\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}{27 \left (2+3 x \right )^{\frac {9}{2}}}-\frac {19885156 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{8751645}-\frac {609304 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{8751645}+\frac {74 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{189 \left (2+3 x \right )^{\frac {7}{2}}}-\frac {8252 \sqrt {1-2 x}\, \sqrt {3+5 x}}{19845 \left (2+3 x \right )^{\frac {5}{2}}}+\frac {280904 \sqrt {1-2 x}\, \sqrt {3+5 x}}{416745 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {19885156 \sqrt {1-2 x}\, \sqrt {3+5 x}}{2917215 \sqrt {2+3 x}} \]
command
integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(11/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (805348818 \, x^{4} + 2174142276 \, x^{3} + 2204875881 \, x^{2} + 993561978 \, x + 167622907\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2917215 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (10 \, x^{2} + x - 3\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64}, x\right ) \]