General mixing formulas are derived for discrete scatterers immersed in a host medium. The inclusion particles are assumed to be ellipsoidal. The electric field inside the scatterers is determined by quasi-static analysis, assuming the diameter of the inclusion particles to be much smaller than one wavelength. The results are applicable to general multiphase mixtures, and the scattering ellipsoids of the different phases can have different sizes and arbitrary ellipticity distribution and axis orientation, i.e. the mixture may be isotropic or anisotropic. The resulting mixing formula is nonlinear and is suitable for iterative solutions. The formula contains a quantity called the apparent permittivity, and with different choices of this quantity, the result leads to the generalized Lorentz-Lorenz formula, the generalized Polder-van Santen formula, and the generalized coherent potential-quasicrystalline approximation formula. The results are applied to calculating the complex effective permittivity of dry and wet snow, and sea ice. >