# Maple integration test file: "1 Algebraic functions\1.2 Trinomial products\1.2.3 General\1.2.3.5 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.txt"

lst:=[

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

# Integrands of the form Pq(x) (a+b x^3+c x^6)^p

# Integrands of the form Pq(x) (a+b x^3+c x^6)^p

# p>0

# p<0
[(d+e*x+f*x^2+g*x^3+h*x^4+j*x^5+k*x^6+l*x^7+m*x^8)/(a+b*x^3+c*x^6),x,37,k*x/c+1/2*l*x^2/c+1/3*m*x^3/c+1/6*(c*j-b*m)*log(a+b*x^3+c*x^6)/c^2+1/3*log(2^(1/3)*c^(1/3)*x+(b-sqrt(b^2-4*a*c))^(1/3))*(g-b*k/c+(2*c^2*d+b^2*k-c*(b*g+2*a*k))/(c*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))*(g-b*k/c+(2*c^2*d+b^2*k-c*(b*g+2*a*k))/(c*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))*(g-b*k/c+(2*c^2*d+b^2*k-c*(b*g+2*a*k))/(c*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))*(h-b*l/c+(2*c^2*e+b^2*l-c*(b*h+2*a*l))/(c*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))*(h-b*l/c+(2*c^2*e+b^2*l-c*(b*h+2*a*l))/(c*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))*(h-b*l/c+(2*c^2*e+b^2*l-c*(b*h+2*a*l))/(c*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*(2*c^2*f-b*c*j+b^2*m-2*a*c*m)*arctanh((b+2*c*x^3)/sqrt(b^2-4*a*c))/(c^2*sqrt(b^2-4*a*c))+1/3*log(2^(1/3)*c^(1/3)*x+(b+sqrt(b^2-4*a*c))^(1/3))*(g-b*k/c+(-2*c^2*d+b*c*g-b^2*k+2*a*c*k)/(c*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))*(g-b*k/c+(-2*c^2*d+b*c*g-b^2*k+2*a*c*k)/(c*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))*(g-b*k/c+(-2*c^2*d+b*c*g-b^2*k+2*a*c*k)/(c*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))*(h-b*l/c+(-2*c^2*e+b*c*h-b^2*l+2*a*c*l)/(c*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))*(h-b*l/c+(-2*c^2*e+b*c*h-b^2*l+2*a*c*l)/(c*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))*(h-b*l/c+(-2*c^2*e+b*c*h-b^2*l+2*a*c*l)/(c*sqrt(b^2-4*a*c)))/(2^(2/3)*c^(2/3)*sqrt(3)*(b+sqrt(b^2-4*a*c))^(1/3))],

