5. Given real numbers $x, y$ satisfy
$$
\left(2015+x^{2}\right)\left(2015+y^{2}\right)=2^{22} \text {. }
$$

Then the maximum value of $x+y$ is $\qquad$
5. $2 \sqrt{33}$.

Notice that, $2^{22}=2048^{2}=(2015+33)^{2}$.
By the Cauchy-Schwarz inequality, we have
$$
\begin{array}{l}
{\left[(2015-33)+33+x^{2}\right]\left[(2015-33)+y^{2}+33\right]} \\
\geqslant[(2015-33)+\sqrt{33} y+x \sqrt{33}]^{2} \\
\Rightarrow(2015+33)^{2} \\
\quad \geqslant[(2015-33)+\sqrt{33}(x+y)]^{2} \\
\Rightarrow x+y \leqslant 2 \sqrt{33} .
\end{array}
$$

When $x=y=\sqrt{33}$, the equality holds.
Therefore, the maximum value of $x+y$ is $2 \sqrt{33}$.