Integrand size = 14, antiderivative size = 21 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^4} \, dx=-\frac {3}{x^3}+\frac {12}{x^2}-\frac {22}{x}+x-8 \log (x) \]
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Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {712} \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^4} \, dx=-\frac {3}{x^3}+\frac {12}{x^2}+x-\frac {22}{x}-8 \log (x) \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (1+\frac {9}{x^4}-\frac {24}{x^3}+\frac {22}{x^2}-\frac {8}{x}\right ) \, dx \\ & = -\frac {3}{x^3}+\frac {12}{x^2}-\frac {22}{x}+x-8 \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^4} \, dx=-\frac {3}{x^3}+\frac {12}{x^2}-\frac {22}{x}+x-8 \log (x) \]
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Time = 16.64 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00
method | result | size |
risch | \(x +\frac {-22 x^{2}+12 x -3}{x^{3}}-8 \ln \left (x \right )\) | \(21\) |
default | \(-\frac {3}{x^{3}}+\frac {12}{x^{2}}-\frac {22}{x}+x -8 \ln \left (x \right )\) | \(22\) |
norman | \(\frac {x^{4}-22 x^{2}+12 x -3}{x^{3}}-8 \ln \left (x \right )\) | \(23\) |
parallelrisch | \(-\frac {8 \ln \left (x \right ) x^{3}-x^{4}+3+22 x^{2}-12 x}{x^{3}}\) | \(28\) |
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^4} \, dx=\frac {x^{4} - 8 \, x^{3} \log \left (x\right ) - 22 \, x^{2} + 12 \, x - 3}{x^{3}} \]
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Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^4} \, dx=x - 8 \log {\left (x \right )} + \frac {- 22 x^{2} + 12 x - 3}{x^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^4} \, dx=x - \frac {22 \, x^{2} - 12 \, x + 3}{x^{3}} - 8 \, \log \left (x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^4} \, dx=x - \frac {22 \, x^{2} - 12 \, x + 3}{x^{3}} - 8 \, \log \left ({\left | x \right |}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^4} \, dx=x-8\,\ln \left (x\right )-\frac {22\,x^2-12\,x+3}{x^3} \]
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