Optimal. Leaf size=25 \[ \frac{(a+x (b+c)) \log (a+x (b+c))}{b+c}-x \]
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Rubi [A] time = 0.0147451, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2444, 2389, 2295} \[ \frac{(a+x (b+c)) \log (a+x (b+c))}{b+c}-x \]
Antiderivative was successfully verified.
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Rule 2444
Rule 2389
Rule 2295
Rubi steps
\begin{align*} \int \log (a+b x+c x) \, dx &=\int \log (a+(b+c) x) \, dx\\ &=\frac{\operatorname{Subst}(\int \log (x) \, dx,x,a+(b+c) x)}{b+c}\\ &=-x+\frac{(a+(b+c) x) \log (a+(b+c) x)}{b+c}\\ \end{align*}
Mathematica [A] time = 0.0068114, size = 25, normalized size = 1. \[ \frac{(a+x (b+c)) \log (a+x (b+c))}{b+c}-x \]
Antiderivative was successfully verified.
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Maple [B] time = 0.057, size = 75, normalized size = 3. \begin{align*}{\frac{\ln \left ( a+ \left ( b+c \right ) x \right ) xb}{b+c}}+{\frac{\ln \left ( a+ \left ( b+c \right ) x \right ) xc}{b+c}}+{\frac{\ln \left ( a+ \left ( b+c \right ) x \right ) a}{b+c}}-{\frac{bx}{b+c}}-{\frac{cx}{b+c}}-{\frac{a}{b+c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24484, size = 46, normalized size = 1.84 \begin{align*} -\frac{b x + c x -{\left (b x + c x + a\right )} \log \left (b x + c x + a\right ) + a}{b + c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00934, size = 80, normalized size = 3.2 \begin{align*} -\frac{{\left (b + c\right )} x -{\left ({\left (b + c\right )} x + a\right )} \log \left ({\left (b + c\right )} x + a\right )}{b + c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.357667, size = 36, normalized size = 1.44 \begin{align*} x \log{\left (a + b x + c x \right )} + \left (- b - c\right ) \left (- \frac{a \log{\left (a + x \left (b + c\right ) \right )}}{\left (b + c\right )^{2}} + \frac{x}{b + c}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19796, size = 46, normalized size = 1.84 \begin{align*} -\frac{b x + c x -{\left (b x + c x + a\right )} \log \left (b x + c x + a\right ) + a}{b + c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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