\[ \int (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx \]
Optimal antiderivative \[ \frac {2 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}}}{55}-\frac {604915631 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{584718750}-\frac {18177329 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{584718750}+\frac {1103 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{259875}+\frac {178 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{7425}-\frac {124891 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{2165625}-\frac {18177329 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{38981250} \]
command
integrate((1-2*x)^(3/2)*(2+3*x)^(5/2)*(3+5*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{38981250} \, {\left (127575000 \, x^{4} + 140805000 \, x^{3} - 48345750 \, x^{2} - 89595360 \, x - 4295257\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]