Integrand size = 30, antiderivative size = 5 \[ \int \frac {(d+e x)^2}{c d^2+2 c d e x+c e^2 x^2} \, dx=\frac {x}{c} \]
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Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {27, 8} \[ \int \frac {(d+e x)^2}{c d^2+2 c d e x+c e^2 x^2} \, dx=\frac {x}{c} \]
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Rule 8
Rule 27
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{c} \, dx \\ & = \frac {x}{c} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{c d^2+2 c d e x+c e^2 x^2} \, dx=\frac {x}{c} \]
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Time = 2.68 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20
method | result | size |
default | \(\frac {x}{c}\) | \(6\) |
risch | \(\frac {x}{c}\) | \(6\) |
norman | \(\frac {\frac {e \,x^{2}}{c}+\frac {d x}{c}}{e x +d}\) | \(24\) |
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none
Time = 0.37 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{c d^2+2 c d e x+c e^2 x^2} \, dx=\frac {x}{c} \]
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Time = 0.02 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.40 \[ \int \frac {(d+e x)^2}{c d^2+2 c d e x+c e^2 x^2} \, dx=\frac {x}{c} \]
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none
Time = 0.19 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{c d^2+2 c d e x+c e^2 x^2} \, dx=\frac {x}{c} \]
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none
Time = 0.29 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{c d^2+2 c d e x+c e^2 x^2} \, dx=\frac {x}{c} \]
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Time = 0.01 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{c d^2+2 c d e x+c e^2 x^2} \, dx=\frac {x}{c} \]
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