What is the integral of (x^2+3x-1)^3(2x+3) ?

The integral of (x^2+3x-1)^3(2x+3) is (x^8)/4+3x^7+(25x^6)/2+18x^5-(21x^4)/4-18x^3+(25x^2)/2-3x+C

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integral of (x^2+3x-1)^3(2x+3)

\int (x^{2} + 3 x - 1)^{3} (2 x + 3) d x

\frac{x ^{8}}{4} + 3 x^{7} + \frac{25 x ^{6}}{2} + 18 x^{5} - \frac{21 x ^{4}}{4} - 18 x^{3} + \frac{25 x ^{2}}{2} - 3 x + C

Solution steps

\int (x^{2} + 3 x - 1)^{3} (2 x + 3) d x

Expand

(x^{2} + 3 x - 1)^{3} (2 x + 3) : 2 x^{7} + 21 x^{6} + 75 x^{5} + 90 x^{4} - 21 x^{3} - 54 x^{2} + 25 x - 3

= \int 2 x^{7} + 21 x^{6} + 75 x^{5} + 90 x^{4} - 21 x^{3} - 54 x^{2} + 25 x - 3 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^{7} d x + \int 21 x^{6} d x + \int 75 x^{5} d x + \int 90 x^{4} d x - \int 21 x^{3} d x - \int 54 x^{2} d x + \int 25 x d x - \int 3 d x

\int 2 x^{7} d x = \frac{x ^{8}}{4}

\int 21 x^{6} d x = 3 x^{7}

\int 75 x^{5} d x = \frac{25 x ^{6}}{2}

\int 90 x^{4} d x = 18 x^{5}

\int 21 x^{3} d x = \frac{21 x ^{4}}{4}

\int 54 x^{2} d x = 18 x^{3}

\int 25 x d x = \frac{25 x ^{2}}{2}

\int 3 d x = 3 x

= \frac{x ^{8}}{4} + 3 x^{7} + \frac{25 x ^{6}}{2} + 18 x^{5} - \frac{21 x ^{4}}{4} - 18 x^{3} + \frac{25 x ^{2}}{2} - 3 x

Add a constant to the solution

= \frac{x ^{8}}{4} + 3 x^{7} + \frac{25 x ^{6}}{2} + 18 x^{5} - \frac{21 x ^{4}}{4} - 18 x^{3} + \frac{25 x ^{2}}{2} - 3 x + C

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\int \frac{1}{x ^{3} ( x ^{2} + x + 1 ) ^{2}} d x

What is the integral of (x^2+3x-1)^3(2x+3) ?
The integral of (x^2+3x-1)^3(2x+3) is (x^8)/4+3x^7+(25x^6)/2+18x^5-(21x^4)/4-18x^3+(25x^2)/2-3x+C