What is the integral of (4x+10)/((x^2+5x+12)^2) ?

The integral of (4x+10)/((x^2+5x+12)^2) is

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integral of (4x+10)/((x^2+5x+12)^2)

\int \frac{4 x + 10}{( x ^{2} + 5 x + 12 ) ^{2}} d x

4 (\frac{2}{23 23} x sin (2 arctan (\frac{1}{23} (2 x + 5))) - \frac{1}{23} ln x^{2} + 5 x + 12 + \frac{1}{23} ln 4 x^{2} + 20 x + 48 - \frac{10}{23 23} arctan (\frac{1}{23} (2 x + 5))) + \frac{20}{23 23} (2 arctan (\frac{2 x + 5}{23}) + sin (2 arctan (\frac{2 x + 5}{23}))) + C

Solution steps

\int \frac{4 x + 10}{( x ^{2} + 5 x + 12 ) ^{2}} d x

Expand

\frac{4 x + 10}{( x ^{2} + 5 x + 12 ) ^{2}} : \frac{4 x}{( x ^{2} + 5 x + 12 ) ^{2}} + \frac{10}{( x ^{2} + 5 x + 12 ) ^{2}}

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{4 x}{( x ^{2} + 5 x + 12 ) ^{2}} d x + \int \frac{10}{( x ^{2} + 5 x + 12 ) ^{2}} d x

\int \frac{4 x}{( x ^{2} + 5 x + 12 ) ^{2}} d x = 4 (\frac{2}{23 23} x sin (2 arctan (\frac{1}{23} (2 x + 5))) - \frac{1}{23} ln x^{2} + 5 x + 12 + \frac{1}{23} ln 4 x^{2} + 20 x + 48 - \frac{10}{23 23} arctan (\frac{1}{23} (2 x + 5)))

\int \frac{10}{( x ^{2} + 5 x + 12 ) ^{2}} d x = \frac{20}{23 23} (2 arctan (\frac{2 x + 5}{23}) + sin (2 arctan (\frac{2 x + 5}{23})))

= 4 (\frac{2}{23 23} x sin (2 arctan (\frac{1}{23} (2 x + 5))) - \frac{1}{23} ln x^{2} + 5 x + 12 + \frac{1}{23} ln 4 x^{2} + 20 x + 48 - \frac{10}{23 23} arctan (\frac{1}{23} (2 x + 5))) + \frac{20}{23 23} (2 arctan (\frac{2 x + 5}{23}) + sin (2 arctan (\frac{2 x + 5}{23})))

Add a constant to the solution

= 4 (\frac{2}{23 23} x sin (2 arctan (\frac{1}{23} (2 x + 5))) - \frac{1}{23} ln x^{2} + 5 x + 12 + \frac{1}{23} ln 4 x^{2} + 20 x + 48 - \frac{10}{23 23} arctan (\frac{1}{23} (2 x + 5))) + \frac{20}{23 23} (2 arctan (\frac{2 x + 5}{23}) + sin (2 arctan (\frac{2 x + 5}{23}))) + C

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What is the integral of (4x+10)/((x^2+5x+12)^2) ?
The integral of (4x+10)/((x^2+5x+12)^2) is