\[ \int \frac {\sqrt [4]{2 b+a x^4} \left (-4 b+a x^8\right )}{x^6 \left (-4 b+c x^4+a x^8\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((2*b + a*x^4)^(1/4)*(-4*b + a*x^8))/(x^6*(-4*b + c*x^4 + a*x^8)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [4]{2 b+a x^4} \left (-4 b-2 a x^4-5 c x^4\right )}{20 b x^5}-\frac {c \text {RootSum}\left [2 a^2-2 a b+a c-4 a \text {$\#$1}^4-c \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {2 a^2 \log (x)-2 a b \log (x)+a c \log (x)-2 a^2 \log \left (\sqrt [4]{2 b+a x^4}-x \text {$\#$1}\right )+2 a b \log \left (\sqrt [4]{2 b+a x^4}-x \text {$\#$1}\right )-a c \log \left (\sqrt [4]{2 b+a x^4}-x \text {$\#$1}\right )-2 a \log (x) \text {$\#$1}^4-c \log (x) \text {$\#$1}^4+2 a \log \left (\sqrt [4]{2 b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+c \log \left (\sqrt [4]{2 b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-4 a \text {$\#$1}^3-c \text {$\#$1}^3+4 \text {$\#$1}^7}\&\right ]}{16 b} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt [4]{2 b+a x^4} \left (-4 b+a x^8\right )}{x^6 \left (-4 b+c x^4+a x^8\right )} \, dx \]________________________________________________________________________________________