\[ \large f\left(\dfrac{x+y}{3}\right)=\dfrac{f(x)+f(y)}{2}\]

If a real function \(f\) satisfies the equation above for all real \(x,y\), then compute the value of \(\displaystyle\int f(2x^9)\;dx\)

If the answer is of the form \(\dfrac{ c^{M}x^{L}}{N}\), find \(\lfloor 1000M+29N+8L\rfloor\)

Take \(c\) as a constant.

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