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