Integrand size = 24, antiderivative size = 22 \[ \int x^{2 n} (c+d x)^n \left (2 c x+3 d x^2\right ) \, dx=\frac {x^{2 (1+n)} (c+d x)^{1+n}}{1+n} \]
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Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {859} \[ \int x^{2 n} (c+d x)^n \left (2 c x+3 d x^2\right ) \, dx=\frac {x^{2 (n+1)} (c+d x)^{n+1}}{n+1} \]
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Rule 859
Rubi steps \begin{align*} \text {integral}& = \frac {x^{2 (1+n)} (c+d x)^{1+n}}{1+n} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int x^{2 n} (c+d x)^n \left (2 c x+3 d x^2\right ) \, dx=\frac {x^{2+2 n} (c+d x)^{1+n}}{1+n} \]
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Time = 1.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05
method | result | size |
gosper | \(\frac {x^{2+2 n} \left (d x +c \right )^{1+n}}{1+n}\) | \(23\) |
risch | \(\frac {\left (d x +c \right )^{n} x^{2 n} x^{2} \left (d x +c \right )}{1+n}\) | \(27\) |
parallelrisch | \(\frac {x^{3} x^{2 n} \left (d x +c \right )^{n} c d +x^{2} x^{2 n} \left (d x +c \right )^{n} c^{2}}{c \left (1+n \right )}\) | \(48\) |
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none
Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int x^{2 n} (c+d x)^n \left (2 c x+3 d x^2\right ) \, dx=\frac {{\left (d x^{3} + c x^{2}\right )} {\left (d x + c\right )}^{n} x^{2 \, n}}{n + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (17) = 34\).
Time = 1.53 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.41 \[ \int x^{2 n} (c+d x)^n \left (2 c x+3 d x^2\right ) \, dx=\begin {cases} \frac {c x^{2} x^{2 n} \left (c + d x\right )^{n}}{n + 1} + \frac {d x^{3} x^{2 n} \left (c + d x\right )^{n}}{n + 1} & \text {for}\: n \neq -1 \\2 \log {\left (x \right )} + \log {\left (\frac {c}{d} + x \right )} & \text {otherwise} \end {cases} \]
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none
Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int x^{2 n} (c+d x)^n \left (2 c x+3 d x^2\right ) \, dx=\frac {{\left (d x^{3} + c x^{2}\right )} e^{\left (n \log \left (d x + c\right ) + 2 \, n \log \left (x\right )\right )}}{n + 1} \]
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Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86 \[ \int x^{2 n} (c+d x)^n \left (2 c x+3 d x^2\right ) \, dx=\frac {{\left (d x + c\right )}^{n} d x^{3} x^{2 \, n} + {\left (d x + c\right )}^{n} c x^{2} x^{2 \, n}}{n + 1} \]
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Time = 9.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int x^{2 n} (c+d x)^n \left (2 c x+3 d x^2\right ) \, dx=\frac {x^{2\,n}\,x^2\,{\left (c+d\,x\right )}^n\,\left (c+d\,x\right )}{n+1} \]
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