The smallest integer x for which the inequality $\frac{x-5}{x^2+5x-14}>0$ is satisfied is given by 

$\begin{array}{l}\frac{x-5}{(x+7)(x-2)}>0\\ \Rightarrow -7<x<2\text{ or }x>5\end{array}$

Thus, the smallest integer for which the inequality is satisfied is $-6.$