Optimal. Leaf size=26 \[ \frac{\log (x) e^{a+b x}}{b}-\frac{e^a \text{Ei}(b x)}{b} \]
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Rubi [A] time = 0.0354785, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2194, 2554, 12, 2178} \[ \frac{\log (x) e^{a+b x}}{b}-\frac{e^a \text{Ei}(b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2554
Rule 12
Rule 2178
Rubi steps
\begin{align*} \int e^{a+b x} \log (x) \, dx &=\frac{e^{a+b x} \log (x)}{b}-\int \frac{e^{a+b x}}{b x} \, dx\\ &=\frac{e^{a+b x} \log (x)}{b}-\frac{\int \frac{e^{a+b x}}{x} \, dx}{b}\\ &=-\frac{e^a \text{Ei}(b x)}{b}+\frac{e^{a+b x} \log (x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0148314, size = 22, normalized size = 0.85 \[ \frac{e^a \left (e^{b x} \log (x)-\text{Ei}(b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 26, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{bx+a}}\ln \left ( x \right ) }{b}}+{\frac{{{\rm e}^{a}}{\it Ei} \left ( 1,-bx \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13359, size = 32, normalized size = 1.23 \begin{align*} -\frac{{\rm Ei}\left (b x\right ) e^{a}}{b} + \frac{e^{\left (b x + a\right )} \log \left (x\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14951, size = 53, normalized size = 2.04 \begin{align*} -\frac{{\rm Ei}\left (b x\right ) e^{a} - e^{\left (b x + a\right )} \log \left (x\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.91723, size = 26, normalized size = 1. \begin{align*} \left (\begin{cases} x & \text{for}\: b = 0 \\\frac{e^{b x}}{b} & \text{otherwise} \end{cases}\right ) e^{a} \log{\left (x \right )} - \left (\begin{cases} x & \text{for}\: b = 0 \\\frac{\operatorname{Ei}{\left (b x \right )}}{b} & \text{otherwise} \end{cases}\right ) e^{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30655, size = 32, normalized size = 1.23 \begin{align*} -\frac{{\rm Ei}\left (b x\right ) e^{a}}{b} + \frac{e^{\left (b x + a\right )} \log \left (x\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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