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