What is the integral of (x^2+x+8)sqrt(x+5) ?

The integral of (x^2+x+8)sqrt(x+5) is 2/7 (x+5)^{7/2}-18/5 (x+5)^{5/2}+56/3 (x+5)^{3/2}+C

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integral of (x^2+x+8)sqrt(x+5)

\int (x^{2} + x + 8) x + 5 d x

\frac{2}{7} (x + 5)^{\frac{7}{2}} - \frac{18}{5} (x + 5)^{\frac{5}{2}} + \frac{56}{3} (x + 5)^{\frac{3}{2}} + C

Solution steps

\int (x^{2} + x + 8) x + 5 d x

Expand

(x^{2} + x + 8) x + 5 : x^{2} x + 5 + x x + 5 + 8 x + 5

= \int x^{2} x + 5 + x x + 5 + 8 x + 5 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^{2} x + 5 d x + \int x x + 5 d x + \int 8 x + 5 d x

\int x^{2} x + 5 d x = \frac{2}{7} (x + 5)^{\frac{7}{2}} - 4 (x + 5)^{\frac{5}{2}} + \frac{50}{3} (x + 5)^{\frac{3}{2}}

\int x x + 5 d x = \frac{2}{5} (x + 5)^{\frac{5}{2}} - \frac{10}{3} (x + 5)^{\frac{3}{2}}

\int 8 x + 5 d x = \frac{16}{3} (x + 5)^{\frac{3}{2}}

= \frac{2}{7} (x + 5)^{\frac{7}{2}} - 4 (x + 5)^{\frac{5}{2}} + \frac{50}{3} (x + 5)^{\frac{3}{2}} + \frac{2}{5} (x + 5)^{\frac{5}{2}} - \frac{10}{3} (x + 5)^{\frac{3}{2}} + \frac{16}{3} (x + 5)^{\frac{3}{2}}

Simplify

\frac{2}{7} (x + 5)^{\frac{7}{2}} - 4 (x + 5)^{\frac{5}{2}} + \frac{50}{3} (x + 5)^{\frac{3}{2}} + \frac{2}{5} (x + 5)^{\frac{5}{2}} - \frac{10}{3} (x + 5)^{\frac{3}{2}} + \frac{16}{3} (x + 5)^{\frac{3}{2}} : \frac{2}{7} (x + 5)^{\frac{7}{2}} - \frac{18}{5} (x + 5)^{\frac{5}{2}} + \frac{56}{3} (x + 5)^{\frac{3}{2}}

= \frac{2}{7} (x + 5)^{\frac{7}{2}} - \frac{18}{5} (x + 5)^{\frac{5}{2}} + \frac{56}{3} (x + 5)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{7} (x + 5)^{\frac{7}{2}} - \frac{18}{5} (x + 5)^{\frac{5}{2}} + \frac{56}{3} (x + 5)^{\frac{3}{2}} + C

What is the integral of (x^2+x+8)sqrt(x+5) ?
The integral of (x^2+x+8)sqrt(x+5) is 2/7 (x+5)^{7/2}-18/5 (x+5)^{5/2}+56/3 (x+5)^{3/2}+C