Optimal. Leaf size=26 \[ \frac {(c x)^{1+m} \left (b x^n\right )^p}{c (1+m+n p)} \]
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Rubi [A]
time = 0.00, antiderivative size = 22, normalized size of antiderivative = 0.85, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {15, 20, 30}
\begin {gather*} \frac {x (c x)^m \left (b x^n\right )^p}{m+n p+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 20
Rule 30
Rubi steps
\begin {align*} \int (c x)^m \left (b x^n\right )^p \, dx &=\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*}
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Mathematica [A]
time = 0.00, size = 22, normalized size = 0.85 \begin {gather*} \frac {x (c x)^m \left (b x^n\right )^p}{1+m+n p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 23, normalized size = 0.88
method | result | size |
gosper | \(\frac {x \left (c x \right )^{m} \left (b \,x^{n}\right )^{p}}{n p +m +1}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 28, normalized size = 1.08 \begin {gather*} \frac {b^{p} c^{m} x e^{\left (m \log \left (x\right ) + p \log \left (x^{n}\right )\right )}}{n p + m + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 29, normalized size = 1.12 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \\\int \left (b x^{n}\right )^{p} \left (c x\right )^{- n p - 1}\, dx & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.72, size = 29, normalized size = 1.12 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.01, size = 22, normalized size = 0.85 \begin {gather*} \frac {x\,{\left (b\,x^n\right )}^p\,{\left (c\,x\right )}^m}{m+n\,p+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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