2-a*e^2)/(e^4*(d+e*x)^5)-1/4*c^3*d^3/(e^4*(d+e*x)^4)],

# p<0
[(d+e*x)^5/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,3,e*(c*d^2-a*e^2)^3*x/(c^4*d^4)+1/2*(c*d^2-a*e^2)^2*(d+e*x)^2/(c^3*d^3)+1/3*(c*d^2-a*e^2)*(d+e*x)^3/(c^2*d^2)+1/4*(d+e*x)^4/(c*d)+(c*d^2-a*e^2)^4*log(a*e+c*d*x)/(c^5*d^5)],
[(d+e*x)^4/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,3,e*(c*d^2-a*e^2)^2*x/(c^3*d^3)+1/2*(c*d^2-a*e^2)*(d+e*x)^2/(c^2*d^2)+1/3*(d+e*x)^3/(c*d)+(c*d^2-a*e^2)^3*log(a*e+c*d*x)/(c^4*d^4)],
[(d+e*x)^3/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,3,e*(c*d^2-a*e^2)*x/(c^2*d^2)+1/2*(d+e*x)^2/(c*d)+(c*d^2-a*e^2)^2*log(a*e+c*d*x)/(c^3*d^3)],
[(d+e*x)^2/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,3,e*x/(c*d)+(c*d^2-a*e^2)*log(a*e+c*d*x)/(c^2*d^2)],
[(d+e*x)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,2,log(a*e+c*d*x)/(c*d)],
[1/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,3,log(a*e+c*d*x)/(c*d^2-a*e^2)-log(d+e*x)/(c*d^2-a*e^2)],
[1/((d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,3,1/((c*d^2-a*e^2)*(d+e*x))+c*d*log(a*e+c*d*x)/(c*d^2-a*e^2)^2-c*d*log(d+e*x)/(c*d^2-a*e^2)^2],
[1/((d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,3,1/2/((c*d^2-a*e^2)*(d+e*x)^2)+c*d/((c*d^2-a*e^2)^2*(d+e*x))+c^2*d^2*log(a*e+c*d*x)/(c*d^2-a*e^2)^3-c^2*d^2*log(d+e*x)/(c*d^2-a*e^2)^3],
[1/((d+e*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,3,1/3/((c*d^2-a*e^2)*(d+e*x)^3)+1/2*c*d/((c*d^2-a*e^2)^2*(d+e*x)^2)+c^2*d^2/((c*d^2-a*e^2)^3*(d+e*x))+c^3*d^3*log(a*e+c*d*x)/(c*d^2-a*e^2)^4-c^3*d^3*log(d+e*x)/(c*d^2-a*e^2)^4],
[(d+e*x)^8/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,3,15*e^2*(c*d^2-a*e^2)^4*x/(c^6*d^6)-(c*d^2-a*e^2)^6/(c^7*d^7*(a*e+c*d*x))+10*e^3*(c*d^2-a*e^2)^3*(a*e+c*d*x)^2/(c^7*d^7)+5*e^4*(c*d^2-a*e^2)^2*(a*e+c*d*x)^3/(c^7*d^7)+3/2*e^5*(c*d^2-a*e^2)*(a*e+c*d*x)^4/(c^7*d^7)+1/5*e^6*(a*e+c*d*x)^5/(c^7*d^7)+6*e*(c*d^2-a*e^2)^5*log(a*e+c*d*x)/(c^7*d^7)],
[(d+e*x)^7/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,3,10*e^2*(c*d^2-a*e^2)^3*x/(c^5*d^5)-(c*d^2-a*e^2)^5/(c^6*d^6*(a*e+c*d*x))+5*e^3*(c*d^2-a*e^2)^2*(a*e+c*d*x)^2/(c^6*d^6)+5/3*e^4*(c*d^2-a*e^2)*(a*e+c*d*x)^3/(c^6*d^6)+1/4*e^5*(a*e+c*d*x)^4/(c^6*d^6)+5*e*(c*d^2-a*e^2)^4*log(a*e+c*d*x)/(c^6*d^6)],
[(d+e*x)^6/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,3,e^2*(6*c^2*d^4-8*a*c*d^2*e^2+3*a^2*e^4)*x/(c^4*d^4)+e^3*(2*c*d^2-a*e^2)*x^2/(c^3*d^3)+1/3*e^4*x^3/(c^2*d^2)-(c*d^2-a*e^2)^4/(c^5*d^5*(a*e+c*d*x))+4*e*(c*d^2-a*e^2)^3*log(a*e+c*d*x)/(c^5*d^5)],
[(d+e*x)^5/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,3,e^2*(3*c*d^2-2*a*e^2)*x/(c^3*d^3)+1/2*e^3*x^2/(c^2*d^2)-(c*d^2-a*e^2)^3/(c^4*d^4*(a*e+c*d*x))+3*e*(c*d^2-a*e^2)^2*log(a*e+c*d*x)/(c^4*d^4)],
[(d+e*x)^4/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,3,e^2*x/(c^2*d^2)-(c*d^2-a*e^2)^2/(c^3*d^3*(a*e+c*d*x))+2*e*(c*d^2-a*e^2)*log(a*e+c*d*x)/(c^3*d^3)],
[(d+e*x)^3/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,3,(-c*d^2+a*e^2)/(c^2*d^2*(a*e+c*d*x))+e*log(a*e+c*d*x)/(c^2*d^2)],
[(d+e*x)^2/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,2,(-1)/(c*d*(a*e+c*d*x))],
[(d+e*x)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,3,(-1)/((c*d^2-a*e^2)*(a*e+c*d*x))-e*log(a*e+c*d*x)/(c*d^2-a*e^2)^2+e*log(d+e*x)/(c*d^2-a*e^2)^2],
[1/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,4,(-c*d^2-a*e^2-2*c*d*e*x)/((c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-2*c*d*e*log(a*e+c*d*x)/(c*d^2-a*e^2)^3+2*c*d*e*log(d+e*x)/(c*d^2-a*e^2)^3],
[1/((d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2),x,3,-c^2*d^2/((c*d^2-a*e^2)^3*(a*e+c*d*x))-1/2*e/((c*d^2-a*e^2)^2*(d+e*x)^2)-2*c*d*e/((c*d^2-a*e^2)^3*(d+e*x))-3*c^2*d^2*e*log(a*e+c*d*x)/(c*d^2-a*e^2)^4+3*c^2*d^2*e*log(d+e*x)/(c*d^2-a*e^2)^4],
[1/((d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2),x,3,-c^3*d^3/((c*d^2-a*e^2)^4*(a*e+c*d*x))-1/3*e/((c*d^2-a*e^2)^2*(d+e*x)^3)-c*d*e/((c*d^2-a*e^2)^3*(d+e*x)^2)-3*c^2*d^2*e/((c*d^2-a*e^2)^4*(d+e*x))-4*c^3*d^3*e*log(a*e+c*d*x)/(c*d^2-a*e^2)^5+4*c^3*d^3*e*log(d+e*x)/(c*d^2-a*e^2)^5],
[(d+e*x)^9/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,3,20*e^3*(c*d^2-a*e^2)^3*x/(c^6*d^6)-1/2*(c*d^2-a*e^2)^6/(c^7*d^7*(a*e+c*d*x)^2)-6*e*(c*d^2-a*e^2)^5/(c^7*d^7*(a*e+c*d*x))+15/2*e^4*(c*d^2-a*e^2)^2*(a*e+c*d*x)^2/(c^7*d^7)+2*e^5*(c*d^2-a*e^2)*(a*e+c*d*x)^3/(c^7*d^7)+1/4*e^6*(a*e+c*d*x)^4/(c^7*d^7)+15*e^2*(c*d^2-a*e^2)^4*log(a*e+c*d*x)/(c^7*d^7)],
[(d+e*x)^8/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,3,e^3*(10*c^2*d^4-15*a*c*d^2*e^2+6*a^2*e^4)*x/(c^5*d^5)+1/2*e^4*(5*c*d^2-3*a*e^2)*x^2/(c^4*d^4)+1/3*e^5*x^3/(c^3*d^3)-1/2*(c*d^2-a*e^2)^5/(c^6*d^6*(a*e+c*d*x)^2)-5*e*(c*d^2-a*e^2)^4/(c^6*d^6*(a*e+c*d*x))+10*e^2*(c*d^2-a*e^2)^3*log(a*e+c*d*x)/(c^6*d^6)],
[(d+e*x)^7/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,3,e^3*(4*c*d^2-3*a*e^2)*x/(c^4*d^4)+1/2*e^4*x^2/(c^3*d^3)-1/2*(c*d^2-a*e^2)^4/(c^5*d^5*(a*e+c*d*x)^2)-4*e*(c*d^2-a*e^2)^3/(c^5*d^5*(a*e+c*d*x))+6*e^2*(c*d^2-a*e^2)^2*log(a*e+c*d*x)/(c^5*d^5)],
[(d+e*x)^6/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,3,e^3*x/(c^3*d^3)-1/2*(c*d^2-a*e^2)^3/(c^4*d^4*(a*e+c*d*x)^2)-3*e*(c*d^2-a*e^2)^2/(c^4*d^4*(a*e+c*d*x))+3*e^2*(c*d^2-a*e^2)*log(a*e+c*d*x)/(c^4*d^4)],
[(d+e*x)^5/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,3,-1/2*(c*d^2-a*e^2)^2/(c^3*d^3*(a*e+c*d*x)^2)-2*e*(c*d^2-a*e^2)/(c^3*d^3*(a*e+c*d*x))+e^2*log(a*e+c*d*x)/(c^3*d^3)],
[(d+e*x)^4/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,2,-1/2*(d+e*x)^2/((c*d^2-a*e^2)*(a*e+c*d*x)^2)],
[(d+e*x)^3/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,2,(-1/2)/(c*d*(a*e+c*d*x)^2)],
[(d+e*x)^2/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,3,(-1/2)/((c*d^2-a*e^2)*(a*e+c*d*x)^2)+e/((c*d^2-a*e^2)^2*(a*e+c*d*x))+e^2*log(a*e+c*d*x)/(c*d^2-a*e^2)^3-e^2*log(d+e*x)/(c*d^2-a*e^2)^3],
[(d+e*x)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,3,-1/2*c*d/((c*d^2-a*e^2)^2*(a*e+c*d*x)^2)+2*c*d*e/((c*d^2-a*e^2)^3*(a*e+c*d*x))+e^2/((c*d^2-a*e^2)^3*(d+e*x))+3*c*d*e^2*log(a*e+c*d*x)/(c*d^2-a*e^2)^4-3*c*d*e^2*log(d+e*x)/(c*d^2-a*e^2)^4],
[1/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,5,1/2*(-c*d^2-a*e^2-2*c*d*e*x)/((c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2)+3*c*d*e*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+6*c^2*d^2*e^2*log(a*e+c*d*x)/(c*d^2-a*e^2)^5-6*c^2*d^2*e^2*log(d+e*x)/(c*d^2-a*e^2)^5],
[1/((d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3),x,3,-1/2*c^3*d^3/((c*d^2-a*e^2)^4*(a*e+c*d*x)^2)+4*c^3*d^3*e/((c*d^2-a*e^2)^5*(a*e+c*d*x))+1/3*e^2/((c*d^2-a*e^2)^3*(d+e*x)^3)+3/2*c*d*e^2/((c*d^2-a*e^2)^4*(d+e*x)^2)+6*c^2*d^2*e^2/((c*d^2-a*e^2)^5*(d+e*x))+10*c^3*d^3*e^2*log(a*e+c*d*x)/(c*d^2-a*e^2)^6-10*c^3*d^3*e^2*log(d+e*x)/(c*d^2-a*e^2)^6],
[(d+e*x)^10/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^4,x,3,e^4*(15*c^2*d^4-24*a*c*d^2*e^2+10*a^2*e^4)*x/(c^6*d^6)+e^5*(3*c*d^2-2*a*e^2)*x^2/(c^5*d^5)+1/3*e^6*x^3/(c^4*d^4)-1/3*(c*d^2-a*e^2)^6/(c^7*d^7*(a*e+c*d*x)^3)-3*e*(c*d^2-a*e^2)^5/(c^7*d^7*(a*e+c*d*x)^2)-15*e^2*(c*d^2-a*e^2)^4/(c^7*d^7*(a*e+c*d*x))+20*e^3*(c*d^2-a*e^2)^3*log(a*e+c*d*x)/(c^7*d^7)],
[(d+e*x)^9/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^4,x,3,e^4*(5*c*d^2-4*a*e^2)*x/(c^5*d^5)+1/2*e^5*x^2/(c^4*d^4)-1/3*(c*d^2-a*e^2)^5/(c^6*d^6*(a*e+c*d*x)^3)-5/2*e*(c*d^2-a*e^2)^4/(c^6*d^6*(a*e+c*d*x)^2)-10*e^2*(c*d^2-a*e^2)^3/(c^6*d^6*(a*e+c*d*x))+10*e^3*(c*d^2-a*e^2)^2*log(a*e+c*d*x)/(c^6*d^6)],
[(d+e*x)^8/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^4,x,3,e^4*x/(c^4*d^4)-1/3*(c*d^2-a*e^2)^4/(c^5*d^5*(a*e+c*d*x)^3)-2*e*(c*d^2-a*e^2)^3/(c^5*d^5*(a*e+c*d*x)^2)-6*e^2*(c*d^2-a*e^2)^2/(c^5*d^5*(a*e+c*d*x))+4*e^3*(c*d^2-a*e^2)*log(a*e+c*d*x)/(c^5*d^5)],
[(d+e*x)^7/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^4,x,3,-1/3*(c*d^2-a*e^2)^3/(c^4*d^4*(a*e+c*d*x)^3)-3/2*e*(c*d^2-a*e^2)^2/(c^4*d^4*(a*e+c*d*x)^2)-3*e^2*(c*d^2-a*e^2)/(c^4*d^4*(a*e+c*d*x))+e^3*log(a*e+c*d*x)/(c^4*d^4)],
[(d+e*x)^6/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^4,x,2,-1/3*(d+e*x)^3/((c*d^2-a*e^2)*(a*e+c*d*x)^3)],
[(d+e*x)^5/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^4,x,3,1/3*(-c*d^2+a*e^2)/(c^2*d^2*(a*e+c*d*x)^3)-1/2*e/(c^2*d^2*(a*e+c*d*x)^2)],
[(d+e*x)^4/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^4,x,2,(-1/3)/(c*d*(a*e+c*d*x)^3)],
[(d+e*x)^3/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^4,x,3,(-1/3)/((c*d^2-a*e^2)*(a*e+c*d*x)^3)+1/2*e/((c*d^2-a*e^2)^2*(a*e+c*d*x)^2)-e^2/((c*d^2-a*e^2)^3*(a*e+c*d*x))-e^3*log(a*e+c*d*x)/(c*d^2-a*e^2)^4+e^3*log(d+e*x)/(c*d^2-a*e^2)^4],
[(d+e*x)^2/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^4,x,3,-1/3*c*d/((c*d^2-a*e^2)^2*(a*e+c*d*x)^3)+c*d*e/((c*d^2-a*e^2)^3*(a*e+c*d*x)^2)-3*c*d*e^2/((c*d^2-a*e^2)^4*(a*e+c*d*x))-e^3/((c*d^2-a*e^2)^4*(d+e*x))-4*c*d*e^3*log(a*e+c*d*x)/(c*d^2-a*e^2)^5+4*c*d*e^3*log(d+e*x)/(c*d^2-a*e^2)^5],
[(d+e*x)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^4,x,3,-1/3*c^2*d^2/((c*d^2-a*e^2)^3*(a*e+c*d*x)^3)+3/2*c^2*d^2*e/((c*d^2-a*e^2)^4*(a*e+c*d*x)^2)-6*c^2*d^2*e^2/((c*d^2-a*e^2)^5*(a*e+c*d*x))-1/2*e^3/((c*d^2-a*e^2)^4*(d+e*x)^2)-4*c*d*e^3/((c*d^2-a*e^2)^5*(d+e*x))-10*c^2*d^2*e^3*log(a*e+c*d*x)/(c*d^2-a*e^2)^6+10*c^2*d^2*e^3*log(d+e*x)/(c*d^2-a*e^2)^6],
[1/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^4,x,6,1/3*(-c*d^2-a*e^2-2*c*d*e*x)/((c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3)+5/3*c*d*e*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2)-10*c^2*d^2*e^2*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^6*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-20*c^3*d^3*e^3*log(a*e+c*d*x)/(c*d^2-a*e^2)^7+20*c^3*d^3*e^3*log(d+e*x)/(c*d^2-a*e^2)^7],

# Integrands of the form (d+e x)^m (a d e+(c d^2+a e^2)x+c d e x^2)^(p/2)

# p>0
[(d+e*x)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,7,7/64*(c*d^2-a*e^2)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^4*d^4)+21/160*(c*d^2-a*e^2)^2*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^3*d^3)+3/20*(c*d^2-a*e^2)*(d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^2*d^2)+1/6*(d+e*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c*d)-21/1024*(c*d^2-a*e^2)^6*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(11/2)*d^(11/2)*e^(3/2))+21/512*(c*d^2-a*e^2)^4*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^5*d^5*e)],
[(d+e*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,6,7/48*(c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^3*d^3)+7/40*(c*d^2-a*e^2)*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^2*d^2)+1/5*(d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c*d)-7/256*(c*d^2-a*e^2)^5*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(9/2)*d^(9/2)*e^(3/2))+7/128*(c*d^2-a*e^2)^3*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^4*d^4*e)],
[(d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,5,5/24*(c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^2*d^2)+1/4*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c*d)-5/128*(c*d^2-a*e^2)^4*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(7/2)*d^(7/2)*e^(3/2))+5/64*(c*d^2-a*e^2)^2*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*e)],
[(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,4,1/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c*d)-1/16*(c*d^2-a*e^2)^3*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(5/2)*d^(5/2)*e^(3/2))+1/8*(c*d^2-a*e^2)*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*e)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,3,-1/8*(c*d^2-a*e^2)^2*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(3/2)*d^(3/2)*e^(3/2))+1/4*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*e)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)/(d+e*x),x,3,-1/2*(c*d^2-a*e^2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(e^(3/2)*sqrt(c)*sqrt(d))+sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/e],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)/(d+e*x)^2,x,3,arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(c)*sqrt(d)/e^(3/2)-2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e*(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)/(d+e*x)^3,x,1,2/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d^2-a*e^2)*(d+e*x)^3)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)/(d+e*x)^4,x,2,2/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d^2-a*e^2)*(d+e*x)^4)+4/15*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d^2-a*e^2)^2*(d+e*x)^3)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)/(d+e*x)^5,x,3,2/7*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d^2-a*e^2)*(d+e*x)^5)+8/35*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d^2-a*e^2)^2*(d+e*x)^4)+16/105*c^2*d^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d^2-a*e^2)^3*(d+e*x)^3)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)/(d+e*x)^6,x,4,2/9*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d^2-a*e^2)*(d+e*x)^6)+4/21*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d^2-a*e^2)^2*(d+e*x)^5)+16/105*c^2*d^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d^2-a*e^2)^3*(d+e*x)^4)+32/315*c^3*d^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d^2-a*e^2)^4*(d+e*x)^3)],
[(d+e*x)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,8,33/2048*(c*d^2-a*e^2)^4*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^5*d^5*e)+33/640*(c*d^2-a*e^2)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^4*d^4)+33/448*(c*d^2-a*e^2)^2*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^3*d^3)+11/112*(c*d^2-a*e^2)*(d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^2*d^2)+1/8*(d+e*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c*d)+99/32768*(c*d^2-a*e^2)^8*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(13/2)*d^(13/2)*e^(5/2))-99/16384*(c*d^2-a*e^2)^6*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^6*d^6*e^2)],
[(d+e*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,7,3/128*(c*d^2-a*e^2)^3*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^4*d^4*e)+3/40*(c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^3*d^3)+3/28*(c*d^2-a*e^2)*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^2*d^2)+1/7*(d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c*d)+9/2048*(c*d^2-a*e^2)^7*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(11/2)*d^(11/2)*e^(5/2))-9/1024*(c*d^2-a*e^2)^5*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^5*d^5*e^2)],
[(d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,6,7/192*(c*d^2-a*e^2)^2*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^3*d^3*e)+7/60*(c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^2*d^2)+1/6*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c*d)+7/1024*(c*d^2-a*e^2)^6*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(9/2)*d^(9/2)*e^(5/2))-7/512*(c*d^2-a*e^2)^4*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^4*d^4*e^2)],
[(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,5,1/16*(c*d^2-a*e^2)*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^2*d^2*e)+1/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c*d)+3/256*(c*d^2-a*e^2)^5*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(7/2)*d^(7/2)*e^(5/2))-3/128*(c*d^2-a*e^2)^3*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*e^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,4,1/8*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c*d*e)+3/128*(c*d^2-a*e^2)^4*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(5/2)*d^(5/2)*e^(5/2))-3/64*(c*d^2-a*e^2)^2*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*e^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x),x,4,1/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/e+1/16*(c*d^2-a*e^2)^3*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(3/2)*d^(3/2)*e^(5/2))+1/8*(a/(c*d)-d/e^2)*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^2,x,4,1/2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(e*(d+e*x))+3/8*(c*d^2-a*e^2)^2*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(e^(5/2)*sqrt(c)*sqrt(d))+3/4*(a-c*d^2/e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^3,x,4,-2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(e*(d+e*x)^2)-3/2*(c*d^2-a*e^2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(c)*sqrt(d)/e^(5/2)+3*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/e^2],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^4,x,4,-2/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(e*(d+e*x)^3)+c^(3/2)*d^(3/2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/e^(5/2)-2*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^2*(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^5,x,1,2/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d^2-a*e^2)*(d+e*x)^5)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^6,x,2,2/7*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d^2-a*e^2)*(d+e*x)^6)+4/35*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d^2-a*e^2)^2*(d+e*x)^5)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^7,x,3,2/9*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d^2-a*e^2)*(d+e*x)^7)+8/63*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d^2-a*e^2)^2*(d+e*x)^6)+16/315*c^2*d^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d^2-a*e^2)^3*(d+e*x)^5)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^8,x,4,2/11*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d^2-a*e^2)*(d+e*x)^8)+4/33*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d^2-a*e^2)^2*(d+e*x)^7)+16/231*c^2*d^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d^2-a*e^2)^3*(d+e*x)^6)+32/1155*c^3*d^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d^2-a*e^2)^4*(d+e*x)^5)],
[(d+e*x)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,9,-143/49152*(c*d^2-a*e^2)^6*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^6*d^6*e^2)+143/15360*(c*d^2-a*e^2)^4*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^5*d^5*e)+143/4480*(c*d^2-a*e^2)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^4*d^4)+143/2880*(c*d^2-a*e^2)^2*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^3*d^3)+13/180*(c*d^2-a*e^2)*(d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^2*d^2)+1/10*(d+e*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c*d)-143/262144*(c*d^2-a*e^2)^10*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(15/2)*d^(15/2)*e^(7/2))+143/131072*(c*d^2-a*e^2)^8*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^7*d^7*e^3)],
[(d+e*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,8,-55/12288*(c*d^2-a*e^2)^5*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^5*d^5*e^2)+11/768*(c*d^2-a*e^2)^3*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^4*d^4*e)+11/224*(c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^3*d^3)+11/144*(c*d^2-a*e^2)*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^2*d^2)+1/9*(d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c*d)-55/65536*(c*d^2-a*e^2)^9*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(13/2)*d^(13/2)*e^(7/2))+55/32768*(c*d^2-a*e^2)^7*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^6*d^6*e^3)],
[(d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,7,-15/2048*(c*d^2-a*e^2)^4*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^4*d^4*e^2)+3/128*(c*d^2-a*e^2)^2*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^3*d^3*e)+9/112*(c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^2*d^2)+1/8*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c*d)-45/32768*(c*d^2-a*e^2)^8*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(11/2)*d^(11/2)*e^(7/2))+45/16384*(c*d^2-a*e^2)^6*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^5*d^5*e^3)],
[(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,6,-5/384*(c*d^2-a*e^2)^3*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^3*d^3*e^2)+1/24*(c*d^2-a*e^2)*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^2*d^2*e)+1/7*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c*d)-5/2048*(c*d^2-a*e^2)^7*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(9/2)*d^(9/2)*e^(7/2))+5/1024*(c*d^2-a*e^2)^5*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^4*d^4*e^3)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,5,-5/192*(c*d^2-a*e^2)^2*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^2*d^2*e^2)+1/12*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c*d*e)-5/1024*(c*d^2-a*e^2)^6*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(7/2)*d^(7/2)*e^(7/2))+5/512*(c*d^2-a*e^2)^4*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*e^3)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x),x,5,1/16*(a/(c*d)-d/e^2)*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)+1/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/e-3/256*(c*d^2-a*e^2)^5*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(5/2)*d^(5/2)*e^(7/2))+3/128*(c*d^2-a*e^2)^3*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*e^3)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^2,x,6,5/24*(a-c*d^2/e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)+1/4*(a*e+c*d*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/e-5/128*(c*d^2-a*e^2)^4*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(3/2)*d^(3/2)*e^(7/2))+5/64*(c*d^2-a*e^2)^2*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*e^3)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^3,x,5,-5/3*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/e^2+2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(e*(d+e*x)^2)-5/16*(c*d^2-a*e^2)^3*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(e^(7/2)*sqrt(c)*sqrt(d))+5/8*(c*d^2-a*e^2)*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/e^3],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^4,x,5,5/2*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(e^2*(d+e*x))-2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(e*(d+e*x)^3)+15/8*(c*d^2-a*e^2)^2*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(c)*sqrt(d)/e^(7/2)+15/4*c*d*(a-c*d^2/e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/e],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^5,x,5,-10/3*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(e^2*(d+e*x)^2)-2/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(e*(d+e*x)^4)-5/2*c^(3/2)*d^(3/2)*(c*d^2-a*e^2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/e^(7/2)+5*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/e^3],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^6,x,5,-2/3*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(e^2*(d+e*x)^3)-2/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(e*(d+e*x)^5)+c^(5/2)*d^(5/2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/e^(7/2)-2*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^3*(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^7,x,1,2/7*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d^2-a*e^2)*(d+e*x)^7)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^8,x,2,2/9*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d^2-a*e^2)*(d+e*x)^8)+4/63*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d^2-a*e^2)^2*(d+e*x)^7)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^9,x,3,2/11*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d^2-a*e^2)*(d+e*x)^9)+8/99*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d^2-a*e^2)^2*(d+e*x)^8)+16/693*c^2*d^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d^2-a*e^2)^3*(d+e*x)^7)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^10,x,4,2/13*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d^2-a*e^2)*(d+e*x)^10)+12/143*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d^2-a*e^2)^2*(d+e*x)^9)+16/429*c^2*d^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d^2-a*e^2)^3*(d+e*x)^8)+32/3003*c^3*d^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d^2-a*e^2)^4*(d+e*x)^7)],

# p<0
[(d+e*x)^3/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,5,5/16*(c*d^2-a*e^2)^3*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(7/2)*d^(7/2)*sqrt(e))+5/8*(c*d^2-a*e^2)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3)+5/12*(c*d^2-a*e^2)*(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2)+1/3*(d+e*x)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d)],
[(d+e*x)^2/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,4,3/8*(c*d^2-a*e^2)^2*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(5/2)*d^(5/2)*sqrt(e))+3/4*(c*d^2-a*e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2)+1/2*(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d)],
[(d+e*x)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,3,1/2*(c*d^2-a*e^2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(3/2)*d^(3/2)*sqrt(e))+sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d)],
[1/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,2,arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(sqrt(c)*sqrt(d)*sqrt(e))],
[1/((d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)),x,1,2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)*(d+e*x))],
[1/((d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)),x,2,2/3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)*(d+e*x)^2)+4/3*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)^2*(d+e*x))],
[1/((d+e*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)),x,3,2/5*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)*(d+e*x)^3)+8/15*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)^2*(d+e*x)^2)+16/15*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)^3*(d+e*x))],
[1/((d+e*x)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)),x,4,2/7*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)*(d+e*x)^4)+12/35*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)^2*(d+e*x)^3)+16/35*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)^3*(d+e*x)^2)+32/35*c^3*d^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)^4*(d+e*x))],
[(d+e*x)^5/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,6,35/16*(c*d^2-a*e^2)^3*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(e)/(c^(9/2)*d^(9/2))-2*(d+e*x)^4/(c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+35/8*e*(c*d^2-a*e^2)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^4*d^4)+35/12*e*(c*d^2-a*e^2)*(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3)+7/3*e*(d+e*x)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2)],
[(d+e*x)^4/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,5,15/8*(c*d^2-a*e^2)^2*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(e)/(c^(7/2)*d^(7/2))-2*(d+e*x)^3/(c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+15/4*e*(c*d^2-a*e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3)+5/2*e*(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2)],
[(d+e*x)^3/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,4,3/2*(c*d^2-a*e^2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(e)/(c^(5/2)*d^(5/2))-2*(d+e*x)^2/(c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+3*e*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2)],
[(d+e*x)^2/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,3,arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(e)/(c^(3/2)*d^(3/2))-2*(d+e*x)/(c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,1,-2*(d+e*x)/((c*d^2-a*e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,1,-2*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/((d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,2,2/3/((c*d^2-a*e^2)*(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-8/3*c*d*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/((d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,3,2/5/((c*d^2-a*e^2)*(d+e*x)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+4/5*c*d/((c*d^2-a*e^2)^2*(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-16/5*c^2*d^2*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/((d+e*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,4,2/7/((c*d^2-a*e^2)*(d+e*x)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+16/35*c*d/((c*d^2-a*e^2)^2*(d+e*x)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+32/35*c^2*d^2/((c*d^2-a*e^2)^3*(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-128/35*c^3*d^3*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^5*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/((d+e*x)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,5,2/9/((c*d^2-a*e^2)*(d+e*x)^4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+20/63*c*d/((c*d^2-a*e^2)^2*(d+e*x)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+32/63*c^2*d^2/((c*d^2-a*e^2)^3*(d+e*x)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+64/63*c^3*d^3/((c*d^2-a*e^2)^4*(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-256/63*c^4*d^4*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^6*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^6/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,6,-2/3*(d+e*x)^5/(c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+35/8*e^(3/2)*(c*d^2-a*e^2)^2*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(9/2)*d^(9/2))-14/3*e*(d+e*x)^3/(c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+35/4*e^2*(c*d^2-a*e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^4*d^4)+35/6*e^2*(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3)],
[(d+e*x)^5/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,5,-2/3*(d+e*x)^4/(c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+5/2*e^(3/2)*(c*d^2-a*e^2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(7/2)*d^(7/2))-10/3*e*(d+e*x)^2/(c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+5*e^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3)],
[(d+e*x)^4/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,4,-2/3*(d+e*x)^3/(c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+e^(3/2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(5/2)*d^(5/2))-2*e*(d+e*x)/(c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^3/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,1,-2/3*(d+e*x)^3/((c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))],
[(d+e*x)^2/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,2,-2/3*(d+e*x)/(c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+2/3*e*(c*d^2+a*e^2+2*c*d*e*x)/(c*d*(c*d^2-a*e^2)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,2,-2/3*(d+e*x)/((c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+8/3*e*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,2,-2/3*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+16/3*c*d*e*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/((d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)),x,3,2/5/((c*d^2-a*e^2)*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-16/15*c*d*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+128/15*c^2*d^2*e*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^5*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/((d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)),x,4,2/7/((c*d^2-a*e^2)*(d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+4/7*c*d/((c*d^2-a*e^2)^2*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-32/21*c^2*d^2*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+256/21*c^3*d^3*e*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^6*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/((d+e*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)),x,5,2/9/((c*d^2-a*e^2)*(d+e*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+8/21*c*d/((c*d^2-a*e^2)^2*(d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+16/21*c^2*d^2/((c*d^2-a*e^2)^3*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-128/63*c^3*d^3*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+1024/63*c^4*d^4*e*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^7*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],

# Integrands of the form (d+e x)^m (a d e+(c d^2+a e^2)x+c d e x^2)^(p/3)

# p>0

# p<0
#  {(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/3), x, 0, 0}

# {(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/3), x, 0, 0} 
[(d+e*x)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/3),x,5,3/4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(2/3)/(c*d)+3/2*(c*d^2-a*e^2)*sqrt((c*d^2+a*e^2+2*c*d*e*x)^2)*sqrt((a*e^2+c*d*(d+2*e*x))^2)/(2^(1/3)*c^(5/3)*d^(5/3)*e^(2/3)*(c*d^2+a*e^2+2*c*d*e*x)*(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1+sqrt(3))))+3^(3/4)*(c*d^2-a*e^2)^(5/3)*((c*d^2-a*e^2)^(2/3)+2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3))*EllipticF((2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1-sqrt(3)))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((c*d^2+a*e^2+2*c*d*e*x)^2)*sqrt(((c*d^2-a*e^2)^(4/3)-2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*(c*d^2-a*e^2)^(2/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+2*2^(1/3)*c^(2/3)*d^(2/3)*e^(2/3)*((a*e+c*d*x)*(d+e*x))^(2/3))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1+sqrt(3)))^2)/(2^(5/6)*c^(5/3)*d^(5/3)*e^(2/3)*(c*d^2+a*e^2+2*c*d*e*x)*sqrt((a*e^2+c*d*(d+2*e*x))^2)*sqrt((c*d^2-a*e^2)^(2/3)*((c*d^2-a*e^2)^(2/3)+2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1+sqrt(3)))^2))-3/4*3^(1/4)*(c*d^2-a*e^2)^(5/3)*((c*d^2-a*e^2)^(2/3)+2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3))*EllipticE((2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1-sqrt(3)))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((c*d^2+a*e^2+2*c*d*e*x)^2)*sqrt(2-sqrt(3))*sqrt(((c*d^2-a*e^2)^(4/3)-2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*(c*d^2-a*e^2)^(2/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+2*2^(1/3)*c^(2/3)*d^(2/3)*e^(2/3)*((a*e+c*d*x)*(d+e*x))^(2/3))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1+sqrt(3)))^2)/(2^(1/3)*c^(5/3)*d^(5/3)*e^(2/3)*(c*d^2+a*e^2+2*c*d*e*x)*sqrt((a*e^2+c*d*(d+2*e*x))^2)*sqrt((c*d^2-a*e^2)^(2/3)*((c*d^2-a*e^2)^(2/3)+2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1+sqrt(3)))^2))],
[1/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/3),x,4,3*sqrt((c*d^2+a*e^2+2*c*d*e*x)^2)*sqrt((a*e^2+c*d*(d+2*e*x))^2)/(2^(1/3)*c^(2/3)*d^(2/3)*e^(2/3)*(c*d^2+a*e^2+2*c*d*e*x)*(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1+sqrt(3))))+2^(1/6)*3^(3/4)*(c*d^2-a*e^2)^(2/3)*((c*d^2-a*e^2)^(2/3)+2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3))*EllipticF((2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1-sqrt(3)))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((c*d^2+a*e^2+2*c*d*e*x)^2)*sqrt(((c*d^2-a*e^2)^(4/3)-2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*(c*d^2-a*e^2)^(2/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+2*2^(1/3)*c^(2/3)*d^(2/3)*e^(2/3)*((a*e+c*d*x)*(d+e*x))^(2/3))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1+sqrt(3)))^2)/(c^(2/3)*d^(2/3)*e^(2/3)*(c*d^2+a*e^2+2*c*d*e*x)*sqrt((a*e^2+c*d*(d+2*e*x))^2)*sqrt((c*d^2-a*e^2)^(2/3)*((c*d^2-a*e^2)^(2/3)+2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1+sqrt(3)))^2))-3/2*3^(1/4)*(c*d^2-a*e^2)^(2/3)*((c*d^2-a*e^2)^(2/3)+2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3))*EllipticE((2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1-sqrt(3)))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((c*d^2+a*e^2+2*c*d*e*x)^2)*sqrt(2-sqrt(3))*sqrt(((c*d^2-a*e^2)^(4/3)-2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*(c*d^2-a*e^2)^(2/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+2*2^(1/3)*c^(2/3)*d^(2/3)*e^(2/3)*((a*e+c*d*x)*(d+e*x))^(2/3))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1+sqrt(3)))^2)/(2^(1/3)*c^(2/3)*d^(2/3)*e^(2/3)*(c*d^2+a*e^2+2*c*d*e*x)*sqrt((a*e^2+c*d*(d+2*e*x))^2)*sqrt((c*d^2-a*e^2)^(2/3)*((c*d^2-a*e^2)^(2/3)+2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3))/(2^(2/3)*c^(1/3)*d^(1/3)*e^(1/3)*((a*e+c*d*x)*(d+e*x))^(1/3)+(c*d^2-a*e^2)^(2/3)*(1+sqrt(3)))^2))],
#  {1/((d + e*x)^1*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/3)), x, 0, 0}
# {1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/3)), x, 0, 0}
# {1/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/3)), x, 0, 0}

# {1/((d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/3)), x, 0, 0} 

# Integrands of the form (d+e x)^(m/2) (a d e+(c d^2+a e^2)x+c d e x^2)^p

# p>0
[(d+e*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,3,2/7*(a-c*d^2/e^2)*(d+e*x)^(7/2)+2/9*c*d*(d+e*x)^(9/2)/e^2],
[(d+e*x)^(1/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,3,2/5*(a-c*d^2/e^2)*(d+e*x)^(5/2)+2/7*c*d*(d+e*x)^(7/2)/e^2],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d+e*x)^(1/2),x,3,2/3*(a-c*d^2/e^2)*(d+e*x)^(3/2)+2/5*c*d*(d+e*x)^(5/2)/e^2],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d+e*x)^(3/2),x,3,2/3*c*d*(d+e*x)^(3/2)/e^2+2*(a-c*d^2/e^2)*sqrt(d+e*x)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d+e*x)^(5/2),x,3,-2*(a-c*d^2/e^2)/sqrt(d+e*x)+2*c*d*sqrt(d+e*x)/e^2],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d+e*x)^(7/2),x,3,-2/3*(a-c*d^2/e^2)/(d+e*x)^(3/2)-2*c*d/(e^2*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d+e*x)^(9/2),x,3,-2/5*(a-c*d^2/e^2)/(d+e*x)^(5/2)-2/3*c*d/(e^2*(d+e*x)^(3/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d+e*x)^(11/2),x,3,-2/7*(a-c*d^2/e^2)/(d+e*x)^(7/2)-2/5*c*d/(e^2*(d+e*x)^(5/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2*sqrt(d+e*x),x,3,2/7*(c*d^2-a*e^2)^2*(d+e*x)^(7/2)/e^3-4/9*c*d*(c*d^2-a*e^2)*(d+e*x)^(9/2)/e^3+2/11*c^2*d^2*(d+e*x)^(11/2)/e^3],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2/sqrt(d+e*x),x,3,2/5*(c*d^2-a*e^2)^2*(d+e*x)^(5/2)/e^3-4/7*c*d*(c*d^2-a*e^2)*(d+e*x)^(7/2)/e^3+2/9*c^2*d^2*(d+e*x)^(9/2)/e^3],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2/(d+e*x)^(3/2),x,3,2/3*(c*d^2-a*e^2)^2*(d+e*x)^(3/2)/e^3-4/5*c*d*(c*d^2-a*e^2)*(d+e*x)^(5/2)/e^3+2/7*c^2*d^2*(d+e*x)^(7/2)/e^3],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2/(d+e*x)^(5/2),x,3,-4/3*c*d*(c*d^2-a*e^2)*(d+e*x)^(3/2)/e^3+2/5*c^2*d^2*(d+e*x)^(5/2)/e^3+2*(c*d^2-a*e^2)^2*sqrt(d+e*x)/e^3],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2/(d+e*x)^(7/2),x,3,2/3*c^2*d^2*(d+e*x)^(3/2)/e^3-2*(c*d^2-a*e^2)^2/(e^3*sqrt(d+e*x))-4*c*d*(c*d^2-a*e^2)*sqrt(d+e*x)/e^3],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2/(d+e*x)^(9/2),x,3,-2/3*(c*d^2-a*e^2)^2/(e^3*(d+e*x)^(3/2))+4*c*d*(c*d^2-a*e^2)/(e^3*sqrt(d+e*x))+2*c^2*d^2*sqrt(d+e*x)/e^3],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2/(d+e*x)^(11/2),x,3,-2/5*(c*d^2-a*e^2)^2/(e^3*(d+e*x)^(5/2))+4/3*c*d*(c*d^2-a*e^2)/(e^3*(d+e*x)^(3/2))-2*c^2*d^2/(e^3*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2/(d+e*x)^(13/2),x,3,-2/7*(c*d^2-a*e^2)^2/(e^3*(d+e*x)^(7/2))+4/5*c*d*(c*d^2-a*e^2)/(e^3*(d+e*x)^(5/2))-2/3*c^2*d^2/(e^3*(d+e*x)^(3/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3*sqrt(d+e*x),x,3,-2/9*(c*d^2-a*e^2)^3*(d+e*x)^(9/2)/e^4+6/11*c*d*(c*d^2-a*e^2)^2*(d+e*x)^(11/2)/e^4-6/13*c^2*d^2*(c*d^2-a*e^2)*(d+e*x)^(13/2)/e^4+2/15*c^3*d^3*(d+e*x)^(15/2)/e^4],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/sqrt(d+e*x),x,3,-2/7*(c*d^2-a*e^2)^3*(d+e*x)^(7/2)/e^4+2/3*c*d*(c*d^2-a*e^2)^2*(d+e*x)^(9/2)/e^4-6/11*c^2*d^2*(c*d^2-a*e^2)*(d+e*x)^(11/2)/e^4+2/13*c^3*d^3*(d+e*x)^(13/2)/e^4],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x)^(3/2),x,3,-2/5*(c*d^2-a*e^2)^3*(d+e*x)^(5/2)/e^4+6/7*c*d*(c*d^2-a*e^2)^2*(d+e*x)^(7/2)/e^4-2/3*c^2*d^2*(c*d^2-a*e^2)*(d+e*x)^(9/2)/e^4+2/11*c^3*d^3*(d+e*x)^(11/2)/e^4],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x)^(5/2),x,3,-2/3*(c*d^2-a*e^2)^3*(d+e*x)^(3/2)/e^4+6/5*c*d*(c*d^2-a*e^2)^2*(d+e*x)^(5/2)/e^4-6/7*c^2*d^2*(c*d^2-a*e^2)*(d+e*x)^(7/2)/e^4+2/9*c^3*d^3*(d+e*x)^(9/2)/e^4],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x)^(7/2),x,3,2*c*d*(c*d^2-a*e^2)^2*(d+e*x)^(3/2)/e^4-6/5*c^2*d^2*(c*d^2-a*e^2)*(d+e*x)^(5/2)/e^4+2/7*c^3*d^3*(d+e*x)^(7/2)/e^4-2*(c*d^2-a*e^2)^3*sqrt(d+e*x)/e^4],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x)^(9/2),x,3,-2*c^2*d^2*(c*d^2-a*e^2)*(d+e*x)^(3/2)/e^4+2/5*c^3*d^3*(d+e*x)^(5/2)/e^4+2*(c*d^2-a*e^2)^3/(e^4*sqrt(d+e*x))+6*c*d*(c*d^2-a*e^2)^2*sqrt(d+e*x)/e^4],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x)^(11/2),x,3,2/3*(c*d^2-a*e^2)^3/(e^4*(d+e*x)^(3/2))+2/3*c^3*d^3*(d+e*x)^(3/2)/e^4-6*c*d*(c*d^2-a*e^2)^2/(e^4*sqrt(d+e*x))-6*c^2*d^2*(c*d^2-a*e^2)*sqrt(d+e*x)/e^4],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3/(d+e*x)^(13/2),x,3,2/5*(c*d^2-a*e^2)^3/(e^4*(d+e*x)^(5/2))-2*c*d*(c*d^2-a*e^2)^2/(e^4*(d+e*x)^(3/2))+6*c^2*d^2*(c*d^2-a*e^2)/(e^4*sqrt(d+e*x))+2*c^3*d^3*sqrt(d+e*x)/e^4],

