\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4-2 x^3+x^6\right )}{x^6 \left (-8+4 x^3+x^6\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((-1 + x^3)^(2/3)*(4 - 2*x^3 + x^6))/(x^6*(-8 + 4*x^3 + x^6)),x]
Mathematica 13.1 output
\[ -\frac {4 \left (-1+x^3\right )^{5/3}+5 x^5 \text {RootSum}\left [3-12 \text {$\#$1}^3+8 \text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-3+4 \text {$\#$1}^3}\&\right ]}{40 x^5} \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4-2 x^3+x^6\right )}{x^6 \left (-8+4 x^3+x^6\right )} \, dx \]________________________________________________________________________________________