\[ \int \frac {\sqrt [4]{-1+x^4} \left (1+x^4+x^8\right )}{x^6 \left (-1+2 x^8\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((-1 + x^4)^(1/4)*(1 + x^4 + x^8))/(x^6*(-1 + 2*x^8)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [4]{-1+x^4} \left (1+4 x^4\right )}{5 x^5}+\frac {1}{8} \text {RootSum}\left [-1-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+\text {$\#$1}^7}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {\sqrt [4]{-1+x^4} \left (1+x^4+x^8\right )}{x^6 \left (-1+2 x^8\right )} \, dx \]________________________________________________________________________________________