Integrand size = 19, antiderivative size = 11 \[ \int \frac {-1+2 x+5 x^2}{\left (1+x+x^2\right )^4} \, dx=-\frac {x}{\left (1+x+x^2\right )^3} \]
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Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {1602} \[ \int \frac {-1+2 x+5 x^2}{\left (1+x+x^2\right )^4} \, dx=-\frac {x}{\left (x^2+x+1\right )^3} \]
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Rule 1602
Rubi steps \begin{align*} \text {integral}& = -\frac {x}{\left (1+x+x^2\right )^3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x+5 x^2}{\left (1+x+x^2\right )^4} \, dx=-\frac {x}{\left (1+x+x^2\right )^3} \]
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Time = 0.76 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
gosper | \(-\frac {x}{\left (x^{2}+x +1\right )^{3}}\) | \(12\) |
default | \(-\frac {x}{\left (x^{2}+x +1\right )^{3}}\) | \(12\) |
norman | \(-\frac {x}{\left (x^{2}+x +1\right )^{3}}\) | \(12\) |
risch | \(-\frac {x}{\left (x^{2}+x +1\right )^{3}}\) | \(12\) |
parallelrisch | \(-\frac {x}{\left (x^{2}+x +1\right )^{3}}\) | \(12\) |
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (11) = 22\).
Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 3.00 \[ \int \frac {-1+2 x+5 x^2}{\left (1+x+x^2\right )^4} \, dx=-\frac {x}{x^{6} + 3 \, x^{5} + 6 \, x^{4} + 7 \, x^{3} + 6 \, x^{2} + 3 \, x + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (10) = 20\).
Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.82 \[ \int \frac {-1+2 x+5 x^2}{\left (1+x+x^2\right )^4} \, dx=- \frac {x}{x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (11) = 22\).
Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 3.00 \[ \int \frac {-1+2 x+5 x^2}{\left (1+x+x^2\right )^4} \, dx=-\frac {x}{x^{6} + 3 \, x^{5} + 6 \, x^{4} + 7 \, x^{3} + 6 \, x^{2} + 3 \, x + 1} \]
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none
Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x+5 x^2}{\left (1+x+x^2\right )^4} \, dx=-\frac {x}{{\left (x^{2} + x + 1\right )}^{3}} \]
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Time = 12.85 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x+5 x^2}{\left (1+x+x^2\right )^4} \, dx=-\frac {x}{{\left (x^2+x+1\right )}^3} \]
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