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