\[ \int \frac {\left (3+2 x^5\right ) \sqrt {x-2 x^4-x^6}}{\left (-1+x^5\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {x \sqrt {-x^{6}-2 x^{4}+x}}{x^{5}-1}-\frac {\arctan \left (\frac {\sqrt {2}\, x \sqrt {-x^{6}-2 x^{4}+x}}{x^{5}+2 x^{3}-1}\right ) \sqrt {2}}{2} \]
command
Integrate[((3 + 2*x^5)*Sqrt[x - 2*x^4 - x^6])/(-1 + x^5)^2,x]
Mathematica 13.1 output
\[ \frac {\sqrt {x-2 x^4-x^6} \left (-\frac {2 x^{3/2}}{-1+x^5}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x^{3/2}}{\sqrt {-1+2 x^3+x^5}}\right )}{\sqrt {-1+2 x^3+x^5}}\right )}{2 \sqrt {x}} \]
Mathematica 12.3 output
\[ \int \frac {\left (3+2 x^5\right ) \sqrt {x-2 x^4-x^6}}{\left (-1+x^5\right )^2} \, dx \]________________________________________________________________________________________