Integrand size = 23, antiderivative size = 4 \[ \int \frac {a c+b c x^2}{x \left (a+b x^2\right )} \, dx=c \log (x) \]
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Time = 0.00 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {21, 29} \[ \int \frac {a c+b c x^2}{x \left (a+b x^2\right )} \, dx=c \log (x) \]
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Rule 21
Rule 29
Rubi steps \begin{align*} \text {integral}& = c \int \frac {1}{x} \, dx \\ & = c \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {a c+b c x^2}{x \left (a+b x^2\right )} \, dx=c \log (x) \]
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Time = 2.48 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.25
method | result | size |
default | \(c \ln \left (x \right )\) | \(5\) |
norman | \(c \ln \left (x \right )\) | \(5\) |
risch | \(c \ln \left (x \right )\) | \(5\) |
parallelrisch | \(c \ln \left (x \right )\) | \(5\) |
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none
Time = 0.25 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {a c+b c x^2}{x \left (a+b x^2\right )} \, dx=c \log \left (x\right ) \]
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Time = 0.02 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.75 \[ \int \frac {a c+b c x^2}{x \left (a+b x^2\right )} \, dx=c \log {\left (x \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.75 \[ \int \frac {a c+b c x^2}{x \left (a+b x^2\right )} \, dx=\frac {1}{2} \, c \log \left (x^{2}\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.25 \[ \int \frac {a c+b c x^2}{x \left (a+b x^2\right )} \, dx=c \log \left ({\left | x \right |}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {a c+b c x^2}{x \left (a+b x^2\right )} \, dx=c\,\ln \left (x\right ) \]
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