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