  Classical quasi-integrable systems are known to have Lyapunov times much
shorter than their ergodicity time -- the most clear example being the Solar
System -- but the situation for their quantum counterparts is less well
understood. As a first example, we examine the quantum Lyapunov exponent,
defined by the evolution of the 4-point out-of-time-order correlator (OTOC), of
integrable systems which are weakly perturbed by an external noise, a setting
that has proven to be illuminating in the classical case. In analogy to the
tangent space in classical systems, we derive a linear superoperator equation
which dictates the OTOC dynamics. We find that i) in the semi-classical limit
the quantum Lyapunov exponent is given by the classical one: it scales as
$\epsilon^{1/3}$, with $\epsilon$ being the variance of the random drive,
leading to short Lyapunov times compared to the diffusion time (which is $\sim
\epsilon^{-1}$). ii) in the highly quantal regime the Lyapunov instability is
suppressed by quantum fluctuations, and iii) for sufficiently small
perturbations the $\epsilon^{1/3}$ dependence is also suppressed -- another
purely quantum effect which we explain. These essential features of the problem
are already present in a rotor that is kicked weakly but randomly. Concerning
quantum limits on chaos, we find that quasi-integrable systems are relatively
good scramblers in the sense that the ratio between the Lyapunov exponent and
$kT/\hbar$ may stay finite at a low temperature $T$.
