Integrand size = 13, antiderivative size = 25 \[ \int \frac {A+B x}{a+b x} \, dx=\frac {B x}{b}+\frac {(A b-a B) \log (a+b x)}{b^2} \]
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Time = 0.01 (sec) , antiderivative size = 25, 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} \[ \int \frac {A+B x}{a+b x} \, dx=\frac {(A b-a B) \log (a+b x)}{b^2}+\frac {B x}{b} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {B}{b}+\frac {A b-a B}{b (a+b x)}\right ) \, dx \\ & = \frac {B x}{b}+\frac {(A b-a B) \log (a+b x)}{b^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x}{a+b x} \, dx=\frac {B x}{b}+\frac {(A b-a B) \log (a+b x)}{b^2} \]
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Time = 2.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04
method | result | size |
default | \(\frac {B x}{b}+\frac {\left (A b -B a \right ) \ln \left (b x +a \right )}{b^{2}}\) | \(26\) |
norman | \(\frac {B x}{b}+\frac {\left (A b -B a \right ) \ln \left (b x +a \right )}{b^{2}}\) | \(26\) |
parallelrisch | \(\frac {A \ln \left (b x +a \right ) b -B \ln \left (b x +a \right ) a +b B x}{b^{2}}\) | \(29\) |
risch | \(\frac {B x}{b}+\frac {\ln \left (b x +a \right ) A}{b}-\frac {\ln \left (b x +a \right ) B a}{b^{2}}\) | \(32\) |
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Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x}{a+b x} \, dx=\frac {B b x - {\left (B a - A b\right )} \log \left (b x + a\right )}{b^{2}} \]
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Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {A+B x}{a+b x} \, dx=\frac {B x}{b} - \frac {\left (- A b + B a\right ) \log {\left (a + b x \right )}}{b^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x}{a+b x} \, dx=\frac {B x}{b} - \frac {{\left (B a - A b\right )} \log \left (b x + a\right )}{b^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x}{a+b x} \, dx=\frac {B x}{b} - \frac {{\left (B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{2}} \]
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Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x}{a+b x} \, dx=\frac {B\,x}{b}+\frac {\ln \left (a+b\,x\right )\,\left (A\,b-B\,a\right )}{b^2} \]
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