\[ \int (1-2 x)^{5/2} \sqrt {2+3 x} \sqrt {3+5 x} \, dx \]
Optimal antiderivative \[ -\frac {6799613 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{15946875}-\frac {110717 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{15946875}+\frac {326 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}} \sqrt {2+3 x}}{4725}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}} \sqrt {2+3 x}}{45}+\frac {10214 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{118125}-\frac {110717 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{1063125} \]
command
integrate((1-2*x)^(5/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{1063125} \, {\left (945000 \, x^{3} - 1111500 \, x^{2} + 55530 \, x + 526861\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]