Integrand size = 22, antiderivative size = 15 \[ \int \frac {1}{c d^2+2 c d e x+c e^2 x^2} \, dx=-\frac {1}{c e (d+e x)} \]
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Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {27, 12, 32} \[ \int \frac {1}{c d^2+2 c d e x+c e^2 x^2} \, dx=-\frac {1}{c e (d+e x)} \]
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Rule 12
Rule 27
Rule 32
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{c (d+e x)^2} \, dx \\ & = \frac {\int \frac {1}{(d+e x)^2} \, dx}{c} \\ & = -\frac {1}{c e (d+e x)} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c d^2+2 c d e x+c e^2 x^2} \, dx=-\frac {1}{c e (d+e x)} \]
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Time = 2.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07
method | result | size |
gosper | \(-\frac {1}{c e \left (e x +d \right )}\) | \(16\) |
default | \(-\frac {1}{c e \left (e x +d \right )}\) | \(16\) |
norman | \(\frac {x}{c d \left (e x +d \right )}\) | \(16\) |
risch | \(-\frac {1}{c e \left (e x +d \right )}\) | \(16\) |
parallelrisch | \(\frac {x}{c d \left (e x +d \right )}\) | \(16\) |
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Time = 0.35 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c d^2+2 c d e x+c e^2 x^2} \, dx=-\frac {1}{c e^{2} x + c d e} \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {1}{c d^2+2 c d e x+c e^2 x^2} \, dx=- \frac {1}{c d e + c e^{2} x} \]
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Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c d^2+2 c d e x+c e^2 x^2} \, dx=-\frac {1}{c e^{2} x + c d e} \]
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Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c d^2+2 c d e x+c e^2 x^2} \, dx=-\frac {1}{{\left (e x + d\right )} c e} \]
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Time = 9.59 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c d^2+2 c d e x+c e^2 x^2} \, dx=-\frac {1}{c\,x\,e^2+c\,d\,e} \]
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