\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {595324 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{195657}-\frac {18016 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{195657}+\frac {4}{77 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}+\frac {186 \sqrt {1-2 x}}{539 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {2+3 x}}-\frac {45040 \sqrt {1-2 x}\, \sqrt {2+3 x}}{17787 \left (3+5 x \right )^{\frac {3}{2}}}+\frac {2976620 \sqrt {1-2 x}\, \sqrt {2+3 x}}{195657 \sqrt {3+5 x}} \]
command
integrate(1/(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
\[ \frac {2 \, {\left (44649300 \, x^{3} + 32744810 \, x^{2} - 10598372 \, x - 8473261\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{195657 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{4500 \, x^{7} + 9600 \, x^{6} + 4685 \, x^{5} - 3083 \, x^{4} - 3181 \, x^{3} - 261 \, x^{2} + 432 \, x + 108}, x\right ) \]