Integrand size = 11, antiderivative size = 29 \[ \int \frac {1}{x (a+b x)^2} \, dx=\frac {1}{a (a+b x)}+\frac {\log (x)}{a^2}-\frac {\log (a+b x)}{a^2} \]
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Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46} \[ \int \frac {1}{x (a+b x)^2} \, dx=-\frac {\log (a+b x)}{a^2}+\frac {\log (x)}{a^2}+\frac {1}{a (a+b x)} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx \\ & = \frac {1}{a (a+b x)}+\frac {\log (x)}{a^2}-\frac {\log (a+b x)}{a^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x (a+b x)^2} \, dx=\frac {\frac {a}{a+b x}+\log (x)-\log (a+b x)}{a^2} \]
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Time = 0.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {1}{a \left (b x +a \right )}+\frac {\ln \left (x \right )}{a^{2}}-\frac {\ln \left (b x +a \right )}{a^{2}}\) | \(30\) |
risch | \(\frac {1}{a \left (b x +a \right )}+\frac {\ln \left (-x \right )}{a^{2}}-\frac {\ln \left (b x +a \right )}{a^{2}}\) | \(32\) |
norman | \(-\frac {b x}{a^{2} \left (b x +a \right )}+\frac {\ln \left (x \right )}{a^{2}}-\frac {\ln \left (b x +a \right )}{a^{2}}\) | \(33\) |
parallelrisch | \(\frac {b \ln \left (x \right ) x -b \ln \left (b x +a \right ) x +a \ln \left (x \right )-a \ln \left (b x +a \right )-b x}{a^{2} \left (b x +a \right )}\) | \(45\) |
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none
Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {1}{x (a+b x)^2} \, dx=-\frac {{\left (b x + a\right )} \log \left (b x + a\right ) - {\left (b x + a\right )} \log \left (x\right ) - a}{a^{2} b x + a^{3}} \]
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Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x (a+b x)^2} \, dx=\frac {1}{a^{2} + a b x} + \frac {\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}}{a^{2}} \]
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none
Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x (a+b x)^2} \, dx=\frac {1}{a b x + a^{2}} - \frac {\log \left (b x + a\right )}{a^{2}} + \frac {\log \left (x\right )}{a^{2}} \]
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none
Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {1}{x (a+b x)^2} \, dx=b {\left (\frac {\log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{2} b} + \frac {1}{{\left (b x + a\right )} a b}\right )} \]
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Time = 0.13 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x (a+b x)^2} \, dx=\frac {1}{a^2+b\,x\,a}-\frac {\ln \left (\frac {a+b\,x}{x}\right )}{a^2} \]
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