Optimal. Leaf size=67 \[ \frac {(e x)^{m+1} \left (a+b \log \left (c \log ^p(d x)\right )\right )}{e (m+1)}-\frac {b p (d x)^{-m-1} (e x)^{m+1} \text {Ei}((m+1) \log (d x))}{e (m+1)} \]
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Rubi [A] time = 0.06, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2522, 2310, 2178} \[ \frac {(e x)^{m+1} \left (a+b \log \left (c \log ^p(d x)\right )\right )}{e (m+1)}-\frac {b p (d x)^{-m-1} (e x)^{m+1} \text {Ei}((m+1) \log (d x))}{e (m+1)} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2310
Rule 2522
Rubi steps
\begin {align*} \int (e x)^m \left (a+b \log \left (c \log ^p(d x)\right )\right ) \, dx &=\frac {(e x)^{1+m} \left (a+b \log \left (c \log ^p(d x)\right )\right )}{e (1+m)}-\frac {(b p) \int \frac {(e x)^m}{\log (d x)} \, dx}{1+m}\\ &=\frac {(e x)^{1+m} \left (a+b \log \left (c \log ^p(d x)\right )\right )}{e (1+m)}-\frac {\left (b p (d x)^{-1-m} (e x)^{1+m}\right ) \operatorname {Subst}\left (\int \frac {e^{(1+m) x}}{x} \, dx,x,\log (d x)\right )}{e (1+m)}\\ &=-\frac {b p (d x)^{-1-m} (e x)^{1+m} \text {Ei}((1+m) \log (d x))}{e (1+m)}+\frac {(e x)^{1+m} \left (a+b \log \left (c \log ^p(d x)\right )\right )}{e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 56, normalized size = 0.84 \[ \frac {(d x)^{-m} (e x)^m \left (d x (d x)^m \left (a+b \log \left (c \log ^p(d x)\right )\right )-b p \text {Ei}((m+1) \log (d x))\right )}{d (m+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 83, normalized size = 1.24 \[ \frac {b d p x e^{\left (m \log \left (d x\right ) + m \log \left (\frac {e}{d}\right )\right )} \log \left (\log \left (d x\right )\right ) - b p \left (\frac {e}{d}\right )^{m} {\rm Ei}\left ({\left (m + 1\right )} \log \left (d x\right )\right ) + {\left (b d x \log \relax (c) + a d x\right )} e^{\left (m \log \left (d x\right ) + m \log \left (\frac {e}{d}\right )\right )}}{d m + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 83, normalized size = 1.24 \[ \frac {b p x x^{m} e^{m} \log \left (\log \relax (d) + \log \relax (x)\right )}{m + 1} + \frac {b x x^{m} e^{m} \log \relax (c)}{m + 1} + \frac {a x x^{m} e^{m}}{m + 1} - \frac {b p {\rm Ei}\left (m \log \relax (d) + m \log \relax (x) + \log \relax (d) + \log \relax (x)\right ) e^{m}}{d d^{m} m + d d^{m}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \ln \left (d x \right )^{p}\right )+a \right ) \left (e x \right )^{m}\, dx \]
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 \[ -{\left (e^{m} p \int \frac {x^{m}}{{\left (m^{2} + 2 \, m + 1\right )} \log \relax (d)^{2} + 2 \, {\left (m^{2} + 2 \, m + 1\right )} \log \relax (d) \log \relax (x) + {\left (m^{2} + 2 \, m + 1\right )} \log \relax (x)^{2}}\,{d x} - \frac {{\left (e^{m} {\left (m + 1\right )} x \log \relax (d) + e^{m} {\left (m + 1\right )} x \log \relax (x)\right )} x^{m} \log \left ({\left (\log \relax (d) + \log \relax (x)\right )}^{p}\right ) + {\left (e^{m} {\left (m + 1\right )} x \log \relax (c) \log \relax (x) + {\left (e^{m} {\left (m + 1\right )} \log \relax (c) \log \relax (d) - e^{m} p\right )} x\right )} x^{m}}{{\left (m^{2} + 2 \, m + 1\right )} \log \relax (d) + {\left (m^{2} + 2 \, m + 1\right )} \log \relax (x)}\right )} b + \frac {\left (e x\right )^{m + 1} a}{e {\left (m + 1\right )}} \]
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 \[ \int \left (a+b\,\ln \left (c\,{\ln \left (d\,x\right )}^p\right )\right )\,{\left (e\,x\right )}^m \,d x \]
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 \[ \int \left (e x\right )^{m} \left (a + b \log {\left (c \log {\left (d x \right )}^{p} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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