\[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^2} \, dx \]
Optimal antiderivative \[ \frac {9 x^{2} \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{7}-\frac {\left (x^{3}+1\right ) \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{x}+\frac {27 \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{7 \left (1+x +\sqrt {3}\right )}+\frac {9 \,3^{\frac {3}{4}} \left (1+x \right )^{\frac {3}{2}} \EllipticF \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \sqrt {2}\, \sqrt {x^{2}-x +1}\, \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}}{7 \left (x^{3}+1\right ) \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}}-\frac {27 \,3^{\frac {1}{4}} \left (1+x \right )^{\frac {3}{2}} \EllipticE \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \sqrt {x^{2}-x +1}\, \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}}{14 \left (x^{3}+1\right ) \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}} \]
command
integrate((1+x)^(3/2)*(x^2-x+1)^(3/2)/x^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (2 \, x^{3} - 7\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} - 27 \, x {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right )}{7 \, x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (x^{3} + 1\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}}{x^{2}}, x\right ) \]