Optimal. Leaf size=18 \[ -c x+\frac {2 a c \log (a+b x)}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45}
\begin {gather*} \frac {2 a c \log (a+b x)}{b}-c x \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {a c-b c x}{a+b x} \, dx &=\int \left (-c+\frac {2 a c}{a+b x}\right ) \, dx\\ &=-c x+\frac {2 a c \log (a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} c \left (-x+\frac {2 a \log (a+b x)}{b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 19, normalized size = 1.06
method | result | size |
default | \(c \left (-x +\frac {2 a \ln \left (b x +a \right )}{b}\right )\) | \(19\) |
norman | \(-c x +\frac {2 a c \ln \left (b x +a \right )}{b}\) | \(19\) |
risch | \(-c x +\frac {2 a c \ln \left (b x +a \right )}{b}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 18, normalized size = 1.00 \begin {gather*} -c x + \frac {2 \, a c \log \left (b x + a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 20, normalized size = 1.11 \begin {gather*} -\frac {b c x - 2 \, a c \log \left (b x + a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 15, normalized size = 0.83 \begin {gather*} \frac {2 a c \log {\left (a + b x \right )}}{b} - c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.01, size = 19, normalized size = 1.06 \begin {gather*} -c x + \frac {2 \, a c \log \left ({\left | b x + a \right |}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 18, normalized size = 1.00 \begin {gather*} \frac {2\,a\,c\,\ln \left (a+b\,x\right )}{b}-c\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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