\[ \int \frac {\left (-2+x^6\right ) \sqrt [3]{2+x^6}}{x^2 \left (2+2 x^3+x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (x^{6}+2\right )^{\frac {1}{3}}}{x}+\frac {2^{\frac {1}{3}} \arctan \! \left (\frac {\sqrt {3}\, x}{-x +2^{\frac {2}{3}} \left (x^{6}+2\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}-\frac {2^{\frac {1}{3}} \ln \! \left (2 x +2^{\frac {2}{3}} \left (x^{6}+2\right )^{\frac {1}{3}}\right )}{3}+\frac {\ln \! \left (-2 x^{2}+2^{\frac {2}{3}} x \left (x^{6}+2\right )^{\frac {1}{3}}-2^{\frac {1}{3}} \left (x^{6}+2\right )^{\frac {2}{3}}\right ) 2^{\frac {1}{3}}}{6} \]
command
Int[((-2 + x^6)*(2 + x^6)^(1/3))/(x^2*(2 + 2*x^3 + x^6)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {\left (-2+x^6\right ) \sqrt [3]{2+x^6}}{x^2 \left (2+2 x^3+x^6\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {i x^5 \operatorname {AppellF1}\left (\frac {5}{6},1,-\frac {1}{3},\frac {11}{6},-\frac {i x^6}{2},-\frac {x^6}{2}\right )}{5\ 2^{2/3}}-\frac {i x^5 \operatorname {AppellF1}\left (\frac {5}{6},1,-\frac {1}{3},\frac {11}{6},\frac {i x^6}{2},-\frac {x^6}{2}\right )}{5\ 2^{2/3}}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) x^2 \operatorname {AppellF1}\left (\frac {1}{3},1,-\frac {1}{3},\frac {4}{3},-\frac {i x^6}{2},-\frac {x^6}{2}\right )}{2^{2/3}}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) x^2 \operatorname {AppellF1}\left (\frac {1}{3},1,-\frac {1}{3},\frac {4}{3},\frac {i x^6}{2},-\frac {x^6}{2}\right )}{2^{2/3}}+\frac {\sqrt [3]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{6},\frac {5}{6},-\frac {x^6}{2}\right )}{x} \]