Optimal. Leaf size=107 \[ \frac {2^{-p} e^{-\frac {2 a (m+1)}{b}} \left (c \sqrt {x}\right )^{-2 (m+1)} (d x)^{m+1} \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (-\frac {(m+1) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 (m+1) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )}{d (m+1)} \]
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Rubi [A] time = 0.08, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2310, 2181} \[ \frac {2^{-p} e^{-\frac {2 a (m+1)}{b}} \left (c \sqrt {x}\right )^{-2 (m+1)} (d x)^{m+1} \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (-\frac {(m+1) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {2 (m+1) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )}{d (m+1)} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 2310
Rubi steps
\begin {align*} \int (d x)^m \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx &=\frac {\left (2 \left (c \sqrt {x}\right )^{-2 (1+m)} (d x)^{1+m}\right ) \operatorname {Subst}\left (\int e^{2 (1+m) x} (a+b x)^p \, dx,x,\log \left (c \sqrt {x}\right )\right )}{d}\\ &=\frac {2^{-p} e^{-\frac {2 a (1+m)}{b}} \left (c \sqrt {x}\right )^{-2 (1+m)} (d x)^{1+m} \Gamma \left (1+p,-\frac {2 (1+m) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (-\frac {(1+m) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )^{-p}}{d (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 103, normalized size = 0.96 \[ \frac {2^{-p} e^{-\frac {2 a (m+1)}{b}} \left (c \sqrt {x}\right )^{-2 m} (d x)^m \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (-\frac {(m+1) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 (m+1) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )}{c^2 (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d x\right )^{m} {\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p}, x\right ) \]
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 \[ \int \left (d x\right )^{m} {\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (b \ln \left (c \sqrt {x}\right )+a \right )^{p}\, dx \]
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 \[ \int \left (d x\right )^{m} {\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p}\,{d x} \]
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 \[ \int {\left (d\,x\right )}^m\,{\left (a+b\,\ln \left (c\,\sqrt {x}\right )\right )}^p \,d x \]
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 \[ \int \left (d x\right )^{m} \left (a + b \log {\left (c \sqrt {x} \right )}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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