Integrand size = 23, antiderivative size = 68 \[ \int \frac {x}{(-a+x) (-b+x) (-c+x)} \, dx=\frac {a \log (a-x)}{(a-b) (a-c)}-\frac {b \log (b-x)}{(a-b) (b-c)}+\frac {c \log (c-x)}{(a-c) (b-c)} \]
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Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {153} \[ \int \frac {x}{(-a+x) (-b+x) (-c+x)} \, dx=\frac {a \log (a-x)}{(a-b) (a-c)}-\frac {b \log (b-x)}{(a-b) (b-c)}+\frac {c \log (c-x)}{(a-c) (b-c)} \]
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Rule 153
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a}{(a-b) (a-c) (a-x)}+\frac {b}{(a-b) (b-c) (b-x)}+\frac {c}{(a-c) (-b+c) (c-x)}\right ) \, dx \\ & = \frac {a \log (a-x)}{(a-b) (a-c)}-\frac {b \log (b-x)}{(a-b) (b-c)}+\frac {c \log (c-x)}{(a-c) (b-c)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91 \[ \int \frac {x}{(-a+x) (-b+x) (-c+x)} \, dx=\frac {a (b-c) \log (-a+x)+b (-a+c) \log (-b+x)+(a-b) c \log (-c+x)}{(a-b) (a-c) (b-c)} \]
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Time = 0.16 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.01
method | result | size |
default | \(\frac {a \ln \left (a -x \right )}{\left (a -b \right ) \left (a -c \right )}-\frac {b \ln \left (b -x \right )}{\left (a -b \right ) \left (b -c \right )}+\frac {c \ln \left (c -x \right )}{\left (a -c \right ) \left (b -c \right )}\) | \(69\) |
norman | \(\frac {c \ln \left (c -x \right )}{a b -a c -b c +c^{2}}+\frac {a \ln \left (a -x \right )}{\left (a -b \right ) \left (a -c \right )}-\frac {b \ln \left (b -x \right )}{\left (a -b \right ) \left (b -c \right )}\) | \(72\) |
risch | \(-\frac {b \ln \left (-b +x \right )}{a b -a c -b^{2}+b c}+\frac {a \ln \left (-a +x \right )}{a^{2}-a b -a c +b c}+\frac {c \ln \left (c -x \right )}{a b -a c -b c +c^{2}}\) | \(79\) |
parallelrisch | \(\frac {\ln \left (-a +x \right ) a b -\ln \left (-a +x \right ) a c -\ln \left (-b +x \right ) a b +\ln \left (-b +x \right ) b c +\ln \left (-c +x \right ) a c -\ln \left (-c +x \right ) b c}{\left (a b -a c -b c +c^{2}\right ) \left (a -b \right )}\) | \(84\) |
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none
Time = 0.31 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.19 \[ \int \frac {x}{(-a+x) (-b+x) (-c+x)} \, dx=\frac {{\left (a - b\right )} c \log \left (-c + x\right ) + {\left (a b - a c\right )} \log \left (-a + x\right ) - {\left (a b - b c\right )} \log \left (-b + x\right )}{a^{2} b - a b^{2} + {\left (a - b\right )} c^{2} - {\left (a^{2} - b^{2}\right )} c} \]
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Timed out. \[ \int \frac {x}{(-a+x) (-b+x) (-c+x)} \, dx=\text {Timed out} \]
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none
Time = 0.19 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.15 \[ \int \frac {x}{(-a+x) (-b+x) (-c+x)} \, dx=\frac {a \log \left (-a + x\right )}{a^{2} - a b - {\left (a - b\right )} c} - \frac {b \log \left (-b + x\right )}{a b - b^{2} - {\left (a - b\right )} c} + \frac {c \log \left (-c + x\right )}{a b - {\left (a + b\right )} c + c^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.19 \[ \int \frac {x}{(-a+x) (-b+x) (-c+x)} \, dx=\frac {a \log \left ({\left | -a + x \right |}\right )}{a^{2} - a b - a c + b c} - \frac {b \log \left ({\left | -b + x \right |}\right )}{a b - b^{2} - a c + b c} + \frac {c \log \left ({\left | -c + x \right |}\right )}{a b - a c - b c + c^{2}} \]
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Time = 0.58 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.28 \[ \int \frac {x}{(-a+x) (-b+x) (-c+x)} \, dx=\ln \left (x-a\right )\,\left (\frac {b}{\left (a-b\right )\,\left (b-c\right )}-\frac {c}{\left (a-c\right )\,\left (b-c\right )}\right )-\frac {b\,\ln \left (x-b\right )}{\left (a-b\right )\,\left (b-c\right )}+\frac {c\,\ln \left (x-c\right )}{\left (a-c\right )\,\left (b-c\right )} \]
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