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