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