Closed form of $\sum\limits_{n=1}^\infty \frac{4^n(x+4)^{2n}}n$

Let's set $A = 4(x+4)^2$. You want to find $$\sum\limits_{n=1}^{\infty} \frac{1}{n}A^n = \sum\limits_{n=1}^{\infty} \int_{0}^A t^{n-1}\,dt = \int_{0}^A \sum\limits_{n=1}^{\infty} t^{n-1} \,dt = \int_0^A \frac{1}{1-t}\,dt = \cdots$$