5. The solution set of the equation $4 \sin x(1+\cos x)=3 \sqrt{3}$ is
5. $x=2 k \pi+\frac{\pi}{3}(k \in \mathbf{Z})$.

Squaring both sides of the original equation, we get
$$
\begin{array}{l}
27=16 \sin ^{2} x(1+\cos x)^{2} \\
=16\left(1-\cos ^{2} x\right)\left(1+2 \cos x+\cos ^{2} x\right) \\
\Rightarrow 16 \cos ^{4} x+32 \cos ^{3} x-32 \cos x+11=0 \\
\Rightarrow(2 \cos x-1)^{2}\left(4 \cos ^{2} x+12 \cos x+11\right)=0 \\
\Rightarrow \cos x=\frac{1}{2} \Rightarrow x=2 k \pi+\frac{\pi}{3}(k \in \mathbf{Z}) .
\end{array}
$$