Calculate mean and variance of a function of random variables

Question

I am working on a problem and I need to compute the mean and variance of $Y$, i.e. $E\{Y\}$ and $E\{Y^{2}\}$ is required, where $$ Y = \frac{A^{2}+B^{2}+AB+CD}{\sqrt{(A+B)^{2}+(C+D)^{2}}}, $$ where A, B, C, and D are Normal random variables with $\mathcal{N}(0,\frac{1}{2})$. First I started with the mean of $Y$. So, I first separated the terms. The first term is $$ Y_{1} = \frac{A^{2}}{\sqrt{(A+B)^{2}+(C+D)^{2}}}=\frac{A^{2}}{\sqrt{X}}. $$ Then, according to the 'law of total expectation', we have $$ E\{Y_{1}\}=E\{E\{Y_{1}|A\}\}=E\{E\{\frac{A^{2}}{\sqrt{X}}|A\}\}=E\{A^{2}E\{\frac{1}{\sqrt{X}}|A\}\}. $$ Is this a right way to continue, or I am totally wrong. I need your hints and help.