Integrand size = 21, antiderivative size = 36 \[ \int \frac {1}{a c+(b c+a d) x+b d x^2} \, dx=\frac {\log (a+b x)}{b c-a d}-\frac {\log (c+d x)}{b c-a d} \]
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Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {630, 31} \[ \int \frac {1}{a c+(b c+a d) x+b d x^2} \, dx=\frac {\log (a+b x)}{b c-a d}-\frac {\log (c+d x)}{b c-a d} \]
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Rule 31
Rule 630
Rubi steps \begin{align*} \text {integral}& = -\frac {(b d) \int \frac {1}{b c+b d x} \, dx}{b c-a d}+\frac {(b d) \int \frac {1}{a d+b d x} \, dx}{b c-a d} \\ & = \frac {\log (a+b x)}{b c-a d}-\frac {\log (c+d x)}{b c-a d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {1}{a c+(b c+a d) x+b d x^2} \, dx=\frac {\log (a+b x)-\log (c+d x)}{b c-a d} \]
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Time = 2.36 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78
method | result | size |
parallelrisch | \(-\frac {\ln \left (b x +a \right )-\ln \left (d x +c \right )}{a d -b c}\) | \(28\) |
default | \(\frac {\ln \left (d x +c \right )}{a d -b c}-\frac {\ln \left (b x +a \right )}{a d -b c}\) | \(37\) |
norman | \(\frac {\ln \left (d x +c \right )}{a d -b c}-\frac {\ln \left (b x +a \right )}{a d -b c}\) | \(37\) |
risch | \(-\frac {\ln \left (b x +a \right )}{a d -b c}+\frac {\ln \left (-d x -c \right )}{a d -b c}\) | \(40\) |
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Time = 0.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {1}{a c+(b c+a d) x+b d x^2} \, dx=\frac {\log \left (b x + a\right ) - \log \left (d x + c\right )}{b c - a d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (26) = 52\).
Time = 0.17 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.56 \[ \int \frac {1}{a c+(b c+a d) x+b d x^2} \, dx=\frac {\log {\left (x + \frac {- \frac {a^{2} d^{2}}{a d - b c} + \frac {2 a b c d}{a d - b c} + a d - \frac {b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{a d - b c} - \frac {\log {\left (x + \frac {\frac {a^{2} d^{2}}{a d - b c} - \frac {2 a b c d}{a d - b c} + a d + \frac {b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{a d - b c} \]
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Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a c+(b c+a d) x+b d x^2} \, dx=\frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d} \]
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.28 \[ \int \frac {1}{a c+(b c+a d) x+b d x^2} \, dx=\frac {b \log \left ({\left | b x + a \right |}\right )}{b^{2} c - a b d} - \frac {d \log \left ({\left | d x + c \right |}\right )}{b c d - a d^{2}} \]
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Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11 \[ \int \frac {1}{a c+(b c+a d) x+b d x^2} \, dx=\frac {\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{a\,d-b\,c} \]
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