# What is #int (-x^3-x-1 ) / (-2x^2+ 3 x +5 )#?

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To find the integral of (\frac{{-x^3 - x - 1}}{{-2x^2 + 3x + 5}}), we can use polynomial long division to simplify the expression first. After simplification, we can integrate each term separately.

After performing polynomial long division, we get:

(\frac{{-x^3 - x - 1}}{{-2x^2 + 3x + 5}} = x + 1 + \frac{{4x - 4}}{{-2x^2 + 3x + 5}})

Now, the integral becomes:

[\int \left( x + 1 + \frac{{4x - 4}}{{-2x^2 + 3x + 5}} \right) dx]

This integral can be split into three separate integrals:

- (\int x , dx)
- (\int 1 , dx)
- (\int \frac{{4x - 4}}{{-2x^2 + 3x + 5}} , dx)

Now, we can integrate each term individually:

- (\int x , dx = \frac{{x^2}}{2} + C_1)
- (\int 1 , dx = x + C_2)
- To integrate (\frac{{4x - 4}}{{-2x^2 + 3x + 5}}), we can perform partial fraction decomposition or use other integration techniques.

After finding the integrals of each term, we can combine them and add the constant of integration to obtain the final result.

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