Assuming a nongravitational interaction among the dark fluids of our Universe - namely, dark matter and dark energy - we study a specific interaction model in the background of a spatially flat Friedmann-Lemaître-Robertson-Walker geometry. We find that the interaction model solves the background evolution in an analytic way when the dark energy takes a constant barotropic equation of state, wx. In particular, we analyze two separate interaction scenarios, namely, when the dark energy is a fluid other than the vacuum energy (i.e., wx≠-1) and when it is vacuum energy itself (i.e., wx=-1). We find that the interacting model with wx≠-1 produces stable perturbations at large scales for wx<-1 with the coupling strength ξ<0. Both scenarios are constrained by the latest astronomical data. The analyses show that a very small interaction with the coupling strength is allowed, and within the 68.3% confidence region ξ=0 is recovered. The analyses further show that a large coupling strength significantly affects the large-scale dynamics of the Universe, while according to the observational data the interaction models are very well consistent with Λ cosmology. Furthermore, we observe that for the vacuum interaction scenario, the tension on H0 is not released while for the interacting dark energy scenario with wx<-1, the tension on H0 seems to be released partially because of the high error bars in H0. Finally, we conclude the work by calculating the Bayesian evidence, which shows that ΛCDM cosmology is favored over the two interacting scenarios.