\[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {62 \left (2+3 x \right ) \sqrt {x}}{21 \sqrt {3 x^{2}+5 x +2}}+\frac {43 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticF \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{14 \sqrt {3 x^{2}+5 x +2}}-\frac {62 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticE \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{21 \sqrt {3 x^{2}+5 x +2}}-\frac {4 \left (1-3 x \right ) \sqrt {3 x^{2}+5 x +2}}{7 x^{\frac {7}{2}}}+\frac {43 \sqrt {3 x^{2}+5 x +2}}{21 x^{\frac {3}{2}}}-\frac {62 \sqrt {3 x^{2}+5 x +2}}{21 \sqrt {x}} \]
command
integrate((2-5*x)*(3*x^2+5*x+2)^(1/2)/x^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {77 \, \sqrt {3} x^{4} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 558 \, \sqrt {3} x^{4} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 9 \, {\left (62 \, x^{3} - 43 \, x^{2} - 36 \, x + 12\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{189 \, x^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )}}{x^{\frac {9}{2}}}, x\right ) \]