# Integrands of the form Pq(x) (a+b x^3+c x^6)^(p/2)

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

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

# p>0

# p<0
[1/(a+b*x^n+c*x^(2*n)),x,3,-2*c*x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(b^2-4*a*c-b*sqrt(b^2-4*a*c))-2*c*x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(b^2-4*a*c+b*sqrt(b^2-4*a*c))],
[(d+e*x)/(a+b*x^n+c*x^(2*n)),x,9,-2*c*d*x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(b^2-4*a*c-b*sqrt(b^2-4*a*c))-c*e*x^2*hypergeom([1,2/n],[(2+n)/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(b^2-4*a*c-b*sqrt(b^2-4*a*c))-2*c*d*x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(b^2-4*a*c+b*sqrt(b^2-4*a*c))-c*e*x^2*hypergeom([1,2/n],[(2+n)/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(b^2-4*a*c+b*sqrt(b^2-4*a*c))],
[(d+e*x+f*x^2)/(a+b*x^n+c*x^(2*n)),x,11,-2*c*d*x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(b^2-4*a*c-b*sqrt(b^2-4*a*c))-c*e*x^2*hypergeom([1,2/n],[(2+n)/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(b^2-4*a*c-b*sqrt(b^2-4*a*c))-2/3*c*f*x^3*hypergeom([1,3/n],[(3+n)/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(b^2-4*a*c-b*sqrt(b^2-4*a*c))-2*c*d*x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(b^2-4*a*c+b*sqrt(b^2-4*a*c))-c*e*x^2*hypergeom([1,2/n],[(2+n)/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(b^2-4*a*c+b*sqrt(b^2-4*a*c))-2/3*c*f*x^3*hypergeom([1,3/n],[(3+n)/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(b^2-4*a*c+b*sqrt(b^2-4*a*c))],
[(d+e*x+f*x^2+g*x^3)/(a+b*x^n+c*x^(2*n)),x,13,-2*c*d*x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(b^2-4*a*c-b*sqrt(b^2-4*a*c))-c*e*x^2*hypergeom([1,2/n],[(2+n)/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(b^2-4*a*c-b*sqrt(b^2-4*a*c))-2/3*c*f*x^3*hypergeom([1,3/n],[(3+n)/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(b^2-4*a*c-b*sqrt(b^2-4*a*c))-1/2*c*g*x^4*hypergeom([1,4/n],[(4+n)/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(b^2-4*a*c-b*sqrt(b^2-4*a*c))-2*c*d*x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(b^2-4*a*c+b*sqrt(b^2-4*a*c))-c*e*x^2*hypergeom([1,2/n],[(2+n)/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(b^2-4*a*c+b*sqrt(b^2-4*a*c))-2/3*c*f*x^3*hypergeom([1,3/n],[(3+n)/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(b^2-4*a*c+b*sqrt(b^2-4*a*c))-1/2*c*g*x^4*hypergeom([1,4/n],[(4+n)/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(b^2-4*a*c+b*sqrt(b^2-4*a*c))],
[1/(a+b*x^n+c*x^(2*n))^2,x,4,x*(b^2-2*a*c+b*c*x^n)/(a*(b^2-4*a*c)*n*(a+b*x^n+c*x^(2*n)))-c*x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))*(4*a*c*(1-2*n)-b^2*(1-n)-b*(1-n)*sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*n*(b^2-4*a*c-b*sqrt(b^2-4*a*c)))-c*x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))*(4*a*c*(1-2*n)-b^2*(1-n)+b*(1-n)*sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*n*(b^2-4*a*c+b*sqrt(b^2-4*a*c)))],
[(d+e*x)/(a+b*x^n+c*x^(2*n))^2,x,15,d*x*(b^2-2*a*c+b*c*x^n)/(a*(b^2-4*a*c)*n*(a+b*x^n+c*x^(2*n)))+e*x^2*(b^2-2*a*c+b*c*x^n)/(a*(b^2-4*a*c)*n*(a+b*x^n+c*x^(2*n)))-2*b*c^2*e*(2-n)*x^(2+n)*hypergeom([1,(2+n)/n],[2*(1+1/n)],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(3/2)*n*(2+n)*(b-sqrt(b^2-4*a*c)))+2*b*c^2*e*(2-n)*x^(2+n)*hypergeom([1,(2+n)/n],[2*(1+1/n)],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(3/2)*n*(2+n)*(b+sqrt(b^2-4*a*c)))-c*e*(4*a*c*(1-n)-b^2*(2-n))*x^2*hypergeom([1,2/n],[(2+n)/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)*n*(b^2-4*a*c-b*sqrt(b^2-4*a*c)))-c*e*(4*a*c*(1-n)-b^2*(2-n))*x^2*hypergeom([1,2/n],[(2+n)/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)*n*(b^2-4*a*c+b*sqrt(b^2-4*a*c)))-c*d*x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))*(4*a*c*(1-2*n)-b^2*(1-n)-b*(1-n)*sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*n*(b^2-4*a*c-b*sqrt(b^2-4*a*c)))-c*d*x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))*(4*a*c*(1-2*n)-b^2*(1-n)+b*(1-n)*sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*n*(b^2-4*a*c+b*sqrt(b^2-4*a*c)))],
