What is the integral of (x^3)/(x^2+10x+25) ?

The integral of (x^3)/(x^2+10x+25) is 1/2 (x+5)^2-15(x+5)+75ln|x+5|+(125)/(x+5)+C

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integral of (x^3)/(x^2+10x+25)

\int \frac{x ^{3}}{x ^{2} + 10 x + 25} d x

\frac{1}{2} (x + 5)^{2} - 15 (x + 5) + 75 ln ∣ x + 5 ∣ + \frac{125}{x + 5} + C

Solution steps

\int \frac{x ^{3}}{x ^{2} + 10 x + 25} d x

Complete the square

x^{2} + 10 x + 25 : (x + 5)^{2}

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

Apply u-substitution

= \int \frac{( u - 5 ) ^{3}}{u ^{2}} d u

Expand

\frac{( u - 5 ) ^{3}}{u ^{2}} : u - 15 + \frac{75}{u} - \frac{125}{u ^{2}}

= \int u - 15 + \frac{75}{u} - \frac{125}{u ^{2}} d u

Apply the Sum Rule:

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

= \int u d u - \int 15 d u + \int \frac{75}{u} d u - \int \frac{125}{u ^{2}} d u

\int u d u = \frac{u ^{2}}{2}

\int 15 d u = 15 u

\int \frac{75}{u} d u = 75 ln ∣ u ∣

\int \frac{125}{u ^{2}} d u = - \frac{125}{u}

= \frac{u ^{2}}{2} - 15 u + 75 ln ∣ u ∣ - (- \frac{125}{u})

Substitute back

u = x + 5

= \frac{( x + 5 ) ^{2}}{2} - 15 (x + 5) + 75 ln ∣ x + 5 ∣ - (- \frac{125}{x + 5})

Simplify

\frac{( x + 5 ) ^{2}}{2} - 15 (x + 5) + 75 ln ∣ x + 5 ∣ - (- \frac{125}{x + 5}) : \frac{1}{2} (x + 5)^{2} - 15 (x + 5) + 75 ln ∣ x + 5 ∣ + \frac{125}{x + 5}

= \frac{1}{2} (x + 5)^{2} - 15 (x + 5) + 75 ln ∣ x + 5 ∣ + \frac{125}{x + 5}

Add a constant to the solution

= \frac{1}{2} (x + 5)^{2} - 15 (x + 5) + 75 ln ∣ x + 5 ∣ + \frac{125}{x + 5} + C

What is the integral of (x^3)/(x^2+10x+25) ?
The integral of (x^3)/(x^2+10x+25) is 1/2 (x+5)^2-15(x+5)+75ln|x+5|+(125)/(x+5)+C