\[ \int \frac {\sqrt [4]{b+a x^4} \left (2 b+3 a x^4\right )}{x^6 \left (b+a x^8\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((b + a*x^4)^(1/4)*(2*b + 3*a*x^4))/(x^6*(b + a*x^8)),x]
Mathematica 13.1 output
\[ \frac {\left (-2 b-17 a x^4\right ) \sqrt [4]{b+a x^4}}{5 b x^5}-\frac {a \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {3 a^2 \log (x)+3 a b \log (x)-3 a^2 \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )-3 a b \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )-3 a \log (x) \text {$\#$1}^4+2 b \log (x) \text {$\#$1}^4+3 a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-2 b \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{8 b} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt [4]{b+a x^4} \left (2 b+3 a x^4\right )}{x^6 \left (b+a x^8\right )} \, dx \]________________________________________________________________________________________