# Maple integration test file: "1 Algebraic functions\1.2 Trinomial products\1.2.3 General\1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.txt"

lst:=[

# Integrands of the form (f x)^m (d+e x^3)^q (a+b x^3+c x^6)^p

# Integrands of the form (f x)^m (d+e x^3)^q (a+b x^3+c x^6)^p with b=0

# Integrands of the form (f x)^m (d+e x^3)^q (a+b x^3+c x^6)^p with b^2-4 a c=0

# Integrands of the form (f x)^m (d+e x^3)^q (a+b x^3+c x^6)^p with c d^2-b d e+a e^2=0

# Integrands of the form (f x)^m (d+e x^3)^q (a+b x^3+c x^6)^p

# Integrands of the form (f x)^m (d+e x^3)^q (a+b x^3+c x^6)^p

# p>0
[(d+e*x^3)^5*(a+b*x^3+c*x^6),x,2,a*d^5*x+1/4*d^4*(b*d+5*a*e)*x^4+1/7*d^3*(c*d^2+5*e*(b*d+2*a*e))*x^7+1/2*d^2*e*(c*d^2+2*e*(b*d+a*e))*x^10+5/13*d*e^2*(2*c*d^2+e*(2*b*d+a*e))*x^13+1/16*e^3*(10*c*d^2+e*(5*b*d+a*e))*x^16+1/19*e^4*(5*c*d+b*e)*x^19+1/22*c*e^5*x^22],
[(d+e*x^3)^4*(a+b*x^3+c*x^6),x,2,a*d^4*x+1/4*d^3*(b*d+4*a*e)*x^4+1/7*d^2*(c*d^2+4*b*d*e+6*a*e^2)*x^7+1/5*d*e*(2*c*d^2+e*(3*b*d+2*a*e))*x^10+1/13*e^2*(6*c*d^2+e*(4*b*d+a*e))*x^13+1/16*e^3*(4*c*d+b*e)*x^16+1/19*c*e^4*x^19],
[(d+e*x^3)^3*(a+b*x^3+c*x^6),x,2,a*d^3*x+1/4*d^2*(b*d+3*a*e)*x^4+1/7*d*(c*d^2+3*e*(b*d+a*e))*x^7+1/10*e*(3*c*d^2+e*(3*b*d+a*e))*x^10+1/13*e^2*(3*c*d+b*e)*x^13+1/16*c*e^3*x^16],
[(d+e*x^3)^2*(a+b*x^3+c*x^6),x,2,a*d^2*x+1/4*d*(b*d+2*a*e)*x^4+1/7*(c*d^2+e*(2*b*d+a*e))*x^7+1/10*e*(2*c*d+b*e)*x^10+1/13*c*e^2*x^13],
[(d+e*x^3)*(a+b*x^3+c*x^6),x,2,a*d*x+1/4*(b*d+a*e)*x^4+1/7*(c*d+b*e)*x^7+1/10*c*e*x^10],
[(a+b*x^3+c*x^6)/(d+e*x^3),x,8,-(c*d-b*e)*x/e^2+1/4*c*x^4/e+1/3*(c*d^2-b*d*e+a*e^2)*log(d^(1/3)+e^(1/3)*x)/(d^(2/3)*e^(7/3))-1/6*(c*d^2-b*d*e+a*e^2)*log(d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(d^(2/3)*e^(7/3))-(c*d^2-b*d*e+a*e^2)*arctan((d^(1/3)-2*e^(1/3)*x)/(d^(1/3)*sqrt(3)))/(d^(2/3)*e^(7/3)*sqrt(3))],
[(a+b*x^3+c*x^6)/(d+e*x^3)^2,x,8,c*x/e^2+1/3*(c*d^2-b*d*e+a*e^2)*x/(d*e^2*(d+e*x^3))-1/9*(4*c*d^2-e*(b*d+2*a*e))*log(d^(1/3)+e^(1/3)*x)/(d^(5/3)*e^(7/3))+1/18*(4*c*d^2-e*(b*d+2*a*e))*log(d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(d^(5/3)*e^(7/3))+1/3*(4*c*d^2-e*(b*d+2*a*e))*arctan((d^(1/3)-2*e^(1/3)*x)/(d^(1/3)*sqrt(3)))/(d^(5/3)*e^(7/3)*sqrt(3))],
[(a+b*x^3+c*x^6)/(d+e*x^3)^3,x,8,1/6*(c*d^2-b*d*e+a*e^2)*x/(d*e^2*(d+e*x^3)^2)-1/18*(7*c*d^2-e*(b*d+5*a*e))*x/(d^2*e^2*(d+e*x^3))+1/27*(2*c*d^2+e*(b*d+5*a*e))*log(d^(1/3)+e^(1/3)*x)/(d^(8/3)*e^(7/3))-1/54*(2*c*d^2+e*(b*d+5*a*e))*log(d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(d^(8/3)*e^(7/3))-1/9*(2*c*d^2+e*(b*d+5*a*e))*arctan((d^(1/3)-2*e^(1/3)*x)/(d^(1/3)*sqrt(3)))/(d^(8/3)*e^(7/3)*sqrt(3))],

