\[ \int \frac {x^3}{(1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {4 x}{27 \sqrt {1+x}\, \sqrt {x^{2}-x +1}}-\frac {2 x}{9 \left (x^{3}+1\right ) \sqrt {1+x}\, \sqrt {x^{2}-x +1}}+\frac {4 \EllipticF \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \sqrt {1+x}\, \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}\, 3^{\frac {3}{4}}}{81 \sqrt {x^{2}-x +1}\, \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}} \]
command
integrate(x^3/(1+x)^(5/2)/(x^2-x+1)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (2 \, x^{4} - x\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} + 2 \, {\left (x^{6} + 2 \, x^{3} + 1\right )} {\rm weierstrassPInverse}\left (0, -4, x\right )\right )}}{27 \, {\left (x^{6} + 2 \, x^{3} + 1\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {x^{2} - x + 1} \sqrt {x + 1} x^{3}}{x^{9} + 3 \, x^{6} + 3 \, x^{3} + 1}, x\right ) \]