\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx \]
Optimal antiderivative \[ -\frac {22738708 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{20420505}-\frac {673072 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{20420505}-\frac {2 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{27 \left (2+3 x \right )^{\frac {9}{2}}}-\frac {214 \sqrt {1-2 x}\, \sqrt {3+5 x}}{3969 \left (2+3 x \right )^{\frac {7}{2}}}+\frac {8842 \sqrt {1-2 x}\, \sqrt {3+5 x}}{138915 \left (2+3 x \right )^{\frac {5}{2}}}+\frac {332372 \sqrt {1-2 x}\, \sqrt {3+5 x}}{972405 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {22738708 \sqrt {1-2 x}\, \sqrt {3+5 x}}{6806835 \sqrt {2+3 x}} \]
command
integrate((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(11/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (920917674 \, x^{4} + 2487189618 \, x^{3} + 2520548433 \, x^{2} + 1134125364 \, x + 190959271\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{6806835 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64}, x\right ) \]