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