\[ \int \frac {\left (-6+x^2\right ) \left (2-x^2+x^3\right )^{2/3}}{x^3 \left (-2+x^2+x^3\right )} \, dx \]
Optimal antiderivative \[ -\frac {3 \left (x^{3}-x^{2}+2\right )^{\frac {2}{3}}}{2 x^{2}}+2^{\frac {2}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{x +2^{\frac {2}{3}} \left (x^{3}-x^{2}+2\right )^{\frac {1}{3}}}\right )-2^{\frac {2}{3}} \ln \left (-2 x +2^{\frac {2}{3}} \left (x^{3}-x^{2}+2\right )^{\frac {1}{3}}\right )+\frac {\ln \left (2 x^{2}+2^{\frac {2}{3}} x \left (x^{3}-x^{2}+2\right )^{\frac {1}{3}}+2^{\frac {1}{3}} \left (x^{3}-x^{2}+2\right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{2} \]
command
Integrate[((-6 + x^2)*(2 - x^2 + x^3)^(2/3))/(x^3*(-2 + x^2 + x^3)),x]
Mathematica 13.1 output
\[ -\frac {3 \left (2-x^2+x^3\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{2-x^2+x^3}}\right )-2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{2-x^2+x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{2-x^2+x^3}+\sqrt [3]{2} \left (2-x^2+x^3\right )^{2/3}\right )}{\sqrt [3]{2}} \]
Mathematica 12.3 output
\[ \int \frac {\left (-6+x^2\right ) \left (2-x^2+x^3\right )^{2/3}}{x^3 \left (-2+x^2+x^3\right )} \, dx \]________________________________________________________________________________________