\[ \int (2-5 x) x^{7/2} \sqrt {2+5 x+3 x^2} \, dx \]
Optimal antiderivative \[ -\frac {21620 x^{\frac {3}{2}} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{34749}+\frac {656 x^{\frac {5}{2}} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{1287}-\frac {10 x^{\frac {7}{2}} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{39}+\frac {157160 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} \sqrt {x}}{243243}+\frac {1543648 \left (2+3 x \right ) \sqrt {x}}{6567561 \sqrt {3 x^{2}+5 x +2}}-\frac {1543648 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticE \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{6567561 \sqrt {3 x^{2}+5 x +2}}+\frac {349240 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticF \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{2189187 \sqrt {3 x^{2}+5 x +2}}-\frac {8 \left (397265+502911 x \right ) \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{2189187} \]
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
\[ -\frac {2}{2189187} \, {\left (841995 \, x^{5} - 270459 \, x^{4} - 185220 \, x^{3} + 167634 \, x^{2} - 162396 \, x + 174620\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} - \frac {204560}{8444007} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - \frac {1543648}{6567561} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (5 \, x^{4} - 2 \, x^{3}\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}, x\right ) \]