To solve the integral provided using partial fraction decomposition:

$\int \frac{5}{x(x^2 + 6)^2} \, dx$

We start by decomposing the fraction into simpler parts. We express $\frac{5}{x(x^2 + 6)^2}$ in terms of partial fractions as follows:

$\frac{5}{x(x^2 + 6)^2} = \frac{A}{x} + \frac{Bx + C}{x^2 + 6} + \frac{Dx + E}{(x^2 + 6)^2}$

To find $A$, $B$, $C$, $D$, and $E$, we multiply both sides by the denominator $x(x^2 + 6)^2$, leading to:

$5 = A(x^2 + 6)^2 + (Bx + C)x(x^2 + 6) + (Dx + E)x$

Now, we can equate the coefficients from each side of the equation for different powers of $x$ to solve for the constants. This process involves setting values for $x$ and equating like terms, or comparing coefficients directly if the expression is expanded fully.

Let's start the process of finding these constants.