\[ \int \frac {(2-5 x) x^{7/2}}{\sqrt {2+5 x+3 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {68920 \left (2+3 x \right ) \sqrt {x}}{15309 \sqrt {3 x^{2}+5 x +2}}+\frac {68920 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticE \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{15309 \sqrt {3 x^{2}+5 x +2}}-\frac {11320 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticF \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{5103 \sqrt {3 x^{2}+5 x +2}}-\frac {820 x^{\frac {3}{2}} \sqrt {3 x^{2}+5 x +2}}{567}+\frac {508 x^{\frac {5}{2}} \sqrt {3 x^{2}+5 x +2}}{567}-\frac {10 x^{\frac {7}{2}} \sqrt {3 x^{2}+5 x +2}}{27}+\frac {11320 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{5103} \]
command
integrate((2-5*x)*x^(7/2)/(3*x^2+5*x+2)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2}{5103} \, {\left (945 \, x^{3} - 2286 \, x^{2} + 3690 \, x - 5660\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} + \frac {20120}{19683} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) + \frac {68920}{15309} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (5 \, x^{4} - 2 \, x^{3}\right )} \sqrt {x}}{\sqrt {3 \, x^{2} + 5 \, x + 2}}, x\right ) \]