\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{3/2}}{\sqrt {3+5 x}} \, dx \]
Optimal antiderivative \[ -\frac {8024546 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{26578125}-\frac {509189 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{26578125}+\frac {106 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {3}{2}} \sqrt {3+5 x}}{1575}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \left (2+3 x \right )^{\frac {3}{2}} \sqrt {3+5 x}}{45}+\frac {8878 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{118125}+\frac {21547 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{1771875} \]
command
integrate((1-2*x)^(5/2)*(2+3*x)^(3/2)/(3+5*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{1771875} \, {\left (945000 \, x^{3} - 1030500 \, x^{2} - 113490 \, x + 683887\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{\sqrt {5 \, x + 3}}, x\right ) \]