What is the integral of (2x^3+9x^2+14x+8)/((x^2+2x)(x^2+2)) ?

The integral of (2x^3+9x^2+14x+8)/((x^2+2x)(x^2+2)) is 5/(sqrt(2))arctan(x/(sqrt(2)))+2ln(x)+C

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integral of (2x^3+9x^2+14x+8)/((x^2+2x)(x^2+2))

\int \frac{2 x ^{3} + 9 x ^{2} + 14 x + 8}{( x ^{2} + 2 x ) ( x ^{2} + 2 )} d x

\frac{5}{2} arctan (\frac{x}{2}) + 2 ln (x) + C

Solution steps

\int \frac{2 x ^{3} + 9 x ^{2} + 14 x + 8}{( x ^{2} + 2 x ) ( x ^{2} + 2 )} d x

Simplify

\frac{2 x ^{3} + 9 x ^{2} + 14 x + 8}{( x ^{2} + 2 x ) ( x ^{2} + 2 )} : \frac{2 x ^{2} + 5 x + 4}{x ( x ^{2} + 2 )}

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

Expand

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

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

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

\int \frac{4}{x ( x ^{2} + 2 )} d x = - ln x^{2} + 2 + 2 ln (x)

= ln x^{2} + 2 + \frac{5}{2} arctan (\frac{x}{2}) - ln x^{2} + 2 + 2 ln (x)

Simplify

= \frac{5}{2} arctan (\frac{x}{2}) + 2 ln (x)

Add a constant to the solution

= \frac{5}{2} arctan (\frac{x}{2}) + 2 ln (x) + C

\frac{1}{x + 1}

\int (6 e^{x} - x^{3} (x + 1)) d x

\int (x^{2} + x + 8) x + 7 d x

\int \frac{tanh ^{2} ( \frac{1}{x} )}{x ^{2}} d x

\int (a - x)^{2} d x

What is the integral of (2x^3+9x^2+14x+8)/((x^2+2x)(x^2+2)) ?
The integral of (2x^3+9x^2+14x+8)/((x^2+2x)(x^2+2)) is 5/(sqrt(2))arctan(x/(sqrt(2)))+2ln(x)+C