What is the integral of (((1+x)^3))/((sqrt(x))) ?

The integral of (((1+x)^3))/((sqrt(x))) is 2sqrt(x)+2x^{3/2}+6/5 x^{5/2}+2/7 x^{7/2}+C

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integral of (((1+x)^3))/((sqrt(x)))

integral

\frac{( ( 1 + x ) ^{3} )}{( x )}

2 x + 2 x^{\frac{3}{2}} + \frac{6}{5} x^{\frac{5}{2}} + \frac{2}{7} x^{\frac{7}{2}} + C

Solution steps

\int \frac{( 1 + x ) ^{3}}{x} d x

Expand

\frac{( 1 + x ) ^{3}}{x} : \frac{1}{x} + 3 x + 3 x^{\frac{3}{2}} + x^{\frac{5}{2}}

= \int \frac{1}{x} + 3 x + 3 x^{\frac{3}{2}} + x^{\frac{5}{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{1}{x} d x + \int 3 x d x + \int 3 x^{\frac{3}{2}} d x + \int x^{\frac{5}{2}} d x

\int \frac{1}{x} d x = 2 x

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

\int 3 x^{\frac{3}{2}} d x = \frac{6}{5} x^{\frac{5}{2}}

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

= 2 x + 2 x^{\frac{3}{2}} + \frac{6}{5} x^{\frac{5}{2}} + \frac{2}{7} x^{\frac{7}{2}}

Add a constant to the solution

= 2 x + 2 x^{\frac{3}{2}} + \frac{6}{5} x^{\frac{5}{2}} + \frac{2}{7} x^{\frac{7}{2}} + C

3 - ∣ x ∣

\frac{1}{( 4 + x ^{2} ) ^{2}}

x^{3} \cdot e^{x}

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

\frac{1}{( 1 + \frac{1}{2} x )}

What is the integral of (((1+x)^3))/((sqrt(x))) ?
The integral of (((1+x)^3))/((sqrt(x))) is 2sqrt(x)+2x^{3/2}+6/5 x^{5/2}+2/7 x^{7/2}+C