\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx \]
Optimal antiderivative \[ -\frac {244604 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{132055}-\frac {7536 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{132055}+\frac {4 \sqrt {3+5 x}}{77 \left (2+3 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}+\frac {138 \sqrt {1-2 x}\, \sqrt {3+5 x}}{2695 \left (2+3 x \right )^{\frac {5}{2}}}+\frac {10308 \sqrt {1-2 x}\, \sqrt {3+5 x}}{18865 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {733812 \sqrt {1-2 x}\, \sqrt {3+5 x}}{132055 \sqrt {2+3 x}} \]
command
integrate(1/(1-2*x)^(3/2)/(2+3*x)^(7/2)/(3+5*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (6604308 \, x^{3} + 5720058 \, x^{2} - 1424784 \, x - 1546591\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{132055 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1620 \, x^{7} + 3672 \, x^{6} + 2025 \, x^{5} - 1077 \, x^{4} - 1312 \, x^{3} - 152 \, x^{2} + 176 \, x + 48}, x\right ) \]