\[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {36968 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{231}-\frac {1112 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{231}+\frac {2 \sqrt {1-2 x}}{3 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}+\frac {416 \sqrt {1-2 x}}{21 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {2+3 x}}-\frac {2780 \sqrt {1-2 x}\, \sqrt {2+3 x}}{21 \left (3+5 x \right )^{\frac {3}{2}}}+\frac {184840 \sqrt {1-2 x}\, \sqrt {2+3 x}}{231 \sqrt {3+5 x}} \]
command
integrate((1-2*x)^(1/2)/(2+3*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (4158900 \, x^{3} + 7902930 \, x^{2} + 4998904 \, x + 1052533\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{231 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3375 \, x^{6} + 12825 \, x^{5} + 20295 \, x^{4} + 17119 \, x^{3} + 8118 \, x^{2} + 2052 \, x + 216}, x\right ) \]