# p<0
[x^8*(d+e*x^3)/(a+b*x^3+c*x^6),x,7,1/3*(c*d-b*e)*x^3/c^2+1/6*e*x^6/c-1/6*(b*c*d-b^2*e+a*c*e)*log(a+b*x^3+c*x^6)/c^3-1/3*(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*e)*arctanh((b+2*c*x^3)/sqrt(b^2-4*a*c))/(c^3*sqrt(b^2-4*a*c))],
[x^5*(d+e*x^3)/(a+b*x^3+c*x^6),x,6,1/3*e*x^3/c+1/6*(c*d-b*e)*log(a+b*x^3+c*x^6)/c^2+1/3*(b*c*d-b^2*e+2*a*c*e)*arctanh((b+2*c*x^3)/sqrt(b^2-4*a*c))/(c^2*sqrt(b^2-4*a*c))],
[x^2*(d+e*x^3)/(a+b*x^3+c*x^6),x,5,1/6*e*log(a+b*x^3+c*x^6)/c-1/3*(2*c*d-b*e)*arctanh((b+2*c*x^3)/sqrt(b^2-4*a*c))/(c*sqrt(b^2-4*a*c))],
[(d+e*x^3)/(x*(a+b*x^3+c*x^6)),x,7,d*log(x)/a-1/6*d*log(a+b*x^3+c*x^6)/a+1/3*(b*d-2*a*e)*arctanh((b+2*c*x^3)/sqrt(b^2-4*a*c))/(a*sqrt(b^2-4*a*c))],
[(d+e*x^3)/(x^4*(a+b*x^3+c*x^6)),x,7,-1/3*d/(a*x^3)-(b*d-a*e)*log(x)/a^2+1/6*(b*d-a*e)*log(a+b*x^3+c*x^6)/a^2-1/3*(b^2*d-2*a*c*d-a*b*e)*arctanh((b+2*c*x^3)/sqrt(b^2-4*a*c))/(a^2*sqrt(b^2-4*a*c))],
[x^4*(d+e*x^3)/(a+b*x^3+c*x^6),x,14,1/2*e*x^2/c-1/3*log(2^(1/3)*c^(1/3)*x+(b-sqrt(b^2-4*a*c))^(1/3))*(c*d-b*e+(-b*c*d+b^2*e-2*a*c*e)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(5/3)*(b-sqrt(b^2-4*a*c))^(1/3))+1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-sqrt(b^2-4*a*c))^(1/3)+(b-sqrt(b^2-4*a*c))^(2/3))*(c*d-b*e+(-b*c*d+b^2*e-2*a*c*e)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(5/3)*(b-sqrt(b^2-4*a*c))^(1/3))-arctan((1-2*2^(1/3)*c^(1/3)*x/(b-sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(c*d-b*e+(-b*c*d+b^2*e-2*a*c*e)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(5/3)*sqrt(3)*(b-sqrt(b^2-4*a*c))^(1/3))-1/3*log(2^(1/3)*c^(1/3)*x+(b+sqrt(b^2-4*a*c))^(1/3))*(c*d-b*e+(b*c*d-b^2*e+2*a*c*e)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(5/3)*(b+sqrt(b^2-4*a*c))^(1/3))+1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+sqrt(b^2-4*a*c))^(1/3)+(b+sqrt(b^2-4*a*c))^(2/3))*(c*d-b*e+(b*c*d-b^2*e+2*a*c*e)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(5/3)*(b+sqrt(b^2-4*a*c))^(1/3))-arctan((1-2*2^(1/3)*c^(1/3)*x/(b+sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(c*d-b*e+(b*c*d-b^2*e+2*a*c*e)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(5/3)*sqrt(3)*(b+sqrt(b^2-4*a*c))^(1/3))],
[x^3*(d+e*x^3)/(a+b*x^3+c*x^6),x,14,e*x/c+1/3*log(2^(1/3)*c^(1/3)*x+(b-sqrt(b^2-4*a*c))^(1/3))*(c*d-b*e+(-b*c*d+b^2*e-2*a*c*e)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*(b-sqrt(b^2-4*a*c))^(2/3))-1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-sqrt(b^2-4*a*c))^(1/3)+(b-sqrt(b^2-4*a*c))^(2/3))*(c*d-b*e+(-b*c*d+b^2*e-2*a*c*e)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*(b-sqrt(b^2-4*a*c))^(2/3))-arctan((1-2*2^(1/3)*c^(1/3)*x/(b-sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(c*d-b*e+(-b*c*d+b^2*e-2*a*c*e)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*sqrt(3)*(b-sqrt(b^2-4*a*c))^(2/3))+1/3*log(2^(1/3)*c^(1/3)*x+(b+sqrt(b^2-4*a*c))^(1/3))*(c*d-b*e+(b*c*d-b^2*e+2*a*c*e)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*(b+sqrt(b^2-4*a*c))^(2/3))-1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+sqrt(b^2-4*a*c))^(1/3)+(b+sqrt(b^2-4*a*c))^(2/3))*(c*d-b*e+(b*c*d-b^2*e+2*a*c*e)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*(b+sqrt(b^2-4*a*c))^(2/3))-arctan((1-2*2^(1/3)*c^(1/3)*x/(b+sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(c*d-b*e+(b*c*d-b^2*e+2*a*c*e)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*sqrt(3)*(b+sqrt(b^2-4*a*c))^(2/3))],
[x*(d+e*x^3)/(a+b*x^3+c*x^6),x,13,-1/3*log(2^(1/3)*c^(1/3)*x+(b-sqrt(b^2-4*a*c))^(1/3))*(e+(2*c*d-b*e)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(2/3)*(b-sqrt(b^2-4*a*c))^(1/3))+1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-sqrt(b^2-4*a*c))^(1/3)+(b-sqrt(b^2-4*a*c))^(2/3))*(e+(2*c*d-b*e)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(2/3)*(b-sqrt(b^2-4*a*c))^(1/3))-arctan((1-2*2^(1/3)*c^(1/3)*x/(b-sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(e+(2*c*d-b*e)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(2/3)*sqrt(3)*(b-sqrt(b^2-4*a*c))^(1/3))-1/3*log(2^(1/3)*c^(1/3)*x+(b+sqrt(b^2-4*a*c))^(1/3))*(e+(-2*c*d+b*e)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(2/3)*(b+sqrt(b^2-4*a*c))^(1/3))+1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+sqrt(b^2-4*a*c))^(1/3)+(b+sqrt(b^2-4*a*c))^(2/3))*(e+(-2*c*d+b*e)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(2/3)*(b+sqrt(b^2-4*a*c))^(1/3))-arctan((1-2*2^(1/3)*c^(1/3)*x/(b+sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(e+(-2*c*d+b*e)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(2/3)*sqrt(3)*(b+sqrt(b^2-4*a*c))^(1/3))],
[(d+e*x^3)/(a+b*x^3+c*x^6),x,13,1/3*log(2^(1/3)*c^(1/3)*x+(b-sqrt(b^2-4*a*c))^(1/3))*(e+(2*c*d-b*e)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(1/3)*(b-sqrt(b^2-4*a*c))^(2/3))-1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-sqrt(b^2-4*a*c))^(1/3)+(b-sqrt(b^2-4*a*c))^(2/3))*(e+(2*c*d-b*e)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(1/3)*(b-sqrt(b^2-4*a*c))^(2/3))-arctan((1-2*2^(1/3)*c^(1/3)*x/(b-sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(e+(2*c*d-b*e)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(1/3)*sqrt(3)*(b-sqrt(b^2-4*a*c))^(2/3))+1/3*log(2^(1/3)*c^(1/3)*x+(b+sqrt(b^2-4*a*c))^(1/3))*(e+(-2*c*d+b*e)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(1/3)*(b+sqrt(b^2-4*a*c))^(2/3))-1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+sqrt(b^2-4*a*c))^(1/3)+(b+sqrt(b^2-4*a*c))^(2/3))*(e+(-2*c*d+b*e)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(1/3)*(b+sqrt(b^2-4*a*c))^(2/3))-arctan((1-2*2^(1/3)*c^(1/3)*x/(b+sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(e+(-2*c*d+b*e)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(1/3)*sqrt(3)*(b+sqrt(b^2-4*a*c))^(2/3))],
[(d+e*x^3)/(x^2*(a+b*x^3+c*x^6)),x,14,-d/(a*x)+1/3*c^(1/3)*log(2^(1/3)*c^(1/3)*x+(b-sqrt(b^2-4*a*c))^(1/3))*(d+(b*d-2*a*e)/sqrt(b^2-4*a*c))/(2^(2/3)*a*(b-sqrt(b^2-4*a*c))^(1/3))-1/6*c^(1/3)*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-sqrt(b^2-4*a*c))^(1/3)+(b-sqrt(b^2-4*a*c))^(2/3))*(d+(b*d-2*a*e)/sqrt(b^2-4*a*c))/(2^(2/3)*a*(b-sqrt(b^2-4*a*c))^(1/3))+c^(1/3)*arctan((1-2*2^(1/3)*c^(1/3)*x/(b-sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(d+(b*d-2*a*e)/sqrt(b^2-4*a*c))/(2^(2/3)*a*sqrt(3)*(b-sqrt(b^2-4*a*c))^(1/3))+1/3*c^(1/3)*log(2^(1/3)*c^(1/3)*x+(b+sqrt(b^2-4*a*c))^(1/3))*(d+(-b*d+2*a*e)/sqrt(b^2-4*a*c))/(2^(2/3)*a*(b+sqrt(b^2-4*a*c))^(1/3))-1/6*c^(1/3)*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+sqrt(b^2-4*a*c))^(1/3)+(b+sqrt(b^2-4*a*c))^(2/3))*(d+(-b*d+2*a*e)/sqrt(b^2-4*a*c))/(2^(2/3)*a*(b+sqrt(b^2-4*a*c))^(1/3))+c^(1/3)*arctan((1-2*2^(1/3)*c^(1/3)*x/(b+sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(d+(-b*d+2*a*e)/sqrt(b^2-4*a*c))/(2^(2/3)*a*sqrt(3)*(b+sqrt(b^2-4*a*c))^(1/3))],
[(d+e*x^3)/(x^3*(a+b*x^3+c*x^6)),x,14,-1/2*d/(a*x^2)-1/3*c^(2/3)*log(2^(1/3)*c^(1/3)*x+(b-sqrt(b^2-4*a*c))^(1/3))*(d+(b*d-2*a*e)/sqrt(b^2-4*a*c))/(2^(1/3)*a*(b-sqrt(b^2-4*a*c))^(2/3))+1/6*c^(2/3)*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-sqrt(b^2-4*a*c))^(1/3)+(b-sqrt(b^2-4*a*c))^(2/3))*(d+(b*d-2*a*e)/sqrt(b^2-4*a*c))/(2^(1/3)*a*(b-sqrt(b^2-4*a*c))^(2/3))+c^(2/3)*arctan((1-2*2^(1/3)*c^(1/3)*x/(b-sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(d+(b*d-2*a*e)/sqrt(b^2-4*a*c))/(2^(1/3)*a*sqrt(3)*(b-sqrt(b^2-4*a*c))^(2/3))-1/3*c^(2/3)*log(2^(1/3)*c^(1/3)*x+(b+sqrt(b^2-4*a*c))^(1/3))*(d+(-b*d+2*a*e)/sqrt(b^2-4*a*c))/(2^(1/3)*a*(b+sqrt(b^2-4*a*c))^(2/3))+1/6*c^(2/3)*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+sqrt(b^2-4*a*c))^(1/3)+(b+sqrt(b^2-4*a*c))^(2/3))*(d+(-b*d+2*a*e)/sqrt(b^2-4*a*c))/(2^(1/3)*a*(b+sqrt(b^2-4*a*c))^(2/3))+c^(2/3)*arctan((1-2*2^(1/3)*c^(1/3)*x/(b+sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(d+(-b*d+2*a*e)/sqrt(b^2-4*a*c))/(2^(1/3)*a*sqrt(3)*(b+sqrt(b^2-4*a*c))^(2/3))],
[x^8*(1-x^3)/(1-x^3+x^6),x,7,-1/6*x^6+1/6*log(1-x^3+x^6)-1/3*arctan((1-2*x^3)/sqrt(3))/sqrt(3)],
[x^5*(1-x^3)/(1-x^3+x^6),x,4,-1/3*x^3-2/3*arctan((1-2*x^3)/sqrt(3))/sqrt(3)],
[x^2*(1-x^3)/(1-x^3+x^6),x,5,-1/6*log(1-x^3+x^6)-1/3*arctan((1-2*x^3)/sqrt(3))/sqrt(3)],
[(1-x^3)/(x*(1-x^3+x^6)),x,7,log(x)-1/6*log(1-x^3+x^6)+1/3*arctan((1-2*x^3)/sqrt(3))/sqrt(3)],
[(1-x^3)/(x^4*(1-x^3+x^6)),x,5,(-1/3)/x^3+2/3*arctan((1-2*x^3)/sqrt(3))/sqrt(3)],
[x^6*(1-x^3)/(1-x^3+x^6),x,15,-1/4*x^4+1/3*arctan((1+2*2^(1/3)*x/(1+I*sqrt(3))^(1/3))/sqrt(3))*(I-sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))+1/9*log(-2^(1/3)*x+(1+I*sqrt(3))^(1/3))*(3-I*sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))-1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1+I*sqrt(3))^(1/3)+(1+I*sqrt(3))^(2/3))*(3-I*sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))+1/9*log(-2^(1/3)*x+(1-I*sqrt(3))^(1/3))*(3+I*sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))-1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1-I*sqrt(3))^(1/3)+(1-I*sqrt(3))^(2/3))*(3+I*sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))-1/3*arctan((1+2*2^(1/3)*x/(1-I*sqrt(3))^(1/3))/sqrt(3))*(I+sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))],
[x^4*(1-x^3)/(1-x^3+x^6),x,15,-1/2*x^2+1/3*I*2^(1/3)*arctan((1+2*2^(1/3)*x/(1-I*sqrt(3))^(1/3))/sqrt(3))/(1-I*sqrt(3))^(1/3)-1/3*I*2^(1/3)*arctan((1+2*2^(1/3)*x/(1+I*sqrt(3))^(1/3))/sqrt(3))/(1+I*sqrt(3))^(1/3)+1/3*I*2^(1/3)*log(-2^(1/3)*x+(1-I*sqrt(3))^(1/3))/((1-I*sqrt(3))^(1/3)*sqrt(3))-1/3*I*log(2^(2/3)*x^2+2^(1/3)*x*(1-I*sqrt(3))^(1/3)+(1-I*sqrt(3))^(2/3))/(2^(2/3)*(1-I*sqrt(3))^(1/3)*sqrt(3))-1/3*I*2^(1/3)*log(-2^(1/3)*x+(1+I*sqrt(3))^(1/3))/((1+I*sqrt(3))^(1/3)*sqrt(3))+1/3*I*log(2^(2/3)*x^2+2^(1/3)*x*(1+I*sqrt(3))^(1/3)+(1+I*sqrt(3))^(2/3))/(2^(2/3)*(1+I*sqrt(3))^(1/3)*sqrt(3))],
[x^3*(1-x^3)/(1-x^3+x^6),x,14,-x-1/3*I*2^(2/3)*arctan((1+2*2^(1/3)*x/(1-I*sqrt(3))^(1/3))/sqrt(3))/(1-I*sqrt(3))^(2/3)+1/3*I*2^(2/3)*arctan((1+2*2^(1/3)*x/(1+I*sqrt(3))^(1/3))/sqrt(3))/(1+I*sqrt(3))^(2/3)+1/3*I*2^(2/3)*log(-2^(1/3)*x+(1-I*sqrt(3))^(1/3))/((1-I*sqrt(3))^(2/3)*sqrt(3))-1/3*I*log(2^(2/3)*x^2+2^(1/3)*x*(1-I*sqrt(3))^(1/3)+(1-I*sqrt(3))^(2/3))/(2^(1/3)*(1-I*sqrt(3))^(2/3)*sqrt(3))-1/3*I*2^(2/3)*log(-2^(1/3)*x+(1+I*sqrt(3))^(1/3))/((1+I*sqrt(3))^(2/3)*sqrt(3))+1/3*I*log(2^(2/3)*x^2+2^(1/3)*x*(1+I*sqrt(3))^(1/3)+(1+I*sqrt(3))^(2/3))/(2^(1/3)*(1+I*sqrt(3))^(2/3)*sqrt(3))],
[x*(1-x^3)/(1-x^3+x^6),x,13,1/3*arctan((1+2*2^(1/3)*x/(1-I*sqrt(3))^(1/3))/sqrt(3))*(I-sqrt(3))/(2^(2/3)*(1-I*sqrt(3))^(1/3))-1/9*log(-2^(1/3)*x+(1-I*sqrt(3))^(1/3))*(3-I*sqrt(3))/(2^(2/3)*(1-I*sqrt(3))^(1/3))+1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1-I*sqrt(3))^(1/3)+(1-I*sqrt(3))^(2/3))*(3-I*sqrt(3))/(2^(2/3)*(1-I*sqrt(3))^(1/3))-1/9*log(-2^(1/3)*x+(1+I*sqrt(3))^(1/3))*(3+I*sqrt(3))/(2^(2/3)*(1+I*sqrt(3))^(1/3))+1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1+I*sqrt(3))^(1/3)+(1+I*sqrt(3))^(2/3))*(3+I*sqrt(3))/(2^(2/3)*(1+I*sqrt(3))^(1/3))-1/3*arctan((1+2*2^(1/3)*x/(1+I*sqrt(3))^(1/3))/sqrt(3))*(I+sqrt(3))/(2^(2/3)*(1+I*sqrt(3))^(1/3))],
[(1-x^3)/(1-x^3+x^6),x,13,-1/3*arctan((1+2*2^(1/3)*x/(1-I*sqrt(3))^(1/3))/sqrt(3))*(I-sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))-1/9*log(-2^(1/3)*x+(1-I*sqrt(3))^(1/3))*(3-I*sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))+1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1-I*sqrt(3))^(1/3)+(1-I*sqrt(3))^(2/3))*(3-I*sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))-1/9*log(-2^(1/3)*x+(1+I*sqrt(3))^(1/3))*(3+I*sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))+1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1+I*sqrt(3))^(1/3)+(1+I*sqrt(3))^(2/3))*(3+I*sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))+1/3*arctan((1+2*2^(1/3)*x/(1+I*sqrt(3))^(1/3))/sqrt(3))*(I+sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))],
[(1-x^3)/(x^2*(1-x^3+x^6)),x,14,(-1)/x+1/3*arctan((1+2*2^(1/3)*x/(1+I*sqrt(3))^(1/3))/sqrt(3))*(I-sqrt(3))/(2^(2/3)*(1+I*sqrt(3))^(1/3))-1/9*log(-2^(1/3)*x+(1+I*sqrt(3))^(1/3))*(3-I*sqrt(3))/(2^(2/3)*(1+I*sqrt(3))^(1/3))+1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1+I*sqrt(3))^(1/3)+(1+I*sqrt(3))^(2/3))*(3-I*sqrt(3))/(2^(2/3)*(1+I*sqrt(3))^(1/3))-1/9*log(-2^(1/3)*x+(1-I*sqrt(3))^(1/3))*(3+I*sqrt(3))/(2^(2/3)*(1-I*sqrt(3))^(1/3))+1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1-I*sqrt(3))^(1/3)+(1-I*sqrt(3))^(2/3))*(3+I*sqrt(3))/(2^(2/3)*(1-I*sqrt(3))^(1/3))-1/3*arctan((1+2*2^(1/3)*x/(1-I*sqrt(3))^(1/3))/sqrt(3))*(I+sqrt(3))/(2^(2/3)*(1-I*sqrt(3))^(1/3))],
[(1-x^3)/(x^3*(1-x^3+x^6)),x,15,(-1/2)/x^2-1/3*arctan((1+2*2^(1/3)*x/(1+I*sqrt(3))^(1/3))/sqrt(3))*(I-sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))-1/9*log(-2^(1/3)*x+(1+I*sqrt(3))^(1/3))*(3-I*sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))+1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1+I*sqrt(3))^(1/3)+(1+I*sqrt(3))^(2/3))*(3-I*sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))-1/9*log(-2^(1/3)*x+(1-I*sqrt(3))^(1/3))*(3+I*sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))+1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1-I*sqrt(3))^(1/3)+(1-I*sqrt(3))^(2/3))*(3+I*sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))+1/3*arctan((1+2*2^(1/3)*x/(1-I*sqrt(3))^(1/3))/sqrt(3))*(I+sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))],
[x^2*(-2+x^3)/(1-x^3+x^6),x,5,1/6*log(1-x^3+x^6)+arctan((1-2*x^3)/sqrt(3))/sqrt(3)],
[(1+x^3)/(x*(1-x^3+x^6)),x,7,log(x)-1/6*log(1-x^3+x^6)-arctan((1-2*x^3)/sqrt(3))/sqrt(3)],
[(1+x^3)/(x-x^4+x^7),x,8,log(x)-1/6*log(1-x^3+x^6)-arctan((1-2*x^3)/sqrt(3))/sqrt(3)],

