To solve the integral using partial fraction decomposition, we first decompose the fraction:

\[
\frac{2x^3 + 5x^2 + 16}{x(5x^4 + 8x^2 + 16)}.
\]

### Step-by-Step Partial Fraction Decomposition

1. **Factor the Denominator:**

   The denominator is already partially factored:

   \[
   x(5x^4 + 8x^2 + 16).
   \]

   Now, we consider the expression \(5x^4 + 8x^2 + 16\) as a quadratic in terms of \(x^2\). Let \(y = x^2\), then we have:

   \[
   5y^2 + 8y + 16.
   \]

   Since this quadratic has no real roots (its discriminant \(8^2 - 4 \cdot 5 \cdot 16 = -264 < 0\)), it is irreducible over the reals. Thus, we treat \(5x^4 + 8x^2 + 16\) as an irreducible quadratic in \(x^2\).

2. **Set Up the Partial Fractions:**

   Now, express the fraction as a sum of partial fractions:

   \[
   \frac{2x^3 + 5x^2 + 16}{x(5x^4 + 8x^2 + 16)} = \frac{A}{x} + \frac{Bx + C}{5x^4 + 8x^2 + 16}.
   \]

   Multiply through by the denominator to clear the fractions:

   \[
   2x^3 + 5x^2 + 16 = A(5x^4 + 8x^2 + 16) + (Bx + C)x.
   \]

   Expand and collect like terms:

   \[
   2x^3 + 5x^2 + 16 = (5A)x^4 + (8A + B)x^3 + (16A + C)x^2.
   \]

3. **Equate Coefficients:**

   Now, match coefficients for each power of \(x\):

   - Coefficient of \(x^4\): \(5A = 0 \implies A = 0\).
   - Coefficient of \(x^3\): \(8A + B = 2 \implies 0 + B = 2 \implies B = 2\).
   - Coefficient of \(x^2\): \(16A + C = 5 \implies 0 + C = 5 \implies C = 5\).

   So, we have \(A = 0\), \(B = 2\), and \(C = 5\).

   The decomposition becomes:

   \[
   \frac{2x^3 + 5x^2 + 16}{x(5x^4 + 8x^2 + 16)} = \frac{2x + 5}{5x^4 + 8x^2 + 16}.
   \]

4. **Rewrite the Integral:**

   The integral now is:

   \[
   \int \frac{2x + 5}{5x^4 + 8x^2 + 16} \, dx.
   \]

### Solving the Integral

Now, let's split the integral into two parts:

\[
\int \frac{2x}{5x^4 + 8x^2 + 16} \, dx + \int \frac{5}{5x^4 + 8x^2 + 16} \, dx.
\]

For the first integral:

- Substitute \(u = 5x^4 + 8x^2 + 16\), \(du = (20x^3 + 16x) dx = 4x(5x^2 + 4) dx \), which simplifies this part.

For the second integral:

- A suitable trigonometric substitution or table integration may apply, depending on its specific form.

Would you like to proceed with solving these integrals, or do you have a preference for a particular approach?

Learn how to solve the integral of (2x^3 + 5x^2 + 16) / (x(5x^4 + 8x^2 + 16)) using partial fraction decomposition. This problem covers advanced calculus concepts, including partial fraction decomposition and integration techniques. Suitable for college-level mathematics.

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