\[ \int \frac {x^3}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \! \left (\frac {\sqrt {3}\, x^{2}}{x^{2}+2 \left (x^{4}+x^{2}\right )^{\frac {2}{3}}}\right ) \sqrt {3}}{6}+\frac {\arctan \! \left (\frac {\sqrt {3}\, x^{2}}{x^{2}+2^{\frac {1}{3}} \left (x^{4}+x^{2}\right )^{\frac {2}{3}}}\right ) 2^{\frac {2}{3}} \sqrt {3}}{24}-\frac {\ln \! \left (-x +\left (x^{4}+x^{2}\right )^{\frac {1}{3}}\right )}{6}-\frac {\ln \! \left (x +\left (x^{4}+x^{2}\right )^{\frac {1}{3}}\right )}{6}+\frac {\ln \! \left (-2 x +2^{\frac {2}{3}} \left (x^{4}+x^{2}\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{24}+\frac {\ln \! \left (2 x +2^{\frac {2}{3}} \left (x^{4}+x^{2}\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{24}+\frac {\ln \! \left (x^{2}-x \left (x^{4}+x^{2}\right )^{\frac {1}{3}}+\left (x^{4}+x^{2}\right )^{\frac {2}{3}}\right )}{12}+\frac {\ln \! \left (x^{2}+x \left (x^{4}+x^{2}\right )^{\frac {1}{3}}+\left (x^{4}+x^{2}\right )^{\frac {2}{3}}\right )}{12}-\frac {\ln \! \left (-2 x^{2}+2^{\frac {2}{3}} x \left (x^{4}+x^{2}\right )^{\frac {1}{3}}-2^{\frac {1}{3}} \left (x^{4}+x^{2}\right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{48}-\frac {\ln \! \left (2 x^{2}+2^{\frac {2}{3}} x \left (x^{4}+x^{2}\right )^{\frac {1}{3}}+2^{\frac {1}{3}} \left (x^{4}+x^{2}\right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{48} \]
command
Int[x^3/((x^2 + x^4)^(1/3)*(-1 + x^6)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {x^3}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -\frac {\sqrt [3]{x^2+1} x^4 \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{3},1,\frac {8}{3},-x^2,-\sqrt [3]{-1} x^2\right )}{10 \sqrt [3]{x^4+x^2}}-\frac {\sqrt [3]{x^2+1} x^4 \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{3},1,\frac {8}{3},-x^2,(-1)^{2/3} x^2\right )}{10 \sqrt [3]{x^4+x^2}}-\frac {\sqrt [3]{x^2+1} x^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^4+x^2}}-\frac {\sqrt [3]{x^2+1} x^{2/3} \arctan \left (\frac {\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^4+x^2}}+\frac {\sqrt [3]{x^2+1} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-x^2\right )}{4 \sqrt [3]{x^4+x^2}}-\frac {\sqrt [3]{x^2+1} x^{2/3} \log \left (\left (1-x^{2/3}\right )^2 \left (x^{2/3}+1\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{x^4+x^2}}-\frac {\sqrt [3]{x^2+1} x^{2/3} \log \left (\frac {2^{2/3} \left (x^{2/3}+1\right )^2}{\left (x^2+1\right )^{2/3}}-\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1\right )}{12 \sqrt [3]{2} \sqrt [3]{x^4+x^2}}+\frac {\sqrt [3]{x^2+1} x^{2/3} \log \left (\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1\right )}{6 \sqrt [3]{2} \sqrt [3]{x^4+x^2}}+\frac {\sqrt [3]{x^2+1} x^{2/3} \log \left (x^{2/3}-2^{2/3} \sqrt [3]{x^2+1}+1\right )}{8 \sqrt [3]{2} \sqrt [3]{x^4+x^2}} \]