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