[(d+e*x+f*x^2)/(a+b*x^n+c*x^(2*n))^2,x,24,d*x*(b^2-2*a*c+b*c*x^n)/(a*(b^2-4*a*c)*n*(a+b*x^n+c*x^(2*n)))+e*x^2*(b^2-2*a*c+b*c*x^n)/(a*(b^2-4*a*c)*n*(a+b*x^n+c*x^(2*n)))+f*x^3*(b^2-2*a*c+b*c*x^n)/(a*(b^2-4*a*c)*n*(a+b*x^n+c*x^(2*n)))-2*b*c^2*e*(2-n)*x^(2+n)*hypergeom([1,(2+n)/n],[2*(1+1/n)],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(3/2)*n*(2+n)*(b-sqrt(b^2-4*a*c)))-2*b*c^2*f*(3-n)*x^(3+n)*hypergeom([1,(3+n)/n],[2+3/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(3/2)*n*(3+n)*(b-sqrt(b^2-4*a*c)))+2*b*c^2*e*(2-n)*x^(2+n)*hypergeom([1,(2+n)/n],[2*(1+1/n)],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(3/2)*n*(2+n)*(b+sqrt(b^2-4*a*c)))+2*b*c^2*f*(3-n)*x^(3+n)*hypergeom([1,(3+n)/n],[2+3/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(3/2)*n*(3+n)*(b+sqrt(b^2-4*a*c)))-c*e*(4*a*c*(1-n)-b^2*(2-n))*x^2*hypergeom([1,2/n],[(2+n)/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)*n*(b^2-4*a*c-b*sqrt(b^2-4*a*c)))-2/3*c*f*(2*a*c*(3-2*n)-b^2*(3-n))*x^3*hypergeom([1,3/n],[(3+n)/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)*n*(b^2-4*a*c-b*sqrt(b^2-4*a*c)))-c*e*(4*a*c*(1-n)-b^2*(2-n))*x^2*hypergeom([1,2/n],[(2+n)/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)*n*(b^2-4*a*c+b*sqrt(b^2-4*a*c)))-2/3*c*f*(2*a*c*(3-2*n)-b^2*(3-n))*x^3*hypergeom([1,3/n],[(3+n)/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)*n*(b^2-4*a*c+b*sqrt(b^2-4*a*c)))-c*d*x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))*(4*a*c*(1-2*n)-b^2*(1-n)-b*(1-n)*sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*n*(b^2-4*a*c-b*sqrt(b^2-4*a*c)))-c*d*x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))*(4*a*c*(1-2*n)-b^2*(1-n)+b*(1-n)*sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*n*(b^2-4*a*c+b*sqrt(b^2-4*a*c)))],
[(d+e*x+f*x^2+g*x^3)/(a+b*x^n+c*x^(2*n))^2,x,33,d*x*(b^2-2*a*c+b*c*x^n)/(a*(b^2-4*a*c)*n*(a+b*x^n+c*x^(2*n)))+e*x^2*(b^2-2*a*c+b*c*x^n)/(a*(b^2-4*a*c)*n*(a+b*x^n+c*x^(2*n)))+f*x^3*(b^2-2*a*c+b*c*x^n)/(a*(b^2-4*a*c)*n*(a+b*x^n+c*x^(2*n)))+g*x^4*(b^2-2*a*c+b*c*x^n)/(a*(b^2-4*a*c)*n*(a+b*x^n+c*x^(2*n)))-2*b*c^2*e*(2-n)*x^(2+n)*hypergeom([1,(2+n)/n],[2*(1+1/n)],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(3/2)*n*(2+n)*(b-sqrt(b^2-4*a*c)))-2*b*c^2*f*(3-n)*x^(3+n)*hypergeom([1,(3+n)/n],[2+3/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(3/2)*n*(3+n)*(b-sqrt(b^2-4*a*c)))-2*b*c^2*g*(4-n)*x^(4+n)*hypergeom([1,(4+n)/n],[2*(1+2/n)],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(3/2)*n*(4+n)*(b-sqrt(b^2-4*a*c)))+2*b*c^2*e*(2-n)*x^(2+n)*hypergeom([1,(2+n)/n],[2*(1+1/n)],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(3/2)*n*(2+n)*(b+sqrt(b^2-4*a*c)))+2*b*c^2*f*(3-n)*x^(3+n)*hypergeom([1,(3+n)/n],[2+3/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(3/2)*n*(3+n)*(b+sqrt(b^2-4*a*c)))+2*b*c^2*g*(4-n)*x^(4+n)*hypergeom([1,(4+n)/n],[2*(1+2/n)],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(3/2)*n*(4+n)*(b+sqrt(b^2-4*a*c)))-c*e*(4*a*c*(1-n)-b^2*(2-n))*x^2*hypergeom([1,2/n],[(2+n)/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)*n*(b^2-4*a*c-b*sqrt(b^2-4*a*c)))-2/3*c*f*(2*a*c*(3-2*n)-b^2*(3-n))*x^3*hypergeom([1,3/n],[(3+n)/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)*n*(b^2-4*a*c-b*sqrt(b^2-4*a*c)))-1/2*c*g*(4*a*c*(2-n)-b^2*(4-n))*x^4*hypergeom([1,4/n],[(4+n)/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)*n*(b^2-4*a*c-b*sqrt(b^2-4*a*c)))-c*e*(4*a*c*(1-n)-b^2*(2-n))*x^2*hypergeom([1,2/n],[(2+n)/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)*n*(b^2-4*a*c+b*sqrt(b^2-4*a*c)))-2/3*c*f*(2*a*c*(3-2*n)-b^2*(3-n))*x^3*hypergeom([1,3/n],[(3+n)/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)*n*(b^2-4*a*c+b*sqrt(b^2-4*a*c)))-1/2*c*g*(4*a*c*(2-n)-b^2*(4-n))*x^4*hypergeom([1,4/n],[(4+n)/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)*n*(b^2-4*a*c+b*sqrt(b^2-4*a*c)))-c*d*x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))*(4*a*c*(1-2*n)-b^2*(1-n)-b*(1-n)*sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*n*(b^2-4*a*c-b*sqrt(b^2-4*a*c)))-c*d*x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))*(4*a*c*(1-2*n)-b^2*(1-n)+b*(1-n)*sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*n*(b^2-4*a*c+b*sqrt(b^2-4*a*c)))],

