\[ \int \frac {x}{(2-x) \sqrt {1+x^3}} \, dx \]
Optimal antiderivative \[ \frac {4 \arctanh \left (\frac {\left (1+x \right )^{2}}{3 \sqrt {x^{3}+1}}\right )}{9}-\frac {2 \left (1+x \right ) \EllipticF \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}\, 3^{\frac {3}{4}}}{9 \sqrt {x^{3}+1}\, \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}} \]
command
integrate(x/(2-x)/(x^3+1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2}{9} \, \log \left (\frac {x^{3} + 12 \, x^{2} + 6 \, \sqrt {x^{3} + 1} {\left (x + 1\right )} - 6 \, x + 10}{x^{3} - 6 \, x^{2} + 12 \, x - 8}\right ) - \frac {2}{3} \, {\rm weierstrassPInverse}\left (0, -4, x\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {x^{3} + 1} x}{x^{4} - 2 \, x^{3} + x - 2}, x\right ) \]