Two major challenges arise in regression analyses of recurrent event data: first, popular existing models, such as the Cox proportional rates model, may not fully capture the covariate effects on the underlying recurrent event process; second, the censoring time remains informative about the risk of experiencing recurrent events after accounting for covariates. We tackle both challenges by a general class of semiparametric scale-change models that allow a scale-change covariate effect as well as a multiplicative covariate effect. The proposed model is flexible and includes several existing models as special cases, such as the popular proportional rates model, the accelerated mean model, and the accelerated rate model. Moreover, it accommodates informative censoring through a subject-level latent frailty whose distribution is left unspecified. A robust estimation procedure which requires neither a parametric assumption on the distribution of the frailty nor a Poisson assumption on the recurrent event process is proposed to estimate the model parameters. The asymptotic properties of the resulting estimator are established, with the asymptotic variance estimated from a novel resampling approach. As a byproduct, the structure of the model provides a model selection approach among the submodels via hypothesis testing of model parameters. Numerical studies show that the proposed estimator and the model selection procedure perform well under both noninformative and informative censoring scenarios. The methods are applied to data from two transplant cohorts to study the risk of infections after transplantation.