\[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} \left (-38 x^{6}-27 x^{5}+3 x^{4}+54 x^{3}-12 x^{2}+12\right )}{4 x^{4} \left (x^{3}+x^{2}-2\right )}-\frac {7 \arctan \left (\frac {\sqrt {3}\, x}{x +2 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}-\frac {7 \ln \left (-x +\left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}\right )}{3}+\frac {7 \ln \left (x^{2}+x \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}+\left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}}\right )}{6} \]
command
Integrate[((-6 + x^2)*(-2 + x^2)*(2 - x^2 + x^3)*(-2 + x^2 + 2*x^3)^(1/3))/(x^5*(-2 + x^2 + x^3)^2),x]
Mathematica 13.1 output
\[ \frac {\sqrt [3]{-2+x^2+2 x^3} \left (12-12 x^2+54 x^3+3 x^4-27 x^5-38 x^6\right )}{4 x^4 \left (-2+x^2+x^3\right )}-\frac {7 \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-2+x^2+2 x^3}}\right )}{\sqrt {3}}-\frac {7}{3} \log \left (-x+\sqrt [3]{-2+x^2+2 x^3}\right )+\frac {7}{6} \log \left (x^2+x \sqrt [3]{-2+x^2+2 x^3}+\left (-2+x^2+2 x^3\right )^{2/3}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx \]________________________________________________________________________________________