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In this paper, we show super-linear propagation in a nonlocal reaction–diffusion–mutation equation modeling the invasion of cane toads in Australia that has attracted attention recently from the mathematical point of view. The population of toads is structured by a phenotypical trait that governs the spatial diffusion. In this paper, we are concerned with the case when the diffusivity can take unbounded values, and we prove that the population spreads as t 3/2. We also get the sharp rate of spreading in a related local model.