\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{13/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (258+367 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{495 \left (3+2 x \right )^{\frac {11}{2}}}-\frac {5861 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{1237500 \sqrt {3 x^{2}+5 x +2}}+\frac {14807 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{1732500 \sqrt {3 x^{2}+5 x +2}}+\frac {14807 \sqrt {3 x^{2}+5 x +2}}{866250 \left (3+2 x \right )^{\frac {3}{2}}}-\frac {\left (15647+14773 x \right ) \sqrt {3 x^{2}+5 x +2}}{57750 \left (3+2 x \right )^{\frac {7}{2}}}+\frac {5861 \sqrt {3 x^{2}+5 x +2}}{618750 \sqrt {3+2 x}} \]
command
integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(13/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {338099 \, \sqrt {6} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 738486 \, \sqrt {6} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 36 \, {\left (1312864 \, x^{5} + 11031040 \, x^{4} + 41848650 \, x^{3} + 65139670 \, x^{2} + 42879355 \, x + 9919671\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{155925000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187}, x\right ) \]