\[ \int \frac {(5-x) (3+2 x)^{5/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (3+2 x \right )^{\frac {3}{2}} \left (121+139 x \right )}{3 \sqrt {3 x^{2}+5 x +2}}+\frac {3830 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{81 \sqrt {3 x^{2}+5 x +2}}-\frac {4150 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{81 \sqrt {3 x^{2}+5 x +2}}+\frac {1660 \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{27} \]
command
integrate((5-x)*(3+2*x)^(5/2)/(3*x^2+5*x+2)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {3355 \, \sqrt {6} {\left (3 \, x^{2} + 5 \, x + 2\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 34470 \, \sqrt {6} {\left (3 \, x^{2} + 5 \, x + 2\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 54 \, {\left (12 \, x^{2} + 1781 \, x + 1607\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{729 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (4 \, x^{3} - 8 \, x^{2} - 51 \, x - 45\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4}, x\right ) \]