Show a function is decreasing

Question

I'm quite new to mathematica, so most of what I know is just the very elementary basics of the language. I'm working on showing that a function is decreasing for $x>0$, but it has two other parameters $m, N$ both of which are integers with some constraints ($m,N\in\mathbb{N}$ and $m>1,N>3$). I quickly realized Solve and NSolve are for polynomial equations so I was figured to use Simplify/FullSimplify. Here's my code below:

d = Log[N, (N^2 + N)/2];


 F[x_] := 
    ((N^m - 2)/(N^m - 1) + 1/((N^m - 1) (N^m)^x))^d
    /(1 - 3/(((N*(N + 1))/2)^m - 2) (1 - (2/((N*(N + 1))/2)^m)^x));

FullSimplify[
    D[F[x], x] < 0, 
    Assumptions -> 
    x > 0 && 
    m \[Element] Integers && m > 1 && 
    N \[Element] Integers && N > 3]

And the output from this FullSimplify is the following large product

(-((-2 + N^m + N^(-m x))/(1 - N^m)))^(Log[1/2 N (1 + N)]/
  Log[N]) 
(2^(1 + m) - (N (1 + N))^m) 
(3 2^(m + x + m x) - 
5 2^m (N (1 + N))^(m x) + (N (1 + N))^(m (1 + x)))^2 
(1 + N^(m x) (-2 + N^m)) 

(m (3 2^(m + x + m x) (1 + N)^(m x) + 
   N^(m (1 + x)) (1 + N)^(m + 2 m x) - 
   5 2^m (N (1 + N)^2)^(m x)) Log[1/2 N (1 + N)] + 
3 2^(m + x + 
  m x) ((1 + N)^(m x) + (N (1 + N))^(m x) (-2 + N^m)) Log[
  2^(1 + m) (N (1 + N))^-m]) < 0

The second half (after line gap) is a single large term which is the place of ambiguity. That is, it is not obvious as to the sign of it (should be positive for the entire expression to be negative).

My question is: is there a better way I should be doing this? So far I just do a FullSimplify and then take the chunk(s) from the output that are not obviously positive/negative and then attempt to further simplify them down to something I can analyze by hand. I successfully did this when the $N$ in the function above was a 3, but I'm trying to generalize to any natural. Again, I'm just trying to show the original function is decreasing so I can do further analysis on it. Thanks for any assistance and sorry the code is so ugly.