\[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {458 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{3087}-\frac {178 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{3087}+\frac {11 \sqrt {3+5 x}}{7 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}-\frac {97 \sqrt {1-2 x}\, \sqrt {3+5 x}}{147 \left (2+3 x \right )^{\frac {3}{2}}}-\frac {458 \sqrt {1-2 x}\, \sqrt {3+5 x}}{1029 \sqrt {2+3 x}} \]
command
integrate((3+5*x)^(3/2)/(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
\[ -\frac {2 \, {\left (1374 \, x^{2} + 908 \, x + 11\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1029 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8}, x\right ) \]