5. The minimum value of the algebraic expression $\sqrt{x^{2}+4}+\sqrt{x^{2}-24 x+153}$ is ( ).
(A) 12
(B) 13
(C) 14
(D) 15
5. B.

As shown in Figure 3, let $AB = 12$, $DA \perp AB$, $EB \perp AB$, and $DA = 2$, $EB = 3$. Then,
$$
\begin{aligned}
DC & =\sqrt{x^{2}+2^{2}}, \\
CE & =\sqrt{(x-12)^{2}+3^{2}} .
\end{aligned}
$$

Construct the point $D'$, which is the reflection of point $D$ over $AB$. Then,
$$
D' C = D C.
$$

Clearly, when points $E$, $C$, and $D'$ are collinear, $D' E$ is minimized, with the minimum value being $\sqrt{12^{2}+(3+2)^{2}}=13$.