A rigorous general formula can be derived for the electron repulsion integral (ERI) over the generally contracted (GC) solid harmonic (SH) Gaussian-type orbitals by the use of the "reducing triply mixed solid harmonics" defined in this article. A general algorithm is obtained inductively from the general formula by the use of the "triply mixed solid harmonics" defined in this article. This algorithm is named as ACEb3k3-SH-GC. This ACEb3k3-SH-GC is rigorous and capable of computing the above SH-ERI very fast. Numerical assessment can be performed for (LL\LL) class of SH-ERIs (L=2-5). It is found that the present ACEb3k3-SH-GC is severalfold to a thousandfold faster than the ACEb3k3 algorithm for the usual segment contraction (which is named as ACEb3k3-SH-SC and is the fastest algorithm of all methods in the literature) for the generally contracted (LL\LL) class of SH-ERIs.