\[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (-2+x^6\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((1 + x^3)^(2/3)*(1 - 2*x^3 + x^6))/(x^6*(-2 + x^6)),x]
Mathematica 13.1 output
\[ \frac {\left (1-4 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+\frac {1}{24} \text {RootSum}\left [1-4 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-2 \log (x)+2 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+\text {$\#$1}^4}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (-2+x^6\right )} \, dx \]________________________________________________________________________________________