What is the integral of (2x+3)sqrt(x^2+3x+5) ?

The integral of (2x+3)sqrt(x^2+3x+5) is 2/3 (x^2+3x+5)^{3/2}+9/4 sqrt(x^2+3x+5)-9/4 sqrt(x^2+3x+5)+C

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integral of (2x+3)sqrt(x^2+3x+5)

\int (2 x + 3) x^{2} + 3 x + 5 d x

\frac{2}{3} (x^{2} + 3 x + 5)^{\frac{3}{2}} + \frac{9}{4} x^{2} + 3 x + 5 - \frac{9}{4} x^{2} + 3 x + 5 + C

Solution steps

\int (2 x + 3) x^{2} + 3 x + 5 d x

Expand

(2 x + 3) x^{2} + 3 x + 5 : 2 x x^{2} + 3 x + 5 + 3 x^{2} + 3 x + 5

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

\int 2 x x^{2} + 3 x + 5 d x = \frac{1}{24} (16 (x^{2} + 3 x + 5)^{\frac{3}{2}} - 9 (4 (x + \frac{3}{2}) x^{2} + 3 x + 5 + 11 (ln 11 \frac{4 x ^{2} + 12 x + 20}{11} + 2 x + 3 - \frac{ln ( 11 )}{2})))

\int 3 x^{2} + 3 x + 5 d x = \frac{3}{8} (2 (2 x + 3) x^{2} + 3 x + 5 + 11 (ln 2 x + 11 \frac{4 x ^{2} + 12 x + 20}{11} + 3 - \frac{ln ( 11 )}{2}))

= \frac{1}{24} (16 (x^{2} + 3 x + 5)^{\frac{3}{2}} - 9 (4 (x + \frac{3}{2}) x^{2} + 3 x + 5 + 11 (ln 11 \frac{4 x ^{2} + 12 x + 20}{11} + 2 x + 3 - \frac{ln ( 11 )}{2}))) + \frac{3}{8} (2 (2 x + 3) x^{2} + 3 x + 5 + 11 (ln 2 x + 11 \frac{4 x ^{2} + 12 x + 20}{11} + 3 - \frac{ln ( 11 )}{2}))

Simplify

\frac{1}{24} (16 (x^{2} + 3 x + 5)^{\frac{3}{2}} - 9 (4 (x + \frac{3}{2}) x^{2} + 3 x + 5 + 11 (ln 11 \frac{4 x ^{2} + 12 x + 20}{11} + 2 x + 3 - \frac{ln ( 11 )}{2}))) + \frac{3}{8} (2 (2 x + 3) x^{2} + 3 x + 5 + 11 (ln 2 x + 11 \frac{4 x ^{2} + 12 x + 20}{11} + 3 - \frac{ln ( 11 )}{2})) : \frac{2}{3} (x^{2} + 3 x + 5)^{\frac{3}{2}} + \frac{9}{4} x^{2} + 3 x + 5 - \frac{9}{4} x^{2} + 3 x + 5

= \frac{2}{3} (x^{2} + 3 x + 5)^{\frac{3}{2}} + \frac{9}{4} x^{2} + 3 x + 5 - \frac{9}{4} x^{2} + 3 x + 5

Add a constant to the solution

= \frac{2}{3} (x^{2} + 3 x + 5)^{\frac{3}{2}} + \frac{9}{4} x^{2} + 3 x + 5 - \frac{9}{4} x^{2} + 3 x + 5 + C

What is the integral of (2x+3)sqrt(x^2+3x+5) ?
The integral of (2x+3)sqrt(x^2+3x+5) is 2/3 (x^2+3x+5)^{3/2}+9/4 sqrt(x^2+3x+5)-9/4 sqrt(x^2+3x+5)+C