Optimal. Leaf size=23 \[ (4+x) \left (-3+x+\frac {x^2}{-1+x}\right )-\log (2 x) \]
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Rubi [A]
time = 0.05, antiderivative size = 22, normalized size of antiderivative = 0.96, number of steps
used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1608, 27, 1634}
\begin {gather*} 2 x^2+6 x-\frac {5}{1-x}-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 1608
Rule 1634
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+3 x-9 x^2-2 x^3+4 x^4}{x \left (1-2 x+x^2\right )} \, dx\\ &=\int \frac {-1+3 x-9 x^2-2 x^3+4 x^4}{(-1+x)^2 x} \, dx\\ &=\int \left (6-\frac {5}{(-1+x)^2}-\frac {1}{x}+4 x\right ) \, dx\\ &=-\frac {5}{1-x}+6 x+2 x^2-\log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.10, size = 20, normalized size = 0.87 \begin {gather*} \frac {5}{-1+x}+6 x+2 x^2-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 21, normalized size = 0.91
method | result | size |
default | \(2 x^{2}+6 x +\frac {5}{x -1}-\ln \left (x \right )\) | \(21\) |
risch | \(2 x^{2}+6 x +\frac {5}{x -1}-\ln \left (x \right )\) | \(21\) |
norman | \(\frac {2 x^{3}+4 x^{2}-1}{x -1}-\ln \left (x \right )\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 20, normalized size = 0.87 \begin {gather*} 2 \, x^{2} + 6 \, x + \frac {5}{x - 1} - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 28, normalized size = 1.22 \begin {gather*} \frac {2 \, x^{3} + 4 \, x^{2} - {\left (x - 1\right )} \log \left (x\right ) - 6 \, x + 5}{x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 15, normalized size = 0.65 \begin {gather*} 2 x^{2} + 6 x - \log {\left (x \right )} + \frac {5}{x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 21, normalized size = 0.91 \begin {gather*} 2 \, x^{2} + 6 \, x + \frac {5}{x - 1} - \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 20, normalized size = 0.87 \begin {gather*} 6\,x-\ln \left (x\right )+\frac {5}{x-1}+2\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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