Optimal. Leaf size=41 \[ \frac{(d+e x) \log ^2(c (d+e x))}{e}-\frac{2 (d+e x) \log (c (d+e x))}{e}+2 x \]
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Rubi [A] time = 0.0184991, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2389, 2296, 2295} \[ \frac{(d+e x) \log ^2(c (d+e x))}{e}-\frac{2 (d+e x) \log (c (d+e x))}{e}+2 x \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2296
Rule 2295
Rubi steps
\begin{align*} \int \log ^2(c (d+e x)) \, dx &=\frac{\operatorname{Subst}\left (\int \log ^2(c x) \, dx,x,d+e x\right )}{e}\\ &=\frac{(d+e x) \log ^2(c (d+e x))}{e}-\frac{2 \operatorname{Subst}(\int \log (c x) \, dx,x,d+e x)}{e}\\ &=2 x-\frac{2 (d+e x) \log (c (d+e x))}{e}+\frac{(d+e x) \log ^2(c (d+e x))}{e}\\ \end{align*}
Mathematica [A] time = 0.0044105, size = 40, normalized size = 0.98 \[ \frac{(d+e x) \log ^2(c (d+e x))-2 (d+e x) \log (c (d+e x))+2 e x}{e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 67, normalized size = 1.6 \begin{align*} \left ( \ln \left ( cex+cd \right ) \right ) ^{2}x+{\frac{ \left ( \ln \left ( cex+cd \right ) \right ) ^{2}d}{e}}-2\,\ln \left ( cex+cd \right ) x-2\,{\frac{\ln \left ( cex+cd \right ) d}{e}}+2\,x+2\,{\frac{d}{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08999, size = 96, normalized size = 2.34 \begin{align*} -2 \, e{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )} c\right ) + x \log \left ({\left (e x + d\right )} c\right )^{2} - \frac{d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8574, size = 99, normalized size = 2.41 \begin{align*} \frac{{\left (e x + d\right )} \log \left (c e x + c d\right )^{2} + 2 \, e x - 2 \,{\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 = 0.433221, size = 46, normalized size = 1.12 \begin{align*} 2 e \left (- \frac{d \log{\left (d + e x \right )}}{e^{2}} + \frac{x}{e}\right ) - 2 x \log{\left (c \left (d + e x\right ) \right )} + \frac{\left (d + e x\right ) \log{\left (c \left (d + e x\right ) \right )}^{2}}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13679, size = 68, normalized size = 1.66 \begin{align*}{\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right )^{2} - 2 \,{\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right ) + 2 \,{\left (x e + d\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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