\[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx \]
Optimal antiderivative \[ \frac {430 \left (3+2 x \right )^{\frac {3}{2}} \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{969}-\frac {2 \left (3+2 x \right )^{\frac {5}{2}} \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{57}-\frac {125 \left (64006+79583 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} \sqrt {3+2 x}}{52378326}+\frac {25 \left (72737+86493 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} \sqrt {3+2 x}}{1247103}+\frac {2350 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} \sqrt {3+2 x}}{2907}-\frac {16503475 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{808122744 \sqrt {3 x^{2}+5 x +2}}+\frac {142149125 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{5656859208 \sqrt {3 x^{2}+5 x +2}}+\frac {25 \left (749099+216603 x \right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{942809868} \]
command
integrate((5-x)*(3+2*x)^(5/2)*(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{942809868} \, {\left (3572753184 \, x^{8} + 5989615632 \, x^{7} - 68880579768 \, x^{6} - 329194523196 \, x^{5} - 650694586500 \, x^{4} - 699517082754 \, x^{3} - 427399643682 \, x^{2} - 139652898507 \, x - 18986144459\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {18691975}{5359129776} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {16503475}{808122744} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (36 \, x^{7} + 48 \, x^{6} - 551 \, x^{5} - 2151 \, x^{4} - 3381 \, x^{3} - 2717 \, x^{2} - 1104 \, x - 180\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}, x\right ) \]