Integrand size = 30, antiderivative size = 17 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{3 c^3 e (d+e x)^3} \]
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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} \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{3 c^3 e (d+e x)^3} \]
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Rule 12
Rule 27
Rule 32
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{c^3 (d+e x)^4} \, dx \\ & = \frac {\int \frac {1}{(d+e x)^4} \, dx}{c^3} \\ & = -\frac {1}{3 c^3 e (d+e x)^3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{3 c^3 e (d+e x)^3} \]
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Time = 2.43 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(-\frac {1}{3 c^{3} e \left (e x +d \right )^{3}}\) | \(16\) |
default | \(-\frac {1}{3 c^{3} e \left (e x +d \right )^{3}}\) | \(16\) |
risch | \(-\frac {1}{3 e \left (e x +d \right ) \left (x^{2} e^{2}+2 d e x +d^{2}\right ) c^{3}}\) | \(34\) |
parallelrisch | \(-\frac {1}{3 e \left (e x +d \right ) \left (x^{2} e^{2}+2 d e x +d^{2}\right ) c^{3}}\) | \(34\) |
norman | \(\frac {-\frac {d^{2}}{3 e c}-\frac {e \,x^{2}}{3 c}-\frac {2 d x}{3 c}}{c^{2} \left (e x +d \right )^{5}}\) | \(40\) |
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Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (15) = 30\).
Time = 0.34 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.76 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{3 \, {\left (c^{3} e^{4} x^{3} + 3 \, c^{3} d e^{3} x^{2} + 3 \, c^{3} d^{2} e^{2} x + c^{3} d^{3} e\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (15) = 30\).
Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 3.00 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=- \frac {1}{3 c^{3} d^{3} e + 9 c^{3} d^{2} e^{2} x + 9 c^{3} d e^{3} x^{2} + 3 c^{3} e^{4} x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (15) = 30\).
Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.76 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{3 \, {\left (c^{3} e^{4} x^{3} + 3 \, c^{3} d e^{3} x^{2} + 3 \, c^{3} d^{2} e^{2} x + c^{3} d^{3} e\right )}} \]
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none
Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{3 \, {\left (e x + d\right )}^{3} c^{3} e} \]
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Time = 9.78 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.88 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=-\frac {1}{3\,c^3\,d^3\,e+9\,c^3\,d^2\,e^2\,x+9\,c^3\,d\,e^3\,x^2+3\,c^3\,e^4\,x^3} \]
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