How to find $E(|X+Y|^3)$?

Question

Supposed that $X,Y$ are independent random variables, and $X\sim N(\mu,\sigma^2),Y\sim N(-\mu,\sigma^2)$. I would like to culculate the value of $E(|X+Y|^3)$.
I have thought of some methods but I was caught into troubles when culculate the integration of $\int\int \vert x+y \vert ^3\, dxdy$.
I also want to find it by some way like $E((X+Y)^3)=E(X^3)+E(Y^3)+3E(X^2Y)+3E(XY^2)$ but I am not sure that it may really work after some trasformation.

Answer 1

If $X\sim N(\mu,\sigma^2)$ and $Y\sim N(-\mu,\sigma^2)$ are independent, then $X+Y\sim N(0,2\sigma^2)$. So we can compute \begin{align} \mathbb{E}[\lvert X+Y\rvert^3]&=2\mathbb{E}[((X+Y)^+)^3]\\ &=2(2\sigma^2)^{3/2}\mathbb{E}[(Z^3)^+]&&Z\sim N(0,1)\\ &=\frac{2(2\sigma^2)^{3/2}}{\sqrt{2\pi}}\int_0^\infty z^3\exp(-z^2/2)\,\mathrm{d}z\\ &=\frac{8}{\sqrt{\pi}}\sigma^3\\ \end{align}