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