\[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {22 \left (2+3 x \right ) \sqrt {x}}{9 \sqrt {3 x^{2}+5 x +2}}-\frac {22 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticE \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{9 \sqrt {3 x^{2}+5 x +2}}+\frac {10 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticF \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{3 \sqrt {3 x^{2}+5 x +2}}-\frac {2 \left (6+5 x \right ) \sqrt {3 x^{2}+5 x +2}}{3 \sqrt {x}} \]
command
integrate((2-5*x)*(3*x^2+5*x+2)^(1/2)/x^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (35 \, \sqrt {3} x {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 99 \, \sqrt {3} x {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 27 \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x + 6\right )} \sqrt {x}\right )}}{81 \, x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )}}{x^{\frac {3}{2}}}, x\right ) \]