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 = 0.04, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 12
Rule 2178
Rule 2194
Rule 2554
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.42, size = 23, normalized size = 0.88 \[ -\frac {{\rm Ei}\left (b x\right ) e^{a} - e^{\left (b x + a\right )} \log \relax (x)}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 24, normalized size = 0.92 \[ -\frac {{\rm Ei}\left (b x\right ) e^{a}}{b} + \frac {e^{\left (b x + a\right )} \log \relax (x)}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 26, normalized size = 1.00 \[ \frac {\Ei \left (1, -b x \right ) {\mathrm e}^{a}}{b}+\frac {{\mathrm e}^{b x +a} \ln \relax (x )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 24, normalized size = 0.92 \[ -\frac {{\rm Ei}\left (b x\right ) e^{a}}{b} + \frac {e^{\left (b x + a\right )} \log \relax (x)}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 20, normalized size = 0.77 \[ -\frac {{\mathrm {e}}^a\,\left (\mathrm {ei}\left (b\,x\right )-{\mathrm {e}}^{b\,x}\,\ln \relax (x)\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.38, size = 26, normalized size = 1.00 \[ \left (\begin {cases} x & \text {for}\: b = 0 \\\frac {e^{b x}}{b} & \text {otherwise} \end {cases}\right ) e^{a} \log {\relax (x )} - \left (\begin {cases} x & \text {for}\: b = 0 \\\frac {\operatorname {Ei}{\left (b x \right )}}{b} & \text {otherwise} \end {cases}\right ) e^{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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