Oddity Using Root Finder with Integrator (long)

[parisse]

Your problem is not well defined, because an integral is defined up to a constant, you should use a integral with boundaries. Should be e.g.
fsolve(int(x^3*(1-x)^5/Beta(4,6),x,0,xx)-.25,xx,.5)
Anyway, I'm unable to reproduce your problem with the latest source build. I get 0.29...
If you want to restrict to an interval, you can enter e.g.
fsolve(int(xx^(p-1)*(1-xx)^(q-1)/Beta(p,q),xx)-.25,xx=-1..1)
then dichotomy is used (and here it will find more than one root), otherwise an iterative Newton-like method is used.