\[ \int \frac {\sqrt {b+a^2 x^2}}{x^2-\sqrt [3]{a x-\sqrt {b+a^2 x^2}}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[Sqrt[b + a^2*x^2]/(x^2 - (a*x - Sqrt[b + a^2*x^2])^(1/3)),x]
Mathematica 13.1 output
\[ -a \log \left (-a x+\sqrt {b+a^2 x^2}\right )+3 a \text {RootSum}\left [b^2-2 b \text {$\#$1}^6-4 a^2 \text {$\#$1}^7+\text {$\#$1}^{12}\&,\frac {b \log \left (\sqrt [3]{a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right )+a^2 \log \left (\sqrt [3]{a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{3 b+7 a^2 \text {$\#$1}-3 \text {$\#$1}^6}\&\right ] \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________