\[ \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (3+2 x \right )^{\frac {5}{2}} \left (121+139 x \right )}{9 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {28 \left (1018+1177 x \right ) \sqrt {3+2 x}}{27 \sqrt {3 x^{2}+5 x +2}}-\frac {31892 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{81 \sqrt {3 x^{2}+5 x +2}}+\frac {41860 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{81 \sqrt {3 x^{2}+5 x +2}} \]
command
integrate((5-x)*(3+2*x)^(7/2)/(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
\[ \frac {2 \, {\left (30401 \, \sqrt {6} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 143514 \, \sqrt {6} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 27 \, {\left (47766 \, x^{3} + 118690 \, x^{2} + 96107 \, x + 25237\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}\right )}}{729 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (8 \, x^{4} - 4 \, x^{3} - 126 \, x^{2} - 243 \, x - 135\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{27 \, x^{6} + 135 \, x^{5} + 279 \, x^{4} + 305 \, x^{3} + 186 \, x^{2} + 60 \, x + 8}, x\right ) \]