9. Given positive real numbers $a, b$ satisfy $a(a+b)=27$, find the maximum value of $a^{2} b$.

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$b=\frac{27}{a}-a \Rightarrow a^{2} b=27 a-a^{3}$, Let $f(x)=27 x-x^{3}, f^{\prime}(x)=27-3 x^{2}$ $=-3(x+3)(x-3)$, then $f(x)$ is monotonically increasing on $(0,3)$ and monotonically decreasing on $(3,+\infty)$, i.e., $f(x)_{\text {max }}=f(3)=54$. Therefore, when $a=3, b=6$, the maximum value of $a^{2} b$ is 54.