Integrand size = 13, antiderivative size = 32 \[ \int \frac {A+B x}{(a+b x)^2} \, dx=-\frac {A b-a B}{b^2 (a+b x)}+\frac {B \log (a+b x)}{b^2} \]
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Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int \frac {A+B x}{(a+b x)^2} \, dx=\frac {B \log (a+b x)}{b^2}-\frac {A b-a B}{b^2 (a+b x)} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A b-a B}{b (a+b x)^2}+\frac {B}{b (a+b x)}\right ) \, dx \\ & = -\frac {A b-a B}{b^2 (a+b x)}+\frac {B \log (a+b x)}{b^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x}{(a+b x)^2} \, dx=\frac {-A b+a B}{b^2 (a+b x)}+\frac {B \log (a+b x)}{b^2} \]
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Time = 2.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {B \ln \left (b x +a \right )}{b^{2}}-\frac {A b -B a}{b^{2} \left (b x +a \right )}\) | \(33\) |
norman | \(\frac {B \ln \left (b x +a \right )}{b^{2}}-\frac {A b -B a}{b^{2} \left (b x +a \right )}\) | \(33\) |
risch | \(\frac {B \ln \left (b x +a \right )}{b^{2}}-\frac {A}{b \left (b x +a \right )}+\frac {B a}{b^{2} \left (b x +a \right )}\) | \(39\) |
parallelrisch | \(-\frac {-B \ln \left (b x +a \right ) x b -B \ln \left (b x +a \right ) a +A b -B a}{b^{2} \left (b x +a \right )}\) | \(42\) |
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none
Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {A+B x}{(a+b x)^2} \, dx=\frac {B a - A b + {\left (B b x + B a\right )} \log \left (b x + a\right )}{b^{3} x + a b^{2}} \]
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Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x}{(a+b x)^2} \, dx=\frac {B \log {\left (a + b x \right )}}{b^{2}} + \frac {- A b + B a}{a b^{2} + b^{3} x} \]
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none
Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x}{(a+b x)^2} \, dx=\frac {B a - A b}{b^{3} x + a b^{2}} + \frac {B \log \left (b x + a\right )}{b^{2}} \]
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none
Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.78 \[ \int \frac {A+B x}{(a+b x)^2} \, dx=-\frac {B {\left (\frac {\log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b} - \frac {a}{{\left (b x + a\right )} b}\right )}}{b} - \frac {A}{{\left (b x + a\right )} b} \]
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Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x}{(a+b x)^2} \, dx=\frac {B\,\ln \left (a+b\,x\right )}{b^2}-\frac {A\,b-B\,a}{b^2\,\left (a+b\,x\right )} \]
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