Integrand size = 10, antiderivative size = 29 \[ \int \frac {\log (x)}{(a+b x)^2} \, dx=\frac {x \log (x)}{a (a+b x)}-\frac {\log (a+b x)}{a b} \]
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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 = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2351, 31} \[ \int \frac {\log (x)}{(a+b x)^2} \, dx=\frac {x \log (x)}{a (a+b x)}-\frac {\log (a+b x)}{a b} \]
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Rule 31
Rule 2351
Rubi steps \begin{align*} \text {integral}& = \frac {x \log (x)}{a (a+b x)}-\frac {\int \frac {1}{a+b x} \, dx}{a} \\ & = \frac {x \log (x)}{a (a+b x)}-\frac {\log (a+b x)}{a b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {\log (x)}{(a+b x)^2} \, dx=\frac {\frac {x \log (x)}{a+b x}-\frac {\log (a+b x)}{b}}{a} \]
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Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {x \ln \left (x \right )}{a \left (b x +a \right )}-\frac {\ln \left (b x +a \right )}{a b}\) | \(30\) |
norman | \(\frac {x \ln \left (x \right )}{a \left (b x +a \right )}-\frac {\ln \left (b x +a \right )}{a b}\) | \(30\) |
parts | \(-\frac {\ln \left (x \right )}{b \left (b x +a \right )}+\frac {\frac {\ln \left (x \right )}{a}-\frac {\ln \left (b x +a \right )}{a}}{b}\) | \(38\) |
parallelrisch | \(\frac {-b \ln \left (b x +a \right ) x +b x \ln \left (x \right )-\ln \left (b x +a \right ) a}{a b \left (b x +a \right )}\) | \(40\) |
risch | \(-\frac {\ln \left (x \right )}{b \left (b x +a \right )}-\frac {\ln \left (b x +a \right )}{a b}+\frac {\ln \left (-x \right )}{a b}\) | \(41\) |
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none
Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {\log (x)}{(a+b x)^2} \, dx=\frac {b x \log \left (x\right ) - {\left (b x + a\right )} \log \left (b x + a\right )}{a b^{2} x + a^{2} b} \]
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Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {\log (x)}{(a+b x)^2} \, dx=- \frac {\log {\left (x \right )}}{a b + b^{2} x} + \frac {\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}}{a b} \]
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none
Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {\log (x)}{(a+b x)^2} \, dx=-\frac {\frac {\log \left (b x + a\right )}{a} - \frac {\log \left (x\right )}{a}}{b} - \frac {\log \left (x\right )}{{\left (b x + a\right )} b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (29) = 58\).
Time = 0.31 (sec) , antiderivative size = 138, normalized size of antiderivative = 4.76 \[ \int \frac {\log (x)}{(a+b x)^2} \, dx=b^{2} {\left (\frac {\log \left (\frac {{\left (b x + a\right )}^{2} {\left | b \right |} {\left | \frac {a}{b x + a} - 1 \right |}}{b^{2} {\left | b x + a \right |}}\right )}{a b^{3}} + \frac {\log \left (-\frac {a + \frac {{\left (b x + a\right )} b {\left (\frac {a}{b x + a} - 1\right )} - a b}{b}}{b}\right )}{{\left ({\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a\right )} b^{3}} - \frac {\log \left ({\left | -{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} + a \right |}\right )}{a b^{3}}\right )} \]
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Time = 0.48 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {\log (x)}{(a+b x)^2} \, dx=\frac {x^2\,\ln \left (x\right )}{a\,\left (b\,x^2+a\,x\right )}-\frac {\ln \left (a+b\,x\right )}{a\,b} \]
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