Integrand size = 30, antiderivative size = 17 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{2 c^3 e (d+e x)^2} \]
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Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32} \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{2 c^3 e (d+e x)^2} \]
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Rule 12
Rule 27
Rule 32
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{c^3 (d+e x)^3} \, dx \\ & = \frac {\int \frac {1}{(d+e x)^3} \, dx}{c^3} \\ & = -\frac {1}{2 c^3 e (d+e x)^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{2 c^3 e (d+e x)^2} \]
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Time = 2.49 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(-\frac {1}{2 c^{3} e \left (e x +d \right )^{2}}\) | \(16\) |
default | \(-\frac {1}{2 c^{3} e \left (e x +d \right )^{2}}\) | \(16\) |
risch | \(-\frac {1}{2 c^{3} e \left (e x +d \right )^{2}}\) | \(16\) |
parallelrisch | \(-\frac {1}{2 c^{3} e \left (e x +d \right )^{2}}\) | \(16\) |
norman | \(\frac {-\frac {d^{3}}{2 e c}-\frac {e^{2} x^{3}}{2 c}-\frac {3 e d \,x^{2}}{2 c}-\frac {3 d^{2} x}{2 c}}{c^{2} \left (e x +d \right )^{5}}\) | \(54\) |
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).
Time = 0.35 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{2 \, {\left (c^{3} e^{3} x^{2} + 2 \, c^{3} d e^{2} x + c^{3} d^{2} e\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (15) = 30\).
Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.12 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=- \frac {1}{2 c^{3} d^{2} e + 4 c^{3} d e^{2} x + 2 c^{3} e^{3} x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).
Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{2 \, {\left (c^{3} e^{3} x^{2} + 2 \, c^{3} d e^{2} x + c^{3} d^{2} e\right )}} \]
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none
Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{2 \, {\left (d^{2} + {\left (e x^{2} + 2 \, d x\right )} e\right )} c^{3} e} \]
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Time = 9.78 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.06 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{2\,c^3\,d^2\,e+4\,c^3\,d\,e^2\,x+2\,c^3\,e^3\,x^2} \]
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