\[ \int (5-x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} \sqrt {3+2 x}}{21}-\frac {697 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{810 \sqrt {3 x^{2}+5 x +2}}+\frac {1039 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{1134 \sqrt {3 x^{2}+5 x +2}}+\frac {\left (2327+2169 x \right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{945} \]
command
integrate((5-x)*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{945} \, {\left (270 \, x^{2} - 1719 \, x - 2147\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {7723}{102060} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {697}{810} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}, x\right ) \]