What is the integral of ((x+x^{-2})^2)/(xsqrt(x)) ?

The integral of ((x+x^{-2})^2)/(xsqrt(x)) is 2/3 x^{3/2}-4/(3x^{3/2)}-2/(9x^{9/2)}+C

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integral of ((x+x^{-2})^2)/(xsqrt(x))

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

\frac{2}{3} x^{\frac{3}{2}} - \frac{4}{3 x ^{\frac{3}{2}}} - \frac{2}{9 x ^{\frac{9}{2}}} + C

Solution steps

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

Expand

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

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

\int x d x = \frac{2}{3} x^{\frac{3}{2}}

\int \frac{2}{x ^{2} x} d x = - \frac{4}{3 x ^{\frac{3}{2}}}

\int \frac{1}{x ^{5} x} d x = - \frac{2}{9 x ^{\frac{9}{2}}}

= \frac{2}{3} x^{\frac{3}{2}} - \frac{4}{3 x ^{\frac{3}{2}}} - \frac{2}{9 x ^{\frac{9}{2}}}

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} - \frac{4}{3 x ^{\frac{3}{2}}} - \frac{2}{9 x ^{\frac{9}{2}}} + C

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What is the integral of ((x+x^{-2})^2)/(xsqrt(x)) ?
The integral of ((x+x^{-2})^2)/(xsqrt(x)) is 2/3 x^{3/2}-4/(3x^{3/2)}-2/(9x^{9/2)}+C