\[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (3+2 x \right )^{\frac {3}{2}} \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{39}+\frac {\left (43822+50771 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} \sqrt {3+2 x}}{27027}+\frac {886 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} \sqrt {3+2 x}}{1287}-\frac {152657 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{2084940 \sqrt {3 x^{2}+5 x +2}}+\frac {332459 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{2918916 \sqrt {3 x^{2}+5 x +2}}-\frac {\left (486863+783711 x \right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{2432430} \]
command
integrate((5-x)*(3+2*x)^(3/2)*(3*x^2+5*x+2)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{2432430} \, {\left (2245320 \, x^{5} - 4218480 \, x^{4} - 43487010 \, x^{3} - 77801130 \, x^{2} - 53083449 \, x - 12602377\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {6411863}{262702440} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {152657}{2084940} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (6 \, x^{4} - 11 \, x^{3} - 76 \, x^{2} - 89 \, x - 30\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}, x\right ) \]