\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {532 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{9}+\frac {16 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{9}+\frac {14 \sqrt {1-2 x}}{9 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {3+5 x}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {2+3 x}\, \sqrt {3+5 x}}-\frac {2660 \sqrt {1-2 x}\, \sqrt {2+3 x}}{9 \sqrt {3+5 x}} \]
command
integrate((1-2*x)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (3990 \, x^{2} + 5188 \, x + 1683\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72}, x\right ) \]