Evaluate the following integration
`int(x^(2)+3)/(x^(6)(x^(2)+1))dx`

Answer

To evaluate the integral

\[ I = \int \frac{x^2 + 3}{x^6 (x^2 + 1)} \, dx, \]

we can start by simplifying the integrand.

### Step 1: Rewrite the integrand
We can express the integrand as:

\[ I = \int \left( \frac{x^2 + 1 + 2}{x^6 (x^2 + 1)} \right) \, dx = \int \left( \frac{x^2 + 1}{x^6 (x^2 + 1)} + \frac{2}{x^6 (x^2 + 1)} \right) \, dx. \]

This simplifies to:

\[ I = \int \frac{1}{x^6} \, dx + 2 \int \frac{1}{x^6 (x^2 + 1)} \, dx. \]

### Step 2: Integrate the first term
The first integral can be computed as:

\[ \int \frac{1}{x^6} \, dx = \int x^{-6} \, dx = \frac{x^{-5}}{-5} = -\frac{1}{5x^5}. \]

### Step 3: Integrate the second term
Now we need to evaluate:

\[ 2 \int \frac{1}{x^6 (x^2 + 1)} \, dx. \]

We can rewrite this integral as:

\[ 2 \int \left( \frac{1}{x^6} - \frac{x^2}{x^6 (x^2 + 1)} \right) \, dx = 2 \int \frac{1}{x^6} \, dx - 2 \int \frac{1}{x^4 (x^2 + 1)} \, dx. \]

### Step 4: Integrate the second part
The first part is already computed as:

\[ 2 \left(-\frac{1}{5x^5}\right) = -\frac{2}{5x^5}. \]

Now we need to compute:

\[ -2 \int \frac{1}{x^4 (x^2 + 1)} \, dx. \]

### Step 5: Use partial fraction decomposition
We can decompose:

\[ \frac{1}{x^4 (x^2 + 1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4} + \frac{Ex + F}{x^2 + 1}. \]

By equating coefficients and solving, we find \(A, B, C, D, E, F\).

### Step 6: Integrate each term
After finding the coefficients, we can integrate each term separately.

1. \(\int \frac{A}{x} \, dx = A \ln |x|\)
2. \(\int \frac{B}{x^2} \, dx = -\frac{B}{x}\)
3. \(\int \frac{C}{x^3} \, dx = -\frac{C}{2x^2}\)
4. \(\int \frac{D}{x^4} \, dx = -\frac{D}{3x^3}\)
5. \(\int \frac{Ex + F}{x^2 + 1} \, dx = E \tan^{-1}(x) + F \ln |x^2 + 1|\)

### Step 7: Combine results
Combine all the results from the integrals to get the final answer:

\[ I = -\frac{2}{5x^5} - 2 \left( \text{result from } -2 \int \frac{1}{x^4 (x^2 + 1)} \, dx \right) + C, \]

where \(C\) is the constant of integration.

### Final Result
The final result will be a combination of logarithmic and arctangent functions along with the polynomial terms from the integration.