# p<0
[(d+e*x)^(9/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,7,2/3*(c*d^2-a*e^2)^2*(d+e*x)^(3/2)/(c^3*d^3)+2/5*(c*d^2-a*e^2)*(d+e*x)^(5/2)/(c^2*d^2)+2/7*(d+e*x)^(7/2)/(c*d)-2*(c*d^2-a*e^2)^(7/2)*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c^(9/2)*d^(9/2))+2*(c*d^2-a*e^2)^3*sqrt(d+e*x)/(c^4*d^4)],
[(d+e*x)^(7/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,6,2/3*(c*d^2-a*e^2)*(d+e*x)^(3/2)/(c^2*d^2)+2/5*(d+e*x)^(5/2)/(c*d)-2*(c*d^2-a*e^2)^(5/2)*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c^(7/2)*d^(7/2))+2*(c*d^2-a*e^2)^2*sqrt(d+e*x)/(c^3*d^3)],
[(d+e*x)^(5/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,5,2/3*(d+e*x)^(3/2)/(c*d)-2*(c*d^2-a*e^2)^(3/2)*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c^(5/2)*d^(5/2))+2*(c*d^2-a*e^2)*sqrt(d+e*x)/(c^2*d^2)],
[(d+e*x)^(3/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,4,-2*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))*sqrt(c*d^2-a*e^2)/(c^(3/2)*d^(3/2))+2*sqrt(d+e*x)/(c*d)],
[(d+e*x)^(1/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,3,-2*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(sqrt(c)*sqrt(d)*sqrt(c*d^2-a*e^2))],
[1/((d+e*x)^(1/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,4,-2*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))*sqrt(c)*sqrt(d)/(c*d^2-a*e^2)^(3/2)+2/((c*d^2-a*e^2)*sqrt(d+e*x))],
[1/((d+e*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,5,2/3/((c*d^2-a*e^2)*(d+e*x)^(3/2))-2*c^(3/2)*d^(3/2)*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c*d^2-a*e^2)^(5/2)+2*c*d/((c*d^2-a*e^2)^2*sqrt(d+e*x))],
[1/((d+e*x)^(5/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,6,2/5/((c*d^2-a*e^2)*(d+e*x)^(5/2))+2/3*c*d/((c*d^2-a*e^2)^2*(d+e*x)^(3/2))-2*c^(5/2)*d^(5/2)*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c*d^2-a*e^2)^(7/2)+2*c^2*d^2/((c*d^2-a*e^2)^3*sqrt(d+e*x))],
[1/((d+e*x)^(7/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,7,2/7/((c*d^2-a*e^2)*(d+e*x)^(7/2))+2/5*c*d/((c*d^2-a*e^2)^2*(d+e*x)^(5/2))+2/3*c^2*d^2/((c*d^2-a*e^2)^3*(d+e*x)^(3/2))-2*c^(7/2)*d^(7/2)*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c*d^2-a*e^2)^(9/2)+2*c^3*d^3/((c*d^2-a*e^2)^4*sqrt(d+e*x))],
[(d+e*x)^(13/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,8,3*e*(c*d^2-a*e^2)^2*(d+e*x)^(3/2)/(c^4*d^4)+9/5*e*(c*d^2-a*e^2)*(d+e*x)^(5/2)/(c^3*d^3)+9/7*e*(d+e*x)^(7/2)/(c^2*d^2)-(d+e*x)^(9/2)/(c*d*(a*e+c*d*x))-9*e*(c*d^2-a*e^2)^(7/2)*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c^(11/2)*d^(11/2))+9*e*(c*d^2-a*e^2)^3*sqrt(d+e*x)/(c^5*d^5)],
[(d+e*x)^(11/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,7,7/3*e*(c*d^2-a*e^2)*(d+e*x)^(3/2)/(c^3*d^3)+7/5*e*(d+e*x)^(5/2)/(c^2*d^2)-(d+e*x)^(7/2)/(c*d*(a*e+c*d*x))-7*e*(c*d^2-a*e^2)^(5/2)*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c^(9/2)*d^(9/2))+7*e*(c*d^2-a*e^2)^2*sqrt(d+e*x)/(c^4*d^4)],
[(d+e*x)^(9/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,6,5/3*e*(d+e*x)^(3/2)/(c^2*d^2)-(d+e*x)^(5/2)/(c*d*(a*e+c*d*x))-5*e*(c*d^2-a*e^2)^(3/2)*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c^(7/2)*d^(7/2))+5*e*(c*d^2-a*e^2)*sqrt(d+e*x)/(c^3*d^3)],
[(d+e*x)^(7/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,5,-(d+e*x)^(3/2)/(c*d*(a*e+c*d*x))-3*e*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))*sqrt(c*d^2-a*e^2)/(c^(5/2)*d^(5/2))+3*e*sqrt(d+e*x)/(c^2*d^2)],
[(d+e*x)^(5/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,4,-e*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c^(3/2)*d^(3/2)*sqrt(c*d^2-a*e^2))-sqrt(d+e*x)/(c*d*(a*e+c*d*x))],
[(d+e*x)^(3/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,4,e*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/((c*d^2-a*e^2)^(3/2)*sqrt(c)*sqrt(d))-sqrt(d+e*x)/((c*d^2-a*e^2)*(a*e+c*d*x))],
[(d+e*x)^(1/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,5,3*e*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))*sqrt(c)*sqrt(d)/(c*d^2-a*e^2)^(5/2)-3*e/((c*d^2-a*e^2)^2*sqrt(d+e*x))+(-1)/((c*d^2-a*e^2)*(a*e+c*d*x)*sqrt(d+e*x))],
[1/((d+e*x)^(1/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2),x,6,-5/3*e/((c*d^2-a*e^2)^2*(d+e*x)^(3/2))+(-1)/((c*d^2-a*e^2)*(a*e+c*d*x)*(d+e*x)^(3/2))+5*c^(3/2)*d^(3/2)*e*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c*d^2-a*e^2)^(7/2)-5*c*d*e/((c*d^2-a*e^2)^3*sqrt(d+e*x))],
[1/((d+e*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2),x,7,-7/5*e/((c*d^2-a*e^2)^2*(d+e*x)^(5/2))+(-1)/((c*d^2-a*e^2)*(a*e+c*d*x)*(d+e*x)^(5/2))-7/3*c*d*e/((c*d^2-a*e^2)^3*(d+e*x)^(3/2))+7*c^(5/2)*d^(5/2)*e*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c*d^2-a*e^2)^(9/2)-7*c^2*d^2*e/((c*d^2-a*e^2)^4*sqrt(d+e*x))],
[(d+e*x)^(15/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,8,21/4*e^2*(c*d^2-a*e^2)*(d+e*x)^(3/2)/(c^4*d^4)+63/20*e^2*(d+e*x)^(5/2)/(c^3*d^3)-9/4*e*(d+e*x)^(7/2)/(c^2*d^2*(a*e+c*d*x))-1/2*(d+e*x)^(9/2)/(c*d*(a*e+c*d*x)^2)-63/4*e^2*(c*d^2-a*e^2)^(5/2)*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c^(11/2)*d^(11/2))+63/4*e^2*(c*d^2-a*e^2)^2*sqrt(d+e*x)/(c^5*d^5)],
[(d+e*x)^(13/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,7,35/12*e^2*(d+e*x)^(3/2)/(c^3*d^3)-7/4*e*(d+e*x)^(5/2)/(c^2*d^2*(a*e+c*d*x))-1/2*(d+e*x)^(7/2)/(c*d*(a*e+c*d*x)^2)-35/4*e^2*(c*d^2-a*e^2)^(3/2)*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c^(9/2)*d^(9/2))+35/4*e^2*(c*d^2-a*e^2)*sqrt(d+e*x)/(c^4*d^4)],
[(d+e*x)^(11/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,6,-5/4*e*(d+e*x)^(3/2)/(c^2*d^2*(a*e+c*d*x))-1/2*(d+e*x)^(5/2)/(c*d*(a*e+c*d*x)^2)-15/4*e^2*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))*sqrt(c*d^2-a*e^2)/(c^(7/2)*d^(7/2))+15/4*e^2*sqrt(d+e*x)/(c^3*d^3)],
[(d+e*x)^(9/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,5,-1/2*(d+e*x)^(3/2)/(c*d*(a*e+c*d*x)^2)-3/4*e^2*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c^(5/2)*d^(5/2)*sqrt(c*d^2-a*e^2))-3/4*e*sqrt(d+e*x)/(c^2*d^2*(a*e+c*d*x))],
[(d+e*x)^(7/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,5,1/4*e^2*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c^(3/2)*d^(3/2)*(c*d^2-a*e^2)^(3/2))-1/2*sqrt(d+e*x)/(c*d*(a*e+c*d*x)^2)-1/4*e*sqrt(d+e*x)/(c*d*(c*d^2-a*e^2)*(a*e+c*d*x))],
[(d+e*x)^(5/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,5,-3/4*e^2*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/((c*d^2-a*e^2)^(5/2)*sqrt(c)*sqrt(d))-1/2*sqrt(d+e*x)/((c*d^2-a*e^2)*(a*e+c*d*x)^2)+3/4*e*sqrt(d+e*x)/((c*d^2-a*e^2)^2*(a*e+c*d*x))],
[(d+e*x)^(3/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,6,-15/4*e^2*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))*sqrt(c)*sqrt(d)/(c*d^2-a*e^2)^(7/2)+15/4*e^2/((c*d^2-a*e^2)^3*sqrt(d+e*x))+(-1/2)/((c*d^2-a*e^2)*(a*e+c*d*x)^2*sqrt(d+e*x))+5/4*e/((c*d^2-a*e^2)^2*(a*e+c*d*x)*sqrt(d+e*x))],
[(d+e*x)^(1/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,7,35/12*e^2/((c*d^2-a*e^2)^3*(d+e*x)^(3/2))+(-1/2)/((c*d^2-a*e^2)*(a*e+c*d*x)^2*(d+e*x)^(3/2))+7/4*e/((c*d^2-a*e^2)^2*(a*e+c*d*x)*(d+e*x)^(3/2))-35/4*c^(3/2)*d^(3/2)*e^2*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c*d^2-a*e^2)^(9/2)+35/4*c*d*e^2/((c*d^2-a*e^2)^4*sqrt(d+e*x))],
[1/((d+e*x)^(1/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3),x,8,63/20*e^2/((c*d^2-a*e^2)^3*(d+e*x)^(5/2))+(-1/2)/((c*d^2-a*e^2)*(a*e+c*d*x)^2*(d+e*x)^(5/2))+9/4*e/((c*d^2-a*e^2)^2*(a*e+c*d*x)*(d+e*x)^(5/2))+21/4*c*d*e^2/((c*d^2-a*e^2)^4*(d+e*x)^(3/2))-63/4*c^(5/2)*d^(5/2)*e^2*arctanh(sqrt(c)*sqrt(d)*sqrt(d+e*x)/sqrt(c*d^2-a*e^2))/(c*d^2-a*e^2)^(11/2)+63/4*c^2*d^2*e^2/((c*d^2-a*e^2)^5*sqrt(d+e*x))],

# Integrands of the form (d+e x)^(m/2) (a d e+(c d^2+a e^2)x+c d e x^2)^(p/2)

# p>0
[(d+e*x)^(7/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,5,256/3465*(c*d^2-a*e^2)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^5*d^5*(d+e*x)^(3/2))+16/99*(c*d^2-a*e^2)*(d+e*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^2*d^2)+2/11*(d+e*x)^(5/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c*d)+128/1155*(c*d^2-a*e^2)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^4*d^4*sqrt(d+e*x))+32/231*(c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)*sqrt(d+e*x)/(c^3*d^3)],
[(d+e*x)^(5/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,4,32/315*(c*d^2-a*e^2)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^4*d^4*(d+e*x)^(3/2))+2/9*(d+e*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c*d)+16/105*(c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^3*d^3*sqrt(d+e*x))+4/21*(c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)*sqrt(d+e*x)/(c^2*d^2)],
[(d+e*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,3,16/105*(c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^3*d^3*(d+e*x)^(3/2))+8/35*(c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^2*d^2*sqrt(d+e*x))+2/7*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)*sqrt(d+e*x)/(c*d)],
[(d+e*x)^(1/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,2,4/15*(c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^2*d^2*(d+e*x)^(3/2))+2/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c*d*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)/(d+e*x)^(1/2),x,1,2/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c*d*(d+e*x)^(3/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)/(d+e*x)^(3/2),x,3,-2*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))*sqrt(c*d^2-a*e^2)/e^(3/2)+2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)/(d+e*x)^(5/2),x,3,c*d*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/(e^(3/2)*sqrt(c*d^2-a*e^2))-sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e*(d+e*x)^(3/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)/(d+e*x)^(7/2),x,4,1/4*c^2*d^2*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/(e^(3/2)*(c*d^2-a*e^2)^(3/2))-1/2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e*(d+e*x)^(5/2))+1/4*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e*(c*d^2-a*e^2)*(d+e*x)^(3/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)/(d+e*x)^(9/2),x,5,1/8*c^3*d^3*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/(e^(3/2)*(c*d^2-a*e^2)^(5/2))-1/3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e*(d+e*x)^(7/2))+1/12*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e*(c*d^2-a*e^2)*(d+e*x)^(5/2))+1/8*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e*(c*d^2-a*e^2)^2*(d+e*x)^(3/2))],
[(d+e*x)^(5/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,5,256/15015*(c*d^2-a*e^2)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^5*d^5*(d+e*x)^(5/2))+128/3003*(c*d^2-a*e^2)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^4*d^4*(d+e*x)^(3/2))+2/13*(d+e*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c*d)+32/429*(c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^3*d^3*sqrt(d+e*x))+16/143*(c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)*sqrt(d+e*x)/(c^2*d^2)],
[(d+e*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,4,32/1155*(c*d^2-a*e^2)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^4*d^4*(d+e*x)^(5/2))+16/231*(c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^3*d^3*(d+e*x)^(3/2))+4/33*(c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^2*d^2*sqrt(d+e*x))+2/11*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)*sqrt(d+e*x)/(c*d)],
[(d+e*x)^(1/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,3,16/315*(c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^3*d^3*(d+e*x)^(5/2))+8/63*(c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^2*d^2*(d+e*x)^(3/2))+2/9*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c*d*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^(1/2),x,2,4/35*(c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^2*d^2*(d+e*x)^(5/2))+2/7*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c*d*(d+e*x)^(3/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^(3/2),x,1,2/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c*d*(d+e*x)^(5/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^(5/2),x,4,2/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(e*(d+e*x)^(3/2))+2*(c*d^2-a*e^2)^(3/2)*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/e^(5/2)+2*(a-c*d^2/e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/sqrt(d+e*x)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^(7/2),x,4,-(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(e*(d+e*x)^(5/2))-3*c*d*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))*sqrt(c*d^2-a*e^2)/e^(5/2)+3*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^2*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^(9/2),x,4,-1/2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(e*(d+e*x)^(7/2))+3/4*c^2*d^2*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/(e^(5/2)*sqrt(c*d^2-a*e^2))-3/4*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^2*(d+e*x)^(3/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^(11/2),x,5,-1/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(e*(d+e*x)^(9/2))+1/8*c^3*d^3*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/(e^(5/2)*(c*d^2-a*e^2)^(3/2))-1/4*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^2*(d+e*x)^(5/2))+1/8*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^2*(c*d^2-a*e^2)*(d+e*x)^(3/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^(13/2),x,6,-1/4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(e*(d+e*x)^(11/2))+3/64*c^4*d^4*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/(e^(5/2)*(c*d^2-a*e^2)^(5/2))-1/8*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^2*(d+e*x)^(7/2))+1/32*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^2*(c*d^2-a*e^2)*(d+e*x)^(5/2))+3/64*c^3*d^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^2*(c*d^2-a*e^2)^2*(d+e*x)^(3/2))],
[(d+e*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,5,256/45045*(c*d^2-a*e^2)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^5*d^5*(d+e*x)^(7/2))+128/6435*(c*d^2-a*e^2)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^4*d^4*(d+e*x)^(5/2))+32/715*(c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^3*d^3*(d+e*x)^(3/2))+16/195*(c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^2*d^2*sqrt(d+e*x))+2/15*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)*sqrt(d+e*x)/(c*d)],
[(d+e*x)^(1/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,4,32/3003*(c*d^2-a*e^2)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^4*d^4*(d+e*x)^(7/2))+16/429*(c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^3*d^3*(d+e*x)^(5/2))+12/143*(c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^2*d^2*(d+e*x)^(3/2))+2/13*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c*d*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(1/2),x,3,16/693*(c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^3*d^3*(d+e*x)^(7/2))+8/99*(c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^2*d^2*(d+e*x)^(5/2))+2/11*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c*d*(d+e*x)^(3/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(3/2),x,2,4/63*(c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^2*d^2*(d+e*x)^(7/2))+2/9*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c*d*(d+e*x)^(5/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(5/2),x,1,2/7*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c*d*(d+e*x)^(7/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(7/2),x,5,2/3*(a-c*d^2/e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^(3/2)+2/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(e*(d+e*x)^(5/2))-2*(c*d^2-a*e^2)^(5/2)*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/e^(7/2)+2*(c*d^2-a*e^2)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^3*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(9/2),x,5,5/3*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(e^2*(d+e*x)^(3/2))-(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(e*(d+e*x)^(7/2))+5*c*d*(c*d^2-a*e^2)^(3/2)*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/e^(7/2)+5*c*d*(a-c*d^2/e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(11/2),x,5,-5/4*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(e^2*(d+e*x)^(5/2))-1/2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(e*(d+e*x)^(9/2))-15/4*c^2*d^2*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))*sqrt(c*d^2-a*e^2)/e^(7/2)+15/4*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^3*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(13/2),x,5,-5/12*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(e^2*(d+e*x)^(7/2))-1/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(e*(d+e*x)^(11/2))+5/8*c^3*d^3*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/(e^(7/2)*sqrt(c*d^2-a*e^2))-5/8*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^3*(d+e*x)^(3/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(15/2),x,6,-5/24*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(e^2*(d+e*x)^(9/2))-1/4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(e*(d+e*x)^(13/2))+5/64*c^4*d^4*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/(e^(7/2)*(c*d^2-a*e^2)^(3/2))-5/32*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^3*(d+e*x)^(5/2))+5/64*c^3*d^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^3*(c*d^2-a*e^2)*(d+e*x)^(3/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(17/2),x,7,-1/8*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(e^2*(d+e*x)^(11/2))-1/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(e*(d+e*x)^(15/2))+3/128*c^5*d^5*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/(e^(7/2)*(c*d^2-a*e^2)^(5/2))-1/16*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^3*(d+e*x)^(7/2))+1/64*c^3*d^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^3*(c*d^2-a*e^2)*(d+e*x)^(5/2))+3/128*c^4*d^4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^3*(c*d^2-a*e^2)^2*(d+e*x)^(3/2))],

# p<0
[(d+e*x)^(7/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,4,12/35*(c*d^2-a*e^2)*(d+e*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2)+2/7*(d+e*x)^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d)+32/35*(c*d^2-a*e^2)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^4*d^4*sqrt(d+e*x))+16/35*(c*d^2-a*e^2)^2*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3)],
[(d+e*x)^(5/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,3,2/5*(d+e*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d)+16/15*(c*d^2-a*e^2)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*sqrt(d+e*x))+8/15*(c*d^2-a*e^2)*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2)],
[(d+e*x)^(3/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,2,4/3*(c*d^2-a*e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*sqrt(d+e*x))+2/3*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d)],
[(d+e*x)^(1/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,1,2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*sqrt(d+e*x))],
[1/((d+e*x)^(1/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)),x,2,2*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/(sqrt(e)*sqrt(c*d^2-a*e^2))],
[1/((d+e*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)),x,3,c*d*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/((c*d^2-a*e^2)^(3/2)*sqrt(e))+sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)*(d+e*x)^(3/2))],
[1/((d+e*x)^(5/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)),x,4,3/4*c^2*d^2*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/((c*d^2-a*e^2)^(5/2)*sqrt(e))+1/2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)*(d+e*x)^(5/2))+3/4*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)^2*(d+e*x)^(3/2))],
[1/((d+e*x)^(7/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)),x,5,5/8*c^3*d^3*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/((c*d^2-a*e^2)^(7/2)*sqrt(e))+1/3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)*(d+e*x)^(7/2))+5/12*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)^2*(d+e*x)^(5/2))+5/8*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)^3*(d+e*x)^(3/2))],
[(d+e*x)^(7/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,3,8/3*(c*d^2-a*e^2)*(d+e*x)^(3/2)/(c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+2/3*(d+e*x)^(5/2)/(c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-16/3*(c*d^2-a*e^2)^2*sqrt(d+e*x)/(c^3*d^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^(5/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,2,2*(d+e*x)^(3/2)/(c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-4*(c*d^2-a*e^2)*sqrt(d+e*x)/(c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^(3/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,1,-2*sqrt(d+e*x)/(c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^(1/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,3,-2*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))*sqrt(e)/(c*d^2-a*e^2)^(3/2)-2*sqrt(d+e*x)/((c*d^2-a*e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/((d+e*x)^(1/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,4,-3*c*d*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))*sqrt(e)/(c*d^2-a*e^2)^(5/2)+1/((c*d^2-a*e^2)*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-3*c*d*sqrt(d+e*x)/((c*d^2-a*e^2)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/((d+e*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,5,-15/4*c^2*d^2*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))*sqrt(e)/(c*d^2-a*e^2)^(7/2)+1/2/((c*d^2-a*e^2)*(d+e*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+5/4*c*d/((c*d^2-a*e^2)^2*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-15/4*c^2*d^2*sqrt(d+e*x)/((c*d^2-a*e^2)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/((d+e*x)^(5/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,6,-35/8*c^3*d^3*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))*sqrt(e)/(c*d^2-a*e^2)^(9/2)+1/3/((c*d^2-a*e^2)*(d+e*x)^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+7/12*c*d/((c*d^2-a*e^2)^2*(d+e*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+35/24*c^2*d^2/((c*d^2-a*e^2)^3*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-35/8*c^3*d^3*sqrt(d+e*x)/((c*d^2-a*e^2)^4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/((d+e*x)^(7/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,7,-315/64*c^4*d^4*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))*sqrt(e)/(c*d^2-a*e^2)^(11/2)+1/4/((c*d^2-a*e^2)*(d+e*x)^(7/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+3/8*c*d/((c*d^2-a*e^2)^2*(d+e*x)^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+21/32*c^2*d^2/((c*d^2-a*e^2)^3*(d+e*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+105/64*c^3*d^3/((c*d^2-a*e^2)^4*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-315/64*c^4*d^4*sqrt(d+e*x)/((c*d^2-a*e^2)^5*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^(7/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,2,4/3*(c*d^2-a*e^2)*(d+e*x)^(3/2)/(c^2*d^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-2*(d+e*x)^(5/2)/(c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))],
[(d+e*x)^(5/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,1,-2/3*(d+e*x)^(3/2)/(c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))],
[(d+e*x)^(3/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,5,2*e^(3/2)*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/(c*d^2-a*e^2)^(5/2)-2/3*sqrt(d+e*x)/(c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-2/3*e/(c*d*(c*d^2-a*e^2)*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+2*e*sqrt(d+e*x)/((c*d^2-a*e^2)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^(1/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,5,5*c*d*e^(3/2)*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/(c*d^2-a*e^2)^(7/2)-2/3*sqrt(d+e*x)/((c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-5/3*e/((c*d^2-a*e^2)^2*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+5*c*d*e*sqrt(d+e*x)/((c*d^2-a*e^2)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/((d+e*x)^(1/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)),x,6,35/4*c^2*d^2*e^(3/2)*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/(c*d^2-a*e^2)^(9/2)+1/2/((c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)*sqrt(d+e*x))-7/6*c*d*sqrt(d+e*x)/((c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-35/12*c*d*e/((c*d^2-a*e^2)^3*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+35/4*c^2*d^2*e*sqrt(d+e*x)/((c*d^2-a*e^2)^4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/((d+e*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)),x,7,1/3/((c*d^2-a*e^2)*(d+e*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+105/8*c^3*d^3*e^(3/2)*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/(c*d^2-a*e^2)^(11/2)+3/4*c*d/((c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)*sqrt(d+e*x))-7/4*c^2*d^2*sqrt(d+e*x)/((c*d^2-a*e^2)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-35/8*c^2*d^2*e/((c*d^2-a*e^2)^4*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+105/8*c^3*d^3*e*sqrt(d+e*x)/((c*d^2-a*e^2)^5*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/((d+e*x)^(5/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)),x,8,1/4/((c*d^2-a*e^2)*(d+e*x)^(5/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+11/24*c*d/((c*d^2-a*e^2)^2*(d+e*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+1155/64*c^4*d^4*e^(3/2)*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/(c*d^2-a*e^2)^(13/2)+33/32*c^2*d^2/((c*d^2-a*e^2)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)*sqrt(d+e*x))-77/32*c^3*d^3*sqrt(d+e*x)/((c*d^2-a*e^2)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-385/64*c^3*d^3*e/((c*d^2-a*e^2)^5*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+1155/64*c^4*d^4*e*sqrt(d+e*x)/((c*d^2-a*e^2)^6*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/((d+e*x)^(7/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)),x,9,1/5/((c*d^2-a*e^2)*(d+e*x)^(7/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+13/40*c*d/((c*d^2-a*e^2)^2*(d+e*x)^(5/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+143/240*c^2*d^2/((c*d^2-a*e^2)^3*(d+e*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+3003/128*c^5*d^5*e^(3/2)*arctan(sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d^2-a*e^2)*sqrt(d+e*x)))/(c*d^2-a*e^2)^(15/2)+429/320*c^3*d^3/((c*d^2-a*e^2)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)*sqrt(d+e*x))-1001/320*c^4*d^4*sqrt(d+e*x)/((c*d^2-a*e^2)^5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-1001/128*c^4*d^4*e/((c*d^2-a*e^2)^6*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+3003/128*c^5*d^5*e*sqrt(d+e*x)/((c*d^2-a*e^2)^7*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/(sqrt(d+e*x)*sqrt(d^2-e^2*x^2)),x,2,-arctanh(sqrt(d^2-e^2*x^2)/(sqrt(2)*sqrt(d)*sqrt(d+e*x)))*sqrt(2)/(e*sqrt(d))],
[1/(sqrt(-d+e*x)*sqrt(d^2-e^2*x^2)),x,2,arctan(sqrt(d^2-e^2*x^2)/(sqrt(2)*sqrt(d)*sqrt(-d+e*x)))*sqrt(2)/(e*sqrt(d))],

# Integrands of the form (d+e x)^(m/3) (a d e+(c d^2+a e^2)x+c d e x^2)^(p/2)
[(d+e*x)^(2/3)/sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,5,3/2*(a*e+c*d*x)*(d+e*x)^(2/3)/(c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+1/4*3^(3/4)*(c*d^2-a*e^2)^(2/3)*(d+e*x)^(2/3)*((c*d^2-a*e^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3))*sqrt(cos(arccos(((c*d^2-a*e^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3)*(1-sqrt(3)))/((c*d^2-a*e^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3)*(1+sqrt(3)))))^2)/cos(arccos(((c*d^2-a*e^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3)*(1-sqrt(3)))/((c*d^2-a*e^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos(((c*d^2-a*e^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3)*(1-sqrt(3)))/((c*d^2-a*e^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt(a*d*e+c*d^2*x)*sqrt(((c*d^2-a*e^2)^(2/3)+c^(1/3)*d^(2/3)*(c*d^2-a*e^2)^(1/3)*(1+e*x/d)^(1/3)+c^(2/3)*d^(4/3)*(1+e*x/d)^(2/3))/((c*d^2-a*e^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3)*(1+sqrt(3)))^2)/(c*d*e*sqrt(d*(a*e+c*d*x))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)*sqrt(-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3)*((c*d^2-a*e^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3))/((c*d^2-a*e^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3)*(1+sqrt(3)))^2))],

# Integrands of the form (d+e x)^m (a d e+(c d^2+a e^2)x+c d e x^2)^p when m symbolic
[(d+e*x)^m*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,3,-(c*d^2-a*e^2)^3*(d+e*x)^(4+m)/(e^4*(4+m))+3*c*d*(c*d^2-a*e^2)^2*(d+e*x)^(5+m)/(e^4*(5+m))-3*c^2*d^2*(c*d^2-a*e^2)*(d+e*x)^(6+m)/(e^4*(6+m))+c^3*d^3*(d+e*x)^(7+m)/(e^4*(7+m))],
[(d+e*x)^m*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,3,(c*d^2-a*e^2)^2*(d+e*x)^(3+m)/(e^3*(3+m))-2*c*d*(c*d^2-a*e^2)*(d+e*x)^(4+m)/(e^3*(4+m))+c^2*d^2*(d+e*x)^(5+m)/(e^3*(5+m))],
[(d+e*x)^m*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,3,-(c*d^2-a*e^2)*(d+e*x)^(2+m)/(e^2*(2+m))+c*d*(d+e*x)^(3+m)/(e^2*(3+m))],
[(d+e*x)^m/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,2,-(d+e*x)^m*hypergeom([1,m],[1+m],c*d*(d+e*x)/(c*d^2-a*e^2))/((c*d^2-a*e^2)*m)],
[(d+e*x)^m/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^2,x,2,-e*(d+e*x)^(-1+m)*hypergeom([2,-1+m],[m],c*d*(d+e*x)/(c*d^2-a*e^2))/((c*d^2-a*e^2)^2*(1-m))],
[(d+e*x)^m/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^3,x,2,e^2*(d+e*x)^(-2+m)*hypergeom([3,-2+m],[-1+m],c*d*(d+e*x)/(c*d^2-a*e^2))/((c*d^2-a*e^2)^3*(2-m))],
[(d+e*x)^m/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^4,x,2,-e^3*(d+e*x)^(-3+m)*hypergeom([4,-3+m],[-2+m],c*d*(d+e*x)/(c*d^2-a*e^2))/((c*d^2-a*e^2)^4*(3-m))],

# Integrands of the form (d+e x)^m (a d e+(c d^2+a e^2)x+c d e x^2)^p when p symbolic
[(d+e*x)^m*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p,x,4,-(a*e+c*d*x)*(d+e*x)^(1+m)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p*hypergeom([1,2+m+2*p],[2+m+p],c*d*(d+e*x)/(c*d^2-a*e^2))/((c*d^2-a*e^2)*(1+m+p)),(a*e+c*d*x)*(d+e*x)^m*(c*d*(d+e*x)/(c*d^2-a*e^2))^(-m-p)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p*hypergeom([-m-p,1+p],[2+p],-e*(a*e+c*d*x)/(c*d^2-a*e^2))/(c*d*(1+p))],
[(d+e*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p,x,3,-(a*e+c*d*x)*(d+e*x)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p*hypergeom([1,5+2*p],[5+p],c*d*(d+e*x)/(c*d^2-a*e^2))/((c*d^2-a*e^2)*(4+p)),(c*d^2-a*e^2)^3*(a*e+c*d*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p*hypergeom([-3-p,1+p],[2+p],-e*(a*e+c*d*x)/(c*d^2-a*e^2))/(c^4*d^4*(1+p)*(c*d*(d+e*x)/(c*d^2-a*e^2))^p)],
[(d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p,x,3,-(a*e+c*d*x)*(d+e*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p*hypergeom([1,2*(2+p)],[4+p],c*d*(d+e*x)/(c*d^2-a*e^2))/((c*d^2-a*e^2)*(3+p)),(c*d^2-a*e^2)^2*(a*e+c*d*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p*hypergeom([-2-p,1+p],[2+p],-e*(a*e+c*d*x)/(c*d^2-a*e^2))/(c^3*d^3*(1+p)*(c*d*(d+e*x)/(c*d^2-a*e^2))^p)],
[(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p,x,2,-(a*e+c*d*x)*(d+e*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p*hypergeom([1,3+2*p],[3+p],c*d*(d+e*x)/(c*d^2-a*e^2))/((c*d^2-a*e^2)*(2+p)),1/2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1+p)/(c*d*(1+p))-1/2*(-e*(a*e+c*d*x)/(c*d^2-a*e^2))^(-1-p)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1+p)*hypergeom([-p,1+p],[2+p],c*d*(d+e*x)/(c*d^2-a*e^2))/(c*d*(1+p))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p,x,1,-(-e*(a*e+c*d*x)/(c*d^2-a*e^2))^(-1-p)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1+p)*hypergeom([-p,1+p],[2+p],c*d*(d+e*x)/(c*d^2-a*e^2))/((c*d^2-a*e^2)*(1+p))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p/(d+e*x),x,3,(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p*hypergeom([-p,p],[1+p],c*d*(d+e*x)/(c*d^2-a*e^2))/(e*p*(-e*(a*e+c*d*x)/(c*d^2-a*e^2))^p)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p/(d+e*x)^2,x,3,(a*e+c*d*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p*hypergeom([1,2*p],[p],c*d*(d+e*x)/(c*d^2-a*e^2))/((c*d^2-a*e^2)*(1-p)*(d+e*x)),c*d*(a*e+c*d*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p*hypergeom([2-p,1+p],[2+p],-e*(a*e+c*d*x)/(c*d^2-a*e^2))/((c*d^2-a*e^2)^2*(1+p)*(c*d*(d+e*x)/(c*d^2-a*e^2))^p)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p/(d+e*x)^3,x,3,(a*e+c*d*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p*hypergeom([1,-1+2*p],[-1+p],c*d*(d+e*x)/(c*d^2-a*e^2))/((c*d^2-a*e^2)*(2-p)*(d+e*x)^2),c^2*d^2*(a*e+c*d*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p*hypergeom([3-p,1+p],[2+p],-e*(a*e+c*d*x)/(c*d^2-a*e^2))/((c*d^2-a*e^2)^3*(1+p)*(c*d*(d+e*x)/(c*d^2-a*e^2))^p)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p/(d+e*x)^(2*p),x,4,(a*e+c*d*x)*(c*d*(d+e*x)/(c*d^2-a*e^2))^p*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p*hypergeom([p,1+p],[2+p],-e*(a*e+c*d*x)/(c*d^2-a*e^2))/(c*d*(1+p)*(d+e*x)^(2*p))],
[(d+e*x)^(-1-2*p)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p,x,4,-(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p*hypergeom([-p,-p],[1-p],c*d*(d+e*x)/(c*d^2-a*e^2))/(e*p*(-e*(a*e+c*d*x)/(c*d^2-a*e^2))^p*(d+e*x)^(2*p))],
[(d+e*x)^(-2-2*p)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p,x,1,(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1+p)/((c*d^2-a*e^2)*(1+p)*(d+e*x)^(2*(1+p)))],
[(d+e*x)^(-3-2*p)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p,x,2,(d+e*x)^(-3-2*p)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1+p)/((c*d^2-a*e^2)*(2+p))+c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1+p)/((c*d^2-a*e^2)^2*(1+p)*(2+p)*(d+e*x)^(2*(1+p)))],
[(d+e*x)^(-4-2*p)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p,x,3,2*c*d*(d+e*x)^(-3-2*p)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1+p)/((c*d^2-a*e^2)^2*(2+p)*(3+p))+2*c^2*d^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1+p)/((c*d^2-a*e^2)^3*(1+p)*(2+p)*(3+p)*(d+e*x)^(2*(1+p)))+(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1+p)/((c*d^2-a*e^2)*(3+p)*(d+e*x)^(2*(2+p)))],
[(d+e*x)^(-5-2*p)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p,x,4,(d+e*x)^(-5-2*p)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1+p)/((c*d^2-a*e^2)*(4+p))+6*c^2*d^2*(d+e*x)^(-3-2*p)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1+p)/((c*d^2-a*e^2)^3*(2+p)*(3+p)*(4+p))+6*c^3*d^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1+p)/((c*d^2-a*e^2)^4*(1+p)*(2+p)*(3+p)*(4+p)*(d+e*x)^(2*(1+p)))+3*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1+p)/((c*d^2-a*e^2)^2*(3+p)*(4+p)*(d+e*x)^(2*(2+p)))],
[(d+e*x)^m/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m,x,1,(d+e*x)^(-1+m)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1-m)/(c*d*(1-m))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^p/(d+e*x)^p,x,1,(d+e*x)^(-1-p)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1+p)/(c*d*(1+p))],

