\[ \int \frac {\left (-8+x^5\right ) \left (2+x^5\right ) \sqrt [4]{2-3 x^4+x^5}}{x^6 \left (4-3 x^4+2 x^5\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (x^{5}-3 x^{4}+2\right )^{\frac {1}{4}} \left (2 x^{5}+9 x^{4}+4\right )}{5 x^{5}}+\frac {3 \,3^{\frac {1}{4}} \arctan \left (\frac {6^{\frac {3}{4}} x \left (x^{5}-3 x^{4}+2\right )^{\frac {1}{4}}}{-3 x^{2}+\sqrt {6}\, \sqrt {x^{5}-3 x^{4}+2}}\right ) 2^{\frac {1}{4}}}{4}-\frac {3 \,3^{\frac {1}{4}} \arctanh \left (\frac {6^{\frac {3}{4}} x \left (x^{5}-3 x^{4}+2\right )^{\frac {1}{4}}}{3 x^{2}+\sqrt {6}\, \sqrt {x^{5}-3 x^{4}+2}}\right ) 2^{\frac {1}{4}}}{4} \]
command
Integrate[((-8 + x^5)*(2 + x^5)*(2 - 3*x^4 + x^5)^(1/4))/(x^6*(4 - 3*x^4 + 2*x^5)),x]
Mathematica 13.1 output
\[ \frac {4 \sqrt [4]{2-3 x^4+x^5} \left (4+9 x^4+2 x^5\right )+15 \sqrt [4]{6} x^5 \text {ArcTan}\left (\frac {\frac {3 \sqrt [4]{6} x^2}{\sqrt [4]{2-3 x^4+x^5}}-6^{3/4} \sqrt [4]{2-3 x^4+x^5}}{6 x}\right )-15 \sqrt [4]{6} x^5 \tanh ^{-1}\left (\frac {6^{3/4} x \sqrt [4]{2-3 x^4+x^5}}{3 x^2+\sqrt {6} \sqrt {2-3 x^4+x^5}}\right )}{20 x^5} \]
Mathematica 12.3 output
\[ \int \frac {\left (-8+x^5\right ) \left (2+x^5\right ) \sqrt [4]{2-3 x^4+x^5}}{x^6 \left (4-3 x^4+2 x^5\right )} \, dx \]________________________________________________________________________________________