What is the integral of (x^2)/(sqrt(x^2+4x-5)) ?

The integral of (x^2)/(sqrt(x^2+4x-5)) is

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integral of (x^2)/(sqrt(x^2+4x-5))

\int \frac{x ^{2}}{x ^{2} + 4 x - 5} d x

9 (\frac{1}{18} (x + 2) x^{2} + 4 x - 5 + \frac{1}{2} ln (\frac{x ^{2} + 4 x - 5 + x + 2}{3})) - 4 x^{2} + 4 x - 5 + 4 ln (\frac{1}{3} x^{2} + 4 x - 5 + x + 2) + C

Solution steps

\int \frac{x ^{2}}{x ^{2} + 4 x - 5} d x

Factor

\frac{x ^{2}}{x ^{2} + 4 x - 5}

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

Complete the square

(x - 1) (x + 5) : (x + 2)^{2} - 9

= \int \frac{x ^{2}}{( x + 2 ) ^{2} - 9} d x

Apply u-substitution

: \int \frac{( u - 2 ) ^{2}}{u ^{2} - 9} d u

= \int \frac{( u - 2 ) ^{2}}{u ^{2} - 9} d u

Expand

\frac{( u - 2 ) ^{2}}{u ^{2} - 9} : \frac{u ^{2}}{u ^{2} - 9} - \frac{4 u}{u ^{2} - 9} + \frac{4}{u ^{2} - 9}

= \int \frac{u ^{2}}{u ^{2} - 9} - \frac{4 u}{u ^{2} - 9} + \frac{4}{u ^{2} - 9} 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 \frac{u ^{2}}{u ^{2} - 9} d u - \int \frac{4 u}{u ^{2} - 9} d u + \int \frac{4}{u ^{2} - 9} d u

\int \frac{u ^{2}}{u ^{2} - 9} d u = 9 (\frac{1}{18} u u^{2} - 9 + \frac{1}{2} ln (\frac{u ^{2} - 9 + u}{3}))

\int \frac{4 u}{u ^{2} - 9} d u = 4 u^{2} - 9

\int \frac{4}{u ^{2} - 9} d u = 4 ln (\frac{1}{3} u^{2} - 9 + u)

= 9 (\frac{1}{18} u u^{2} - 9 + \frac{1}{2} ln (\frac{u ^{2} - 9 + u}{3})) - 4 u^{2} - 9 + 4 ln (\frac{1}{3} u^{2} - 9 + u)

Substitute back

u = x + 2

= 9 \frac{1}{18} (x + 2) (x + 2)^{2} - 9 + \frac{1}{2} ln \frac{( x + 2 ) ^{2} - 9 + x + 2}{3} - 4 (x + 2)^{2} - 9 + 4 ln (\frac{1}{3} (x + 2)^{2} - 9 + x + 2)

Simplify

9 \frac{1}{18} (x + 2) (x + 2)^{2} - 9 + \frac{1}{2} ln \frac{( x + 2 ) ^{2} - 9 + x + 2}{3} - 4 (x + 2)^{2} - 9 + 4 ln (\frac{1}{3} (x + 2)^{2} - 9 + x + 2) : 9 (\frac{1}{18} (x + 2) x^{2} + 4 x - 5 + \frac{1}{2} ln (\frac{x ^{2} + 4 x - 5 + x + 2}{3})) - 4 x^{2} + 4 x - 5 + 4 ln (\frac{1}{3} x^{2} + 4 x - 5 + x + 2)

= 9 (\frac{1}{18} (x + 2) x^{2} + 4 x - 5 + \frac{1}{2} ln (\frac{x ^{2} + 4 x - 5 + x + 2}{3})) - 4 x^{2} + 4 x - 5 + 4 ln (\frac{1}{3} x^{2} + 4 x - 5 + x + 2)

Add a constant to the solution

= 9 (\frac{1}{18} (x + 2) x^{2} + 4 x - 5 + \frac{1}{2} ln (\frac{x ^{2} + 4 x - 5 + x + 2}{3})) - 4 x^{2} + 4 x - 5 + 4 ln (\frac{1}{3} x^{2} + 4 x - 5 + x + 2) + C

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