\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2209 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{2025}+\frac {494 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{2025}-\frac {2 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}{3 \sqrt {2+3 x}}-\frac {8 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{15}+\frac {494 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{135} \]
command
integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (90 \, x^{2} - 102 \, x - 143\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{135 \, \sqrt {3 \, x + 2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (10 \, x^{2} + x - 3\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{9 \, x^{2} + 12 \, x + 4}, x\right ) \]