As the name suggests, Blake/Rapid construction is a whole lot like Blake construction. There is a row of Blake stitching along the insole; but instead of attaching the insole to the outsole, it attaches the insole to a midsole. The midsole is attached to the outsole by a row of stitching (that's the Rapid part of the combination) running outside the shoe. Conceptually, it's a bit like a combination of Goodyear welting and Blake construction. Because the row of Blake stitching doesn't go all the way from the interior of the sole to the outsole, it doesn't have the problem with ground moisture that Blake-constructed shoes; but this increased degree of waterproofing comes at a price. The presence of the midsole and the necessity for a row of stitching on the outside of the shoe attaching the midsole to the outsole mean that Blake/Rapid shoes can neither be as flexible nor have soles that are as close-cut as Blake-constructed shoes. In addition, all other things being equal, Blake/Rapid shoes will have a more rugged appearance than equivalent shoes made with Blake construction. This can either be an advantage or a disadvantage, depending on the look that you're seeking.
Like Blake construction, Blake/Rapid construction is a mainstay for most Italian manufacturers. Most manufacturers who do Blake also do Blake/Rapid and will switch between the two depending on the shoes that they are making. The diagram above is courtesy of Ron Rider, who is the US agent for Romano Martegani, a prominent manufacturer in Tradate in Italy that is something of a Blake/Rapid specialist.