# Integrands of the form (f x)^m (d+e x^3)^(q/2) (a+b x^3+c x^6)^p
[(d+e*x^3)^(5/2)*(a+b*x^3+c*x^6),x,6,30/124729*d*(16*c*d^2-58*b*d*e+667*a*e^2)*x*(d+e*x^3)^(3/2)/e^2+2/11339*(16*c*d^2-58*b*d*e+667*a*e^2)*x*(d+e*x^3)^(5/2)/e^2-2/667*(8*c*d-29*b*e)*x*(d+e*x^3)^(7/2)/e^2+2/29*c*x^4*(d+e*x^3)^(7/2)/e+54/124729*d^2*(16*c*d^2-58*b*d*e+667*a*e^2)*x*sqrt(d+e*x^3)/e^2+54/124729*3^(3/4)*d^3*(16*c*d^2-58*b*d*e+667*a*e^2)*(d^(1/3)+e^(1/3)*x)*EllipticF((e^(1/3)*x+d^(1/3)*(1-sqrt(3)))/(e^(1/3)*x+d^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(e^(1/3)*x+d^(1/3)*(1+sqrt(3)))^2)/(e^(7/3)*sqrt(d+e*x^3)*sqrt(d^(1/3)*(d^(1/3)+e^(1/3)*x)/(e^(1/3)*x+d^(1/3)*(1+sqrt(3)))^2))],
[(d+e*x^3)^(3/2)*(a+b*x^3+c*x^6),x,5,2/4301*(16*c*d^2-46*b*d*e+391*a*e^2)*x*(d+e*x^3)^(3/2)/e^2-2/391*(8*c*d-23*b*e)*x*(d+e*x^3)^(5/2)/e^2+2/23*c*x^4*(d+e*x^3)^(5/2)/e+18/21505*d*(16*c*d^2-46*b*d*e+391*a*e^2)*x*sqrt(d+e*x^3)/e^2+18/21505*3^(3/4)*d^2*(16*c*d^2-46*b*d*e+391*a*e^2)*(d^(1/3)+e^(1/3)*x)*EllipticF((e^(1/3)*x+d^(1/3)*(1-sqrt(3)))/(e^(1/3)*x+d^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(e^(1/3)*x+d^(1/3)*(1+sqrt(3)))^2)/(e^(7/3)*sqrt(d+e*x^3)*sqrt(d^(1/3)*(d^(1/3)+e^(1/3)*x)/(e^(1/3)*x+d^(1/3)*(1+sqrt(3)))^2))],
[(d+e*x^3)^(1/2)*(a+b*x^3+c*x^6),x,4,-2/187*(8*c*d-17*b*e)*x*(d+e*x^3)^(3/2)/e^2+2/17*c*x^4*(d+e*x^3)^(3/2)/e+2/935*(16*c*d^2-34*b*d*e+187*a*e^2)*x*sqrt(d+e*x^3)/e^2+2/935*3^(3/4)*d*(16*c*d^2-34*b*d*e+187*a*e^2)*(d^(1/3)+e^(1/3)*x)*EllipticF((e^(1/3)*x+d^(1/3)*(1-sqrt(3)))/(e^(1/3)*x+d^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(e^(1/3)*x+d^(1/3)*(1+sqrt(3)))^2)/(e^(7/3)*sqrt(d+e*x^3)*sqrt(d^(1/3)*(d^(1/3)+e^(1/3)*x)/(e^(1/3)*x+d^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3+c*x^6)/(d+e*x^3)^(1/2),x,3,-2/55*(8*c*d-11*b*e)*x*sqrt(d+e*x^3)/e^2+2/11*c*x^4*sqrt(d+e*x^3)/e+2/55*(16*c*d^2-22*b*d*e+55*a*e^2)*(d^(1/3)+e^(1/3)*x)*EllipticF((e^(1/3)*x+d^(1/3)*(1-sqrt(3)))/(e^(1/3)*x+d^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(e^(1/3)*x+d^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*e^(7/3)*sqrt(d+e*x^3)*sqrt(d^(1/3)*(d^(1/3)+e^(1/3)*x)/(e^(1/3)*x+d^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3+c*x^6)/(d+e*x^3)^(3/2),x,3,2/3*(c*d^2-b*d*e+a*e^2)*x/(d*e^2*sqrt(d+e*x^3))+2/5*c*x*sqrt(d+e*x^3)/e^2-2/15*(16*c*d^2-5*e*(2*b*d+a*e))*(d^(1/3)+e^(1/3)*x)*EllipticF((e^(1/3)*x+d^(1/3)*(1-sqrt(3)))/(e^(1/3)*x+d^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(e^(1/3)*x+d^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*d*e^(7/3)*sqrt(d+e*x^3)*sqrt(d^(1/3)*(d^(1/3)+e^(1/3)*x)/(e^(1/3)*x+d^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3+c*x^6)/(d+e*x^3)^(5/2),x,3,2/9*(c*d^2-b*d*e+a*e^2)*x/(d*e^2*(d+e*x^3)^(3/2))-2/27*(11*c*d^2-2*b*d*e-7*a*e^2)*x/(d^2*e^2*sqrt(d+e*x^3))+2/27*(16*c*d^2+e*(2*b*d+7*a*e))*(d^(1/3)+e^(1/3)*x)*EllipticF((e^(1/3)*x+d^(1/3)*(1-sqrt(3)))/(e^(1/3)*x+d^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(e^(1/3)*x+d^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*d^2*e^(7/3)*sqrt(d+e*x^3)*sqrt(d^(1/3)*(d^(1/3)+e^(1/3)*x)/(e^(1/3)*x+d^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3+c*x^6)/(d+e*x^3)^(7/2),x,4,2/15*(c*d^2-b*d*e+a*e^2)*x/(d*e^2*(d+e*x^3)^(5/2))-2/135*(17*c*d^2-2*b*d*e-13*a*e^2)*x/(d^2*e^2*(d+e*x^3)^(3/2))+2/405*(16*c*d^2+14*b*d*e+91*a*e^2)*x/(d^3*e^2*sqrt(d+e*x^3))+2/405*(16*c*d^2+14*b*d*e+91*a*e^2)*(d^(1/3)+e^(1/3)*x)*EllipticF((e^(1/3)*x+d^(1/3)*(1-sqrt(3)))/(e^(1/3)*x+d^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(e^(1/3)*x+d^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*d^3*e^(7/3)*sqrt(d+e*x^3)*sqrt(d^(1/3)*(d^(1/3)+e^(1/3)*x)/(e^(1/3)*x+d^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3+c*x^6)/(d+e*x^3)^(9/2),x,5,2/21*(c*d^2-b*d*e+a*e^2)*x/(d*e^2*(d+e*x^3)^(7/2))-2/315*(23*c*d^2-2*b*d*e-19*a*e^2)*x/(d^2*e^2*(d+e*x^3)^(5/2))+2/2835*(16*c*d^2+26*b*d*e+247*a*e^2)*x/(d^3*e^2*(d+e*x^3)^(3/2))+2/1215*(16*c*d^2+26*b*d*e+247*a*e^2)*x/(d^4*e^2*sqrt(d+e*x^3))+2/1215*(16*c*d^2+26*b*d*e+247*a*e^2)*(d^(1/3)+e^(1/3)*x)*EllipticF((e^(1/3)*x+d^(1/3)*(1-sqrt(3)))/(e^(1/3)*x+d^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(e^(1/3)*x+d^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*d^4*e^(7/3)*sqrt(d+e*x^3)*sqrt(d^(1/3)*(d^(1/3)+e^(1/3)*x)/(e^(1/3)*x+d^(1/3)*(1+sqrt(3)))^2))],

