\[ \int \frac {\left (-2+2 x^4+5 x^7\right ) \sqrt [3]{x-x^3+x^5+x^8}}{\left (2+x^2+2 x^4+2 x^7\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {x \left (x^{8}+x^{5}-x^{3}+x \right )^{\frac {1}{3}}}{4 x^{7}+4 x^{4}+2 x^{2}+4}+\frac {\arctan \left (\frac {\sqrt {3}\, \left (x^{8}+x^{5}-x^{3}+x \right )^{\frac {1}{3}}}{2^{\frac {2}{3}} 3^{\frac {1}{3}} x -\left (x^{8}+x^{5}-x^{3}+x \right )^{\frac {1}{3}}}\right ) 2^{\frac {2}{3}} 3^{\frac {5}{6}}}{36}+\frac {\ln \left (2^{\frac {2}{3}} 3^{\frac {1}{3}} x +2 \left (x^{8}+x^{5}-x^{3}+x \right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}}}{36}-\frac {\ln \left (2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2}-2^{\frac {2}{3}} 3^{\frac {1}{3}} x \left (x^{8}+x^{5}-x^{3}+x \right )^{\frac {1}{3}}+2 \left (x^{8}+x^{5}-x^{3}+x \right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}}}{72} \]
command
Integrate[((-2 + 2*x^4 + 5*x^7)*(x - x^3 + x^5 + x^8)^(1/3))/(2 + x^2 + 2*x^4 + 2*x^7)^2,x]
Mathematica 13.1 output
\[ \frac {\sqrt [3]{x-x^3+x^5+x^8} \left (-\frac {36 x^{4/3}}{2+x^2+2 x^4+2 x^7}+\frac {2\ 2^{2/3} 3^{5/6} \text {ArcTan}\left (\frac {3^{5/6} x^{2/3}}{\sqrt [3]{3} x^{2/3}-2 \sqrt [3]{2} \sqrt [3]{1-x^2+x^4+x^7}}\right )}{\sqrt [3]{1-x^2+x^4+x^7}}+\frac {2\ 2^{2/3} \sqrt [3]{3} \log \left (3 x^{2/3}+\sqrt [3]{2} 3^{2/3} \sqrt [3]{1-x^2+x^4+x^7}\right )}{\sqrt [3]{1-x^2+x^4+x^7}}-\frac {2^{2/3} \sqrt [3]{3} \log \left (3 x^{4/3}-\sqrt [3]{2} 3^{2/3} x^{2/3} \sqrt [3]{1-x^2+x^4+x^7}+2^{2/3} \sqrt [3]{3} \left (1-x^2+x^4+x^7\right )^{2/3}\right )}{\sqrt [3]{1-x^2+x^4+x^7}}\right )}{72 \sqrt [3]{x}} \]
Mathematica 12.3 output
\[ \int \frac {\left (-2+2 x^4+5 x^7\right ) \sqrt [3]{x-x^3+x^5+x^8}}{\left (2+x^2+2 x^4+2 x^7\right )^2} \, dx \]________________________________________________________________________________________