For the purpose of this post, let f(n) = the left expression, g(n) = the center expression, and h(n) = the right expression.
It is clear that f(n) < h(n), so it's all about establishing where g(n) lies.
Maybe you should turn this into an initial-value problem. Specifically, show that f'(n) <= g'(n) < h'(n), and f(1) <= g(1) < h(1). Of course, g'(n) is really gonna get messy, so maybe this isn't the way to go. But remember, you don't necessarily have to solve for g'(n); you just have to show that it lies between f'(n) and h'(n).