Fractional parts in frenzy

Looking at the top integral:

$\displaystyle A_1 =\int_{0}^{n}x\lfloor x \rfloor dx = \sum_{i =1}^{n-1} i^2+\frac{i}{2}$ , by looking at the graph. For $n = 12$ , $A_1 = 539$ .

Looking at the bottom integral and using $\{x\} = x - \lfloor x \rfloor$ :

$\displaystyle\int_{0}^{n}x^2\lfloor x \rfloor - x \lfloor x \rfloor ^2 dx$

$\displaystyle A_2 = \int_{0}^{n}x^2\lfloor x \rfloor dx = \sum_{i=1}^{n-1}\int_{i}^{i+1}ix^2dx =$ , by looking at the graph.

$\displaystyle\frac{1}{3}\sum_{i=1}^{n-1}i(i+1)^3 - i^4$ For $n = 12$ , $A_2 = 4884$ .

$\displaystyle A_3 = \int_{0}^{n}x \lfloor x \rfloor ^2 dx = \sum_{i = 1}^{n-1}i^3+\frac{i^2}{2}$ , by looking at the graph. For $n = 12$ , $A_3 = 4609$

Finally, $\displaystyle\frac{A_1}{A_2-A_3} = \boxed{\frac{49}{25}}$