\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (3354+2531 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{210 \left (3+2 x \right )^{\frac {5}{2}}}-\frac {\left (43+7 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{35 \left (3+2 x \right )^{\frac {7}{2}}}-\frac {4091 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{120 \sqrt {3 x^{2}+5 x +2}}+\frac {2505 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{56 \sqrt {3 x^{2}+5 x +2}}+\frac {\left (6292+1823 x \right ) \sqrt {3 x^{2}+5 x +2}}{140 \sqrt {3+2 x}} \]
command
integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {109079 \, \sqrt {6} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 515466 \, \sqrt {6} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 36 \, {\left (756 \, x^{5} - 6216 \, x^{4} - 192878 \, x^{3} - 730460 \, x^{2} - 998487 \, x - 467340\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{15120 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243}, x\right ) \]