What is the integral of (x^2+3x-7)^5(2x+3) ?

The integral of (x^2+3x-7)^5(2x+3) is

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integral of (x^2+3x-7)^5(2x+3)

\int (x^{2} + 3 x - 7)^{5} (2 x + 3) d x

\frac{x ^{12}}{6} + 3 x^{11} + \frac{31 x ^{10}}{2} - 15 x^{9} - 305 x^{8} - 177 x^{7} + \frac{16549 x ^{6}}{6} + 1239 x^{5} - 14945 x^{4} + 5145 x^{3} + \frac{74431 x ^{2}}{2} - 50421 x + C

Solution steps

\int (x^{2} + 3 x - 7)^{5} (2 x + 3) d x

Expand

(x^{2} + 3 x - 7)^{5} (2 x + 3) : 2 x^{11} + 33 x^{10} + 155 x^{9} - 135 x^{8} - 2440 x^{7} - 1239 x^{6} + 16549 x^{5} + 6195 x^{4} - 59780 x^{3} + 15435 x^{2} + 74431 x - 50421

= \int 2 x^{11} + 33 x^{10} + 155 x^{9} - 135 x^{8} - 2440 x^{7} - 1239 x^{6} + 16549 x^{5} + 6195 x^{4} - 59780 x^{3} + 15435 x^{2} + 74431 x - 50421 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 33 x^{10} d x + \int 155 x^{9} d x - \int 135 x^{8} d x - \int 2440 x^{7} d x - \int 1239 x^{6} d x + \int 16549 x^{5} d x + \int 6195 x^{4} d x - \int 59780 x^{3} d x + \int 15435 x^{2} d x + \int 74431 x d x - \int 50421 d x

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

\int 33 x^{10} d x = 3 x^{11}

\int 155 x^{9} d x = \frac{31 x ^{10}}{2}

\int 135 x^{8} d x = 15 x^{9}

\int 2440 x^{7} d x = 305 x^{8}

\int 1239 x^{6} d x = 177 x^{7}

\int 16549 x^{5} d x = \frac{16549 x ^{6}}{6}

\int 6195 x^{4} d x = 1239 x^{5}

\int 59780 x^{3} d x = 14945 x^{4}

\int 15435 x^{2} d x = 5145 x^{3}

\int 74431 x d x = \frac{74431 x ^{2}}{2}

\int 50421 d x = 50421 x

= \frac{x ^{12}}{6} + 3 x^{11} + \frac{31 x ^{10}}{2} - 15 x^{9} - 305 x^{8} - 177 x^{7} + \frac{16549 x ^{6}}{6} + 1239 x^{5} - 14945 x^{4} + 5145 x^{3} + \frac{74431 x ^{2}}{2} - 50421 x

Add a constant to the solution

= \frac{x ^{12}}{6} + 3 x^{11} + \frac{31 x ^{10}}{2} - 15 x^{9} - 305 x^{8} - 177 x^{7} + \frac{16549 x ^{6}}{6} + 1239 x^{5} - 14945 x^{4} + 5145 x^{3} + \frac{74431 x ^{2}}{2} - 50421 x + C

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What is the integral of (x^2+3x-7)^5(2x+3) ?
The integral of (x^2+3x-7)^5(2x+3) is