\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {556 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{15}-\frac {184 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{165}+\frac {14 \sqrt {1-2 x}}{3 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {2+3 x}}-\frac {92 \sqrt {1-2 x}\, \sqrt {2+3 x}}{3 \left (3+5 x \right )^{\frac {3}{2}}}+\frac {556 \sqrt {1-2 x}\, \sqrt {2+3 x}}{3 \sqrt {3+5 x}} \]
command
integrate((1-2*x)^(3/2)/(2+3*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (4170 \, x^{2} + 5144 \, x + 1583\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{1125 \, x^{5} + 3525 \, x^{4} + 4415 \, x^{3} + 2763 \, x^{2} + 864 \, x + 108}, x\right ) \]