# Integrands of the form (f x)^m (d+e x^4)^q (a+b x^4+c x^8)^p

# Integrands of the form (f x)^m (d+e x^4)^q (a+b x^4+c x^8)^p with b=0

# Integrands of the form (f x)^m (d+e x^4)^q (a+b x^4+c x^8)^p with b^2-4 a c=0

# Integrands of the form (f x)^m (d+e x^4)^q (a+b x^4+c x^8)^p with c d^2-b d e+a e^2=0

# Integrands of the form (f x)^m (d+e x^4)^q (a+b x^4+c x^8)^p

# Integrands of the form (f x)^m (d+e x^4)^q (a+b x^4+c x^8)^p

# p>0

# p<0
[x^4*(d+e*x^4)/(a+b*x^4+c*x^8),x,8,e*x/c-1/2*arctan(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(c*d-b*e+(b*c*d-b^2*e+2*a*c*e)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(-b-sqrt(b^2-4*a*c))^(3/4))-1/2*arctanh(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(c*d-b*e+(b*c*d-b^2*e+2*a*c*e)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(-b-sqrt(b^2-4*a*c))^(3/4))-1/2*arctan(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(c*d-b*e+(-b*c*d+b^2*e-2*a*c*e)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(-b+sqrt(b^2-4*a*c))^(3/4))-1/2*arctanh(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(c*d-b*e+(-b*c*d+b^2*e-2*a*c*e)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(-b+sqrt(b^2-4*a*c))^(3/4))],
[x^3*(d+e*x^4)/(a+b*x^4+c*x^8),x,5,1/8*e*log(a+b*x^4+c*x^8)/c-1/4*(2*c*d-b*e)*arctanh((b+2*c*x^4)/sqrt(b^2-4*a*c))/(c*sqrt(b^2-4*a*c))],
[x^2*(d+e*x^4)/(a+b*x^4+c*x^8),x,7,1/2*arctan(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(e+(-2*c*d+b*e)/sqrt(b^2-4*a*c))/(2^(3/4)*c^(3/4)*(-b-sqrt(b^2-4*a*c))^(1/4))-1/2*arctanh(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(e+(-2*c*d+b*e)/sqrt(b^2-4*a*c))/(2^(3/4)*c^(3/4)*(-b-sqrt(b^2-4*a*c))^(1/4))+1/2*arctan(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(e+(2*c*d-b*e)/sqrt(b^2-4*a*c))/(2^(3/4)*c^(3/4)*(-b+sqrt(b^2-4*a*c))^(1/4))-1/2*arctanh(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(e+(2*c*d-b*e)/sqrt(b^2-4*a*c))/(2^(3/4)*c^(3/4)*(-b+sqrt(b^2-4*a*c))^(1/4))],
[x*(d+e*x^4)/(a+b*x^4+c*x^8),x,4,1/2*arctan(x^2*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(e+(2*c*d-b*e)/sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan(x^2*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(e+(-2*c*d+b*e)/sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^4)/(a+b*x^4+c*x^8),x,7,-1/2*arctan(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(e+(-2*c*d+b*e)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(1/4)*(-b-sqrt(b^2-4*a*c))^(3/4))-1/2*arctanh(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(e+(-2*c*d+b*e)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(1/4)*(-b-sqrt(b^2-4*a*c))^(3/4))-1/2*arctan(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(e+(2*c*d-b*e)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(1/4)*(-b+sqrt(b^2-4*a*c))^(3/4))-1/2*arctanh(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(e+(2*c*d-b*e)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(1/4)*(-b+sqrt(b^2-4*a*c))^(3/4))],
[(d+e*x^4)/(x*(a+b*x^4+c*x^8)),x,7,d*log(x)/a-1/8*d*log(a+b*x^4+c*x^8)/a+1/4*(b*d-2*a*e)*arctanh((b+2*c*x^4)/sqrt(b^2-4*a*c))/(a*sqrt(b^2-4*a*c))],
[(d+e*x^4)/(x^2*(a+b*x^4+c*x^8)),x,8,-d/(a*x)-1/2*c^(1/4)*arctan(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(d+(-b*d+2*a*e)/sqrt(b^2-4*a*c))/(2^(3/4)*a*(-b-sqrt(b^2-4*a*c))^(1/4))+1/2*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(d+(-b*d+2*a*e)/sqrt(b^2-4*a*c))/(2^(3/4)*a*(-b-sqrt(b^2-4*a*c))^(1/4))-1/2*c^(1/4)*arctan(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(d+(b*d-2*a*e)/sqrt(b^2-4*a*c))/(2^(3/4)*a*(-b+sqrt(b^2-4*a*c))^(1/4))+1/2*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(d+(b*d-2*a*e)/sqrt(b^2-4*a*c))/(2^(3/4)*a*(-b+sqrt(b^2-4*a*c))^(1/4))],
[(d+e*x^4)/(x^3*(a+b*x^4+c*x^8)),x,5,-1/2*d/(a*x^2)-1/2*arctan(x^2*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(d+(b*d-2*a*e)/sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-1/2*arctan(x^2*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(d+(-b*d+2*a*e)/sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^4)/(x^4*(a+b*x^4+c*x^8)),x,8,-1/3*d/(a*x^3)+1/2*c^(3/4)*arctan(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(d+(-b*d+2*a*e)/sqrt(b^2-4*a*c))/(2^(1/4)*a*(-b-sqrt(b^2-4*a*c))^(3/4))+1/2*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(d+(-b*d+2*a*e)/sqrt(b^2-4*a*c))/(2^(1/4)*a*(-b-sqrt(b^2-4*a*c))^(3/4))+1/2*c^(3/4)*arctan(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(d+(b*d-2*a*e)/sqrt(b^2-4*a*c))/(2^(1/4)*a*(-b+sqrt(b^2-4*a*c))^(3/4))+1/2*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(d+(b*d-2*a*e)/sqrt(b^2-4*a*c))/(2^(1/4)*a*(-b+sqrt(b^2-4*a*c))^(3/4))],
[x^4*(1-x^4)/(1-x^4+x^8),x,20,-x-1/2*arctan((-2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(6)+1/2*arctan((2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(6)-1/2*arctan((-2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(6)+1/2*arctan((2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(6)-1/4*log(1+x^2-x*sqrt(2-sqrt(3)))/sqrt(6)+1/4*log(1+x^2+x*sqrt(2-sqrt(3)))/sqrt(6)-1/4*log(1+x^2-x*sqrt(2+sqrt(3)))/sqrt(6)+1/4*log(1+x^2+x*sqrt(2+sqrt(3)))/sqrt(6)],
[x^3*(1-x^4)/(1-x^4+x^8),x,5,-1/8*log(1-x^4+x^8)-1/4*arctan((1-2*x^4)/sqrt(3))/sqrt(3)],
[x^2*(1-x^4)/(1-x^4+x^8),x,21,1/8*log(1+x^2-x*sqrt(2-sqrt(3)))*sqrt(1/3*(2-sqrt(3)))-1/8*log(1+x^2+x*sqrt(2-sqrt(3)))*sqrt(1/3*(2-sqrt(3)))+1/4*arctan((-2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(3*(2-sqrt(3)))-1/4*arctan((2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(3*(2-sqrt(3)))-1/8*log(1+x^2-x*sqrt(2+sqrt(3)))*sqrt(1/3*(2+sqrt(3)))+1/8*log(1+x^2+x*sqrt(2+sqrt(3)))*sqrt(1/3*(2+sqrt(3)))-1/4*arctan((-2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(3*(2+sqrt(3)))+1/4*arctan((2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(3*(2+sqrt(3)))],
[x*(1-x^4)/(1-x^4+x^8),x,4,-1/4*log(1+x^4-x^2*sqrt(3))/sqrt(3)+1/4*log(1+x^4+x^2*sqrt(3))/sqrt(3)],
[(1-x^4)/(1-x^4+x^8),x,19,1/8*log(1+x^2-x*sqrt(2-sqrt(3)))*sqrt(1/3*(2-sqrt(3)))-1/8*log(1+x^2+x*sqrt(2-sqrt(3)))*sqrt(1/3*(2-sqrt(3)))-1/4*arctan((-2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(3*(2-sqrt(3)))+1/4*arctan((2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(3*(2-sqrt(3)))-1/8*log(1+x^2-x*sqrt(2+sqrt(3)))*sqrt(1/3*(2+sqrt(3)))+1/8*log(1+x^2+x*sqrt(2+sqrt(3)))*sqrt(1/3*(2+sqrt(3)))+1/4*arctan((-2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(3*(2+sqrt(3)))-1/4*arctan((2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(3*(2+sqrt(3)))],
[(1-x^4)/(x*(1-x^4+x^8)),x,7,log(x)-1/8*log(1-x^4+x^8)+1/4*arctan((1-2*x^4)/sqrt(3))/sqrt(3)],
[(1-x^4)/(x^2*(1-x^4+x^8)),x,20,(-1)/x+1/2*arctan((-2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(6)-1/2*arctan((2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(6)+1/2*arctan((-2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(6)-1/2*arctan((2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(6)-1/4*log(1+x^2-x*sqrt(2-sqrt(3)))/sqrt(6)+1/4*log(1+x^2+x*sqrt(2-sqrt(3)))/sqrt(6)-1/4*log(1+x^2-x*sqrt(2+sqrt(3)))/sqrt(6)+1/4*log(1+x^2+x*sqrt(2+sqrt(3)))/sqrt(6)],
[(1-x^4)/(x^3*(1-x^4+x^8)),x,11,(-1/2)/x^2+1/4*arctan(-2*x^2+sqrt(3))-1/4*arctan(2*x^2+sqrt(3))-1/8*log(1+x^4-x^2*sqrt(3))/sqrt(3)+1/8*log(1+x^4+x^2*sqrt(3))/sqrt(3)],
[(1-x^4)/(x^4*(1-x^4+x^8)),x,21,(-1/3)/x^3-1/4*arctan((-2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))*sqrt(1/3*(2-sqrt(3)))+1/4*arctan((2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))*sqrt(1/3*(2-sqrt(3)))-1/8*log(1+x^2-x*sqrt(2+sqrt(3)))*sqrt(1/3*(2-sqrt(3)))+1/8*log(1+x^2+x*sqrt(2+sqrt(3)))*sqrt(1/3*(2-sqrt(3)))+1/4*arctan((-2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))*sqrt(1/3*(2+sqrt(3)))-1/4*arctan((2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))*sqrt(1/3*(2+sqrt(3)))+1/8*log(1+x^2-x*sqrt(2-sqrt(3)))*sqrt(1/3*(2+sqrt(3)))-1/8*log(1+x^2+x*sqrt(2-sqrt(3)))*sqrt(1/3*(2+sqrt(3)))],

