\[ \int \frac {x}{\left (2^{2/3}+x\right ) \sqrt {1+x^3}} \, dx \]
Optimal antiderivative \[ -\frac {2 \,2^{\frac {2}{3}} \arctan \left (\frac {\left (1+2^{\frac {1}{3}} x \right ) \sqrt {3}}{\sqrt {x^{3}+1}}\right ) \sqrt {3}}{9}+\frac {2 \left (1+x \right ) \EllipticF \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}\, 3^{\frac {3}{4}}}{9 \sqrt {x^{3}+1}\, \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}} \]
command
integrate(x/(2^(2/3)+x)/(x^3+1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{9} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (-\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2 \, x^{5} + 2 \, x^{2} - 2^{\frac {2}{3}} {\left (7 \, x^{4} + 4 \, x\right )} - 2^{\frac {1}{3}} {\left (5 \, x^{3} + 2\right )}\right )} \sqrt {x^{3} + 1}}{12 \, {\left (2 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) + \frac {2}{3} \, {\rm weierstrassPInverse}\left (0, -4, x\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (x^{3} - 2^{\frac {2}{3}} x^{2} + 2 \cdot 2^{\frac {1}{3}} x\right )} \sqrt {x^{3} + 1}}{x^{6} + 5 \, x^{3} + 4}, x\right ) \]