Integrand size = 30, antiderivative size = 17 \[ \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{7 c^3 e (d+e x)^7} \]
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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} \[ \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{7 c^3 e (d+e x)^7} \]
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Rule 12
Rule 27
Rule 32
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{c^3 (d+e x)^8} \, dx \\ & = \frac {\int \frac {1}{(d+e x)^8} \, dx}{c^3} \\ & = -\frac {1}{7 c^3 e (d+e x)^7} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{7 c^3 e (d+e x)^7} \]
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Time = 2.47 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {1}{7 c^{3} e \left (e x +d \right )^{7}}\) | \(16\) |
norman | \(-\frac {1}{7 c^{3} e \left (e x +d \right )^{7}}\) | \(16\) |
gosper | \(-\frac {1}{7 \left (e x +d \right ) e \,c^{3} \left (x^{2} e^{2}+2 d e x +d^{2}\right )^{3}}\) | \(34\) |
risch | \(-\frac {1}{7 e \left (e x +d \right )^{3} c^{3} \left (x^{2} e^{2}+2 d e x +d^{2}\right )^{2}}\) | \(34\) |
parallelrisch | \(-\frac {1}{7 e \left (e x +d \right )^{3} c^{3} \left (x^{2} e^{2}+2 d e x +d^{2}\right )^{2}}\) | \(34\) |
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Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (15) = 30\).
Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 6.06 \[ \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{7 \, {\left (c^{3} e^{8} x^{7} + 7 \, c^{3} d e^{7} x^{6} + 21 \, c^{3} d^{2} e^{6} x^{5} + 35 \, c^{3} d^{3} e^{5} x^{4} + 35 \, c^{3} d^{4} e^{4} x^{3} + 21 \, c^{3} d^{5} e^{3} x^{2} + 7 \, c^{3} d^{6} e^{2} x + c^{3} d^{7} e\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (15) = 30\).
Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 6.59 \[ \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=- \frac {1}{7 c^{3} d^{7} e + 49 c^{3} d^{6} e^{2} x + 147 c^{3} d^{5} e^{3} x^{2} + 245 c^{3} d^{4} e^{4} x^{3} + 245 c^{3} d^{3} e^{5} x^{4} + 147 c^{3} d^{2} e^{6} x^{5} + 49 c^{3} d e^{7} x^{6} + 7 c^{3} e^{8} x^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (15) = 30\).
Time = 0.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 6.06 \[ \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{7 \, {\left (c^{3} e^{8} x^{7} + 7 \, c^{3} d e^{7} x^{6} + 21 \, c^{3} d^{2} e^{6} x^{5} + 35 \, c^{3} d^{3} e^{5} x^{4} + 35 \, c^{3} d^{4} e^{4} x^{3} + 21 \, c^{3} d^{5} e^{3} x^{2} + 7 \, c^{3} d^{6} e^{2} x + c^{3} d^{7} e\right )}} \]
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none
Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{7 \, {\left (e x + d\right )}^{7} c^{3} e} \]
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Time = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 6.18 \[ \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{7\,c^3\,d^7\,e+49\,c^3\,d^6\,e^2\,x+147\,c^3\,d^5\,e^3\,x^2+245\,c^3\,d^4\,e^4\,x^3+245\,c^3\,d^3\,e^5\,x^4+147\,c^3\,d^2\,e^6\,x^5+49\,c^3\,d\,e^7\,x^6+7\,c^3\,e^8\,x^7} \]
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