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

# Integrands of the form x^m (d+e x)^q (a+b/x+c/x^2)^p
[x^3/((a+c/x^2+b/x)*(d+e*x)),x,7,(a^2*d^2+b^2*e^2+a*e*(b*d-c*e))*x/(a^3*e^3)-1/2*(a*d+b*e)*x^2/(a^2*e^2)+1/3*x^3/(a*e)-d^5*log(d+e*x)/(e^4*(a*d^2-e*(b*d-c*e)))+1/2*(b^4*d-3*a*b^2*c*d+a^2*c^2*d-b^3*c*e+2*a*b*c^2*e)*log(c+b*x+a*x^2)/(a^4*(a*d^2-e*(b*d-c*e)))+(b^5*d-5*a*b^3*c*d+5*a^2*b*c^2*d-b^4*c*e+4*a*b^2*c^2*e-2*a^2*c^3*e)*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/(a^4*(a*d^2-e*(b*d-c*e))*sqrt(b^2-4*a*c))],
[x^2/((a+c/x^2+b/x)*(d+e*x)),x,7,-(a*d+b*e)*x/(a^2*e^2)+1/2*x^2/(a*e)+d^4*log(d+e*x)/(e^3*(a*d^2-e*(b*d-c*e)))-1/2*(b^3*d-2*a*b*c*d-b^2*c*e+a*c^2*e)*log(c+b*x+a*x^2)/(a^3*(a*d^2-e*(b*d-c*e)))-(b^4*d-4*a*b^2*c*d+2*a^2*c^2*d-b^3*c*e+3*a*b*c^2*e)*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/(a^3*(a*d^2-e*(b*d-c*e))*sqrt(b^2-4*a*c))],
[x/((a+c/x^2+b/x)*(d+e*x)),x,7,x/(a*e)-d^3*log(d+e*x)/(e^2*(a*d^2-e*(b*d-c*e)))+1/2*(b^2*d-a*c*d-b*c*e)*log(c+b*x+a*x^2)/(a^2*(a*d^2-e*(b*d-c*e)))+(b^3*d-3*a*b*c*d-b^2*c*e+2*a*c^2*e)*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/(a^2*(a*d^2-e*(b*d-c*e))*sqrt(b^2-4*a*c))],
[1/((a+c/x^2+b/x)*(d+e*x)),x,7,d^2*log(d+e*x)/(e*(a*d^2-b*d*e+c*e^2))-1/2*(b*d-c*e)*log(c+b*x+a*x^2)/(a*(a*d^2-e*(b*d-c*e)))-(b^2*d-2*a*c*d-b*c*e)*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/(a*(a*d^2-e*(b*d-c*e))*sqrt(b^2-4*a*c))],
[1/((a+c/x^2+b/x)*x*(d+e*x)),x,7,-d*log(d+e*x)/(a*d^2-e*(b*d-c*e))+1/2*d*log(c+b*x+a*x^2)/(a*d^2-e*(b*d-c*e))+(b*d-2*c*e)*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/((a*d^2-e*(b*d-c*e))*sqrt(b^2-4*a*c))],
[1/((a+c/x^2+b/x)*x^2*(d+e*x)),x,7,e*log(d+e*x)/(a*d^2-b*d*e+c*e^2)-1/2*e*log(c+b*x+a*x^2)/(a*d^2-b*d*e+c*e^2)-(2*a*d-b*e)*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/((a*d^2-e*(b*d-c*e))*sqrt(b^2-4*a*c))],
[1/((a+c/x^2+b/x)*x^3*(d+e*x)),x,7,log(x)/(c*d)-e^2*log(d+e*x)/(d*(a*d^2-b*d*e+c*e^2))-1/2*(a*d-b*e)*log(c+b*x+a*x^2)/(c*(a*d^2-e*(b*d-c*e)))+(a*b*d-b^2*e+2*a*c*e)*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/(c*(a*d^2-e*(b*d-c*e))*sqrt(b^2-4*a*c))],
[1/((a+c/x^2+b/x)*x^4*(d+e*x)),x,7,(-1)/(c*d*x)-(b*d+c*e)*log(x)/(c^2*d^2)+e^3*log(d+e*x)/(d^2*(a*d^2-e*(b*d-c*e)))+1/2*(a*b*d-b^2*e+a*c*e)*log(c+b*x+a*x^2)/(c^2*(a*d^2-e*(b*d-c*e)))+(2*a^2*c*d+b^3*e-a*b*(b*d+3*c*e))*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/(c^2*(a*d^2-e*(b*d-c*e))*sqrt(b^2-4*a*c))],
[1/((a+c/x^2+b/x)*x^5*(d+e*x)),x,7,(-1/2)/(c*d*x^2)+(b*d+c*e)/(c^2*d^2*x)+(b^2*d^2+b*c*d*e-c*(a*d^2-c*e^2))*log(x)/(c^3*d^3)-e^4*log(d+e*x)/(d^3*(a*d^2-e*(b*d-c*e)))+1/2*(a^2*c*d+b^3*e-a*b*(b*d+2*c*e))*log(c+b*x+a*x^2)/(c^3*(a*d^2-e*(b*d-c*e)))-(b^4*e+a^2*c*(3*b*d+2*c*e)-a*b^2*(b*d+4*c*e))*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/(c^3*(a*d^2-e*(b*d-c*e))*sqrt(b^2-4*a*c))],
[x^3/((a+c/x^2+b/x)*(d+e*x)^2),x,7,-(2*a*d+b*e)*x/(a^2*e^3)+1/2*x^2/(a*e^2)+d^5/(e^4*(a*d^2-e*(b*d-c*e))*(d+e*x))+d^4*(3*a*d^2-e*(4*b*d-5*c*e))*log(d+e*x)/(e^4*(a*d^2-e*(b*d-c*e))^2)+1/2*(b^4*d^2-2*b^3*c*d*e+4*a*b*c^2*d*e+a*c^2*(a*d^2-c*e^2)-b^2*c*(3*a*d^2-c*e^2))*log(c+b*x+a*x^2)/(a^3*(a*d^2-e*(b*d-c*e))^2)+(b^5*d^2-2*b^4*c*d*e+8*a*b^2*c^2*d*e-4*a^2*c^3*d*e+a*b*c^2*(5*a*d^2-3*c*e^2)-b^3*c*(5*a*d^2-c*e^2))*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/(a^3*(a*d^2-e*(b*d-c*e))^2*sqrt(b^2-4*a*c))],
[x^2/((a+c/x^2+b/x)*(d+e*x)^2),x,7,x/(a*e^2)-d^4/(e^3*(a*d^2-e*(b*d-c*e))*(d+e*x))-d^3*(2*a*d^2-e*(3*b*d-4*c*e))*log(d+e*x)/(e^3*(a*d^2-e*(b*d-c*e))^2)-1/2*(b*d-c*e)*(b^2*d-2*a*c*d-b*c*e)*log(c+b*x+a*x^2)/(a^2*(a*d^2-e*(b*d-c*e))^2)-(b^4*d^2-2*b^3*c*d*e+6*a*b*c^2*d*e+2*a*c^2*(a*d^2-c*e^2)-b^2*c*(4*a*d^2-c*e^2))*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/(a^2*(a*d^2-e*(b*d-c*e))^2*sqrt(b^2-4*a*c))],
[x/((a+c/x^2+b/x)*(d+e*x)^2),x,7,d^3/(e^2*(a*d^2-e*(b*d-c*e))*(d+e*x))+d^2*(a*d^2-e*(2*b*d-3*c*e))*log(d+e*x)/(e^2*(a*d^2-e*(b*d-c*e))^2)+1/2*(b^2*d^2-2*b*c*d*e-c*(a*d^2-c*e^2))*log(c+b*x+a*x^2)/(a*(a*d^2-e*(b*d-c*e))^2)+(b^3*d^2-2*b^2*c*d*e+4*a*c^2*d*e-b*c*(3*a*d^2-c*e^2))*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/(a*(a*d^2-e*(b*d-c*e))^2*sqrt(b^2-4*a*c))],
[1/((a+c/x^2+b/x)*(d+e*x)^2),x,7,-d^2/(e*(a*d^2-b*d*e+c*e^2)*(d+e*x))+d*(b*d-2*c*e)*log(d+e*x)/(a*d^2-e*(b*d-c*e))^2-1/2*d*(b*d-2*c*e)*log(c+b*x+a*x^2)/(a*d^2-e*(b*d-c*e))^2-(b^2*d^2-2*b*c*d*e-2*c*(a*d^2-c*e^2))*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/((a*d^2-e*(b*d-c*e))^2*sqrt(b^2-4*a*c))],
[1/((a+c/x^2+b/x)*x*(d+e*x)^2),x,7,d/((a*d^2-b*d*e+c*e^2)*(d+e*x))-(a*d^2-c*e^2)*log(d+e*x)/(a*d^2-e*(b*d-c*e))^2+1/2*(a*d^2-c*e^2)*log(c+b*x+a*x^2)/(a*d^2-e*(b*d-c*e))^2+(b*c*e^2+a*d*(b*d-4*c*e))*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/((a*d^2-e*(b*d-c*e))^2*sqrt(b^2-4*a*c))],
[1/((a+c/x^2+b/x)*x^2*(d+e*x)^2),x,8,-e/((a*d^2-b*d*e+c*e^2)*(d+e*x))+e*(2*a*d-b*e)*log(d+e*x)/(a*d^2-e*(b*d-c*e))^2-1/2*e*(2*a*d-b*e)*log(c+b*x+a*x^2)/(a*d^2-e*(b*d-c*e))^2-(2*a^2*d^2+b^2*e^2-2*a*e*(b*d+c*e))*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/((a*d^2-e*(b*d-c*e))^2*sqrt(b^2-4*a*c))],
[1/((a+c/x^2+b/x)*x^3*(d+e*x)^2),x,7,e^2/(d*(a*d^2-b*d*e+c*e^2)*(d+e*x))+log(x)/(c*d^2)-e^2*(3*a*d^2-e*(2*b*d-c*e))*log(d+e*x)/(d^2*(a*d^2-e*(b*d-c*e))^2)-1/2*(a^2*d^2+b^2*e^2-a*e*(2*b*d+c*e))*log(c+b*x+a*x^2)/(c*(a*d^2-e*(b*d-c*e))^2)+(b^3*e^2-a*b*e*(2*b*d+3*c*e)+a^2*d*(b*d+4*c*e))*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/(c*(a*d^2-e*(b*d-c*e))^2*sqrt(b^2-4*a*c))],
[1/((a+c/x^2+b/x)*x^4*(d+e*x)^2),x,7,(-1)/(c*d^2*x)-e^3/(d^2*(a*d^2-e*(b*d-c*e))*(d+e*x))-(b*d+2*c*e)*log(x)/(c^2*d^3)+e^3*(4*a*d^2-e*(3*b*d-2*c*e))*log(d+e*x)/(d^3*(a*d^2-e*(b*d-c*e))^2)+1/2*(a*d-b*e)*(a*b*d-b^2*e+2*a*c*e)*log(c+b*x+a*x^2)/(c^2*(a*d^2-e*(b*d-c*e))^2)+(2*a^3*c*d^2-b^4*e^2+2*a*b^2*e*(b*d+2*c*e)-a^2*(b^2*d^2+6*b*c*d*e+2*c^2*e^2))*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/(c^2*(a*d^2-e*(b*d-c*e))^2*sqrt(b^2-4*a*c))],
[1/((a+c/x^2+b/x)*x^5*(d+e*x)^2),x,7,(-1/2)/(c*d^2*x^2)+(b*d+2*c*e)/(c^2*d^3*x)+e^4/(d^3*(a*d^2-e*(b*d-c*e))*(d+e*x))+(b^2*d^2+2*b*c*d*e-c*(a*d^2-3*c*e^2))*log(x)/(c^3*d^4)-e^4*(5*a*d^2-e*(4*b*d-3*c*e))*log(d+e*x)/(d^4*(a*d^2-e*(b*d-c*e))^2)+1/2*(a^3*c*d^2-b^4*e^2+a*b^2*e*(2*b*d+3*c*e)-a^2*(b^2*d^2+4*b*c*d*e+c^2*e^2))*log(c+b*x+a*x^2)/(c^3*(a*d^2-e*(b*d-c*e))^2)+(b^5*e^2-a^3*c*d*(3*b*d+4*c*e)-a*b^3*e*(2*b*d+5*c*e)+a^2*b*(b^2*d^2+8*b*c*d*e+5*c^2*e^2))*arctanh((b+2*a*x)/sqrt(b^2-4*a*c))/(c^3*(a*d^2-e*(b*d-c*e))^2*sqrt(b^2-4*a*c))],

