Integrand size = 9, antiderivative size = 24 \[ \int \frac {x}{(a+b x)^2} \, dx=-\frac {x}{b (a+b x)}+\frac {\log (a+b x)}{b^2} \]
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Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int \frac {x}{(a+b x)^2} \, dx=\frac {a}{b^2 (a+b x)}+\frac {\log (a+b x)}{b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx \\ & = \frac {a}{b^2 (a+b x)}+\frac {\log (a+b x)}{b^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {x}{(a+b x)^2} \, dx=\frac {\frac {a}{a+b x}+\log (a+b x)}{b^2} \]
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Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {a}{b^{2} \left (b x +a \right )}+\frac {\ln \left (b x +a \right )}{b^{2}}\) | \(24\) |
norman | \(\frac {a}{b^{2} \left (b x +a \right )}+\frac {\ln \left (b x +a \right )}{b^{2}}\) | \(24\) |
risch | \(\frac {a}{b^{2} \left (b x +a \right )}+\frac {\ln \left (b x +a \right )}{b^{2}}\) | \(24\) |
parallelrisch | \(\frac {\ln \left (b x +a \right ) x b +a \ln \left (b x +a \right )+a}{b^{2} \left (b x +a \right )}\) | \(31\) |
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none
Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {x}{(a+b x)^2} \, dx=\frac {{\left (b x + a\right )} \log \left (b x + a\right ) + a}{b^{3} x + a b^{2}} \]
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Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {x}{(a+b x)^2} \, dx=\frac {a}{a b^{2} + b^{3} x} + \frac {\log {\left (a + b x \right )}}{b^{2}} \]
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none
Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x}{(a+b x)^2} \, dx=\frac {a}{b^{3} x + a b^{2}} + \frac {\log \left (b x + a\right )}{b^{2}} \]
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none
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {x}{(a+b x)^2} \, dx=-\frac {\frac {\log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b} - \frac {a}{{\left (b x + a\right )} b}}{b} \]
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Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {x}{(a+b x)^2} \, dx=\frac {\ln \left (a+b\,x\right )}{b^2}+\frac {a}{b^2\,\left (a+b\,x\right )} \]
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