# Integrands of the form (d+e x)^m (a+b x+c x^2)^p

# Integrands of the form (d+e x)^m (a+b x+c x^2)^p

# p>0
[(d+e*x)^4*(a+b*x+c*x^2),x,2,1/5*(c*d^2-b*d*e+a*e^2)*(d+e*x)^5/e^3-1/6*(2*c*d-b*e)*(d+e*x)^6/e^3+1/7*c*(d+e*x)^7/e^3],
[(d+e*x)^3*(a+b*x+c*x^2),x,2,1/4*(c*d^2-b*d*e+a*e^2)*(d+e*x)^4/e^3-1/5*(2*c*d-b*e)*(d+e*x)^5/e^3+1/6*c*(d+e*x)^6/e^3],
[(d+e*x)^2*(a+b*x+c*x^2),x,2,1/3*(c*d^2-b*d*e+a*e^2)*(d+e*x)^3/e^3-1/4*(2*c*d-b*e)*(d+e*x)^4/e^3+1/5*c*(d+e*x)^5/e^3],
[(d+e*x)*(a+b*x+c*x^2),x,2,a*d*x+1/2*(b*d+a*e)*x^2+1/3*(c*d+b*e)*x^3+1/4*c*e*x^4],
[a+b*x+c*x^2,x,1,a*x+1/2*b*x^2+1/3*c*x^3],
[(a+b*x+c*x^2)/(d+e*x),x,2,-(c*d-b*e)*x/e^2+1/2*c*x^2/e+(c*d^2-b*d*e+a*e^2)*log(d+e*x)/e^3],
[(a+b*x+c*x^2)/(d+e*x)^2,x,2,c*x/e^2+(-c*d^2+b*d*e-a*e^2)/(e^3*(d+e*x))-(2*c*d-b*e)*log(d+e*x)/e^3],
[(a+b*x+c*x^2)/(d+e*x)^3,x,2,1/2*(-c*d^2+b*d*e-a*e^2)/(e^3*(d+e*x)^2)+(2*c*d-b*e)/(e^3*(d+e*x))+c*log(d+e*x)/e^3],
[(a+b*x+c*x^2)/(d+e*x)^4,x,2,1/3*(-c*d^2+b*d*e-a*e^2)/(e^3*(d+e*x)^3)+1/2*(2*c*d-b*e)/(e^3*(d+e*x)^2)-c/(e^3*(d+e*x))],
[(a+b*x+c*x^2)/(d+e*x)^5,x,2,1/4*(-c*d^2+b*d*e-a*e^2)/(e^3*(d+e*x)^4)+1/3*(2*c*d-b*e)/(e^3*(d+e*x)^3)-1/2*c/(e^3*(d+e*x)^2)],
[(a+b*x+c*x^2)/(d+e*x)^6,x,2,1/5*(-c*d^2+b*d*e-a*e^2)/(e^3*(d+e*x)^5)+1/4*(2*c*d-b*e)/(e^3*(d+e*x)^4)-1/3*c/(e^3*(d+e*x)^3)],
[(d+e*x)^4*(a+b*x+c*x^2)^2,x,2,1/5*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^5/e^5-1/3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(d+e*x)^6/e^5+1/7*(6*c^2*d^2+b^2*e^2-2*c*e*(3*b*d-a*e))*(d+e*x)^7/e^5-1/4*c*(2*c*d-b*e)*(d+e*x)^8/e^5+1/9*c^2*(d+e*x)^9/e^5],
[(d+e*x)^3*(a+b*x+c*x^2)^2,x,2,1/4*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^4/e^5-2/5*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(d+e*x)^5/e^5+1/6*(6*c^2*d^2+b^2*e^2-2*c*e*(3*b*d-a*e))*(d+e*x)^6/e^5-2/7*c*(2*c*d-b*e)*(d+e*x)^7/e^5+1/8*c^2*(d+e*x)^8/e^5],
[(d+e*x)^2*(a+b*x+c*x^2)^2,x,2,1/3*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^3/e^5-1/2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(d+e*x)^4/e^5+1/5*(6*c^2*d^2+b^2*e^2-2*c*e*(3*b*d-a*e))*(d+e*x)^5/e^5-1/3*c*(2*c*d-b*e)*(d+e*x)^6/e^5+1/7*c^2*(d+e*x)^7/e^5],
[(d+e*x)*(a+b*x+c*x^2)^2,x,2,a^2*d*x+1/2*a*(2*b*d+a*e)*x^2+1/3*(b^2*d+2*a*c*d+2*a*b*e)*x^3+1/4*(2*b*c*d+b^2*e+2*a*c*e)*x^4+1/5*c*(c*d+2*b*e)*x^5+1/6*c^2*e*x^6],
[(a+b*x+c*x^2)^2,x,2,a^2*x+a*b*x^2+1/3*(b^2+2*a*c)*x^3+1/2*b*c*x^4+1/5*c^2*x^5],
[(a+b*x+c*x^2)^2/(d+e*x),x,2,-(c*d-b*e)*(c*d^2-e*(b*d-2*a*e))*x/e^4+1/2*(c^2*d^2+b^2*e^2-2*c*e*(b*d-a*e))*x^2/e^3-1/3*c*(c*d-2*b*e)*x^3/e^2+1/4*c^2*x^4/e+(c*d^2-b*d*e+a*e^2)^2*log(d+e*x)/e^5],
[(a+b*x+c*x^2)^2/(d+e*x)^2,x,2,(3*c^2*d^2+b^2*e^2-2*c*e*(2*b*d-a*e))*x/e^4-c*(c*d-b*e)*x^2/e^3+1/3*c^2*x^3/e^2-(c*d^2-b*d*e+a*e^2)^2/(e^5*(d+e*x))-2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*log(d+e*x)/e^5],
[(a+b*x+c*x^2)^2/(d+e*x)^3,x,2,-c*(3*c*d-2*b*e)*x/e^4+1/2*c^2*x^2/e^3-1/2*(c*d^2-b*d*e+a*e^2)^2/(e^5*(d+e*x)^2)+2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)/(e^5*(d+e*x))+(6*c^2*d^2+b^2*e^2-2*c*e*(3*b*d-a*e))*log(d+e*x)/e^5],
[(a+b*x+c*x^2)^2/(d+e*x)^4,x,2,c^2*x/e^4-1/3*(c*d^2-b*d*e+a*e^2)^2/(e^5*(d+e*x)^3)+(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)/(e^5*(d+e*x)^2)+(-6*c^2*d^2-b^2*e^2+2*c*e*(3*b*d-a*e))/(e^5*(d+e*x))-2*c*(2*c*d-b*e)*log(d+e*x)/e^5],
[(a+b*x+c*x^2)^2/(d+e*x)^5,x,2,-1/4*(c*d^2-b*d*e+a*e^2)^2/(e^5*(d+e*x)^4)+2/3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)/(e^5*(d+e*x)^3)+1/2*(-6*c^2*d^2-b^2*e^2+2*c*e*(3*b*d-a*e))/(e^5*(d+e*x)^2)+2*c*(2*c*d-b*e)/(e^5*(d+e*x))+c^2*log(d+e*x)/e^5],
[(a+b*x+c*x^2)^2/(d+e*x)^6,x,2,-1/5*(c*d^2-b*d*e+a*e^2)^2/(e^5*(d+e*x)^5)+1/2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)/(e^5*(d+e*x)^4)+1/3*(-6*c^2*d^2-b^2*e^2+2*c*e*(3*b*d-a*e))/(e^5*(d+e*x)^3)+c*(2*c*d-b*e)/(e^5*(d+e*x)^2)-c^2/(e^5*(d+e*x))],
[(a+b*x+c*x^2)^2/(d+e*x)^7,x,2,-1/6*(c*d^2-b*d*e+a*e^2)^2/(e^5*(d+e*x)^6)+2/5*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)/(e^5*(d+e*x)^5)+1/4*(-6*c^2*d^2-b^2*e^2+2*c*e*(3*b*d-a*e))/(e^5*(d+e*x)^4)+2/3*c*(2*c*d-b*e)/(e^5*(d+e*x)^3)-1/2*c^2/(e^5*(d+e*x)^2)],
[(a+b*x+c*x^2)^2/(d+e*x)^8,x,2,-1/7*(c*d^2-b*d*e+a*e^2)^2/(e^5*(d+e*x)^7)+1/3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)/(e^5*(d+e*x)^6)+1/5*(-6*c^2*d^2-b^2*e^2+2*c*e*(3*b*d-a*e))/(e^5*(d+e*x)^5)+1/2*c*(2*c*d-b*e)/(e^5*(d+e*x)^4)-1/3*c^2/(e^5*(d+e*x)^3)],
[(d+e*x)^4*(a+b*x+c*x^2)^3,x,2,1/5*(c*d^2-b*d*e+a*e^2)^3*(d+e*x)^5/e^7-1/2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^6/e^7+3/7*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^7/e^7-1/8*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*(d+e*x)^8/e^7+1/3*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^9/e^7-3/10*c^2*(2*c*d-b*e)*(d+e*x)^10/e^7+1/11*c^3*(d+e*x)^11/e^7],
[(d+e*x)^3*(a+b*x+c*x^2)^3,x,2,1/4*(c*d^2-b*d*e+a*e^2)^3*(d+e*x)^4/e^7-3/5*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^5/e^7+1/2*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^6/e^7-1/7*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*(d+e*x)^7/e^7+3/8*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^8/e^7-1/3*c^2*(2*c*d-b*e)*(d+e*x)^9/e^7+1/10*c^3*(d+e*x)^10/e^7],
[(d+e*x)^2*(a+b*x+c*x^2)^3,x,2,1/3*(c*d^2-b*d*e+a*e^2)^3*(d+e*x)^3/e^7-3/4*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^4/e^7+3/5*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^5/e^7-1/6*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*(d+e*x)^6/e^7+3/7*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^7/e^7-3/8*c^2*(2*c*d-b*e)*(d+e*x)^8/e^7+1/9*c^3*(d+e*x)^9/e^7],
[(d+e*x)*(a+b*x+c*x^2)^3,x,2,a^3*d*x+1/2*a^2*(3*b*d+a*e)*x^2+a*(b^2*d+a*c*d+a*b*e)*x^3+1/4*(b^3*d+6*a*b*c*d+3*a*b^2*e+3*a^2*c*e)*x^4+1/5*(3*b^2*c*d+3*a*c^2*d+b^3*e+6*a*b*c*e)*x^5+1/2*c*(b*c*d+b^2*e+a*c*e)*x^6+1/7*c^2*(c*d+3*b*e)*x^7+1/8*c^3*e*x^8],
[(a+b*x+c*x^2)^3,x,2,a^3*x+3/2*a^2*b*x^2+a*(b^2+a*c)*x^3+1/4*b*(b^2+6*a*c)*x^4+3/5*c*(b^2+a*c)*x^5+1/2*b*c^2*x^6+1/7*c^3*x^7],
[(a+b*x+c*x^2)^3/(d+e*x),x,2,-3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2*x/e^6+3/2*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^2/e^7-1/3*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*(d+e*x)^3/e^7+3/4*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^4/e^7-3/5*c^2*(2*c*d-b*e)*(d+e*x)^5/e^7+1/6*c^3*(d+e*x)^6/e^7+(c*d^2-b*d*e+a*e^2)^3*log(d+e*x)/e^7],
[(a+b*x+c*x^2)^3/(d+e*x)^2,x,2,3*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*x/e^6-(c*d^2-b*d*e+a*e^2)^3/(e^7*(d+e*x))-1/2*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*(d+e*x)^2/e^7+c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^3/e^7-3/4*c^2*(2*c*d-b*e)*(d+e*x)^4/e^7+1/5*c^3*(d+e*x)^5/e^7-3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2*log(d+e*x)/e^7],
[(a+b*x+c*x^2)^3/(d+e*x)^3,x,2,-(10*c^3*d^3-b^3*e^3+3*b*c*e^2*(3*b*d-2*a*e)-9*c^2*d*e*(2*b*d-a*e))*x/e^6+3/2*c*(2*c^2*d^2+b^2*e^2-c*e*(3*b*d-a*e))*x^2/e^5-c^2*(c*d-b*e)*x^3/e^4+1/4*c^3*x^4/e^3-1/2*(c*d^2-b*d*e+a*e^2)^3/(e^7*(d+e*x)^2)+3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2/(e^7*(d+e*x))+3*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*log(d+e*x)/e^7],
[(a+b*x+c*x^2)^3/(d+e*x)^4,x,2,c*(10*c^2*d^2+3*b^2*e^2-3*c*e*(4*b*d-a*e))*x/e^6-1/2*c^2*(4*c*d-3*b*e)*x^2/e^5+1/3*c^3*x^3/e^4-1/3*(c*d^2-b*d*e+a*e^2)^3/(e^7*(d+e*x)^3)+3/2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2/(e^7*(d+e*x)^2)-3*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))/(e^7*(d+e*x))-(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*log(d+e*x)/e^7],
[(a+b*x+c*x^2)^3/(d+e*x)^5,x,2,-c^2*(5*c*d-3*b*e)*x/e^6+1/2*c^3*x^2/e^5-1/4*(c*d^2-b*d*e+a*e^2)^3/(e^7*(d+e*x)^4)+(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2/(e^7*(d+e*x)^3)-3/2*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))/(e^7*(d+e*x)^2)+(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))/(e^7*(d+e*x))+3*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*log(d+e*x)/e^7],
[(a+b*x+c*x^2)^3/(d+e*x)^6,x,2,c^3*x/e^6-1/5*(c*d^2-b*d*e+a*e^2)^3/(e^7*(d+e*x)^5)+3/4*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2/(e^7*(d+e*x)^4)-(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))/(e^7*(d+e*x)^3)+1/2*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))/(e^7*(d+e*x)^2)-3*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))/(e^7*(d+e*x))-3*c^2*(2*c*d-b*e)*log(d+e*x)/e^7],
[(a+b*x+c*x^2)^3/(d+e*x)^7,x,2,-1/6*(c*d^2-b*d*e+a*e^2)^3/(e^7*(d+e*x)^6)+3/5*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2/(e^7*(d+e*x)^5)-3/4*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))/(e^7*(d+e*x)^4)+1/3*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))/(e^7*(d+e*x)^3)-3/2*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))/(e^7*(d+e*x)^2)+3*c^2*(2*c*d-b*e)/(e^7*(d+e*x))+c^3*log(d+e*x)/e^7],
[(a+b*x+c*x^2)^3/(d+e*x)^8,x,2,-1/7*(c*d^2-b*d*e+a*e^2)^3/(e^7*(d+e*x)^7)+1/2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2/(e^7*(d+e*x)^6)-3/5*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))/(e^7*(d+e*x)^5)+1/4*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))/(e^7*(d+e*x)^4)-c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))/(e^7*(d+e*x)^3)+3/2*c^2*(2*c*d-b*e)/(e^7*(d+e*x)^2)-c^3/(e^7*(d+e*x))],
[(a+b*x+c*x^2)^3/(d+e*x)^9,x,2,-1/8*(c*d^2-b*d*e+a*e^2)^3/(e^7*(d+e*x)^8)+3/7*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2/(e^7*(d+e*x)^7)-1/2*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))/(e^7*(d+e*x)^6)+1/5*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))/(e^7*(d+e*x)^5)-3/4*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))/(e^7*(d+e*x)^4)+c^2*(2*c*d-b*e)/(e^7*(d+e*x)^3)-1/2*c^3/(e^7*(d+e*x)^2)],
[(a+b*x+c*x^2)^3/(d+e*x)^10,x,2,-1/9*(c*d^2-b*d*e+a*e^2)^3/(e^7*(d+e*x)^9)+3/8*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2/(e^7*(d+e*x)^8)-3/7*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))/(e^7*(d+e*x)^7)+1/6*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))/(e^7*(d+e*x)^6)-3/5*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))/(e^7*(d+e*x)^5)+3/4*c^2*(2*c*d-b*e)/(e^7*(d+e*x)^4)-1/3*c^3/(e^7*(d+e*x)^3)],
[(d+e*x)^4*(a+b*x+c*x^2)^4,x,2,1/5*(c*d^2-b*d*e+a*e^2)^4*(d+e*x)^5/e^9-2/3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^3*(d+e*x)^6/e^9+2/7*(c*d^2-b*d*e+a*e^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*(d+e*x)^7/e^9-1/2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*(d+e*x)^8/e^9+1/9*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(5*b*d-3*a*e)-20*c^3*d^2*e*(7*b*d-3*a*e)+6*c^2*e^2*(15*b^2*d^2-10*a*b*d*e+a^2*e^2))*(d+e*x)^9/e^9-2/5*c*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*(d+e*x)^10/e^9+2/11*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*(d+e*x)^11/e^9-1/3*c^3*(2*c*d-b*e)*(d+e*x)^12/e^9+1/13*c^4*(d+e*x)^13/e^9],
[(d+e*x)^3*(a+b*x+c*x^2)^4,x,2,1/4*(c*d^2-b*d*e+a*e^2)^4*(d+e*x)^4/e^9-4/5*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^3*(d+e*x)^5/e^9+1/3*(c*d^2-b*d*e+a*e^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*(d+e*x)^6/e^9-4/7*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*(d+e*x)^7/e^9+1/8*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(5*b*d-3*a*e)-20*c^3*d^2*e*(7*b*d-3*a*e)+6*c^2*e^2*(15*b^2*d^2-10*a*b*d*e+a^2*e^2))*(d+e*x)^8/e^9-4/9*c*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*(d+e*x)^9/e^9+1/5*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*(d+e*x)^10/e^9-4/11*c^3*(2*c*d-b*e)*(d+e*x)^11/e^9+1/12*c^4*(d+e*x)^12/e^9],
[(d+e*x)^2*(a+b*x+c*x^2)^4,x,2,1/3*(c*d^2-b*d*e+a*e^2)^4*(d+e*x)^3/e^9-(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^3*(d+e*x)^4/e^9+2/5*(c*d^2-b*d*e+a*e^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*(d+e*x)^5/e^9-2/3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*(d+e*x)^6/e^9+1/7*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(5*b*d-3*a*e)-20*c^3*d^2*e*(7*b*d-3*a*e)+6*c^2*e^2*(15*b^2*d^2-10*a*b*d*e+a^2*e^2))*(d+e*x)^7/e^9-1/2*c*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*(d+e*x)^8/e^9+2/9*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*(d+e*x)^9/e^9-2/5*c^3*(2*c*d-b*e)*(d+e*x)^10/e^9+1/11*c^4*(d+e*x)^11/e^9],
[(d+e*x)*(a+b*x+c*x^2)^4,x,2,a^4*d*x+1/2*a^3*(4*b*d+a*e)*x^2+2/3*a^2*(3*b^2*d+2*a*c*d+2*a*b*e)*x^3+1/2*a*(2*b^3*d+6*a*b*c*d+3*a*b^2*e+2*a^2*c*e)*x^4+1/5*(b^4*d+12*a*b^2*c*d+6*a^2*c^2*d+4*a*b^3*e+12*a^2*b*c*e)*x^5+1/6*(4*b^3*c*d+12*a*b*c^2*d+b^4*e+12*a*b^2*c*e+6*a^2*c^2*e)*x^6+2/7*c*(3*b^2*c*d+2*a*c^2*d+2*b^3*e+6*a*b*c*e)*x^7+1/4*c^2*(2*b*c*d+3*b^2*e+2*a*c*e)*x^8+1/9*c^3*(c*d+4*b*e)*x^9+1/10*c^4*e*x^10],
[(a+b*x+c*x^2)^4,x,2,a^4*x+2*a^3*b*x^2+2/3*a^2*(3*b^2+2*a*c)*x^3+a*b*(b^2+3*a*c)*x^4+1/5*(b^4+12*a*b^2*c+6*a^2*c^2)*x^5+2/3*b*c*(b^2+3*a*c)*x^6+2/7*c^2*(3*b^2+2*a*c)*x^7+1/2*b*c^3*x^8+1/9*c^4*x^9],
[(a+b*x+c*x^2)^4/(d+e*x),x,2,-4*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^3*x/e^8+(c*d^2-b*d*e+a*e^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*(d+e*x)^2/e^9-4/3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*(d+e*x)^3/e^9+1/4*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(5*b*d-3*a*e)-20*c^3*d^2*e*(7*b*d-3*a*e)+6*c^2*e^2*(15*b^2*d^2-10*a*b*d*e+a^2*e^2))*(d+e*x)^4/e^9-4/5*c*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*(d+e*x)^5/e^9+1/3*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*(d+e*x)^6/e^9-4/7*c^3*(2*c*d-b*e)*(d+e*x)^7/e^9+1/8*c^4*(d+e*x)^8/e^9+(c*d^2-b*d*e+a*e^2)^4*log(d+e*x)/e^9],
[(a+b*x+c*x^2)^4/(d+e*x)^2,x,2,2*(c*d^2-b*d*e+a*e^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*x/e^8-(c*d^2-b*d*e+a*e^2)^4/(e^9*(d+e*x))-2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*(d+e*x)^2/e^9+1/3*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(5*b*d-3*a*e)-20*c^3*d^2*e*(7*b*d-3*a*e)+6*c^2*e^2*(15*b^2*d^2-10*a*b*d*e+a^2*e^2))*(d+e*x)^3/e^9-c*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*(d+e*x)^4/e^9+2/5*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*(d+e*x)^5/e^9-2/3*c^3*(2*c*d-b*e)*(d+e*x)^6/e^9+1/7*c^4*(d+e*x)^7/e^9-4*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^3*log(d+e*x)/e^9],
[(a+b*x+c*x^2)^4/(d+e*x)^3,x,2,-4*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*x/e^8-1/2*(c*d^2-b*d*e+a*e^2)^4/(e^9*(d+e*x)^2)+4*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^3/(e^9*(d+e*x))+1/2*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(5*b*d-3*a*e)-20*c^3*d^2*e*(7*b*d-3*a*e)+6*c^2*e^2*(15*b^2*d^2-10*a*b*d*e+a^2*e^2))*(d+e*x)^2/e^9-4/3*c*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*(d+e*x)^3/e^9+1/2*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*(d+e*x)^4/e^9-4/5*c^3*(2*c*d-b*e)*(d+e*x)^5/e^9+1/6*c^4*(d+e*x)^6/e^9+2*(c*d^2-b*d*e+a*e^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*log(d+e*x)/e^9],
[(a+b*x+c*x^2)^4/(d+e*x)^4,x,2,(35*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(4*b*d-3*a*e)-40*c^3*d^2*e*(2*b*d-a*e)+6*c^2*e^2*(10*b^2*d^2-8*a*b*d*e+a^2*e^2))*x/e^8-2*c*(5*c^3*d^3-b^3*e^3-2*c^2*d*e*(5*b*d-2*a*e)+3*b*c*e^2*(2*b*d-a*e))*x^2/e^7+2/3*c^2*(5*c^2*d^2+3*b^2*e^2-2*c*e*(4*b*d-a*e))*x^3/e^6-c^3*(c*d-b*e)*x^4/e^5+1/5*c^4*x^5/e^4-1/3*(c*d^2-b*d*e+a*e^2)^4/(e^9*(d+e*x)^3)+2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^3/(e^9*(d+e*x)^2)-2*(c*d^2-b*d*e+a*e^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))/(e^9*(d+e*x))-4*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*log(d+e*x)/e^9],
[(a+b*x+c*x^2)^4/(d+e*x)^5,x,2,-c*(35*c^3*d^3-4*b^3*e^3+6*b*c*e^2*(5*b*d-2*a*e)-20*c^2*d*e*(3*b*d-a*e))*x/e^8+1/2*c^2*(15*c^2*d^2+6*b^2*e^2-4*c*e*(5*b*d-a*e))*x^2/e^7-1/3*c^3*(5*c*d-4*b*e)*x^3/e^6+1/4*c^4*x^4/e^5-1/4*(c*d^2-b*d*e+a*e^2)^4/(e^9*(d+e*x)^4)+4/3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^3/(e^9*(d+e*x)^3)-(c*d^2-b*d*e+a*e^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))/(e^9*(d+e*x)^2)+4*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))/(e^9*(d+e*x))+(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(5*b*d-3*a*e)-20*c^3*d^2*e*(7*b*d-3*a*e)+6*c^2*e^2*(15*b^2*d^2-10*a*b*d*e+a^2*e^2))*log(d+e*x)/e^9],
[(a+b*x+c*x^2)^4/(d+e*x)^6,x,2,c^2*(21*c^2*d^2+6*b^2*e^2-4*c*e*(6*b*d-a*e))*x/e^8-c^3*(3*c*d-2*b*e)*x^2/e^7+1/3*c^4*x^3/e^6-1/5*(c*d^2-b*d*e+a*e^2)^4/(e^9*(d+e*x)^5)+(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^3/(e^9*(d+e*x)^4)-2/3*(c*d^2-b*d*e+a*e^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))/(e^9*(d+e*x)^3)+2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))/(e^9*(d+e*x)^2)+(-70*c^4*d^4-b^4*e^4+4*b^2*c*e^3*(5*b*d-3*a*e)+20*c^3*d^2*e*(7*b*d-3*a*e)-6*c^2*e^2*(15*b^2*d^2-10*a*b*d*e+a^2*e^2))/(e^9*(d+e*x))-4*c*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*log(d+e*x)/e^9],
[(a+b*x+c*x^2)^4/(d+e*x)^7,x,2,-c^3*(7*c*d-4*b*e)*x/e^8+1/2*c^4*x^2/e^7-1/6*(c*d^2-b*d*e+a*e^2)^4/(e^9*(d+e*x)^6)+4/5*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^3/(e^9*(d+e*x)^5)-1/2*(c*d^2-b*d*e+a*e^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))/(e^9*(d+e*x)^4)+4/3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))/(e^9*(d+e*x)^3)+1/2*(-70*c^4*d^4-b^4*e^4+4*b^2*c*e^3*(5*b*d-3*a*e)+20*c^3*d^2*e*(7*b*d-3*a*e)-6*c^2*e^2*(15*b^2*d^2-10*a*b*d*e+a^2*e^2))/(e^9*(d+e*x)^2)+4*c*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))/(e^9*(d+e*x))+2*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*log(d+e*x)/e^9],
[(a+b*x+c*x^2)^4/(d+e*x)^8,x,2,c^4*x/e^8-1/7*(c*d^2-b*d*e+a*e^2)^4/(e^9*(d+e*x)^7)+2/3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^3/(e^9*(d+e*x)^6)-2/5*(c*d^2-b*d*e+a*e^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))/(e^9*(d+e*x)^5)+(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))/(e^9*(d+e*x)^4)+1/3*(-70*c^4*d^4-b^4*e^4+4*b^2*c*e^3*(5*b*d-3*a*e)+20*c^3*d^2*e*(7*b*d-3*a*e)-6*c^2*e^2*(15*b^2*d^2-10*a*b*d*e+a^2*e^2))/(e^9*(d+e*x)^3)+2*c*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))/(e^9*(d+e*x)^2)-2*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))/(e^9*(d+e*x))-4*c^3*(2*c*d-b*e)*log(d+e*x)/e^9],
[(a+b*x+c*x^2)^4/(d+e*x)^9,x,2,-1/8*(c*d^2-b*d*e+a*e^2)^4/(e^9*(d+e*x)^8)+4/7*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^3/(e^9*(d+e*x)^7)-1/3*(c*d^2-b*d*e+a*e^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))/(e^9*(d+e*x)^6)+4/5*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))/(e^9*(d+e*x)^5)+1/4*(-70*c^4*d^4-b^4*e^4+4*b^2*c*e^3*(5*b*d-3*a*e)+20*c^3*d^2*e*(7*b*d-3*a*e)-6*c^2*e^2*(15*b^2*d^2-10*a*b*d*e+a^2*e^2))/(e^9*(d+e*x)^4)+4/3*c*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))/(e^9*(d+e*x)^3)-c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))/(e^9*(d+e*x)^2)+4*c^3*(2*c*d-b*e)/(e^9*(d+e*x))+c^4*log(d+e*x)/e^9],
[(a+b*x+c*x^2)^4/(d+e*x)^10,x,2,-1/9*(c*d^2-b*d*e+a*e^2)^4/(e^9*(d+e*x)^9)+1/2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^3/(e^9*(d+e*x)^8)-2/7*(c*d^2-b*d*e+a*e^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))/(e^9*(d+e*x)^7)+2/3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))/(e^9*(d+e*x)^6)+1/5*(-70*c^4*d^4-b^4*e^4+4*b^2*c*e^3*(5*b*d-3*a*e)+20*c^3*d^2*e*(7*b*d-3*a*e)-6*c^2*e^2*(15*b^2*d^2-10*a*b*d*e+a^2*e^2))/(e^9*(d+e*x)^5)+c*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))/(e^9*(d+e*x)^4)-2/3*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))/(e^9*(d+e*x)^3)+2*c^3*(2*c*d-b*e)/(e^9*(d+e*x)^2)-c^4/(e^9*(d+e*x))],
[(a+b*x+c*x^2)^4/(d+e*x)^11,x,2,-1/10*(c*d^2-b*d*e+a*e^2)^4/(e^9*(d+e*x)^10)+4/9*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^3/(e^9*(d+e*x)^9)-1/4*(c*d^2-b*d*e+a*e^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))/(e^9*(d+e*x)^8)+4/7*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))/(e^9*(d+e*x)^7)+1/6*(-70*c^4*d^4-b^4*e^4+4*b^2*c*e^3*(5*b*d-3*a*e)+20*c^3*d^2*e*(7*b*d-3*a*e)-6*c^2*e^2*(15*b^2*d^2-10*a*b*d*e+a^2*e^2))/(e^9*(d+e*x)^6)+4/5*c*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))/(e^9*(d+e*x)^5)-1/2*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))/(e^9*(d+e*x)^4)+4/3*c^3*(2*c*d-b*e)/(e^9*(d+e*x)^3)-1/2*c^4/(e^9*(d+e*x)^2)],
[(a+b*x+c*x^2)^4/(d+e*x)^12,x,2,-1/11*(c*d^2-b*d*e+a*e^2)^4/(e^9*(d+e*x)^11)+2/5*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^3/(e^9*(d+e*x)^10)-2/9*(c*d^2-b*d*e+a*e^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))/(e^9*(d+e*x)^9)+1/2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))/(e^9*(d+e*x)^8)+1/7*(-70*c^4*d^4-b^4*e^4+4*b^2*c*e^3*(5*b*d-3*a*e)+20*c^3*d^2*e*(7*b*d-3*a*e)-6*c^2*e^2*(15*b^2*d^2-10*a*b*d*e+a^2*e^2))/(e^9*(d+e*x)^7)+2/3*c*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))/(e^9*(d+e*x)^6)-2/5*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))/(e^9*(d+e*x)^5)+c^3*(2*c*d-b*e)/(e^9*(d+e*x)^4)-1/3*c^4/(e^9*(d+e*x)^3)],
[x^4*(3-4*x+x^2)^2,x,2,9/5*x^5-4*x^6+22/7*x^7-x^8+1/9*x^9],
[x^3*(3-4*x+x^2)^2,x,2,9/4*x^4-24/5*x^5+11/3*x^6-8/7*x^7+1/8*x^8],
[x^2*(3-4*x+x^2)^2,x,2,3*x^3-6*x^4+22/5*x^5-4/3*x^6+1/7*x^7],
[x*(3-4*x+x^2)^2,x,2,9/2*x^2-8*x^3+11/2*x^4-8/5*x^5+1/6*x^6],
[(3-4*x+x^2)^2,x,3,-4/3*(3-x)^3-1/5*(3-x)^5+(-3+x)^4],
[(3-4*x+x^2)^2/x,x,2,-24*x+11*x^2-8/3*x^3+1/4*x^4+9*log(x)],
[(3-4*x+x^2)^2/x^2,x,2,(-9)/x+22*x-4*x^2+1/3*x^3-24*log(x)],
[(3-4*x+x^2)^2/x^3,x,2,(-9/2)/x^2+24/x-8*x+1/2*x^2+22*log(x)],
[(3-4*x+x^2)^2/x^4,x,2,(-3)/x^3+12/x^2+(-22)/x+x-8*log(x)],
[(3-4*x+x^2)^2/x^5,x,2,(-9/4)/x^4+8/x^3+(-11)/x^2+8/x+log(x)],
[(3-4*x+x^2)^2/x^6,x,2,(-9/5)/x^5+6/x^4+(-22/3)/x^3+4/x^2+(-1)/x],
[(3-4*x+x^2)^2/x^7,x,2,(-3/2)/x^6+24/5/x^5+(-11/2)/x^4+8/3/x^3+(-1/2)/x^2],
[(2+2*x+x^2)/(2+x),x,2,1/2*x^2+2*log(2+x)],
[(5+4*x+x^2)/(-2+x),x,2,6*x+1/2*x^2+17*log(2-x)],
[(2+2*x+x^2)/(1+x)^3,x,2,(-1/2)/(1+x)^2+log(1+x)],
[(3+3*x+2*x^2)/(1+x)^3,x,2,(-1)/(1+x)^2+1/(1+x)+2*log(1+x)],
[(1+x+x^2)/x,x,2,x+1/2*x^2+log(x)],
[(9+6*x+x^2)/x^2,x,2,(-9)/x+x+6*log(x)],
[(1+2*x+x^2)/x^4,x,2,(-1/3)/x^3+(-1)/x^2+(-1)/x],

