What is the integral of x(sqrt(1+x^2)+sqrt(x)) ?

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

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integral of x(sqrt(1+x^2)+sqrt(x))

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

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