\[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {16732 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{924385}+\frac {3946 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{252105}+\frac {11 \sqrt {3+5 x}}{21 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {5}{2}}}+\frac {124 \sqrt {3+5 x}}{147 \left (2+3 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}-\frac {779 \sqrt {1-2 x}\, \sqrt {3+5 x}}{1715 \left (2+3 x \right )^{\frac {5}{2}}}-\frac {2264 \sqrt {1-2 x}\, \sqrt {3+5 x}}{12005 \left (2+3 x \right )^{\frac {3}{2}}}-\frac {3946 \sqrt {1-2 x}\, \sqrt {3+5 x}}{84035 \sqrt {2+3 x}} \]
command
integrate((3+5*x)^(3/2)/(1-2*x)^(5/2)/(2+3*x)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (213084 \, x^{4} + 356292 \, x^{3} - 2199 \, x^{2} - 158902 \, x - 43881\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{252105 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{648 \, x^{7} + 756 \, x^{6} - 378 \, x^{5} - 609 \, x^{4} + 56 \, x^{3} + 168 \, x^{2} - 16}, x\right ) \]