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