This work has been motivated by a longitudinal data set on HIV CD4 T+ cell counts from Livingstone district, Zambia. The corresponding histogram plots indicate lack of symmetry in the marginal distributions and the pairwise scatter plots show non-elliptical dependence patterns. The standard linear mixed model for longitudinal data fails to capture these features. Thus it seems appropriate to consider a more general framework for modeling such data. In this article, we consider generalized linear mixed models (GLMM) for the marginals (e.g. Gamma mixed model), and temporal dependency of the repeated measurements is modeled by the copula corresponding to some skew-elliptical distributions (like skew-normal/skew-t). Our proposed class of copula based mixed models simultaneously takes into account asymmetry, between-subject variability and non-standard temporal dependence, and hence can be considered extensions to the standard linear mixed model based on multivariate normality. We estimate the model parameters using the IFM (inference function of margins) method, and also describe how to obtain standard errors of the parameter estimates. We investigate the finite sample performance of our procedure with extensive simulation studies involving skewed and symmetric marginal distributions and several choices of the copula. We finally apply our models to the HIV data set and report the findings.