Integrand size = 13, antiderivative size = 26 \[ \int (c x)^m \left (b x^n\right )^p \, dx=\frac {(c x)^{1+m} \left (b x^n\right )^p}{c (1+m+n p)} \]
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Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {15, 20, 30} \[ \int (c x)^m \left (b x^n\right )^p \, dx=\frac {x (c x)^m \left (b x^n\right )^p}{m+n p+1} \]
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Rule 15
Rule 20
Rule 30
Rubi steps \begin{align*} \text {integral}& = \left (x^{-n p} \left (b x^n\right )^p\right ) \int x^{n p} (c x)^m \, dx \\ & = \left (x^{-m-n p} (c x)^m \left (b x^n\right )^p\right ) \int x^{m+n p} \, dx \\ & = \frac {x (c x)^m \left (b x^n\right )^p}{1+m+n p} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int (c x)^m \left (b x^n\right )^p \, dx=\frac {x (c x)^m \left (b x^n\right )^p}{1+m+n p} \]
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Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
gosper | \(\frac {x \left (c x \right )^{m} \left (b \,x^{n}\right )^{p}}{n p +m +1}\) | \(23\) |
parallelrisch | \(\frac {x \left (c x \right )^{m} \left (b \,x^{n}\right )^{p}}{n p +m +1}\) | \(23\) |
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Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int (c x)^m \left (b x^n\right )^p \, dx=\frac {x e^{\left (n p \log \left (x\right ) + p \log \left (b\right ) + m \log \left (c\right ) + m \log \left (x\right )\right )}}{n p + m + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (20) = 40\).
Time = 0.98 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77 \[ \int (c x)^m \left (b x^n\right )^p \, dx=\begin {cases} \frac {x \left (b x^{n}\right )^{p} \left (c x\right )^{m}}{m + n p + 1} & \text {for}\: m \neq - n p - 1 \\x \left (b x^{n}\right )^{p} \left (c x\right )^{- n p - 1} \log {\left (x \right )} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int (c x)^m \left (b x^n\right )^p \, dx=\frac {b^{p} c^{m} x e^{\left (m \log \left (x\right ) + p \log \left (x^{n}\right )\right )}}{n p + m + 1} \]
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Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int (c x)^m \left (b x^n\right )^p \, dx=\frac {x e^{\left (n p \log \left (x\right ) + p \log \left (b\right ) + m \log \left (c\right ) + m \log \left (x\right )\right )}}{n p + m + 1} \]
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Time = 5.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int (c x)^m \left (b x^n\right )^p \, dx=\frac {x\,{\left (b\,x^n\right )}^p\,{\left (c\,x\right )}^m}{m+n\,p+1} \]
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