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