jmc

We consider the separate cases by the dimensions of each type of rectangle. There are 7 of the $1\times 1$ squares. There are 4 vertical $1\times 2$ rectangles, and 4 horizontal $1\times 2$ rectangles. There are also 1 each of vertical and horizontal $1\times 3$ rectangles. And, finally, there are the two $2\times 2$ squares. In total, there are $7+4+4+1+1+2=\boxed{19}$ rectangles.