\[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {595324 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{138915}-\frac {18016 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{138915}-\frac {2 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {3+5 x}}{21 \left (2+3 x \right )^{\frac {7}{2}}}+\frac {82 \sqrt {1-2 x}\, \sqrt {3+5 x}}{315 \left (2+3 x \right )^{\frac {5}{2}}}+\frac {8516 \sqrt {1-2 x}\, \sqrt {3+5 x}}{6615 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {595324 \sqrt {1-2 x}\, \sqrt {3+5 x}}{46305 \sqrt {2+3 x}} \]
command
integrate((1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (8036874 \, x^{3} + 16342002 \, x^{2} + 11095995 \, x + 2510369\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{46305 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32}, x\right ) \]