An interesting infinitely nested radical

While perusing the 2018 MIT Integration Bee Qualifying Exam, I happened upon the somewhat interesting expression:

Now, recursive methods for evaluating infinitely nested radicals have been well known for some time, and are accessible to many first-year university students. For instance, it's been known for some time that the golden ratio, , is equal to:

Our astute readers may notice though, that the expression for contains only square roots, and involves addition, whilst the problem posed above involves increasing indices and multiplication. But fear not! This problem is actually solvable with little more than a knowledge of basic algebra (plus a knowledge of series).

Let’s begin by considering the first few finite cases:

At this point, my gut instinct was to combine the exponents, but then I noticed something: For the nth term of the sequence, powers of x correspond to the reciprocals of n+1. Therefore, I could write recursively:

Which means that would be:

Now, it's also well-known that Euler's number can be expressed as:

Which means that, finally:

The ultimate goal of the question, of course, was to integrate the entire radical expression. Once it's simplified, it's rather easy to write that: