A rigorous general formula for calculating the electron repulsion integral (ERI) over the solid harmonic (SH) Gaussian-type orbitals (GTOs) can be derived, A general algorithm can be obtained from this formula named as : accompanying coordinate expansion (ACE) b3k3. This algorithm is capable of computing very fast SH-ERIs, especially for SH contracted GTOs. Numerical assessment is shown for the (LL\LL) class of SH-ERIs (L=2-5). It is found that the present ACE-b3k3 algorithm is the fastest among all algorithms in the literature in the floating-point-opration (FLOP) count assessment when the degree of contraction is large.