\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{17/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (8901+8399 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{64350 \left (3+2 x \right )^{\frac {11}{2}}}+\frac {\left (94+119 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{195 \left (3+2 x \right )^{\frac {15}{2}}}-\frac {335723 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{160875000 \sqrt {3 x^{2}+5 x +2}}+\frac {594851 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{225225000 \sqrt {3 x^{2}+5 x +2}}+\frac {594851 \sqrt {3 x^{2}+5 x +2}}{112612500 \left (3+2 x \right )^{\frac {3}{2}}}-\frac {\left (386846+328339 x \right ) \sqrt {3 x^{2}+5 x +2}}{7507500 \left (3+2 x \right )^{\frac {7}{2}}}+\frac {335723 \sqrt {3 x^{2}+5 x +2}}{80437500 \sqrt {3+2 x}} \]
command
integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(17/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {7967807 \, \sqrt {6} {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 42301098 \, \sqrt {6} {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 36 \, {\left (300807808 \, x^{7} + 3348834304 \, x^{6} + 17742950508 \, x^{5} + 46830142120 \, x^{4} + 67557035830 \, x^{3} + 55283449932 \, x^{2} + 24502214271 \, x + 4641518352\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{20270250000 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683}, x\right ) \]