Gamma Under Sum

$\large S_x = \sum_{k=0}^\infty \frac {(-1)^k 7^k k x^{k+1}}{(e^x-1)\Gamma (k+2)}$

Given an equation above, let $I = \int_{0}^{\infty} {S_x \, dx}$

If $I$ can also be expressed as $I = \int_{0}^{1}{\dfrac{t^A}{t-1} (\ln(t))^B \, dt},$

find $A+B$ .

This is an Original Problem.