# Integrands of the form Pq(x) (a+b x^n+c x^(2 n))^(p/2)
[(-a*h*x^(-1+1/2*n)+c*f*x^(-1+n)+c*g*x^(-1+2*n)+c*h*x^(-1+5/2*n))/(a+b*x^n+c*x^(2*n))^(3/2),x,2,-2*(c*(b*f-2*a*g)+(b^2-4*a*c)*h*x^(1/2*n)+c*(2*c*f-b*g)*x^n)/((b^2-4*a*c)*n*sqrt(a+b*x^n+c*x^(2*n)))],

# Integrands of the form Pq(x) (a+b x^n+c x^(2 n))^p with p symbolic
[(a+b*x^n+c*x^(2*n))^p*(a+b*(1+n+n*p)*x^n+c*(1+2*n*(1+p))*x^(2*n)),x,1,x*(a+b*x^n+c*x^(2*n))^(1+p)],

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

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

# Integrands of the form (d x)^m Pq(x) (a+b x^n+c x^(2 n))^p with b=0
[x^(-1+1/4*n)*(-a*h+c*f*x^(1/4*n)+c*g*x^(3/4*n)+c*h*x^n)/(a+c*x^n)^(3/2),x,1,-2*(a*g+2*a*h*x^(1/4*n)-c*f*x^(1/2*n))/(a*n*sqrt(a+c*x^n))],
[(d*x)^(-1+1/4*n)*(-a*h+c*f*x^(1/4*n)+c*g*x^(3/4*n)+c*h*x^n)/(a+c*x^n)^(3/2),x,2,-2*x^(1-1/4*n)*(d*x)^(1/4*(-4+n))*(a*g+2*a*h*x^(1/4*n)-c*f*x^(1/2*n))/(a*n*sqrt(a+c*x^n))],

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

# Integrands of the form (d x)^m Pq(x) (a+b x^n+c x^(2 n))^(p/2)
[x^(-1+1/2*n)*(-a*h+c*f*x^(1/2*n)+c*g*x^(3/2*n)+c*h*x^(2*n))/(a+b*x^n+c*x^(2*n))^(3/2),x,1,-2*(c*(b*f-2*a*g)+(b^2-4*a*c)*h*x^(1/2*n)+c*(2*c*f-b*g)*x^n)/((b^2-4*a*c)*n*sqrt(a+b*x^n+c*x^(2*n)))],
[(d*x)^(-1+1/2*n)*(-a*h+c*f*x^(1/2*n)+c*g*x^(3/2*n)+c*h*x^(2*n))/(a+b*x^n+c*x^(2*n))^(3/2),x,2,-2*x^(1-1/2*n)*(d*x)^(1/2*(-2+n))*(c*(b*f-2*a*g)+(b^2-4*a*c)*h*x^(1/2*n)+c*(2*c*f-b*g)*x^n)/((b^2-4*a*c)*n*sqrt(a+b*x^n+c*x^(2*n)))],

# Integrands of the form (d x)^m Pq(x) (a+b x^n+c x^(2 n))^p with p symbolic
[(g*x)^m*(a+b*x^n+c*x^(2*n))^p*(a*(1+m)+b*(1+m+n+n*p)*x^n+c*(1+m+2*n*(1+p))*x^(2*n)),x,1,(g*x)^(1+m)*(a+b*x^n+c*x^(2*n))^(1+p)/g],

# Integrands of the form Pq(x^n) (a+b x^n+c x^(2 n))^p
[(A+B*x^n+C*x^(2*n)+D*x^(3*n))/(a+b*x^n+c*x^(2*n))^2,x,4,x*(A*c*(b^2-2*a*c)-a*(b*B*c-2*a*c*C+a*b*D)+(b*c*(A*c+a*C)-a*b^2*D-2*a*c*(B*c-a*D))*x^n)/(a*c*(b^2-4*a*c)*n*(a+b*x^n+c*x^(2*n)))+x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))*(a*b^2*D-b*c*(A*c+a*C)*(1-n)+2*a*c*(B*c*(1-n)-a*D*(1+n))+(A*c^2*(4*a*c*(1-2*n)-b^2*(1-n))-a*(4*a*c^2*C+b^3*D-b^2*c*C*(1-n)-2*b*c*(B*c*n+a*D*(2+n))))/sqrt(b^2-4*a*c))/(a*c*(b^2-4*a*c)*n*(b-sqrt(b^2-4*a*c)))+x*hypergeom([1,1/n],[1+1/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))*(a*b^2*D-b*c*(A*c+a*C)*(1-n)+2*a*c*(B*c*(1-n)-a*D*(1+n))+(-A*c^2*(4*a*c*(1-2*n)-b^2*(1-n))+a*(4*a*c^2*C+b^3*D-b^2*c*C*(1-n)-2*b*c*(B*c*n+a*D*(2+n))))/sqrt(b^2-4*a*c))/(a*c*(b^2-4*a*c)*n*(b+sqrt(b^2-4*a*c)))]]:
