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