\[ \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {\left (15076+34643 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} \sqrt {3+2 x}}{162162}+\frac {\left (15467+17193 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} \sqrt {3+2 x}}{19305}-\frac {2 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} \sqrt {3+2 x}}{45}-\frac {2742319 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{12509640 \sqrt {3 x^{2}+5 x +2}}+\frac {5021353 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{17513496 \sqrt {3 x^{2}+5 x +2}}+\frac {\left (287729-2667537 x \right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{14594580} \]
command
integrate((5-x)*(3*x^2+5*x+2)^(5/2)*(3+2*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{14594580} \, {\left (17513496 \, x^{6} - 29413692 \, x^{5} - 314201916 \, x^{4} - 624522906 \, x^{3} - 552292686 \, x^{2} - 231246315 \, x - 39157073\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {72391021}{1576214640} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {2742319}{12509640} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}, x\right ) \]