\[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx \]
Optimal antiderivative \[ \frac {4 \left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}}}{x}-2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {3}{4}} x \left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}}}{x^{2} \sqrt {2}-\sqrt {x^{5}-2 x^{4}+1}}\right )-2^{\frac {3}{4}} \arctanh \left (\frac {2 \,2^{\frac {1}{4}} x \left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}}}{2 x^{2}+\sqrt {2}\, \sqrt {x^{5}-2 x^{4}+1}}\right ) \]
command
Integrate[((-4 + x^5)*(1 - 2*x^4 + x^5)^(1/4))/(x^2*(1 + x^5)),x]
Mathematica 13.1 output
\[ \frac {4 \sqrt [4]{1-2 x^4+x^5}}{x}-2^{3/4} \text {ArcTan}\left (\frac {2^{3/4} x \sqrt [4]{1-2 x^4+x^5}}{\sqrt {2} x^2-\sqrt {1-2 x^4+x^5}}\right )-2^{3/4} \tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{1-2 x^4+x^5}}{2 x^2+\sqrt {2} \sqrt {1-2 x^4+x^5}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx \]________________________________________________________________________________________