\[ \int (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{55}-\frac {90397364 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{58471875}-\frac {5442127 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{116943750}+\frac {62 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}{2475}-\frac {40703 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{433125}-\frac {23 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{9625}-\frac {5442127 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{7796250} \]
command
integrate((1-2*x)^(3/2)*(2+3*x)^(3/2)*(3+5*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{7796250} \, {\left (42525000 \, x^{4} + 43470000 \, x^{3} - 17237250 \, x^{2} - 27227430 \, x - 810641\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]