\[ \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (3+2 x \right )^{\frac {3}{2}} \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{51}-\frac {\left (949997+1332121 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} \sqrt {3+2 x}}{8270262}+\frac {\left (1063774+1253571 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} \sqrt {3+2 x}}{984555}+\frac {1166 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} \sqrt {3+2 x}}{2295}-\frac {34355693 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{637991640 \sqrt {3 x^{2}+5 x +2}}+\frac {62005241 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{893188296 \sqrt {3 x^{2}+5 x +2}}+\frac {\left (12174838-22593339 x \right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{744323580} \]
command
integrate((5-x)*(3+2*x)^(3/2)*(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{744323580} \, {\left (1576214640 \, x^{7} + 35026992 \, x^{6} - 31471976844 \, x^{5} - 98939331792 \, x^{4} - 136604504862 \, x^{3} - 98401767552 \, x^{2} - 36153819495 \, x - 5382782386\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {866317037}{80386946640} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {34355693}{637991640} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (18 \, x^{6} - 3 \, x^{5} - 271 \, x^{4} - 669 \, x^{3} - 687 \, x^{2} - 328 \, x - 60\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}, x\right ) \]