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