Let $|x| = k$. The equation transforms to: $2k(k^2 + 1) = 5k^2$. Expanding yields: $2k^3 + 2k = 5k^2$. Rearranging terms: $2k^3 - 5k^2 + 2k = 0$. Factoring out $k$: $k(2k^2 - 5k + 2) = 0$. Solving the quadratic $2k^2 - 5k + 2 = 0$ by factorization: $2k^2 - 4k - k + 2 = 0 \Rightarrow 2k(k - 2) -1(k - 2) = 0 \Rightarrow (2k - 1)(k - 2) = 0$. The solutions for $k$ are $k = 0$, $k = \dfrac{1}{2}$, or $k = 2$. Substituting back $|x| = k$: $|x| = 0 \Rightarrow x = 0$; $|x| = \dfrac{1}{2} \Rightarrow x = \pm \dfrac{1}{2}$; $|x| = 2 \Rightarrow x = \pm 2$. The possible values for $x$ are: $x = 0, \dfrac{1}{2}, -\dfrac{1}{2}, 2, -2$. The integral values among these are: $x = 0, 2, -2$. Therefore, there are 3 integral solutions to the equation $2|x|(x^2+1) = 5x^2$.