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