More generally, a cubic equation $f(x)=c$ has three real roots for all values of $c$ between $f(x_1)$ and $f(x_2)$, where $x_1$ and $x_2$ are the critical points of $f$.
For $f(x)=3x^3-25x$, the critical points are $\pm 5/3$ and the interval is $(-250/9, 250/9) \approx (-27.7, 27.7)$, which contains $55$ integers.
Alternatively, you can use this fact
A cubic equation has three distinct real roots iff its discriminant is positive.
The discriminant of $3x^3-25x+n$ is $\Delta=-3 (81 n^2 - 62500)$. Therefore, $\Delta>0$ iff $81 n^2 - 62500<0$, that is, iff $n \in (-250/9, 250/9)$, as before.