\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (73-33 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{6930 \left (3+2 x \right )^{\frac {9}{2}}}+\frac {\left (8+9 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{11 \left (3+2 x \right )^{\frac {13}{2}}}+\frac {5083 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{495000 \sqrt {3 x^{2}+5 x +2}}-\frac {9421 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{693000 \sqrt {3 x^{2}+5 x +2}}+\frac {\left (21492+17833 x \right ) \sqrt {3 x^{2}+5 x +2}}{346500 \left (3+2 x \right )^{\frac {5}{2}}}-\frac {5083 \sqrt {3 x^{2}+5 x +2}}{247500 \sqrt {3+2 x}} \]
command
integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(15/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {139297 \, \sqrt {6} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 640458 \, \sqrt {6} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 36 \, {\left (2277184 \, x^{6} + 6409516 \, x^{5} + 12953760 \, x^{4} + 33648370 \, x^{3} + 54318160 \, x^{2} + 41339721 \, x + 11865789\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{62370000 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561}, x\right ) \]