Let $X$ and $Y$ i.i.d standardized normally distributed random variables.

Calculate the conditional expectation of : 
$$ \mathbb{E}[(X+Y)^{3}  | \mathscr{G}] $$

where $\mathscr{G} = \sigma(X)$ ($\sigma$-field generated from $X$) 


**Proposal**

$$X,Y \sim N(0,1)$$


now using $(a+b)^3 = a^3+3a^2b+3ab^2+b^3$

we have : 


\begin{align*}
    \mathbb{E}[(X+Y)^3| \mathscr{G}] &=\mathbb{E}[X^3+3X^2Y+3XY^2+Y^3|\mathscr{G} ]  \\
    &= \mathbb{E}[X^3|\mathscr{G}] +\mathbb{E}[3X^2Y|\mathscr{G}]+3\mathbb{E}[XY^2|\mathscr{G}]+\mathbb{E}[Y^3|\mathscr{G}] 
\end{align*}

But how can I proceed further?Am I right so far ?