Oddity Using Root Finder with Integrator (long)

[Werner]

Wolframalpha confirms the emulator's result:
(integrate (x^3*(1-x)^5) dx from x=0 to y)*beta(4,6)-0.25=0
y = -1.62561, and it's the only root.

UPDATE: and that shows the error you (and I) made...
If I change it to *divide* by beta(4,6):

(integrate (x^3*(1-x)^5) dx from x=0 to y)/beta(4,6)-0.25=0
There are three roots,
y = -0.17836
y = 0.290986
y = 1.3936