Integrand size = 38, antiderivative size = 25 \[ \int \frac {4+26 x+x^2+72 x^6-72 x^7+18 x^8}{4 x-4 x^2+x^3} \, dx=-3 \left (\frac {5 x}{-2+x}-x^6+(3+\log (2))^2\right )+\log (x) \]
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Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {1608, 27, 1634} \[ \int \frac {4+26 x+x^2+72 x^6-72 x^7+18 x^8}{4 x-4 x^2+x^3} \, dx=3 x^6+\frac {30}{2-x}+\log (x) \]
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Rule 27
Rule 1608
Rule 1634
Rubi steps \begin{align*} \text {integral}& = \int \frac {4+26 x+x^2+72 x^6-72 x^7+18 x^8}{x \left (4-4 x+x^2\right )} \, dx \\ & = \int \frac {4+26 x+x^2+72 x^6-72 x^7+18 x^8}{(-2+x)^2 x} \, dx \\ & = \int \left (\frac {30}{(-2+x)^2}+\frac {1}{x}+18 x^5\right ) \, dx \\ & = \frac {30}{2-x}+3 x^6+\log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.60 \[ \int \frac {4+26 x+x^2+72 x^6-72 x^7+18 x^8}{4 x-4 x^2+x^3} \, dx=-\frac {30}{-2+x}+3 x^6+\log (x) \]
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Time = 0.07 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.64
method | result | size |
default | \(3 x^{6}+\ln \left (x \right )-\frac {30}{-2+x}\) | \(16\) |
risch | \(3 x^{6}+\ln \left (x \right )-\frac {30}{-2+x}\) | \(16\) |
norman | \(\frac {3 x^{7}-6 x^{6}-30}{-2+x}+\ln \left (x \right )\) | \(22\) |
parallelrisch | \(\frac {3 x^{7}-6 x^{6}+x \ln \left (x \right )-30-2 \ln \left (x \right )}{-2+x}\) | \(27\) |
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Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {4+26 x+x^2+72 x^6-72 x^7+18 x^8}{4 x-4 x^2+x^3} \, dx=\frac {3 \, x^{7} - 6 \, x^{6} + {\left (x - 2\right )} \log \left (x\right ) - 30}{x - 2} \]
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Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.48 \[ \int \frac {4+26 x+x^2+72 x^6-72 x^7+18 x^8}{4 x-4 x^2+x^3} \, dx=3 x^{6} + \log {\left (x \right )} - \frac {30}{x - 2} \]
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Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.60 \[ \int \frac {4+26 x+x^2+72 x^6-72 x^7+18 x^8}{4 x-4 x^2+x^3} \, dx=3 \, x^{6} - \frac {30}{x - 2} + \log \left (x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.64 \[ \int \frac {4+26 x+x^2+72 x^6-72 x^7+18 x^8}{4 x-4 x^2+x^3} \, dx=3 \, x^{6} - \frac {30}{x - 2} + \log \left ({\left | x \right |}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.60 \[ \int \frac {4+26 x+x^2+72 x^6-72 x^7+18 x^8}{4 x-4 x^2+x^3} \, dx=\ln \left (x\right )-\frac {30}{x-2}+3\,x^6 \]
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