To evaluate the integral [tex]\(\int \frac{x+4}{\left(x^2+8 x\right)^6} \, dx\)[/tex], let's proceed with the following steps:
Step 1: Identify the integral structure
We are looking at the integral:
[tex]\[
\int \frac{x+4}{(x^2 + 8x)^6} \, dx
\][/tex]
Step 2: Simplify the denominator
Note that the denominator [tex]\( (x^2 + 8x)^6 \)[/tex] can be simplified by factoring out [tex]\( x \)[/tex]:
[tex]\[
x^2 + 8x = x(x + 8)
\][/tex]
Thus,
[tex]\[
(x^2 + 8x)^6 = (x(x + 8))^6 = x^6 (x + 8)^6
\][/tex]
Step 3: Simplify the integral
Rewriting the integral, we get:
[tex]\[
\int \frac{x+4}{x^6 (x + 8)^6} \, dx
\][/tex]
Step 4: Identify the substitution
Let [tex]\( u = x^2 + 8x \)[/tex]. Then, [tex]\( du = (2x + 8) \, dx \)[/tex]. Notice that [tex]\( x + 4 \)[/tex] is half of [tex]\( 2x + 8 \)[/tex]:
[tex]\[
2x + 8 = 2(x + 4) \implies x + 4 = \frac{1}{2} (2x + 8) = \frac{1}{2} du \implies (2x + 8) \, dx \Rightarrow du = 2(x + 4) \, dx \Rightarrow dx = \frac{du}{2(x + 4)}
\][/tex]
Step 5: Switch to the new variable
Using the substitution [tex]\( u = x^2 + 8x \)[/tex], the integral becomes:
[tex]\[
\int \frac{\frac{1}{2} du}{ u^6} = \frac{1}{2} \int u^{-6} \, du
\][/tex]
Step 6: Integrate with respect to [tex]\( u \)[/tex]
[tex]\[
\frac{1}{2} \int u^{-6} \, du = \frac{1}{2} \left( \frac{u^{-5}}{-5} \right) = -\frac{1}{10} u^{-5} = -\frac{1}{10} \frac{1}{u^5}
\][/tex]
Step 7: Substitute back to [tex]\( x \)[/tex]
Recalling that [tex]\( u = x^2 + 8x \)[/tex], we substitute back:
[tex]\[
- \frac{1}{10 (x^2 + 8x)^5}
\][/tex]
Thus, the integral evaluates to:
[tex]\[
- \frac{1}{10 (x^2 + 8x)^5} + C
\][/tex]
The correct choice is therefore:
[tex]\[
\boxed{A}
\][/tex]
