Integrand size = 30, antiderivative size = 17 \[ \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )} \, dx=-\frac {1}{3 c e (d+e x)^3} \]
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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} \[ \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )} \, dx=-\frac {1}{3 c e (d+e x)^3} \]
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Rule 12
Rule 27
Rule 32
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{c (d+e x)^4} \, dx \\ & = \frac {\int \frac {1}{(d+e x)^4} \, dx}{c} \\ & = -\frac {1}{3 c e (d+e x)^3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )} \, dx=-\frac {1}{3 c e (d+e x)^3} \]
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Time = 2.48 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {1}{3 c e \left (e x +d \right )^{3}}\) | \(16\) |
norman | \(-\frac {1}{3 c e \left (e x +d \right )^{3}}\) | \(16\) |
gosper | \(-\frac {1}{3 \left (e x +d \right ) e c \left (x^{2} e^{2}+2 d e x +d^{2}\right )}\) | \(34\) |
risch | \(-\frac {1}{3 \left (e x +d \right ) e c \left (x^{2} e^{2}+2 d e x +d^{2}\right )}\) | \(34\) |
parallelrisch | \(-\frac {1}{3 \left (e x +d \right ) e c \left (x^{2} e^{2}+2 d e x +d^{2}\right )}\) | \(34\) |
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (15) = 30\).
Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.29 \[ \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )} \, dx=-\frac {1}{3 \, {\left (c e^{4} x^{3} + 3 \, c d e^{3} x^{2} + 3 \, c d^{2} e^{2} x + c d^{3} e\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (14) = 28\).
Time = 0.13 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.59 \[ \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )} \, dx=- \frac {1}{3 c d^{3} e + 9 c d^{2} e^{2} x + 9 c d e^{3} x^{2} + 3 c e^{4} x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (15) = 30\).
Time = 0.21 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.29 \[ \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )} \, dx=-\frac {1}{3 \, {\left (c e^{4} x^{3} + 3 \, c d e^{3} x^{2} + 3 \, c d^{2} e^{2} x + c d^{3} e\right )}} \]
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none
Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )} \, dx=-\frac {1}{3 \, {\left (e x + d\right )}^{3} c e} \]
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Time = 9.64 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.41 \[ \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )} \, dx=-\frac {1}{3\,c\,d^3\,e+9\,c\,d^2\,e^2\,x+9\,c\,d\,e^3\,x^2+3\,c\,e^4\,x^3} \]
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