What is the integral of sqrt(x)sqrt((8x)^{3/2)+7} ?

The integral of sqrt(x)sqrt((8x)^{3/2)+7} is 1/(36sqrt(2))(16sqrt(2)x^{3/2}+7)^{3/2}+C

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integral of sqrt(x)sqrt((8x)^{3/2)+7}

\int x (8 x)^{\frac{3}{2}} + 7 d x

\frac{1}{36 2} (162 x^{\frac{3}{2}} + 7)^{\frac{3}{2}} + C

Solution steps

\int x (8 x)^{\frac{3}{2}} + 7 d x

Apply u-substitution

: \int \frac{u}{24 \cdot 2 ^{\frac{1}{2}}} d u

= \int \frac{u}{24 \cdot 2 ^{\frac{1}{2}}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{24 \cdot 2 ^{\frac{1}{2}}} \cdot \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{1}{24 \cdot 2 ^{\frac{1}{2}}} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = (8 x)^{\frac{3}{2}} + 7

= \frac{1}{24 \cdot 2 ^{\frac{1}{2}}} \cdot \frac{2}{3} ((8 x)^{\frac{3}{2}} + 7)^{\frac{3}{2}}

Simplify

\frac{1}{24 \cdot 2 ^{\frac{1}{2}}} \cdot \frac{2}{3} ((8 x)^{\frac{3}{2}} + 7)^{\frac{3}{2}} : \frac{1}{36 2} (162 x^{\frac{3}{2}} + 7)^{\frac{3}{2}}

= \frac{1}{36 2} (162 x^{\frac{3}{2}} + 7)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{36 2} (162 x^{\frac{3}{2}} + 7)^{\frac{3}{2}} + C

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What is the integral of sqrt(x)sqrt((8x)^{3/2)+7} ?
The integral of sqrt(x)sqrt((8x)^{3/2)+7} is 1/(36sqrt(2))(16sqrt(2)x^{3/2}+7)^{3/2}+C