\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (2291+879 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{210 \left (3+2 x \right )^{\frac {3}{2}}}-\frac {\left (53+5 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{35 \left (3+2 x \right )^{\frac {5}{2}}}-\frac {12857 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{168 \sqrt {3 x^{2}+5 x +2}}+\frac {2333 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{40 \sqrt {3 x^{2}+5 x +2}}-\frac {\left (10763+3117 x \right ) \sqrt {3 x^{2}+5 x +2}}{140 \sqrt {3+2 x}} \]
command
integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {62207 \, \sqrt {6} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 293958 \, \sqrt {6} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 12 \, {\left (540 \, x^{5} - 3024 \, x^{4} - 2190 \, x^{3} + 145640 \, x^{2} + 386981 \, x + 265653\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{5040 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81}, x\right ) \]