\[ \int \frac {\left (-4+x^3\right ) \left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^6 \left (-2+x^3+x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (-13 x^{3}+8\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{10 x^{5}}+2^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {5}{6}} x}{3^{\frac {1}{3}} x +2 \,2^{\frac {1}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}}\right )-\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (-3 x +2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3}+\frac {\ln \left (3 x^{2}+2^{\frac {1}{3}} 3^{\frac {2}{3}} x \left (x^{3}-1\right )^{\frac {1}{3}}+2^{\frac {2}{3}} 3^{\frac {1}{3}} \left (x^{3}-1\right )^{\frac {2}{3}}\right ) 2^{\frac {1}{3}} 3^{\frac {2}{3}}}{6} \]
command
Integrate[((-4 + x^3)*(-2 + x^3)*(-1 + x^3)^(2/3))/(x^6*(-2 + x^3 + x^6)),x]
Mathematica 13.1 output
\[ \frac {\left (8-13 x^3\right ) \left (-1+x^3\right )^{2/3}}{10 x^5}+\sqrt [3]{2} \sqrt [6]{3} \text {ArcTan}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )-\sqrt [3]{\frac {2}{3}} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )+\frac {\log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )}{2^{2/3} \sqrt [3]{3}} \]
Mathematica 12.3 output
\[ \int \frac {\left (-4+x^3\right ) \left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^6 \left (-2+x^3+x^6\right )} \, dx \]________________________________________________________________________________________