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 = 0.02, antiderivative size = 32, 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} \begin {gather*} \frac {2 a}{b c^2 (a-b x)}+\frac {\log (a-b x)}{b c^2} \end {gather*}
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*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 0.88 \begin {gather*} \frac {\log (c (a-b x))+\frac {2 a}{a-b x}}{b c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x}{(a c-b c x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.25, size = 39, normalized size = 1.22 \begin {gather*} \frac {{\left (b x - a\right )} \log \left (b x - a\right ) - 2 \, a}{b^{2} c^{2} x - a b c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.11, size = 81, normalized size = 2.53 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 1.09 \begin {gather*} -\frac {2 a}{\left (b x -a \right ) b \,c^{2}}+\frac {\ln \left (b x -a \right )}{b \,c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 37, normalized size = 1.16 \begin {gather*} -\frac {2 \, a}{b^{2} c^{2} x - a b c^{2}} + \frac {\log \left (b x - a\right )}{b c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 37, normalized size = 1.16 \begin {gather*} \frac {\ln \left (b\,x-a\right )}{b\,c^2}+\frac {2\,a}{b\,\left (a\,c^2-b\,c^2\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 29, normalized size = 0.91 \begin {gather*} - \frac {2 a}{- a b c^{2} + b^{2} c^{2} x} + \frac {\log {\left (- a + b x \right )}}{b c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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