\[ \int \frac {(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {592 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{43923}-\frac {230 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{43923}+\frac {7 \sqrt {2+3 x}}{33 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}+\frac {26 \sqrt {2+3 x}}{121 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}-\frac {575 \sqrt {1-2 x}\, \sqrt {2+3 x}}{3993 \left (3+5 x \right )^{\frac {3}{2}}}-\frac {2960 \sqrt {1-2 x}\, \sqrt {2+3 x}}{43923 \sqrt {3+5 x}} \]
command
integrate((2+3*x)^(3/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (29600 \, x^{3} + 810 \, x^{2} - 13572 \, x - 1775\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{43923 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}}{1000 \, x^{6} + 300 \, x^{5} - 870 \, x^{4} - 179 \, x^{3} + 261 \, x^{2} + 27 \, x - 27}, x\right ) \]