8. Let the function be
$$
\begin{array}{l} 
f(x)= \sqrt{10-6 \cos x}+\sqrt{\frac{17}{8}-\frac{3 \sqrt{2}}{2} \sin x}+ \\
\sqrt{19-2 \sqrt{2} \cos x-8 \sin x} .
\end{array}
$$

Then the minimum value of $f(x)$ is $\qquad$
8. $\frac{21 \sqrt{2}}{4}-1$.

Notice
$$
\begin{aligned}
f(x)= & \sqrt{(\cos x-3)^{2}+\sin ^{2} x}+ \\
& \sqrt{\cos ^{2} x+\left(\sin x-\frac{3 \sqrt{2}}{4}\right)^{2}}+ \\
& \sqrt{(\cos x-\sqrt{2})^{2}+(\sin x-4)^{2}} .
\end{aligned}
$$

Let $A(3,0) 、 B\left(0, \frac{3 \sqrt{2}}{4}\right) 、 C(\sqrt{2}, 4)$ 、 $P(\cos x, \sin x)$. Then,
$$
f(x)=|P A|+|P B|+|P C|,
$$

where point $P$ is on the circle $\odot O: x^{2}+y^{2}=1$.
Since the circle $\odot O: x^{2}+y^{2}=1$ is tangent to $A B$ at point $D\left(\frac{1}{3}, \frac{2 \sqrt{2}}{3}\right)$, and points $O 、 D 、 C$ are collinear, we have
$$
P C \geqslant D C \text {. }
$$

Also, $P B+P A \geqslant D B+D A=A B$, so
$$
\begin{array}{l}
P A+P B+P C \geqslant A B+C D \\
=\frac{9 \sqrt{2}}{4}+(3 \sqrt{2}-1)=\frac{21 \sqrt{2}}{4}-1 .
\end{array}
$$