# p<0
[(d+e*x)^4/(a+b*x+c*x^2),x,6,e^2*(6*c^2*d^2+b^2*e^2-c*e*(4*b*d+a*e))*x/c^3+1/2*e^3*(4*c*d-b*e)*x^2/c^2+1/3*e^4*x^3/c+1/2*e*(2*c*d-b*e)*(2*c^2*d^2+b^2*e^2-2*c*e*(b*d+a*e))*log(a+b*x+c*x^2)/c^4-(2*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(b*d+a*e)-4*c^3*d^2*e*(b*d+3*a*e)+2*c^2*e^2*(3*b^2*d^2+6*a*b*d*e+a^2*e^2))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^4*sqrt(b^2-4*a*c))],
[(d+e*x)^3/(a+b*x+c*x^2),x,6,e^2*(3*c*d-b*e)*x/c^2+1/2*e^3*x^2/c+1/2*e*(3*c^2*d^2+b^2*e^2-c*e*(3*b*d+a*e))*log(a+b*x+c*x^2)/c^3-(2*c*d-b*e)*(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^3*sqrt(b^2-4*a*c))],
[(d+e*x)^2/(a+b*x+c*x^2),x,6,e^2*x/c+1/2*e*(2*c*d-b*e)*log(a+b*x+c*x^2)/c^2-(2*c^2*d^2+b^2*e^2-2*c*e*(b*d+a*e))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^2*sqrt(b^2-4*a*c))],
[(d+e*x)/(a+b*x+c*x^2),x,4,1/2*e*log(a+b*x+c*x^2)/c-(2*c*d-b*e)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c*sqrt(b^2-4*a*c))],
[1/(a+b*x+c*x^2),x,2,-2*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c)],
[1/((d+e*x)*(a+b*x+c*x^2)),x,6,e*log(d+e*x)/(c*d^2-b*d*e+a*e^2)-1/2*e*log(a+b*x+c*x^2)/(c*d^2-b*d*e+a*e^2)-(2*c*d-b*e)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)*sqrt(b^2-4*a*c))],
[1/((d+e*x)^2*(a+b*x+c*x^2)),x,7,-e/((c*d^2-b*d*e+a*e^2)*(d+e*x))+e*(2*c*d-b*e)*log(d+e*x)/(c*d^2-b*d*e+a*e^2)^2-1/2*e*(2*c*d-b*e)*log(a+b*x+c*x^2)/(c*d^2-b*d*e+a*e^2)^2-(2*c^2*d^2+b^2*e^2-2*c*e*(b*d+a*e))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^2*sqrt(b^2-4*a*c))],
[1/((d+e*x)^3*(a+b*x+c*x^2)),x,7,-1/2*e/((c*d^2-b*d*e+a*e^2)*(d+e*x)^2)-e*(2*c*d-b*e)/((c*d^2-b*d*e+a*e^2)^2*(d+e*x))+e*(3*c^2*d^2+b^2*e^2-c*e*(3*b*d+a*e))*log(d+e*x)/(c*d^2-b*d*e+a*e^2)^3-1/2*e*(3*c^2*d^2+b^2*e^2-c*e*(3*b*d+a*e))*log(a+b*x+c*x^2)/(c*d^2-b*d*e+a*e^2)^3-(2*c*d-b*e)*(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^3*sqrt(b^2-4*a*c))],
[(d+e*x)^5/(a+b*x+c*x^2)^2,x,7,e^2*(12*c^3*d^3-3*b^3*e^3-10*c^2*d*e*(b*d+3*a*e)+b*c*e^2*(10*b*d+11*a*e))*x/(c^3*(b^2-4*a*c))+1/2*e^3*(16*c^2*d^2+3*b^2*e^2-2*c*e*(5*b*d+4*a*e))*x^2/(c^2*(b^2-4*a*c))+e^4*(2*c*d-b*e)*x^3/(c*(b^2-4*a*c))-(d+e*x)^4*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2))+(2*c*d-b*e)*(2*c^4*d^4-3*b^4*e^4-4*c^3*d^2*e*(b*d-5*a*e)+4*b^2*c*e^3*(b*d+5*a*e)-2*c^2*e^2*(b^2*d^2+10*a*b*d*e+15*a^2*e^2))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^4*(b^2-4*a*c)^(3/2))+1/2*e^3*(10*c^2*d^2+3*b^2*e^2-2*c*e*(5*b*d+a*e))*log(a+b*x+c*x^2)/c^4],
[(d+e*x)^4/(a+b*x+c*x^2)^2,x,7,2*e^2*(3*c^2*d^2+b^2*e^2-c*e*(2*b*d+3*a*e))*x/(c^2*(b^2-4*a*c))+e^3*(2*c*d-b*e)*x^2/(c*(b^2-4*a*c))-(d+e*x)^3*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2))+2*(2*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(b*d-3*a*e)-6*a*c^2*e^3*(2*b*d+a*e)+2*b^2*c*e^3*(b*d+3*a*e))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^3*(b^2-4*a*c)^(3/2))+e^3*(2*c*d-b*e)*log(a+b*x+c*x^2)/c^3],
[(d+e*x)^3/(a+b*x+c*x^2)^2,x,6,e^2*(2*c*d-b*e)*x/(c*(b^2-4*a*c))-(d+e*x)^2*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2))+(2*c*d-b*e)*(2*c^2*d^2-b^2*e^2-2*c*e*(b*d-3*a*e))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^2*(b^2-4*a*c)^(3/2))+1/2*e^3*log(a+b*x+c*x^2)/c^2],
[(d+e*x)^2/(a+b*x+c*x^2)^2,x,3,-(d+e*x)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2))+4*(c*d^2-b*d*e+a*e^2)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)],
[(d+e*x)/(a+b*x+c*x^2)^2,x,3,(-b*d+2*a*e-(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2))+2*(2*c*d-b*e)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)],
[1/(a+b*x+c*x^2)^2,x,3,(-b-2*c*x)/((b^2-4*a*c)*(a+b*x+c*x^2))+4*c*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)],
[1/((d+e*x)*(a+b*x+c*x^2)^2),x,7,(-b*c*d+b^2*e-2*a*c*e-c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(a+b*x+c*x^2))+(2*c*d-b*e)*(2*c^2*d^2-b^2*e^2-2*c*e*(b*d-3*a*e))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)^2)+e^3*log(d+e*x)/(c*d^2-b*d*e+a*e^2)^2-1/2*e^3*log(a+b*x+c*x^2)/(c*d^2-b*d*e+a*e^2)^2],
[1/((d+e*x)^2*(a+b*x+c*x^2)^2),x,7,-2*e*(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^2*(d+e*x))+(-b*c*d+b^2*e-2*a*c*e-c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(d+e*x)*(a+b*x+c*x^2))+2*(2*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(b*d-3*a*e)-6*a*c^2*e^3*(2*b*d+a*e)+2*b^2*c*e^3*(b*d+3*a*e))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)^3)+2*e^3*(2*c*d-b*e)*log(d+e*x)/(c*d^2-b*d*e+a*e^2)^3-e^3*(2*c*d-b*e)*log(a+b*x+c*x^2)/(c*d^2-b*d*e+a*e^2)^3],
[1/((d+e*x)^3*(a+b*x+c*x^2)^2),x,7,-1/2*e*(4*c^2*d^2+3*b^2*e^2-4*c*e*(b*d+2*a*e))/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^2)-e*(2*c*d-b*e)*(c^2*d^2+3*b^2*e^2-c*e*(b*d+11*a*e))/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^3*(d+e*x))+(-b*c*d+b^2*e-2*a*c*e-c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(d+e*x)^2*(a+b*x+c*x^2))+(2*c*d-b*e)*(2*c^4*d^4-3*b^4*e^4-4*c^3*d^2*e*(b*d-5*a*e)+4*b^2*c*e^3*(b*d+5*a*e)-2*c^2*e^2*(b^2*d^2+10*a*b*d*e+15*a^2*e^2))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)^4)+e^3*(10*c^2*d^2+3*b^2*e^2-2*c*e*(5*b*d+a*e))*log(d+e*x)/(c*d^2-b*d*e+a*e^2)^4-1/2*e^3*(10*c^2*d^2+3*b^2*e^2-2*c*e*(5*b*d+a*e))*log(a+b*x+c*x^2)/(c*d^2-b*d*e+a*e^2)^4],
[x^7/(a+b*x+c*x^2)^3,x,8,-3*b*(2*b^2-9*a*c)*(b^2-3*a*c)*x/(c^4*(b^2-4*a*c)^2)+3/2*(2*b^4-13*a*b^2*c+16*a^2*c^2)*x^2/(c^3*(b^2-4*a*c)^2)-b*(2*b^2-11*a*c)*x^3/(c^2*(b^2-4*a*c)^2)+1/2*x^6*(2*a+b*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)+3/2*x^4*(a*(b^2-8*a*c)+b*(b^2-6*a*c)*x)/(c*(b^2-4*a*c)^2*(a+b*x+c*x^2))+3*b*(2*b^6-21*a*b^4*c+70*a^2*b^2*c^2-70*a^3*c^3)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^5*(b^2-4*a*c)^(5/2))+3/2*(2*b^2-a*c)*log(a+b*x+c*x^2)/c^5],
[x^6/(a+b*x+c*x^2)^3,x,8,3*(b^4-7*a*b^2*c+10*a^2*c^2)*x/(c^3*(b^2-4*a*c)^2)-3/2*b*(b^2-6*a*c)*x^2/(c^2*(b^2-4*a*c)^2)+1/2*x^5*(2*a+b*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)+x^3*(a*(b^2-10*a*c)+b*(b^2-7*a*c)*x)/(c*(b^2-4*a*c)^2*(a+b*x+c*x^2))-3*(b^6-10*a*b^4*c+30*a^2*b^2*c^2-20*a^3*c^3)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^4*(b^2-4*a*c)^(5/2))-3/2*b*log(a+b*x+c*x^2)/c^4],
[(d+e*x)^5/(a+b*x+c*x^2)^3,x,7,-e^2*(2*c*d-b*e)*(3*c^2*d^2-b^2*e^2-c*e*(3*b*d-7*a*e))*x/(c^2*(b^2-4*a*c)^2)-1/2*(d+e*x)^4*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)-1/2*(d+e*x)^2*(8*a*c*e*(c*d^2+2*a*e^2)-6*b*c*d*(c*d^2+3*a*e^2)+b^2*(7*c*d^2*e-a*e^3)-(2*c*d-b*e)*(6*c^2*d^2-b^2*e^2-2*c*e*(3*b*d-5*a*e))*x)/(c*(b^2-4*a*c)^2*(a+b*x+c*x^2))-(12*c^5*d^5-b^5*e^5+10*a*b^3*c*e^5-30*a^2*b*c^2*e^5-10*c^4*d^3*e*(3*b*d-4*a*e)+20*c^3*d*e^2*(b^2*d^2-3*a*b*d*e+3*a^2*e^2))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^3*(b^2-4*a*c)^(5/2))+1/2*e^5*log(a+b*x+c*x^2)/c^3],
[(d+e*x)^4/(a+b*x+c*x^2)^3,x,4,-1/2*(d+e*x)^3*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)+3*(c*d^2-b*d*e+a*e^2)*(d+e*x)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2))-12*(c*d^2-b*d*e+a*e^2)^2*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[(d+e*x)^3/(a+b*x+c*x^2)^3,x,4,-1/2*(b+2*c*x)*(d+e*x)^3/((b^2-4*a*c)*(a+b*x+c*x^2)^2)+3/2*(2*c*d-b*e)*(d+e*x)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2))-6*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[(d+e*x)^2/(a+b*x+c*x^2)^3,x,4,-1/2*(d+e*x)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)+(-2*b^2*d*e-4*a*c*d*e+3*b*(c*d^2+a*e^2)+(6*c^2*d^2+b^2*e^2-2*c*e*(3*b*d-a*e))*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2))-2*(6*c^2*d^2+b^2*e^2-2*c*e*(3*b*d-a*e))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[(d+e*x)/(a+b*x+c*x^2)^3,x,4,1/2*(-b*d+2*a*e-(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)+3/2*(2*c*d-b*e)*(b+2*c*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2))-6*c*(2*c*d-b*e)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[1/(a+b*x+c*x^2)^3,x,4,1/2*(-b-2*c*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)+3*c*(b+2*c*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2))-12*c^2*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[1/((d+e*x)*(a+b*x+c*x^2)^3),x,8,1/2*(-b*c*d+b^2*e-2*a*c*e-c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(a+b*x+c*x^2)^2)+1/2*(-3*a*c*e*(2*c*d-b*e)^2+(b*c*d-b^2*e+2*a*c*e)*(6*c^2*d^2-2*b^2*e^2-c*e*(3*b*d-8*a*e))+2*c*(2*c*d-b*e)*(3*c^2*d^2-b^2*e^2-c*e*(3*b*d-7*a*e))*x)/((b^2-4*a*c)^2*(c*d^2-b*d*e+a*e^2)^2*(a+b*x+c*x^2))-(12*c^5*d^5-b^5*e^5+10*a*b^3*c*e^5-30*a^2*b*c^2*e^5-10*c^4*d^3*e*(3*b*d-4*a*e)+20*c^3*d*e^2*(b^2*d^2-3*a*b*d*e+3*a^2*e^2))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(5/2)*(c*d^2-b*d*e+a*e^2)^3)+e^5*log(d+e*x)/(c*d^2-b*d*e+a*e^2)^3-1/2*e^5*log(a+b*x+c*x^2)/(c*d^2-b*d*e+a*e^2)^3],
[1/(x^2*(a+b*x+c*x^2)^3),x,8,-3*(b^2-5*a*c)*(b^2-2*a*c)/(a^3*(b^2-4*a*c)^2*x)+1/2*(b^2-2*a*c+b*c*x)/(a*(b^2-4*a*c)*x*(a+b*x+c*x^2)^2)+1/2*(3*b^4-20*a*b^2*c+20*a^2*c^2+3*b*c*(b^2-6*a*c)*x)/(a^2*(b^2-4*a*c)^2*x*(a+b*x+c*x^2))-3*(b^6-10*a*b^4*c+30*a^2*b^2*c^2-20*a^3*c^3)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(a^4*(b^2-4*a*c)^(5/2))-3*b*log(x)/a^4+3/2*b*log(a+b*x+c*x^2)/a^4],
[1/(x^3*(a+b*x+c*x^2)^3),x,8,-3/2*(2*b^4-13*a*b^2*c+16*a^2*c^2)/(a^3*(b^2-4*a*c)^2*x^2)+3*b*(2*b^2-9*a*c)*(b^2-3*a*c)/(a^4*(b^2-4*a*c)^2*x)+1/2*(b^2-2*a*c+b*c*x)/(a*(b^2-4*a*c)*x^2*(a+b*x+c*x^2)^2)+1/2*(4*b^4-25*a*b^2*c+24*a^2*c^2+2*b*c*(2*b^2-11*a*c)*x)/(a^2*(b^2-4*a*c)^2*x^2*(a+b*x+c*x^2))+3*b*(2*b^6-21*a*b^4*c+70*a^2*b^2*c^2-70*a^3*c^3)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(a^5*(b^2-4*a*c)^(5/2))+3*(2*b^2-a*c)*log(x)/a^5-3/2*(2*b^2-a*c)*log(a+b*x+c*x^2)/a^5],
[x^8/(a+b*x+c*x^2)^4,x,9,4*(b^6-11*a*b^4*c+38*a^2*b^2*c^2-35*a^3*c^3)*x/(c^4*(b^2-4*a*c)^3)-2*b*(b^4-10*a*b^2*c+29*a^2*c^2)*x^2/(c^3*(b^2-4*a*c)^3)+1/3*x^7*(2*a+b*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^3)+1/3*x^5*(a*(b^2-14*a*c)+b*(b^2-9*a*c)*x)/(c*(b^2-4*a*c)^2*(a+b*x+c*x^2)^2)+1/3*x^3*(4*a*(b^4-9*a*b^2*c+35*a^2*c^2)+b*(4*b^4-39*a*b^2*c+122*a^2*c^2)*x)/(c^2*(b^2-4*a*c)^3*(a+b*x+c*x^2))-4*(b^8-14*a*b^6*c+70*a^2*b^4*c^2-140*a^3*b^2*c^3+70*a^4*c^4)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^5*(b^2-4*a*c)^(7/2))-2*b*log(a+b*x+c*x^2)/c^5],
[x^7/(a+b*x+c*x^2)^4,x,8,-b*(b^4-11*a*b^2*c+38*a^2*c^2)*x/(c^3*(b^2-4*a*c)^3)+1/3*x^6*(2*a+b*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^3)+1/6*x^4*(a*(b^2-24*a*c)+b*(b^2-14*a*c)*x)/(c*(b^2-4*a*c)^2*(a+b*x+c*x^2)^2)+1/6*x^2*(3*a*(b^4-10*a*b^2*c+64*a^2*c^2)+b*(3*b^4-32*a*b^2*c+140*a^2*c^2)*x)/(c^2*(b^2-4*a*c)^3*(a+b*x+c*x^2))+b*(b^6-14*a*b^4*c+70*a^2*b^2*c^2-140*a^3*c^3)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^4*(b^2-4*a*c)^(7/2))+1/2*log(a+b*x+c*x^2)/c^4],
[x^6/(a+b*x+c*x^2)^4,x,5,1/3*x^5*(2*a+b*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^3)-5/3*a*x^3*(2*a+b*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2)^2)+10*a^2*x*(2*a+b*x)/((b^2-4*a*c)^3*(a+b*x+c*x^2))+40*a^3*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(7/2)],
[x^5/(a+b*x+c*x^2)^4,x,5,-1/3*x^5*(b+2*c*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^3)+5/6*b*x^3*(2*a+b*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2)^2)-5*a*b*x*(2*a+b*x)/((b^2-4*a*c)^3*(a+b*x+c*x^2))-20*a^2*b*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(7/2)],
[(d+e*x)^4/(a+b*x+c*x^2)^4,x,5,-1/3*(b+2*c*x)*(d+e*x)^4/((b^2-4*a*c)*(a+b*x+c*x^2)^3)+1/3*(d+e*x)^3*(5*b*c*d-2*b^2*e-2*a*c*e+5*c*(2*c*d-b*e)*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2)^2)-2*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)^3*(a+b*x+c*x^2))+8*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(7/2)],
[(d+e*x)^3/(a+b*x+c*x^2)^4,x,5,-1/3*(b+2*c*x)*(d+e*x)^3/((b^2-4*a*c)*(a+b*x+c*x^2)^3)+1/6*(d+e*x)^2*(10*b*c*d-3*b^2*e-8*a*c*e+10*c*(2*c*d-b*e)*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2)^2)+1/3*(-3*b^3*d*e^2+16*a*c*e*(5*c*d^2+a*e^2)-6*b*c*d*(5*c*d^2+13*a*e^2)+b^2*(25*c*d^2*e+11*a*e^3)-2*(2*c*d-b*e)*(15*c^2*d^2+4*b^2*e^2-c*e*(15*b*d+a*e))*x)/((b^2-4*a*c)^3*(a+b*x+c*x^2))+2*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(7/2)],
[(d+e*x)^2/(a+b*x+c*x^2)^4,x,5,-1/3*(d+e*x)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^3)+1/3*(-3*b^2*d*e-8*a*c*d*e+5*b*(c*d^2+a*e^2)+2*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2)^2)-2*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(b+2*c*x)/((b^2-4*a*c)^3*(a+b*x+c*x^2))+8*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(7/2)],
[(d+e*x)/(a+b*x+c*x^2)^4,x,5,1/3*(-b*d+2*a*e-(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^3)+5/6*(2*c*d-b*e)*(b+2*c*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2)^2)-5*c*(2*c*d-b*e)*(b+2*c*x)/((b^2-4*a*c)^3*(a+b*x+c*x^2))+20*c^2*(2*c*d-b*e)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(7/2)],
[1/(a+b*x+c*x^2)^4,x,5,1/3*(-b-2*c*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^3)+5/3*c*(b+2*c*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2)^2)-10*c^2*(b+2*c*x)/((b^2-4*a*c)^3*(a+b*x+c*x^2))+40*c^3*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(7/2)],
[1/((d+e*x)*(a+b*x+c*x^2)^4),x,9,1/3*(-b*c*d+b^2*e-2*a*c*e-c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(a+b*x+c*x^2)^3)+1/6*(-5*a*c*e*(2*c*d-b*e)^2+(b*c*d-b^2*e+2*a*c*e)*(10*c^2*d^2-3*b^2*e^2-c*e*(5*b*d-12*a*e))+c*(2*c*d-b*e)*(10*c^2*d^2-3*b^2*e^2-2*c*e*(5*b*d-11*a*e))*x)/((b^2-4*a*c)^2*(c*d^2-b*d*e+a*e^2)^2*(a+b*x+c*x^2)^2)+1/2*(b^5*c*d*e^4+2*b^6*e^5-64*a^3*c^3*e^5+b^4*c*e^3*(c*d^2-23*a*e^2)-2*b^3*c^2*d*e^2*(17*c*d^2+5*a*e^2)-4*b*c^3*d*(5*c^2*d^4+16*a*c*d^2*e^2+19*a^2*e^4)+2*b^2*c^2*e*(25*c^2*d^4+48*a*c*d^2*e^2+43*a^2*e^4)-2*c*(2*c*d-b*e)*(10*c^4*d^4+b^4*e^4+b^2*c*e^3*(3*b*d-11*a*e)-4*c^3*d^2*e*(5*b*d-8*a*e)+c^2*e^2*(7*b^2*d^2-32*a*b*d*e+38*a^2*e^2))*x)/((b^2-4*a*c)^3*(c*d^2-b*d*e+a*e^2)^3*(a+b*x+c*x^2))+(40*c^7*d^7+b^7*e^7-14*a*b^5*c*e^7+70*a^2*b^3*c^2*e^7-140*a^3*b*c^3*e^7-28*c^6*d^5*e*(5*b*d-6*a*e)+28*c^5*d^3*e^2*(6*b^2*d^2-15*a*b*d*e+10*a^2*e^2)-70*c^4*d*e^3*(b^3*d^3-4*a*b^2*d^2*e+6*a^2*b*d*e^2-4*a^3*e^3))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(7/2)*(c*d^2-b*d*e+a*e^2)^4)+e^7*log(d+e*x)/(c*d^2-b*d*e+a*e^2)^4-1/2*e^7*log(a+b*x+c*x^2)/(c*d^2-b*d*e+a*e^2)^4],
[1/(x^2*(a+b*x+c*x^2)^4),x,9,-4*(b^6-11*a*b^4*c+38*a^2*b^2*c^2-35*a^3*c^3)/(a^4*(b^2-4*a*c)^3*x)+1/3*(b^2-2*a*c+b*c*x)/(a*(b^2-4*a*c)*x*(a+b*x+c*x^2)^3)+1/3*(2*(b^4-7*a*b^2*c+7*a^2*c^2)+b*c*(2*b^2-13*a*c)*x)/(a^2*(b^2-4*a*c)^2*x*(a+b*x+c*x^2)^2)+2/3*(3*b^6-32*a*b^4*c+105*a^2*b^2*c^2-70*a^3*c^3+3*b*c*(b^4-10*a*b^2*c+29*a^2*c^2)*x)/(a^3*(b^2-4*a*c)^3*x*(a+b*x+c*x^2))-4*(b^8-14*a*b^6*c+70*a^2*b^4*c^2-140*a^3*b^2*c^3+70*a^4*c^4)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(a^5*(b^2-4*a*c)^(7/2))-4*b*log(x)/a^5+2*b*log(a+b*x+c*x^2)/a^5],
[(d+e*x)^5/(a+b*x+c*x^2)^5,x,6,-1/4*(b+2*c*x)*(d+e*x)^5/((b^2-4*a*c)*(a+b*x+c*x^2)^4)+1/12*(d+e*x)^4*(14*b*c*d-5*b^2*e-8*a*c*e+14*c*(2*c*d-b*e)*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2)^3)+1/12*(d+e*x)^3*(63*b^2*c*d*e+28*a*c^2*d*e-10*b^3*e^2-10*b*c*(7*c*d^2+3*a*e^2)-c*(140*c^2*d^2+27*b^2*e^2-4*c*e*(35*b*d-8*a*e))*x)/((b^2-4*a*c)^3*(a+b*x+c*x^2)^2)+5/2*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*(d+e*x)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)^4*(a+b*x+c*x^2))-10*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(9/2)],
[(d+e*x)^4/(a+b*x+c*x^2)^5,x,6,-1/4*(b+2*c*x)*(d+e*x)^4/((b^2-4*a*c)*(a+b*x+c*x^2)^4)+1/6*(d+e*x)^3*(7*b*c*d-2*b^2*e-6*a*c*e+7*c*(2*c*d-b*e)*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2)^3)+1/6*(d+e*x)^2*(28*b^2*c*d*e+28*a*c^2*d*e-3*b^3*e^2-b*c*(35*c*d^2+23*a*e^2)-c*(70*c^2*d^2+13*b^2*e^2-2*c*e*(35*b*d-9*a*e))*x)/((b^2-4*a*c)^3*(a+b*x+c*x^2)^2)+1/6*(-6*b^4*d*e^3-16*a*c^2*d*e*(35*c*d^2+16*a*e^2)-4*b^2*c*d*e*(70*c*d^2+83*a*e^2)+5*b^3*(19*c*d^2*e^2+5*a*e^4)+10*b*c*(21*c^2*d^4+88*a*c*d^2*e^2+11*a^2*e^4)+(420*c^4*d^4+19*b^4*e^4-40*c^3*d^2*e*(21*b*d-2*a*e)-38*b^2*c*e^3*(5*b*d-a*e)+2*c^2*e^2*(305*b^2*d^2-40*a*b*d*e-18*a^2*e^2))*x)/((b^2-4*a*c)^4*(a+b*x+c*x^2))-2*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(5*b*d-3*a*e)-20*c^3*d^2*e*(7*b*d-3*a*e)+6*c^2*e^2*(15*b^2*d^2-10*a*b*d*e+a^2*e^2))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(9/2)],
[(d+e*x)^3/(a+b*x+c*x^2)^5,x,6,-1/4*(b+2*c*x)*(d+e*x)^3/((b^2-4*a*c)*(a+b*x+c*x^2)^4)+1/12*(d+e*x)^2*(14*b*c*d-3*b^2*e-16*a*c*e+14*c*(2*c*d-b*e)*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2)^3)+1/12*(-3*b^3*d*e^2+32*a*c*e*(7*c*d^2+a*e^2)-2*b*c*d*(35*c*d^2+99*a*e^2)+b^2*(49*c*d^2*e+27*a*e^3)-2*(2*c*d-b*e)*(35*c^2*d^2+12*b^2*e^2-c*e*(35*b*d+13*a*e))*x)/((b^2-4*a*c)^3*(a+b*x+c*x^2)^2)+5/2*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*(b+2*c*x)/((b^2-4*a*c)^4*(a+b*x+c*x^2))-10*c*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(9/2)],
[(d+e*x)^2/(a+b*x+c*x^2)^5,x,6,-1/4*(d+e*x)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^4)+1/6*(-4*b^2*d*e-12*a*c*d*e+7*b*(c*d^2+a*e^2)+(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2)^3)-5/12*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*(b+2*c*x)/((b^2-4*a*c)^3*(a+b*x+c*x^2)^2)+5/2*c*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*(b+2*c*x)/((b^2-4*a*c)^4*(a+b*x+c*x^2))-10*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(9/2)],
[(d+e*x)/(a+b*x+c*x^2)^5,x,6,1/4*(-b*d+2*a*e-(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^4)+7/12*(2*c*d-b*e)*(b+2*c*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2)^3)-35/12*c*(2*c*d-b*e)*(b+2*c*x)/((b^2-4*a*c)^3*(a+b*x+c*x^2)^2)+35/2*c^2*(2*c*d-b*e)*(b+2*c*x)/((b^2-4*a*c)^4*(a+b*x+c*x^2))-70*c^3*(2*c*d-b*e)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(9/2)],
[1/(a+b*x+c*x^2)^5,x,6,1/4*(-b-2*c*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^4)+7/6*c*(b+2*c*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2)^3)-35/6*c^2*(b+2*c*x)/((b^2-4*a*c)^3*(a+b*x+c*x^2)^2)+35*c^3*(b+2*c*x)/((b^2-4*a*c)^4*(a+b*x+c*x^2))-140*c^4*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(9/2)],
[1/((d+e*x)*(a+b*x+c*x^2)^5),x,10,1/4*(-b*c*d+b^2*e-2*a*c*e-c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(a+b*x+c*x^2)^4)+1/12*(-7*a*c*e*(2*c*d-b*e)^2+(b*c*d-b^2*e+2*a*c*e)*(14*c^2*d^2-4*b^2*e^2-c*e*(7*b*d-16*a*e))+2*c*(2*c*d-b*e)*(7*c^2*d^2-2*b^2*e^2-c*e*(7*b*d-15*a*e))*x)/((b^2-4*a*c)^2*(c*d^2-b*d*e+a*e^2)^2*(a+b*x+c*x^2)^3)+1/12*(5*a*c*e*(2*c*d-b*e)^2*(7*c^2*d^2-2*b^2*e^2-c*e*(7*b*d-15*a*e))-(b*c*d-b^2*e+2*a*c*e)*(70*c^4*d^4+6*b^4*e^4+2*b^2*c*e^3*(5*b*d-24*a*e)-15*c^3*d^2*e*(7*b*d-10*a*e)+3*c^2*e^2*(5*b^2*d^2-25*a*b*d*e+32*a^2*e^2))-2*c*(2*c*d-b*e)*(35*c^4*d^4+3*b^4*e^4+2*b^2*c*e^3*(5*b*d-17*a*e)-10*c^3*d^2*e*(7*b*d-11*a*e)+c^2*e^2*(25*b^2*d^2-110*a*b*d*e+123*a^2*e^2))*x)/((b^2-4*a*c)^3*(c*d^2-b*d*e+a*e^2)^3*(a+b*x+c*x^2)^2)+1/2*(b^7*c*d*e^6+2*b^8*e^7+256*a^4*c^4*e^7+b^6*c*e^5*(c*d^2-31*a*e^2)+b^5*c^2*d*e^4*(c*d^2-14*a*e^2)-b^4*c^2*e^3*(125*c^2*d^4+13*a*c*d^2*e^2-178*a^2*e^4)+b^3*c^3*d*e^2*(295*c^2*d^4+492*a*c*d^2*e^2+69*a^2*e^4)+2*b*c^4*d*(35*c^3*d^6+145*a*c^2*d^4*e^2+233*a^2*c*d^2*e^4+187*a^3*e^6)-b^2*c^3*e*(245*c^3*d^6+725*a*c^2*d^4*e^2+699*a^2*c*d^2*e^4+443*a^3*e^6)+2*c*(2*c*d-b*e)*(35*c^6*d^6-b^6*e^6-5*c^5*d^4*e*(21*b*d-29*a*e)-3*b^4*c*e^5*(b*d-5*a*e)-b^2*c^2*e^4*(7*b^2*d^2-44*a*b*d*e+82*a^2*e^2)+c^4*d^2*e^2*(95*b^2*d^2-290*a*b*d*e+233*a^2*e^2)-c^3*e^3*(15*b^3*d^3-101*a*b^2*d^2*e+233*a^2*b*d*e^2-187*a^3*e^3))*x)/((b^2-4*a*c)^4*(c*d^2-b*d*e+a*e^2)^4*(a+b*x+c*x^2))-(140*c^9*d^9-b^9*e^9+18*a*b^7*c*e^9-126*a^2*b^5*c^2*e^9+420*a^3*b^3*c^3*e^9-630*a^4*b*c^4*e^9-90*c^8*d^7*e*(7*b*d-8*a*e)+72*c^7*d^5*e^2*(15*b^2*d^2-35*a*b*d*e+21*a^2*e^2)-84*c^6*d^3*e^3*(10*b^3*d^3-36*a*b^2*d^2*e+45*a^2*b*d*e^2-20*a^3*e^3)+252*c^5*d*e^4*(b^4*d^4-5*a*b^3*d^3*e+10*a^2*b^2*d^2*e^2-10*a^3*b*d*e^3+5*a^4*e^4))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(9/2)*(c*d^2-e*(b*d-a*e))^5)+e^9*log(d+e*x)/(c*d^2-b*d*e+a*e^2)^5-1/2*e^9*log(a+b*x+c*x^2)/(c*d^2-b*d*e+a*e^2)^5],
[1/((d+e*x)^2*(a+b*x+c*x^2)^5),x,10,5*e*(14*c^8*d^8-b^8*e^8-4*c^7*d^6*e*(14*b*d-19*a*e)+b^6*c*e^7*(b*d+15*a*e)+b^4*c^2*e^6*(b^2*d^2-16*a*b*d*e-82*a^2*e^2)+c^6*d^4*e^2*(79*b^2*d^2-228*a*b*d*e+176*a^2*e^2)-c^5*d^2*e^3*(41*b^3*d^3-197*a*b^2*d^2*e+352*a^2*b*d*e^2-244*a^3*e^3)+b^2*c^3*e^5*(b^3*d^3-15*a*b^2*d^2*e+95*a^2*b*d*e^2+187*a^3*e^3)+c^4*e^4*(b^4*d^4-14*a*b^3*d^3*e+81*a^2*b^2*d^2*e^2-244*a^3*b*d*e^3-126*a^4*e^4))/((b^2-4*a*c)^4*(c*d^2-b*d*e+a*e^2)^5*(d+e*x))+1/4*(-b*c*d+b^2*e-2*a*c*e-c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(d+e*x)*(a+b*x+c*x^2)^4)+1/12*(-8*a*c*e*(2*c*d-b*e)^2+(b*c*d-b^2*e+2*a*c*e)*(14*c^2*d^2-5*b^2*e^2-6*c*e*(b*d-3*a*e))+c*(2*c*d-b*e)*(14*c^2*d^2-5*b^2*e^2-2*c*e*(7*b*d-17*a*e))*x)/((b^2-4*a*c)^2*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)*(a+b*x+c*x^2)^3)+1/12*(3*a*c*e*(2*c*d-b*e)^2*(14*c^2*d^2-5*b^2*e^2-2*c*e*(7*b*d-17*a*e))-(b*c*d-b^2*e+2*a*c*e)*(70*c^4*d^4+10*b^4*e^4-2*c^3*d^2*e*(49*b*d-78*a*e)+5*b^2*c*e^3*(2*b*d-15*a*e)+3*c^2*e^2*(b^2*d^2-18*a*b*d*e+42*a^2*e^2))-5*c*(2*c*d-b*e)*(14*c^4*d^4+2*b^4*e^4+b^2*c*e^3*(5*b*d-21*a*e)-4*c^3*d^2*e*(7*b*d-12*a*e)+3*c^2*e^2*(3*b^2*d^2-16*a*b*d*e+22*a^2*e^2))*x)/((b^2-4*a*c)^3*(c*d^2-b*d*e+a*e^2)^3*(d+e*x)*(a+b*x+c*x^2)^2)-5/6*(2*a*c*e*(2*c*d-b*e)^2*(14*c^4*d^4+2*b^4*e^4+b^2*c*e^3*(5*b*d-21*a*e)-4*c^3*d^2*e*(7*b*d-12*a*e)+3*c^2*e^2*(3*b^2*d^2-16*a*b*d*e+22*a^2*e^2))-(b*c*d-b^2*e+2*a*c*e)*(42*c^6*d^6-3*b^6*e^6-2*c^5*d^4*e*(49*b*d-65*a*e)-2*b^4*c*e^5*(b*d-17*a*e)+b^2*c^2*e^4*(b^2*d^2+16*a*b*d*e-123*a^2*e^2)+c^4*d^2*e^2*(55*b^2*d^2-164*a*b*d*e+150*a^2*e^2)+6*c^3*e^3*(b^3*d^3-4*a*b^2*d^2*e-3*a^2*b*d*e^2+21*a^3*e^3))-3*c*(2*c*d-b*e)*(14*c^6*d^6-b^6*e^6-2*c^5*d^4*e*(21*b*d-31*a*e)-2*b^4*c*e^5*(b*d-7*a*e)-b^2*c^2*e^4*(3*b^2*d^2-26*a*b*d*e+69*a^2*e^2)+c^4*d^2*e^2*(37*b^2*d^2-124*a*b*d*e+114*a^2*e^2)-2*c^3*e^3*(2*b^3*d^3-18*a*b^2*d^2*e+57*a^2*b*d*e^2-65*a^3*e^3))*x)/((b^2-4*a*c)^4*(c*d^2-b*d*e+a*e^2)^4*(d+e*x)*(a+b*x+c*x^2))-5*(28*c^10*d^10+b^10*e^10-20*c^9*d^8*e*(7*b*d-9*a*e)-252*a^4*c^5*e^9*(5*b*d+a*e)+210*a^3*b^2*c^4*e^9*(4*b*d+3*a*e)-84*a^2*b^4*c^3*e^9*(3*b*d+5*a*e)+18*a*b^6*c^2*e^9*(2*b*d+7*a*e)-2*b^8*c*e^9*(b*d+9*a*e)+18*c^8*d^6*e^2*(15*b^2*d^2-40*a*b*d*e+28*a^2*e^2)-24*c^7*d^4*e^3*(10*b^3*d^3-42*a*b^2*d^2*e+63*a^2*b*d*e^2-35*a^3*e^3)+84*c^6*d^2*e^4*(b^4*d^4-6*a*b^3*d^3*e+15*a^2*b^2*d^2*e^2-20*a^3*b*d*e^3+15*a^4*e^4))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(9/2)*(c*d^2-e*(b*d-a*e))^6)+5*e^9*(2*c*d-b*e)*log(d+e*x)/(c*d^2-b*d*e+a*e^2)^6-5/2*e^9*(2*c*d-b*e)*log(a+b*x+c*x^2)/(c*d^2-b*d*e+a*e^2)^6],
[1/((1+2*x)*(2+3*x+5*x^2)^3),x,8,1/434*(37+20*x)/(2+3*x+5*x^2)^2+2/47089*(2609+2290*x)/(2+3*x+5*x^2)+32/343*log(1+2*x)-16/343*log(2+3*x+5*x^2)+125624/329623*arctan((3+10*x)/sqrt(31))/sqrt(31)],
[1/((1+2*x)^2*(2+3*x+5*x^2)^3),x,8,(-51516/329623)/(1+2*x)+1/434*(37+20*x)/((1+2*x)*(2+3*x+5*x^2)^2)+1/47089*(6427+5820*x)/((1+2*x)*(2+3*x+5*x^2))+384/2401*log(1+2*x)-192/2401*log(2+3*x+5*x^2)-1065012/2307361*arctan((3+10*x)/sqrt(31))/sqrt(31)],
[1/((1+2*x)*(2+3*x+5*x^2)^4),x,9,1/651*(37+20*x)/(2+3*x+5*x^2)^3+4/141267*(1983+1805*x)/(2+3*x+5*x^2)^2+4/10218313*(180133+203230*x)/(2+3*x+5*x^2)+128/2401*log(1+2*x)-64/2401*log(2+3*x+5*x^2)+19007376/71528191*arctan((3+10*x)/sqrt(31))/sqrt(31)],
[1/((1+2*x)^2*(2+3*x+5*x^2)^4),x,9,(-6802312/71528191)/(1+2*x)+1/651*(37+20*x)/((1+2*x)*(2+3*x+5*x^2)^3)+1/47089*(3047+2820*x)/((1+2*x)*(2+3*x+5*x^2)^2)+2/10218313*(504757+603620*x)/((1+2*x)*(2+3*x+5*x^2))+2048/16807*log(1+2*x)-1024/16807*log(2+3*x+5*x^2)-116056984/500697337*arctan((3+10*x)/sqrt(31))/sqrt(31)],
[(7-3*x)/(-5+2*x+x^2),x,3,-1/6*log(1+x-sqrt(6))*(9-5*sqrt(6))-1/6*log(1+x+sqrt(6))*(9+5*sqrt(6))],
[1/((-1+x)*(1+x+x^2)),x,6,1/3*log(1-x)-1/6*log(1+x+x^2)-arctan((1+2*x)/sqrt(3))/sqrt(3)],
[2*((a/b)^(1/n)-x*cos((-1+2*k)*Pi/n))/((a/b)^(2/n)+x^2-2*(a/b)^(1/n)*x*cos((-1+2*k)*Pi/n)),x,5,-cos((Pi-2*k*Pi)/n)*log((a/b)^(2/n)+x^2-2*(a/b)^(1/n)*x*cos((Pi-2*k*Pi)/n))+2*arctan((x-(a/b)^(1/n)*cos((Pi-2*k*Pi)/n))*csc((Pi-2*k*Pi)/n)/(a/b)^(1/n))*sin((Pi-2*k*Pi)/n)],
[x^4/(2+13*x+15*x^2),x,5,139/3375*x-13/450*x^2+1/45*x^3-16/567*log(2+3*x)+1/4375*log(1+5*x)],
[x^3/(2+13*x+15*x^2),x,5,-13/225*x+1/30*x^2+8/189*log(2+3*x)-1/875*log(1+5*x)],
[x^2/(2+13*x+15*x^2),x,4,1/15*x-4/63*log(2+3*x)+1/175*log(1+5*x)],
[x/(2+13*x+15*x^2),x,3,2/21*log(2+3*x)-1/35*log(1+5*x)],
[1/(2+13*x+15*x^2),x,3,-1/7*log(2+3*x)+1/7*log(1+5*x)],
[1/(x*(2+13*x+15*x^2)),x,5,1/2*log(x)+3/14*log(2+3*x)-5/7*log(1+5*x)],
[1/(x^2*(2+13*x+15*x^2)),x,3,(-1/2)/x-13/4*log(x)-9/28*log(2+3*x)+25/7*log(1+5*x)],
[1/(x^3*(2+13*x+15*x^2)),x,3,(-1/4)/x^2+13/4/x+139/8*log(x)+27/56*log(2+3*x)-125/7*log(1+5*x)],
[1/(x^4*(2+13*x+15*x^2)),x,3,(-1/6)/x^3+13/8/x^2+(-139/8)/x-1417/16*log(x)-81/112*log(2+3*x)+625/7*log(1+5*x)],
[x^5/(2*x+13*x^2+15*x^3),x,6,139/3375*x-13/450*x^2+1/45*x^3-16/567*log(2+3*x)+1/4375*log(1+5*x)],
[x^4/(2*x+13*x^2+15*x^3),x,6,-13/225*x+1/30*x^2+8/189*log(2+3*x)-1/875*log(1+5*x)],
[x^3/(2*x+13*x^2+15*x^3),x,5,1/15*x-4/63*log(2+3*x)+1/175*log(1+5*x)],
[x^2/(2*x+13*x^2+15*x^3),x,4,2/21*log(2+3*x)-1/35*log(1+5*x)],
[x/(2*x+13*x^2+15*x^3),x,4,-1/7*log(2+3*x)+1/7*log(1+5*x)],
[1/(2*x+13*x^2+15*x^3),x,6,1/2*log(x)+3/14*log(2+3*x)-5/7*log(1+5*x)],
[1/(x*(2*x+13*x^2+15*x^3)),x,4,(-1/2)/x-13/4*log(x)-9/28*log(2+3*x)+25/7*log(1+5*x)],
[1/(x^2*(2*x+13*x^2+15*x^3)),x,4,(-1/4)/x^2+13/4/x+139/8*log(x)+27/56*log(2+3*x)-125/7*log(1+5*x)],
[1/(x^3*(2*x+13*x^2+15*x^3)),x,4,(-1/6)/x^3+13/8/x^2+(-139/8)/x-1417/16*log(x)-81/112*log(2+3*x)+625/7*log(1+5*x)],
[x/(4+4*x+x^2),x,3,2/(2+x)+log(2+x)],
[x/(5+2*x+x^2),x,4,-1/2*arctan(1/2*(1+x))+1/2*log(5+2*x+x^2)],
[x/(6-5*x+x^2),x,3,-2*log(2-x)+3*log(3-x)],
[x/(2+2*x+x^2)^2,x,3,1/2*(-2-x)/(2+2*x+x^2)-1/2*arctan(1+x)],
[x/(1+x+x^2)^3,x,4,1/6*(-2-x)/(1+x+x^2)^2+1/6*(-1-2*x)/(1+x+x^2)-2/3*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[x^2/(1+x+x^2),x,5,x-1/2*log(1+x+x^2)-arctan((1+2*x)/sqrt(3))/sqrt(3)],
[x^2/(2-3*x+x^2),x,4,x-log(1-x)+4*log(2-x)],
[x^2/(-6+x+x^2),x,4,x+4/5*log(2-x)-9/5*log(3+x)],
[x^2/(2+2*x+x^2)^2,x,3,1/(2+2*x+x^2)+arctan(1+x),-1/2*x*(2+x)/(2+2*x+x^2)+arctan(1+x)],
[x^3/(2-3*x+x^2),x,5,3*x+1/2*x^2-log(1-x)+8*log(2-x)],
[x^3/(1+2*x+x^2),x,3,-2*x+1/2*x^2+1/(1+x)+3*log(1+x)],
[x^3/(1-2*x+x^2),x,3,1/(1-x)+2*x+1/2*x^2+3*log(1-x)],
[x^4/(4+4*x+x^2),x,3,12*x-2*x^2+1/3*x^3+(-16)/(2+x)-32*log(2+x)],
[1/(x*(1+x+x^2)),x,6,log(x)-1/2*log(1+x+x^2)-arctan((1+2*x)/sqrt(3))/sqrt(3)],

# Integrands of the form (d+e x)^(m/2) (a+b x+c x^2)^p

# p>0
[(d+e*x)^(5/2)*(a+b*x+c*x^2),x,2,2/7*(c*d^2-b*d*e+a*e^2)*(d+e*x)^(7/2)/e^3-2/9*(2*c*d-b*e)*(d+e*x)^(9/2)/e^3+2/11*c*(d+e*x)^(11/2)/e^3],
[(d+e*x)^(3/2)*(a+b*x+c*x^2),x,2,2/5*(c*d^2-b*d*e+a*e^2)*(d+e*x)^(5/2)/e^3-2/7*(2*c*d-b*e)*(d+e*x)^(7/2)/e^3+2/9*c*(d+e*x)^(9/2)/e^3],
[(a+b*x+c*x^2)*sqrt(d+e*x),x,2,2/3*(c*d^2-b*d*e+a*e^2)*(d+e*x)^(3/2)/e^3-2/5*(2*c*d-b*e)*(d+e*x)^(5/2)/e^3+2/7*c*(d+e*x)^(7/2)/e^3],
[(a+b*x+c*x^2)/sqrt(d+e*x),x,2,-2/3*(2*c*d-b*e)*(d+e*x)^(3/2)/e^3+2/5*c*(d+e*x)^(5/2)/e^3+2*(c*d^2-b*d*e+a*e^2)*sqrt(d+e*x)/e^3],
[(a+b*x+c*x^2)/(d+e*x)^(3/2),x,2,2/3*c*(d+e*x)^(3/2)/e^3-2*(c*d^2-b*d*e+a*e^2)/(e^3*sqrt(d+e*x))-2*(2*c*d-b*e)*sqrt(d+e*x)/e^3],
[(a+b*x+c*x^2)/(d+e*x)^(5/2),x,2,-2/3*(c*d^2-b*d*e+a*e^2)/(e^3*(d+e*x)^(3/2))+2*(2*c*d-b*e)/(e^3*sqrt(d+e*x))+2*c*sqrt(d+e*x)/e^3],
[(a+b*x+c*x^2)/(d+e*x)^(7/2),x,2,-2/5*(c*d^2-b*d*e+a*e^2)/(e^3*(d+e*x)^(5/2))+2/3*(2*c*d-b*e)/(e^3*(d+e*x)^(3/2))-2*c/(e^3*sqrt(d+e*x))],
[(d+e*x)^(5/2)*(a+b*x+c*x^2)^2,x,2,2/7*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^(7/2)/e^5-4/9*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(d+e*x)^(9/2)/e^5+2/11*(6*c^2*d^2+b^2*e^2-2*c*e*(3*b*d-a*e))*(d+e*x)^(11/2)/e^5-4/13*c*(2*c*d-b*e)*(d+e*x)^(13/2)/e^5+2/15*c^2*(d+e*x)^(15/2)/e^5],
[(d+e*x)^(3/2)*(a+b*x+c*x^2)^2,x,2,2/5*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^(5/2)/e^5-4/7*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(d+e*x)^(7/2)/e^5+2/9*(6*c^2*d^2+b^2*e^2-2*c*e*(3*b*d-a*e))*(d+e*x)^(9/2)/e^5-4/11*c*(2*c*d-b*e)*(d+e*x)^(11/2)/e^5+2/13*c^2*(d+e*x)^(13/2)/e^5],
[(a+b*x+c*x^2)^2*sqrt(d+e*x),x,2,2/3*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^(3/2)/e^5-4/5*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(d+e*x)^(5/2)/e^5+2/7*(6*c^2*d^2+b^2*e^2-2*c*e*(3*b*d-a*e))*(d+e*x)^(7/2)/e^5-4/9*c*(2*c*d-b*e)*(d+e*x)^(9/2)/e^5+2/11*c^2*(d+e*x)^(11/2)/e^5],
[(a+b*x+c*x^2)^2/sqrt(d+e*x),x,2,-4/3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(d+e*x)^(3/2)/e^5+2/5*(6*c^2*d^2+b^2*e^2-2*c*e*(3*b*d-a*e))*(d+e*x)^(5/2)/e^5-4/7*c*(2*c*d-b*e)*(d+e*x)^(7/2)/e^5+2/9*c^2*(d+e*x)^(9/2)/e^5+2*(c*d^2-b*d*e+a*e^2)^2*sqrt(d+e*x)/e^5],
[(a+b*x+c*x^2)^2/(d+e*x)^(3/2),x,2,2/3*(6*c^2*d^2+b^2*e^2-2*c*e*(3*b*d-a*e))*(d+e*x)^(3/2)/e^5-4/5*c*(2*c*d-b*e)*(d+e*x)^(5/2)/e^5+2/7*c^2*(d+e*x)^(7/2)/e^5-2*(c*d^2-b*d*e+a*e^2)^2/(e^5*sqrt(d+e*x))-4*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*sqrt(d+e*x)/e^5],
[(a+b*x+c*x^2)^2/(d+e*x)^(5/2),x,2,-2/3*(c*d^2-b*d*e+a*e^2)^2/(e^5*(d+e*x)^(3/2))-4/3*c*(2*c*d-b*e)*(d+e*x)^(3/2)/e^5+2/5*c^2*(d+e*x)^(5/2)/e^5+4*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)/(e^5*sqrt(d+e*x))+2*(6*c^2*d^2+b^2*e^2-2*c*e*(3*b*d-a*e))*sqrt(d+e*x)/e^5],
[(a+b*x+c*x^2)^2/(d+e*x)^(7/2),x,2,-2/5*(c*d^2-b*d*e+a*e^2)^2/(e^5*(d+e*x)^(5/2))+4/3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)/(e^5*(d+e*x)^(3/2))+2/3*c^2*(d+e*x)^(3/2)/e^5-2*(6*c^2*d^2+b^2*e^2-2*c*e*(3*b*d-a*e))/(e^5*sqrt(d+e*x))-4*c*(2*c*d-b*e)*sqrt(d+e*x)/e^5],
[(d+e*x)^(5/2)*(a+b*x+c*x^2)^3,x,2,2/7*(c*d^2-b*d*e+a*e^2)^3*(d+e*x)^(7/2)/e^7-2/3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^(9/2)/e^7+6/11*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^(11/2)/e^7-2/13*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*(d+e*x)^(13/2)/e^7+2/5*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^(15/2)/e^7-6/17*c^2*(2*c*d-b*e)*(d+e*x)^(17/2)/e^7+2/19*c^3*(d+e*x)^(19/2)/e^7],
[(d+e*x)^(3/2)*(a+b*x+c*x^2)^3,x,2,2/5*(c*d^2-b*d*e+a*e^2)^3*(d+e*x)^(5/2)/e^7-6/7*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^(7/2)/e^7+2/3*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^(9/2)/e^7-2/11*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*(d+e*x)^(11/2)/e^7+6/13*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^(13/2)/e^7-2/5*c^2*(2*c*d-b*e)*(d+e*x)^(15/2)/e^7+2/17*c^3*(d+e*x)^(17/2)/e^7],
[(a+b*x+c*x^2)^3*sqrt(d+e*x),x,2,2/3*(c*d^2-b*d*e+a*e^2)^3*(d+e*x)^(3/2)/e^7-6/5*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^(5/2)/e^7+6/7*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^(7/2)/e^7-2/9*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*(d+e*x)^(9/2)/e^7+6/11*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^(11/2)/e^7-6/13*c^2*(2*c*d-b*e)*(d+e*x)^(13/2)/e^7+2/15*c^3*(d+e*x)^(15/2)/e^7],
[(a+b*x+c*x^2)^3/sqrt(d+e*x),x,2,-2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^(3/2)/e^7+6/5*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^(5/2)/e^7-2/7*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*(d+e*x)^(7/2)/e^7+2/3*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^(9/2)/e^7-6/11*c^2*(2*c*d-b*e)*(d+e*x)^(11/2)/e^7+2/13*c^3*(d+e*x)^(13/2)/e^7+2*(c*d^2-b*d*e+a*e^2)^3*sqrt(d+e*x)/e^7],
[(a+b*x+c*x^2)^3/(d+e*x)^(3/2),x,2,2*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^(3/2)/e^7-2/5*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*(d+e*x)^(5/2)/e^7+6/7*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^(7/2)/e^7-2/3*c^2*(2*c*d-b*e)*(d+e*x)^(9/2)/e^7+2/11*c^3*(d+e*x)^(11/2)/e^7-2*(c*d^2-b*d*e+a*e^2)^3/(e^7*sqrt(d+e*x))-6*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2*sqrt(d+e*x)/e^7],
[(a+b*x+c*x^2)^3/(d+e*x)^(5/2),x,2,-2/3*(c*d^2-b*d*e+a*e^2)^3/(e^7*(d+e*x)^(3/2))-2/3*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*(d+e*x)^(3/2)/e^7+6/5*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^(5/2)/e^7-6/7*c^2*(2*c*d-b*e)*(d+e*x)^(7/2)/e^7+2/9*c^3*(d+e*x)^(9/2)/e^7+6*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2/(e^7*sqrt(d+e*x))+6*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*sqrt(d+e*x)/e^7],
[(a+b*x+c*x^2)^3/(d+e*x)^(7/2),x,2,-2/5*(c*d^2-b*d*e+a*e^2)^3/(e^7*(d+e*x)^(5/2))+2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2/(e^7*(d+e*x)^(3/2))+2*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^(3/2)/e^7-6/5*c^2*(2*c*d-b*e)*(d+e*x)^(5/2)/e^7+2/7*c^3*(d+e*x)^(7/2)/e^7-6*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))/(e^7*sqrt(d+e*x))-2*(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*sqrt(d+e*x)/e^7],

# p<0
[(d+e*x)^(5/2)/(a+b*x+c*x^2),x,6,2/3*e*(d+e*x)^(3/2)/c+2*e*(2*c*d-b*e)*sqrt(d+e*x)/c^2-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*(2*c^3*d^3-b^2*e^3*(b-sqrt(b^2-4*a*c))-3*c^2*d*e*(b*d+2*a*e-d*sqrt(b^2-4*a*c))+c*e^2*(3*b^2*d+3*a*b*e-3*b*d*sqrt(b^2-4*a*c)-a*e*sqrt(b^2-4*a*c)))/(c^(5/2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(2*c^3*d^3-b^2*e^3*(b+sqrt(b^2-4*a*c))-3*c^2*d*e*(b*d+2*a*e+d*sqrt(b^2-4*a*c))+c*e^2*(3*b^2*d+a*e*sqrt(b^2-4*a*c)+3*b*(a*e+d*sqrt(b^2-4*a*c))))/(c^(5/2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x)^(3/2)/(a+b*x+c*x^2),x,5,2*e*sqrt(d+e*x)/c-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*(2*c^2*d^2+b*e^2*(b-sqrt(b^2-4*a*c))-2*c*e*(b*d+a*e-d*sqrt(b^2-4*a*c)))/(c^(3/2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(2*c^2*d^2+b*e^2*(b+sqrt(b^2-4*a*c))-2*c*e*(b*d+a*e+d*sqrt(b^2-4*a*c)))/(c^(3/2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x)^(1/2)/(a+b*x+c*x^2),x,4,-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(c)*sqrt(b^2-4*a*c))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(c)*sqrt(b^2-4*a*c))],
[1/((d+e*x)^(1/2)*(a+b*x+c*x^2)),x,4,-2*arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)/(sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+2*arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)/(sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[1/((d+e*x)^(3/2)*(a+b*x+c*x^2)),x,5,-2*e/((c*d^2-b*d*e+a*e^2)*sqrt(d+e*x))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)*(2*c*d-e*(b+sqrt(b^2-4*a*c)))/((c*d^2-b*d*e+a*e^2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))/((c*d^2-b*d*e+a*e^2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[1/((d+e*x)^(5/2)*(a+b*x+c*x^2)),x,6,-2/3*e/((c*d^2-b*d*e+a*e^2)*(d+e*x)^(3/2))-2*e*(2*c*d-b*e)/((c*d^2-b*d*e+a*e^2)^2*sqrt(d+e*x))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)*(2*c^2*d^2+b*e^2*(b+sqrt(b^2-4*a*c))-2*c*e*(b*d+a*e+d*sqrt(b^2-4*a*c)))/((c*d^2-b*d*e+a*e^2)^2*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)*(2*c^2*d^2+b*e^2*(b-sqrt(b^2-4*a*c))-2*c*e*(b*d+a*e-d*sqrt(b^2-4*a*c)))/((c*d^2-b*d*e+a*e^2)^2*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x)^(7/2)/(a+b*x+c*x^2)^2,x,7,e*(2*c*d-b*e)*(d+e*x)^(3/2)/(c*(b^2-4*a*c))-(d+e*x)^(5/2)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2))+e*(2*c^2*d^2+3*b^2*e^2-2*c*e*(b*d+5*a*e))*sqrt(d+e*x)/(c^2*(b^2-4*a*c))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(8*c^4*d^4-3*b^3*e^4*(b-sqrt(b^2-4*a*c))-2*c^3*d^2*e*(8*b*d-18*a*e-d*sqrt(b^2-4*a*c))+b*c*e^3*(5*b^2*d+19*a*b*e-5*b*d*sqrt(b^2-4*a*c)-13*a*e*sqrt(b^2-4*a*c))+c^2*e^2*(3*b^2*d^2-3*b*d*(12*a*e+d*sqrt(b^2-4*a*c))+2*a*e*(-10*a*e+13*d*sqrt(b^2-4*a*c))))/(c^(5/2)*(b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(8*c^4*d^4-3*b^3*e^4*(b+sqrt(b^2-4*a*c))-2*c^3*d^2*e*(8*b*d-18*a*e+d*sqrt(b^2-4*a*c))+b*c*e^3*(5*b^2*d+19*a*b*e+5*b*d*sqrt(b^2-4*a*c)+13*a*e*sqrt(b^2-4*a*c))+c^2*e^2*(3*b^2*d^2+3*b*d*(-12*a*e+d*sqrt(b^2-4*a*c))-2*a*e*(10*a*e+13*d*sqrt(b^2-4*a*c))))/(c^(5/2)*(b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x)^(5/2)/(a+b*x+c*x^2)^2,x,6,-(d+e*x)^(3/2)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2))+e*(2*c*d-b*e)*sqrt(d+e*x)/(c*(b^2-4*a*c))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(8*c^3*d^3+b^2*e^3*(b-sqrt(b^2-4*a*c))-2*c^2*d*e*(6*b*d-8*a*e-d*sqrt(b^2-4*a*c))+2*c*e^2*(b^2*d+3*a*e*sqrt(b^2-4*a*c)-b*(4*a*e+d*sqrt(b^2-4*a*c))))/(c^(3/2)*(b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(8*c^3*d^3+b^2*e^3*(b+sqrt(b^2-4*a*c))-2*c^2*d*e*(6*b*d-8*a*e+d*sqrt(b^2-4*a*c))+2*c*e^2*(b^2*d-4*a*b*e+b*d*sqrt(b^2-4*a*c)-3*a*e*sqrt(b^2-4*a*c)))/(c^(3/2)*(b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x)^(3/2)/(a+b*x+c*x^2)^2,x,5,-(b*d-2*a*e+(2*c*d-b*e)*x)*sqrt(d+e*x)/((b^2-4*a*c)*(a+b*x+c*x^2))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(8*c^2*d^2+b*e^2*(b-sqrt(b^2-4*a*c))-2*c*e*(4*b*d-2*a*e-d*sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(8*c^2*d^2+b*e^2*(b+sqrt(b^2-4*a*c))-2*c*e*(4*b*d-2*a*e+d*sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x)^(1/2)/(a+b*x+c*x^2)^2,x,5,-(b+2*c*x)*sqrt(d+e*x)/((b^2-4*a*c)*(a+b*x+c*x^2))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)*(4*c*d-e*(2*b-sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(3/2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)*(4*c*d-e*(2*b+sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(3/2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[1/((d+e*x)^(1/2)*(a+b*x+c*x^2)^2),x,5,-(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)*sqrt(d+e*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(a+b*x+c*x^2))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(c)*(8*c^2*d^2-b*e^2*(b+sqrt(b^2-4*a*c))-2*c*e*(4*b*d-6*a*e-d*sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(c)*(8*c^2*d^2-b*e^2*(b-sqrt(b^2-4*a*c))-2*c*e*(4*b*d-6*a*e+d*sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[1/((d+e*x)^(3/2)*(a+b*x+c*x^2)^2),x,6,-e*(2*c^2*d^2+3*b^2*e^2-2*c*e*(b*d+5*a*e))/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^2*sqrt(d+e*x))+(-b*c*d+b^2*e-2*a*c*e-c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(a+b*x+c*x^2)*sqrt(d+e*x))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(c)*(8*c^3*d^3+3*b^2*e^3*(b+sqrt(b^2-4*a*c))-2*c^2*d*e*(6*b*d-16*a*e-d*sqrt(b^2-4*a*c))-2*c*e^2*(b^2*d+8*a*b*e+b*d*sqrt(b^2-4*a*c)+5*a*e*sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)^2*sqrt(2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(c)*(8*c^3*d^3+3*b^2*e^3*(b-sqrt(b^2-4*a*c))-2*c^2*d*e*(6*b*d-16*a*e+d*sqrt(b^2-4*a*c))-2*c*e^2*(b^2*d+8*a*b*e-b*d*sqrt(b^2-4*a*c)-5*a*e*sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)^2*sqrt(2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[1/(x^(5/2)*(a+b*x+c*x^2)^2),x,7,1/3*(-5*b^2+14*a*c)/(a^2*(b^2-4*a*c)*x^(3/2))+(b^2-2*a*c+b*c*x)/(a*(b^2-4*a*c)*x^(3/2)*(a+b*x+c*x^2))+b*(5*b^2-19*a*c)/(a^3*(b^2-4*a*c)*sqrt(x))+arctan(sqrt(2)*sqrt(c)*sqrt(x)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(5*b^4-29*a*b^2*c+28*a^2*c^2+b*(5*b^2-19*a*c)*sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-arctan(sqrt(2)*sqrt(c)*sqrt(x)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(5*b^4-29*a*b^2*c+28*a^2*c^2-b*(5*b^2-19*a*c)*sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x)^(7/2)/(a+b*x+c*x^2)^3,x,6,-1/2*(d+e*x)^(5/2)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)+1/4*(12*b*c*d*(c*d^2+3*a*e^2)-4*a*c*e*(7*c*d^2+5*a*e^2)-b^2*(11*c*d^2*e+a*e^3)+(2*c*d-b*e)*(12*c^2*d^2+b^2*e^2-4*c*e*(3*b*d-2*a*e))*x)*sqrt(d+e*x)/(c*(b^2-4*a*c)^2*(a+b*x+c*x^2))-1/4*arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(96*c^4*d^4-b^3*e^4*(b-sqrt(b^2-4*a*c))-8*c^3*d^2*e*(24*b*d-19*a*e-3*d*sqrt(b^2-4*a*c))-2*b*c*e^3*(5*b^2*d-9*a*b*e-5*b*d*sqrt(b^2-4*a*c)+8*a*e*sqrt(b^2-4*a*c))+2*c^2*e^2*(53*b^2*d^2+4*a*e*(5*a*e+4*d*sqrt(b^2-4*a*c))-2*b*d*(38*a*e+9*d*sqrt(b^2-4*a*c))))/(c^(3/2)*(b^2-4*a*c)^(5/2)*sqrt(2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+1/4*arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(96*c^4*d^4-b^3*e^4*(b+sqrt(b^2-4*a*c))-8*c^3*d^2*e*(24*b*d-19*a*e+3*d*sqrt(b^2-4*a*c))-2*b*c*e^3*(5*b^2*d-9*a*b*e+5*b*d*sqrt(b^2-4*a*c)-8*a*e*sqrt(b^2-4*a*c))+2*c^2*e^2*(53*b^2*d^2-4*a*e*(-5*a*e+4*d*sqrt(b^2-4*a*c))+2*b*d*(-38*a*e+9*d*sqrt(b^2-4*a*c))))/(c^(3/2)*(b^2-4*a*c)^(5/2)*sqrt(2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x)^(5/2)/(a+b*x+c*x^2)^3,x,6,-1/2*(d+e*x)^(3/2)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)-3/4*(3*b^2*d*e+4*a*c*d*e-4*b*(c*d^2+a*e^2)-(8*c^2*d^2+b^2*e^2-4*c*e*(2*b*d-a*e))*x)*sqrt(d+e*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2))-3/4*arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(32*c^3*d^3-b^2*e^3*(b-sqrt(b^2-4*a*c))-8*c^2*d*e*(6*b*d-3*a*e-d*sqrt(b^2-4*a*c))+2*c*e^2*(9*b^2*d-6*a*b*e-4*b*d*sqrt(b^2-4*a*c)+2*a*e*sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(5/2)*sqrt(2)*sqrt(c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+3/4*arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(32*c^3*d^3-b^2*e^3*(b+sqrt(b^2-4*a*c))-8*c^2*d*e*(6*b*d-3*a*e+d*sqrt(b^2-4*a*c))+2*c*e^2*(9*b^2*d-6*a*b*e+4*b*d*sqrt(b^2-4*a*c)-2*a*e*sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(5/2)*sqrt(2)*sqrt(c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x)^(3/2)/(a+b*x+c*x^2)^3,x,6,-1/2*(b*d-2*a*e+(2*c*d-b*e)*x)*sqrt(d+e*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)+1/4*(12*b*c*d-7*b^2*e+4*a*c*e+12*c*(2*c*d-b*e)*x)*sqrt(d+e*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2))-3/2*arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(c)*(16*c^2*d^2+b*e^2*(3*b-2*sqrt(b^2-4*a*c))-4*c*e*(4*b*d-a*e-d*sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(5/2)*sqrt(2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+3/2*arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(c)*(16*c^2*d^2+b*e^2*(3*b+2*sqrt(b^2-4*a*c))-4*c*e*(4*b*d-a*e+d*sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(5/2)*sqrt(2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x)^(1/2)/(a+b*x+c*x^2)^3,x,6,-1/2*(b+2*c*x)*sqrt(d+e*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)-1/4*(13*b^2*c*d*e-4*a*c^2*d*e-b^3*e^2-4*b*c*(3*c*d^2+2*a*e^2)-c*(24*c^2*d^2+b^2*e^2-4*c*e*(6*b*d-5*a*e))*x)*sqrt(d+e*x)/((b^2-4*a*c)^2*(c*d^2-b*d*e+a*e^2)*(a+b*x+c*x^2))-1/4*arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(c)*(96*c^3*d^3+b^2*e^3*(b+sqrt(b^2-4*a*c))-8*c^2*d*e*(18*b*d-13*a*e-3*d*sqrt(b^2-4*a*c))+2*c*e^2*(23*b^2*d+10*a*e*sqrt(b^2-4*a*c)-2*b*(13*a*e+6*d*sqrt(b^2-4*a*c))))/((b^2-4*a*c)^(5/2)*(c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+1/4*arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(c)*(96*c^3*d^3+b^2*e^3*(b-sqrt(b^2-4*a*c))-8*c^2*d*e*(18*b*d-13*a*e+3*d*sqrt(b^2-4*a*c))+2*c*e^2*(23*b^2*d-26*a*b*e+12*b*d*sqrt(b^2-4*a*c)-10*a*e*sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(5/2)*(c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[1/((d+e*x)^(1/2)*(a+b*x+c*x^2)^3),x,6,-1/2*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)*sqrt(d+e*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(a+b*x+c*x^2)^2)-1/4*(5*a*c*e*(2*c*d-b*e)^2-(b*c*d-b^2*e+2*a*c*e)*(12*c^2*d^2-3*b^2*e^2-7*c*e*(b*d-2*a*e))-3*c*(2*c*d-b*e)*(4*c^2*d^2-b^2*e^2-4*c*e*(b*d-2*a*e))*x)*sqrt(d+e*x)/((b^2-4*a*c)^2*(c*d^2-b*d*e+a*e^2)^2*(a+b*x+c*x^2))-3/4*arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(c)*(32*c^4*d^4+b^3*e^4*(b+sqrt(b^2-4*a*c))-8*c^3*d^2*e*(8*b*d-9*a*e-d*sqrt(b^2-4*a*c))+2*b*c*e^3*(b^2*d-5*a*b*e+b*d*sqrt(b^2-4*a*c)-4*a*e*sqrt(b^2-4*a*c))+2*c^2*e^2*(15*b^2*d^2-6*b*d*(6*a*e+d*sqrt(b^2-4*a*c))+4*a*e*(7*a*e+2*d*sqrt(b^2-4*a*c))))/((b^2-4*a*c)^(5/2)*(c*d^2-b*d*e+a*e^2)^2*sqrt(2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+3/4*arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(c)*(32*c^4*d^4+b^3*e^4*(b-sqrt(b^2-4*a*c))-8*c^3*d^2*e*(8*b*d-9*a*e+d*sqrt(b^2-4*a*c))+2*b*c*e^3*(b^2*d+4*a*e*sqrt(b^2-4*a*c)-b*(5*a*e+d*sqrt(b^2-4*a*c)))+2*c^2*e^2*(15*b^2*d^2+6*b*d*(-6*a*e+d*sqrt(b^2-4*a*c))-4*a*e*(-7*a*e+2*d*sqrt(b^2-4*a*c))))/((b^2-4*a*c)^(5/2)*(c*d^2-b*d*e+a*e^2)^2*sqrt(2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x)^(1/2)/(a+I*b*x+c*x^2),x,10,e*arctanh((-2*sqrt(c)*sqrt(d+e*x)+sqrt(2*c*d-I*b*e+2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))/sqrt(2*c*d-I*b*e-2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))/(sqrt(c)*sqrt(2*c*d-I*b*e-2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))-e*arctanh((2*sqrt(c)*sqrt(d+e*x)+sqrt(2*c*d-I*b*e+2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))/sqrt(2*c*d-I*b*e-2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))/(sqrt(c)*sqrt(2*c*d-I*b*e-2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))+1/2*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2-e*(I*b*d-a*e))-sqrt(d+e*x)*sqrt(2*c*d-I*b*e+2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))/(sqrt(c)*sqrt(2*c*d-I*b*e+2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))-1/2*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2-e*(I*b*d-a*e))+sqrt(d+e*x)*sqrt(2*c*d-I*b*e+2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))/(sqrt(c)*sqrt(2*c*d-I*b*e+2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))],
[1/((d+e*x)^(1/2)*(a+I*b*x+c*x^2)),x,10,e*arctanh((-2*sqrt(c)*sqrt(d+e*x)+sqrt(2*c*d-I*b*e+2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))/sqrt(2*c*d-I*b*e-2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))/(sqrt(c*d^2-e*(I*b*d-a*e))*sqrt(2*c*d-I*b*e-2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))-e*arctanh((2*sqrt(c)*sqrt(d+e*x)+sqrt(2*c*d-I*b*e+2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))/sqrt(2*c*d-I*b*e-2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))/(sqrt(c*d^2-e*(I*b*d-a*e))*sqrt(2*c*d-I*b*e-2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))-1/2*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2-e*(I*b*d-a*e))-sqrt(d+e*x)*sqrt(2*c*d-I*b*e+2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))/(sqrt(c*d^2-e*(I*b*d-a*e))*sqrt(2*c*d-I*b*e+2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))+1/2*e*log((d+e*x)*sqrt(c)+sqrt(c*d^2-e*(I*b*d-a*e))+sqrt(d+e*x)*sqrt(2*c*d-I*b*e+2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))/(sqrt(c*d^2-e*(I*b*d-a*e))*sqrt(2*c*d-I*b*e+2*sqrt(c)*sqrt(c*d^2-e*(I*b*d-a*e))))],

