\[ \int \sqrt {1+x} (a+b x) \sqrt {1-x+x^2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (5 b \,x^{2}+7 a x \right ) \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{35}+\frac {6 b \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{7 \left (1+x +\sqrt {3}\right )}-\frac {3 \,3^{\frac {1}{4}} b \left (1+x \right )^{\frac {3}{2}} \EllipticE \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \sqrt {x^{2}-x +1}\, \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}}{7 \left (x^{3}+1\right ) \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}}+\frac {2 \,3^{\frac {3}{4}} \left (1+x \right )^{\frac {3}{2}} \EllipticF \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \left (7 a -5 b \left (1-\sqrt {3}\right )\right ) \sqrt {x^{2}-x +1}\, \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}}{35 \left (x^{3}+1\right ) \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}} \]
command
integrate((1+x)^(1/2)*(b*x+a)*(x^2-x+1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2}{35} \, {\left (5 \, b x^{2} + 7 \, a x\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} + \frac {6}{5} \, a {\rm weierstrassPInverse}\left (0, -4, x\right ) - \frac {6}{7} \, b {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b x + a\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}, x\right ) \]