\[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (121+139 x \right ) \sqrt {3+2 x}}{9 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {4 \left (1390+1689 x \right ) \sqrt {3+2 x}}{9 \sqrt {3 x^{2}+5 x +2}}-\frac {2252 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{9 \sqrt {3 x^{2}+5 x +2}}+\frac {2956 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{9 \sqrt {3 x^{2}+5 x +2}} \]
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
\[ \frac {2 \, {\left (2147 \, \sqrt {6} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 10134 \, \sqrt {6} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 27 \, {\left (3378 \, x^{3} + 8410 \, x^{2} + 6839 \, x + 1813\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}\right )}}{81 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (2 \, x^{2} - 7 \, x - 15\right )} \sqrt {2 \, x + 3}}{27 \, x^{6} + 135 \, x^{5} + 279 \, x^{4} + 305 \, x^{3} + 186 \, x^{2} + 60 \, x + 8}, x\right ) \]