Optimal. Leaf size=41 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b c}-\frac{1}{2 a b c (a+b x)} \]
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Rubi [A] time = 0.0279136, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {44, 208} \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b c}-\frac{1}{2 a b c (a+b x)} \]
Antiderivative was successfully verified.
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Rule 44
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^2 (a c-b c x)} \, dx &=\int \left (\frac{1}{2 a c (a+b x)^2}+\frac{1}{2 a c \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=-\frac{1}{2 a b c (a+b x)}+\frac{\int \frac{1}{a^2-b^2 x^2} \, dx}{2 a c}\\ &=-\frac{1}{2 a b c (a+b x)}+\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b c}\\ \end{align*}
Mathematica [A] time = 0.0136945, size = 50, normalized size = 1.22 \[ \frac{-(a+b x) \log (a-b x)+(a+b x) \log (a+b x)-2 a}{4 a^2 b c (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 56, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( bx+a \right ) }{4\,c{a}^{2}b}}-{\frac{1}{2\,abc \left ( bx+a \right ) }}-{\frac{\ln \left ( bx-a \right ) }{4\,c{a}^{2}b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01729, size = 74, normalized size = 1.8 \begin{align*} -\frac{1}{2 \,{\left (a b^{2} c x + a^{2} b c\right )}} + \frac{\log \left (b x + a\right )}{4 \, a^{2} b c} - \frac{\log \left (b x - a\right )}{4 \, a^{2} b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53883, size = 115, normalized size = 2.8 \begin{align*} \frac{{\left (b x + a\right )} \log \left (b x + a\right ) -{\left (b x + a\right )} \log \left (b x - a\right ) - 2 \, a}{4 \,{\left (a^{2} b^{2} c x + a^{3} b c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.421321, size = 44, normalized size = 1.07 \begin{align*} - \frac{1}{2 a^{2} b c + 2 a b^{2} c x} - \frac{\frac{\log{\left (- \frac{a}{b} + x \right )}}{4} - \frac{\log{\left (\frac{a}{b} + x \right )}}{4}}{a^{2} b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06203, size = 59, normalized size = 1.44 \begin{align*} -\frac{\log \left ({\left | -\frac{2 \, a}{b x + a} + 1 \right |}\right )}{4 \, a^{2} b c} - \frac{1}{2 \,{\left (b x + a\right )} a b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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