\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (73+x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{11 \sqrt {3+2 x}}+\frac {5 \left (218+3031 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} \sqrt {3+2 x}}{1386}+\frac {451331 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{21384 \sqrt {3 x^{2}+5 x +2}}-\frac {4145485 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{149688 \sqrt {3 x^{2}+5 x +2}}-\frac {\left (21871-471213 x \right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{24948} \]
command
integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {12034829 \, \sqrt {6} {\left (2 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 56867706 \, \sqrt {6} {\left (2 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 108 \, {\left (20412 \, x^{5} - 78624 \, x^{4} - 249894 \, x^{3} - 326988 \, x^{2} - 59065 \, x + 610149\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{2694384 \, {\left (2 \, x + 3\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{4 \, x^{2} + 12 \, x + 9}, x\right ) \]