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In this paper, we present a new 3-D full electromagnetic relativistic hybrid plasma code, the hybrid virtual laser plasma laboratory. The full kinetic particle-in-cell method is used to simulate low-density hot plasmas while the hydrodynamic model applies to the high-density cold background plasma. To simulate the linear electromagnetic response of the high-density plasma, we use a newly developed form of an exponential integrator method. It allows us to simulate plasmas of arbitrary densities using large time steps. The model reproduces the plasma dispersion and gives the correct spatial scales like the plasma skin depth even for large grid cell sizes. We test the hybrid model validity by applying it to some physical examples.