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