\[ \int \frac {\left (4+x^5\right ) \sqrt [4]{-2+x^4+2 x^5} \left (2-4 x^5+x^8+2 x^{10}\right )}{x^{10} \left (-1+x^5\right )} \, dx \]
Optimal antiderivative \[ \frac {4 \left (2 x^{5}+x^{4}-2\right )^{\frac {1}{4}} \left (10 x^{10}+x^{9}+43 x^{8}-20 x^{5}-x^{4}+10\right )}{45 x^{9}}+2 \arctan \left (\frac {x}{\left (2 x^{5}+x^{4}-2\right )^{\frac {1}{4}}}\right )-2 \arctanh \left (\frac {x}{\left (2 x^{5}+x^{4}-2\right )^{\frac {1}{4}}}\right ) \]
command
Integrate[((4 + x^5)*(-2 + x^4 + 2*x^5)^(1/4)*(2 - 4*x^5 + x^8 + 2*x^10))/(x^10*(-1 + x^5)),x]
Mathematica 13.1 output
\[ \frac {4 \sqrt [4]{-2+x^4+2 x^5} \left (10-x^4-20 x^5+43 x^8+x^9+10 x^{10}\right )}{45 x^9}+2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{-2+x^4+2 x^5}}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-2+x^4+2 x^5}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (4+x^5\right ) \sqrt [4]{-2+x^4+2 x^5} \left (2-4 x^5+x^8+2 x^{10}\right )}{x^{10} \left (-1+x^5\right )} \, dx \]________________________________________________________________________________________