Optimal. Leaf size=63 \[ \frac {2^{-p-1} e^{-\frac {2 a}{b}} (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 (a+b \log (c x))}{b}\right )}{c^2} \]
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Rubi [A] time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2309, 2181} \[ \frac {2^{-p-1} e^{-\frac {2 a}{b}} (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {2 (a+b \log (c x))}{b}\right )}{c^2} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 2309
Rubi steps
\begin {align*} \int x (a+b \log (c x))^p \, dx &=\frac {\operatorname {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log (c x)\right )}{c^2}\\ &=\frac {2^{-1-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 (a+b \log (c x))}{b}\right ) (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p}}{c^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 63, normalized size = 1.00 \[ \frac {2^{-p-1} e^{-\frac {2 a}{b}} (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 (a+b \log (c x))}{b}\right )}{c^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \log \left (c x\right ) + a\right )}^{p} x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c x\right ) + a\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int x \left (b \ln \left (c x \right )+a \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 44, normalized size = 0.70 \[ -\frac {{\left (b \log \left (c x\right ) + a\right )}^{p + 1} e^{\left (-\frac {2 \, a}{b}\right )} E_{-p}\left (-\frac {2 \, {\left (b \log \left (c x\right ) + a\right )}}{b}\right )}{b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\left (a+b\,\ln \left (c\,x\right )\right )}^p \,d x \]
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 \[ \int x \left (a + b \log {\left (c x \right )}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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