#  b^2-4*a*c<0 
[(1+2*x)^(7/2)/(2+3*x+5*x^2),x,13,16/75*(1+2*x)^(3/2)+4/25*(1+2*x)^(5/2)-76/125*sqrt(1+2*x)+1/125*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/155*(-168698+42875*sqrt(35)))-1/125*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/155*(-168698+42875*sqrt(35)))-1/125*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(168698+42875*sqrt(35)))+1/125*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(168698+42875*sqrt(35)))],
[(1+2*x)^(5/2)/(2+3*x+5*x^2),x,12,4/15*(1+2*x)^(3/2)+16/25*sqrt(1+2*x)+1/25*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(-7162+1225*sqrt(35)))-1/25*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(-7162+1225*sqrt(35)))+1/25*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/155*(7162+1225*sqrt(35)))-1/25*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/155*(7162+1225*sqrt(35)))],
[(1+2*x)^(3/2)/(2+3*x+5*x^2),x,11,4/5*sqrt(1+2*x)+1/5*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/155*(-178+35*sqrt(35)))-1/5*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/155*(-178+35*sqrt(35)))+1/5*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(178+35*sqrt(35)))-1/5*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(178+35*sqrt(35)))],
[(1+2*x)^(1/2)/(2+3*x+5*x^2),x,10,-arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/5/(-2+sqrt(35)))+arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/5/(-2+sqrt(35)))+log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))/sqrt(10*(2+sqrt(35)))-log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))/sqrt(10*(2+sqrt(35)))],
[1/((1+2*x)^(1/2)*(2+3*x+5*x^2)),x,10,-arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/217*(2+sqrt(35)))+arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/217*(2+sqrt(35)))-log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))/sqrt(14*(2+sqrt(35)))+log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))/sqrt(14*(2+sqrt(35)))],
[1/((1+2*x)^(3/2)*(2+3*x+5*x^2)),x,11,(-4/7)/sqrt(1+2*x)+1/7*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/217*(-178+35*sqrt(35)))-1/7*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/217*(-178+35*sqrt(35)))-1/7*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(178+35*sqrt(35)))+1/7*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(178+35*sqrt(35)))],
[1/((1+2*x)^(5/2)*(2+3*x+5*x^2)),x,12,(-4/21)/(1+2*x)^(3/2)+(-16/49)/sqrt(1+2*x)-1/49*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(-7162+1225*sqrt(35)))+1/49*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(-7162+1225*sqrt(35)))+1/49*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/217*(7162+1225*sqrt(35)))-1/49*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/217*(7162+1225*sqrt(35)))],
[(1+2*x)^(7/2)/(2+3*x+5*x^2)^2,x,13,-8/155*(1+2*x)^(3/2)-1/31*(5-4*x)*(1+2*x)^(5/2)/(2+3*x+5*x^2)+604/775*sqrt(1+2*x)+1/775*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/155*(-5682718+968975*sqrt(35)))-1/775*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/155*(-5682718+968975*sqrt(35)))+1/775*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(5682718+968975*sqrt(35)))-1/775*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(5682718+968975*sqrt(35)))],
[(1+2*x)^(5/2)/(2+3*x+5*x^2)^2,x,12,-1/31*(5-4*x)*(1+2*x)^(3/2)/(2+3*x+5*x^2)-8/155*sqrt(1+2*x)+1/155*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(-32678+10325*sqrt(35)))-1/155*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(-32678+10325*sqrt(35)))-1/155*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/155*(32678+10325*sqrt(35)))+1/155*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/155*(32678+10325*sqrt(35)))],
[(1+2*x)^(3/2)/(2+3*x+5*x^2)^2,x,11,-1/31*(5-4*x)*sqrt(1+2*x)/(2+3*x+5*x^2)-1/31*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(-218+47*sqrt(35)))+1/31*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(-218+47*sqrt(35)))-1/31*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/155*(218+47*sqrt(35)))+1/31*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/155*(218+47*sqrt(35)))],
[(1+2*x)^(1/2)/(2+3*x+5*x^2)^2,x,11,1/31*(3+10*x)*sqrt(1+2*x)/(2+3*x+5*x^2)+1/31*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(-218+47*sqrt(35)))-1/31*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(-218+47*sqrt(35)))-1/31*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/217*(218+47*sqrt(35)))+1/31*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/217*(218+47*sqrt(35)))],
[1/((1+2*x)^(1/2)*(2+3*x+5*x^2)^2),x,11,1/217*(37+20*x)*sqrt(1+2*x)/(2+3*x+5*x^2)-1/217*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(-32678+10325*sqrt(35)))+1/217*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(-32678+10325*sqrt(35)))-1/217*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/217*(32678+10325*sqrt(35)))+1/217*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/217*(32678+10325*sqrt(35)))],
[1/((1+2*x)^(3/2)*(2+3*x+5*x^2)^2),x,12,(-604/1519)/sqrt(1+2*x)+1/217*(37+20*x)/((2+3*x+5*x^2)*sqrt(1+2*x))+1/1519*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/217*(-5682718+968975*sqrt(35)))-1/1519*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/217*(-5682718+968975*sqrt(35)))-1/1519*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(5682718+968975*sqrt(35)))+1/1519*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(5682718+968975*sqrt(35)))],
[1/((1+2*x)^(5/2)*(2+3*x+5*x^2)^2),x,13,(-820/4557)/(1+2*x)^(3/2)+1/217*(37+20*x)/((1+2*x)^(3/2)*(2+3*x+5*x^2))+(-4680/10633)/sqrt(1+2*x)-5/10633*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(-12504542+2632525*sqrt(35)))+5/10633*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(-12504542+2632525*sqrt(35)))+5/10633*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/217*(12504542+2632525*sqrt(35)))-5/10633*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(2/217*(12504542+2632525*sqrt(35)))],
[(1+2*x)^(9/2)/(2+3*x+5*x^2)^3,x,13,-1/62*(5-4*x)*(1+2*x)^(7/2)/(2+3*x+5*x^2)^2-1/9610*(1143-1088*x)*(1+2*x)^(3/2)/(2+3*x+5*x^2)-1584/24025*sqrt(1+2*x)+3/48050*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(-250141922+64681225*sqrt(35)))-3/48050*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(-250141922+64681225*sqrt(35)))-3/24025*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(1/310*(250141922+64681225*sqrt(35)))+3/24025*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(1/310*(250141922+64681225*sqrt(35)))],
[(1+2*x)^(7/2)/(2+3*x+5*x^2)^3,x,12,-1/62*(5-4*x)*(1+2*x)^(5/2)/(2+3*x+5*x^2)^2-1/9610*(957-592*x)*sqrt(1+2*x)/(2+3*x+5*x^2)-1/9610*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(-9651062+1806875*sqrt(35)))+1/9610*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(-9651062+1806875*sqrt(35)))-1/4805*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(1/310*(9651062+1806875*sqrt(35)))+1/4805*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(1/310*(9651062+1806875*sqrt(35)))],
[(1+2*x)^(5/2)/(2+3*x+5*x^2)^3,x,12,-1/62*(5-4*x)*(1+2*x)^(3/2)/(2+3*x+5*x^2)^2+3/1922*(11+78*x)*sqrt(1+2*x)/(2+3*x+5*x^2)+3/1922*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(-15082+2705*sqrt(35)))-3/1922*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/310*(-15082+2705*sqrt(35)))-3/961*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(1/310*(15082+2705*sqrt(35)))+3/961*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(1/310*(15082+2705*sqrt(35)))],
[(1+2*x)^(3/2)/(2+3*x+5*x^2)^3,x,12,-1/62*(5-4*x)*sqrt(1+2*x)/(2+3*x+5*x^2)^2+1/1922*(67+120*x)*sqrt(1+2*x)/(2+3*x+5*x^2)-3/1922*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(-15082+2705*sqrt(35)))+3/1922*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(-15082+2705*sqrt(35)))-3/961*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(1/434*(15082+2705*sqrt(35)))+3/961*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(1/434*(15082+2705*sqrt(35)))],
[(1+2*x)^(1/2)/(2+3*x+5*x^2)^3,x,12,1/62*(3+10*x)*sqrt(1+2*x)/(2+3*x+5*x^2)^2+1/13454*(599+1790*x)*sqrt(1+2*x)/(2+3*x+5*x^2)+1/13454*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(-9651062+1806875*sqrt(35)))-1/13454*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(-9651062+1806875*sqrt(35)))-1/6727*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(1/434*(9651062+1806875*sqrt(35)))+1/6727*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(1/434*(9651062+1806875*sqrt(35)))],
[1/((1+2*x)^(1/2)*(2+3*x+5*x^2)^3),x,12,1/434*(37+20*x)*sqrt(1+2*x)/(2+3*x+5*x^2)^2+1/94178*(9227+7920*x)*sqrt(1+2*x)/(2+3*x+5*x^2)-3/47089*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*(7379+264*sqrt(35))*sqrt(1/434*(2+sqrt(35)))+3/47089*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*(7379+264*sqrt(35))*sqrt(1/434*(2+sqrt(35)))-3/94178*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(-250141922+64681225*sqrt(35)))+3/94178*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(-250141922+64681225*sqrt(35)))],
[1/((1+2*x)^(3/2)*(2+3*x+5*x^2)^3),x,13,(-81090/329623)/sqrt(1+2*x)+1/434*(37+20*x)/((2+3*x+5*x^2)^2*sqrt(1+2*x))+5/94178*(2329+2080*x)/((2+3*x+5*x^2)*sqrt(1+2*x))-15/329623*arctan((-10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(1/434*(-2257111762+387427075*sqrt(35)))+15/329623*arctan((10*sqrt(1+2*x)+sqrt(10*(2+sqrt(35))))/sqrt(10*(-2+sqrt(35))))*sqrt(1/434*(-2257111762+387427075*sqrt(35)))-15/659246*log(5*(1+2*x)+sqrt(35)-sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(2257111762+387427075*sqrt(35)))+15/659246*log(5*(1+2*x)+sqrt(35)+sqrt(1+2*x)*sqrt(10*(2+sqrt(35))))*sqrt(1/434*(2257111762+387427075*sqrt(35)))],
[x^(9/2)/(a+b*x+c*x^2)^3,x,7,1/2*x^(7/2)*(2*a+b*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)+1/4*x^(3/2)*(a*(b^2-28*a*c)+b*(b^2-16*a*c)*x)/(c*(b^2-4*a*c)^2*(a+b*x+c*x^2))-3/4*b*(b^2-8*a*c)*sqrt(x)/(c^2*(b^2-4*a*c)^2)+3/4*arctan(sqrt(2)*sqrt(c)*sqrt(x)/sqrt(b-sqrt(b^2-4*a*c)))*(b^4-9*a*b^2*c+28*a^2*c^2+(-b^5+11*a*b^3*c-44*a^2*b*c^2)/sqrt(b^2-4*a*c))/(c^(5/2)*(b^2-4*a*c)^2*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+3/4*arctan(sqrt(2)*sqrt(c)*sqrt(x)/sqrt(b+sqrt(b^2-4*a*c)))*(b^4-9*a*b^2*c+28*a^2*c^2+(b^5-11*a*b^3*c+44*a^2*b*c^2)/sqrt(b^2-4*a*c))/(c^(5/2)*(b^2-4*a*c)^2*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/(x^(3/2)*(a+b*x+c*x^2)^3),x,7,-3/4*(5*b^2-12*a*c)*(b^2-5*a*c)/(a^3*(b^2-4*a*c)^2*sqrt(x))+1/2*(b^2-2*a*c+b*c*x)/(a*(b^2-4*a*c)*(a+b*x+c*x^2)^2*sqrt(x))+1/4*(5*b^4-35*a*b^2*c+36*a^2*c^2+b*c*(5*b^2-32*a*c)*x)/(a^2*(b^2-4*a*c)^2*(a+b*x+c*x^2)*sqrt(x))-3/4*arctan(sqrt(2)*sqrt(c)*sqrt(x)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(5*b^5-47*a*b^3*c+124*a^2*b*c^2+(5*b^4-37*a*b^2*c+60*a^2*c^2)*sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(5/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+3/4*arctan(sqrt(2)*sqrt(c)*sqrt(x)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(5*b^5-47*a*b^3*c+124*a^2*b*c^2-5*b^4*sqrt(b^2-4*a*c)+37*a*b^2*c*sqrt(b^2-4*a*c)-60*a^2*c^2*sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(5/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],

# Integrands of the form (d+e x)^(m/3) (a+b x+c x^2)^p
[(3-x+x^2)/x^(1/3),x,2,9/2*x^(2/3)-3/5*x^(5/3)+3/8*x^(8/3)],

# Integrands of the form (d+e x)^m (a+b x+c x^2)^(p/2)

# p>0
[(d+e*x)^3*sqrt(a+b*x+c*x^2),x,5,1/5*e*(d+e*x)^2*(a+b*x+c*x^2)^(3/2)/c+1/240*e*(192*c^2*d^2+35*b^2*e^2-2*c*e*(75*b*d+16*a*e)+42*c*e*(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(3/2)/c^3-1/256*(b^2-4*a*c)*(2*c*d-b*e)*(16*c^2*d^2+7*b^2*e^2-4*c*e*(4*b*d+3*a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(9/2)+1/128*(2*c*d-b*e)*(16*c^2*d^2+7*b^2*e^2-4*c*e*(4*b*d+3*a*e))*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c^4],
[(d+e*x)^2*sqrt(a+b*x+c*x^2),x,5,5/24*e*(2*c*d-b*e)*(a+b*x+c*x^2)^(3/2)/c^2+1/4*e*(d+e*x)*(a+b*x+c*x^2)^(3/2)/c-1/128*(b^2-4*a*c)*(16*c^2*d^2+5*b^2*e^2-4*c*e*(4*b*d+a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(7/2)+1/64*(16*c^2*d^2+5*b^2*e^2-4*c*e*(4*b*d+a*e))*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c^3],
[(d+e*x)*sqrt(a+b*x+c*x^2),x,4,1/3*e*(a+b*x+c*x^2)^(3/2)/c-1/16*(b^2-4*a*c)*(2*c*d-b*e)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(5/2)+1/8*(2*c*d-b*e)*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c^2],
[sqrt(a+b*x+c*x^2),x,3,-1/8*(b^2-4*a*c)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(3/2)+1/4*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c],
[sqrt(a+b*x+c*x^2)/(d+e*x),x,6,-1/2*(2*c*d-b*e)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(e^2*sqrt(c))+arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*d^2-b*d*e+a*e^2)/e^2+sqrt(a+b*x+c*x^2)/e],
[sqrt(a+b*x+c*x^2)/(d+e*x)^2,x,6,arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))*sqrt(c)/e^2-1/2*(2*c*d-b*e)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(e^2*sqrt(c*d^2-b*d*e+a*e^2))-sqrt(a+b*x+c*x^2)/(e*(d+e*x))],
[sqrt(a+b*x+c*x^2)/(d+e*x)^3,x,3,-1/8*(b^2-4*a*c)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(3/2)+1/4*(b*d-2*a*e+(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*(d+e*x)^2)],
[sqrt(a+b*x+c*x^2)/(d+e*x)^4,x,4,-1/3*e*(a+b*x+c*x^2)^(3/2)/((c*d^2-b*d*e+a*e^2)*(d+e*x)^3)-1/16*(b^2-4*a*c)*(2*c*d-b*e)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(5/2)+1/8*(2*c*d-b*e)*(b*d-2*a*e+(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^2*(d+e*x)^2)],
[sqrt(a+b*x+c*x^2)/(d+e*x)^5,x,5,-1/4*e*(a+b*x+c*x^2)^(3/2)/((c*d^2-b*d*e+a*e^2)*(d+e*x)^4)-5/24*e*(2*c*d-b*e)*(a+b*x+c*x^2)^(3/2)/((c*d^2-b*d*e+a*e^2)^2*(d+e*x)^3)-1/128*(b^2-4*a*c)*(16*c^2*d^2+5*b^2*e^2-4*c*e*(4*b*d+a*e))*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(7/2)+1/64*(16*c^2*d^2+5*b^2*e^2-4*c*e*(4*b*d+a*e))*(b*d-2*a*e+(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^3*(d+e*x)^2)],
[sqrt(a+b*x+c*x^2)/(d+e*x)^6,x,6,-1/5*e*(a+b*x+c*x^2)^(3/2)/((c*d^2-b*d*e+a*e^2)*(d+e*x)^5)-7/40*e*(2*c*d-b*e)*(a+b*x+c*x^2)^(3/2)/((c*d^2-b*d*e+a*e^2)^2*(d+e*x)^4)-1/240*e*(108*c^2*d^2+35*b^2*e^2-4*c*e*(27*b*d+8*a*e))*(a+b*x+c*x^2)^(3/2)/((c*d^2-b*d*e+a*e^2)^3*(d+e*x)^3)-1/256*(b^2-4*a*c)*(2*c*d-b*e)*(16*c^2*d^2+7*b^2*e^2-4*c*e*(4*b*d+3*a*e))*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(9/2)+1/128*(2*c*d-b*e)*(16*c^2*d^2+7*b^2*e^2-4*c*e*(4*b*d+3*a*e))*(b*d-2*a*e+(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^4*(d+e*x)^2)],
[(d+e*x)^3*(a+b*x+c*x^2)^(3/2),x,6,1/128*(2*c*d-b*e)*(8*c^2*d^2+3*b^2*e^2-4*c*e*(2*b*d+a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(3/2)/c^4+1/7*e*(d+e*x)^2*(a+b*x+c*x^2)^(5/2)/c+1/280*e*(128*c^2*d^2+21*b^2*e^2-2*c*e*(49*b*d+8*a*e)+30*c*e*(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(5/2)/c^3+3/2048*(b^2-4*a*c)^2*(2*c*d-b*e)*(8*c^2*d^2+3*b^2*e^2-4*c*e*(2*b*d+a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(11/2)-3/1024*(b^2-4*a*c)*(2*c*d-b*e)*(8*c^2*d^2+3*b^2*e^2-4*c*e*(2*b*d+a*e))*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c^5],
[(d+e*x)^2*(a+b*x+c*x^2)^(3/2),x,6,1/192*(24*c^2*d^2+7*b^2*e^2-4*c*e*(6*b*d+a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(3/2)/c^3+7/60*e*(2*c*d-b*e)*(a+b*x+c*x^2)^(5/2)/c^2+1/6*e*(d+e*x)*(a+b*x+c*x^2)^(5/2)/c+1/1024*(b^2-4*a*c)^2*(24*c^2*d^2+7*b^2*e^2-4*c*e*(6*b*d+a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(9/2)-1/512*(b^2-4*a*c)*(24*c^2*d^2+7*b^2*e^2-4*c*e*(6*b*d+a*e))*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c^4],
[(d+e*x)*(a+b*x+c*x^2)^(3/2),x,5,1/16*(2*c*d-b*e)*(b+2*c*x)*(a+b*x+c*x^2)^(3/2)/c^2+1/5*e*(a+b*x+c*x^2)^(5/2)/c+3/256*(b^2-4*a*c)^2*(2*c*d-b*e)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(7/2)-3/128*(b^2-4*a*c)*(2*c*d-b*e)*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c^3],
[(a+b*x+c*x^2)^(3/2),x,4,1/8*(b+2*c*x)*(a+b*x+c*x^2)^(3/2)/c+3/128*(b^2-4*a*c)^2*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(5/2)-3/64*(b^2-4*a*c)*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c^2],
[(a+b*x+c*x^2)^(3/2)/(d+e*x),x,7,1/3*(a+b*x+c*x^2)^(3/2)/e-1/16*(2*c*d-b*e)*(8*c^2*d^2-b^2*e^2-4*c*e*(2*b*d-3*a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*e^4)+(c*d^2-b*d*e+a*e^2)^(3/2)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/e^4+1/8*(8*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-4*a*e)-2*c*e*(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/(c*e^3)],
[(a+b*x+c*x^2)^(3/2)/(d+e*x)^2,x,7,-(a+b*x+c*x^2)^(3/2)/(e*(d+e*x))+3/8*(8*c^2*d^2+b^2*e^2-4*c*e*(2*b*d-a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(e^4*sqrt(c))-3/2*(2*c*d-b*e)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*d^2-b*d*e+a*e^2)/e^4-3/4*(4*c*d-3*b*e-2*c*e*x)*sqrt(a+b*x+c*x^2)/e^3],
[(a+b*x+c*x^2)^(3/2)/(d+e*x)^3,x,7,-1/2*(a+b*x+c*x^2)^(3/2)/(e*(d+e*x)^2)-3/2*(2*c*d-b*e)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))*sqrt(c)/e^4+3/8*(8*c^2*d^2+b^2*e^2-4*c*e*(2*b*d-a*e))*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(e^4*sqrt(c*d^2-b*d*e+a*e^2))+3/4*(4*c*d-b*e+2*c*e*x)*sqrt(a+b*x+c*x^2)/(e^3*(d+e*x))],
[(a+b*x+c*x^2)^(3/2)/(d+e*x)^4,x,7,-1/3*(a+b*x+c*x^2)^(3/2)/(e*(d+e*x)^3)+c^(3/2)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/e^4-1/16*(2*c*d-b*e)*(8*c^2*d^2-b^2*e^2-4*c*e*(2*b*d-3*a*e))*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(e^4*(c*d^2-b*d*e+a*e^2)^(3/2))-1/8*(8*c^2*d^3-b*e^2*(b*d-2*a*e)-2*c*d*e*(3*b*d-2*a*e)+e*(12*c^2*d^2+b^2*e^2-4*c*e*(3*b*d-2*a*e))*x)*sqrt(a+b*x+c*x^2)/(e^3*(c*d^2-b*d*e+a*e^2)*(d+e*x)^2)],
[(a+b*x+c*x^2)^(3/2)/(d+e*x)^5,x,4,1/8*(b*d-2*a*e+(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(3/2)/((c*d^2-b*d*e+a*e^2)*(d+e*x)^4)+3/128*(b^2-4*a*c)^2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(5/2)-3/64*(b^2-4*a*c)*(b*d-2*a*e+(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^2*(d+e*x)^2)],
[(a+b*x+c*x^2)^(3/2)/(d+e*x)^6,x,5,1/16*(2*c*d-b*e)*(b*d-2*a*e+(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(3/2)/((c*d^2-b*d*e+a*e^2)^2*(d+e*x)^4)-1/5*e*(a+b*x+c*x^2)^(5/2)/((c*d^2-b*d*e+a*e^2)*(d+e*x)^5)+3/256*(b^2-4*a*c)^2*(2*c*d-b*e)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(7/2)-3/128*(b^2-4*a*c)*(2*c*d-b*e)*(b*d-2*a*e+(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^3*(d+e*x)^2)],
[(a+b*x+c*x^2)^(3/2)/(d+e*x)^7,x,6,1/192*(24*c^2*d^2+7*b^2*e^2-4*c*e*(6*b*d+a*e))*(b*d-2*a*e+(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(3/2)/((c*d^2-b*d*e+a*e^2)^3*(d+e*x)^4)-1/6*e*(a+b*x+c*x^2)^(5/2)/((c*d^2-b*d*e+a*e^2)*(d+e*x)^6)-7/60*e*(2*c*d-b*e)*(a+b*x+c*x^2)^(5/2)/((c*d^2-b*d*e+a*e^2)^2*(d+e*x)^5)+1/1024*(b^2-4*a*c)^2*(24*c^2*d^2+7*b^2*e^2-4*c*e*(6*b*d+a*e))*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(9/2)-1/512*(b^2-4*a*c)*(24*c^2*d^2+7*b^2*e^2-4*c*e*(6*b*d+a*e))*(b*d-2*a*e+(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^4*(d+e*x)^2)],
[(a+b*x+c*x^2)^(3/2)/(d+e*x)^8,x,7,1/128*(2*c*d-b*e)*(8*c^2*d^2+3*b^2*e^2-4*c*e*(2*b*d+a*e))*(b*d-2*a*e+(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(3/2)/((c*d^2-b*d*e+a*e^2)^4*(d+e*x)^4)-1/7*e*(a+b*x+c*x^2)^(5/2)/((c*d^2-b*d*e+a*e^2)*(d+e*x)^7)-3/28*e*(2*c*d-b*e)*(a+b*x+c*x^2)^(5/2)/((c*d^2-b*d*e+a*e^2)^2*(d+e*x)^6)-1/280*e*(68*c^2*d^2+21*b^2*e^2-4*c*e*(17*b*d+4*a*e))*(a+b*x+c*x^2)^(5/2)/((c*d^2-b*d*e+a*e^2)^3*(d+e*x)^5)+3/2048*(b^2-4*a*c)^2*(2*c*d-b*e)*(8*c^2*d^2+3*b^2*e^2-4*c*e*(2*b*d+a*e))*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(11/2)-3/1024*(b^2-4*a*c)*(2*c*d-b*e)*(8*c^2*d^2+3*b^2*e^2-4*c*e*(2*b*d+a*e))*(b*d-2*a*e+(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^5*(d+e*x)^2)],
[(d+e*x)^3*(a+b*x+c*x^2)^(5/2),x,7,-5/12288*(b^2-4*a*c)*(2*c*d-b*e)*(32*c^2*d^2+11*b^2*e^2-4*c*e*(8*b*d+3*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(3/2)/c^5+1/768*(2*c*d-b*e)*(32*c^2*d^2+11*b^2*e^2-4*c*e*(8*b*d+3*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(5/2)/c^4+1/9*e*(d+e*x)^2*(a+b*x+c*x^2)^(7/2)/c+1/2016*e*(640*c^2*d^2+99*b^2*e^2-2*c*e*(243*b*d+32*a*e)+154*c*e*(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(7/2)/c^3-5/65536*(b^2-4*a*c)^3*(2*c*d-b*e)*(32*c^2*d^2+11*b^2*e^2-4*c*e*(8*b*d+3*a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(13/2)+5/32768*(b^2-4*a*c)^2*(2*c*d-b*e)*(32*c^2*d^2+11*b^2*e^2-4*c*e*(8*b*d+3*a*e))*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c^6],
[(d+e*x)^2*(a+b*x+c*x^2)^(5/2),x,7,-5/6144*(b^2-4*a*c)*(32*c^2*d^2+9*b^2*e^2-4*c*e*(8*b*d+a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(3/2)/c^4+1/384*(32*c^2*d^2+9*b^2*e^2-4*c*e*(8*b*d+a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(5/2)/c^3+9/112*e*(2*c*d-b*e)*(a+b*x+c*x^2)^(7/2)/c^2+1/8*e*(d+e*x)*(a+b*x+c*x^2)^(7/2)/c-5/32768*(b^2-4*a*c)^3*(32*c^2*d^2+9*b^2*e^2-4*c*e*(8*b*d+a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(11/2)+5/16384*(b^2-4*a*c)^2*(32*c^2*d^2+9*b^2*e^2-4*c*e*(8*b*d+a*e))*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c^5],
[(d+e*x)*(a+b*x+c*x^2)^(5/2),x,6,-5/384*(b^2-4*a*c)*(2*c*d-b*e)*(b+2*c*x)*(a+b*x+c*x^2)^(3/2)/c^3+1/24*(2*c*d-b*e)*(b+2*c*x)*(a+b*x+c*x^2)^(5/2)/c^2+1/7*e*(a+b*x+c*x^2)^(7/2)/c-5/2048*(b^2-4*a*c)^3*(2*c*d-b*e)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(9/2)+5/1024*(b^2-4*a*c)^2*(2*c*d-b*e)*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c^4],
[(a+b*x+c*x^2)^(5/2),x,5,-5/192*(b^2-4*a*c)*(b+2*c*x)*(a+b*x+c*x^2)^(3/2)/c^2+1/12*(b+2*c*x)*(a+b*x+c*x^2)^(5/2)/c-5/1024*(b^2-4*a*c)^3*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(7/2)+5/512*(b^2-4*a*c)^2*(b+2*c*x)*sqrt(a+b*x+c*x^2)/c^3],
[(a+b*x+c*x^2)^(5/2)/(d+e*x),x,8,1/48*(16*c^2*d^2+3*b^2*e^2-2*c*e*(11*b*d-8*a*e)-6*c*e*(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(3/2)/(c*e^3)+1/5*(a+b*x+c*x^2)^(5/2)/e-1/256*(2*c*d-b*e)*(128*c^4*d^4+3*b^4*e^4+8*b^2*c*e^3*(2*b*d-5*a*e)-64*c^3*d^2*e*(4*b*d-5*a*e)+16*c^2*e^2*(7*b^2*d^2-20*a*b*d*e+15*a^2*e^2))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(5/2)*e^6)+(c*d^2-b*d*e+a*e^2)^(5/2)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/e^6+1/128*(128*c^4*d^4-3*b^4*e^4-2*b^2*c*e^3*(5*b*d-14*a*e)-32*c^3*d^2*e*(9*b*d-8*a*e)+8*c^2*e^2*(22*b^2*d^2-39*a*b*d*e+16*a^2*e^2)-2*c*e*(2*c*d-b*e)*(16*c^2*d^2-3*b^2*e^2-4*c*e*(4*b*d-7*a*e))*x)*sqrt(a+b*x+c*x^2)/(c^2*e^5)],
[(a+b*x+c*x^2)^(5/2)/(d+e*x)^2,x,8,-5/24*(8*c*d-7*b*e-6*c*e*x)*(a+b*x+c*x^2)^(3/2)/e^3-(a+b*x+c*x^2)^(5/2)/(e*(d+e*x))+5/128*(128*c^4*d^4-b^4*e^4-8*b^2*c*e^3*(2*b*d-3*a*e)-64*c^3*d^2*e*(4*b*d-3*a*e)+48*c^2*e^2*(3*b^2*d^2-4*a*b*d*e+a^2*e^2))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*e^6)-5/2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^(3/2)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/e^6-5/64*(64*c^3*d^3-b^3*e^3+4*b*c*e^2*(12*b*d-11*a*e)-16*c^2*d*e*(7*b*d-4*a*e)-2*c*e*(16*c^2*d^2+b^2*e^2-4*c*e*(4*b*d-3*a*e))*x)*sqrt(a+b*x+c*x^2)/(c*e^5)],
[(a+b*x+c*x^2)^(5/2)/(d+e*x)^3,x,8,5/12*(8*c*d-3*b*e+2*c*e*x)*(a+b*x+c*x^2)^(3/2)/(e^3*(d+e*x))-1/2*(a+b*x+c*x^2)^(5/2)/(e*(d+e*x)^2)-5/16*(2*c*d-b*e)*(16*c^2*d^2+b^2*e^2-4*c*e*(4*b*d-3*a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(e^6*sqrt(c))+5/8*(16*c^2*d^2+3*b^2*e^2-4*c*e*(4*b*d-a*e))*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*d^2-b*d*e+a*e^2)/e^6+5/8*(16*c^2*d^2+5*b^2*e^2-4*c*e*(5*b*d-a*e)-4*c*e*(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/e^5],
[(a+b*x+c*x^2)^(5/2)/(d+e*x)^4,x,8,5/12*(4*c*d-b*e+2*c*e*x)*(a+b*x+c*x^2)^(3/2)/(e^3*(d+e*x)^2)-1/3*(a+b*x+c*x^2)^(5/2)/(e*(d+e*x)^3)+5/8*(16*c^2*d^2+3*b^2*e^2-4*c*e*(4*b*d-a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))*sqrt(c)/e^6-5/16*(2*c*d-b*e)*(16*c^2*d^2+b^2*e^2-4*c*e*(4*b*d-3*a*e))*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(e^6*sqrt(c*d^2-b*d*e+a*e^2))-5/8*(16*c^2*d^2+b^2*e^2-4*c*e*(3*b*d-a*e)+4*c*e*(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/(e^5*(d+e*x))],
[(a+b*x+c*x^2)^(5/2)/(d+e*x)^5,x,8,-5/96*(16*c^2*d^3-b*e^2*(b*d-4*a*e)-4*c*d*e*(3*b*d-a*e)+3*e*(8*c^2*d^2+b^2*e^2-4*c*e*(2*b*d-a*e))*x)*(a+b*x+c*x^2)^(3/2)/(e^3*(c*d^2-b*d*e+a*e^2)*(d+e*x)^3)-1/4*(a+b*x+c*x^2)^(5/2)/(e*(d+e*x)^4)-5/2*c^(3/2)*(2*c*d-b*e)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/e^6+5/128*(128*c^4*d^4-b^4*e^4-8*b^2*c*e^3*(2*b*d-3*a*e)-64*c^3*d^2*e*(4*b*d-3*a*e)+48*c^2*e^2*(3*b^2*d^2-4*a*b*d*e+a^2*e^2))*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(e^6*(c*d^2-b*d*e+a*e^2)^(3/2))+5/64*(64*c^3*d^3+b^3*e^3+4*b*c*e^2*(4*b*d-5*a*e)-16*c^2*d*e*(5*b*d-4*a*e)+2*c*e*(16*c^2*d^2+b^2*e^2-4*c*e*(4*b*d-3*a*e))*x)*sqrt(a+b*x+c*x^2)/(e^5*(c*d^2-b*d*e+a*e^2)*(d+e*x))],
[sqrt(-2-3*x+5*x^2)/x,x,6,arctan(1/2*(4+3*x)/(sqrt(2)*sqrt(-2-3*x+5*x^2)))*sqrt(2)+3/2*arctanh(1/2*(3-10*x)/(sqrt(5)*sqrt(-2-3*x+5*x^2)))/sqrt(5)+sqrt(-2-3*x+5*x^2)],
[sqrt(2-x-x^2)/x^2,x,6,arcsin(1/3*(-1-2*x))+1/2*arctanh(1/2*(4-x)/(sqrt(2)*sqrt(2-x-x^2)))/sqrt(2)-sqrt(2-x-x^2)/x],
[(1+x)^3*sqrt(2+2*x+x^2),x,2,-2/15*(2+2*x+x^2)^(3/2)+1/5*(1+x)^2*(2+2*x+x^2)^(3/2)],
[(-2+3*x)*sqrt(8+12*x+9*x^2),x,4,1/9*(8+12*x+9*x^2)^(3/2)-8/3*arcsinh(1+3/2*x)-2/3*(2+3*x)*sqrt(8+12*x+9*x^2)],
[(7-2*x)*sqrt(9+16*x-4*x^2),x,4,1/6*(9+16*x-4*x^2)^(3/2)-75/4*arcsin(2/5*(2-x))-3/2*(2-x)*sqrt(9+16*x-4*x^2)],
[sqrt(-1-x+x^2)/(1+x),x,6,3/2*arctanh(1/2*(1-2*x)/sqrt(-1-x+x^2))+arctanh(1/2*(1+3*x)/sqrt(-1-x+x^2))+sqrt(-1-x+x^2)],
[sqrt(-1-x+x^2)/(1-x),x,6,-arctan(1/2*(3-x)/sqrt(-1-x+x^2))+1/2*arctanh(1/2*(1-2*x)/sqrt(-1-x+x^2))-sqrt(-1-x+x^2)],

