What is the integral of x^2sqrt(8x+9) ?

The integral of x^2sqrt(8x+9) is 1/1120 (40x^2-36x+27)(8x+9)^{3/2}+C

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integral of x^2sqrt(8x+9)

\int x^{2} 8 x + 9 d x

\frac{1}{1120} (40 x^{2} - 36 x + 27) (8 x + 9)^{\frac{3}{2}} + C

Solution steps

\int x^{2} 8 x + 9 d x

Apply u-substitution

: \int \frac{u ( u - 9 ) ^{2}}{512} d u

= \int \frac{u ( u - 9 ) ^{2}}{512} d u

Take the constant out:

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

= \frac{1}{512} \cdot \int u (u - 9)^{2} d u

Expand

u (u - 9)^{2} : u^{\frac{5}{2}} - 18 u^{\frac{3}{2}} + 81 u

= \frac{1}{512} \cdot \int u^{\frac{5}{2}} - 18 u^{\frac{3}{2}} + 81 u d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{512} (\int u^{\frac{5}{2}} d u - \int 18 u^{\frac{3}{2}} d u + \int 81 u d u)

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

\int 18 u^{\frac{3}{2}} d u = \frac{36}{5} u^{\frac{5}{2}}

\int 81 u d u = 54 u^{\frac{3}{2}}

= \frac{1}{512} (\frac{2}{7} u^{\frac{7}{2}} - \frac{36}{5} u^{\frac{5}{2}} + 54 u^{\frac{3}{2}})

Substitute back

u = 8 x + 9

= \frac{1}{512} (\frac{2}{7} (8 x + 9)^{\frac{7}{2}} - \frac{36}{5} (8 x + 9)^{\frac{5}{2}} + 54 (8 x + 9)^{\frac{3}{2}})

Simplify

\frac{1}{512} (\frac{2}{7} (8 x + 9)^{\frac{7}{2}} - \frac{36}{5} (8 x + 9)^{\frac{5}{2}} + 54 (8 x + 9)^{\frac{3}{2}}) : \frac{1}{1120} (40 x^{2} - 36 x + 27) (8 x + 9)^{\frac{3}{2}}

= \frac{1}{1120} (40 x^{2} - 36 x + 27) (8 x + 9)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{1120} (40 x^{2} - 36 x + 27) (8 x + 9)^{\frac{3}{2}} + C

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What is the integral of x^2sqrt(8x+9) ?
The integral of x^2sqrt(8x+9) is 1/1120 (40x^2-36x+27)(8x+9)^{3/2}+C