\[ \int \frac {\sqrt {-1+x^5} \left (2+3 x^5\right )}{x^2 \left (-1-a x^2+x^5\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \sqrt {x^{5}-1}}{x}-2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {a}\, x}{\sqrt {x^{5}-1}}\right ) \]
command
Integrate[(Sqrt[-1 + x^5]*(2 + 3*x^5))/(x^2*(-1 - a*x^2 + x^5)),x]
Mathematica 13.1 output
\[ \frac {2 \sqrt {-1+x^5}}{x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {-1+x^5}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {-1+x^5} \left (2+3 x^5\right )}{x^2 \left (-1-a x^2+x^5\right )} \, dx \]________________________________________________________________________________________