Integrand size = 11, antiderivative size = 28 \[ \int \frac {1}{x^2 (a+b x)} \, dx=-\frac {1}{a x}-\frac {b \log (x)}{a^2}+\frac {b \log (a+b x)}{a^2} \]
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Time = 0.01 (sec) , antiderivative size = 28, 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} \[ \int \frac {1}{x^2 (a+b x)} \, dx=-\frac {b \log (x)}{a^2}+\frac {b \log (a+b x)}{a^2}-\frac {1}{a x} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx \\ & = -\frac {1}{a x}-\frac {b \log (x)}{a^2}+\frac {b \log (a+b x)}{a^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 (a+b x)} \, dx=-\frac {1}{a x}-\frac {b \log (x)}{a^2}+\frac {b \log (a+b x)}{a^2} \]
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Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
method | result | size |
parallelrisch | \(-\frac {b \ln \left (x \right ) x -b \ln \left (b x +a \right ) x +a}{a^{2} x}\) | \(26\) |
default | \(-\frac {1}{a x}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (b x +a \right )}{a^{2}}\) | \(29\) |
norman | \(-\frac {1}{a x}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (b x +a \right )}{a^{2}}\) | \(29\) |
risch | \(-\frac {1}{a x}+\frac {b \ln \left (-b x -a \right )}{a^{2}}-\frac {b \ln \left (x \right )}{a^{2}}\) | \(32\) |
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Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^2 (a+b x)} \, dx=\frac {b x \log \left (b x + a\right ) - b x \log \left (x\right ) - a}{a^{2} x} \]
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Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x^2 (a+b x)} \, dx=- \frac {1}{a x} + \frac {b \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{2}} \]
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Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 (a+b x)} \, dx=\frac {b \log \left (b x + a\right )}{a^{2}} - \frac {b \log \left (x\right )}{a^{2}} - \frac {1}{a x} \]
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Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 (a+b x)} \, dx=\frac {b \log \left ({\left | b x + a \right |}\right )}{a^{2}} - \frac {b \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {1}{a x} \]
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Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^2 (a+b x)} \, dx=\frac {2\,b\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^2}-\frac {1}{a\,x} \]
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