Integrand size = 28, antiderivative size = 25 \[ \int x^n \left (b+c x+d x^2\right )^n \left (b+2 c x+3 d x^2\right ) \, dx=\frac {x^{1+n} \left (b+c x+d x^2\right )^{1+n}}{1+n} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {1604} \[ \int x^n \left (b+c x+d x^2\right )^n \left (b+2 c x+3 d x^2\right ) \, dx=\frac {x^{n+1} \left (b+c x+d x^2\right )^{n+1}}{n+1} \]
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Rule 1604
Rubi steps \begin{align*} \text {integral}& = \frac {x^{1+n} \left (b+c x+d x^2\right )^{1+n}}{1+n} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int x^n \left (b+c x+d x^2\right )^n \left (b+2 c x+3 d x^2\right ) \, dx=\frac {x^{1+n} (b+x (c+d x))^{1+n}}{1+n} \]
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Time = 1.87 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04
method | result | size |
gosper | \(\frac {x^{1+n} \left (d \,x^{2}+c x +b \right )^{1+n}}{1+n}\) | \(26\) |
risch | \(\frac {x \left (d \,x^{2}+c x +b \right ) x^{n} \left (d \,x^{2}+c x +b \right )^{n}}{1+n}\) | \(33\) |
parallelrisch | \(\frac {x^{3} x^{n} \left (d \,x^{2}+c x +b \right )^{n} d^{2}+x^{2} x^{n} \left (d \,x^{2}+c x +b \right )^{n} c d +x \,x^{n} \left (d \,x^{2}+c x +b \right )^{n} b d}{d \left (1+n \right )}\) | \(73\) |
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none
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int x^n \left (b+c x+d x^2\right )^n \left (b+2 c x+3 d x^2\right ) \, dx=\frac {{\left (d x^{3} + c x^{2} + b x\right )} {\left (d x^{2} + c x + b\right )}^{n} x^{n}}{n + 1} \]
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Timed out. \[ \int x^n \left (b+c x+d x^2\right )^n \left (b+2 c x+3 d x^2\right ) \, dx=\text {Timed out} \]
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none
Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int x^n \left (b+c x+d x^2\right )^n \left (b+2 c x+3 d x^2\right ) \, dx=\frac {{\left (d x^{3} + c x^{2} + b x\right )} e^{\left (n \log \left (d x^{2} + c x + b\right ) + n \log \left (x\right )\right )}}{n + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (25) = 50\).
Time = 0.30 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.60 \[ \int x^n \left (b+c x+d x^2\right )^n \left (b+2 c x+3 d x^2\right ) \, dx=\frac {{\left (d x^{2} + c x + b\right )}^{n} d x^{3} x^{n} + {\left (d x^{2} + c x + b\right )}^{n} c x^{2} x^{n} + {\left (d x^{2} + c x + b\right )}^{n} b x x^{n}}{n + 1} \]
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Time = 9.11 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04 \[ \int x^n \left (b+c x+d x^2\right )^n \left (b+2 c x+3 d x^2\right ) \, dx=\left (\frac {c\,x^n\,x^2}{n+1}+\frac {d\,x^n\,x^3}{n+1}+\frac {b\,x\,x^n}{n+1}\right )\,{\left (d\,x^2+c\,x+b\right )}^n \]
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