\[ \int (2-5 x) x^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {136 x^{\frac {3}{2}} \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{351}-\frac {2 x^{\frac {5}{2}} \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{9}+\frac {8 \left (27010+32921 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} \sqrt {x}}{243243}-\frac {4660 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} \sqrt {x}}{11583}-\frac {497824 \left (2+3 x \right ) \sqrt {x}}{32837805 \sqrt {3 x^{2}+5 x +2}}+\frac {497824 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticE \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{32837805 \sqrt {3 x^{2}+5 x +2}}-\frac {61736 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticF \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{2189187 \sqrt {3 x^{2}+5 x +2}}-\frac {8 \left (190465+205407 x \right ) \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{10945935} \]
command
integrate((2-5*x)*x^(5/2)*(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
\[ -\frac {2}{10945935} \, {\left (10945935 \, x^{6} + 17401230 \, x^{5} + 1199205 \, x^{4} - 5859000 \, x^{3} - 292590 \, x^{2} + 215748 \, x - 154340\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} - \frac {87632}{8444007} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) + \frac {497824}{32837805} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (15 \, x^{5} + 19 \, x^{4} - 4 \, x^{2}\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}, x\right ) \]