\[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {6 \left (37+47 x \right )}{5 \left (3+2 x \right )^{\frac {5}{2}} \sqrt {3 x^{2}+5 x +2}}+\frac {213374 \EllipticE \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{9375 \sqrt {3 x^{2}+5 x +2}}-\frac {30734 \EllipticF \left (\sqrt {1+x}\, \sqrt {3}, \frac {i \sqrt {6}}{3}\right ) \sqrt {-3 x^{2}-5 x -2}\, \sqrt {3}}{1875 \sqrt {3 x^{2}+5 x +2}}-\frac {4124 \sqrt {3 x^{2}+5 x +2}}{125 \left (3+2 x \right )^{\frac {5}{2}}}-\frac {61468 \sqrt {3 x^{2}+5 x +2}}{1875 \left (3+2 x \right )^{\frac {3}{2}}}-\frac {426748 \sqrt {3 x^{2}+5 x +2}}{9375 \sqrt {3+2 x}} \]
command
integrate((5-x)/(3+2*x)^(7/2)/(3*x^2+5*x+2)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {161981 \, \sqrt {6} {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) - 1920366 \, \sqrt {6} {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 18 \, {\left (2560488 \, x^{4} + 12870964 \, x^{3} + 23654210 \, x^{2} + 18680161 \, x + 5280177\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{84375 \, {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}}{144 \, x^{8} + 1344 \, x^{7} + 5416 \, x^{6} + 12296 \, x^{5} + 17185 \, x^{4} + 15126 \, x^{3} + 8181 \, x^{2} + 2484 \, x + 324}, x\right ) \]