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