What is the integral of ((x^3+4x^2+8x+5)/(x^2+2x+1)) ?

The integral of ((x^3+4x^2+8x+5)/(x^2+2x+1)) is 1/2 x^2+6x+4/(x+1)+3ln|x+1|-(4x^2)/(x+1)+5+1/2+C

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integral of ((x^3+4x^2+8x+5)/(x^2+2x+1))

\int (\frac{x ^{3} + 4 x ^{2} + 8 x + 5}{x ^{2} + 2 x + 1}) d x

\frac{1}{2} x^{2} + 6 x + \frac{4}{x + 1} + 3 ln ∣ x + 1 ∣ - \frac{4 x ^{2}}{x + 1} + 5 + \frac{1}{2} + C

Solution steps

\int \frac{x ^{3} + 4 x ^{2} + 8 x + 5}{x ^{2} + 2 x + 1} d x

Expand

\frac{x ^{3} + 4 x ^{2} + 8 x + 5}{x ^{2} + 2 x + 1} : \frac{x ^{3}}{x ^{2} + 2 x + 1} + \frac{4 x ^{2}}{x ^{2} + 2 x + 1} + \frac{8 x}{x ^{2} + 2 x + 1} + \frac{5}{x ^{2} + 2 x + 1}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x ^{3}}{x ^{2} + 2 x + 1} d x + \int \frac{4 x ^{2}}{x ^{2} + 2 x + 1} d x + \int \frac{8 x}{x ^{2} + 2 x + 1} d x + \int \frac{5}{x ^{2} + 2 x + 1} d x

\int \frac{x ^{3}}{x ^{2} + 2 x + 1} d x = \frac{1}{2} (x + 1)^{2} - 3 (x + 1) + 3 ln ∣ x + 1 ∣ + \frac{1}{x + 1}

\int \frac{4 x ^{2}}{x ^{2} + 2 x + 1} d x = 4 (- \frac{x ^{2}}{x + 1} + 2 (x + 1 - ln ∣ x + 1 ∣))

\int \frac{8 x}{x ^{2} + 2 x + 1} d x = 8 (ln ∣ x + 1 ∣ + \frac{1}{x + 1})

\int \frac{5}{x ^{2} + 2 x + 1} d x = - \frac{5}{x + 1}

= \frac{1}{2} (x + 1)^{2} - 3 (x + 1) + 3 ln ∣ x + 1 ∣ + \frac{1}{x + 1} + 4 (- \frac{x ^{2}}{x + 1} + 2 (x + 1 - ln ∣ x + 1 ∣)) + 8 (ln ∣ x + 1 ∣ + \frac{1}{x + 1}) - \frac{5}{x + 1}

Simplify

\frac{1}{2} (x + 1)^{2} - 3 (x + 1) + 3 ln ∣ x + 1 ∣ + \frac{1}{x + 1} + 4 (- \frac{x ^{2}}{x + 1} + 2 (x + 1 - ln ∣ x + 1 ∣)) + 8 (ln ∣ x + 1 ∣ + \frac{1}{x + 1}) - \frac{5}{x + 1} : \frac{1}{2} x^{2} + 6 x + \frac{4}{x + 1} + 3 ln ∣ x + 1 ∣ - \frac{4 x ^{2}}{x + 1} + 5 + \frac{1}{2}

= \frac{1}{2} x^{2} + 6 x + \frac{4}{x + 1} + 3 ln ∣ x + 1 ∣ - \frac{4 x ^{2}}{x + 1} + 5 + \frac{1}{2}

Add a constant to the solution

= \frac{1}{2} x^{2} + 6 x + \frac{4}{x + 1} + 3 ln ∣ x + 1 ∣ - \frac{4 x ^{2}}{x + 1} + 5 + \frac{1}{2} + C

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What is the integral of ((x^3+4x^2+8x+5)/(x^2+2x+1)) ?
The integral of ((x^3+4x^2+8x+5)/(x^2+2x+1)) is 1/2 x^2+6x+4/(x+1)+3ln|x+1|-(4x^2)/(x+1)+5+1/2+C