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Jacobi-Fourier moments are useful tools in pattern recognition and image analysis due to their perfect feature capability and high noise resistance. However, direct computation of these moments is very expensive, limiting their use as feature descriptors especially at high orders. The existing methods by employing quantized polar coordinate systems not only save the computational time, but also reduce the accuracy of the moments. In this paper, we propose a hybrid algorithm, which re-organize Jacobi-Fourier moments with any order and repetition as a linear combination of generalized Fourier-Mellin moments, to calculate Jacobi-Fourier moments at high orders fast and accurately. First, arbitrary precision arithmetic is employed to preserve accuracy. Second, the property of symmetry is applied to the generalized Fourier-Mellin moments to reduce their computational cost. Third, the recursive relations of Jacobi polynomial coefficients are used to speed up their computation. Experimental results reveal that the proposed method is more efficient than the other methods.