5. Let $a_{n}=2^{n}, b_{n}=n \ (n=1,2, \cdots), A_{n} 、 B_{n}$ be the sums of the first $n$ terms of the sequences $\left\{a_{n}\right\} 、\left\{b_{n}\right\}$, respectively. Define
$$
c_{n}=a_{n} B_{n}+b_{n} A_{n}-a_{n} b_{n} \text {. }
$$

Then the sum of the first 10 terms of the sequence $\left\{c_{n}\right\}$ is ( ).
(A) $2^{10}+53$
(B) $2^{11}+53$
(C) $110\left(2^{9}-1\right)$
(D) $110\left(2^{10}-1\right)$
5. D.

When $n \geqslant 2$,
$$
\begin{aligned}
c_{n}= & \left(A_{n}-A_{n-1}\right) B_{n}+\left(B_{n}-B_{n-1}\right) A_{n}- \\
& \left(A_{n}-A_{n-1}\right)\left(B_{n}-B_{n-1}\right) \\
\Rightarrow & c_{n}=A_{n} B_{n}-A_{n-1} B_{n-1} .
\end{aligned}
$$

Also, $c_{1}=a_{1} b_{1}=A_{1} B_{1}$, then the sum of the first $n$ terms of $\left\{c_{n}\right\}$
is $\sum_{i=1}^{n} c_{i}=A_{1} B_{1}+\sum_{i=2}^{n}\left(A_{i} B_{i}-A_{i-1} B_{i-1}\right)$
$$
=A_{n} B_{n} \text {. }
$$

Therefore, the sum of the first 10 terms of the sequence $\left\{c_{n}\right\}$ is
$$
2\left(2^{10}-1\right) \frac{10 \times 11}{2}=110\left(2^{10}-1\right) \text {. }
$$