\[ \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\frac {76}{3}+30 x}{x^{\frac {5}{2}} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {-1541-1965 x}{3 x^{\frac {5}{2}} \sqrt {3 x^{2}+5 x +2}}-\frac {9521 \left (2+3 x \right ) \sqrt {x}}{30 \sqrt {3 x^{2}+5 x +2}}+\frac {9521 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticE \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{30 \sqrt {3 x^{2}+5 x +2}}-\frac {1733 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticF \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{4 \sqrt {3 x^{2}+5 x +2}}+\frac {1252 \sqrt {3 x^{2}+5 x +2}}{5 x^{\frac {5}{2}}}-\frac {1733 \sqrt {3 x^{2}+5 x +2}}{6 x^{\frac {3}{2}}}+\frac {9521 \sqrt {3 x^{2}+5 x +2}}{30 \sqrt {x}} \]
command
integrate((2-5*x)/x^(7/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {30380 \, \sqrt {3} {\left (9 \, x^{7} + 30 \, x^{6} + 37 \, x^{5} + 20 \, x^{4} + 4 \, x^{3}\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 85689 \, \sqrt {3} {\left (9 \, x^{7} + 30 \, x^{6} + 37 \, x^{5} + 20 \, x^{4} + 4 \, x^{3}\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 9 \, {\left (85689 \, x^{6} + 207645 \, x^{5} + 159935 \, x^{4} + 36225 \, x^{3} - 1752 \, x^{2} + 130 \, x - 12\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{270 \, {\left (9 \, x^{7} + 30 \, x^{6} + 37 \, x^{5} + 20 \, x^{4} + 4 \, x^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )} \sqrt {x}}{27 \, x^{10} + 135 \, x^{9} + 279 \, x^{8} + 305 \, x^{7} + 186 \, x^{6} + 60 \, x^{5} + 8 \, x^{4}}, x\right ) \]