\[ \int (2-5 x) \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {4 \left (65+84 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} \sqrt {x}}{231}-\frac {10 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} \sqrt {x}}{33}-\frac {424 \left (2+3 x \right ) \sqrt {x}}{1155 \sqrt {3 x^{2}+5 x +2}}+\frac {424 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticE \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{1155 \sqrt {3 x^{2}+5 x +2}}-\frac {36 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticF \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{77 \sqrt {3 x^{2}+5 x +2}}-\frac {4 \left (55+39 x \right ) \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{385} \]
command
integrate((2-5*x)*(3*x^2+5*x+2)^(3/2)*x^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2}{1155} \, {\left (1575 \, x^{4} + 2730 \, x^{3} + 325 \, x^{2} - 1196 \, x - 270\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} - \frac {32}{297} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) + \frac {424}{1155} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (15 \, x^{3} + 19 \, x^{2} - 4\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}, x\right ) \]