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