# Integrands of the form x^m (d+e x)^(q/2) (a+b/x+c/x^2)^(p/2)
[x^4*sqrt(a+c/x^2+b/x)*sqrt(d+e*x),x,11,2/3465*(233*a^3*d^3+48*b^3*e^3+a*b*e^2*(67*b*d-157*c*e)+4*a^2*d*e*(18*b*d-37*c*e))*x*(d+e*x)^(3/2)*sqrt(a+c/x^2+b/x)/(a^3*e^4)-2/693*(29*a^2*d^2+8*b^2*e^2+a*e*(19*b*d-18*c*e))*x*(d+e*x)^(5/2)*sqrt(a+c/x^2+b/x)/(a^2*e^4)+2/99*(a*d+b*e)*x*(d+e*x)^(7/2)*sqrt(a+c/x^2+b/x)/(a*e^4)-2/3465*(187*a^4*d^4+64*b^4*e^4+4*a*b^2*e^3*(7*b*d-69*c*e)-4*a^3*d^2*e*(2*b*d+3*c*e)+3*a^2*e^2*(3*b^2*d^2-29*b*c*d*e+50*c^2*e^2))*x*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)/(a^4*e^4)+2/11*x^5*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)+1/3465*(128*a^5*d^5+128*b^5*e^5-4*a^4*d^3*e*(14*b*d-27*c*e)-8*a*b^3*e^4*(7*b*d+87*c*e)-a^2*b*e^3*(37*b^2*d^2-258*b*c*d*e-771*c^2*e^2)-a^3*d*e^2*(37*b^2*d^2-135*b*c*d*e+156*c^2*e^2))*x*EllipticE(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))/(a^5*e^5*(c+b*x+a*x^2)*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c)))))-2/3465*(a*d^2-e*(b*d-c*e))*(128*a^4*d^4-64*b^4*e^4-4*a*b^2*e^3*(7*b*d-69*c*e)+4*a^3*d^2*e*(2*b*d+3*c*e)-3*a^2*e^2*(3*b^2*d^2-29*b*c*d*e+50*c^2*e^2))*x*EllipticF(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c))))/(a^5*e^5*(c+b*x+a*x^2)*sqrt(d+e*x))],
[x^3*sqrt(a+c/x^2+b/x)*sqrt(d+e*x),x,10,-4/315*(8*a^2*d^2+3*b^2*e^2+a*e*(4*b*d-7*c*e))*x*(d+e*x)^(3/2)*sqrt(a+c/x^2+b/x)/(a^2*e^3)+2/63*(a*d+b*e)*x*(d+e*x)^(5/2)*sqrt(a+c/x^2+b/x)/(a*e^3)+2/315*(19*a^3*d^3-6*a^2*c*d*e^2+8*b^3*e^3+3*a*b*e^2*(b*d-9*c*e))*x*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)/(a^3*e^3)+2/9*x^4*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)-2/315*(8*a^4*d^4+8*b^4*e^4-a^3*d^2*e*(4*b*d-9*c*e)-4*a*b^2*e^3*(b*d+9*c*e)-3*a^2*e^2*(b^2*d^2-5*b*c*d*e-7*c^2*e^2))*x*EllipticE(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))/(a^4*e^4*(c+b*x+a*x^2)*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c)))))+2/315*(16*a^3*d^3+6*a^2*c*d*e^2-8*b^3*e^3-3*a*b*e^2*(b*d-9*c*e))*(a*d^2-e*(b*d-c*e))*x*EllipticF(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c))))/(a^4*e^4*(c+b*x+a*x^2)*sqrt(d+e*x))],
[x^2*sqrt(a+c/x^2+b/x)*sqrt(d+e*x),x,8,-2/105*x*(4*a^2*d^2+4*b^2*e^2-a*e*(2*b*d-5*c*e)-3*a*e*(a*d-4*b*e)*x)*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)/(a^2*e^2)+2/7*x*(c+b*x+a*x^2)*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)/a+1/105*(8*a^3*d^3+8*b^3*e^3-a^2*d*e*(5*b*d-16*c*e)-a*b*e^2*(5*b*d+29*c*e))*x*EllipticE(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))/(a^3*e^3*(c+b*x+a*x^2)*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c)))))-2/105*(8*a^2*d^2-4*b^2*e^2-a*e*(b*d-10*c*e))*(a*d^2-e*(b*d-c*e))*x*EllipticF(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c))))/(a^3*e^3*(c+b*x+a*x^2)*sqrt(d+e*x))],
[x*sqrt(a+c/x^2+b/x)*sqrt(d+e*x),x,8,2/5*x*(d+e*x)^(3/2)*sqrt(a+c/x^2+b/x)/e-2/15*(2*a*d-b*e)*x*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)/(a*e)-2/15*(a^2*d^2+b^2*e^2-a*e*(b*d+3*c*e))*x*EllipticE(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))/(a^2*e^2*(c+b*x+a*x^2)*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c)))))+2/15*(2*a*d-b*e)*(a*d^2-e*(b*d-c*e))*x*EllipticF(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c))))/(a^2*e^2*(c+b*x+a*x^2)*sqrt(d+e*x))],
[sqrt(a+c/x^2+b/x)*sqrt(d+e*x),x,16,2/3*x*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)+1/3*(a*d+b*e)*x*EllipticE(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))/(a*e*(c+b*x+a*x^2)*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c)))))-2/3*d*(a*d+b*e)*x*EllipticF(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c))))/(a*e*(c+b*x+a*x^2)*sqrt(d+e*x))+4/3*(b*d+c*e)*x*EllipticF(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c))))/(a*(c+b*x+a*x^2)*sqrt(d+e*x))-c*x*EllipticPi(sqrt(2)*sqrt(a)*sqrt(d+e*x)/sqrt(2*a*d-e*(b-sqrt(b^2-4*a*c))),1/2*(2*a*d-b*e+e*sqrt(b^2-4*a*c))/(a*d),sqrt((b-2*a*d/e-sqrt(b^2-4*a*c))/(b-2*a*d/e+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(a+c/x^2+b/x)*sqrt(1-2*a*(d+e*x)/(2*a*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2*a*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c))))/((c+b*x+a*x^2)*sqrt(a))],
[sqrt(a+c/x^2+b/x)*sqrt(d+e*x)/x,x,16,-sqrt(a+c/x^2+b/x)*sqrt(d+e*x)+3*x*EllipticE(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))/((c+b*x+a*x^2)*sqrt(2)*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c)))))-3*d*x*EllipticF(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c))))/((c+b*x+a*x^2)*sqrt(d+e*x))+2*(a*d+b*e)*x*EllipticF(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c))))/(a*(c+b*x+a*x^2)*sqrt(d+e*x))-(b*d+c*e)*x*EllipticPi(sqrt(2)*sqrt(a)*sqrt(d+e*x)/sqrt(2*a*d-e*(b-sqrt(b^2-4*a*c))),1/2*(2*a*d-b*e+e*sqrt(b^2-4*a*c))/(a*d),sqrt((b-2*a*d/e-sqrt(b^2-4*a*c))/(b-2*a*d/e+sqrt(b^2-4*a*c))))*sqrt(a+c/x^2+b/x)*sqrt(1-2*a*(d+e*x)/(2*a*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2*a*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c))))/(d*(c+b*x+a*x^2)*sqrt(2)*sqrt(a))],
[sqrt(a+c/x^2+b/x)*sqrt(d+e*x)/x^2,x,24,-1/4*(b*d+c*e)*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)/(c*d)-1/2*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)/x+1/4*(b*d+c*e)*x*EllipticE(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(d+e*x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))/(c*d*(c+b*x+a*x^2)*sqrt(2)*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c)))))+3*e*x*EllipticF(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c))))/((c+b*x+a*x^2)*sqrt(2)*sqrt(d+e*x))-1/2*(b*d+c*e)*x*EllipticF(sqrt((b+2*a*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*a*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(a+c/x^2+b/x)*sqrt(-a*(c+b*x+a*x^2)/(b^2-4*a*c))*sqrt(a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c))))/(c*(c+b*x+a*x^2)*sqrt(2)*sqrt(d+e*x))-(a*d+b*e)*x*EllipticPi(sqrt(2)*sqrt(a)*sqrt(d+e*x)/sqrt(2*a*d-e*(b-sqrt(b^2-4*a*c))),1/2*(2*a*d-b*e+e*sqrt(b^2-4*a*c))/(a*d),sqrt((b-2*a*d/e-sqrt(b^2-4*a*c))/(b-2*a*d/e+sqrt(b^2-4*a*c))))*sqrt(a+c/x^2+b/x)*sqrt(1-2*a*(d+e*x)/(2*a*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2*a*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c))))/(d*(c+b*x+a*x^2)*sqrt(2)*sqrt(a))+1/4*(b*d+c*e)^2*x*EllipticPi(sqrt(2)*sqrt(a)*sqrt(d+e*x)/sqrt(2*a*d-e*(b-sqrt(b^2-4*a*c))),1/2*(2*a*d-b*e+e*sqrt(b^2-4*a*c))/(a*d),sqrt((b-2*a*d/e-sqrt(b^2-4*a*c))/(b-2*a*d/e+sqrt(b^2-4*a*c))))*sqrt(a+c/x^2+b/x)*sqrt(1-2*a*(d+e*x)/(2*a*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2*a*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*a*(d+e*x)/(2*a*d-e*(b+sqrt(b^2-4*a*c))))/(c*d^2*(c+b*x+a*x^2)*sqrt(2)*sqrt(a))],

# Integrands of the form (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p with symbolic n

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

# Integrands of the form (f x)^m (d+e x^n)^q (a+c x^(2 n))^p with p symbolic
[(f*x)^m*(d+e*x^n)^q*(a+c*x^(2*n))^p,x,0,Unintegrable((f*x)^m*(d+e*x^n)^q*(a+c*x^(2*n))^p,x)],
[(f*x)^m*(d+e*x^n)^3*(a+c*x^(2*n))^p,x,13,d^3*(f*x)^(1+m)*(a+c*x^(2*n))^p*hypergeom([1/2*(1+m)/n,-p],[1+1/2*(1+m)/n],-c*x^(2*n)/a)/(f*(1+m)*(1+c*x^(2*n)/a)^p)+3*d^2*e*x^(1+n)*(f*x)^m*(a+c*x^(2*n))^p*hypergeom([1/2*(1+m+n)/n,-p],[1/2*(1+m+3*n)/n],-c*x^(2*n)/a)/((1+m+n)*(1+c*x^(2*n)/a)^p)+3*d*e^2*x^(1+2*n)*(f*x)^m*(a+c*x^(2*n))^p*hypergeom([1/2*(1+m+2*n)/n,-p],[1/2*(1+m+4*n)/n],-c*x^(2*n)/a)/((1+m+2*n)*(1+c*x^(2*n)/a)^p)+e^3*x^(1+3*n)*(f*x)^m*(a+c*x^(2*n))^p*hypergeom([1/2*(1+m+3*n)/n,-p],[1/2*(1+m+5*n)/n],-c*x^(2*n)/a)/((1+m+3*n)*(1+c*x^(2*n)/a)^p)],
[(f*x)^m*(d+e*x^n)^2*(a+c*x^(2*n))^p,x,10,d^2*(f*x)^(1+m)*(a+c*x^(2*n))^p*hypergeom([1/2*(1+m)/n,-p],[1+1/2*(1+m)/n],-c*x^(2*n)/a)/(f*(1+m)*(1+c*x^(2*n)/a)^p)+2*d*e*x^(1+n)*(f*x)^m*(a+c*x^(2*n))^p*hypergeom([1/2*(1+m+n)/n,-p],[1/2*(1+m+3*n)/n],-c*x^(2*n)/a)/((1+m+n)*(1+c*x^(2*n)/a)^p)+e^2*x^(1+2*n)*(f*x)^m*(a+c*x^(2*n))^p*hypergeom([1/2*(1+m+2*n)/n,-p],[1/2*(1+m+4*n)/n],-c*x^(2*n)/a)/((1+m+2*n)*(1+c*x^(2*n)/a)^p)],
[(f*x)^m*(d+e*x^n)*(a+c*x^(2*n))^p,x,7,d*(f*x)^(1+m)*(a+c*x^(2*n))^p*hypergeom([1/2*(1+m)/n,-p],[1+1/2*(1+m)/n],-c*x^(2*n)/a)/(f*(1+m)*(1+c*x^(2*n)/a)^p)+e*x^(1+n)*(f*x)^m*(a+c*x^(2*n))^p*hypergeom([1/2*(1+m+n)/n,-p],[1/2*(1+m+3*n)/n],-c*x^(2*n)/a)/((1+m+n)*(1+c*x^(2*n)/a)^p)],
[(f*x)^m*(a+c*x^(2*n))^p/(d+e*x^n),x,6,x*(f*x)^m*(a+c*x^(2*n))^p*AppellF1(1/2*(1+m)/n,-p,1,1+1/2*(1+m)/n,-c*x^(2*n)/a,e^2*x^(2*n)/d^2)/(d*(1+m)*(1+c*x^(2*n)/a)^p)-e*x^(1+n)*(f*x)^m*(a+c*x^(2*n))^p*AppellF1(1/2*(1+m+n)/n,-p,1,1/2*(1+m+3*n)/n,-c*x^(2*n)/a,e^2*x^(2*n)/d^2)/(d^2*(1+m+n)*(1+c*x^(2*n)/a)^p)],
[(f*x)^m*(a+c*x^(2*n))^p/(d+e*x^n)^2,x,8,x*(f*x)^m*(a+c*x^(2*n))^p*AppellF1(1/2*(1+m)/n,-p,2,1+1/2*(1+m)/n,-c*x^(2*n)/a,e^2*x^(2*n)/d^2)/(d^2*(1+m)*(1+c*x^(2*n)/a)^p)-2*e*x^(1+n)*(f*x)^m*(a+c*x^(2*n))^p*AppellF1(1/2*(1+m+n)/n,-p,2,1/2*(1+m+3*n)/n,-c*x^(2*n)/a,e^2*x^(2*n)/d^2)/(d^3*(1+m+n)*(1+c*x^(2*n)/a)^p)+e^2*x^(1+2*n)*(f*x)^m*(a+c*x^(2*n))^p*AppellF1(1/2*(1+m+2*n)/n,-p,2,1/2*(1+m+4*n)/n,-c*x^(2*n)/a,e^2*x^(2*n)/d^2)/(d^4*(1+m+2*n)*(1+c*x^(2*n)/a)^p)],
[(f*x)^m*(a+c*x^(2*n))^p/(d+e*x^n)^3,x,10,x*(f*x)^m*(a+c*x^(2*n))^p*AppellF1(1/2*(1+m)/n,-p,3,1+1/2*(1+m)/n,-c*x^(2*n)/a,e^2*x^(2*n)/d^2)/(d^3*(1+m)*(1+c*x^(2*n)/a)^p)-3*e*x^(1+n)*(f*x)^m*(a+c*x^(2*n))^p*AppellF1(1/2*(1+m+n)/n,-p,3,1/2*(1+m+3*n)/n,-c*x^(2*n)/a,e^2*x^(2*n)/d^2)/(d^4*(1+m+n)*(1+c*x^(2*n)/a)^p)+3*e^2*x^(1+2*n)*(f*x)^m*(a+c*x^(2*n))^p*AppellF1(1/2*(1+m+2*n)/n,-p,3,1/2*(1+m+4*n)/n,-c*x^(2*n)/a,e^2*x^(2*n)/d^2)/(d^5*(1+m+2*n)*(1+c*x^(2*n)/a)^p)-e^3*x^(1+3*n)*(f*x)^m*(a+c*x^(2*n))^p*AppellF1(1/2*(1+m+3*n)/n,-p,3,1/2*(1+m+5*n)/n,-c*x^(2*n)/a,e^2*x^(2*n)/d^2)/(d^6*(1+m+3*n)*(1+c*x^(2*n)/a)^p)],

