\[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \! \left (\frac {\sqrt {3}\, x}{-x +2 \left (x^{6}-1\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}+\frac {\ln \! \left (x +\left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3}-\frac {\ln \! \left (x^{2}-x \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right )}{6} \]
command
Int[(x + x^7)/((-1 + x^6)^(2/3)*(-1 + x^3 + x^6)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {\left (5-\sqrt {5}\right ) \left (1-x^6\right )^{2/3} x^5 \operatorname {AppellF1}\left (\frac {5}{6},\frac {2}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{25 \left (3-\sqrt {5}\right ) \left (x^6-1\right )^{2/3}}-\frac {4 \left (1-x^6\right )^{2/3} x^5 \operatorname {AppellF1}\left (\frac {5}{6},\frac {2}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{5 \sqrt {5} \left (3-\sqrt {5}\right ) \left (x^6-1\right )^{2/3}}+\frac {\left (5+\sqrt {5}\right ) \left (1-x^6\right )^{2/3} x^5 \operatorname {AppellF1}\left (\frac {5}{6},\frac {2}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{25 \left (3+\sqrt {5}\right ) \left (x^6-1\right )^{2/3}}+\frac {4 \left (1-x^6\right )^{2/3} x^5 \operatorname {AppellF1}\left (\frac {5}{6},\frac {2}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{5 \sqrt {5} \left (3+\sqrt {5}\right ) \left (x^6-1\right )^{2/3}}-\frac {\left (5-\sqrt {5}\right ) \left (1-x^6\right )^{2/3} x^2 \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{5 \left (3-\sqrt {5}\right ) \left (x^6-1\right )^{2/3}}+\frac {\left (1-x^6\right )^{2/3} x^2 \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{2 \sqrt {5} \left (x^6-1\right )^{2/3}}-\frac {\left (5+\sqrt {5}\right ) \left (1-x^6\right )^{2/3} x^2 \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {2}{3},\frac {4}{3},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{5 \left (3+\sqrt {5}\right ) \left (x^6-1\right )^{2/3}}-\frac {\left (1-x^6\right )^{2/3} x^2 \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {2}{3},\frac {4}{3},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{2 \sqrt {5} \left (x^6-1\right )^{2/3}}+\frac {\left (1-x^6\right )^{2/3} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^6\right )}{2 \left (x^6-1\right )^{2/3}} \]