\[ \int \frac {(8+3 x) \sqrt [4]{-2-x+2 x^4}}{x^2 \left (2+x+x^4\right )} \, dx \]
Optimal antiderivative \[ -\frac {4 \left (2 x^{4}-x -2\right )^{\frac {1}{4}}}{x}-2 \,3^{\frac {1}{4}} \arctan \left (\frac {3^{\frac {1}{4}} x}{\left (2 x^{4}-x -2\right )^{\frac {1}{4}}}\right )+2 \,3^{\frac {1}{4}} \arctanh \left (\frac {3^{\frac {1}{4}} x}{\left (2 x^{4}-x -2\right )^{\frac {1}{4}}}\right ) \]
command
Integrate[((8 + 3*x)*(-2 - x + 2*x^4)^(1/4))/(x^2*(2 + x + x^4)),x]
Mathematica 13.1 output
\[ -\frac {4 \sqrt [4]{-2-x+2 x^4}}{x}-2 \sqrt [4]{3} \text {ArcTan}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-2-x+2 x^4}}\right )+2 \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-2-x+2 x^4}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {(8+3 x) \sqrt [4]{-2-x+2 x^4}}{x^2 \left (2+x+x^4\right )} \, dx \]________________________________________________________________________________________