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To solve the integral:

\[
\int \frac{9x + 48}{x^3 + 8x^2 + 16x} \, dx
\]

We first express the integrand as a sum of partial fractions. The denominator is \( x^3 + 8x^2 + 16x \), which can be factored. Factor \( x^3 + 8x^2 + 16x \):

\[
x^3 + 8x^2 + 16x = x(x^2 + 8x + 16) = x(x + 4)^2
\]

So, the original expression becomes:

\[
\frac{9x + 48}{x(x + 4)^2}
\]

Now, let's decompose this into partial fractions. We assume the form:

\[
\frac{9x + 48}{x(x + 4)^2} = \frac{A}{x} + \frac{B}{x+4} + \frac{C}{(x+4)^2}
\]

Multiply both sides by \( x(x + 4)^2 \) to eliminate the denominator:

\[
9x + 48 = A(x + 4)^2 + Bx(x + 4) + Cx
\]

Expand both sides:

\[
9x + 48 = A(x^2 + 8x + 16) + Bx(x + 4) + Cx
\]
\[
9x + 48 = A(x^2 + 8x + 16) + B(x^2 + 4x) + Cx
\]
\[
9x + 48 = (A + B)x^2 + (8A + 4B + C)x + 16A
\]

Now, equate the coefficients of powers of \(x\):

- Coefficient of \(x^2\): \( A + B = 0 \)
- Coefficient of \(x\): \( 8A + 4B + C = 9 \)
- Constant term: \( 16A = 48 \)

From \( 16A = 48 \), we get \( A = 3 \).

Substitute \( A = 3 \) into the first equation:

\[
3 + B = 0 \quad \Rightarrow \quad B = -3
\]

Now substitute \( A = 3 \) and \( B = -3 \) into the second equation:

\[
8(3) + 4(-3) + C = 9 \quad \Rightarrow \quad 24 - 12 + C = 9 \quad \Rightarrow \quad C = -3
\]

Thus, the partial fraction decomposition is:

\[
\frac{9x + 48}{x(x + 4)^2} = \frac{3}{x} - \frac{3}{x + 4} - \frac{3}{(x + 4)^2}
\]

Now, we integrate each term:

\[
\int \left( \frac{3}{x} - \frac{3}{x + 4} - \frac{3}{(x + 4)^2} \right) dx
\]

The integrals are straightforward:

\[
\int \frac{3}{x} \, dx = 3 \ln |x| + C_1
\]
\[
\int \frac{3}{x + 4} \, dx = 3 \ln |x + 4| + C_2
\]
\[
\int \frac{3}{(x + 4)^2} \, dx = \frac{3}{x + 4} + C_3
\]

So, the final result is:

\[
3 \ln |x| - 3 \ln |x + 4| - \frac{3}{x + 4} + C
\]

Which simplifies to:

\[
3 \ln \left(\frac{x}{x + 4}\right) - \frac{3}{x + 4} + C
\]

Comparing this with the options given, **Option D** is the correct answer.

Would you like more details or have any questions? 

Here are some related questions for further practice:
1. How do you factor cubic expressions in general?
2. What are the rules for finding partial fractions for rational functions?
3. Can partial fraction decomposition be applied to non-polynomial functions?
4. What is the technique for integrating rational functions after decomposition?
5. How do logarithmic properties simplify results in integrals?

**Tip:** Always check if a denominator can be factored to simplify complex rational functions before attempting integration.

Express the integrand as a sum of partial fractions and evaluate the integral. Given the integral is \( \int \frac{9x + 48}{x^3 + 8x^2 + 16x} \, dx \)

Learn how to express an integrand as a sum of partial fractions and evaluate the integral of \( \frac{9x + 48}{x^3 + 8x^2 + 16x} \). This tutorial covers polynomial factorization, partial fraction decomposition, and integration techniques.

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