Integrand size = 33, antiderivative size = 16 \[ \int \frac {d+e x}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\log (a e+c d x)}{c d} \]
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Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {640, 31} \[ \int \frac {d+e x}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\log (a e+c d x)}{c d} \]
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Rule 31
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{a e+c d x} \, dx \\ & = \frac {\log (a e+c d x)}{c d} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {d+e x}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\log (a e+c d x)}{c d} \]
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Time = 2.66 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06
method | result | size |
default | \(\frac {\ln \left (c d x +a e \right )}{c d}\) | \(17\) |
norman | \(\frac {\ln \left (c d x +a e \right )}{c d}\) | \(17\) |
risch | \(\frac {\ln \left (c d x +a e \right )}{c d}\) | \(17\) |
parallelrisch | \(\frac {\ln \left (c d x +a e \right )}{c d}\) | \(17\) |
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none
Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {d+e x}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\log \left (c d x + a e\right )}{c d} \]
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Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {d+e x}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\log {\left (a e + c d x \right )}}{c d} \]
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none
Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {d+e x}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\log \left (c d x + a e\right )}{c d} \]
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none
Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {d+e x}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\log \left ({\left | c d x + a e \right |}\right )}{c d} \]
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Time = 9.67 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {d+e x}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\ln \left (a\,e+c\,d\,x\right )}{c\,d} \]
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