Optimal. Leaf size=25 \[ \frac {(a+x (b+c)) \log (a+x (b+c))}{b+c}-x \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 2295
Rule 2389
Rule 2444
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*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ \frac {(a+x (b+c)) \log (a+x (b+c))}{b+c}-x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 30, normalized size = 1.20 \[ -\frac {{\left (b + c\right )} x - {\left ({\left (b + c\right )} x + a\right )} \log \left ({\left (b + c\right )} x + a\right )}{b + c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 34, normalized size = 1.36 \[ -\frac {b x + c x - {\left (b x + c x + a\right )} \log \left (b x + c x + a\right ) + a}{b + c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 75, normalized size = 3.00 \[ \frac {b x \ln \left (a +\left (b +c \right ) x \right )}{b +c}+\frac {c x \ln \left (a +\left (b +c \right ) x \right )}{b +c}+\frac {a \ln \left (a +\left (b +c \right ) x \right )}{b +c}-\frac {b x}{b +c}-\frac {c x}{b +c}-\frac {a}{b +c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 34, normalized size = 1.36 \[ -\frac {b x + c x - {\left (b x + c x + a\right )} \log \left (b x + c x + a\right ) + a}{b + c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 31, normalized size = 1.24 \[ x\,\ln \left (a+b\,x+c\,x\right )-x+\frac {a\,\ln \left (a+b\,x+c\,x\right )}{b+c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 36, normalized size = 1.44 \[ x \log {\left (a + b x + c x \right )} + \left (- b - c\right ) \left (- \frac {a \log {\left (a + x \left (b + c\right ) \right )}}{\left (b + c\right )^{2}} + \frac {x}{b + c}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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