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The Spreading Projection Algorithm for Rapid K-space sampLING, or SPARKLING, is an optimization-driven method that has been recently introduced for accelerated 2D MRI using compressed sensing. It has then been extended to address 3D imaging using either stacks of 2D sampling patterns or a local 3D strategy that optimizes a single sampling trajectory at a time. 2D SPARKLING actually performs variable density sampling (VDS) along a prescribed target density while maximizing sampling efficiency and meeting the gradient-based hardware constraints. However, 3D SPARKLING has remained limited in terms of acceleration factors along the third dimension if one wants to preserve a peaky point spread function (PSF) and thus good image quality. In this paper, in order to achieve higher acceleration factors in 3D imaging while preserving image quality, we propose a new efficient algorithm that performs optimization on full 3D SPARKLING. The proposed implementation based on fast multipole methods (FMM) allows us to design sampling patterns with up to ${10}^{{7}}$ k-space samples, thus opening the door to 3D VDS. We compare multi-CPU and GPU implementations and demonstrate that the latter is optimal for 3D imaging in the high-resolution acquisition regime ($600\mu $ m isotropic). Finally, we show that this novel optimization for full 3D SPARKLING outperforms stacking strategies or 3D twisted projection imaging through retrospective and prospective studies on NIST phantom and in vivo brain scans at 3 Tesla taking the particular case of ${T}_{{2}}$ *-w imaging. Overall the proposed method allows for 2.5-3.75x shorter scan times compared to GRAPPA-4 parallel imaging acquisition at 3 Tesla without compromising image quality. [ABSTRACT FROM AUTHOR]