\[ \int \frac {5-x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {6 \left (37+47 x \right )}{5 \left (3+2 x \right )^{\frac {3}{2}} \sqrt {3 x^{2}+5 x +2}}+\frac {7438 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{375 \sqrt {3 x^{2}+5 x +2}}-\frac {1258 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{75 \sqrt {3 x^{2}+5 x +2}}-\frac {2516 \sqrt {3 x^{2}+5 x +2}}{75 \left (3+2 x \right )^{\frac {3}{2}}}-\frac {14876 \sqrt {3 x^{2}+5 x +2}}{375 \sqrt {3+2 x}} \]
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
\[ \frac {1447 \, \sqrt {6} {\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) - 66942 \, \sqrt {6} {\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 18 \, {\left (44628 \, x^{3} + 160192 \, x^{2} + 183347 \, x + 65533\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{3375 \, {\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}}{72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108}, x\right ) \]