Fractional polynomials (FP) have been shown to be much more flexible than polynomials for fitting continuous outcomes in the biological and health sciences. Despite their increasing popularity, design issues for FP models have never been addressed. D- and I-optimal experimental designs will be computed for prediction using FP models. Their properties will be evaluated and a catalogue of design points useful for FP models will be provided. As applications, we consider linear mixed effects models for longitudinal studies. To provide greater flexibility in modeling the shape of the response, we use fractional polynomials and not polynomials to approximate the mean response. An example using gene expression data will be considered comparing the designs used in practice. An additional an interesting problem is finding designs for effective model discrimination for FP models. This will be explored from the KL-optimality point of view.