\[ \int \frac {a+b x}{(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 x \left (b x +a \right )}{3 \sqrt {1+x}\, \sqrt {x^{2}-x +1}}-\frac {2 b \left (x^{3}+1\right )}{3 \left (1+x +\sqrt {3}\right ) \sqrt {1+x}\, \sqrt {x^{2}-x +1}}+\frac {3^{\frac {1}{4}} b \EllipticE \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \sqrt {1+x}\, \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}}{3 \sqrt {x^{2}-x +1}\, \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}}+\frac {2 \EllipticF \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \left (a +b -b \sqrt {3}\right ) \sqrt {1+x}\, \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}\, 3^{\frac {3}{4}}}{9 \sqrt {x^{2}-x +1}\, \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}} \]
command
integrate((b*x+a)/(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
\[ \frac {2 \, {\left ({\left (b x^{2} + a x\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} + {\left (a x^{3} + a\right )} {\rm weierstrassPInverse}\left (0, -4, x\right ) + {\left (b x^{3} + b\right )} {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right )\right )}}{3 \, {\left (x^{3} + 1\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b x + a\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}}{x^{6} + 2 \, x^{3} + 1}, x\right ) \]