IsImageUrl extension method written in F#
1.
F# has a good system of type inference. You don't need to specify the type explicitly, the compiler in many cases will do it for you.
You can read more about it here.

2.
As @JoelMueller said in comment, F# there's such an amazing opportunity as partial application.


  The idea of partial application is that if you fix the first N
  parameters of the function, you get a function of the remaining
  parameters


Taken from Partial application

So, you can rewrite the functions hasAnImageFileExtension, hasAnNonImageFileExtension:

let private hasAnImageFileExtension =
  hasAFileExtensionInList ImageFileExtensions 

let private hasAnNonImageFileExtension =
  hasAFileExtensionInList NonImageFileExtensions


3.
Several very small comments:

a) In hasAFileExtensionInList instead urlSplitList.[urlSplitList.Length - 1] you can use Seq.last

b) for me, construction like:

if Path.HasExtension url = false then
         false
      else ...


better, rerwite as:

  if Path.HasExtension url then
     ...
  else  
     false


4.
In function IsImageUrl it is better to use if-then-else instead pattern matching

5.
Question. 

Why not use Path.GetExtension ?

When string not valid will throw the exception. It is better to provide this option.



I don't post a new version of the code, as I plan to edit the answer.