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

The integral of (x^2+3x-1)/(x^3+8) is

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

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

\frac{1}{3} ln x^{3} + 8 - \frac{7}{12} ln ∣ x + 2 ∣ + \frac{1}{4} ln x^{2} - 2 x + 4 + \frac{1}{24} ln \frac{1}{3} (x - 1)^{2} + 1 + \frac{5}{4 3} arctan (\frac{1}{3} (x - 1)) + C

Solution steps

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

Expand

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

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

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

\int \frac{3 x}{x ^{3} + 8} d x = 3 (- \frac{1}{6} ln ∣ x + 2 ∣ + \frac{1}{12} (ln x^{2} - 2 x + 4 + 23 arctan (\frac{1}{3} (x - 1))))

\int \frac{1}{x ^{3} + 8} d x = \frac{1}{12} ln ∣ x + 2 ∣ + \frac{1}{12} (- \frac{1}{2} ln \frac{1}{3} (x - 1)^{2} + 1 + 3 arctan (\frac{1}{3} (x - 1)))

= \frac{1}{3} ln x^{3} + 8 + 3 (- \frac{1}{6} ln ∣ x + 2 ∣ + \frac{1}{12} (ln x^{2} - 2 x + 4 + 23 arctan (\frac{1}{3} (x - 1)))) - (\frac{1}{12} ln ∣ x + 2 ∣ + \frac{1}{12} (- \frac{1}{2} ln \frac{1}{3} (x - 1)^{2} + 1 + 3 arctan (\frac{1}{3} (x - 1))))

Simplify

\frac{1}{3} ln x^{3} + 8 + 3 (- \frac{1}{6} ln ∣ x + 2 ∣ + \frac{1}{12} (ln x^{2} - 2 x + 4 + 23 arctan (\frac{1}{3} (x - 1)))) - (\frac{1}{12} ln ∣ x + 2 ∣ + \frac{1}{12} (- \frac{1}{2} ln \frac{1}{3} (x - 1)^{2} + 1 + 3 arctan (\frac{1}{3} (x - 1)))) : \frac{1}{3} ln x^{3} + 8 - \frac{7}{12} ln ∣ x + 2 ∣ + \frac{1}{4} ln x^{2} - 2 x + 4 + \frac{1}{24} ln \frac{1}{3} (x - 1)^{2} + 1 + \frac{5}{4 3} arctan (\frac{1}{3} (x - 1))

= \frac{1}{3} ln x^{3} + 8 - \frac{7}{12} ln ∣ x + 2 ∣ + \frac{1}{4} ln x^{2} - 2 x + 4 + \frac{1}{24} ln \frac{1}{3} (x - 1)^{2} + 1 + \frac{5}{4 3} arctan (\frac{1}{3} (x - 1))

Add a constant to the solution

= \frac{1}{3} ln x^{3} + 8 - \frac{7}{12} ln ∣ x + 2 ∣ + \frac{1}{4} ln x^{2} - 2 x + 4 + \frac{1}{24} ln \frac{1}{3} (x - 1)^{2} + 1 + \frac{5}{4 3} arctan (\frac{1}{3} (x - 1)) + C

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