The main contribution of this paper is a new H2 gain-scheduled state-feedback synthesis condition for continuous-time linear parameter-varying (LPV) systems where the parameters have bounded rates of variation. The matrices of the model as well as the control gain can have arbitrary polynomial dependence on the time-varying parameters. The synthesis conditions are formulated in terms of LMI relaxations combined with a search in a bounded scalar parameter, which can be used to obtain controllers with improved performance. As an experimental validation of the results, gain-scheduled controllers are designed and applied to a control moment gyroscope modeled as an LPV system with polynomial dependence on the time-varying parameters. Comparisons with a time-invariant controller, designed with the same performance criterion, are presented to illustrate the advantages of the proposed approach.