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

The integral of ((2+3x)^2)/(sqrt(x)) is 8sqrt(x)+8x^{3/2}+18/5 x^{5/2}+C

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integral of ((2+3x)^2)/(sqrt(x))

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

8 x + 8 x^{\frac{3}{2}} + \frac{18}{5} x^{\frac{5}{2}} + C

Solution steps

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

Expand

\frac{( 2 + 3 x ) ^{2}}{x} : \frac{4}{x} + 12 x + 9 x^{\frac{3}{2}}

= \int \frac{4}{x} + 12 x + 9 x^{\frac{3}{2}} 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 \frac{4}{x} d x + \int 12 x d x + \int 9 x^{\frac{3}{2}} d x

\int \frac{4}{x} d x = 8 x

\int 12 x d x = 8 x^{\frac{3}{2}}

\int 9 x^{\frac{3}{2}} d x = \frac{18}{5} x^{\frac{5}{2}}

= 8 x + 8 x^{\frac{3}{2}} + \frac{18}{5} x^{\frac{5}{2}}

Add a constant to the solution

= 8 x + 8 x^{\frac{3}{2}} + \frac{18}{5} x^{\frac{5}{2}} + C

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What is the integral of ((2+3x)^2)/(sqrt(x)) ?
The integral of ((2+3x)^2)/(sqrt(x)) is 8sqrt(x)+8x^{3/2}+18/5 x^{5/2}+C