It is obvious that both $f(x)= 4^x+9^x+25^x$ and $g(x)=6^x+10^x+15^x$ are strict monotonic increasing functions. It is also easy to check that 0 is a solution of the equation. Also I chart the functions, and it looks that for any $x$, $f(x)>g(x)$, which can be somehow proof by studying the derivative of the $h(x)=f(x)-g(x)$ and showing that $(0,0)$ it's an absolute minimum point for $h(x)$. However $h(x)$ it is a function with a messy derivative, and is not looking easy(for me) to find the zeroes of this derivative.
Does anyone, know an elegant proof(maybe an elementary one, without derivatives) for this problem?