\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}}}{21 \left (2+3 x \right )^{\frac {7}{2}}}+\frac {46 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}{63 \left (2+3 x \right )^{\frac {5}{2}}}-\frac {11576 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{11907}-\frac {4244 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{11907}+\frac {608 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{189 \left (2+3 x \right )^{\frac {3}{2}}}-\frac {4244 \sqrt {1-2 x}\, \sqrt {3+5 x}}{3969 \sqrt {2+3 x}} \]
command
integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (182736 \, x^{3} + 409005 \, x^{2} + 292578 \, x + 67759\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3969 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32}, x\right ) \]