\[ \int \frac {x^4 \sqrt {b+a^2 x^2}}{x^2-\sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(x^4*Sqrt[b + a^2*x^2])/(x^2 - Sqrt[a*x - Sqrt[b + a^2*x^2]]),x]
Mathematica 13.1 output
\[ -a^2 x+a \sqrt {b+a^2 x^2}+\frac {b^2}{6 a \left (a x-\sqrt {b+a^2 x^2}\right )^{3/2}}-\frac {b \sqrt {a x-\sqrt {b+a^2 x^2}}}{a}-\frac {\left (a x-\sqrt {b+a^2 x^2}\right )^{5/2}}{10 a}+\frac {b^4}{64 a^3 \left (-a x+\sqrt {b+a^2 x^2}\right )^4}-\frac {\left (-a x+\sqrt {b+a^2 x^2}\right )^4}{64 a^3}+\frac {b^2 \log \left (-a x+\sqrt {b+a^2 x^2}\right )}{8 a^3}+2 a \text {RootSum}\left [b^2-2 b \text {$\#$1}^4-4 a^2 \text {$\#$1}^5+\text {$\#$1}^8\&,\frac {b \log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^2+a^2 \log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3}{2 b+5 a^2 \text {$\#$1}-2 \text {$\#$1}^4}\&\right ] \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________