\[ \int \frac {(2-5 x) x^{5/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 x^{\frac {3}{2}} \left (74+95 x \right )}{9 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {3464 \left (2+3 x \right ) \sqrt {x}}{27 \sqrt {3 x^{2}+5 x +2}}+\frac {4 \left (715+866 x \right ) \sqrt {x}}{9 \sqrt {3 x^{2}+5 x +2}}+\frac {3464 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticE \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{27 \sqrt {3 x^{2}+5 x +2}}-\frac {1430 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticF \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{9 \sqrt {3 x^{2}+5 x +2}} \]
command
integrate((2-5*x)*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
\[ -\frac {2 \, {\left (4210 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 15588 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 27 \, {\left (5196 \, x^{3} + 13045 \, x^{2} + 10688 \, x + 2860\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}\right )}}{243 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (5 \, x^{3} - 2 \, x^{2}\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{27 \, x^{6} + 135 \, x^{5} + 279 \, x^{4} + 305 \, x^{3} + 186 \, x^{2} + 60 \, x + 8}, x\right ) \]