Optimal. Leaf size=86 \[ \frac {e^{-\frac {a (m+1)}{b}} (c x)^{-m-1} (d x)^{m+1} (a+b \log (c x))^p \left (-\frac {(m+1) (a+b \log (c x))}{b}\right )^{-p} \Gamma \left (p+1,-\frac {(m+1) (a+b \log (c x))}{b}\right )}{d (m+1)} \]
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Rubi [A] time = 0.07, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2310, 2181} \[ \frac {e^{-\frac {a (m+1)}{b}} (c x)^{-m-1} (d x)^{m+1} (a+b \log (c x))^p \left (-\frac {(m+1) (a+b \log (c x))}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {(m+1) (a+b \log (c x))}{b}\right )}{d (m+1)} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 2310
Rubi steps
\begin {align*} \int (d x)^m (a+b \log (c x))^p \, dx &=\frac {\left ((c x)^{-1-m} (d x)^{1+m}\right ) \operatorname {Subst}\left (\int e^{(1+m) x} (a+b x)^p \, dx,x,\log (c x)\right )}{d}\\ &=\frac {e^{-\frac {a (1+m)}{b}} (c x)^{-1-m} (d x)^{1+m} \Gamma \left (1+p,-\frac {(1+m) (a+b \log (c x))}{b}\right ) (a+b \log (c x))^p \left (-\frac {(1+m) (a+b \log (c x))}{b}\right )^{-p}}{d (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 82, normalized size = 0.95 \[ \frac {e^{-\frac {a (m+1)}{b}} (c x)^{-m} (d x)^m (a+b \log (c x))^p \left (-\frac {(m+1) (a+b \log (c x))}{b}\right )^{-p} \Gamma \left (p+1,-\frac {(m+1) (a+b \log (c x))}{b}\right )}{c (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d x\right )^{m} {\left (b \log \left (c x\right ) + a\right )}^{p}, x\right ) \]
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 \[ \int \left (d x\right )^{m} {\left (b \log \left (c x\right ) + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (b \ln \left (c x \right )+a \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{m} {\left (b \log \left (c x\right ) + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\ln \left (c\,x\right )\right )}^p\,{\left (d\,x\right )}^m \,d x \]
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 \[ \int \left (d x\right )^{m} \left (a + b \log {\left (c x \right )}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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