Integration of rational functions
`I = int(x^(4)dx)/((2+x)(x^(2)-1))`

Answer

To solve the integral

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

we can start by simplifying the integrand using polynomial long division and partial fraction decomposition.

### Step 1: Polynomial Long Division
Since the degree of the numerator \(x^4\) is greater than the degree of the denominator \((2+x)(x^2-1)\), we first perform polynomial long division.

1. Divide \(x^4\) by \(x^3\) (the leading term of the denominator):
- The first term is \(x\).
- Multiply \(x\) by \((2+x)(x^2-1)\) and subtract from \(x^4\).

\[ x^4 - x(2+x)(x^2-1) = x^4 - x(2x^2 - 1) = x^4 - (2x^3 - x) = x^4 - 2x^3 + x. \]

2. The result is \(x^4 - 2x^3 + x\). Now we need to divide \(-2x^3 + x\) by \((2+x)(x^2-1)\).

### Step 2: Partial Fraction Decomposition
We can express the integrand as:

\[ \frac{x^4}{(2+x)(x^2-1)} = x + \frac{Ax + B}{(2+x)(x^2-1)}, \]

where \(A\) and \(B\) are constants to be determined.

### Step 3: Finding A and B
To find \(A\) and \(B\), we equate:

\[ x^4 = (2+x)(x^2-1)(Ax + B) + x(2+x)(x^2-1). \]

Expanding and simplifying will give us a system of equations to solve for \(A\) and \(B\).

### Step 4: Integration
Once we have the partial fractions, we can integrate each term separately. The integral will look like:

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

### Step 5: Evaluate the Integrals
1. The first integral is straightforward:

\[ \int x \, dx = \frac{x^2}{2} + C_1. \]

2. The second integral can be solved using partial fractions again, leading to logarithmic terms.

### Final Result
After performing all the integrations and combining the results, we will arrive at:

\[ I = \frac{x^2}{2} + \text{(logarithmic terms)} + C, \]

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