# p<0
[x^6/sqrt(a+b*x+c*x^2),x,7,1/1024*(231*b^6-1260*a*b^4*c+1680*a^2*b^2*c^2-320*a^3*c^3)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(13/2)-1/320*b*(77*b^2-156*a*c)*x^2*sqrt(a+b*x+c*x^2)/c^4+1/480*(99*b^2-100*a*c)*x^3*sqrt(a+b*x+c*x^2)/c^3-11/60*b*x^4*sqrt(a+b*x+c*x^2)/c^2+1/6*x^5*sqrt(a+b*x+c*x^2)/c-1/2560*(7*b*(165*b^4-680*a*b^2*c+528*a^2*c^2)-2*c*(385*b^4-1176*a*b^2*c+400*a^2*c^2)*x)*sqrt(a+b*x+c*x^2)/c^6],
[x^5/sqrt(a+b*x+c*x^2),x,6,-1/256*b*(63*b^4-280*a*b^2*c+240*a^2*c^2)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(11/2)+1/240*(63*b^2-64*a*c)*x^2*sqrt(a+b*x+c*x^2)/c^3-9/40*b*x^3*sqrt(a+b*x+c*x^2)/c^2+1/5*x^4*sqrt(a+b*x+c*x^2)/c+1/1920*(945*b^4-2940*a*b^2*c+1024*a^2*c^2-14*b*c*(45*b^2-92*a*c)*x)*sqrt(a+b*x+c*x^2)/c^5],
[(d+e*x)^4/sqrt(a+b*x+c*x^2),x,5,1/128*(128*c^4*d^4+35*b^4*e^4-128*c^3*d^2*e*(2*b*d+3*a*e)-40*b^2*c*e^3*(4*b*d+3*a*e)+48*c^2*e^2*(6*b^2*d^2+8*a*b*d*e+a^2*e^2))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(9/2)+7/24*e*(2*c*d-b*e)*(d+e*x)^2*sqrt(a+b*x+c*x^2)/c^2+1/4*e*(d+e*x)^3*sqrt(a+b*x+c*x^2)/c+1/192*e*(608*c^3*d^3-105*b^3*e^3+20*b*c*e^2*(24*b*d+11*a*e)-8*c^2*d*e*(101*b*d+64*a*e)+2*c*e*(104*c^2*d^2+35*b^2*e^2-4*c*e*(26*b*d+9*a*e))*x)*sqrt(a+b*x+c*x^2)/c^4],
[(d+e*x)^3/sqrt(a+b*x+c*x^2),x,4,1/16*(2*c*d-b*e)*(8*c^2*d^2+5*b^2*e^2-4*c*e*(2*b*d+3*a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(7/2)+1/3*e*(d+e*x)^2*sqrt(a+b*x+c*x^2)/c+1/24*e*(64*c^2*d^2+15*b^2*e^2-2*c*e*(27*b*d+8*a*e)+10*c*e*(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/c^3],
[(d+e*x)^2/sqrt(a+b*x+c*x^2),x,4,1/8*(8*c^2*d^2+3*b^2*e^2-4*c*e*(2*b*d+a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(5/2)+3/4*e*(2*c*d-b*e)*sqrt(a+b*x+c*x^2)/c^2+1/2*e*(d+e*x)*sqrt(a+b*x+c*x^2)/c],
[(d+e*x)/sqrt(a+b*x+c*x^2),x,3,1/2*(2*c*d-b*e)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(3/2)+e*sqrt(a+b*x+c*x^2)/c],
[1/sqrt(a+b*x+c*x^2),x,2,arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/sqrt(c)],
[1/((d+e*x)*sqrt(a+b*x+c*x^2)),x,2,arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/sqrt(c*d^2-b*d*e+a*e^2)],
[1/((d+e*x)^2*sqrt(a+b*x+c*x^2)),x,3,1/2*(2*c*d-b*e)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(3/2)-e*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*(d+e*x))],
[1/((d+e*x)^3*sqrt(a+b*x+c*x^2)),x,4,1/8*(8*c^2*d^2+3*b^2*e^2-4*c*e*(2*b*d+a*e))*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(5/2)-1/2*e*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*(d+e*x)^2)-3/4*e*(2*c*d-b*e)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^2*(d+e*x))],
[1/((d+e*x)^4*sqrt(a+b*x+c*x^2)),x,5,1/16*(2*c*d-b*e)*(8*c^2*d^2+5*b^2*e^2-4*c*e*(2*b*d+3*a*e))*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(7/2)-1/3*e*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*(d+e*x)^3)-5/12*e*(2*c*d-b*e)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^2*(d+e*x)^2)-1/24*e*(44*c^2*d^2+15*b^2*e^2-4*c*e*(11*b*d+4*a*e))*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^3*(d+e*x))],
[(d+e*x)^4/(a+b*x+c*x^2)^(3/2),x,5,3/8*e^2*(16*c^2*d^2+5*b^2*e^2-4*c*e*(4*b*d+a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(7/2)-2*(d+e*x)^3*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*sqrt(a+b*x+c*x^2))+2*e*(2*c*d-b*e)*(d+e*x)^2*sqrt(a+b*x+c*x^2)/(c*(b^2-4*a*c))+1/4*e*(32*c^3*d^3-15*b^3*e^3+4*b*c*e^2*(12*b*d+13*a*e)-8*c^2*d*e*(5*b*d+16*a*e)+2*c*e*(8*c^2*d^2+5*b^2*e^2-4*c*e*(2*b*d+3*a*e))*x)*sqrt(a+b*x+c*x^2)/(c^3*(b^2-4*a*c))],
[(d+e*x)^3/(a+b*x+c*x^2)^(3/2),x,4,3/2*e^2*(2*c*d-b*e)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(5/2)-2*(d+e*x)^2*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*sqrt(a+b*x+c*x^2))+e*(8*c^2*d^2+3*b^2*e^2-2*c*e*(3*b*d+4*a*e)+2*c*e*(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/(c^2*(b^2-4*a*c))],
[(d+e*x)^2/(a+b*x+c*x^2)^(3/2),x,4,e^2*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(3/2)-2*(d+e*x)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*sqrt(a+b*x+c*x^2))+2*e*(2*c*d-b*e)*sqrt(a+b*x+c*x^2)/(c*(b^2-4*a*c))],
[(d+e*x)/(a+b*x+c*x^2)^(3/2),x,1,-2*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*sqrt(a+b*x+c*x^2))],
[1/(a+b*x+c*x^2)^(3/2),x,1,-2*(b+2*c*x)/((b^2-4*a*c)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)*(a+b*x+c*x^2)^(3/2)),x,4,e^2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(3/2)-2*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)^2*(a+b*x+c*x^2)^(3/2)),x,4,3/2*e^2*(2*c*d-b*e)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(5/2)-2*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(d+e*x)*sqrt(a+b*x+c*x^2))-e*(4*c^2*d^2+3*b^2*e^2-4*c*e*(b*d+2*a*e))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^2*(d+e*x))],
[1/((d+e*x)^3*(a+b*x+c*x^2)^(3/2)),x,5,3/8*e^2*(16*c^2*d^2+5*b^2*e^2-4*c*e*(4*b*d+a*e))*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(7/2)-2*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(d+e*x)^2*sqrt(a+b*x+c*x^2))-1/2*e*(8*c^2*d^2+5*b^2*e^2-4*c*e*(2*b*d+3*a*e))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^2)-1/4*e*(2*c*d-b*e)*(8*c^2*d^2+15*b^2*e^2-4*c*e*(2*b*d+13*a*e))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^3*(d+e*x))],
[1/((d+e*x)^4*(a+b*x+c*x^2)^(3/2)),x,6,5/16*e^2*(2*c*d-b*e)*(16*c^2*d^2+7*b^2*e^2-4*c*e*(4*b*d+3*a*e))*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(9/2)-2*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(d+e*x)^3*sqrt(a+b*x+c*x^2))-1/3*e*(12*c^2*d^2+7*b^2*e^2-4*c*e*(3*b*d+4*a*e))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^3)-1/12*e*(2*c*d-b*e)*(24*c^2*d^2+35*b^2*e^2-4*c*e*(6*b*d+29*a*e))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^3*(d+e*x)^2)-1/24*e*(96*c^4*d^4+105*b^4*e^4-20*b^2*c*e^3*(19*b*d+23*a*e)-16*c^3*d^2*e*(12*b*d+83*a*e)+4*c^2*e^2*(119*b^2*d^2+332*a*b*d*e+64*a^2*e^2))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^4*(d+e*x))],
[(d+e*x)^5/(a+b*x+c*x^2)^(5/2),x,5,-2/3*(d+e*x)^4*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(3/2))+5/2*e^4*(2*c*d-b*e)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(7/2)-8/3*(d+e*x)^2*(8*a^2*c*e^3-2*b*c*d*(c*d^2+3*a*e^2)+b^2*(3*c*d^2*e-a*e^3)-(2*c*d-b*e)*(2*c^2*d^2-b^2*e^2-2*c*e*(b*d-3*a*e))*x)/(c*(b^2-4*a*c)^2*sqrt(a+b*x+c*x^2))-1/3*e*(64*c^4*d^4-15*b^4*e^4-16*c^3*d^2*e*(7*b*d-16*a*e)+10*b^2*c*e^3*(3*b*d+10*a*e)+8*c^2*e^2*(b^2*d^2-25*a*b*d*e-16*a^2*e^2)+2*c*e*(2*c*d-b*e)*(8*c^2*d^2-5*b^2*e^2-4*c*e*(2*b*d-7*a*e))*x)*sqrt(a+b*x+c*x^2)/(c^3*(b^2-4*a*c)^2)],
[(d+e*x)^4/(a+b*x+c*x^2)^(5/2),x,5,-2/3*(d+e*x)^3*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(3/2))+e^4*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/c^(5/2)+4/3*(d+e*x)*(4*b*c*d*(c*d^2+3*a*e^2)-4*a*c*e*(c*d^2+3*a*e^2)-b^2*(5*c*d^2*e-a*e^3)+(2*c*d-b*e)*(4*c^2*d^2-b^2*e^2-4*c*e*(b*d-2*a*e))*x)/(c*(b^2-4*a*c)^2*sqrt(a+b*x+c*x^2))-2/3*e*(2*c*d-b*e)*(8*c^2*d^2-3*b^2*e^2-4*c*e*(2*b*d-5*a*e))*sqrt(a+b*x+c*x^2)/(c^2*(b^2-4*a*c)^2)],
[(d+e*x)^3/(a+b*x+c*x^2)^(5/2),x,2,-2/3*(d+e*x)^2*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(3/2))+16/3*(c*d^2-b*d*e+a*e^2)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)^2*sqrt(a+b*x+c*x^2))],
[(d+e*x)^2/(a+b*x+c*x^2)^(5/2),x,2,-2/3*(b+2*c*x)*(d+e*x)^2/((b^2-4*a*c)*(a+b*x+c*x^2)^(3/2))+8/3*(2*c*d-b*e)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)^2*sqrt(a+b*x+c*x^2))],
[(d+e*x)/(a+b*x+c*x^2)^(5/2),x,2,-2/3*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(3/2))+8/3*(2*c*d-b*e)*(b+2*c*x)/((b^2-4*a*c)^2*sqrt(a+b*x+c*x^2))],
[1/(a+b*x+c*x^2)^(5/2),x,2,-2/3*(b+2*c*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(3/2))+16/3*c*(b+2*c*x)/((b^2-4*a*c)^2*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)*(a+b*x+c*x^2)^(5/2)),x,5,-2/3*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(a+b*x+c*x^2)^(3/2))+e^4*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(5/2)-2/3*(4*a*c*e*(2*c*d-b*e)^2-(b*c*d-b^2*e+2*a*c*e)*(8*c^2*d^2-3*b^2*e^2-4*c*e*(b*d-3*a*e))-c*(2*c*d-b*e)*(8*c^2*d^2-3*b^2*e^2-4*c*e*(2*b*d-5*a*e))*x)/((b^2-4*a*c)^2*(c*d^2-b*d*e+a*e^2)^2*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)^2*(a+b*x+c*x^2)^(5/2)),x,5,-2/3*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(d+e*x)*(a+b*x+c*x^2)^(3/2))+5/2*e^4*(2*c*d-b*e)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(7/2)-2/3*(6*a*c*e*(2*c*d-b*e)^2-(b*c*d-b^2*e+2*a*c*e)*(8*c^2*d^2-5*b^2*e^2-2*c*e*(b*d-8*a*e))-c*(2*c*d-b*e)*(8*c^2*d^2-5*b^2*e^2-4*c*e*(2*b*d-7*a*e))*x)/((b^2-4*a*c)^2*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)*sqrt(a+b*x+c*x^2))+1/3*e*(32*c^4*d^4-15*b^4*e^4-16*c^3*d^2*e*(4*b*d-9*a*e)+20*b^2*c*e^3*(b*d+5*a*e)+4*c^2*e^2*(3*b^2*d^2-36*a*b*d*e-32*a^2*e^2))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^2*(c*d^2-b*d*e+a*e^2)^3*(d+e*x))],
[1/((d+e*x)^3*(a+b*x+c*x^2)^(5/2)),x,6,-2/3*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(d+e*x)^2*(a+b*x+c*x^2)^(3/2))+5/8*e^4*(24*c^2*d^2+7*b^2*e^2-4*c*e*(6*b*d+a*e))*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(9/2)-2/3*(8*a*c*e*(2*c*d-b*e)^2-(b*c*d-b^2*e+2*a*c*e)*(8*c^2*d^2-7*b^2*e^2+20*a*c*e^2)-c*(2*c*d-b*e)*(8*c^2*d^2-7*b^2*e^2-4*c*e*(2*b*d-9*a*e))*x)/((b^2-4*a*c)^2*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^2*sqrt(a+b*x+c*x^2))+1/6*e*(64*c^4*d^4-35*b^4*e^4-128*c^3*d^2*e*(b*d-3*a*e)-48*a*c^2*e^3*(8*b*d+5*a*e)+8*b^2*c*e^3*(8*b*d+27*a*e))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^2*(c*d^2-b*d*e+a*e^2)^3*(d+e*x)^2)+1/12*e*(2*c*d-b*e)*(64*c^4*d^4-105*b^4*e^4-64*c^3*d^2*e*(2*b*d-7*a*e)+40*b^2*c*e^3*(2*b*d+19*a*e)-16*c^2*e^2*(b^2*d^2+28*a*b*d*e+81*a^2*e^2))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^2*(c*d^2-b*d*e+a*e^2)^4*(d+e*x))],
[(3+x)/sqrt(5-4*x-x^2),x,3,-arcsin(1/3*(-2-x))-sqrt(5-4*x-x^2)],
[(5-4*x)/sqrt(-8+12*x-4*x^2),x,3,1/2*arcsin(3-2*x)+sqrt(-8+12*x-4*x^2)],
[(3+2*x)/sqrt(5+2*x+x^2),x,3,arcsinh(1/2*(1+x))+2*sqrt(5+2*x+x^2)],
[(-1+x)/sqrt(3-4*x+x^2),x,3,-arctanh((2-x)/sqrt(3-4*x+x^2))+sqrt(3-4*x+x^2)],
[1/((1-x)*sqrt(-4+2*x+x^2)),x,2,arctan((3-2*x)/sqrt(-4+2*x+x^2))],
[1/((-2+x)*sqrt(3-4*x+x^2)),x,2,arctan(sqrt(3-4*x+x^2))],
[(1+x)/(2+3*x+x^2)^(3/2),x,1,2*(1+x)/sqrt(2+3*x+x^2)],
[1/((d+e*x)*sqrt(1/4*b^2/c+b*x+c*x^2)),x,4,2*(b+2*c*x)*log(b+2*c*x)/((2*c*d-b*e)*sqrt(b^2/c+4*b*x+4*c*x^2))-2*(b+2*c*x)*log(d+e*x)/((2*c*d-b*e)*sqrt(b^2/c+4*b*x+4*c*x^2))],
[1/((1/2*b*e/c+e*x)*sqrt(a+b*x+c*x^2)),x,2,2*arctan(2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt(c)/(e*sqrt(b^2-4*a*c))],
[1/((d+e*x)*sqrt((-c*d^2+b*d*e)/e^2+b*x+c*x^2)),x,1,2*e*sqrt(-d*(c*d-b*e)/e^2+b*x+c*x^2)/((2*c*d-b*e)*(d+e*x))],
[1/((1/2*b*e/c+e*x)*sqrt(1/4*b^2/c+b*x+c*x^2)),x,2,(-2)/(e*sqrt(b^2/c+4*b*x+4*c*x^2))],
[x/sqrt(2+4*x+3*x^2),x,3,-2/3*arcsinh((2+3*x)/sqrt(2))/sqrt(3)+1/3*sqrt(2+4*x+3*x^2)],
[x/sqrt(2+4*x-3*x^2),x,3,-2/3*arcsin((2-3*x)/sqrt(10))/sqrt(3)-1/3*sqrt(2+4*x-3*x^2)],
[x/sqrt(2+5*x+3*x^2),x,3,-5/6*arctanh(1/2*(5+6*x)/(sqrt(3)*sqrt(2+5*x+3*x^2)))/sqrt(3)+1/3*sqrt(2+5*x+3*x^2)],
[x/sqrt(2+5*x-3*x^2),x,3,-5/6*arcsin(1/7*(5-6*x))/sqrt(3)-1/3*sqrt(2+5*x-3*x^2)],
[x/sqrt(-2+4*x+3*x^2),x,3,-2/3*arctanh((2+3*x)/(sqrt(3)*sqrt(-2+4*x+3*x^2)))/sqrt(3)+1/3*sqrt(-2+4*x+3*x^2)],
[x/sqrt(-2+4*x-3*x^2),x,3,-2/3*arctan((2-3*x)/(sqrt(3)*sqrt(-2+4*x-3*x^2)))/sqrt(3)-1/3*sqrt(-2+4*x-3*x^2)],
[x/sqrt(-2+5*x+3*x^2),x,3,-5/6*arctanh(1/2*(5+6*x)/(sqrt(3)*sqrt(-2+5*x+3*x^2)))/sqrt(3)+1/3*sqrt(-2+5*x+3*x^2)],
[x/sqrt(-2+5*x-3*x^2),x,3,-5/6*arcsin(5-6*x)/sqrt(3)-1/3*sqrt(-2+5*x-3*x^2)],
[1/(x*sqrt(4+12*x+9*x^2)),x,4,-(2+3*x)*arctanh(1+3*x)/sqrt(4+12*x+9*x^2),1/2*(2+3*x)*log(x)/sqrt(4+12*x+9*x^2)-1/2*(2+3*x)*log(2+3*x)/sqrt(4+12*x+9*x^2)],
[1/(x*sqrt(4-12*x+9*x^2)),x,4,-(2-3*x)*arctanh(1-3*x)/sqrt(4-12*x+9*x^2),-1/2*(2-3*x)*log(2-3*x)/sqrt(4-12*x+9*x^2)+1/2*(2-3*x)*log(x)/sqrt(4-12*x+9*x^2)],
[1/(x*sqrt(-4+12*x-9*x^2)),x,4,-(2-3*x)*arctanh(1-3*x)/sqrt(-4+12*x-9*x^2),-1/2*(2-3*x)*log(2-3*x)/sqrt(-4+12*x-9*x^2)+1/2*(2-3*x)*log(x)/sqrt(-4+12*x-9*x^2)],
[1/(x*sqrt(-4-12*x-9*x^2)),x,4,-(2+3*x)*arctanh(1+3*x)/sqrt(-4-12*x-9*x^2),1/2*(2+3*x)*log(x)/sqrt(-4-12*x-9*x^2)-1/2*(2+3*x)*log(2+3*x)/sqrt(-4-12*x-9*x^2)],
[1/(x*sqrt(a^2+2*a*b*x+b^2*x^2)),x,4,(a+b*x)*log(x)/(a*sqrt(a^2+2*a*b*x+b^2*x^2))-(a+b*x)*log(a+b*x)/(a*sqrt(a^2+2*a*b*x+b^2*x^2))],
[1/(x*sqrt(a^2-2*a*b*x+b^2*x^2)),x,4,(a-b*x)*log(x)/(a*sqrt(a^2-2*a*b*x+b^2*x^2))-(a-b*x)*log(a-b*x)/(a*sqrt(a^2-2*a*b*x+b^2*x^2))],
[1/(x*sqrt(-a^2+2*a*b*x-b^2*x^2)),x,4,(a-b*x)*log(x)/(a*sqrt(-a^2+2*a*b*x-b^2*x^2))-(a-b*x)*log(a-b*x)/(a*sqrt(-a^2+2*a*b*x-b^2*x^2))],
[1/(x*sqrt(-a^2-2*a*b*x-b^2*x^2)),x,4,(a+b*x)*log(x)/(a*sqrt(-a^2-2*a*b*x-b^2*x^2))-(a+b*x)*log(a+b*x)/(a*sqrt(-a^2-2*a*b*x-b^2*x^2))],
[x*sqrt(3-2*x-x^2),x,4,-1/3*(3-2*x-x^2)^(3/2)+2*arcsin(1/2*(-1-x))-1/2*(1+x)*sqrt(3-2*x-x^2)],
[x*sqrt(8+2*x-x^2),x,4,-1/3*(8+2*x-x^2)^(3/2)-9/2*arcsin(1/3*(1-x))-1/2*(1-x)*sqrt(8+2*x-x^2)],
[x*sqrt(4+2*x+x^2),x,4,1/3*(4+2*x+x^2)^(3/2)-3/2*arcsinh((1+x)/sqrt(3))-1/2*(1+x)*sqrt(4+2*x+x^2)],
[1/(x*sqrt(2+4*x+3*x^2)),x,2,-arctanh((1+x)*sqrt(2)/sqrt(2+4*x+3*x^2))/sqrt(2)],
[1/(x*sqrt(2+4*x-3*x^2)),x,2,-arctanh((1+x)*sqrt(2)/sqrt(2+4*x-3*x^2))/sqrt(2)],
[1/(x*sqrt(2+5*x+3*x^2)),x,2,-arctanh(1/2*(4+5*x)/(sqrt(2)*sqrt(2+5*x+3*x^2)))/sqrt(2)],
[1/(x*sqrt(2+5*x-3*x^2)),x,2,-arctanh(1/2*(4+5*x)/(sqrt(2)*sqrt(2+5*x-3*x^2)))/sqrt(2)],
[1/(x*sqrt(-2+4*x+3*x^2)),x,2,-arctan((1-x)*sqrt(2)/sqrt(-2+4*x+3*x^2))/sqrt(2)],
[1/(x*sqrt(-2+4*x-3*x^2)),x,2,-arctan((1-x)*sqrt(2)/sqrt(-2+4*x-3*x^2))/sqrt(2)],
[1/(x*sqrt(-2+5*x+3*x^2)),x,2,-arctan(1/2*(4-5*x)/(sqrt(2)*sqrt(-2+5*x+3*x^2)))/sqrt(2)],
[1/(x*sqrt(-2+5*x-3*x^2)),x,2,-arctan(1/2*(4-5*x)/(sqrt(2)*sqrt(-2+5*x-3*x^2)))/sqrt(2)],
[1/(x^3*sqrt(1+x+x^2)),x,4,1/8*arctanh(1/2*(2+x)/sqrt(1+x+x^2))-1/2*sqrt(1+x+x^2)/x^2+3/4*sqrt(1+x+x^2)/x],
[1/x+(-1)/(x*sqrt(1+b*x+c*x^2)),x,3,log(-2-b*x-2*sqrt(1+b*x+c*x^2)),arctanh(1/2*(2+b*x)/sqrt(1+b*x+c*x^2))+log(x)],

# Integrands of the form (d+e x)^(m/2) (a+b x+c x^2)^(p/2)

# p>0
[(d*x)^(5/2)*sqrt(a+b*x+c*x^2),x,8,2/9*d*(d*x)^(3/2)*(a+b*x+c*x^2)^(3/2)/c-4/21*b*d^2*(a+b*x+c*x^2)^(3/2)*sqrt(d*x)/c^2-4/315*(8*b^4-36*a*b^2*c+21*a^2*c^2)*d^3*x*sqrt(a+b*x+c*x^2)/(c^(7/2)*(sqrt(a)+x*sqrt(c))*sqrt(d*x))+2/315*d^2*(b*(8*b^2+3*a*c)+3*c*(8*b^2-7*a*c)*x)*sqrt(d*x)*sqrt(a+b*x+c*x^2)/c^3+4/315*a^(1/4)*(8*b^4-36*a*b^2*c+21*a^2*c^2)*d^3*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x*sqrt(c))*sqrt(x)*sqrt((a+b*x+c*x^2)/(sqrt(a)+x*sqrt(c))^2)/(c^(15/4)*sqrt(d*x)*sqrt(a+b*x+c*x^2))-1/315*a^(1/4)*d^3*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x*sqrt(c))*(16*b^4-72*a*b^2*c+42*a^2*c^2+b*(8*b^2-27*a*c)*sqrt(a)*sqrt(c))*sqrt(x)*sqrt((a+b*x+c*x^2)/(sqrt(a)+x*sqrt(c))^2)/(c^(15/4)*sqrt(d*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^(3/2)*sqrt(a+b*x+c*x^2),x,7,2/7*e*(a+b*x+c*x^2)^(3/2)*sqrt(d+e*x)/c+2/105*(3*c^2*d^2-4*b^2*e^2+c*e*(9*b*d-5*a*e)+12*c*e*(2*c*d-b*e)*x)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/(c^2*e)-1/105*(2*c*d-b*e)*(3*c^2*d^2+8*b^2*e^2-c*e*(3*b*d+29*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*e^2*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+4/105*(c*d^2-b*d*e+a*e^2)*(3*c^2*d^2+2*b^2*e^2-c*e*(3*b*d+5*a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^3*e^2*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^(1/2)*sqrt(a+b*x+c*x^2),x,7,2/5*(d+e*x)^(3/2)*sqrt(a+b*x+c*x^2)/e-2/15*(2*c*d-b*e)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/(c*e)-2/15*(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*e^2*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+2/15*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^2*e^2*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[sqrt(a+b*x+c*x^2)/(d+e*x)^(1/2),x,6,2/3*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/e-1/3*(2*c*d-b*e)*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*e^2*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+4/3*(c*d^2-b*d*e+a*e^2)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c*e^2*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[sqrt(a+b*x+c*x^2)/(d+e*x)^(3/2),x,6,-2*sqrt(a+b*x+c*x^2)/(e*sqrt(d+e*x))+2*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e^2*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-2*(2*c*d-b*e)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c*e^2*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[sqrt(a+b*x+c*x^2)/(d+e*x)^(5/2),x,7,-2/3*sqrt(a+b*x+c*x^2)/(e*(d+e*x)^(3/2))+2/3*(2*c*d-b*e)*sqrt(a+b*x+c*x^2)/(e*(c*d^2-b*d*e+a*e^2)*sqrt(d+e*x))-1/3*(2*c*d-b*e)*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e^2*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+4/3*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e^2*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[sqrt(a+b*x+c*x^2)/(d+e*x)^(7/2),x,8,-2/5*sqrt(a+b*x+c*x^2)/(e*(d+e*x)^(5/2))+2/15*(2*c*d-b*e)*sqrt(a+b*x+c*x^2)/(e*(c*d^2-b*d*e+a*e^2)*(d+e*x)^(3/2))+4/15*(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*sqrt(a+b*x+c*x^2)/(e*(c*d^2-b*d*e+a*e^2)^2*sqrt(d+e*x))-2/15*(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e^2*(c*d^2-b*d*e+a*e^2)^2*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+2/15*(2*c*d-b*e)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e^2*(c*d^2-b*d*e+a*e^2)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^(3/2)*(a+b*x+c*x^2)^(3/2),x,8,2/231*(c^2*d^2-6*b^2*e^2+c*e*(13*b*d-3*a*e)+14*c*e*(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(3/2)*sqrt(d+e*x)/(c^2*e)+2/11*e*(a+b*x+c*x^2)^(5/2)*sqrt(d+e*x)/c+2/1155*(8*c^4*d^4+8*b^4*e^4-c^3*d^2*e*(19*b*d-42*a*e)-b^2*c*e^3*(19*b*d+21*a*e)+3*c^2*e^2*(2*b^2*d^2+17*a*b*d*e-10*a^2*e^2)-3*c*e*(2*c*d-b*e)*(c^2*d^2+8*b^2*e^2-c*e*(b*d+31*a*e))*x)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/(c^3*e^3)-8/1155*(2*c*d-b*e)*(c^2*d^2-2*b^2*e^2-c*e*(b*d-9*a*e))*(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*e^4*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+2/1155*(c*d^2-b*d*e+a*e^2)*(16*c^4*d^4-8*b^4*e^4-4*c^3*d^2*e*(8*b*d-21*a*e)+b^2*c*e^3*(13*b*d+51*a*e)+3*c^2*e^2*(b^2*d^2-28*a*b*d*e-20*a^2*e^2))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^4*e^4*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^(1/2)*(a+b*x+c*x^2)^(3/2),x,8,2/9*(d+e*x)^(3/2)*(a+b*x+c*x^2)^(3/2)/e-2/21*(2*c*d-b*e)*(a+b*x+c*x^2)^(3/2)*sqrt(d+e*x)/(c*e)+2/315*(8*c^3*d^3-4*b^3*e^3-3*c^2*d*e*(5*b*d-8*a*e)+3*b*c*e^2*(b*d+3*a*e)-6*c*e*(c^2*d^2+2*b^2*e^2-c*e*(b*d+7*a*e))*x)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/(c^2*e^3)-1/315*(16*c^4*d^4-8*b^4*e^4-4*c^3*d^2*e*(8*b*d-15*a*e)+b^2*c*e^3*(7*b*d+57*a*e)+3*c^2*e^2*(3*b^2*d^2-20*a*b*d*e-28*a^2*e^2))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*e^4*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+8/315*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(2*c^2*d^2-b^2*e^2-2*c*e*(b*d-3*a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^3*e^4*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(3/2)/(d+e*x)^(1/2),x,7,2/7*(a+b*x+c*x^2)^(3/2)*sqrt(d+e*x)/e+2/35*(8*c^2*d^2+b^2*e^2-c*e*(11*b*d-10*a*e)-3*c*e*(2*c*d-b*e)*x)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/(c*e^3)-2/35*(2*c*d-b*e)*(4*c^2*d^2-b^2*e^2-4*c*e*(b*d-2*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*e^4*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+2/35*(c*d^2-b*d*e+a*e^2)*(16*c^2*d^2-b^2*e^2-4*c*e*(4*b*d-5*a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^2*e^4*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(3/2)/(d+e*x)^(3/2),x,7,-2*(a+b*x+c*x^2)^(3/2)/(e*sqrt(d+e*x))-2/5*(8*c*d-7*b*e-6*c*e*x)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/e^3+1/5*(16*c^2*d^2+b^2*e^2-4*c*e*(4*b*d-3*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*e^4*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-16/5*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c*e^4*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(3/2)/(d+e*x)^(5/2),x,7,-2/3*(a+b*x+c*x^2)^(3/2)/(e*(d+e*x)^(3/2))+2/3*(8*c*d-3*b*e+2*c*e*x)*sqrt(a+b*x+c*x^2)/(e^3*sqrt(d+e*x))-8/3*(2*c*d-b*e)*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e^4*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+2/3*(16*c^2*d^2+3*b^2*e^2-4*c*e*(4*b*d-a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c*e^4*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(3/2)/(d+e*x)^(7/2),x,7,-2/5*(a+b*x+c*x^2)^(3/2)/(e*(d+e*x)^(5/2))-2/5*(8*c^2*d^3+a*b*e^3-c*d*e*(7*b*d-4*a*e)+e*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*x)*sqrt(a+b*x+c*x^2)/(e^3*(c*d^2-b*d*e+a*e^2)*(d+e*x)^(3/2))+1/5*(16*c^2*d^2+b^2*e^2-4*c*e*(4*b*d-3*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e^4*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-16/5*(2*c*d-b*e)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e^4*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(3/2)/(d+e*x)^(9/2),x,8,-2/7*(a+b*x+c*x^2)^(3/2)/(e*(d+e*x)^(7/2))-2/35*(8*c^2*d^3-c*d*e*(5*b*d-4*a*e)-b*e^2*(2*b*d-3*a*e)+e*(14*c^2*d^2+b^2*e^2-2*c*e*(7*b*d-5*a*e))*x)*sqrt(a+b*x+c*x^2)/(e^3*(c*d^2-b*d*e+a*e^2)*(d+e*x)^(5/2))+4/35*(2*c*d-b*e)*(4*c^2*d^2-b^2*e^2-4*c*e*(b*d-2*a*e))*sqrt(a+b*x+c*x^2)/(e^3*(c*d^2-b*d*e+a*e^2)^2*sqrt(d+e*x))-2/35*(2*c*d-b*e)*(4*c^2*d^2-b^2*e^2-4*c*e*(b*d-2*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e^4*(c*d^2-b*d*e+a*e^2)^2*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+2/35*(16*c^2*d^2-b^2*e^2-4*c*e*(4*b*d-5*a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e^4*(c*d^2-b*d*e+a*e^2)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(d*x)^(1/2)*(a+b*x+c*x^2)^(5/2),x,9,2/13*(d*x)^(3/2)*(a+b*x+c*x^2)^(5/2)/d-10/9009*(3*b*(6*b^2-19*a*c)+14*c*(3*b^2-11*a*c)*x)*(a+b*x+c*x^2)^(3/2)*sqrt(d*x)/c^2+10/143*b*(a+b*x+c*x^2)^(5/2)*sqrt(d*x)/c-4/9009*(24*b^6-268*a*b^4*c+951*a^2*b^2*c^2-924*a^3*c^3)*d*x*sqrt(a+b*x+c*x^2)/(c^(7/2)*(sqrt(a)+x*sqrt(c))*sqrt(d*x))+2/9009*(b*(24*b^4-151*a*b^2*c+108*a^2*c^2)+3*c*(24*b^4-181*a*b^2*c+308*a^2*c^2)*x)*sqrt(d*x)*sqrt(a+b*x+c*x^2)/c^3+4/9009*a^(1/4)*(24*b^6-268*a*b^4*c+951*a^2*b^2*c^2-924*a^3*c^3)*d*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x*sqrt(c))*sqrt(x)*sqrt((a+b*x+c*x^2)/(sqrt(a)+x*sqrt(c))^2)/(c^(15/4)*sqrt(d*x)*sqrt(a+b*x+c*x^2))-1/9009*a^(1/4)*d*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x*sqrt(c))*(2*(24*b^6-268*a*b^4*c+951*a^2*b^2*c^2-924*a^3*c^3)+b*(24*b^4-241*a*b^2*c+708*a^2*c^2)*sqrt(a)*sqrt(c))*sqrt(x)*sqrt((a+b*x+c*x^2)/(sqrt(a)+x*sqrt(c))^2)/(c^(15/4)*sqrt(d*x)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(5/2)/(d+e*x)^(1/2),x,8,10/693*(16*c^2*d^2+3*b^2*e^2-c*e*(23*b*d-18*a*e)-7*c*e*(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(3/2)*sqrt(d+e*x)/(c*e^3)+2/11*(a+b*x+c*x^2)^(5/2)*sqrt(d+e*x)/e+2/693*(128*c^4*d^4-4*b^4*e^4-4*c^3*d^2*e*(76*b*d-69*a*e)-b^2*c*e^3*(7*b*d-27*a*e)+3*c^2*e^2*(65*b^2*d^2-124*a*b*d*e+60*a^2*e^2)-12*c*e*(2*c*d-b*e)*(4*c^2*d^2-b^2*e^2-4*c*e*(b*d-2*a*e))*x)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/(c^2*e^5)-1/693*(2*c*d-b*e)*(128*c^4*d^4+8*b^4*e^4+b^2*c*e^3*(29*b*d-93*a*e)-4*c^3*d^2*e*(64*b*d-93*a*e)+3*c^2*e^2*(33*b^2*d^2-124*a*b*d*e+124*a^2*e^2))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*e^6*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+4/693*(c*d^2-b*d*e+a*e^2)*(128*c^4*d^4+2*b^4*e^4-4*c^3*d^2*e*(64*b*d-69*a*e)+b^2*c*e^3*(5*b*d-21*a*e)+3*c^2*e^2*(41*b^2*d^2-92*a*b*d*e+60*a^2*e^2))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^3*e^6*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(5/2)/(d+e*x)^(3/2),x,8,-2*(a+b*x+c*x^2)^(5/2)/(e*sqrt(d+e*x))-10/63*(16*c*d-15*b*e-14*c*e*x)*(a+b*x+c*x^2)^(3/2)*sqrt(d+e*x)/e^3-2/63*(128*c^3*d^3-b^3*e^3+3*b*c*e^2*(37*b*d-36*a*e)-12*c^2*d*e*(20*b*d-11*a*e)-3*c*e*(32*c^2*d^2+b^2*e^2-4*c*e*(8*b*d-7*a*e))*x)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/(c*e^5)+2/63*(128*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(64*b*d-57*a*e)-b^2*c*e^3*(7*b*d-15*a*e)+3*c^2*e^2*(45*b^2*d^2-76*a*b*d*e+28*a^2*e^2))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*e^6*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-2/63*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(128*c^2*d^2-b^2*e^2-4*c*e*(32*b*d-33*a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^2*e^6*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(5/2)/(d+e*x)^(5/2),x,8,-2/3*(a+b*x+c*x^2)^(5/2)/(e*(d+e*x)^(3/2))+10/21*(16*c*d-7*b*e+2*c*e*x)*(a+b*x+c*x^2)^(3/2)/(e^3*sqrt(d+e*x))+2/21*(128*c^2*d^2+51*b^2*e^2-4*c*e*(44*b*d-5*a*e)-48*c*e*(2*c*d-b*e)*x)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/e^5-1/21*(2*c*d-b*e)*(128*c^2*d^2+3*b^2*e^2-4*c*e*(32*b*d-29*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*e^6*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+4/21*(c*d^2-b*d*e+a*e^2)*(128*c^2*d^2+27*b^2*e^2-4*c*e*(32*b*d-5*a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c*e^6*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(5/2)/(d+e*x)^(7/2),x,8,2/15*(16*c*d-5*b*e+6*c*e*x)*(a+b*x+c*x^2)^(3/2)/(e^3*(d+e*x)^(3/2))-2/5*(a+b*x+c*x^2)^(5/2)/(e*(d+e*x)^(5/2))-2/15*(128*c^2*d^2+15*b^2*e^2-4*c*e*(28*b*d-9*a*e)+16*c*e*(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/(e^5*sqrt(d+e*x))+2/15*(128*c^2*d^2+23*b^2*e^2-4*c*e*(32*b*d-9*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e^6*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-2/15*(2*c*d-b*e)*(128*c^2*d^2+15*b^2*e^2-4*c*e*(32*b*d-17*a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c*e^6*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(5/2)/(d+e*x)^(9/2),x,8,-2/21*(16*c^2*d^3+3*a*b*e^3-c*d*e*(13*b*d-4*a*e)+e*(22*c^2*d^2+3*b^2*e^2-2*c*e*(11*b*d-5*a*e))*x)*(a+b*x+c*x^2)^(3/2)/(e^3*(c*d^2-b*d*e+a*e^2)*(d+e*x)^(5/2))-2/7*(a+b*x+c*x^2)^(5/2)/(e*(d+e*x)^(7/2))+2/21*c*(128*c^2*d^3-4*c*d*e*(44*b*d-29*a*e)+3*b*e^2*(17*b*d-16*a*e)+e*(32*c^2*d^2+3*b^2*e^2-4*c*e*(8*b*d-5*a*e))*x)*sqrt(a+b*x+c*x^2)/(e^5*(c*d^2-b*d*e+a*e^2)*sqrt(d+e*x))-1/21*(2*c*d-b*e)*(128*c^2*d^2+3*b^2*e^2-4*c*e*(32*b*d-29*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e^6*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+4/21*(128*c^2*d^2+27*b^2*e^2-4*c*e*(32*b*d-5*a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e^6*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(a+b*x+c*x^2)^(5/2)/(d+e*x)^(11/2),x,8,-2/63*(16*c^2*d^3-b*e^2*(2*b*d-5*a*e)-c*d*e*(11*b*d-4*a*e)+e*(26*c^2*d^2+3*b^2*e^2-2*c*e*(13*b*d-7*a*e))*x)*(a+b*x+c*x^2)^(3/2)/(e^3*(c*d^2-b*d*e+a*e^2)*(d+e*x)^(7/2))-2/9*(a+b*x+c*x^2)^(5/2)/(e*(d+e*x)^(9/2))-2/63*(128*c^4*d^5-2*a*b^3*e^5-4*c^3*d^3*e*(60*b*d-49*a*e)-b*c*e^3*(b^2*d^2+9*a*b*d*e-24*a^2*e^2)+3*c^2*d*e^2*(37*b^2*d^2-52*a*b*d*e+12*a^2*e^2)+e*(160*c^4*d^4-2*b^4*e^4-4*c^3*d^2*e*(80*b*d-69*a*e)-b^2*c*e^3*(11*b*d-27*a*e)+3*c^2*e^2*(57*b^2*d^2-92*a*b*d*e+28*a^2*e^2))*x)*sqrt(a+b*x+c*x^2)/(e^5*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^(3/2))+2/63*(128*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(64*b*d-57*a*e)-b^2*c*e^3*(7*b*d-15*a*e)+3*c^2*e^2*(45*b^2*d^2-76*a*b*d*e+28*a^2*e^2))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e^6*(c*d^2-b*d*e+a*e^2)^2*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-2/63*(2*c*d-b*e)*(128*c^2*d^2-b^2*e^2-4*c*e*(32*b*d-33*a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e^6*(c*d^2-b*d*e+a*e^2)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],

# p<0
[(d+e*x)^(7/2)/sqrt(a+b*x+c*x^2),x,8,12/35*e*(2*c*d-b*e)*(d+e*x)^(3/2)*sqrt(a+b*x+c*x^2)/c^2+2/7*e*(d+e*x)^(5/2)*sqrt(a+b*x+c*x^2)/c+2/105*e*(71*c^2*d^2+24*b^2*e^2-c*e*(71*b*d+25*a*e))*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/c^3+8/105*(2*c*d-b*e)*(11*c^2*d^2+6*b^2*e^2-c*e*(11*b*d+13*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-2/105*(c*d^2-b*d*e+a*e^2)*(71*c^2*d^2+24*b^2*e^2-c*e*(71*b*d+25*a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^4*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^(5/2)/sqrt(a+b*x+c*x^2),x,7,2/5*e*(d+e*x)^(3/2)*sqrt(a+b*x+c*x^2)/c+8/15*e*(2*c*d-b*e)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/c^2+1/15*(23*c^2*d^2+8*b^2*e^2-c*e*(23*b*d+9*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-8/15*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^3*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^(3/2)/sqrt(a+b*x+c*x^2),x,6,2/3*e*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/c+2/3*(2*c*d-b*e)*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-2/3*(c*d^2-b*d*e+a*e^2)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^2*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^(1/2)/sqrt(a+b*x+c*x^2),x,2,EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))],
[1/((d+e*x)^(1/2)*sqrt(a+b*x+c*x^2)),x,2,2*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)^(3/2)*sqrt(a+b*x+c*x^2)),x,4,-2*e*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*sqrt(d+e*x))+EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))],
[1/((d+e*x)^(5/2)*sqrt(a+b*x+c*x^2)),x,7,-2/3*e*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*(d+e*x)^(3/2))-4/3*e*(2*c*d-b*e)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^2*sqrt(d+e*x))+2/3*(2*c*d-b*e)*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^2*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-2/3*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/((c*d^2-b*d*e+a*e^2)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)^(7/2)*sqrt(a+b*x+c*x^2)),x,8,-2/5*e*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*(d+e*x)^(5/2))-8/15*e*(2*c*d-b*e)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^2*(d+e*x)^(3/2))-2/15*e*(23*c^2*d^2+8*b^2*e^2-c*e*(23*b*d+9*a*e))*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^3*sqrt(d+e*x))+1/15*(23*c^2*d^2+8*b^2*e^2-c*e*(23*b*d+9*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^3*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-8/15*(2*c*d-b*e)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/((c*d^2-b*d*e+a*e^2)^2*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^(7/2)/(a+b*x+c*x^2)^(3/2),x,8,-2*(d+e*x)^(5/2)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*sqrt(a+b*x+c*x^2))+2*e*(2*c*d-b*e)*(d+e*x)^(3/2)*sqrt(a+b*x+c*x^2)/(c*(b^2-4*a*c))+4/3*e*(3*c^2*d^2+2*b^2*e^2-c*e*(3*b*d+5*a*e))*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/(c^2*(b^2-4*a*c))+1/3*(2*c*d-b*e)*(3*c^2*d^2+8*b^2*e^2-c*e*(3*b*d+29*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*sqrt(b^2-4*a*c)*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-4/3*(c*d^2-b*d*e+a*e^2)*(3*c^2*d^2+2*b^2*e^2-c*e*(3*b*d+5*a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^3*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^(5/2)/(a+b*x+c*x^2)^(3/2),x,7,-2*(d+e*x)^(3/2)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*sqrt(a+b*x+c*x^2))+2*e*(2*c*d-b*e)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/(c*(b^2-4*a*c))+2*(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*sqrt(b^2-4*a*c)*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^2*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^(3/2)/(a+b*x+c*x^2)^(3/2),x,6,-2*(b*d-2*a*e+(2*c*d-b*e)*x)*sqrt(d+e*x)/((b^2-4*a*c)*sqrt(a+b*x+c*x^2))+(2*c*d-b*e)*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*sqrt(b^2-4*a*c)*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-4*(c*d^2-b*d*e+a*e^2)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^(1/2)/(a+b*x+c*x^2)^(3/2),x,6,-2*(b+2*c*x)*sqrt(d+e*x)/((b^2-4*a*c)*sqrt(a+b*x+c*x^2))+2*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(sqrt(b^2-4*a*c)*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-2*(2*c*d-b*e)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)^(1/2)*(a+b*x+c*x^2)^(3/2)),x,6,-2*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)*sqrt(d+e*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2))+(2*c*d-b*e)*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)*sqrt(b^2-4*a*c)*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-4*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)^(3/2)*(a+b*x+c*x^2)^(3/2)),x,7,-2*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))-4*e*(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^2*sqrt(d+e*x))+2*(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^2*sqrt(b^2-4*a*c)*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-2*(2*c*d-b*e)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/((c*d^2-b*d*e+a*e^2)*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)^(5/2)*(a+b*x+c*x^2)^(3/2)),x,8,-2*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(d+e*x)^(3/2)*sqrt(a+b*x+c*x^2))-4/3*e*(3*c^2*d^2+2*b^2*e^2-c*e*(3*b*d+5*a*e))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^(3/2))-2/3*e*(2*c*d-b*e)*(3*c^2*d^2+8*b^2*e^2-c*e*(3*b*d+29*a*e))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^3*sqrt(d+e*x))+1/3*(2*c*d-b*e)*(3*c^2*d^2+8*b^2*e^2-c*e*(3*b*d+29*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^3*sqrt(b^2-4*a*c)*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-4/3*(3*c^2*d^2+2*b^2*e^2-c*e*(3*b*d+5*a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/((c*d^2-b*d*e+a*e^2)^2*sqrt(b^2-4*a*c)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^(7/2)/(a+b*x+c*x^2)^(5/2),x,7,-2/3*(d+e*x)^(5/2)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(3/2))+2/3*(8*b*c*d*(c*d^2+3*a*e^2)-4*a*c*e*(3*c*d^2+5*a*e^2)-b^2*(9*c*d^2*e-a*e^3)+(2*c*d-b*e)*(8*c^2*d^2-b^2*e^2-4*c*e*(2*b*d-3*a*e))*x)*sqrt(d+e*x)/(c*(b^2-4*a*c)^2*sqrt(a+b*x+c*x^2))-2/3*(2*c*d-b*e)*(4*c^2*d^2-b^2*e^2-4*c*e*(b*d-2*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*(b^2-4*a*c)^(3/2)*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+2/3*(c*d^2-b*d*e+a*e^2)*(16*c^2*d^2-b^2*e^2-4*c*e*(4*b*d-5*a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^2*(b^2-4*a*c)^(3/2)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^(5/2)/(a+b*x+c*x^2)^(5/2),x,7,-2/3*(d+e*x)^(3/2)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(3/2))-2/3*(7*b^2*d*e+4*a*c*d*e-8*b*(c*d^2+a*e^2)-(16*c^2*d^2+b^2*e^2-4*c*e*(4*b*d-3*a*e))*x)*sqrt(d+e*x)/((b^2-4*a*c)^2*sqrt(a+b*x+c*x^2))-1/3*(16*c^2*d^2+b^2*e^2-4*c*e*(4*b*d-3*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*(b^2-4*a*c)^(3/2)*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+16/3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c*(b^2-4*a*c)^(3/2)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^(3/2)/(a+b*x+c*x^2)^(5/2),x,7,-2/3*(b*d-2*a*e+(2*c*d-b*e)*x)*sqrt(d+e*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(3/2))+2/3*(8*b*c*d-5*b^2*e+4*a*c*e+8*c*(2*c*d-b*e)*x)*sqrt(d+e*x)/((b^2-4*a*c)^2*sqrt(a+b*x+c*x^2))-8/3*(2*c*d-b*e)*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+2/3*(16*c^2*d^2+3*b^2*e^2-4*c*e*(4*b*d-a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c*(b^2-4*a*c)^(3/2)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^(1/2)/(a+b*x+c*x^2)^(5/2),x,7,-2/3*(b+2*c*x)*sqrt(d+e*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(3/2))-2/3*(9*b^2*c*d*e-4*a*c^2*d*e-b^3*e^2-4*b*c*(2*c*d^2+a*e^2)-c*(16*c^2*d^2+b^2*e^2-4*c*e*(4*b*d-3*a*e))*x)*sqrt(d+e*x)/((b^2-4*a*c)^2*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2))-1/3*(16*c^2*d^2+b^2*e^2-4*c*e*(4*b*d-3*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+16/3*(2*c*d-b*e)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/((b^2-4*a*c)^(3/2)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)^(1/2)*(a+b*x+c*x^2)^(5/2)),x,7,-2/3*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)*sqrt(d+e*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(a+b*x+c*x^2)^(3/2))-2/3*(3*a*c*e*(2*c*d-b*e)^2-(b*c*d-b^2*e+2*a*c*e)*(8*c^2*d^2-2*b^2*e^2-5*c*e*(b*d-2*a*e))-2*c*(2*c*d-b*e)*(4*c^2*d^2-b^2*e^2-4*c*e*(b*d-2*a*e))*x)*sqrt(d+e*x)/((b^2-4*a*c)^2*(c*d^2-b*d*e+a*e^2)^2*sqrt(a+b*x+c*x^2))-2/3*(2*c*d-b*e)*(4*c^2*d^2-b^2*e^2-4*c*e*(b*d-2*a*e))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)^2*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+2/3*(16*c^2*d^2-b^2*e^2-4*c*e*(4*b*d-5*a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/((b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)^(3/2)*(a+b*x+c*x^2)^(5/2)),x,8,-2/3*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(a+b*x+c*x^2)^(3/2)*sqrt(d+e*x))-2/3*(5*a*c*e*(2*c*d-b*e)^2-(b*c*d-b^2*e+2*a*c*e)*(8*c^2*d^2-4*b^2*e^2-c*e*(3*b*d-14*a*e))-4*c*(2*c*d-b*e)*(2*c^2*d^2-b^2*e^2-2*c*e*(b*d-3*a*e))*x)/((b^2-4*a*c)^2*(c*d^2-b*d*e+a*e^2)^2*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))+2/3*e*(16*c^4*d^4-8*b^4*e^4-4*c^3*d^2*e*(8*b*d-15*a*e)+b^2*c*e^3*(7*b*d+57*a*e)+3*c^2*e^2*(3*b^2*d^2-20*a*b*d*e-28*a^2*e^2))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)^2*(c*d^2-b*d*e+a*e^2)^3*sqrt(d+e*x))-1/3*(16*c^4*d^4-8*b^4*e^4-4*c^3*d^2*e*(8*b*d-15*a*e)+b^2*c*e^3*(7*b*d+57*a*e)+3*c^2*e^2*(3*b^2*d^2-20*a*b*d*e-28*a^2*e^2))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(d+e*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)^3*sqrt(a+b*x+c*x^2)*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+8/3*(2*c*d-b*e)*(2*c^2*d^2-b^2*e^2-2*c*e*(b*d-3*a*e))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*e*sqrt(b^2-4*a*c)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/((b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)^2*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))],
[sqrt(3+5*x)/sqrt(2+5*x-12*x^2),x,2,-1/3*EllipticE(2*sqrt(2-3*x)/sqrt(11),sqrt(55/76))*sqrt(19)],

