A Hard Way And A Hard Way

Algebra Level 5

\[\large \frac{9a^{2}+9ab}{c^{2}+bc}+\frac{12b^{2}+12bc}{a^{2}+ac}+\frac{16c^{2}+16ac}{b^{2}+ab}\]

Let \(a,b,c\) be positive real numbers. The ratio of \(c:a\) such that the above expression is minimized can be expressed as a fraction in the form

\[\large \frac{m+\sqrt{n}}{p}\]

where \(m,n,p\) are integers and \(p\) is not divisible by the square of any prime.

Find \(m+n+p\).

Inspired by the comments of Pi Han Goh and Otto Bretscher on my problem An Easy Way And A Hard Way II.

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