We are given: $|\vec a|=\sqrt{3},\quad |\vec b|=5,\quad \vec b\cdot \vec c=10$ and the angle between $\vec b$ and $\vec c$ is $\theta=\frac{\pi}{3}.$

First find $|\vec c|$ using the dot product formula: $\vec b\cdot \vec c=|\vec b|\,|\vec c|\cos\theta.$ So, $10=5\cdot |\vec c|\cdot \cos\frac{\pi}{3}.$ Since $\cos\frac{\pi}{3}=\frac12,$ we get $10=5\cdot |\vec c|\cdot \frac12=\frac{5|\vec c|}{2}.$ Hence, $|\vec c|=4.$

Now find $|\vec b\times \vec c|$: $|\vec b\times \vec c|=|\vec b|\,|\vec c|\sin\theta$ $=5\cdot 4\cdot \sin\frac{\pi}{3}$ $=20\cdot \frac{\sqrt{3}}{2}=10\sqrt{3}.$

We are told that $\vec a$ is perpendicular to $\vec b\times \vec c$. Therefore, the angle between $\vec a$ and $(\vec b\times \vec c)$ is $\frac{\pi}{2}$.

Use the magnitude formula for cross product: $|\vec a\times (\vec b\times \vec c)|=|\vec a|\,|\vec b\times \vec c|\sin\frac{\pi}{2}.$ Since $\sin\frac{\pi}{2}=1$, $|\vec a\times (\vec b\times \vec c)|=\sqrt{3}\cdot 10\sqrt{3}=30.$

Therefore, the required integer is: $\boxed{30}$