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 = 0.0572072, antiderivative size = 67, normalized size of antiderivative = 1., 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 2522
Rule 2310
Rule 2178
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*}
Mathematica [A] time = 0.13092, 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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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( a+b\ln \left ( c \left ( \ln \left ( dx \right ) \right ) ^{p} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05197, size = 204, normalized size = 3.04 \begin{align*} \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 \left (c\right ) + a d x\right )} e^{\left (m \log \left (d x\right ) + m \log \left (\frac{e}{d}\right )\right )}}{d m + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \left (a + b \log{\left (c \log{\left (d x \right )}^{p} \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31031, size = 112, normalized size = 1.67 \begin{align*} \frac{b p x x^{m} e^{m} \log \left (\log \left (d\right ) + \log \left (x\right )\right )}{m + 1} + \frac{b x x^{m} e^{m} \log \left (c\right )}{m + 1} + \frac{a x x^{m} e^{m}}{m + 1} - \frac{b p{\rm Ei}\left (m \log \left (d\right ) + m \log \left (x\right ) + \log \left (d\right ) + \log \left (x\right )\right ) e^{m}}{d d^{m} m + d d^{m}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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