Help with inequality estimate, in $H^1$, Given a bilinear form on $H^1 \times H^1$, where $H^1 = W^{1,2}$
\begin{align*}
B[u,v] = \int_U \sum_{i,j}a^{i,j}(x)u_{x_i}v_{x_j} + \sum_ib^i(x)u_{x_i}v + c(x)uv \, \mathrm{d}x
\end{align*}
the book gives the inequality bound $\vert B[u,v] \vert \leq \alpha \Vert u \Vert_{H^1} \Vert v \Vert_{H^1}$. 
I'm trying to verify this:
\begin{align*}
\vert B[u,v] \vert &\leq \sum_{i,j} \vert a^{i,j} \int u_{x_i}v_{x_j} \vert + \sum_i \vert b^i \int u_{x_i} v \vert + \vert \int c u v \vert \\
&\leq \sum_{i,j} \Vert a^{i,j} \Vert_{\infty}\Vert u_{x_i} \Vert_{L^2} \Vert v_{x_j}\Vert_{L^2} + \sum_i \Vert b^{i} \Vert_{\infty}\Vert u_{x_i} \Vert_{L^2} \Vert v\Vert_{L^2} + \Vert c \Vert_{\infty} \Vert u \Vert_{L^2} \Vert v \Vert_{L^2}
\end{align*}
And from here, I'm not sure how I should proceed. I know I need to eventually get something of the form
\begin{align*}
\Vert u \Vert_{H^1} \Vert v \Vert_{H^1} = (\Vert u \Vert_{L^2}^2 + \sum_i \Vert u_{x_i} \Vert_{L^2}^2)^{1/2}(\Vert v \Vert_{L^2}^2 + \sum_i \Vert v_{x_i} \Vert_{L^2}^2)^{1/2}
\end{align*}
but I'm just not sure how. If someone could help me with the missing steps, I'd really appreciate it.

The claim follows since both $\|u\|_{L^2}$ and $\|u_{x_i}\|_{L^2}$ are bounded by $\|u\|_{H^1}$.
