\[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {5594 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{46305}-\frac {1196 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{46305}+\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}}{105 \left (2+3 x \right )^{\frac {5}{2}}}-\frac {404 \sqrt {1-2 x}\, \sqrt {3+5 x}}{2205 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {5594 \sqrt {1-2 x}\, \sqrt {3+5 x}}{15435 \sqrt {2+3 x}} \]
command
integrate((3+5*x)^(3/2)/(2+3*x)^(7/2)/(1-2*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (25173 \, x^{2} + 29322 \, x + 8507\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{15435 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16}, x\right ) \]