\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {106558 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{5294205}-\frac {220028 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{5294205}+\frac {11 \left (3+5 x \right )^{\frac {3}{2}}}{7 \left (2+3 x \right )^{\frac {7}{2}} \sqrt {1-2 x}}+\frac {229 \sqrt {1-2 x}\, \sqrt {3+5 x}}{1029 \left (2+3 x \right )^{\frac {7}{2}}}-\frac {37117 \sqrt {1-2 x}\, \sqrt {3+5 x}}{36015 \left (2+3 x \right )^{\frac {5}{2}}}-\frac {106772 \sqrt {1-2 x}\, \sqrt {3+5 x}}{252105 \left (2+3 x \right )^{\frac {3}{2}}}-\frac {106558 \sqrt {1-2 x}\, \sqrt {3+5 x}}{1764735 \sqrt {2+3 x}} \]
command
integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (2877066 \, x^{4} + 11042235 \, x^{3} + 12020751 \, x^{2} + 4889131 \, x + 616327\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1764735 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32}, x\right ) \]