\[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+x^3+2 x^6\right )}{x^6 \left (-1+2 x^6\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((1 + x^3)^(2/3)*(1 + x^3 + 2*x^6))/(x^6*(-1 + 2*x^6)),x]
Mathematica 13.1 output
\[ \frac {\left (1+x^3\right )^{2/3} \left (2+7 x^3\right )}{10 x^5}-\frac {1}{6} \text {RootSum}\left [-1-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log (x)-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+5 \log (x) \text {$\#$1}^3-5 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+\text {$\#$1}^4}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+x^3+2 x^6\right )}{x^6 \left (-1+2 x^6\right )} \, dx \]________________________________________________________________________________________