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

The integral of (2x+5)(x^2+5x+3)^5 is

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

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

\frac{x ^{12}}{6} + 5 x^{11} + \frac{131 x ^{10}}{2} + \frac{1475 x ^{9}}{3} + 2335 x^{8} + 7325 x^{7} + \frac{92665 x ^{6}}{6} + 21975 x^{5} + 21015 x^{4} + 13275 x^{3} + \frac{10611 x ^{2}}{2} + 1215 x + C

Solution steps

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

Expand

(2 x + 5) (x^{2} + 5 x + 3)^{5} : 2 x^{11} + 55 x^{10} + 655 x^{9} + 4425 x^{8} + 18680 x^{7} + 51275 x^{6} + 92665 x^{5} + 109875 x^{4} + 84060 x^{3} + 39825 x^{2} + 10611 x + 1215

= \int 2 x^{11} + 55 x^{10} + 655 x^{9} + 4425 x^{8} + 18680 x^{7} + 51275 x^{6} + 92665 x^{5} + 109875 x^{4} + 84060 x^{3} + 39825 x^{2} + 10611 x + 1215 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^{11} d x + \int 55 x^{10} d x + \int 655 x^{9} d x + \int 4425 x^{8} d x + \int 18680 x^{7} d x + \int 51275 x^{6} d x + \int 92665 x^{5} d x + \int 109875 x^{4} d x + \int 84060 x^{3} d x + \int 39825 x^{2} d x + \int 10611 x d x + \int 1215 d x

\int 2 x^{11} d x = \frac{x ^{12}}{6}

\int 55 x^{10} d x = 5 x^{11}

\int 655 x^{9} d x = \frac{131 x ^{10}}{2}

\int 4425 x^{8} d x = \frac{1475 x ^{9}}{3}

\int 18680 x^{7} d x = 2335 x^{8}

\int 51275 x^{6} d x = 7325 x^{7}

\int 92665 x^{5} d x = \frac{92665 x ^{6}}{6}

\int 109875 x^{4} d x = 21975 x^{5}

\int 84060 x^{3} d x = 21015 x^{4}

\int 39825 x^{2} d x = 13275 x^{3}

\int 10611 x d x = \frac{10611 x ^{2}}{2}

\int 1215 d x = 1215 x

= \frac{x ^{12}}{6} + 5 x^{11} + \frac{131 x ^{10}}{2} + \frac{1475 x ^{9}}{3} + 2335 x^{8} + 7325 x^{7} + \frac{92665 x ^{6}}{6} + 21975 x^{5} + 21015 x^{4} + 13275 x^{3} + \frac{10611 x ^{2}}{2} + 1215 x

Add a constant to the solution

= \frac{x ^{12}}{6} + 5 x^{11} + \frac{131 x ^{10}}{2} + \frac{1475 x ^{9}}{3} + 2335 x^{8} + 7325 x^{7} + \frac{92665 x ^{6}}{6} + 21975 x^{5} + 21015 x^{4} + 13275 x^{3} + \frac{10611 x ^{2}}{2} + 1215 x + C

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