\[ \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {\left (119+127 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} \sqrt {3+2 x}}{99}-\frac {2 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} \sqrt {3+2 x}}{33}-\frac {15283 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{53460 \sqrt {3 x^{2}+5 x +2}}+\frac {4153 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{10692 \sqrt {3 x^{2}+5 x +2}}-\frac {\left (1246+3987 x \right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{8910} \]
command
integrate((5-x)*(3*x^2+5*x+2)^(3/2)*(3+2*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{8910} \, {\left (4860 \, x^{4} - 18090 \, x^{3} - 69300 \, x^{2} - 61623 \, x - 18014\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {64621}{962280} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {15283}{53460} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}, x\right ) \]