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

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

# p>0
[(b+2*c*x)*(a+b*x+c*x^2)^13,x,1,1/14*(a+b*x+c*x^2)^14],
[x*(b+2*c*x^2)*(a+b*x^2+c*x^4)^13,x,2,1/28*(a+b*x^2+c*x^4)^14],
[x^2*(b+2*c*x^3)*(a+b*x^3+c*x^6)^13,x,2,1/42*(a+b*x^3+c*x^6)^14],
[x^(-1+n)*(b+2*c*x^n)*(a+b*x^n+c*x^(2*n))^13,x,2,1/14*(a+b*x^n+c*x^(2*n))^14/n],
[(b+2*c*x)*(-a+b*x+c*x^2)^13,x,1,1/14*(a-b*x-c*x^2)^14],
[x*(b+2*c*x^2)*(-a+b*x^2+c*x^4)^13,x,2,1/28*(a-b*x^2-c*x^4)^14],
[x^2*(b+2*c*x^3)*(-a+b*x^3+c*x^6)^13,x,2,1/42*(a-b*x^3-c*x^6)^14],
[x^(-1+n)*(b+2*c*x^n)*(-a+b*x^n+c*x^(2*n))^13,x,2,1/14*(a-b*x^n-c*x^(2*n))^14/n],
[(b+2*c*x)*(b*x+c*x^2)^13,x,1,1/14*(b*x+c*x^2)^14],
[x*(b+2*c*x^2)*(b*x^2+c*x^4)^13,x,3,1/28*x^28*(b+c*x^2)^14],
[x^2*(b+2*c*x^3)*(b*x^3+c*x^6)^13,x,3,1/42*x^42*(b+c*x^3)^14],
[x^(-1+n)*(b+2*c*x^n)*(b*x^n+c*x^(2*n))^13,x,3,1/14*x^(14*n)*(b+c*x^n)^14/n],

# p<0
[(b+2*c*x)/(a+b*x+c*x^2),x,1,log(a+b*x+c*x^2)],
[x*(b+2*c*x^2)/(a+b*x^2+c*x^4),x,2,1/2*log(a+b*x^2+c*x^4)],
[x^2*(b+2*c*x^3)/(a+b*x^3+c*x^6),x,2,1/3*log(a+b*x^3+c*x^6)],
[x^(-1+n)*(b+2*c*x^n)/(a+b*x^n+c*x^(2*n)),x,2,log(a+b*x^n+c*x^(2*n))/n],
[(b+2*c*x)/(a+b*x+c*x^2)^8,x,1,(-1/7)/(a+b*x+c*x^2)^7],
[x*(b+2*c*x^2)/(a+b*x^2+c*x^4)^8,x,2,(-1/14)/(a+b*x^2+c*x^4)^7],
[x^2*(b+2*c*x^3)/(a+b*x^3+c*x^6)^8,x,2,(-1/21)/(a+b*x^3+c*x^6)^7],
[x^(-1+n)*(b+2*c*x^n)/(a+b*x^n+c*x^(2*n))^8,x,2,(-1/7)/(n*(a+b*x^n+c*x^(2*n))^7)],
[(b+2*c*x)/(-a+b*x+c*x^2),x,1,log(a-b*x-c*x^2)],
[x*(b+2*c*x^2)/(-a+b*x^2+c*x^4),x,2,1/2*log(a-b*x^2-c*x^4)],
[x^2*(b+2*c*x^3)/(-a+b*x^3+c*x^6),x,2,1/3*log(a-b*x^3-c*x^6)],
[x^(-1+n)*(b+2*c*x^n)/(-a+b*x^n+c*x^(2*n)),x,2,log(a-b*x^n-c*x^(2*n))/n],
[(b+2*c*x)/(-a+b*x+c*x^2)^8,x,1,1/7/(a-b*x-c*x^2)^7],
[x*(b+2*c*x^2)/(-a+b*x^2+c*x^4)^8,x,2,1/14/(a-b*x^2-c*x^4)^7],
[x^2*(b+2*c*x^3)/(-a+b*x^3+c*x^6)^8,x,2,1/21/(a-b*x^3-c*x^6)^7],
[x^(-1+n)*(b+2*c*x^n)/(-a+b*x^n+c*x^(2*n))^8,x,2,1/7/(n*(a-b*x^n-c*x^(2*n))^7)],
[(b+2*c*x)/(b*x+c*x^2),x,1,log(b*x+c*x^2)],
[x*(b+2*c*x^2)/(b*x^2+c*x^4),x,4,1/2*log(b*x^2+c*x^4),log(x)+1/2*log(b+c*x^2)],
[x^2*(b+2*c*x^3)/(b*x^3+c*x^6),x,4,1/3*log(b*x^3+c*x^6),log(x)+1/3*log(b+c*x^3)],
[x^(-1+n)*(b+2*c*x^n)/(b*x^n+c*x^(2*n)),x,4,log(x)+log(b+c*x^n)/n],
[(b+2*c*x)/(b*x+c*x^2)^8,x,1,(-1/7)/(b*x+c*x^2)^7],
[x*(b+2*c*x^2)/(b*x^2+c*x^4)^8,x,3,(-1/14)/(x^14*(b+c*x^2)^7)],
[x^2*(b+2*c*x^3)/(b*x^3+c*x^6)^8,x,3,(-1/21)/(x^21*(b+c*x^3)^7)],
[x^(-1+n)*(b+2*c*x^n)/(b*x^n+c*x^(2*n))^8,x,3,(-1/7)/(n*x^(7*n)*(b+c*x^n)^7)],

# Integrands of the form x^m (b + 2 c x^n) (a + b x^n + c x^(2 n))^p with p symbolic
[(b+2*c*x)*(a+b*x+c*x^2)^p,x,1,(a+b*x+c*x^2)^(1+p)/(1+p)],
[x*(b+2*c*x^2)*(a+b*x^2+c*x^4)^p,x,2,1/2*(a+b*x^2+c*x^4)^(1+p)/(1+p)],
[x^2*(b+2*c*x^3)*(a+b*x^3+c*x^6)^p,x,2,1/3*(a+b*x^3+c*x^6)^(1+p)/(1+p)],
[x^(-1+n)*(b+2*c*x^n)*(a+b*x^n+c*x^(2*n))^p,x,2,(a+b*x^n+c*x^(2*n))^(1+p)/(n*(1+p))],
[(b+2*c*x)*(-a+b*x+c*x^2)^p,x,1,(-a+b*x+c*x^2)^(1+p)/(1+p)],
[x*(b+2*c*x^2)*(-a+b*x^2+c*x^4)^p,x,2,1/2*(-a+b*x^2+c*x^4)^(1+p)/(1+p)],
[x^2*(b+2*c*x^3)*(-a+b*x^3+c*x^6)^p,x,2,1/3*(-a+b*x^3+c*x^6)^(1+p)/(1+p)],
[x^(-1+n)*(b+2*c*x^n)*(-a+b*x^n+c*x^(2*n))^p,x,2,(-a+b*x^n+c*x^(2*n))^(1+p)/(n*(1+p))],
[(b+2*c*x)*(b*x+c*x^2)^p,x,1,(b*x+c*x^2)^(1+p)/(1+p)],
[x*(b+2*c*x^2)*(b*x^2+c*x^4)^p,x,1,1/2*(b*x^2+c*x^4)^(1+p)/(1+p)],
[x^2*(b+2*c*x^3)*(b*x^3+c*x^6)^p,x,1,1/3*(b*x^3+c*x^6)^(1+p)/(1+p)],
[x^(-1+n)*(b+2*c*x^n)*(b*x^n+c*x^(2*n))^p,x,2,(b*x^n+c*x^(2*n))^(1+p)/(n*(1+p))],

