\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {159 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{1250 \sqrt {3 x^{2}+5 x +2}}+\frac {183 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{1750 \sqrt {3 x^{2}+5 x +2}}+\frac {183 \sqrt {3 x^{2}+5 x +2}}{875 \left (3+2 x \right )^{\frac {3}{2}}}+\frac {\left (46+139 x \right ) \sqrt {3 x^{2}+5 x +2}}{175 \left (3+2 x \right )^{\frac {7}{2}}}+\frac {159 \sqrt {3 x^{2}+5 x +2}}{625 \sqrt {3+2 x}} \]
command
integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {223 \, \sqrt {6} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) - 6678 \, \sqrt {6} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 12 \, {\left (8904 \, x^{3} + 43728 \, x^{2} + 74557 \, x + 39436\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{52500 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}}{32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243}, x\right ) \]