\[ \int \frac {2 d+c x^4}{\sqrt [4]{-b+a x^4} \left (-2 f+e x^8\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(2*d + c*x^4)/((-b + a*x^4)^(1/4)*(-2*f + e*x^8)),x]
Mathematica 13.1 output
\[ \frac {\text {RootSum}\left [b^2 e-2 a^2 f+4 a f \text {$\#$1}^4-2 f \text {$\#$1}^8\&,\frac {b c \log (x)+2 a d \log (x)-b c \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-2 a d \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-2 d \log (x) \text {$\#$1}^4+2 d \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}+\text {$\#$1}^5}\&\right ]}{16 f} \]
Mathematica 12.3 output
\[ \int \frac {2 d+c x^4}{\sqrt [4]{-b+a x^4} \left (-2 f+e x^8\right )} \, dx \]________________________________________________________________________________________