The integral $\int \sqrt{\ x \sqrt[3]{\ x \sqrt[4]{\ x \sqrt[5]{\ x \cdots}}}} dx$

Question

I have no idea how to solve the second question from 2018 MIT integration bee,

$$\int \sqrt{\ x \sqrt[3]{\ x \sqrt[4]{\ x \sqrt[5]{\ x \cdots}}}} dx = \frac{x^{e-1}}{e-1}.$$

I tried to to find a value for the roots and i tried converting it to a power, and I tried making a $u$ substitution but nothing worked out, while i think that the right approach to this question is to convert the roots to a power, i cant find the right approach to the question.

Answer 1

\begin{align} I&=\int \sqrt{\ x\sqrt[3]{\ x\sqrt[4]{\ x\cdots}}}\,dx\\ I&=\int \sqrt{\ x}\cdot \sqrt{\ \sqrt[3]{\ x}}\cdot \sqrt{\ \sqrt[3]{\ \sqrt[4]{x}}}\cdots\,dx\\ I&=\int x^{1/2}\cdot x^{1/6}\cdot x^{1/24}\cdots\,dx\\ I&=\int x^{\sum_{n=2}^{\infty} \frac{1}{n!}}\,dx\\ I&=\int x^{e-2}\,dx\\ I&=\frac{x^{e-1}}{e-1}+C \end{align}