Optimal. Leaf size=21 \[ \frac {x^{1+m} \left (b x^n\right )^p}{1+m+n p} \]
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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {15, 30}
\begin {gather*} \frac {x^{m+1} \left (b x^n\right )^p}{m+n p+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin {align*} \int x^m \left (b x^n\right )^p \, dx &=\left (x^{-n p} \left (b x^n\right )^p\right ) \int x^{m+n p} \, dx\\ &=\frac {x^{1+m} \left (b x^n\right )^p}{1+m+n p}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} \frac {x^{1+m} \left (b x^n\right )^p}{1+m+n p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 22, normalized size = 1.05
method | result | size |
gosper | \(\frac {x^{1+m} \left (b \,x^{n}\right )^{p}}{n p +m +1}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 25, normalized size = 1.19 \begin {gather*} \frac {b^{p} 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 = 0.38, size = 24, normalized size = 1.14 \begin {gather*} \frac {x x^{m} e^{\left (n p \log \left (x\right ) + p \log \left (b\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 x^{m} \left (b x^{n}\right )^{p}}{m + n p + 1} & \text {for}\: m \neq - n p - 1 \\\int x^{- n p - 1} \left (b x^{n}\right )^{p}\, dx & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.07, size = 24, normalized size = 1.14 \begin {gather*} \frac {x x^{m} e^{\left (n p \log \left (x\right ) + p \log \left (b\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 = 1.14, size = 21, normalized size = 1.00 \begin {gather*} \frac {x^{m+1}\,{\left (b\,x^n\right )}^p}{m+n\,p+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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