Integrand size = 30, antiderivative size = 17 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {c^2}{3 e (d+e x)^3} \]
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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} \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {c^2}{3 e (d+e x)^3} \]
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Rule 12
Rule 27
Rule 32
Rubi steps \begin{align*} \text {integral}& = \int \frac {c^2}{(d+e x)^4} \, dx \\ & = c^2 \int \frac {1}{(d+e x)^4} \, dx \\ & = -\frac {c^2}{3 e (d+e x)^3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {c^2}{3 e (d+e x)^3} \]
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Time = 2.14 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(-\frac {c^{2}}{3 e \left (e x +d \right )^{3}}\) | \(16\) |
default | \(-\frac {c^{2}}{3 e \left (e x +d \right )^{3}}\) | \(16\) |
risch | \(-\frac {c^{2}}{3 e \left (e x +d \right )^{3}}\) | \(16\) |
parallelrisch | \(-\frac {c^{2}}{3 e \left (e x +d \right )^{3}}\) | \(16\) |
norman | \(\frac {-\frac {c^{2} d^{4}}{3 e}-\frac {c^{2} x^{4} e^{3}}{3}-\frac {4 d \,e^{2} c^{2} x^{3}}{3}-2 c^{2} d^{2} e \,x^{2}-\frac {4 c^{2} d^{3} x}{3}}{\left (e x +d \right )^{7}}\) | \(65\) |
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Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (15) = 30\).
Time = 0.31 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.24 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {c^{2}}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (14) = 28\).
Time = 0.13 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.29 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^8} \, dx=- \frac {c^{2}}{3 d^{3} e + 9 d^{2} e^{2} x + 9 d e^{3} x^{2} + 3 e^{4} x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (15) = 30\).
Time = 0.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.24 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {c^{2}}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]
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none
Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {c^{2}}{3 \, {\left (e x + d\right )}^{3} e} \]
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Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.18 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {c^2}{3\,e\,\left (d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3\right )} \]
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