\[ \int \frac {5-x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {6 \left (37+47 x \right ) \sqrt {3+2 x}}{5 \sqrt {3 x^{2}+5 x +2}}-\frac {70 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{3 \sqrt {3 x^{2}+5 x +2}}+\frac {94 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{5 \sqrt {3 x^{2}+5 x +2}} \]
command
integrate((5-x)/(3*x^2+5*x+2)^(3/2)/(3+2*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {149 \, \sqrt {6} {\left (3 \, x^{2} + 5 \, x + 2\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 846 \, \sqrt {6} {\left (3 \, x^{2} + 5 \, x + 2\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 54 \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (47 \, x + 37\right )} \sqrt {2 \, x + 3}}{45 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}}{18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12}, x\right ) \]