Solution:

For $x \in [n, n + 1)$ the right-hand side is $\sqrt{P(n)},$ which must be an integer, so $P(n) = n^2 - 3n - 9$ must be a perfect square. For $n = 5, 6, \ldots, 14$ the values are $1, 9, 19, 31, 45, 61, 79, 99, 121, 145:$ only $n = 5, 6, 13$ give squares, with $\sqrt{P(n)} = 1, 3, 11$ respectively.

$P$ is increasing on $[5, 15],$ so for $x \in [n, n + 1)$ we automatically have $\sqrt{P(x)} \ge \sqrt{P(n)},$ and $\left\lfloor \sqrt{P(x)} \right\rfloor = \sqrt{P(n)} = m$ holds exactly when $P(x) \lt (m + 1)^2,$ i.e. $x \lt \frac{3 + \sqrt{45 + 4(m+1)^2}}{2}.$ For $m = 1, 3, 11$ the cutoffs are $\frac{3 + \sqrt{61}}{2},$ $\frac{3 + \sqrt{109}}{2},$ $\frac{3 + \sqrt{621}}{2},$ each lying inside the corresponding unit interval, so the successful subintervals have lengths $\frac{\sqrt{61} - 7}{2},$ $\frac{\sqrt{109} - 9}{2},$ $\frac{\sqrt{621} - 23}{2}.$

The interval $[5, 15]$ has length $10,$ so the probability is $$\frac{1}{10} \cdot \frac{\sqrt{61} + \sqrt{109} + \sqrt{621} - 39}{2} = \frac{\sqrt{61} + \sqrt{109} + \sqrt{621} - 39}{20},$$ giving $a + b + c + d + e = 61 + 109 + 621 + 39 + 20 = 850.$