Integrand size = 18, antiderivative size = 53 \[ \int \frac {1}{x (a+b x) (c+d x)} \, dx=\frac {\log (x)}{a c}-\frac {b \log (a+b x)}{a (b c-a d)}+\frac {d \log (c+d x)}{c (b c-a d)} \]
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Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {84} \[ \int \frac {1}{x (a+b x) (c+d x)} \, dx=-\frac {b \log (a+b x)}{a (b c-a d)}+\frac {d \log (c+d x)}{c (b c-a d)}+\frac {\log (x)}{a c} \]
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Rule 84
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x (a+b x) (c+d x)} \, dx=\frac {b c \log (x)-a d \log (x)-b c \log (a+b x)+a d \log (c+d x)}{a b c^2-a^2 c d} \]
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Time = 1.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.92
method | result | size |
parallelrisch | \(\frac {\ln \left (x \right ) a d -c \ln \left (x \right ) b +\ln \left (b x +a \right ) b c -d \ln \left (d x +c \right ) a}{a c \left (a d -b c \right )}\) | \(49\) |
default | \(\frac {\ln \left (x \right )}{a c}-\frac {d \ln \left (d x +c \right )}{c \left (a d -b c \right )}+\frac {b \ln \left (b x +a \right )}{a \left (a d -b c \right )}\) | \(54\) |
norman | \(\frac {\ln \left (x \right )}{a c}-\frac {d \ln \left (d x +c \right )}{c \left (a d -b c \right )}+\frac {b \ln \left (b x +a \right )}{a \left (a d -b c \right )}\) | \(54\) |
risch | \(\frac {b \ln \left (b x +a \right )}{a \left (a d -b c \right )}-\frac {d \ln \left (d x +c \right )}{c \left (a d -b c \right )}+\frac {\ln \left (-x \right )}{a c}\) | \(56\) |
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none
Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x (a+b x) (c+d x)} \, dx=-\frac {b c \log \left (b x + a\right ) - a d \log \left (d x + c\right ) - {\left (b c - a d\right )} \log \left (x\right )}{a b c^{2} - a^{2} c d} \]
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Timed out. \[ \int \frac {1}{x (a+b x) (c+d x)} \, dx=\text {Timed out} \]
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none
Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (a+b x) (c+d x)} \, dx=-\frac {b \log \left (b x + a\right )}{a b c - a^{2} d} + \frac {d \log \left (d x + c\right )}{b c^{2} - a c d} + \frac {\log \left (x\right )}{a c} \]
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Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.25 \[ \int \frac {1}{x (a+b x) (c+d x)} \, dx=-\frac {b^{2} \log \left ({\left | b x + a \right |}\right )}{a b^{2} c - a^{2} b d} + \frac {d^{2} \log \left ({\left | d x + c \right |}\right )}{b c^{2} d - a c d^{2}} + \frac {\log \left ({\left | x \right |}\right )}{a c} \]
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Time = 0.65 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x (a+b x) (c+d x)} \, dx=\frac {\ln \left (x\right )}{a\,c}+\frac {b\,\ln \left (a+b\,x\right )}{a^2\,d-a\,b\,c}+\frac {d\,\ln \left (c+d\,x\right )}{b\,c^2-a\,c\,d} \]
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