\[ \int \frac {\left (-1+x^4+2 x^6\right ) \sqrt [3]{x+x^5+x^7}}{\left (1+x^4+x^6\right ) \left (1-x^2+x^4+x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{x +2 \left (x^{7}+x^{5}+x \right )^{\frac {1}{3}}}\right )}{2}+\frac {\ln \left (-x +\left (x^{7}+x^{5}+x \right )^{\frac {1}{3}}\right )}{2}-\frac {\ln \left (x^{2}+x \left (x^{7}+x^{5}+x \right )^{\frac {1}{3}}+\left (x^{7}+x^{5}+x \right )^{\frac {2}{3}}\right )}{4} \]
command
Integrate[((-1 + x^4 + 2*x^6)*(x + x^5 + x^7)^(1/3))/((1 + x^4 + x^6)*(1 - x^2 + x^4 + x^6)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [3]{x+x^5+x^7} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^4+x^6}}\right )+2 \log \left (-x^{2/3}+\sqrt [3]{1+x^4+x^6}\right )-\log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^4+x^6}+\left (1+x^4+x^6\right )^{2/3}\right )\right )}{4 \sqrt [3]{x} \sqrt [3]{1+x^4+x^6}} \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+x^4+2 x^6\right ) \sqrt [3]{x+x^5+x^7}}{\left (1+x^4+x^6\right ) \left (1-x^2+x^4+x^6\right )} \, dx \]________________________________________________________________________________________