Integrand size = 29, antiderivative size = 26 \[ \int \frac {(a+b x)^2}{a c+(b c+a d) x+b d x^2} \, dx=\frac {b x}{d}-\frac {(b c-a d) \log (c+d x)}{d^2} \]
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Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 45} \[ \int \frac {(a+b x)^2}{a c+(b c+a d) x+b d x^2} \, dx=\frac {b x}{d}-\frac {(b c-a d) \log (c+d x)}{d^2} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {a+b x}{c+d x} \, dx \\ & = \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx \\ & = \frac {b x}{d}-\frac {(b c-a d) \log (c+d x)}{d^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^2}{a c+(b c+a d) x+b d x^2} \, dx=\frac {b x}{d}+\frac {(-b c+a d) \log (c+d x)}{d^2} \]
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Time = 2.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {b x}{d}+\frac {\left (a d -b c \right ) \ln \left (d x +c \right )}{d^{2}}\) | \(26\) |
norman | \(\frac {b x}{d}+\frac {\left (a d -b c \right ) \ln \left (d x +c \right )}{d^{2}}\) | \(26\) |
parallelrisch | \(\frac {\ln \left (d x +c \right ) a d -\ln \left (d x +c \right ) b c +b d x}{d^{2}}\) | \(29\) |
risch | \(\frac {b x}{d}+\frac {\ln \left (d x +c \right ) a}{d}-\frac {\ln \left (d x +c \right ) b c}{d^{2}}\) | \(32\) |
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Time = 0.38 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^2}{a c+(b c+a d) x+b d x^2} \, dx=\frac {b d x - {\left (b c - a d\right )} \log \left (d x + c\right )}{d^{2}} \]
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Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b x)^2}{a c+(b c+a d) x+b d x^2} \, dx=\frac {b x}{d} + \frac {\left (a d - b c\right ) \log {\left (c + d x \right )}}{d^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^2}{a c+(b c+a d) x+b d x^2} \, dx=\frac {b x}{d} - \frac {{\left (b c - a d\right )} \log \left (d x + c\right )}{d^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b x)^2}{a c+(b c+a d) x+b d x^2} \, dx=\frac {b x}{d} - \frac {{\left (b c - a d\right )} \log \left ({\left | d x + c \right |}\right )}{d^{2}} \]
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Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^2}{a c+(b c+a d) x+b d x^2} \, dx=\frac {\ln \left (c+d\,x\right )\,\left (a\,d-b\,c\right )}{d^2}+\frac {b\,x}{d} \]
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