Integrand size = 30, antiderivative size = 5 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {x}{c^2} \]
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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} \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {x}{c^2} \]
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Rule 8
Rule 27
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{c^2} \, dx \\ & = \frac {x}{c^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {x}{c^2} \]
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Time = 2.80 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20
method | result | size |
default | \(\frac {x}{c^{2}}\) | \(6\) |
risch | \(\frac {x}{c^{2}}\) | \(6\) |
norman | \(\frac {\frac {e^{3} x^{4}}{c}+\frac {d^{3} x}{c}+\frac {3 e^{2} d \,x^{3}}{c}+\frac {3 e \,d^{2} x^{2}}{c}}{c \left (e x +d \right )^{3}}\) | \(55\) |
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none
Time = 0.25 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {x}{c^{2}} \]
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Time = 0.04 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.60 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {x}{c^{2}} \]
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none
Time = 0.19 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {x}{c^{2}} \]
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none
Time = 0.29 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {x}{c^{2}} \]
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Time = 0.01 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {x}{c^2} \]
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