# Integrands of the form (d+e x)^m (a+b x+c x^2)^(p/3)

# p>0
[(d+e*x)^2*(a+b*x+c*x^2)^(4/3),x,6,-3/935*(b^2-4*a*c)*(17*c^2*d^2+5*b^2*e^2-c*e*(17*b*d+3*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(1/3)/c^4+3/374*(17*c^2*d^2+5*b^2*e^2-c*e*(17*b*d+3*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(4/3)/c^3+15/119*e*(2*c*d-b*e)*(a+b*x+c*x^2)^(7/3)/c^2+3/17*e*(d+e*x)*(a+b*x+c*x^2)^(7/3)/c+1/935*2^(1/3)*3^(3/4)*(b^2-4*a*c)^2*(17*c^2*d^2+5*b^2*e^2-c*e*(17*b*d+3*a*e))*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))*EllipticF((2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1-sqrt(3)))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt(((b^2-4*a*c)^(2/3)-2^(2/3)*c^(1/3)*(b^2-4*a*c)^(1/3)*(a+b*x+c*x^2)^(1/3)+2*2^(1/3)*c^(2/3)*(a+b*x+c*x^2)^(2/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2)/(c^(13/3)*(b+2*c*x)*sqrt((b^2-4*a*c)^(1/3)*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2))],
[(d+e*x)*(a+b*x+c*x^2)^(4/3),x,5,-3/110*(b^2-4*a*c)*(2*c*d-b*e)*(b+2*c*x)*(a+b*x+c*x^2)^(1/3)/c^3+3/44*(2*c*d-b*e)*(b+2*c*x)*(a+b*x+c*x^2)^(4/3)/c^2+3/14*e*(a+b*x+c*x^2)^(7/3)/c+1/55*3^(3/4)*(b^2-4*a*c)^2*(2*c*d-b*e)*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))*EllipticF((2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1-sqrt(3)))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt(((b^2-4*a*c)^(2/3)-2^(2/3)*c^(1/3)*(b^2-4*a*c)^(1/3)*(a+b*x+c*x^2)^(1/3)+2*2^(1/3)*c^(2/3)*(a+b*x+c*x^2)^(2/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2)/(2^(2/3)*c^(10/3)*(b+2*c*x)*sqrt((b^2-4*a*c)^(1/3)*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x+c*x^2)^(4/3),x,4,-3/55*(b^2-4*a*c)*(b+2*c*x)*(a+b*x+c*x^2)^(1/3)/c^2+3/22*(b+2*c*x)*(a+b*x+c*x^2)^(4/3)/c+1/55*2^(1/3)*3^(3/4)*(b^2-4*a*c)^2*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))*EllipticF((2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1-sqrt(3)))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt(((b^2-4*a*c)^(2/3)-2^(2/3)*c^(1/3)*(b^2-4*a*c)^(1/3)*(a+b*x+c*x^2)^(1/3)+2*2^(1/3)*c^(2/3)*(a+b*x+c*x^2)^(2/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2)/(c^(7/3)*(b+2*c*x)*sqrt((b^2-4*a*c)^(1/3)*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x+c*x^2)^(4/3)/(d+e*x),x,2,3*(a+b*x+c*x^2)^(4/3)*AppellF1(-8/3,-4/3,-4/3,-5/3,1/2*(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(c*(d+e*x)),1/2*(2*d-e*(b+sqrt(b^2-4*a*c))/c)/(d+e*x))/(2^(1/3)*e*(e*(b+2*c*x-sqrt(b^2-4*a*c))/(c*(d+e*x)))^(4/3)*(e*(b+2*c*x+sqrt(b^2-4*a*c))/(c*(d+e*x)))^(4/3))],
[(a+b*x+c*x^2)^(4/3)/(d+e*x)^2,x,2,12/5*2^(2/3)*(a+b*x+c*x^2)^(4/3)*AppellF1(-5/3,-4/3,-4/3,-2/3,1/2*(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(c*(d+e*x)),1/2*(2*d-e*(b+sqrt(b^2-4*a*c))/c)/(d+e*x))/(e*(d+e*x)*(e*(b+2*c*x-sqrt(b^2-4*a*c))/(c*(d+e*x)))^(4/3)*(e*(b+2*c*x+sqrt(b^2-4*a*c))/(c*(d+e*x)))^(4/3))],
[(a+b*x+c*x^2)^(4/3)/(d+e*x)^3,x,2,6*2^(2/3)*(a+b*x+c*x^2)^(4/3)*AppellF1(-2/3,-4/3,-4/3,1/3,1/2*(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(c*(d+e*x)),1/2*(2*d-e*(b+sqrt(b^2-4*a*c))/c)/(d+e*x))/(e*(d+e*x)^2*(e*(b+2*c*x-sqrt(b^2-4*a*c))/(c*(d+e*x)))^(4/3)*(e*(b+2*c*x+sqrt(b^2-4*a*c))/(c*(d+e*x)))^(4/3))],

# p<0
[(d+e*x)^3/(a+b*x+c*x^2)^(7/3),x,6,-3/4*(d+e*x)^2*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(4/3))+3/4*(10*b*c*d*(c*d^2+3*a*e^2)-8*a*c*e*(2*c*d^2+3*a*e^2)-b^2*(11*c*d^2*e-a*e^3)+(2*c*d-b*e)*(10*c^2*d^2-b^2*e^2-2*c*e*(5*b*d-7*a*e))*x)/(c*(b^2-4*a*c)^2*(a+b*x+c*x^2)^(1/3))-3/2*(2*c*d-b*e)*(5*c^2*d^2-b^2*e^2-c*e*(5*b*d-9*a*e))*(b+2*c*x)/(2^(1/3)*c^(5/3)*(b^2-4*a*c)^2*(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3))))-3^(3/4)*(2*c*d-b*e)*(5*c^2*d^2-b^2*e^2-c*e*(5*b*d-9*a*e))*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))*EllipticF((2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1-sqrt(3)))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(((b^2-4*a*c)^(2/3)-2^(2/3)*c^(1/3)*(b^2-4*a*c)^(1/3)*(a+b*x+c*x^2)^(1/3)+2*2^(1/3)*c^(2/3)*(a+b*x+c*x^2)^(2/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2)/(2^(5/6)*c^(5/3)*(b^2-4*a*c)^(5/3)*(b+2*c*x)*sqrt((b^2-4*a*c)^(1/3)*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2))+3/4*3^(1/4)*(2*c*d-b*e)*(5*c^2*d^2-b^2*e^2-c*e*(5*b*d-9*a*e))*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))*EllipticE((2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1-sqrt(3)))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt(((b^2-4*a*c)^(2/3)-2^(2/3)*c^(1/3)*(b^2-4*a*c)^(1/3)*(a+b*x+c*x^2)^(1/3)+2*2^(1/3)*c^(2/3)*(a+b*x+c*x^2)^(2/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2)/(2^(1/3)*c^(5/3)*(b^2-4*a*c)^(5/3)*(b+2*c*x)*sqrt((b^2-4*a*c)^(1/3)*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2))],
[(d+e*x)^2/(a+b*x+c*x^2)^(7/3),x,6,-3/4*(d+e*x)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(4/3))-3/2*(4*b^2*d*e+4*a*c*d*e-5*b*(c*d^2+a*e^2)-(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2)^(1/3))-3/2*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*(b+2*c*x)/(2^(1/3)*c^(2/3)*(b^2-4*a*c)^2*(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3))))-3^(3/4)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))*EllipticF((2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1-sqrt(3)))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(((b^2-4*a*c)^(2/3)-2^(2/3)*c^(1/3)*(b^2-4*a*c)^(1/3)*(a+b*x+c*x^2)^(1/3)+2*2^(1/3)*c^(2/3)*(a+b*x+c*x^2)^(2/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2)/(2^(5/6)*c^(2/3)*(b^2-4*a*c)^(5/3)*(b+2*c*x)*sqrt((b^2-4*a*c)^(1/3)*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2))+3/4*3^(1/4)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))*EllipticE((2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1-sqrt(3)))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt(((b^2-4*a*c)^(2/3)-2^(2/3)*c^(1/3)*(b^2-4*a*c)^(1/3)*(a+b*x+c*x^2)^(1/3)+2*2^(1/3)*c^(2/3)*(a+b*x+c*x^2)^(2/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2)/(2^(1/3)*c^(2/3)*(b^2-4*a*c)^(5/3)*(b+2*c*x)*sqrt((b^2-4*a*c)^(1/3)*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2))],
[(d+e*x)/(a+b*x+c*x^2)^(7/3),x,6,-3/4*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(4/3))+15/4*(2*c*d-b*e)*(b+2*c*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2)^(1/3))-15/2*c^(1/3)*(2*c*d-b*e)*(b+2*c*x)/(2^(1/3)*(b^2-4*a*c)^2*(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3))))-5*3^(3/4)*c^(1/3)*(2*c*d-b*e)*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))*EllipticF((2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1-sqrt(3)))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(((b^2-4*a*c)^(2/3)-2^(2/3)*c^(1/3)*(b^2-4*a*c)^(1/3)*(a+b*x+c*x^2)^(1/3)+2*2^(1/3)*c^(2/3)*(a+b*x+c*x^2)^(2/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2)/(2^(5/6)*(b^2-4*a*c)^(5/3)*(b+2*c*x)*sqrt((b^2-4*a*c)^(1/3)*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2))+15/4*3^(1/4)*c^(1/3)*(2*c*d-b*e)*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))*EllipticE((2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1-sqrt(3)))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt(((b^2-4*a*c)^(2/3)-2^(2/3)*c^(1/3)*(b^2-4*a*c)^(1/3)*(a+b*x+c*x^2)^(1/3)+2*2^(1/3)*c^(2/3)*(a+b*x+c*x^2)^(2/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2)/(2^(1/3)*(b^2-4*a*c)^(5/3)*(b+2*c*x)*sqrt((b^2-4*a*c)^(1/3)*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2))],
[1/(a+b*x+c*x^2)^(7/3),x,6,-3/4*(b+2*c*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(4/3))+15/2*c*(b+2*c*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2)^(1/3))-15*c^(4/3)*(b+2*c*x)/(2^(1/3)*(b^2-4*a*c)^2*(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3))))-5*2^(1/6)*3^(3/4)*c^(4/3)*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))*EllipticF((2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1-sqrt(3)))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(((b^2-4*a*c)^(2/3)-2^(2/3)*c^(1/3)*(b^2-4*a*c)^(1/3)*(a+b*x+c*x^2)^(1/3)+2*2^(1/3)*c^(2/3)*(a+b*x+c*x^2)^(2/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2)/((b^2-4*a*c)^(5/3)*(b+2*c*x)*sqrt((b^2-4*a*c)^(1/3)*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2))+15/2*3^(1/4)*c^(4/3)*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))*EllipticE((2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1-sqrt(3)))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt(((b^2-4*a*c)^(2/3)-2^(2/3)*c^(1/3)*(b^2-4*a*c)^(1/3)*(a+b*x+c*x^2)^(1/3)+2*2^(1/3)*c^(2/3)*(a+b*x+c*x^2)^(2/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2)/(2^(1/3)*(b^2-4*a*c)^(5/3)*(b+2*c*x)*sqrt((b^2-4*a*c)^(1/3)*((b^2-4*a*c)^(1/3)+2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*(a+b*x+c*x^2)^(1/3)+(b^2-4*a*c)^(1/3)*(1+sqrt(3)))^2))],
[1/((d+e*x)*(a+b*x+c*x^2)^(7/3)),x,2,-3/224*AppellF1(14/3,7/3,7/3,17/3,1/2*(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(c*(d+e*x)),1/2*(2*d-e*(b+sqrt(b^2-4*a*c))/c)/(d+e*x))*(e*(b+2*c*x-sqrt(b^2-4*a*c))/(c*(d+e*x)))^(7/3)*(e*(b+2*c*x+sqrt(b^2-4*a*c))/(c*(d+e*x)))^(7/3)/(2^(2/3)*e*(a+b*x+c*x^2)^(7/3))],
[1/((d+e*x)^2*(a+b*x+c*x^2)^(7/3)),x,2,-3/272*AppellF1(17/3,7/3,7/3,20/3,1/2*(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(c*(d+e*x)),1/2*(2*d-e*(b+sqrt(b^2-4*a*c))/c)/(d+e*x))*(e*(b+2*c*x-sqrt(b^2-4*a*c))/(c*(d+e*x)))^(7/3)*(e*(b+2*c*x+sqrt(b^2-4*a*c))/(c*(d+e*x)))^(7/3)/(2^(2/3)*e*(d+e*x)*(a+b*x+c*x^2)^(7/3))],
[1/((d+e*x)^3*(a+b*x+c*x^2)^(7/3)),x,2,-3/320*AppellF1(20/3,7/3,7/3,23/3,1/2*(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(c*(d+e*x)),1/2*(2*d-e*(b+sqrt(b^2-4*a*c))/c)/(d+e*x))*(e*(b+2*c*x-sqrt(b^2-4*a*c))/(c*(d+e*x)))^(7/3)*(e*(b+2*c*x+sqrt(b^2-4*a*c))/(c*(d+e*x)))^(7/3)/(2^(2/3)*e*(d+e*x)^2*(a+b*x+c*x^2)^(7/3))],
[1/((d+e*x)*(c^2*d^2-b*c*d*e+b^2*e^2+3*b*c*e^2*x+3*c^2*e^2*x^2)^(1/3)),x,1,-1/2*log(d+e*x)/(e*(2*c*d-b*e)^(2/3))+1/2*log(3*c*e^2*(c*d-b*e)-3*c^2*e^3*x-3*c*e^2*(2*c*d-b*e)^(1/3)*(c^2*d^2-b*c*d*e+b^2*e^2+3*b*c*e^2*x+3*c^2*e^2*x^2)^(1/3))/(e*(2*c*d-b*e)^(2/3))-arctan(1/sqrt(3)+2*(c*d-b*e-c*e*x)/((2*c*d-b*e)^(1/3)*(c^2*d^2-b*c*d*e+b^2*e^2+3*b*c*e^2*x+3*c^2*e^2*x^2)^(1/3)*sqrt(3)))/(e*(2*c*d-b*e)^(2/3)*sqrt(3))],
[(2+3*x)^3/(52-54*x+27*x^2)^(1/3),x,7,1/30*(2+3*x)^2*(52-54*x+27*x^2)^(2/3)+1/7*(27+8*x)*(52-54*x+27*x^2)^(2/3)+9000/7*5^(1/3)*(1-x)/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))+50/189*5^(5/6)*(30-10^(1/3)*(2700+(-54+54*x)^2)^(1/3))*EllipticF((-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1+sqrt(3)))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((900+30*10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+10^(2/3)*(2700+(-54+54*x)^2)^(2/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2)/(3^(3/4)*(1-x)*sqrt((-30+10^(1/3)*(2700+(-54+54*x)^2)^(1/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2))-25/189*5^(5/6)*(30-10^(1/3)*(2700+(-54+54*x)^2)^(1/3))*EllipticE((-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1+sqrt(3)))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((900+30*10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+10^(2/3)*(2700+(-54+54*x)^2)^(2/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2)*sqrt(1/2*(2+sqrt(3)))/(3^(1/4)*(1-x)*sqrt((-30+10^(1/3)*(2700+(-54+54*x)^2)^(1/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2))],
[(2+3*x)^2/(52-54*x+27*x^2)^(1/3),x,8,25/42*(52-54*x+27*x^2)^(2/3)+1/21*(2+3*x)*(52-54*x+27*x^2)^(2/3)+2700/7*5^(1/3)*(1-x)/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))+5/63*5^(5/6)*(30-10^(1/3)*(2700+(-54+54*x)^2)^(1/3))*EllipticF((-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1+sqrt(3)))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((900+30*10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+10^(2/3)*(2700+(-54+54*x)^2)^(2/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2)/(3^(3/4)*(1-x)*sqrt((-30+10^(1/3)*(2700+(-54+54*x)^2)^(1/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2))-5/126*5^(5/6)*(30-10^(1/3)*(2700+(-54+54*x)^2)^(1/3))*EllipticE((-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1+sqrt(3)))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((900+30*10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+10^(2/3)*(2700+(-54+54*x)^2)^(2/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2)*sqrt(1/2*(2+sqrt(3)))/(3^(1/4)*(1-x)*sqrt((-30+10^(1/3)*(2700+(-54+54*x)^2)^(1/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2))],
[(2+3*x)/(52-54*x+27*x^2)^(1/3),x,6,1/12*(52-54*x+27*x^2)^(2/3)+90*5^(1/3)*(1-x)/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))+1/54*5^(5/6)*(30-10^(1/3)*(2700+(-54+54*x)^2)^(1/3))*EllipticF((-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1+sqrt(3)))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((900+30*10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+10^(2/3)*(2700+(-54+54*x)^2)^(2/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2)/(3^(3/4)*(1-x)*sqrt((-30+10^(1/3)*(2700+(-54+54*x)^2)^(1/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2))-1/108*5^(5/6)*(30-10^(1/3)*(2700+(-54+54*x)^2)^(1/3))*EllipticE((-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1+sqrt(3)))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((900+30*10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+10^(2/3)*(2700+(-54+54*x)^2)^(2/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2)*sqrt(1/2*(2+sqrt(3)))/(3^(1/4)*(1-x)*sqrt((-30+10^(1/3)*(2700+(-54+54*x)^2)^(1/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2))],
[1/((2+3*x)*(52-54*x+27*x^2)^(1/3)),x,1,-1/6*log(2+3*x)/10^(2/3)+1/6*log(216-81*x-27*10^(1/3)*(52-54*x+27*x^2)^(1/3))/10^(2/3)-1/3*arctan(1/sqrt(3)+2^(2/3)*(8-3*x)/(5^(1/3)*(52-54*x+27*x^2)^(1/3)*sqrt(3)))/(10^(2/3)*sqrt(3))],
[1/((2+3*x)^2*(52-54*x+27*x^2)^(1/3)),x,8,-1/300*(52-54*x+27*x^2)^(2/3)/(2+3*x)-1/60*log(2+3*x)/10^(2/3)+1/60*log(216-81*x-27*10^(1/3)*(52-54*x+27*x^2)^(1/3))/10^(2/3)+9/10*(1-x)/(5^(2/3)*(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3))))-1/30*arctan(1/sqrt(3)+2^(2/3)*(8-3*x)/(5^(1/3)*(52-54*x+27*x^2)^(1/3)*sqrt(3)))/(10^(2/3)*sqrt(3))+1/5400*(30-10^(1/3)*(2700+(-54+54*x)^2)^(1/3))*EllipticF((-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1+sqrt(3)))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((900+30*10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+10^(2/3)*(2700+(-54+54*x)^2)^(2/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2)/(3^(3/4)*5^(1/6)*(1-x)*sqrt((-30+10^(1/3)*(2700+(-54+54*x)^2)^(1/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2))-1/10800*(30-10^(1/3)*(2700+(-54+54*x)^2)^(1/3))*EllipticE((-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1+sqrt(3)))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((900+30*10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+10^(2/3)*(2700+(-54+54*x)^2)^(2/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2)*sqrt(1/2*(2+sqrt(3)))/(3^(1/4)*5^(1/6)*(1-x)*sqrt((-30+10^(1/3)*(2700+(-54+54*x)^2)^(1/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2))],
[1/((2+3*x)^3*(52-54*x+27*x^2)^(1/3)),x,9,-1/600*(52-54*x+27*x^2)^(2/3)/(2+3*x)^2-1/1500*(52-54*x+27*x^2)^(2/3)/(2+3*x)-1/600*log(2+3*x)/10^(2/3)+1/600*log(216-81*x-27*10^(1/3)*(52-54*x+27*x^2)^(1/3))/10^(2/3)+9/50*(1-x)/(5^(2/3)*(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3))))-1/300*arctan(1/sqrt(3)+2^(2/3)*(8-3*x)/(5^(1/3)*(52-54*x+27*x^2)^(1/3)*sqrt(3)))/(10^(2/3)*sqrt(3))+1/27000*(30-10^(1/3)*(2700+(-54+54*x)^2)^(1/3))*EllipticF((-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1+sqrt(3)))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((900+30*10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+10^(2/3)*(2700+(-54+54*x)^2)^(2/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2)/(3^(3/4)*5^(1/6)*(1-x)*sqrt((-30+10^(1/3)*(2700+(-54+54*x)^2)^(1/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2))-1/54000*(30-10^(1/3)*(2700+(-54+54*x)^2)^(1/3))*EllipticE((-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1+sqrt(3)))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((900+30*10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+10^(2/3)*(2700+(-54+54*x)^2)^(2/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2)*sqrt(1/2*(2+sqrt(3)))/(3^(1/4)*5^(1/6)*(1-x)*sqrt((-30+10^(1/3)*(2700+(-54+54*x)^2)^(1/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30*(1-sqrt(3)))^2))],
[(2+3*x)^3/(28+54*x+27*x^2)^(1/3),x,7,1/30*(2+3*x)^2*(28+54*x+27*x^2)^(2/3)-1/35*(1+8*x)*(28+54*x+27*x^2)^(2/3)+72/7*(1+x)/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))+4/63*(6-2^(1/3)*(108+(54+54*x)^2)^(1/3))*EllipticF((-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1+sqrt(3)))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((1+(28+54*x+27*x^2)^(1/3)+(28+54*x+27*x^2)^(2/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2)/(3^(3/4)*(1+x)*sqrt((-6+2^(1/3)*(108+(54+54*x)^2)^(1/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2))-1/63*(6-2^(1/3)*(108+(54+54*x)^2)^(1/3))*EllipticE((-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1+sqrt(3)))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((1+(28+54*x+27*x^2)^(1/3)+(28+54*x+27*x^2)^(2/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2)*sqrt(2*(2+sqrt(3)))/(3^(1/4)*(1+x)*sqrt((-6+2^(1/3)*(108+(54+54*x)^2)^(1/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2))],
[(2+3*x)^2/(28+54*x+27*x^2)^(1/3),x,7,-5/42*(28+54*x+27*x^2)^(2/3)+1/21*(2+3*x)*(28+54*x+27*x^2)^(2/3)-108/7*(1+x)/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))-2/21*(6-2^(1/3)*(108+(54+54*x)^2)^(1/3))*EllipticF((-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1+sqrt(3)))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((1+(28+54*x+27*x^2)^(1/3)+(28+54*x+27*x^2)^(2/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2)/(3^(3/4)*(1+x)*sqrt((-6+2^(1/3)*(108+(54+54*x)^2)^(1/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2))+1/21*(6-2^(1/3)*(108+(54+54*x)^2)^(1/3))*EllipticE((-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1+sqrt(3)))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((1+(28+54*x+27*x^2)^(1/3)+(28+54*x+27*x^2)^(2/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(1/4)*(1+x)*sqrt(2)*sqrt((-6+2^(1/3)*(108+(54+54*x)^2)^(1/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2))],
[(2+3*x)/(28+54*x+27*x^2)^(1/3),x,6,1/12*(28+54*x+27*x^2)^(2/3)+18*(1+x)/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))+1/9*(6-2^(1/3)*(108+(54+54*x)^2)^(1/3))*EllipticF((-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1+sqrt(3)))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((1+(28+54*x+27*x^2)^(1/3)+(28+54*x+27*x^2)^(2/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2)/(3^(3/4)*(1+x)*sqrt((-6+2^(1/3)*(108+(54+54*x)^2)^(1/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2))-1/18*(6-2^(1/3)*(108+(54+54*x)^2)^(1/3))*EllipticE((-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1+sqrt(3)))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((1+(28+54*x+27*x^2)^(1/3)+(28+54*x+27*x^2)^(2/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(1/4)*(1+x)*sqrt(2)*sqrt((-6+2^(1/3)*(108+(54+54*x)^2)^(1/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2))],
[1/((2+3*x)*(28+54*x+27*x^2)^(1/3)),x,1,-1/6*log(2+3*x)/2^(2/3)+1/6*log(-108-81*x+27*2^(1/3)*(28+54*x+27*x^2)^(1/3))/2^(2/3)-1/3*arctan(1/sqrt(3)+2^(2/3)*(4+3*x)/((28+54*x+27*x^2)^(1/3)*sqrt(3)))/(2^(2/3)*sqrt(3))],
[1/((2+3*x)^2*(28+54*x+27*x^2)^(1/3)),x,9,-1/12*(28+54*x+27*x^2)^(2/3)/(2+3*x)+1/12*log(2+3*x)/2^(2/3)-1/12*log(-108-81*x+27*2^(1/3)*(28+54*x+27*x^2)^(1/3))/2^(2/3)-9/2*(1+x)/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))+1/6*arctan(1/sqrt(3)+2^(2/3)*(4+3*x)/((28+54*x+27*x^2)^(1/3)*sqrt(3)))/(2^(2/3)*sqrt(3))-1/36*(6-2^(1/3)*(108+(54+54*x)^2)^(1/3))*EllipticF((-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1+sqrt(3)))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((1+(28+54*x+27*x^2)^(1/3)+(28+54*x+27*x^2)^(2/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2)/(3^(3/4)*(1+x)*sqrt((-6+2^(1/3)*(108+(54+54*x)^2)^(1/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2))+1/72*(6-2^(1/3)*(108+(54+54*x)^2)^(1/3))*EllipticE((-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1+sqrt(3)))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((1+(28+54*x+27*x^2)^(1/3)+(28+54*x+27*x^2)^(2/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(1/4)*(1+x)*sqrt(2)*sqrt((-6+2^(1/3)*(108+(54+54*x)^2)^(1/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2))],
[1/((2+3*x)^3*(28+54*x+27*x^2)^(1/3)),x,9,-1/24*(28+54*x+27*x^2)^(2/3)/(2+3*x)^2+1/12*(28+54*x+27*x^2)^(2/3)/(2+3*x)-1/24*log(2+3*x)/2^(2/3)+1/24*log(-108-81*x+27*2^(1/3)*(28+54*x+27*x^2)^(1/3))/2^(2/3)+9/2*(1+x)/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))-1/12*arctan(1/sqrt(3)+2^(2/3)*(4+3*x)/((28+54*x+27*x^2)^(1/3)*sqrt(3)))/(2^(2/3)*sqrt(3))+1/36*(6-2^(1/3)*(108+(54+54*x)^2)^(1/3))*EllipticF((-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1+sqrt(3)))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((1+(28+54*x+27*x^2)^(1/3)+(28+54*x+27*x^2)^(2/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2)/(3^(3/4)*(1+x)*sqrt((-6+2^(1/3)*(108+(54+54*x)^2)^(1/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2))-1/72*(6-2^(1/3)*(108+(54+54*x)^2)^(1/3))*EllipticE((-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1+sqrt(3)))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((1+(28+54*x+27*x^2)^(1/3)+(28+54*x+27*x^2)^(2/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(1/4)*(1+x)*sqrt(2)*sqrt((-6+2^(1/3)*(108+(54+54*x)^2)^(1/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6*(1-sqrt(3)))^2))],

# Integrands of the form (d+e x)^m (a+b x+c x^2)^(p/3) when c^2 d^2-b c d e-2 b^2 e^2+9 a c e^2=0
[1/((d+e*x)*(-c^2*d^2+b*c*d*e+2*b^2*e^2+9*b*c*e^2*x+9*c^2*e^2*x^2)^(1/3)),x,2,-1/2*(-c*e*(c*d-2*b*e)+3*c^2*e^2*x)^(1/3)*(c*e*(c*d+b*e)+3*c^2*e^2*x)^(1/3)*log(d+e*x)/(2^(1/3)*c^(2/3)*e^(5/3)*(2*c*d-b*e)^(2/3)*(-(c*d-2*b*e)*(c*d+b*e)+9*b*c*e^2*x+9*c^2*e^2*x^2)^(1/3))+3/4*(-c*e*(c*d-2*b*e)+3*c^2*e^2*x)^(1/3)*(c*e*(c*d+b*e)+3*c^2*e^2*x)^(1/3)*log(-(3/2)^(1/3)*(-c*e*(c*d-2*b*e)+3*c^2*e^2*x)^(2/3)/(c^(1/3)*e^(1/3)*(2*c*d-b*e)^(1/3))-6^(1/3)*(c*e*(c*d+b*e)+3*c^2*e^2*x)^(1/3))/(2^(1/3)*c^(2/3)*e^(5/3)*(2*c*d-b*e)^(2/3)*(-(c*d-2*b*e)*(c*d+b*e)+9*b*c*e^2*x+9*c^2*e^2*x^2)^(1/3))-1/2*(-c*e*(c*d-2*b*e)+3*c^2*e^2*x)^(1/3)*(c*e*(c*d+b*e)+3*c^2*e^2*x)^(1/3)*arctan(1/sqrt(3)-2^(1/3)*(-c*e*(c*d-2*b*e)+3*c^2*e^2*x)^(2/3)/(c^(1/3)*e^(1/3)*(2*c*d-b*e)^(1/3)*(c*e*(c*d+b*e)+3*c^2*e^2*x)^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*c^(2/3)*e^(5/3)*(2*c*d-b*e)^(2/3)*(-(c*d-2*b*e)*(c*d+b*e)+9*b*c*e^2*x+9*c^2*e^2*x^2)^(1/3))],

# Integrands of the form (d+e x)^m (a+b x+c x^2)^(p/4)

# p>0
[(d+e*x)^3*(a+b*x+c*x^2)^(1/4),x,5,1/168*(2*c*d-b*e)*(28*c^2*d^2+13*b^2*e^2-4*c*e*(7*b*d+6*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)/c^4+2/9*e*(d+e*x)^2*(a+b*x+c*x^2)^(5/4)/c+1/630*e*(616*c^2*d^2+117*b^2*e^2-2*c*e*(243*b*d+56*a*e)+130*c*e*(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(5/4)/c^3-1/336*(b^2-4*a*c)^(5/4)*(2*c*d-b*e)*(28*c^2*d^2+13*b^2*e^2-4*c*e*(7*b*d+6*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(17/4)*(b+2*c*x)*sqrt(2))],
[(d+e*x)^2*(a+b*x+c*x^2)^(1/4),x,5,1/84*(28*c^2*d^2+9*b^2*e^2-4*c*e*(7*b*d+2*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)/c^3+9/35*e*(2*c*d-b*e)*(a+b*x+c*x^2)^(5/4)/c^2+2/7*e*(d+e*x)*(a+b*x+c*x^2)^(5/4)/c-1/168*(b^2-4*a*c)^(5/4)*(28*c^2*d^2+9*b^2*e^2-4*c*e*(7*b*d+2*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(13/4)*(b+2*c*x)*sqrt(2))],
[(d+e*x)*(a+b*x+c*x^2)^(1/4),x,4,1/6*(2*c*d-b*e)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)/c^2+2/5*e*(a+b*x+c*x^2)^(5/4)/c-1/12*(b^2-4*a*c)^(5/4)*(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(9/4)*(b+2*c*x)*sqrt(2))],
[(a+b*x+c*x^2)^(1/4),x,3,1/3*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)/c-1/6*(b^2-4*a*c)^(5/4)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(5/4)*(b+2*c*x)*sqrt(2))],
[(a+b*x+c*x^2)^(1/4)/(d+e*x),x,19,2*(a+b*x+c*x^2)^(1/4)/e-(-b^2+4*a*c)^(3/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*arctan((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(3/4)*e^(3/2)*(a+b*x+c*x^2)^(3/4))-(-b^2+4*a*c)^(3/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*arctanh((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(3/4)*e^(3/2)*(a+b*x+c*x^2)^(3/4))-(b^2-4*a*c)*(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),-1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(c*e^2*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)*sqrt(2))-(b^2-4*a*c)*(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(c*e^2*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)*sqrt(2))-(b^2-4*a*c)^(1/4)*(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(1/4)*e^2*(b+2*c*x)*sqrt(2))],
[(a+b*x+c*x^2)^(1/4)/(d+e*x)^2,x,19,-(a+b*x+c*x^2)^(1/4)/(e*(d+e*x))+1/4*(-b^2+4*a*c)^(3/4)*(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*arctan((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(3/4)*e^(3/2)*(c*d^2-b*d*e+a*e^2)^(3/4)*(a+b*x+c*x^2)^(3/4))+1/4*(-b^2+4*a*c)^(3/4)*(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*arctanh((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(3/4)*e^(3/2)*(c*d^2-b*d*e+a*e^2)^(3/4)*(a+b*x+c*x^2)^(3/4))+1/4*(b^2-4*a*c)*(2*c*d-b*e)^2*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),-1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(c*e^2*(c*d^2-b*d*e+a*e^2)*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)*sqrt(2))+1/4*(b^2-4*a*c)*(2*c*d-b*e)^2*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(c*e^2*(c*d^2-b*d*e+a*e^2)*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)*sqrt(2))+c^(3/4)*(b^2-4*a*c)^(1/4)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(e^2*(b+2*c*x)*sqrt(2))],
[(d+e*x)^3*(a+b*x+c*x^2)^(3/4),x,7,1/120*(2*c*d-b*e)*(12*c^2*d^2+5*b^2*e^2-4*c*e*(3*b*d+2*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)/c^4+2/11*e*(d+e*x)^2*(a+b*x+c*x^2)^(7/4)/c+1/462*e*(312*c^2*d^2+55*b^2*e^2-2*c*e*(121*b*d+24*a*e)+70*c*e*(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(7/4)/c^3-1/80*(2*c*d-b*e)*(12*c^2*d^2+5*b^2*e^2-4*c*e*(3*b*d+2*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(b^2-4*a*c)/(c^(9/2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))+1/80*(b^2-4*a*c)^(7/4)*(2*c*d-b*e)*(12*c^2*d^2+5*b^2*e^2-4*c*e*(3*b*d+2*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(19/4)*(b+2*c*x)*sqrt(2))-1/160*(b^2-4*a*c)^(7/4)*(2*c*d-b*e)*(12*c^2*d^2+5*b^2*e^2-4*c*e*(3*b*d+2*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(19/4)*(b+2*c*x)*sqrt(2))],
[(d+e*x)^2*(a+b*x+c*x^2)^(3/4),x,7,1/180*(36*c^2*d^2+11*b^2*e^2-4*c*e*(9*b*d+2*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)/c^3+11/63*e*(2*c*d-b*e)*(a+b*x+c*x^2)^(7/4)/c^2+2/9*e*(d+e*x)*(a+b*x+c*x^2)^(7/4)/c-1/120*(36*c^2*d^2+11*b^2*e^2-4*c*e*(9*b*d+2*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(b^2-4*a*c)/(c^(7/2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))+1/120*(b^2-4*a*c)^(7/4)*(36*c^2*d^2+11*b^2*e^2-4*c*e*(9*b*d+2*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(15/4)*(b+2*c*x)*sqrt(2))-1/240*(b^2-4*a*c)^(7/4)*(36*c^2*d^2+11*b^2*e^2-4*c*e*(9*b*d+2*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(15/4)*(b+2*c*x)*sqrt(2))],
[(d+e*x)*(a+b*x+c*x^2)^(3/4),x,6,1/10*(2*c*d-b*e)*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)/c^2+2/7*e*(a+b*x+c*x^2)^(7/4)/c-3/20*(2*c*d-b*e)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(b^2-4*a*c)/(c^(5/2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))+3/20*(b^2-4*a*c)^(7/4)*(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(11/4)*(b+2*c*x)*sqrt(2))-3/40*(b^2-4*a*c)^(7/4)*(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(11/4)*(b+2*c*x)*sqrt(2))],
[(a+b*x+c*x^2)^(3/4),x,5,1/5*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)/c-3/10*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(b^2-4*a*c)/(c^(3/2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))+3/10*(b^2-4*a*c)^(7/4)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(7/4)*(b+2*c*x)*sqrt(2))-3/20*(b^2-4*a*c)^(7/4)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(7/4)*(b+2*c*x)*sqrt(2))],
[(a+b*x+c*x^2)^(3/4)/(d+e*x),x,20,2/3*(a+b*x+c*x^2)^(3/4)/e+(-b^2+4*a*c)^(1/4)*(c*d^2-b*d*e+a*e^2)^(3/4)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*arctan((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(1/4)*e^(5/2)*(a+b*x+c*x^2)^(1/4))-(-b^2+4*a*c)^(1/4)*(c*d^2-b*d*e+a*e^2)^(3/4)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*arctanh((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(1/4)*e^(5/2)*(a+b*x+c*x^2)^(1/4))-(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),-1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt(-b^2+4*a*c)*sqrt(c*d^2-b*d*e+a*e^2)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(e^3*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(2)*sqrt(c))+(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt(-b^2+4*a*c)*sqrt(c*d^2-b*d*e+a*e^2)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(e^3*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(2)*sqrt(c))-(2*c*d-b*e)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)/(e^2*sqrt(c)*sqrt(b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))+(b^2-4*a*c)^(3/4)*(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(3/4)*e^2*(b+2*c*x)*sqrt(2))-1/2*(b^2-4*a*c)^(3/4)*(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(3/4)*e^2*(b+2*c*x)*sqrt(2))],
[(a+b*x+c*x^2)^(3/4)/(d+e*x)^2,x,20,-(a+b*x+c*x^2)^(3/4)/(e*(d+e*x))-3/4*(-b^2+4*a*c)^(1/4)*(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*arctan((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(1/4)*e^(5/2)*(c*d^2-b*d*e+a*e^2)^(1/4)*(a+b*x+c*x^2)^(1/4))+3/4*(-b^2+4*a*c)^(1/4)*(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*arctanh((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(1/4)*e^(5/2)*(c*d^2-b*d*e+a*e^2)^(1/4)*(a+b*x+c*x^2)^(1/4))+3/4*(2*c*d-b*e)^2*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),-1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt(-b^2+4*a*c)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(e^3*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(2)*sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2))-3/4*(2*c*d-b*e)^2*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt(-b^2+4*a*c)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(e^3*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(2)*sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2))+3*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(c)/(e^2*sqrt(b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))-3*c^(1/4)*(b^2-4*a*c)^(3/4)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(e^2*(b+2*c*x)*sqrt(2))+3/2*c^(1/4)*(b^2-4*a*c)^(3/4)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(e^2*(b+2*c*x)*sqrt(2))],
[(d+e*x)^3*(a+b*x+c*x^2)^(5/4),x,6,-5/7392*(b^2-4*a*c)*(2*c*d-b*e)*(44*c^2*d^2+17*b^2*e^2-4*c*e*(11*b*d+6*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)/c^5+1/616*(2*c*d-b*e)*(44*c^2*d^2+17*b^2*e^2-4*c*e*(11*b*d+6*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(5/4)/c^4+2/13*e*(d+e*x)^2*(a+b*x+c*x^2)^(9/4)/c+1/2574*e*(1320*c^2*d^2+221*b^2*e^2-2*c*e*(507*b*d+88*a*e)+306*c*e*(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(9/4)/c^3+5/14784*(b^2-4*a*c)^(9/4)*(2*c*d-b*e)*(44*c^2*d^2+17*b^2*e^2-4*c*e*(11*b*d+6*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(21/4)*(b+2*c*x)*sqrt(2))],
[(d+e*x)^2*(a+b*x+c*x^2)^(5/4),x,6,-5/3696*(b^2-4*a*c)*(44*c^2*d^2+13*b^2*e^2-4*c*e*(11*b*d+2*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)/c^4+1/308*(44*c^2*d^2+13*b^2*e^2-4*c*e*(11*b*d+2*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(5/4)/c^3+13/99*e*(2*c*d-b*e)*(a+b*x+c*x^2)^(9/4)/c^2+2/11*e*(d+e*x)*(a+b*x+c*x^2)^(9/4)/c+5/7392*(b^2-4*a*c)^(9/4)*(44*c^2*d^2+13*b^2*e^2-4*c*e*(11*b*d+2*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(17/4)*(b+2*c*x)*sqrt(2))],
[(d+e*x)*(a+b*x+c*x^2)^(5/4),x,5,-5/168*(b^2-4*a*c)*(2*c*d-b*e)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)/c^3+1/14*(2*c*d-b*e)*(b+2*c*x)*(a+b*x+c*x^2)^(5/4)/c^2+2/9*e*(a+b*x+c*x^2)^(9/4)/c+5/336*(b^2-4*a*c)^(9/4)*(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(13/4)*(b+2*c*x)*sqrt(2))],
[(a+b*x+c*x^2)^(5/4),x,4,-5/84*(b^2-4*a*c)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)/c^2+1/7*(b+2*c*x)*(a+b*x+c*x^2)^(5/4)/c+5/168*(b^2-4*a*c)^(9/4)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(9/4)*(b+2*c*x)*sqrt(2))],
[(a+b*x+c*x^2)^(5/4)/(d+e*x),x,20,1/6*(12*c^2*d^2+b^2*e^2-2*c*e*(7*b*d-6*a*e)-2*c*e*(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(1/4)/(c*e^3)+2/5*(a+b*x+c*x^2)^(5/4)/e-(-b^2+4*a*c)^(3/4)*(c*d^2-b*d*e+a*e^2)^(5/4)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*arctan((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(3/4)*e^(7/2)*(a+b*x+c*x^2)^(3/4))-(-b^2+4*a*c)^(3/4)*(c*d^2-b*d*e+a*e^2)^(5/4)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*arctanh((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(3/4)*e^(7/2)*(a+b*x+c*x^2)^(3/4))-(b^2-4*a*c)*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),-1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(c*e^4*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)*sqrt(2))-(b^2-4*a*c)*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(c*e^4*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)*sqrt(2))-1/12*(b^2-4*a*c)^(1/4)*(2*c*d-b*e)*(12*c^2*d^2-b^2*e^2-4*c*e*(3*b*d-4*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(5/4)*e^4*(b+2*c*x)*sqrt(2))],
[(a+b*x+c*x^2)^(5/4)/(d+e*x)^2,x,20,-5/3*(3*c*d-2*b*e-c*e*x)*(a+b*x+c*x^2)^(1/4)/e^3-(a+b*x+c*x^2)^(5/4)/(e*(d+e*x))+5/4*(-b^2+4*a*c)^(3/4)*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^(1/4)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*arctan((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(3/4)*e^(7/2)*(a+b*x+c*x^2)^(3/4))+5/4*(-b^2+4*a*c)^(3/4)*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^(1/4)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*arctanh((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(3/4)*e^(7/2)*(a+b*x+c*x^2)^(3/4))+5/4*(b^2-4*a*c)*(2*c*d-b*e)^2*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),-1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(c*e^4*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)*sqrt(2))+5/4*(b^2-4*a*c)*(2*c*d-b*e)^2*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(c*e^4*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)*sqrt(2))+5/6*(b^2-4*a*c)^(1/4)*(6*c^2*d^2+b^2*e^2-2*c*e*(3*b*d-a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(1/4)*e^4*(b+2*c*x)*sqrt(2))],

