\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (2+3 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{3 x^{\frac {3}{2}}}-\frac {34 \left (2+3 x \right ) \sqrt {x}}{3 \sqrt {3 x^{2}+5 x +2}}+\frac {34 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticE \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{3 \sqrt {3 x^{2}+5 x +2}}-\frac {14 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticF \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{\sqrt {3 x^{2}+5 x +2}}+\frac {2 \left (2-x \right ) \sqrt {3 x^{2}+5 x +2}}{\sqrt {x}} \]
command
integrate((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (41 \, \sqrt {3} x^{2} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 153 \, \sqrt {3} x^{2} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) + 9 \, {\left (9 \, x^{3} + 24 \, x^{2} + 10 \, x + 4\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}\right )}}{27 \, x^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (15 \, x^{3} + 19 \, x^{2} - 4\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{x^{\frac {5}{2}}}, x\right ) \]