Optimal. Leaf size=35 \[ 5+\frac {1}{2} (-2+x)+\frac {1}{4} \left (3+\frac {5}{x}+\frac {1}{4} \log \left (5-x-x^2\right )\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 28, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1594, 1628, 628} \begin {gather*} \frac {1}{16} \log \left (-x^2-x+5\right )+\frac {x}{2}+\frac {5}{4 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 1594
Rule 1628
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {100-20 x-59 x^2+10 x^3+8 x^4}{x^2 \left (-80+16 x+16 x^2\right )} \, dx\\ &=\int \left (\frac {1}{2}-\frac {5}{4 x^2}+\frac {1+2 x}{16 \left (-5+x+x^2\right )}\right ) \, dx\\ &=\frac {5}{4 x}+\frac {x}{2}+\frac {1}{16} \int \frac {1+2 x}{-5+x+x^2} \, dx\\ &=\frac {5}{4 x}+\frac {x}{2}+\frac {1}{16} \log \left (5-x-x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.69 \begin {gather*} \frac {1}{16} \left (\frac {20}{x}+8 x+\log \left (5-x-x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 21, normalized size = 0.60 \begin {gather*} \frac {8 \, x^{2} + x \log \left (x^{2} + x - 5\right ) + 20}{16 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.32, size = 19, normalized size = 0.54 \begin {gather*} \frac {1}{2} \, x + \frac {5}{4 \, x} + \frac {1}{16} \, \log \left ({\left | x^{2} + x - 5 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 19, normalized size = 0.54
method | result | size |
default | \(\frac {x}{2}+\frac {\ln \left (x^{2}+x -5\right )}{16}+\frac {5}{4 x}\) | \(19\) |
risch | \(\frac {x}{2}+\frac {\ln \left (x^{2}+x -5\right )}{16}+\frac {5}{4 x}\) | \(19\) |
norman | \(\frac {\frac {5}{4}+\frac {x^{2}}{2}}{x}+\frac {\ln \left (x^{2}+x -5\right )}{16}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 18, normalized size = 0.51 \begin {gather*} \frac {1}{2} \, x + \frac {5}{4 \, x} + \frac {1}{16} \, \log \left (x^{2} + x - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 18, normalized size = 0.51 \begin {gather*} \frac {x}{2}+\frac {\ln \left (x^2+x-5\right )}{16}+\frac {5}{4\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 17, normalized size = 0.49 \begin {gather*} \frac {x}{2} + \frac {\log {\left (x^{2} + x - 5 \right )}}{16} + \frac {5}{4 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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