5. Let $f(x)=x^{3}+\log _{2}\left(x+\sqrt{x^{2}+1}\right)$, then for any real numbers $a, b, a+b \geqslant 0$ is the condition for $f(a)+f(b) \geqslant 0$
A. Sufficient and necessary condition
B. Sufficient but not necessary condition
C. Necessary but not sufficient condition
D. Neither sufficient nor necessary condition
$$
\begin{array}{l}
\text { Since } f(-x)=-x^{3}+\log _{2}\left(-x+\sqrt{x^{2}+1}\right)=-f(x), \\
f^{\prime}(x)=3 x^{2}+\frac{1+\frac{x}{\sqrt{x^{2}+1}}}{x+\sqrt{x^{2}+1}}=3 x^{2}+\frac{1}{\sqrt{x^{2}+1}}>0,
\end{array}
$$

then $f(x)$ is an odd function and an increasing function on $\mathbf{R}$. Therefore, $a+b \geqslant 0 \Leftrightarrow a \geqslant-b$ $\Leftrightarrow f(a) \geqslant f(-b)=-f(b) \Leftrightarrow f(a)+f(b) \geqslant 0$. So the answer is $A$.