5. Arrange the coefficients of the function $f(x)=\frac{a x+b}{c x+d}$ in their natural positions to form a 2x2 matrix, denoted as $A=$ $\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$, where each number is called an element of the 2x2 matrix. Also, denote
$$
\begin{array}{l}
f(f(x))=\frac{a f(x)+b}{c f(x)+d} \\
=\frac{\left(a^{2}+b c\right) x+(a b+b d)}{(a c+c d) x+\left(b c+d^{2}\right)}
\end{array}
$$

the coefficients of which form the 2x2 matrix
$$
\left(\begin{array}{ll}
a^{2}+a b & a b+b d \\
a c+c d & b c+d^{2}
\end{array}\right)
$$

as the square of $A$, i.e.,
$$
A^{2}=A \times A=\left(\begin{array}{ll}
a^{2}+b c & a b+b d \\
a c+c d & b c+d^{2}
\end{array}\right) .
$$

Observe the pattern of 2x2 matrix multiplication and write out
$$
A^{3}=A^{2} \times A=\left(\begin{array}{ll}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}\right)
$$

where the element $a_{21}=$
5. $a^{2} c+a c d+b c^{2}+c d^{2}$.

According to the rules of second-order matrix multiplication, we know that
$$
A^{3}=\left(\begin{array}{ll}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}\right)
$$

the $a_{ij}$ should be the algebraic sum of the products of the elements in the $i$-th row of $A^{2}$ and the corresponding elements in the $j$-th column of $A$, thus
$$
\begin{array}{l}
a_{21}=(a c+c d) a+\left(b c+d^{2}\right) c \\
=a^{2} c+a c d+b c^{2}+c d^{2} .
\end{array}
$$