Optimal. Leaf size=27 \[ -\frac {2 a c}{b (a+b x)}-\frac {c \log (a+b x)}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 27, 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 = {43} \[ -\frac {2 a c}{b (a+b x)}-\frac {c \log (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {a c-b c x}{(a+b x)^2} \, dx &=\int \left (\frac {2 a c}{(a+b x)^2}-\frac {c}{a+b x}\right ) \, dx\\ &=-\frac {2 a c}{b (a+b x)}-\frac {c \log (a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.85 \[ -\frac {c \left (\frac {2 a}{a+b x}+\log (a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 33, normalized size = 1.22 \[ -\frac {2 \, a c + {\left (b c x + a c\right )} \log \left (b x + a\right )}{b^{2} x + a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 54, normalized size = 2.00 \[ c {\left (\frac {\log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b} - \frac {a}{{\left (b x + a\right )} b}\right )} - \frac {a c}{{\left (b x + a\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 1.04 \[ -\frac {2 a c}{\left (b x +a \right ) b}-\frac {c \ln \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 28, normalized size = 1.04 \[ -\frac {2 \, a c}{b^{2} x + a b} - \frac {c \log \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 27, normalized size = 1.00 \[ -\frac {c\,\ln \left (a+b\,x\right )}{b}-\frac {2\,a\,c}{b\,\left (a+b\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 24, normalized size = 0.89 \[ - \frac {2 a c}{a b + b^{2} x} - \frac {c \log {\left (a + b x \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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