Partial Fraction Decomposition:
Partial fraction decomposition is a method used to simplify a rational function by expressing it as a sum of simpler fractions. It is particularly useful when integrating rational functions, as it allows us to integrate each term separately.

Step 1: Factorize the denominator
In order to perform partial fraction decomposition, we need to factorize the denominator of the rational function. In this case, the denominator is (x²-1)(x+1)³.

Step 2: Express the rational function as a sum of simpler fractions
We need to find the partial fraction decomposition of the given rational function: 5x³ + 6x² + 5x / (x²-1)(x+1)³.

Step 3: Decompose the fractions
We will decompose the rational function into three simpler fractions as follows:

1. Decompose the fraction with the factor (x+1)³:
The denominator (x+1)³ can be written as (x+1)(x+1)(x+1). So, we can write the fraction as:
A / (x+1) + B / (x+1)² + C / (x+1)³.

2. Decompose the fraction with the factor (x-1):
The denominator (x²-1) can be written as (x+1)(x-1). So, we can write the fraction as:
D / (x+1) + E / (x-1).

Step 4: Find the values of the unknown coefficients
To find the values of the unknown coefficients A, B, C, D, and E, we need to equate the numerators of the original rational function and the decomposed fractions.

For A / (x+1), the numerator is 5x³ + 6x² + 5x. As the denominator is (x+1), we can equate the numerators:
5x³ + 6x² + 5x = A(x²-1)(x-1).

For B / (x+1)², the numerator is 5x³ + 6x² + 5x. As the denominator is (x+1)², we can equate the numerators:
5x³ + 6x² + 5x = B(x²-1).

For C / (x+1)³, the numerator is 5x³ + 6x² + 5x. As the denominator is (x+1)³, we can equate the numerators:
5x³ + 6x² + 5x = C.

For D / (x+1), the numerator is 5x³ + 6x² + 5x. As the denominator is (x-1), we can equate the numerators:
5x³ + 6x² + 5x = D(x+1)(x-1).

For E / (x-1), the numerator is 5x³ + 6x² + 5x. As the denominator is (x-1), we can equate the numerators:
5x³ + 6x² + 5x = E(x+1)(x+1).

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