\[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {38 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{1029}-\frac {212 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{11319}+\frac {2 \sqrt {3+5 x}}{7 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}-\frac {8 \sqrt {1-2 x}\, \sqrt {3+5 x}}{49 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {38 \sqrt {1-2 x}\, \sqrt {3+5 x}}{343 \sqrt {2+3 x}} \]
command
integrate((3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (114 \, x^{2} - 37 \, x - 59\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{343 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8}, x\right ) \]