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

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

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

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

# p>0

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

# Integrands of the form (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p with q symbolic

# p>0

# p<0
[(f*x)^m*(d+e*x^n)^q/(a+b*x^n+c*x^(2*n)),x,5,2*c*(f*x)^(1+m)*(d+e*x^n)^q*AppellF1((1+m)/n,1,-q,(1+m+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-e*x^n/d)/(f*(1+m)*(1+e*x^n/d)^q*(b-sqrt(b^2-4*a*c))*sqrt(b^2-4*a*c))-2*c*(f*x)^(1+m)*(d+e*x^n)^q*AppellF1((1+m)/n,1,-q,(1+m+n)/n,-2*c*x^n/(b+sqrt(b^2-4*a*c)),-e*x^n/d)/(f*(1+m)*(1+e*x^n/d)^q*sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c)))],
[x^2*(d+e*x^n)^q/(a+b*x^n+c*x^(2*n)),x,5,-2/3*c*x^3*(d+e*x^n)^q*AppellF1(3/n,1,-q,(3+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-e*x^n/d)/((1+e*x^n/d)^q*(b^2-4*a*c-b*sqrt(b^2-4*a*c)))-2/3*c*x^3*(d+e*x^n)^q*AppellF1(3/n,1,-q,(3+n)/n,-2*c*x^n/(b+sqrt(b^2-4*a*c)),-e*x^n/d)/((1+e*x^n/d)^q*(b^2-4*a*c+b*sqrt(b^2-4*a*c)))],
[x*(d+e*x^n)^q/(a+b*x^n+c*x^(2*n)),x,5,-c*x^2*(d+e*x^n)^q*AppellF1(2/n,1,-q,(2+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-e*x^n/d)/((1+e*x^n/d)^q*(b^2-4*a*c-b*sqrt(b^2-4*a*c)))-c*x^2*(d+e*x^n)^q*AppellF1(2/n,1,-q,(2+n)/n,-2*c*x^n/(b+sqrt(b^2-4*a*c)),-e*x^n/d)/((1+e*x^n/d)^q*(b^2-4*a*c+b*sqrt(b^2-4*a*c)))],
[(d+e*x^n)^q/(a+b*x^n+c*x^(2*n)),x,5,-2*c*x*(d+e*x^n)^q*AppellF1(1/n,1,-q,1+1/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-e*x^n/d)/((1+e*x^n/d)^q*(b^2-4*a*c-b*sqrt(b^2-4*a*c)))-2*c*x*(d+e*x^n)^q*AppellF1(1/n,1,-q,1+1/n,-2*c*x^n/(b+sqrt(b^2-4*a*c)),-e*x^n/d)/((1+e*x^n/d)^q*(b^2-4*a*c+b*sqrt(b^2-4*a*c)))],
[(d+e*x^n)^q/(x*(a+b*x^n+c*x^(2*n))),x,8,-(d+e*x^n)^(1+q)*hypergeom([1,1+q],[2+q],1+e*x^n/d)/(a*d*n*(1+q))+c*(d+e*x^n)^(1+q)*hypergeom([1,1+q],[2+q],2*c*(d+e*x^n)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(1+b/sqrt(b^2-4*a*c))/(a*n*(1+q)*(2*c*d-e*(b-sqrt(b^2-4*a*c))))+c*(d+e*x^n)^(1+q)*hypergeom([1,1+q],[2+q],2*c*(d+e*x^n)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(1-b/sqrt(b^2-4*a*c))/(a*n*(1+q)*(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x^n)^q/(x^2*(a+b*x^n+c*x^(2*n))),x,5,2*c*(d+e*x^n)^q*AppellF1((-1)/n,1,-q,(-1+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-e*x^n/d)/(x*(1+e*x^n/d)^q*(b^2-4*a*c-b*sqrt(b^2-4*a*c)))+2*c*(d+e*x^n)^q*AppellF1((-1)/n,1,-q,(-1+n)/n,-2*c*x^n/(b+sqrt(b^2-4*a*c)),-e*x^n/d)/(x*(1+e*x^n/d)^q*(b^2-4*a*c+b*sqrt(b^2-4*a*c)))],
[(d+e*x^n)^q/(x^3*(a+b*x^n+c*x^(2*n))),x,5,c*(d+e*x^n)^q*AppellF1((-2)/n,1,-q,(-2+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-e*x^n/d)/(x^2*(1+e*x^n/d)^q*(b^2-4*a*c-b*sqrt(b^2-4*a*c)))+c*(d+e*x^n)^q*AppellF1((-2)/n,1,-q,(-2+n)/n,-2*c*x^n/(b+sqrt(b^2-4*a*c)),-e*x^n/d)/(x^2*(1+e*x^n/d)^q*(b^2-4*a*c+b*sqrt(b^2-4*a*c)))],
#  {x^m*(d + e*x^n)^q/(a + b*x^n + c*x^(2*n))^2, x, 0, -((4*c^2*x^(1 + m)*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))*(d + e*x^n)^q*AppellF1[(1 + m)/n, 1, -q, (1 + m + n)/n, -((2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])), -((e*x^n)/d)])/((1 + (e*x^n)/d)^q*((b^2 - 4*a*c)^(3/2)*(1 + m)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)))) + (4*c^2*x^(1 + m)*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))*(d + e*x^n)^q*AppellF1[(1 + m)/n, 1, -q, (1 + m + n)/n, -((2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])), -((e*x^n)/d)])/((1 + (e*x^n)/d)^q*((b^2 - 4*a*c)^(3/2)*(1 + m)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))) + (4*c^2*x^(1 + m)*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^2*(d + e*x^n)^q*AppellF1[(1 + m)/n, 2, -q, (1 + m + n)/n, -((2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])), -((e*x^n)/d)])/((1 + (e*x^n)/d)^q*((b^2 - 4*a*c)*(1 + m)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)^2)) + (4*c^2*x^(1 + m)*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^2*(d + e*x^n)^q*AppellF1[(1 + m)/n, 2, -q, (1 + m + n)/n, -((2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])), -((e*x^n)/d)])/((1 + (e*x^n)/d)^q*((b^2 - 4*a*c)*(1 + m)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)^2))}
# {x^2*(d + e*x^n)^q/(a + b*x^n + c*x^(2*n))^2, x, 0, 0}
# {x^1*(d + e*x^n)^q/(a + b*x^n + c*x^(2*n))^2, x, 0, -((2*c^2*x^2*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))*(d + e*x^n)^q*AppellF1[2/n, 1, -q, (2 + n)/n, -((2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])), -((e*x^n)/d)])/((1 + (e*x^n)/d)^q*((b^2 - 4*a*c)^(3/2)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)))) + (2*c^2*x^2*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))*(d + e*x^n)^q*AppellF1[2/n, 1, -q, (2 + n)/n, -((2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])), -((e*x^n)/d)])/((1 + (e*x^n)/d)^q*((b^2 - 4*a*c)^(3/2)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))) + (2*c^2*x^2*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^2*(d + e*x^n)^q*AppellF1[2/n, 2, -q, (2 + n)/n, -((2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])), -((e*x^n)/d)])/((1 + (e*x^n)/d)^q*((b^2 - 4*a*c)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)^2)) + (2*c^2*x^2*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^2*(d + e*x^n)^q*AppellF1[2/n, 2, -q, (2 + n)/n, -((2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])), -((e*x^n)/d)])/((1 + (e*x^n)/d)^q*((b^2 - 4*a*c)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)^2))}
# {x^0*(d + e*x^n)^q/(a + b*x^n + c*x^(2*n))^2, x, 0, -((4*c^2*x*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))*(d + e*x^n)^q*AppellF1[1/n, 1, -q, (1 + n)/n, -((2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])), -((e*x^n)/d)])/((1 + (e*x^n)/d)^q*((b^2 - 4*a*c)^(3/2)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)))) + (4*c^2*x*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))*(d + e*x^n)^q*AppellF1[1/n, 1, -q, (1 + n)/n, -((2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])), -((e*x^n)/d)])/((1 + (e*x^n)/d)^q*((b^2 - 4*a*c)^(3/2)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))) + (4*c^2*x*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^2*(d + e*x^n)^q*AppellF1[1/n, 2, -q, (1 + n)/n, -((2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])), -((e*x^n)/d)])/((1 + (e*x^n)/d)^q*((b^2 - 4*a*c)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)^2)) + (4*c^2*x*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^2*(d + e*x^n)^q*AppellF1[1/n, 2, -q, (1 + n)/n, -((2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])), -((e*x^n)/d)])/((1 + (e*x^n)/d)^q*((b^2 - 4*a*c)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)^2))}
# {(d + e*x^n)^q/(x^1*(a + b*x^n + c*x^(2*n))^2), x, 0, ((d + e*x)^(1 + q)*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e + c*(b*c*d - b^2*e + 2*a*c*e)*x))/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)) - (c*(1 + b/Sqrt[b^2 - 4*a*c])*(d + e*x)^(1 + q)*Hypergeometric2F1[1, 1, 1 - q, -((2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)/(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)))])/(a^2*e*q*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)) - (c*(e*(b*c*d - b^2*e + 2*a*c*e)*q - (2*b*c*(c*d^2 + a*e^2*(1 - 2*q)) + 4*a*c^2*d*e*q + b^3*e^2*q - b^2*c*d*e*(2 + q))/Sqrt[b^2 - 4*a*c])*(d + e*x)^(1 + q)*Hypergeometric2F1[1, 1, 1 - q, -((2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)/(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)))])/(a*(b^2 - 4*a*c)*e*(c*d^2 - b*d*e + a*e^2)*q*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)) - (c*(1 - b/Sqrt[b^2 - 4*a*c])*(d + e*x)^(1 + q)*Hypergeometric2F1[1, 1, 1 - q, -((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)))])/(a^2*e*q*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)) - (c*(e*(b*c*d - b^2*e + 2*a*c*e)*q + (2*b*c*(c*d^2 + a*e^2*(1 - 2*q)) + 4*a*c^2*d*e*q + b^3*e^2*q - b^2*c*d*e*(2 + q))/Sqrt[b^2 - 4*a*c])*(d + e*x)^(1 + q)*Hypergeometric2F1[1, 1, 1 - q, -((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)))])/(a*(b^2 - 4*a*c)*e*(c*d^2 - b*d*e + a*e^2)*q*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)) - ((d + e*x)^(1 + q)*Hypergeometric2F1[1, 1 + q, 2 + q, (d + e*x)/d])/(a^2*d*(1 + q))}
# {(d + e*x^n)^q/(x^2*(a + b*x^n + c*x^(2*n))^2), x, 0, 0}

# {(d + e*x^n)^q/(x^3*(a + b*x^n + c*x^(2*n))^2), x, 0, 0} 

# Integrands of the form (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p with p symbolic
[(f*x)^m*(d+e*x^n)^2*(a+b*x^n+c*x^(2*n))^p,x,10,d^2*(f*x)^(1+m)*(a+b*x^n+c*x^(2*n))^p*AppellF1((1+m)/n,-p,-p,(1+m+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(f*(1+m)*(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))^p)+2*d*e*x^(1+n)*(f*x)^m*(a+b*x^n+c*x^(2*n))^p*AppellF1((1+m+n)/n,-p,-p,(1+m+2*n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))/((1+m+n)*(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))^p)+e^2*x^(1+2*n)*(f*x)^m*(a+b*x^n+c*x^(2*n))^p*AppellF1((1+m+2*n)/n,-p,-p,(1+m+3*n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))/((1+m+2*n)*(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))^p)],
[(f*x)^m*(d+e*x^n)*(a+b*x^n+c*x^(2*n))^p,x,7,d*(f*x)^(1+m)*(a+b*x^n+c*x^(2*n))^p*AppellF1((1+m)/n,-p,-p,(1+m+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(f*(1+m)*(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))^p)+e*x^(1+n)*(f*x)^m*(a+b*x^n+c*x^(2*n))^p*AppellF1((1+m+n)/n,-p,-p,(1+m+2*n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))/((1+m+n)*(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))^p)],
[(f*x)^m*(a+b*x^n+c*x^(2*n))^p,x,2,(f*x)^(1+m)*(a+b*x^n+c*x^(2*n))^p*AppellF1((1+m)/n,-p,-p,(1+m+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(f*(1+m)*(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))^p)],
[(f*x)^m*(a+b*x^n+c*x^(2*n))^p/(d+e*x^n),x,0,Unintegrable((f*x)^m*(a+b*x^n+c*x^(2*n))^p/(d+e*x^n),x)],
[(f*x)^m*(a+b*x^n+c*x^(2*n))^p/(d+e*x^n)^2,x,0,Unintegrable((f*x)^m*(a+b*x^n+c*x^(2*n))^p/(d+e*x^n)^2,x)]]:
