\[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((-4*b + a*x^4)*(-b + a*x^4)^(1/4))/(x^6*(-8*b + a*x^8)),x]
Mathematica 13.1 output
\[ \frac {\frac {64 \sqrt [4]{-b+a x^4} \left (-4 b+9 a x^4\right )}{x^5}+5 a \text {RootSum}\left [8 a^2-a b-16 a \text {$\#$1}^4+8 \text {$\#$1}^8\&,\frac {8 a^2 \log (x)-a b \log (x)-8 a^2 \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+a b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-8 a \log (x) \text {$\#$1}^4-4 b \log (x) \text {$\#$1}^4+8 a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+4 b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{2560 b} \]
Mathematica 12.3 output
\[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx \]________________________________________________________________________________________