\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx \]
Optimal antiderivative \[ -\frac {1752 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{3773}-\frac {68 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{3773}+\frac {4 \sqrt {3+5 x}}{77 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}+\frac {54 \sqrt {1-2 x}\, \sqrt {3+5 x}}{539 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {5256 \sqrt {1-2 x}\, \sqrt {3+5 x}}{3773 \sqrt {2+3 x}} \]
command
integrate(1/(1-2*x)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (15768 \, x^{2} + 3006 \, x - 5543\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3773 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24}, x\right ) \]