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