\[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx \]
Optimal antiderivative \[ \frac {3 \left (2 x^{7}+1\right )^{\frac {1}{3}}}{x}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{-x +2 \left (2 x^{7}+1\right )^{\frac {1}{3}}}\right )-\ln \left (x +\left (2 x^{7}+1\right )^{\frac {1}{3}}\right )+\frac {\ln \left (x^{2}-x \left (2 x^{7}+1\right )^{\frac {1}{3}}+\left (2 x^{7}+1\right )^{\frac {2}{3}}\right )}{2} \]
command
Integrate[((1 + 2*x^7)^(1/3)*(-3 + 8*x^7))/(x^2*(1 + x^3 + 2*x^7)),x]
Mathematica 13.1 output
\[ \frac {3 \sqrt [3]{1+2 x^7}}{x}+\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+2 x^7}}\right )-\log \left (x+\sqrt [3]{1+2 x^7}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+2 x^7}+\left (1+2 x^7\right )^{2/3}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx \]________________________________________________________________________________________