We report the first detailed fully atomistic molecular dynamics study of the encapsulation of symmetric (C(60)) and asymmetric fullerenes (C(70) and C(78)) inside single-walled carbon nanotubes of different diameters. Different ordered phases have been found and shown to be tube diameter dependent. Rotational structural disorder significantly affecting the volume fraction of the packing was observed for the molecular arrangements of asymmetric fullerenes. Although these effects make more difficult the existence of ordered phases, our results showed that complex packing arrangements (very similar to the ones obtained for C(60)) are also possible for C(70) and C(78). Comparisons with results from continuum and hard-sphere models, ab initio electronic structure calculations, and simulations of the high-resolution transmission electron microscopy images of the obtained fullerene packing phases are also presented.