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