\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (44+51 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{45 \left (3+2 x \right )^{\frac {9}{2}}}+\frac {23 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{22500 \sqrt {3 x^{2}+5 x +2}}+\frac {7 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{4500 \sqrt {3 x^{2}+5 x +2}}-\frac {\left (189+211 x \right ) \sqrt {3 x^{2}+5 x +2}}{2250 \left (3+2 x \right )^{\frac {5}{2}}}-\frac {23 \sqrt {3 x^{2}+5 x +2}}{11250 \sqrt {3+2 x}} \]
command
integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(11/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {499 \, \sqrt {6} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) - 414 \, \sqrt {6} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 36 \, {\left (368 \, x^{4} - 31822 \, x^{3} - 75342 \, x^{2} - 54697 \, x - 11632\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{405000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729}, x\right ) \]