  We derive an exact formula for the scaled cumulant generating function of the
time-integrated current associated to an arbitrary ballistically transported
conserved charge. Our results rely on the Euler-scale description of
interacting, many-body, integrable models out of equilibrium given by the
generalized hydrodynamics, and on the large deviation theory. Crucially, our
findings extend previous studies by accounting for inhomogeneous and dynamical
initial states in interacting systems. We present exact expressions for the
first three cumulants of the time-integrated current. Considering the
non-interacting limit of our general expression for the scaled cumulant
generating function, we further show that for the partitioning protocol initial
state our result coincides with previous results of the literature. Given the
universality of the generalized hydrodynamics, the expression obtained for the
scaled cumulant generating function is applicable to any interacting integrable
model obeying the hydrodynamic equations, both classical and quantum.
