Let me separate the radial integration from the angular integration,
$$\int_{|\mathbf{x}|\leq 1}f(\mathbf{x})d\mathbf{x}=\frac{2\pi^{n/2}}{\Gamma(n/2)}\int_0^1 r^{n-1}\bar{f}(r)\,dr,$$
where $\bar{f}(r)$ is the average of $f$ over the surface of the $n$-dimensional hypersphere of radius $r$. In our case
$$f(\mathbf{r})=\bigg(\sum_{j=1}^{n}a_{j}x_{j}^2\bigg)^m.$$
For $n\gg 1$ the concentration of measure allows us to replace $x_j^2$ by $r^2/n$, so 
$$\bar{f}(r)\approx (r^2/n)^m\bigg(\sum_{j=1}^{n}a_{j}\bigg)^m,$$
and thus we estimate
$$\int_{B(0,1)}\bigg(\sum_{j=1}^{n}a_{j}x_{j}^2\bigg)^m \mbox{d}x\approx
\frac{2\pi^{n/2}}{n^m(2m+n)\Gamma(n/2)}\bigg(\sum_{j=1}^{n}a_{j}\bigg)^m,\;\;n\gg 1.$$