What is the integral of (3x^2+6x-2)(x^3+3x^2-2x+5)^4 ?

The integral of (3x^2+6x-2)(x^3+3x^2-2x+5)^4 is

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integral of (3x^2+6x-2)(x^3+3x^2-2x+5)^4

\int (3 x^{2} + 6 x - 2) (x^{3} + 3 x^{2} - 2 x + 5)^{4} d x

\frac{x ^{15}}{5} + 3 x^{14} + 16 x^{13} + 35 x^{12} + 41 x^{11} + \frac{673 x ^{10}}{5} + 268 x^{9} + 15 x^{8} + 562 x^{7} + 768 x^{6} - \frac{5432 x ^{5}}{5} + 3130 x^{4} - 2775 x^{3} + 2875 x^{2} - 1250 x + C

Solution steps

\int (3 x^{2} + 6 x - 2) (x^{3} + 3 x^{2} - 2 x + 5)^{4} d x

Expand

(3 x^{2} + 6 x - 2) (x^{3} + 3 x^{2} - 2 x + 5)^{4} : 3 x^{14} + 42 x^{13} + 208 x^{12} + 420 x^{11} + 451 x^{10} + 1346 x^{9} + 2412 x^{8} + 120 x^{7} + 3934 x^{6} + 4608 x^{5} - 5432 x^{4} + 12520 x^{3} - 8325 x^{2} + 5750 x - 1250

= \int 3 x^{14} + 42 x^{13} + 208 x^{12} + 420 x^{11} + 451 x^{10} + 1346 x^{9} + 2412 x^{8} + 120 x^{7} + 3934 x^{6} + 4608 x^{5} - 5432 x^{4} + 12520 x^{3} - 8325 x^{2} + 5750 x - 1250 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 3 x^{14} d x + \int 42 x^{13} d x + \int 208 x^{12} d x + \int 420 x^{11} d x + \int 451 x^{10} d x + \int 1346 x^{9} d x + \int 2412 x^{8} d x + \int 120 x^{7} d x + \int 3934 x^{6} d x + \int 4608 x^{5} d x - \int 5432 x^{4} d x + \int 12520 x^{3} d x - \int 8325 x^{2} d x + \int 5750 x d x - \int 1250 d x

\int 3 x^{14} d x = \frac{x ^{15}}{5}

\int 42 x^{13} d x = 3 x^{14}

\int 208 x^{12} d x = 16 x^{13}

\int 420 x^{11} d x = 35 x^{12}

\int 451 x^{10} d x = 41 x^{11}

\int 1346 x^{9} d x = \frac{673 x ^{10}}{5}

\int 2412 x^{8} d x = 268 x^{9}

\int 120 x^{7} d x = 15 x^{8}

\int 3934 x^{6} d x = 562 x^{7}

\int 4608 x^{5} d x = 768 x^{6}

\int 5432 x^{4} d x = \frac{5432 x ^{5}}{5}

\int 12520 x^{3} d x = 3130 x^{4}

\int 8325 x^{2} d x = 2775 x^{3}

\int 5750 x d x = 2875 x^{2}

\int 1250 d x = 1250 x

= \frac{x ^{15}}{5} + 3 x^{14} + 16 x^{13} + 35 x^{12} + 41 x^{11} + \frac{673 x ^{10}}{5} + 268 x^{9} + 15 x^{8} + 562 x^{7} + 768 x^{6} - \frac{5432 x ^{5}}{5} + 3130 x^{4} - 2775 x^{3} + 2875 x^{2} - 1250 x

Add a constant to the solution

= \frac{x ^{15}}{5} + 3 x^{14} + 16 x^{13} + 35 x^{12} + 41 x^{11} + \frac{673 x ^{10}}{5} + 268 x^{9} + 15 x^{8} + 562 x^{7} + 768 x^{6} - \frac{5432 x ^{5}}{5} + 3130 x^{4} - 2775 x^{3} + 2875 x^{2} - 1250 x + C

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What is the integral of (3x^2+6x-2)(x^3+3x^2-2x+5)^4 ?
The integral of (3x^2+6x-2)(x^3+3x^2-2x+5)^4 is