\[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {582 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{26411}+\frac {496 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{7203}+\frac {11 \sqrt {3+5 x}}{21 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {3}{2}}}+\frac {58 \sqrt {3+5 x}}{147 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}-\frac {89 \sqrt {1-2 x}\, \sqrt {3+5 x}}{343 \left (2+3 x \right )^{\frac {3}{2}}}-\frac {496 \sqrt {1-2 x}\, \sqrt {3+5 x}}{2401 \sqrt {2+3 x}} \]
command
integrate((3+5*x)^(3/2)/(1-2*x)^(5/2)/(2+3*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (8928 \, x^{3} + 762 \, x^{2} - 4616 \, x - 885\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{7203 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{216 \, x^{6} + 108 \, x^{5} - 198 \, x^{4} - 71 \, x^{3} + 66 \, x^{2} + 12 \, x - 8}, x\right ) \]