Find the minimum value of
\[\frac{(a^2 + b^2)^2}{a^3 b}\]for positive real numbers $a$ and $b.$  Enter the answer in the simplified form $\frac{m \sqrt{n}}{p},$ where $m,$ $n,$ and $p$ are positive integers.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the minimum value of
\[\frac{(a^2 + b^2)^2}{a^3 b}\]for positive real numbers $a$ and $b.$  Enter the answer in the simplified form $\frac{m \sqrt{n}}{p},$ where $m,$ $n,$ and $p$ are positive integers.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{16 \sqrt{3}}{9}
Find the minimum value of
\[\frac{(a^2 + b^2)^2}{a^3 b}\]for positive real numbers $a$ and $b.$  Enter the answer in the simplified form $\frac{m \sqrt{n}}{p},$ where $m,$ $n,$ and $p$ are positive integers.
train