Optimal. Leaf size=21 \[ \frac{(d+e x) \log (c (d+e x))}{e}-x \]
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Rubi [A] time = 0.0080155, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2389, 2295} \[ \frac{(d+e x) \log (c (d+e x))}{e}-x \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2295
Rubi steps
\begin{align*} \int \log (c (d+e x)) \, dx &=\frac{\operatorname{Subst}(\int \log (c x) \, dx,x,d+e x)}{e}\\ &=-x+\frac{(d+e x) \log (c (d+e x))}{e}\\ \end{align*}
Mathematica [A] time = 0.0044039, size = 21, normalized size = 1. \[ \frac{(d+e x) \log (c (d+e x))}{e}-x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.064, size = 36, normalized size = 1.7 \begin{align*} \ln \left ( cex+cd \right ) x+{\frac{\ln \left ( cex+cd \right ) d}{e}}-x-{\frac{d}{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12578, size = 42, normalized size = 2. \begin{align*} \frac{{\left (e x + d\right )} c \log \left ({\left (e x + d\right )} c\right ) -{\left (e x + d\right )} c}{c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8341, size = 53, normalized size = 2.52 \begin{align*} -\frac{e x -{\left (e x + d\right )} \log \left (c e x + c d\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.372492, size = 26, normalized size = 1.24 \begin{align*} - e \left (- \frac{d \log{\left (d + e x \right )}}{e^{2}} + \frac{x}{e}\right ) + x \log{\left (c \left (d + e x\right ) \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17423, size = 45, normalized size = 2.14 \begin{align*} \frac{{\left ({\left (x e + d\right )} c \log \left ({\left (x e + d\right )} c\right ) -{\left (x e + d\right )} c\right )} e^{\left (-1\right )}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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