Optimal. Leaf size=22 \[ \frac {(b x)^{p+1} (c x)^m}{b (m+p+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 22, 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 = {20, 32} \[ \frac {(b x)^{p+1} (c x)^m}{b (m+p+1)} \]
Antiderivative was successfully verified.
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Rule 20
Rule 32
Rubi steps
\begin {align*} \int (b x)^p (c x)^m \, dx &=\left ((b x)^{-m} (c x)^m\right ) \int (b x)^{m+p} \, dx\\ &=\frac {(b x)^{1+p} (c x)^m}{b (1+m+p)}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 0.82 \[ \frac {x (b x)^p (c x)^m}{m+p+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 29, normalized size = 1.32 \[ \frac {\left (b x\right )^{p} x e^{\left (m \log \left (b x\right ) + m \log \left (\frac {c}{b}\right )\right )}}{m + p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 26, normalized size = 1.18 \[ \frac {x e^{\left (p \log \relax (b) + m \log \relax (c) + m \log \relax (x) + p \log \relax (x)\right )}}{m + p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 0.86 \[ \frac {x \left (b x \right )^{p} \left (c x \right )^{m}}{m +p +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 24, normalized size = 1.09 \[ \frac {b^{p} c^{m} x e^{\left (m \log \relax (x) + p \log \relax (x)\right )}}{m + p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 18, normalized size = 0.82 \[ \frac {x\,{\left (b\,x\right )}^p\,{\left (c\,x\right )}^m}{m+p+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.13, size = 60, normalized size = 2.73 \[ \begin {cases} \frac {b^{p} c^{m} x x^{m} x^{p}}{m + p + 1} & \text {for}\: m \neq - p - 1 \\\begin {cases} \frac {b^{p} c^{- p} \log {\relax (x )}}{c} & \text {for}\: \left |{x}\right | < 1 \\- \frac {b^{p} c^{- p} {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )}}{c} + \frac {b^{p} c^{- p} {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )}}{c} & \text {otherwise} \end {cases} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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