\[ \int \frac {(2-5 x) \sqrt {x}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (30+37 x \right ) \sqrt {x}}{3 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {198 \left (2+3 x \right ) \sqrt {x}}{\sqrt {3 x^{2}+5 x +2}}+\frac {2 \left (250+297 x \right ) \sqrt {x}}{\sqrt {3 x^{2}+5 x +2}}+\frac {198 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticE \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{\sqrt {3 x^{2}+5 x +2}}-\frac {245 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticF \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{\sqrt {3 x^{2}+5 x +2}} \]
command
integrate((2-5*x)*x^(1/2)/(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
\[ -\frac {2 \, {\left (80 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 297 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - {\left (2673 \, x^{3} + 6705 \, x^{2} + 5495 \, x + 1470\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}\right )}}{3 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )} \sqrt {x}}{27 \, x^{6} + 135 \, x^{5} + 279 \, x^{4} + 305 \, x^{3} + 186 \, x^{2} + 60 \, x + 8}, x\right ) \]