What is the integral of (x^3+69x+84)/(x^3+22x) ?

The integral of (x^3+69x+84)/(x^3+22x) is

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integral of (x^3+69x+84)/(x^3+22x)

\int \frac{x ^{3} + 69 x + 84}{x ^{3} + 22 x} d x

- 22 arctan (\frac{x}{22}) + x + \frac{69}{22} arctan (\frac{x}{22}) + 84 (\frac{1}{22} ln ∣ x ∣ - \frac{1}{44} ln x^{2} + 22) + C

Solution steps

\int \frac{x ^{3} + 69 x + 84}{x ^{3} + 22 x} d x

Expand

\frac{x ^{3} + 69 x + 84}{x ^{3} + 22 x} : \frac{x ^{3}}{x ^{3} + 22 x} + \frac{69 x}{x ^{3} + 22 x} + \frac{84}{x ^{3} + 22 x}

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 ^{3} + 22 x} d x + \int \frac{69 x}{x ^{3} + 22 x} d x + \int \frac{84}{x ^{3} + 22 x} d x

\int \frac{x ^{3}}{x ^{3} + 22 x} d x = - 22 arctan (\frac{x}{22}) + x

\int \frac{69 x}{x ^{3} + 22 x} d x = \frac{69}{22} arctan (\frac{x}{22})

\int \frac{84}{x ^{3} + 22 x} d x = 84 (\frac{1}{22} ln ∣ x ∣ - \frac{1}{44} ln x^{2} + 22)

= - 22 arctan (\frac{x}{22}) + x + \frac{69}{22} arctan (\frac{x}{22}) + 84 (\frac{1}{22} ln ∣ x ∣ - \frac{1}{44} ln x^{2} + 22)

Add a constant to the solution

= - 22 arctan (\frac{x}{22}) + x + \frac{69}{22} arctan (\frac{x}{22}) + 84 (\frac{1}{22} ln ∣ x ∣ - \frac{1}{44} ln x^{2} + 22) + C

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y^{^{''}} - 2 y^{^{'}} - 3 y = t

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What is the integral of (x^3+69x+84)/(x^3+22x) ?
The integral of (x^3+69x+84)/(x^3+22x) is