smc

The sides of the triangle are $a,$ $\sqrt{c^2-a^2},$ and $c.$ We know that the hypotenuse and one leg are consecutive integers, so we can rewrite the side lengths as $a,$ $\sqrt{2a+1},$ and $a+1.$ Squaring the middle length gets $2a+1=a+c,$ so the answer is $\boxed{c+a}.$