\[ \int \frac {2-5 x}{x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {76+90 x}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}+\frac {39 \left (2+3 x \right ) \sqrt {x}}{\sqrt {3 x^{2}+5 x +2}}-\frac {39 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticE \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{\sqrt {3 x^{2}+5 x +2}}+\frac {45 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticF \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{\sqrt {3 x^{2}+5 x +2}}-\frac {39 \sqrt {3 x^{2}+5 x +2}}{\sqrt {x}} \]
command
integrate((2-5*x)/x^(3/2)/(3*x^2+5*x+2)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {25 \, \sqrt {3} {\left (3 \, x^{3} + 5 \, x^{2} + 2 \, x\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 117 \, \sqrt {3} {\left (3 \, x^{3} + 5 \, x^{2} + 2 \, x\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 3 \, {\left (117 \, x^{2} + 105 \, x + 2\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{3 \, {\left (3 \, x^{3} + 5 \, x^{2} + 2 \, x\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )} \sqrt {x}}{9 \, x^{6} + 30 \, x^{5} + 37 \, x^{4} + 20 \, x^{3} + 4 \, x^{2}}, x\right ) \]