\[ \int \frac {\sqrt {2+3 x}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx \]
Optimal antiderivative \[ \frac {37 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{2541}-\frac {2 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{2541}+\frac {2 \sqrt {2+3 x}\, \sqrt {3+5 x}}{33 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {74 \sqrt {2+3 x}\, \sqrt {3+5 x}}{2541 \sqrt {1-2 x}} \]
command
integrate((2+3*x)^(1/2)/(1-2*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
\[ -\frac {4 \, {\left (37 \, x - 57\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2541 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{40 \, x^{4} - 36 \, x^{3} - 6 \, x^{2} + 13 \, x - 3}, x\right ) \]