4.14 ** Solve the inequality
$$
\frac{\sqrt{x^{2}+1}+2^{x}}{\sqrt{x^{2}+1}+2^{x+1}}+\frac{\sqrt{x^{2}+2}+3^{2 x}}{\sqrt{x^{2}+2}+3^{2 x+1}}+\frac{\sqrt{x^{2}+3}+6^{3 x}}{\sqrt{x^{2}+3}+6^{3 x+1}}>1 .
$$
Let $a, b, m>0$, and $a\frac{a}{b}$ holds, from which we can infer: for any real number $x$, the value of the first fraction in the original inequality is greater than $\frac{1}{2}$; similarly, the value of the second fraction is greater than $\frac{1}{3}$; the value of the third fraction is greater than $\frac{1}{6}$, and $\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1$, so the original inequality holds for any real number $x$.
In conclusion, the solution set of the original inequality is $\mathbf{R}$.