5. On the coordinate plane, a point $(x, y)$ where both the x-coordinate and y-coordinate are integers is called an integer point. If the area enclosed by the quadratic function $y=-x^{2}+8 x-\frac{39}{4}$ and the x-axis is colored red, then the number of integer points inside this red region and on its boundary is ( ) points.
(A) 16
(B) 20
(C) 25
(D) 30
5.C.
$y=-\left(x-\frac{3}{2}\right)\left(x-\frac{13}{2}\right)$ intersects the $x$-axis at two points $M\left(\frac{3}{2}, 0\right)$ and $N\left(\frac{13}{2}, 0\right)$. Between $x=\frac{3}{2}$ and $x=\frac{13}{2}$, there are 5 integers: $2, 3, 4, 5, 6$.
Rewrite the function as
$$
y=-(x-4)^{2}+\frac{25}{4}.
$$

When $x=2$ or $6$, $y=\frac{9}{4}$, and the integers satisfying $0 \leqslant y \leqslant \frac{9}{4}$ are $0, 1, 2$, a total of 6 integers;

When $x=3$ or $5$, $y=\frac{21}{4}$, and the integers satisfying $0 \leqslant y \leqslant \frac{21}{4}$ are $0, 1, 2, 3, 4, 5$, a total of 12 integers;

When $x=4$, $y=\frac{25}{4}$, and the integers satisfying $0 \leqslant y \leqslant \frac{25}{4}$ are $0, 1, 2, 3, 4, 5, 6$, a total of 7 integers.
In total, there are 25 integer points.