5. Given $P$ is a point on the parabola $y^{2}=4 x$, let the distance from $P$ to the directrix of this parabola be $d_{1}$, and the distance from $P$ to the line $x+2 y-12=0$ be $d_{2}$. Then the minimum value of $d_{1}+d_{2}$ is ( ).
(A) $\frac{11 \sqrt{5}}{5}$
(B) $\frac{12 \sqrt{5}}{5}+1$
(C) $2 \sqrt{5}$
(D) does not exist
5.A.

Since point $P$ is on $y^{2}=4 x$, let's set $P\left(t^{2}, 2 t\right)$. Therefore, we have $d_{1}+d_{2}=t^{2}+1+\frac{\left|t^{2}+4 t-12\right|}{\sqrt{5}}$.
(1) When $t \leqslant-6$ or $t \geqslant 2$,
$$
d_{1}+d_{2}=\left(1+\frac{1}{\sqrt{5}}\right) t^{2}+\frac{4}{\sqrt{5}} t+1-\frac{12}{\sqrt{5}} \text {. }
$$

When $t=2$, equation (1) achieves the minimum value of 5.
(2) When $-6<t<2$,
$$
d_{1}+d_{2}=\left(1-\frac{1}{\sqrt{5}}\right) t^{2}-\frac{4}{\sqrt{5}} t+1+\frac{12}{\sqrt{5}} \text {. }
$$

When $t=\frac{2}{\sqrt{5}-1}$, equation (2) achieves the minimum value of $\frac{11 \sqrt{5}}{5}$.
Combining (1) and (2), the minimum value of $d_{1}+d_{2}$ is $\frac{11 \sqrt{5}}{5}$.