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