\[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right ) \left (a q+b x^2+a p x^5\right )}{x^4 \left (c q+d x^2+c p x^5\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \sqrt {p \,x^{5}+q}\, \left (a c p \,x^{5}-3 a d \,x^{2}+3 b c \,x^{2}+a c q \right )}{3 c^{2} x^{3}}+\frac {2 \left (b c \sqrt {d}-a \,d^{\frac {3}{2}}\right ) \arctan \left (\frac {\sqrt {d}\, x}{\sqrt {c}\, \sqrt {p \,x^{5}+q}}\right )}{c^{\frac {5}{2}}} \]
command
Integrate[(Sqrt[q + p*x^5]*(-2*q + 3*p*x^5)*(a*q + b*x^2 + a*p*x^5))/(x^4*(c*q + d*x^2 + c*p*x^5)),x]
Mathematica 13.1 output
\[ \frac {2 \sqrt {q+p x^5} \left (a c q+3 b c x^2-3 a d x^2+a c p x^5\right )}{3 c^2 x^3}+\frac {2 \sqrt {d} (b c-a d) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c} \sqrt {q+p x^5}}\right )}{c^{5/2}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right ) \left (a q+b x^2+a p x^5\right )}{x^4 \left (c q+d x^2+c p x^5\right )} \, dx \]________________________________________________________________________________________