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