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