\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx \]
Optimal antiderivative \[ -\frac {886499 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{765450}-\frac {11908 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{382725}+\frac {2 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}} \sqrt {2+3 x}}{27}-\frac {499 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{2835}+\frac {46 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{567}-\frac {11908 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{25515} \]
command
integrate((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{25515} \, {\left (94500 \, x^{3} + 14400 \, x^{2} - 62325 \, x - 10259\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{\sqrt {3 \, x + 2}}, x\right ) \]