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

The integral of xsqrt((x+1)^3) is 2/5 x(x+1)^{5/2}-4/35 (x+1)^{7/2}+C

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integral of xsqrt((x+1)^3)

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

\frac{2}{5} x (x + 1)^{\frac{5}{2}} - \frac{4}{35} (x + 1)^{\frac{7}{2}} + C

Solution steps

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

(x + 1)^{3} = (x + 1)^{\frac{3}{2}},

assuming

(x + 1) \geq 0

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

Apply Integration By Parts

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

\int \frac{2}{5} (x + 1)^{\frac{5}{2}} d x = \frac{4}{35} (x + 1)^{\frac{7}{2}}

= \frac{2}{5} x (x + 1)^{\frac{5}{2}} - \frac{4}{35} (x + 1)^{\frac{7}{2}}

Add a constant to the solution

= \frac{2}{5} x (x + 1)^{\frac{5}{2}} - \frac{4}{35} (x + 1)^{\frac{7}{2}} + C

What is the integral of xsqrt((x+1)^3) ?
The integral of xsqrt((x+1)^3) is 2/5 x(x+1)^{5/2}-4/35 (x+1)^{7/2}+C