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