Solution:

The diagonal of the rectangle from the upper-left corner to the lower-right corner divides the shaded region into four triangles, all meeting at the center of the rectangle.

Two of these triangles have a horizontal base of length $1$ and altitude $\frac12\cdot 5=\frac52,$ and the other two have a vertical base of length $1$ and altitude $\frac12\cdot 8=4.$ The total area is $$2\cdot\dfrac12\cdot 1\cdot\dfrac52+2\cdot\dfrac12\cdot 1\cdot 4=\dfrac52+4=\dfrac{13}{2}.$$

Thus, the correct answer is D.