\[ \int \frac {x^5 \left (-4 b+5 a x^2\right )}{\sqrt [4]{-b+a x^2} \left (b-b x^8+a x^{10}\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(x^5*(-4*b + 5*a*x^2))/((-b + a*x^2)^(1/4)*(b - b*x^8 + a*x^10)),x]
Mathematica 13.1 output
\[ \frac {1}{2} a \text {RootSum}\left [a^4 b+b^4 \text {$\#$1}^4+4 b^3 \text {$\#$1}^8+6 b^2 \text {$\#$1}^{12}+4 b \text {$\#$1}^{16}+\text {$\#$1}^{20}\&,\frac {\log \left (\sqrt [4]{-b+a x^2}-\text {$\#$1}\right )}{b \text {$\#$1}+\text {$\#$1}^5}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {x^5 \left (-4 b+5 a x^2\right )}{\sqrt [4]{-b+a x^2} \left (b-b x^8+a x^{10}\right )} \, dx \]________________________________________________________________________________________