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