24.19 Problem number 515

\[ \int \frac {1}{x^2 (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx \]

Optimal antiderivative \[ \frac {2}{3 x \sqrt {1+x}\, \sqrt {x^{2}-x +1}}-\frac {5 \left (x^{3}+1\right )}{3 x \sqrt {1+x}\, \sqrt {x^{2}-x +1}}+\frac {\frac {5 x^{3}}{3}+\frac {5}{3}}{\left (1+x +\sqrt {3}\right ) \sqrt {1+x}\, \sqrt {x^{2}-x +1}}+\frac {5 \EllipticF \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \sqrt {2}\, \sqrt {1+x}\, \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}\, 3^{\frac {3}{4}}}{9 \sqrt {x^{2}-x +1}\, \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}}-\frac {5 \,3^{\frac {1}{4}} \EllipticE \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \sqrt {1+x}\, \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}}{6 \sqrt {x^{2}-x +1}\, \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}} \]

command

integrate(1/x^2/(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

\[ -\frac {{\left (5 \, x^{3} + 3\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} + 5 \, {\left (x^{4} + x\right )} {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right )}{3 \, {\left (x^{4} + x\right )}} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left (\frac {\sqrt {x^{2} - x + 1} \sqrt {x + 1}}{x^{8} + 2 \, x^{5} + x^{2}}, x\right ) \]