\[ \int x (1+x)^{3/2} \left (1-x+x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {18 x^{2} \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{91}+\frac {2 x^{2} \left (x^{3}+1\right ) \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{13}+\frac {54 \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{91 \left (1+x +\sqrt {3}\right )}+\frac {18 \,3^{\frac {3}{4}} \left (1+x \right )^{\frac {3}{2}} \EllipticF \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \sqrt {2}\, \sqrt {x^{2}-x +1}\, \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}}{91 \left (x^{3}+1\right ) \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}}-\frac {27 \,3^{\frac {1}{4}} \left (1+x \right )^{\frac {3}{2}} \EllipticE \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \sqrt {x^{2}-x +1}\, \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}}{91 \left (x^{3}+1\right ) \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}} \]
command
integrate(x*(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
\[ \frac {2}{91} \, {\left (7 \, x^{5} + 16 \, x^{2}\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} - \frac {54}{91} \, {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (x^{4} + x\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}, x\right ) \]