\[ \int (2-5 x) x^{3/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {10 x^{\frac {3}{2}} \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{39}-\frac {4 \left (6959+8575 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} \sqrt {x}}{27027}+\frac {556 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} \sqrt {x}}{1287}+\frac {55112 \left (2+3 x \right ) \sqrt {x}}{729729 \sqrt {3 x^{2}+5 x +2}}-\frac {55112 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticE \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{729729 \sqrt {3 x^{2}+5 x +2}}+\frac {25448 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticF \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{243243 \sqrt {3 x^{2}+5 x +2}}+\frac {8 \left (6908+6381 x \right ) \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{243243} \]
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
\[ -\frac {2}{243243} \, {\left (280665 \, x^{5} + 462672 \, x^{4} + 40635 \, x^{3} - 172818 \, x^{2} - 16614 \, x + 12724\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} + \frac {26072}{938223} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - \frac {55112}{729729} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (15 \, x^{4} + 19 \, x^{3} - 4 \, x\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}, x\right ) \]