\[ \int \frac {\sqrt {-1+x^5} \left (2+3 x^5\right )}{1-a x^4-2 x^5+x^{10}} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {a^{\frac {1}{4}} x}{\sqrt {x^{5}-1}}\right )}{a^{\frac {1}{4}}}-\frac {\arctanh \left (\frac {a^{\frac {1}{4}} x}{\sqrt {x^{5}-1}}\right )}{a^{\frac {1}{4}}} \]
command
Integrate[(Sqrt[-1 + x^5]*(2 + 3*x^5))/(1 - a*x^4 - 2*x^5 + x^10),x]
Mathematica 13.1 output
\[ -\frac {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt {-1+x^5}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {-1+x^5}}\right )}{\sqrt [4]{a}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {-1+x^5} \left (2+3 x^5\right )}{1-a x^4-2 x^5+x^{10}} \, dx \]________________________________________________________________________________________