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

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

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

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

x^{3} - 2 x + \frac{2}{3} ln x^{3} + 1 + 1 + C

Solution steps

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

Expand

\frac{3 x ^{5} - 2 x ^{3} + 5 x ^{2} - 2}{x ^{3} + 1} : \frac{3 x ^{5}}{x ^{3} + 1} - \frac{2 x ^{3}}{x ^{3} + 1} + \frac{5 x ^{2}}{x ^{3} + 1} - \frac{2}{x ^{3} + 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{3 x ^{5}}{x ^{3} + 1} d x - \int \frac{2 x ^{3}}{x ^{3} + 1} d x + \int \frac{5 x ^{2}}{x ^{3} + 1} d x - \int \frac{2}{x ^{3} + 1} d x

\int \frac{3 x ^{5}}{x ^{3} + 1} d x = x^{3} + 1 - ln x^{3} + 1

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

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

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

= x^{3} + 1 - ln x^{3} + 1 - 2 (- \frac{1}{3} ln ∣ x + 1 ∣ - \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3})) + x) + \frac{5}{3} ln x^{3} + 1 - 2 (\frac{1}{3} ln ∣ x + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} - 4 x}{3} + 23 arctan (\frac{2 x - 1}{3})))

Simplify

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

= x^{3} - 2 x + \frac{2}{3} ln x^{3} + 1 + 1

Add a constant to the solution

= x^{3} - 2 x + \frac{2}{3} ln x^{3} + 1 + 1 + C

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