\[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {5636 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{36015}-\frac {4364 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{396165}+\frac {2 \sqrt {3+5 x}}{7 \left (2+3 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}-\frac {36 \sqrt {1-2 x}\, \sqrt {3+5 x}}{245 \left (2+3 x \right )^{\frac {5}{2}}}-\frac {26 \sqrt {1-2 x}\, \sqrt {3+5 x}}{1715 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {5636 \sqrt {1-2 x}\, \sqrt {3+5 x}}{12005 \sqrt {2+3 x}} \]
command
integrate((3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (50724 \, x^{3} + 41724 \, x^{2} - 13127 \, x - 11923\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{12005 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16}, x\right ) \]