# p<0
[(d+e*x)^3/(a+b*x+c*x^2)^(1/4),x,6,2/7*e*(d+e*x)^2*(a+b*x+c*x^2)^(3/4)/c+1/210*e*(360*c^2*d^2+77*b^2*e^2-2*c*e*(147*b*d+40*a*e)+66*c*e*(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(3/4)/c^3+1/20*(2*c*d-b*e)*(20*c^2*d^2+11*b^2*e^2-4*c*e*(5*b*d+6*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)/(c^(7/2)*sqrt(b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))-1/20*(b^2-4*a*c)^(3/4)*(2*c*d-b*e)*(20*c^2*d^2+11*b^2*e^2-4*c*e*(5*b*d+6*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(15/4)*(b+2*c*x)*sqrt(2))+1/40*(b^2-4*a*c)^(3/4)*(2*c*d-b*e)*(20*c^2*d^2+11*b^2*e^2-4*c*e*(5*b*d+6*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(15/4)*(b+2*c*x)*sqrt(2))],
[(d+e*x)^2/(a+b*x+c*x^2)^(1/4),x,6,7/15*e*(2*c*d-b*e)*(a+b*x+c*x^2)^(3/4)/c^2+2/5*e*(d+e*x)*(a+b*x+c*x^2)^(3/4)/c+1/10*(20*c^2*d^2+7*b^2*e^2-4*c*e*(5*b*d+2*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)/(c^(5/2)*sqrt(b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))-1/10*(b^2-4*a*c)^(3/4)*(20*c^2*d^2+7*b^2*e^2-4*c*e*(5*b*d+2*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(11/4)*(b+2*c*x)*sqrt(2))+1/20*(b^2-4*a*c)^(3/4)*(20*c^2*d^2+7*b^2*e^2-4*c*e*(5*b*d+2*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(11/4)*(b+2*c*x)*sqrt(2))],
[(d+e*x)/(a+b*x+c*x^2)^(1/4),x,5,2/3*e*(a+b*x+c*x^2)^(3/4)/c+(2*c*d-b*e)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)/(c^(3/2)*sqrt(b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))-(b^2-4*a*c)^(3/4)*(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(7/4)*(b+2*c*x)*sqrt(2))+1/2*(b^2-4*a*c)^(3/4)*(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(7/4)*(b+2*c*x)*sqrt(2))],
[1/(a+b*x+c*x^2)^(1/4),x,4,2*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)/(sqrt(c)*sqrt(b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))+(b^2-4*a*c)^(3/4)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(3/4)*(b+2*c*x)*sqrt(2))-(b^2-4*a*c)^(3/4)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*sqrt(2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(3/4)*(b+2*c*x))],
[1/((d+e*x)*(a+b*x+c*x^2)^(1/4)),x,14,(-b^2+4*a*c)^(1/4)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*arctan((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(e))-(-b^2+4*a*c)^(1/4)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*arctanh((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(e))-(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),-1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt(-b^2+4*a*c)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(e*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(2)*sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2))+(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt(-b^2+4*a*c)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(e*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(2)*sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2))],
[1/((d+e*x)^2*(a+b*x+c*x^2)^(1/4)),x,20,-e*(a+b*x+c*x^2)^(3/4)/((c*d^2-b*d*e+a*e^2)*(d+e*x))+1/4*(-b^2+4*a*c)^(1/4)*(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*arctan((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(5/4)*(a+b*x+c*x^2)^(1/4)*sqrt(e))-1/4*(-b^2+4*a*c)^(1/4)*(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*arctanh((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(5/4)*(a+b*x+c*x^2)^(1/4)*sqrt(e))-1/4*(2*c*d-b*e)^2*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),-1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt(-b^2+4*a*c)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(e*(c*d^2-b*d*e+a*e^2)^(3/2)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(2)*sqrt(c))+1/4*(2*c*d-b*e)^2*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt(-b^2+4*a*c)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(e*(c*d^2-b*d*e+a*e^2)^(3/2)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(2)*sqrt(c))+(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(c)/((c*d^2-b*d*e+a*e^2)*sqrt(b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))-c^(1/4)*(b^2-4*a*c)^(3/4)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/((c*d^2-b*d*e+a*e^2)*(b+2*c*x)*sqrt(2))+1/2*c^(1/4)*(b^2-4*a*c)^(3/4)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/((c*d^2-b*d*e+a*e^2)*(b+2*c*x)*sqrt(2))],
[1/((d+e*x)^3*(a+b*x+c*x^2)^(1/4)),x,21,-1/2*e*(a+b*x+c*x^2)^(3/4)/((c*d^2-b*d*e+a*e^2)*(d+e*x)^2)-5/8*e*(2*c*d-b*e)*(a+b*x+c*x^2)^(3/4)/((c*d^2-b*d*e+a*e^2)^2*(d+e*x))+1/32*(-b^2+4*a*c)^(1/4)*(12*c^2*d^2+5*b^2*e^2-4*c*e*(3*b*d+2*a*e))*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*arctan((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(9/4)*(a+b*x+c*x^2)^(1/4)*sqrt(e))-1/32*(-b^2+4*a*c)^(1/4)*(12*c^2*d^2+5*b^2*e^2-4*c*e*(3*b*d+2*a*e))*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*arctanh((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(9/4)*(a+b*x+c*x^2)^(1/4)*sqrt(e))-1/32*(2*c*d-b*e)*(12*c^2*d^2+5*b^2*e^2-4*c*e*(3*b*d+2*a*e))*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),-1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt(-b^2+4*a*c)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(e*(c*d^2-b*d*e+a*e^2)^(5/2)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(2)*sqrt(c))+1/32*(2*c*d-b*e)*(12*c^2*d^2+5*b^2*e^2-4*c*e*(3*b*d+2*a*e))*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt(-b^2+4*a*c)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(e*(c*d^2-b*d*e+a*e^2)^(5/2)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(2)*sqrt(c))+5/8*(2*c*d-b*e)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(c)/((c*d^2-b*d*e+a*e^2)^2*sqrt(b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))-5/8*c^(1/4)*(b^2-4*a*c)^(3/4)*(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/((c*d^2-b*d*e+a*e^2)^2*(b+2*c*x)*sqrt(2))+5/16*c^(1/4)*(b^2-4*a*c)^(3/4)*(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/((c*d^2-b*d*e+a*e^2)^2*(b+2*c*x)*sqrt(2))],
[(d+e*x)^3/(a+b*x+c*x^2)^(3/4),x,4,2/5*e*(d+e*x)^2*(a+b*x+c*x^2)^(1/4)/c+1/10*e*(56*c^2*d^2+15*b^2*e^2-2*c*e*(25*b*d+8*a*e)+6*c*e*(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(1/4)/c^3+1/4*(b^2-4*a*c)^(1/4)*(2*c*d-b*e)*(4*c^2*d^2+3*b^2*e^2-4*c*e*(b*d+2*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(13/4)*(b+2*c*x)*sqrt(2))],
[(d+e*x)^2/(a+b*x+c*x^2)^(3/4),x,4,5/3*e*(2*c*d-b*e)*(a+b*x+c*x^2)^(1/4)/c^2+2/3*e*(d+e*x)*(a+b*x+c*x^2)^(1/4)/c+1/6*(b^2-4*a*c)^(1/4)*(12*c^2*d^2+5*b^2*e^2-4*c*e*(3*b*d+2*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(9/4)*(b+2*c*x)*sqrt(2))],
[(d+e*x)/(a+b*x+c*x^2)^(3/4),x,3,2*e*(a+b*x+c*x^2)^(1/4)/c+(b^2-4*a*c)^(1/4)*(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(5/4)*(b+2*c*x)*sqrt(2))],
[1/(a+b*x+c*x^2)^(3/4),x,2,(b^2-4*a*c)^(1/4)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*sqrt(2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(1/4)*(b+2*c*x))],
[1/((d+e*x)*(a+b*x+c*x^2)^(3/4)),x,15,-(-b^2+4*a*c)^(3/4)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*arctan((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))*sqrt(e)/(c^(3/4)*(c*d^2-b*d*e+a*e^2)^(3/4)*(a+b*x+c*x^2)^(3/4))-(-b^2+4*a*c)^(3/4)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*arctanh((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))*sqrt(e)/(c^(3/4)*(c*d^2-b*d*e+a*e^2)^(3/4)*(a+b*x+c*x^2)^(3/4))-(b^2-4*a*c)*(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),-1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(c*(c*d^2-b*d*e+a*e^2)*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)*sqrt(2))-(b^2-4*a*c)*(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(c*(c*d^2-b*d*e+a*e^2)*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)*sqrt(2))],
[1/((d+e*x)^2*(a+b*x+c*x^2)^(3/4)),x,19,-e*(a+b*x+c*x^2)^(1/4)/((c*d^2-b*d*e+a*e^2)*(d+e*x))-3/4*(-b^2+4*a*c)^(3/4)*(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*arctan((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))*sqrt(e)/(c^(3/4)*(c*d^2-b*d*e+a*e^2)^(7/4)*(a+b*x+c*x^2)^(3/4))-3/4*(-b^2+4*a*c)^(3/4)*(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*arctanh((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))*sqrt(e)/(c^(3/4)*(c*d^2-b*d*e+a*e^2)^(7/4)*(a+b*x+c*x^2)^(3/4))-3/4*(b^2-4*a*c)*(2*c*d-b*e)^2*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),-1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(c*(c*d^2-b*d*e+a*e^2)^2*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)*sqrt(2))-3/4*(b^2-4*a*c)*(2*c*d-b*e)^2*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(c*(c*d^2-b*d*e+a*e^2)^2*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)*sqrt(2))-c^(3/4)*(b^2-4*a*c)^(1/4)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/((c*d^2-b*d*e+a*e^2)*(b+2*c*x)*sqrt(2))],
[1/((d+e*x)^3*(a+b*x+c*x^2)^(3/4)),x,20,-1/2*e*(a+b*x+c*x^2)^(1/4)/((c*d^2-b*d*e+a*e^2)*(d+e*x)^2)-7/8*e*(2*c*d-b*e)*(a+b*x+c*x^2)^(1/4)/((c*d^2-b*d*e+a*e^2)^2*(d+e*x))-3/32*(-b^2+4*a*c)^(3/4)*(20*c^2*d^2+7*b^2*e^2-4*c*e*(5*b*d+2*a*e))*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*arctan((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))*sqrt(e)/(c^(3/4)*(c*d^2-b*d*e+a*e^2)^(11/4)*(a+b*x+c*x^2)^(3/4))-3/32*(-b^2+4*a*c)^(3/4)*(20*c^2*d^2+7*b^2*e^2-4*c*e*(5*b*d+2*a*e))*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*arctanh((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))*sqrt(e)/(c^(3/4)*(c*d^2-b*d*e+a*e^2)^(11/4)*(a+b*x+c*x^2)^(3/4))-3/32*(b^2-4*a*c)*(2*c*d-b*e)*(20*c^2*d^2+7*b^2*e^2-4*c*e*(5*b*d+2*a*e))*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),-1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(c*(c*d^2-b*d*e+a*e^2)^3*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)*sqrt(2))-3/32*(b^2-4*a*c)*(2*c*d-b*e)*(20*c^2*d^2+7*b^2*e^2-4*c*e*(5*b*d+2*a*e))*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(3/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/(c*(c*d^2-b*d*e+a*e^2)^3*(b+2*c*x)*(a+b*x+c*x^2)^(3/4)*sqrt(2))-7/8*c^(3/4)*(b^2-4*a*c)^(1/4)*(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/((c*d^2-b*d*e+a*e^2)^2*(b+2*c*x)*sqrt(2))],
[(d+e*x)^3/(a+b*x+c*x^2)^(5/4),x,6,-4*(d+e*x)^2*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(1/4))+2/3*e*(24*c^2*d^2+7*b^2*e^2-2*c*e*(9*b*d+8*a*e)+6*c*e*(2*c*d-b*e)*x)*(a+b*x+c*x^2)^(3/4)/(c^2*(b^2-4*a*c))+(2*c*d-b*e)*(4*c^2*d^2+7*b^2*e^2-4*c*e*(b*d+6*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)/(c^(5/2)*(b^2-4*a*c)^(3/2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))-(2*c*d-b*e)*(4*c^2*d^2+7*b^2*e^2-4*c*e*(b*d+6*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(11/4)*(b^2-4*a*c)^(1/4)*(b+2*c*x)*sqrt(2))+1/2*(2*c*d-b*e)*(4*c^2*d^2+7*b^2*e^2-4*c*e*(b*d+6*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(11/4)*(b^2-4*a*c)^(1/4)*(b+2*c*x)*sqrt(2))],
[(d+e*x)^2/(a+b*x+c*x^2)^(5/4),x,6,-4*(d+e*x)*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(1/4))+4*e*(2*c*d-b*e)*(a+b*x+c*x^2)^(3/4)/(c*(b^2-4*a*c))+2*(4*c^2*d^2+3*b^2*e^2-4*c*e*(b*d+2*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)/(c^(3/2)*(b^2-4*a*c)^(3/2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))+(4*c^2*d^2+3*b^2*e^2-4*c*e*(b*d+2*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(7/4)*(b^2-4*a*c)^(1/4)*(b+2*c*x)*sqrt(2))-(4*c^2*d^2+3*b^2*e^2-4*c*e*(b*d+2*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*sqrt(2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(7/4)*(b^2-4*a*c)^(1/4)*(b+2*c*x))],
[(d+e*x)/(a+b*x+c*x^2)^(5/4),x,5,-4*(b*d-2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(1/4))+4*(2*c*d-b*e)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)/((b^2-4*a*c)^(3/2)*sqrt(c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))-2*(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*sqrt(2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(3/4)*(b^2-4*a*c)^(1/4)*(b+2*c*x))+(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*sqrt(2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/(c^(3/4)*(b^2-4*a*c)^(1/4)*(b+2*c*x))],
[1/(a+b*x+c*x^2)^(5/4),x,5,-4*(b+2*c*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^(1/4))+8*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(c)/((b^2-4*a*c)^(3/2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))-4*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*sqrt(2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/((b^2-4*a*c)^(1/4)*(b+2*c*x))+2*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*sqrt(2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/((b^2-4*a*c)^(1/4)*(b+2*c*x))],
[1/((d+e*x)*(a+b*x+c*x^2)^(5/4)),x,20,-4*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(a+b*x+c*x^2)^(1/4))+(-b^2+4*a*c)^(1/4)*e^(3/2)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*arctan((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(5/4)*(a+b*x+c*x^2)^(1/4))-(-b^2+4*a*c)^(1/4)*e^(3/2)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*arctanh((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(5/4)*(a+b*x+c*x^2)^(1/4))-e*(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),-1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt(-b^2+4*a*c)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^(3/2)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(2)*sqrt(c))+e*(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt(-b^2+4*a*c)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^(3/2)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(2)*sqrt(c))+4*(2*c*d-b*e)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(c)/((b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))-2*c^(1/4)*(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*sqrt(2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/((b^2-4*a*c)^(1/4)*(c*d^2-b*d*e+a*e^2)*(b+2*c*x))+c^(1/4)*(2*c*d-b*e)*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*sqrt(2)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/((b^2-4*a*c)^(1/4)*(c*d^2-b*d*e+a*e^2)*(b+2*c*x))],
[1/((d+e*x)^2*(a+b*x+c*x^2)^(5/4)),x,21,-4*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(d+e*x)*(a+b*x+c*x^2)^(1/4))-e*(8*c^2*d^2+5*b^2*e^2-4*c*e*(2*b*d+3*a*e))*(a+b*x+c*x^2)^(3/4)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^2*(d+e*x))+5/4*(-b^2+4*a*c)^(1/4)*e^(3/2)*(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*arctan((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(9/4)*(a+b*x+c*x^2)^(1/4))-5/4*(-b^2+4*a*c)^(1/4)*e^(3/2)*(2*c*d-b*e)*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*arctanh((-b^2+4*a*c)^(1/4)*(1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4)*sqrt(e)/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(2)))/(c^(1/4)*(c*d^2-b*d*e+a*e^2)^(9/4)*(a+b*x+c*x^2)^(1/4))-5/4*e*(2*c*d-b*e)^2*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),-1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt(-b^2+4*a*c)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^(5/2)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(2)*sqrt(c))+5/4*e*(2*c*d-b*e)^2*(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^(1/4)*EllipticPi((1-(b+2*c*x)^2/(b^2-4*a*c))^(1/4),1/2*e*sqrt(-b^2+4*a*c)/(sqrt(c)*sqrt(c*d^2-b*d*e+a*e^2)),I)*sqrt(-b^2+4*a*c)*sqrt((b+2*c*x)^2/(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^(5/2)*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(2)*sqrt(c))+(8*c^2*d^2+5*b^2*e^2-4*c*e*(2*b*d+3*a*e))*(b+2*c*x)*(a+b*x+c*x^2)^(1/4)*sqrt(c)/((b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)^2*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c)))-c^(1/4)*(8*c^2*d^2+5*b^2*e^2-4*c*e*(2*b*d+3*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/((b^2-4*a*c)^(1/4)*(c*d^2-b*d*e+a*e^2)^2*(b+2*c*x)*sqrt(2))+1/2*c^(1/4)*(8*c^2*d^2+5*b^2*e^2-4*c*e*(2*b*d+3*a*e))*sqrt(cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))^2)/cos(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(a+b*x+c*x^2)^(1/4)*sqrt(2)/(b^2-4*a*c)^(1/4))),sqrt(1/2))*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))*sqrt((b+2*c*x)^2/((b^2-4*a*c)*(1+2*sqrt(c)*sqrt(a+b*x+c*x^2)/sqrt(b^2-4*a*c))^2))/((b^2-4*a*c)^(1/4)*(c*d^2-b*d*e+a*e^2)^2*(b+2*c*x)*sqrt(2))],

# Integrands of the form (d+e x)^(m/2) (a+b x+c x^2)^(p/4)
[1/((d+e*x)^(3/2)*(a+b*x+c*x^2)^(1/4)),x,1,2*hypergeom([-1/2,1/4],[1/2],-4*c*(d+e*x)*sqrt(b^2-4*a*c)/((b+2*c*x-sqrt(b^2-4*a*c))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*(b+2*c*x-sqrt(b^2-4*a*c))*((2*c*d-e*(b-sqrt(b^2-4*a*c)))*(b+2*c*x+sqrt(b^2-4*a*c))/((b+2*c*x-sqrt(b^2-4*a*c))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))))^(1/4)/((a+b*x+c*x^2)^(1/4)*(2*c*d-b*e+e*sqrt(b^2-4*a*c))*sqrt(d+e*x))],

# Integrands of the form (d+e x)^m (a+b x+c x^2)^p when m symbolic
[(d+e*x)^m*(a+b*x+c*x^2)^4,x,2,(c*d^2-b*d*e+a*e^2)^4*(d+e*x)^(1+m)/(e^9*(1+m))-4*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^3*(d+e*x)^(2+m)/(e^9*(2+m))+2*(c*d^2-b*d*e+a*e^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*(d+e*x)^(3+m)/(e^9*(3+m))-4*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*(d+e*x)^(4+m)/(e^9*(4+m))+(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(5*b*d-3*a*e)-20*c^3*d^2*e*(7*b*d-3*a*e)+6*c^2*e^2*(15*b^2*d^2-10*a*b*d*e+a^2*e^2))*(d+e*x)^(5+m)/(e^9*(5+m))-4*c*(2*c*d-b*e)*(7*c^2*d^2+b^2*e^2-c*e*(7*b*d-3*a*e))*(d+e*x)^(6+m)/(e^9*(6+m))+2*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(7*b*d-a*e))*(d+e*x)^(7+m)/(e^9*(7+m))-4*c^3*(2*c*d-b*e)*(d+e*x)^(8+m)/(e^9*(8+m))+c^4*(d+e*x)^(9+m)/(e^9*(9+m))],
[(d+e*x)^m*(a+b*x+c*x^2)^3,x,2,(c*d^2-b*d*e+a*e^2)^3*(d+e*x)^(1+m)/(e^7*(1+m))-3*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^(2+m)/(e^7*(2+m))+3*(c*d^2-b*d*e+a*e^2)*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^(3+m)/(e^7*(3+m))-(2*c*d-b*e)*(10*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-3*a*e))*(d+e*x)^(4+m)/(e^7*(4+m))+3*c*(5*c^2*d^2+b^2*e^2-c*e*(5*b*d-a*e))*(d+e*x)^(5+m)/(e^7*(5+m))-3*c^2*(2*c*d-b*e)*(d+e*x)^(6+m)/(e^7*(6+m))+c^3*(d+e*x)^(7+m)/(e^7*(7+m))],
[(d+e*x)^m*(a+b*x+c*x^2)^2,x,2,(c*d^2-b*d*e+a*e^2)^2*(d+e*x)^(1+m)/(e^5*(1+m))-2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(d+e*x)^(2+m)/(e^5*(2+m))+(6*c^2*d^2+b^2*e^2-2*c*e*(3*b*d-a*e))*(d+e*x)^(3+m)/(e^5*(3+m))-2*c*(2*c*d-b*e)*(d+e*x)^(4+m)/(e^5*(4+m))+c^2*(d+e*x)^(5+m)/(e^5*(5+m))],
[(d+e*x)^m*(a+b*x+c*x^2),x,2,(c*d^2-b*d*e+a*e^2)*(d+e*x)^(1+m)/(e^3*(1+m))-(2*c*d-b*e)*(d+e*x)^(2+m)/(e^3*(2+m))+c*(d+e*x)^(3+m)/(e^3*(3+m))],
[(d+e*x)^m/(a+b*x+c*x^2),x,4,-2*c*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))/((1+m)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b^2-4*a*c))+2*c*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/((1+m)*sqrt(b^2-4*a*c)*(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x)^m/(a+b*x+c*x^2)^2,x,5,-(d+e*x)^(1+m)*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(a+b*x+c*x^2))-c*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(e*(2*c*d-b*e)*m+(4*c^2*d^2-4*c*e*(b*d-a*e*(1-m))+b^2*e^2*m)/sqrt(b^2-4*a*c))/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(1+m)*(2*c*d-e*(b+sqrt(b^2-4*a*c))))+c*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(4*c^2*d^2+b*e^2*m*(b+sqrt(b^2-4*a*c))-2*c*e*(2*b*d-2*a*e*(1-m)+d*m*sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)*(1+m)*(2*c*d-e*(b-sqrt(b^2-4*a*c))))],
[(d+e*x)^m*(a+b*x+c*x^2)^(5/2),x,2,(d+e*x)^(1+m)*(a+b*x+c*x^2)^(5/2)*AppellF1(1+m,-5/2,-5/2,2+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e*(1+m)*(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))^(5/2)*(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))^(5/2))],
[(d+e*x)^m*(a+b*x+c*x^2)^(3/2),x,2,(d+e*x)^(1+m)*(a+b*x+c*x^2)^(3/2)*AppellF1(1+m,-3/2,-3/2,2+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e*(1+m)*(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))^(3/2)*(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))^(3/2))],
[(d+e*x)^m*(a+b*x+c*x^2)^(1/2),x,2,(d+e*x)^(1+m)*AppellF1(1+m,-1/2,-1/2,2+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(a+b*x+c*x^2)/(e*(1+m)*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))],
[(d+e*x)^m/(a+b*x+c*x^2)^(1/2),x,2,(d+e*x)^(1+m)*AppellF1(1+m,1/2,1/2,2+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e*(1+m)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^m/(a+b*x+c*x^2)^(3/2),x,2,(d+e*x)^(1+m)*AppellF1(1+m,3/2,3/2,2+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))^(3/2)*(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))^(3/2)/(e*(1+m)*(a+b*x+c*x^2)^(3/2))],
[(d+e*x)^m/(a+b*x+c*x^2)^(5/2),x,2,(d+e*x)^(1+m)*AppellF1(1+m,5/2,5/2,2+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))^(5/2)*(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))^(5/2)/(e*(1+m)*(a+b*x+c*x^2)^(5/2))],

# Integrands of the form (d+e x)^m (a+b x+c x^2)^p when p symbolic
[(d*x)^m*(a+b*x+c*x^2)^p,x,2,(d*x)^(1+m)*(a+b*x+c*x^2)^p*AppellF1(1+m,-p,-p,2+m,-2*c*x/(b-sqrt(b^2-4*a*c)),-2*c*x/(b+sqrt(b^2-4*a*c)))/(d*(1+m)*(1+2*c*x/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x/(b+sqrt(b^2-4*a*c)))^p)],
[(d+e*x)^m*(a+b*x+c*x^2)^p,x,2,(d+e*x)^(1+m)*(a+b*x+c*x^2)^p*AppellF1(1+m,-p,-p,2+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e*(1+m)*(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))^p*(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))^p)],
[(d+e*x)^3*(a+b*x+c*x^2)^p,x,3,1/2*e*(d+e*x)^2*(a+b*x+c*x^2)^(1+p)/(c*(2+p))-1/4*e*(b*e*(2*c*d-b*e)*(2+p)*(3+p)-2*c*(3+2*p)*(c*d^2*(5+2*p)-e*(a*e+b*d*(2+p)))-2*c*e*(2*c*d-b*e)*(1+p)*(3+p)*x)*(a+b*x+c*x^2)^(1+p)/(c^3*(2+p)*(3+5*p+2*p^2))-2^(-1+p)*(2*c*d-b*e)*(b^2*e^2*(3+p)+2*c^2*d^2*(3+2*p)-2*c*e*(3*a*e+b*d*(3+2*p)))*(a+b*x+c*x^2)^(1+p)*hypergeom([-p,1+p],[2+p],1/2*(b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))*((-b-2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))^(-1-p)/(c^3*(1+p)*(3+2*p)*sqrt(b^2-4*a*c))],
[(d+e*x)^2*(a+b*x+c*x^2)^p,x,3,1/2*e*(2*c*d-b*e)*(2+p)*(a+b*x+c*x^2)^(1+p)/(c^2*(1+p)*(3+2*p))+e*(d+e*x)*(a+b*x+c*x^2)^(1+p)/(c*(3+2*p))-2^p*(b^2*e^2*(2+p)+2*c^2*d^2*(3+2*p)-2*c*e*(a*e+b*d*(3+2*p)))*(a+b*x+c*x^2)^(1+p)*hypergeom([-p,1+p],[2+p],1/2*(b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))*((-b-2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))^(-1-p)/(c^2*(1+p)*(3+2*p)*sqrt(b^2-4*a*c))],
[(d+e*x)*(a+b*x+c*x^2)^p,x,2,1/2*e*(a+b*x+c*x^2)^(1+p)/(c*(1+p))-2^p*(2*c*d-b*e)*(a+b*x+c*x^2)^(1+p)*hypergeom([-p,1+p],[2+p],1/2*(b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))*((-b-2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))^(-1-p)/(c*(1+p)*sqrt(b^2-4*a*c))],
[(a+b*x+c*x^2)^p,x,1,-2^(1+p)*(a+b*x+c*x^2)^(1+p)*hypergeom([-p,1+p],[2+p],1/2*(b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))*((-b-2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))^(-1-p)/((1+p)*sqrt(b^2-4*a*c))],
[(a+b*x+c*x^2)^p/(d+e*x),x,2,2^(-1+2*p)*(a+b*x+c*x^2)^p*AppellF1(-2*p,-p,-p,1-2*p,1/2*(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(c*(d+e*x)),1/2*(2*d-e*(b+sqrt(b^2-4*a*c))/c)/(d+e*x))/(e*p*(e*(b+2*c*x-sqrt(b^2-4*a*c))/(c*(d+e*x)))^p*(e*(b+2*c*x+sqrt(b^2-4*a*c))/(c*(d+e*x)))^p)],
[(a+b*x+c*x^2)^p/(d+e*x)^2,x,2,-4^p*(a+b*x+c*x^2)^p*AppellF1(1-2*p,-p,-p,2*(1-p),1/2*(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(c*(d+e*x)),1/2*(2*d-e*(b+sqrt(b^2-4*a*c))/c)/(d+e*x))/(e*(1-2*p)*(d+e*x)*(e*(b+2*c*x-sqrt(b^2-4*a*c))/(c*(d+e*x)))^p*(e*(b+2*c*x+sqrt(b^2-4*a*c))/(c*(d+e*x)))^p)],
[(a+b*x+c*x^2)^p/(d+e*x)^3,x,2,-2^(-1+2*p)*(a+b*x+c*x^2)^p*AppellF1(2*(1-p),-p,-p,3-2*p,1/2*(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(c*(d+e*x)),1/2*(2*d-e*(b+sqrt(b^2-4*a*c))/c)/(d+e*x))/(e*(1-p)*(d+e*x)^2*(e*(b+2*c*x-sqrt(b^2-4*a*c))/(c*(d+e*x)))^p*(e*(b+2*c*x+sqrt(b^2-4*a*c))/(c*(d+e*x)))^p)],
[(d+e*x)^(3/2)*(a+b*x+c*x^2)^p,x,2,2/5*(d+e*x)^(5/2)*(a+b*x+c*x^2)^p*AppellF1(5/2,-p,-p,7/2,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e*(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))^p*(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))^p)],
[(d+e*x)^(1/2)*(a+b*x+c*x^2)^p,x,2,2/3*(d+e*x)^(3/2)*(a+b*x+c*x^2)^p*AppellF1(3/2,-p,-p,5/2,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e*(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))^p*(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))^p)],
[(a+b*x+c*x^2)^p/(d+e*x)^(1/2),x,2,2*(a+b*x+c*x^2)^p*AppellF1(1/2,-p,-p,3/2,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(d+e*x)/(e*(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))^p*(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))^p)],
[(a+b*x+c*x^2)^p/(d+e*x)^(3/2),x,2,-2*(a+b*x+c*x^2)^p*AppellF1(-1/2,-p,-p,1/2,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e*(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))^p*(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))^p*sqrt(d+e*x))],
[(a+b*x+c*x^2)^p/(d+e*x)^(2*p),x,2,(d+e*x)^(1-2*p)*(a+b*x+c*x^2)^p*AppellF1(1-2*p,-p,-p,2-2*p,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e*(1-2*p)*(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))^p*(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))^p)],
[(d+e*x)^(-1-2*p)*(a+b*x+c*x^2)^p,x,2,-1/2*(a+b*x+c*x^2)^p*AppellF1(-2*p,-p,-p,1-2*p,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e*p*(d+e*x)^(2*p)*(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))^p*(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))^p)],
[(d+e*x)^(-2-2*p)*(a+b*x+c*x^2)^p,x,1,(d+e*x)^(-1-2*p)*(a+b*x+c*x^2)^p*hypergeom([-1-2*p,-p],[-2*p],-4*c*(d+e*x)*sqrt(b^2-4*a*c)/((b+2*c*x-sqrt(b^2-4*a*c))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*(b+2*c*x-sqrt(b^2-4*a*c))/((1+2*p)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*((2*c*d-e*(b-sqrt(b^2-4*a*c)))*(b+2*c*x+sqrt(b^2-4*a*c))/((b+2*c*x-sqrt(b^2-4*a*c))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))))^p)],
[(d+e*x)^(-3-2*p)*(a+b*x+c*x^2)^p,x,2,-1/2*e*(a+b*x+c*x^2)^(1+p)/((c*d^2-b*d*e+a*e^2)*(1+p)*(d+e*x)^(2*(1+p)))+1/2*(2*c*d-b*e)*(d+e*x)^(-1-2*p)*(a+b*x+c*x^2)^p*hypergeom([-1-2*p,-p],[-2*p],-4*c*(d+e*x)*sqrt(b^2-4*a*c)/((b+2*c*x-sqrt(b^2-4*a*c))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*(b+2*c*x-sqrt(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)*(1+2*p)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*((2*c*d-e*(b-sqrt(b^2-4*a*c)))*(b+2*c*x+sqrt(b^2-4*a*c))/((b+2*c*x-sqrt(b^2-4*a*c))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))))^p)],
[(d+e*x)^(-4-2*p)*(a+b*x+c*x^2)^p,x,3,-e*(d+e*x)^(-3-2*p)*(a+b*x+c*x^2)^(1+p)/((c*d^2-b*d*e+a*e^2)*(3+2*p))-1/2*e*(2*c*d-b*e)*(2+p)*(a+b*x+c*x^2)^(1+p)/((c*d^2-b*d*e+a*e^2)^2*(1+p)*(3+2*p)*(d+e*x)^(2*(1+p)))+1/2*(b^2*e^2*(2+p)+2*c^2*d^2*(3+2*p)-2*c*e*(a*e+b*d*(3+2*p)))*(d+e*x)^(-1-2*p)*(a+b*x+c*x^2)^p*hypergeom([-1-2*p,-p],[-2*p],-4*c*(d+e*x)*sqrt(b^2-4*a*c)/((b+2*c*x-sqrt(b^2-4*a*c))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*(b+2*c*x-sqrt(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^2*(1+2*p)*(3+2*p)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*((2*c*d-e*(b-sqrt(b^2-4*a*c)))*(b+2*c*x+sqrt(b^2-4*a*c))/((b+2*c*x-sqrt(b^2-4*a*c))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))))^p)],
[(d+e*x)^(-5-2*p)*(a+b*x+c*x^2)^p,x,4,-1/2*e*(2*c*d-b*e)*(3+p)*(d+e*x)^(-3-2*p)*(a+b*x+c*x^2)^(1+p)/((c*d^2-b*d*e+a*e^2)^2*(2+p)*(3+2*p))-1/4*e*(b^2*e^2*(6+5*p+p^2)+2*c^2*d^2*(9+8*p+2*p^2)-2*c*e*(a*e*(3+2*p)+b*d*(9+8*p+2*p^2)))*(a+b*x+c*x^2)^(1+p)/((c*d^2-b*d*e+a*e^2)^3*(1+p)*(2+p)*(3+2*p)*(d+e*x)^(2*(1+p)))-1/2*e*(a+b*x+c*x^2)^(1+p)/((c*d^2-b*d*e+a*e^2)*(2+p)*(d+e*x)^(2*(2+p)))+1/4*(2*c*d-b*e)*(b^2*e^2*(3+p)+2*c^2*d^2*(3+2*p)-2*c*e*(3*a*e+b*d*(3+2*p)))*(d+e*x)^(-1-2*p)*(a+b*x+c*x^2)^p*hypergeom([-1-2*p,-p],[-2*p],-4*c*(d+e*x)*sqrt(b^2-4*a*c)/((b+2*c*x-sqrt(b^2-4*a*c))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*(b+2*c*x-sqrt(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^3*(1+2*p)*(3+2*p)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*((2*c*d-e*(b-sqrt(b^2-4*a*c)))*(b+2*c*x+sqrt(b^2-4*a*c))/((b+2*c*x-sqrt(b^2-4*a*c))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))))^p)],
[(d+e*x)^(-6-2*p)*(a+b*x+c*x^2)^p,x,5,-e*(d+e*x)^(-5-2*p)*(a+b*x+c*x^2)^(1+p)/((c*d^2-b*d*e+a*e^2)*(5+2*p))-1/2*e*(b^2*e^2*(12+7*p+p^2)+2*c^2*d^2*(18+11*p+2*p^2)-2*c*e*(3*a*e*(2+p)+b*d*(18+11*p+2*p^2)))*(d+e*x)^(-3-2*p)*(a+b*x+c*x^2)^(1+p)/((c*d^2-b*d*e+a*e^2)^3*(2+p)*(3+2*p)*(5+2*p))-1/4*e*(2*c*d-b*e)*(3+p)*(b^2*e^2*(8+6*p+p^2)+2*c^2*d^2*(8+7*p+2*p^2)-2*c*e*(a*e*(8+5*p)+b*d*(8+7*p+2*p^2)))*(a+b*x+c*x^2)^(1+p)/((c*d^2-b*d*e+a*e^2)^4*(1+p)*(2+p)*(3+2*p)*(5+2*p)*(d+e*x)^(2*(1+p)))-1/2*e*(2*c*d-b*e)*(4+p)*(a+b*x+c*x^2)^(1+p)/((c*d^2-b*d*e+a*e^2)^2*(2+p)*(5+2*p)*(d+e*x)^(2*(2+p)))+1/4*(b^4*e^4*(12+7*p+p^2)+4*c^4*d^4*(15+16*p+4*p^2)-8*c^3*d^2*e*(5+2*p)*(3*a*e+b*d*(3+2*p))-4*b^2*c*e^3*(3+p)*(3*a*e+b*d*(5+2*p))+12*c^2*e^2*(a^2*e^2+2*a*b*d*e*(5+2*p)+b^2*d^2*(10+9*p+2*p^2)))*(d+e*x)^(-1-2*p)*(a+b*x+c*x^2)^p*hypergeom([-1-2*p,-p],[-2*p],-4*c*(d+e*x)*sqrt(b^2-4*a*c)/((b+2*c*x-sqrt(b^2-4*a*c))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*(b+2*c*x-sqrt(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^4*(1+2*p)*(3+2*p)*(5+2*p)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*((2*c*d-e*(b-sqrt(b^2-4*a*c)))*(b+2*c*x+sqrt(b^2-4*a*c))/((b+2*c*x-sqrt(b^2-4*a*c))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))))^p)],
[(d+e*x)^m*(a+b*x+c*x^2)^(-2-1/2*m),x,3,e*(d+e*x)^(1+m)*(a+b*x+c*x^2)^(-1-1/2*m)/((c*d^2-b*d*e+a*e^2)*(1+m))+1/2*e*(2*c*d-b*e)*m*(d+e*x)^(2+m)*(a+b*x+c*x^2)^(-1-1/2*m)/((c*d^2-b*d*e+a*e^2)^2*(1+m)*(2+m))-1/4*(b^2*e^2*m+4*c^2*d^2*(1+m)+4*c*e*(a*e-b*d*(1+m)))*(d+e*x)^(3+m)*(a+b*x+c*x^2)^(-2-1/2*m)*hypergeom([3+m,1/2*(4+m)],[4+m],-4*c*(d+e*x)*sqrt(b^2-4*a*c)/((b+2*c*x-sqrt(b^2-4*a*c))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*(b+2*c*x-sqrt(b^2-4*a*c))*((2*c*d-e*(b-sqrt(b^2-4*a*c)))*(b+2*c*x+sqrt(b^2-4*a*c))/((b+2*c*x-sqrt(b^2-4*a*c))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))))^(1/2*(4+m))/((c*d^2-b*d*e+a*e^2)^2*(1+m)*(3+m)*(2*c*d-e*(b-sqrt(b^2-4*a*c))))],
[1/((1+x)^(1/3)*(1-x+x^2)^(1/3)),x,2,-1/2*(1+x^3)^(1/3)*log(-x+(1+x^3)^(1/3))/((1+x)^(1/3)*(1-x+x^2)^(1/3))+(1+x^3)^(1/3)*arctan((1+2*x/(1+x^3)^(1/3))/sqrt(3))/((1+x)^(1/3)*(1-x+x^2)^(1/3)*sqrt(3))],
[1/((1+x)^(2/3)*(1-x+x^2)^(2/3)),x,2,x*(1+x^3)^(2/3)*hypergeom([1/3,2/3],[4/3],-x^3)/((1+x)^(2/3)*(1-x+x^2)^(2/3))],
[(1+x)^p*(1-x+x^2)^p,x,2,x*(1+x)^p*(1-x+x^2)^p*hypergeom([1/3,-p],[4/3],-x^3)/(1+x^3)^p],
[1/((1-x)^(1/3)*(1+x+x^2)^(1/3)),x,2,1/2*(1-x^3)^(1/3)*log(x+(1-x^3)^(1/3))/((1-x)^(1/3)*(1+x+x^2)^(1/3))-(1-x^3)^(1/3)*arctan((1-2*x/(1-x^3)^(1/3))/sqrt(3))/((1-x)^(1/3)*(1+x+x^2)^(1/3)*sqrt(3))],
[1/((1-x)^(2/3)*(1+x+x^2)^(2/3)),x,2,x*(1-x^3)^(2/3)*hypergeom([1/3,2/3],[4/3],x^3)/((1-x)^(2/3)*(1+x+x^2)^(2/3))],
[(1-x)^p*(1+x+x^2)^p,x,2,(1-x)^p*x*(1+x+x^2)^p*hypergeom([1/3,-p],[4/3],x^3)/(1-x^3)^p],
[1/((b*e-c*e*x)^(1/3)*(b^2+b*c*x+c^2*x^2)^(1/3)),x,2,1/2*(b^3*e-c^3*e*x^3)^(1/3)*log(c*e^(1/3)*x+(b^3*e-c^3*e*x^3)^(1/3))/(c*e^(1/3)*(b*e-c*e*x)^(1/3)*(b^2+b*c*x+c^2*x^2)^(1/3))-(b^3*e-c^3*e*x^3)^(1/3)*arctan((1-2*c*e^(1/3)*x/(b^3*e-c^3*e*x^3)^(1/3))/sqrt(3))/(c*e^(1/3)*(b*e-c*e*x)^(1/3)*(b^2+b*c*x+c^2*x^2)^(1/3)*sqrt(3))],
[1/((b*e-c*e*x)^(2/3)*(b^2+b*c*x+c^2*x^2)^(2/3)),x,3,x*(1-c^3*x^3/b^3)^(2/3)*hypergeom([1/3,2/3],[4/3],c^3*x^3/b^3)/((b*e-c*e*x)^(2/3)*(b^2+b*c*x+c^2*x^2)^(2/3))],
[(b*e-c*e*x)^p*(b^2+b*c*x+c^2*x^2)^p,x,3,x*(b*e-c*e*x)^p*(b^2+b*c*x+c^2*x^2)^p*hypergeom([1/3,-p],[4/3],c^3*x^3/b^3)/(1-c^3*x^3/b^3)^p]]:
