\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (37+3 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{15 \left (3+2 x \right )^{\frac {3}{2}}}+\frac {289 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{12 \sqrt {3 x^{2}+5 x +2}}-\frac {367 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{20 \sqrt {3 x^{2}+5 x +2}}+\frac {\left (241+69 x \right ) \sqrt {3 x^{2}+5 x +2}}{10 \sqrt {3+2 x}} \]
command
integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1399 \, \sqrt {6} {\left (4 \, x^{2} + 12 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 6606 \, \sqrt {6} {\left (4 \, x^{2} + 12 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 12 \, {\left (18 \, x^{3} - 162 \, x^{2} - 1685 \, x - 2021\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{360 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27}, x\right ) \]