\int \frac{2x^3+4x^2+17x+16}{\left(x^2+4\right)^2}dx

We are tasked with evaluating the integral:

\[
I = \int \frac{2x^3 + 4x^2 + 17x + 16}{\left(x^2 + 4\right)^2} \, dx
\]

### Step-by-Step Solution

1. **Simplifying the integrand**:
   We notice that the denominator \(\left(x^2 + 4\right)^2\) suggests that the problem can be approached by decomposing the numerator using a combination of terms that fit well with the structure of the denominator. 

2. **Decomposition**:
   Let's attempt to split the numerator into parts that can be easier to integrate. We rewrite the numerator as:

   \[
   2x^3 + 4x^2 + 17x + 16 = A \cdot \frac{d}{dx}(x^2 + 4) + \frac{Bx + C}{x^2 + 4}
   \]
   
   This form breaks the problem into two parts: one that involves the derivative of the denominator and another part that might require a simple arctangent-type substitution.

   Now, \( \frac{d}{dx}(x^2 + 4) = 2x \), so we equate:

   \[
   2x^3 + 4x^2 + 17x + 16 = A(2x)(x^2 + 4) + (Bx + C)(x^2 + 4)
   \]

   Expanding both terms on the right-hand side:

   \[
   A(2x)(x^2 + 4) = 2Ax^3 + 8Ax
   \]
   \[
   (Bx + C)(x^2 + 4) = Bx^3 + 4Bx + Cx^2 + 4C
   \]

   Combining terms:

   \[
   2Ax^3 + 8Ax + Bx^3 + 4Bx + Cx^2 + 4C = (2A + B)x^3 + Cx^2 + (8A + 4B)x + 4C
   \]

3. **Matching coefficients**:
   We now match the coefficients of \(x^3\), \(x^2\), \(x\), and the constant term from both sides of the equation:

   \[
   2A + B = 2 \quad \text{(coefficient of \(x^3\))} 
   \]
   \[
   C = 4 \quad \text{(coefficient of \(x^2\))} 
   \]
   \[
   8A + 4B = 17 \quad \text{(coefficient of \(x\))} 
   \]
   \[
   4C = 16 \quad \text{(constant term)}
   \]

4. **Solving the system**:
   From the equation \(4C = 16\), we get \(C = 4\).

   Substituting \(C = 4\) into the equation \(C = 4\), we get:

   \[
   C = 4
   \]

   From the equation \(2A + B = 2\), and using \(C = 4\9

 `

This problem involves the integration of a rational function \(\frac{2x^3 + 4x^2 + 17x + 16}{(x^2 + 4)^2}\), utilizing partial fraction decomposition and arctangent substitution. Suitable for undergraduate calculus students, this solution includes step-by-step instructions and detailed mathematical explanations.

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