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Recent laser wakefield acceleration experiments have demonstrated the generation of femtosecond, nano-Coulomb, low emittance, nearly monokinetic relativistic electron bunches of sufficient quality to produce bright, tunable, ultrafast x-rays via Compton scattering. Design parameters for a proof-of-concept experiment are presented using a three-dimensional Compton scattering code and a laser-plasma interaction particle-in-cell code modeling the wakefield acceleration process; x-ray fluxes exceeding 10^{21}   s^{-1} are predicted, with a peak brightness >10^{19}   photons/(mm^{2} mrad^{2} s   0.1%   bandwidth)).