\[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {220076 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{108045}-\frac {6584 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{108045}-\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}}{21 \left (2+3 x \right )^{\frac {7}{2}}}+\frac {74 \sqrt {1-2 x}\, \sqrt {3+5 x}}{735 \left (2+3 x \right )^{\frac {5}{2}}}+\frac {3184 \sqrt {1-2 x}\, \sqrt {3+5 x}}{5145 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {220076 \sqrt {1-2 x}\, \sqrt {3+5 x}}{36015 \sqrt {2+3 x}} \]
command
integrate((1-2*x)^(1/2)*(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
\[ \frac {2 \, {\left (2971026 \, x^{3} + 6042348 \, x^{2} + 4100535 \, x + 926791\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{36015 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32}, x\right ) \]