What is the integral of (2x+5)(x^2+5x+8)^6 ?

The integral of (2x+5)(x^2+5x+8)^6 is

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integral of (2x+5)(x^2+5x+8)^6

\int (2 x + 5) (x^{2} + 5 x + 8)^{6} d x

\frac{x ^{14}}{7} + 5 x^{13} + 83 x^{12} + 865 x^{11} + 6317 x^{10} + 34175 x^{9} + 141185 x^{8} + \frac{3166525 x ^{7}}{7} + 1129480 x^{6} + 2187200 x^{5} + 3234304 x^{4} + 3543040 x^{3} + 2719744 x^{2} + 1310720 x + C

Solution steps

\int (2 x + 5) (x^{2} + 5 x + 8)^{6} d x

Expand

(2 x + 5) (x^{2} + 5 x + 8)^{6} : 2 x^{13} + 65 x^{12} + 996 x^{11} + 9515 x^{10} + 63170 x^{9} + 307575 x^{8} + 1129480 x^{7} + 3166525 x^{6} + 6776880 x^{5} + 10936000 x^{4} + 12937216 x^{3} + 10629120 x^{2} + 5439488 x + 1310720

= \int 2 x^{13} + 65 x^{12} + 996 x^{11} + 9515 x^{10} + 63170 x^{9} + 307575 x^{8} + 1129480 x^{7} + 3166525 x^{6} + 6776880 x^{5} + 10936000 x^{4} + 12937216 x^{3} + 10629120 x^{2} + 5439488 x + 1310720 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^{13} d x + \int 65 x^{12} d x + \int 996 x^{11} d x + \int 9515 x^{10} d x + \int 63170 x^{9} d x + \int 307575 x^{8} d x + \int 1129480 x^{7} d x + \int 3166525 x^{6} d x + \int 6776880 x^{5} d x + \int 10936000 x^{4} d x + \int 12937216 x^{3} d x + \int 10629120 x^{2} d x + \int 5439488 x d x + \int 1310720 d x

\int 2 x^{13} d x = \frac{x ^{14}}{7}

\int 65 x^{12} d x = 5 x^{13}

\int 996 x^{11} d x = 83 x^{12}

\int 9515 x^{10} d x = 865 x^{11}

\int 63170 x^{9} d x = 6317 x^{10}

\int 307575 x^{8} d x = 34175 x^{9}

\int 1129480 x^{7} d x = 141185 x^{8}

\int 3166525 x^{6} d x = \frac{3166525 x ^{7}}{7}

\int 6776880 x^{5} d x = 1129480 x^{6}

\int 10936000 x^{4} d x = 2187200 x^{5}

\int 12937216 x^{3} d x = 3234304 x^{4}

\int 10629120 x^{2} d x = 3543040 x^{3}

\int 5439488 x d x = 2719744 x^{2}

\int 1310720 d x = 1310720 x

= \frac{x ^{14}}{7} + 5 x^{13} + 83 x^{12} + 865 x^{11} + 6317 x^{10} + 34175 x^{9} + 141185 x^{8} + \frac{3166525 x ^{7}}{7} + 1129480 x^{6} + 2187200 x^{5} + 3234304 x^{4} + 3543040 x^{3} + 2719744 x^{2} + 1310720 x

Add a constant to the solution

= \frac{x ^{14}}{7} + 5 x^{13} + 83 x^{12} + 865 x^{11} + 6317 x^{10} + 34175 x^{9} + 141185 x^{8} + \frac{3166525 x ^{7}}{7} + 1129480 x^{6} + 2187200 x^{5} + 3234304 x^{4} + 3543040 x^{3} + 2719744 x^{2} + 1310720 x + C

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What is the integral of (2x+5)(x^2+5x+8)^6 ?
The integral of (2x+5)(x^2+5x+8)^6 is