Optimal. Leaf size=32 \[ \frac{2 a}{b c^2 (a-b x)}+\frac{\log (a-b x)}{b c^2} \]
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Rubi [A] time = 0.017297, antiderivative size = 32, normalized size of antiderivative = 1., 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} \[ \frac{2 a}{b c^2 (a-b x)}+\frac{\log (a-b x)}{b c^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{a+b x}{(a c-b c x)^2} \, dx &=\int \left (\frac{2 a}{c^2 (a-b x)^2}-\frac{1}{c^2 (a-b x)}\right ) \, dx\\ &=\frac{2 a}{b c^2 (a-b x)}+\frac{\log (a-b x)}{b c^2}\\ \end{align*}
Mathematica [A] time = 0.0166424, size = 28, normalized size = 0.88 \[ \frac{\log (c (a-b x))+\frac{2 a}{a-b x}}{b c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 35, normalized size = 1.1 \begin{align*} -2\,{\frac{a}{{c}^{2}b \left ( bx-a \right ) }}+{\frac{\ln \left ( bx-a \right ) }{{c}^{2}b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02122, size = 50, normalized size = 1.56 \begin{align*} -\frac{2 \, a}{b^{2} c^{2} x - a b c^{2}} + \frac{\log \left (b x - a\right )}{b c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41035, size = 73, normalized size = 2.28 \begin{align*} \frac{{\left (b x - a\right )} \log \left (b x - a\right ) - 2 \, a}{b^{2} c^{2} x - a b c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.338311, size = 29, normalized size = 0.91 \begin{align*} - \frac{2 a}{- a b c^{2} + b^{2} c^{2} x} + \frac{\log{\left (- a + b x \right )}}{b c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.07027, size = 109, normalized size = 3.41 \begin{align*} -\frac{\frac{a}{{\left (b c x - a c\right )} b} + \frac{\log \left (\frac{{\left | b c x - a c \right |}}{{\left (b c x - a c\right )}^{2}{\left | b \right |}{\left | c \right |}}\right )}{b c}}{c} - \frac{a}{{\left (b c x - a c\right )} b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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