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