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