Integrand size = 18, antiderivative size = 19 \[ \int (b+2 c x) \left (b x+c x^2\right )^p \, dx=\frac {\left (b x+c x^2\right )^{1+p}}{1+p} \]
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Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {643} \[ \int (b+2 c x) \left (b x+c x^2\right )^p \, dx=\frac {\left (b x+c x^2\right )^{p+1}}{p+1} \]
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Rule 643
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b x+c x^2\right )^{1+p}}{1+p} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int (b+2 c x) \left (b x+c x^2\right )^p \, dx=\frac {(x (b+c x))^{1+p}}{1+p} \]
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Time = 0.67 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(\frac {\left (c \,x^{2}+b x \right )^{1+p}}{1+p}\) | \(20\) |
default | \(\frac {\left (c \,x^{2}+b x \right )^{1+p}}{1+p}\) | \(20\) |
risch | \(\frac {x \left (c x +b \right ) \left (x \left (c x +b \right )\right )^{p}}{1+p}\) | \(22\) |
gosper | \(\frac {x \left (c x +b \right ) \left (c \,x^{2}+b x \right )^{p}}{1+p}\) | \(24\) |
parallelrisch | \(\frac {x^{2} \left (x \left (c x +b \right )\right )^{p} b c +x \left (x \left (c x +b \right )\right )^{p} b^{2}}{b \left (1+p \right )}\) | \(40\) |
norman | \(\frac {b x \,{\mathrm e}^{p \ln \left (c \,x^{2}+b x \right )}}{1+p}+\frac {c \,x^{2} {\mathrm e}^{p \ln \left (c \,x^{2}+b x \right )}}{1+p}\) | \(46\) |
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none
Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37 \[ \int (b+2 c x) \left (b x+c x^2\right )^p \, dx=\frac {{\left (c x^{2} + b x\right )} {\left (c x^{2} + b x\right )}^{p}}{p + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (14) = 28\).
Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.42 \[ \int (b+2 c x) \left (b x+c x^2\right )^p \, dx=\begin {cases} \frac {b x \left (b x + c x^{2}\right )^{p}}{p + 1} + \frac {c x^{2} \left (b x + c x^{2}\right )^{p}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (x \right )} + \log {\left (\frac {b}{c} + x \right )} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int (b+2 c x) \left (b x+c x^2\right )^p \, dx=\frac {{\left (c x^{2} + b x\right )}^{p + 1}}{p + 1} \]
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Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int (b+2 c x) \left (b x+c x^2\right )^p \, dx=\frac {{\left (c x^{2} + b x\right )}^{p + 1}}{p + 1} \]
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Time = 8.56 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int (b+2 c x) \left (b x+c x^2\right )^p \, dx=\frac {x\,{\left (c\,x^2+b\,x\right )}^p\,\left (b+c\,x\right )}{p+1} \]
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