What is this distribution called by? I have pdf of a continuous distribution as below
$$f\left(x\right)=\frac{1}{\sqrt{2 \pi x^3}} e^{-\frac{1}{2} {\left( \frac{x-\mu}{\mu \sqrt{x}} \right)}^2}$$
Is there any specific name for this distribution? Apparently it looks like a normal distribution, but it is not the case

[ Revised: the original question had an unfortunate \times notation that looked like an $x$. That has been removed, and the answer below now reflects the above. Notation matters! ]
The stated functional form is Inverse Gaussian with pdf:
$$f(x) = \frac{1}{\sqrt{2 \pi  x^3}} {\exp \left[-\frac{(x-\mu )^2}{2 \mu ^2 x}\right]}  \quad \quad \text{for} \quad  x> 0$$
As the OP notes, the pdf does not look like a Normal distribution and nor should it generally do so. The following diagram plots the above Inverse Gaussian when $\mu$ is 1 , 2 and 3 and contrasts with a typical Normal pdf (shown in orange):

