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