\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{\sqrt {3+2 x}} \, dx \]
Optimal antiderivative \[ -\frac {5 \left (563+4669 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} \sqrt {3+2 x}}{18018}+\frac {\left (224-33 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} \sqrt {3+2 x}}{429}-\frac {651617 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{277992 \sqrt {3 x^{2}+5 x +2}}+\frac {5983645 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{1945944 \sqrt {3 x^{2}+5 x +2}}+\frac {\left (34372-676791 x \right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{324324} \]
command
integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{324324} \, {\left (224532 \, x^{5} - 775656 \, x^{4} - 2896614 \, x^{3} - 3513708 \, x^{2} - 1516527 \, x - 610408\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {17362253}{35026992} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {651617}{277992} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{\sqrt {2 \, x + 3}}, x\right ) \]