\[ \int \frac {\sqrt [3]{-x+x^3}}{b+a x^6} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(-x + x^3)^(1/3)/(b + a*x^6),x]
Mathematica 13.1 output
\[ \frac {\sqrt [3]{x \left (-1+x^2\right )} \text {RootSum}\left [a+b-3 b \text {$\#$1}^3+3 b \text {$\#$1}^6-b \text {$\#$1}^9\&,\frac {-2 \log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{-1+x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ]}{6 b \sqrt [3]{x} \sqrt [3]{-1+x^2}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt [3]{-x+x^3}}{b+a x^6} \, dx \]________________________________________________________________________________________