Optimal. Leaf size=53 \[ -\frac {b \log (a+b x)}{a (b c-a d)}+\frac {d \log (c+d x)}{c (b c-a d)}+\frac {\log (x)}{a c} \]
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Rubi [A] time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {72} \begin {gather*} -\frac {b \log (a+b x)}{a (b c-a d)}+\frac {d \log (c+d x)}{c (b c-a d)}+\frac {\log (x)}{a c} \end {gather*}
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin {align*} \int \frac {1}{x (a+b x) (c+d x)} \, dx &=\int \left (\frac {1}{a c x}+\frac {b^2}{a (-b c+a d) (a+b x)}+\frac {d^2}{c (b c-a d) (c+d x)}\right ) \, dx\\ &=\frac {\log (x)}{a c}-\frac {b \log (a+b x)}{a (b c-a d)}+\frac {d \log (c+d x)}{c (b c-a d)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 48, normalized size = 0.91 \begin {gather*} \frac {-b c \log (a+b x)+a d \log (c+d x)-a d \log (x)+b c \log (x)}{a b c^2-a^2 c d} \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 {1}{x (a+b x) (c+d x)} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.55, size = 50, normalized size = 0.94 \begin {gather*} -\frac {b c \log \left (b x + a\right ) - a d \log \left (d x + c\right ) - {\left (b c - a d\right )} \log \relax (x)}{a b c^{2} - a^{2} c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.74, size = 66, normalized size = 1.25 \begin {gather*} -\frac {b^{2} \log \left ({\left | b x + a \right |}\right )}{a b^{2} c - a^{2} b d} + \frac {d^{2} \log \left ({\left | d x + c \right |}\right )}{b c^{2} d - a c d^{2}} + \frac {\log \left ({\left | x \right |}\right )}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 1.02 \begin {gather*} \frac {b \ln \left (b x +a \right )}{\left (a d -b c \right ) a}-\frac {d \ln \left (d x +c \right )}{\left (a d -b c \right ) c}+\frac {\ln \relax (x )}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 53, normalized size = 1.00 \begin {gather*} -\frac {b \log \left (b x + a\right )}{a b c - a^{2} d} + \frac {d \log \left (d x + c\right )}{b c^{2} - a c d} + \frac {\log \relax (x)}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 52, normalized size = 0.98 \begin {gather*} \frac {\ln \relax (x)}{a\,c}+\frac {b\,\ln \left (a+b\,x\right )}{a^2\,d-a\,b\,c}+\frac {d\,\ln \left (c+d\,x\right )}{b\,c^2-a\,c\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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