Optimal. Leaf size=22 \[ \frac {x}{1+\frac {x}{-5+\left (25 x^2+\log (x)\right )^2}} \]
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Rubi [F] time = 1.55, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {25+50 x^3-6250 x^4+2500 x^5+390625 x^8+\left (2 x-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)}{25-10 x+x^2-6250 x^4+1250 x^5+390625 x^8+\left (-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+2 x+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 \left (1+2 x^3-250 x^4+100 x^5+15625 x^8\right )+2 x \left (1-250 x+50 x^2+31250 x^5\right ) \log (x)+10 \left (-1+375 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)}{\left (5-x-625 x^4-50 x^2 \log (x)-\log ^2(x)\right )^2} \, dx\\ &=\int \left (1+\frac {x \left (x+50 x^2+2500 x^4+2 \log (x)+100 x^2 \log (x)\right )}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2}-\frac {2 x}{-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)}\right ) \, dx\\ &=x-2 \int \frac {x}{-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)} \, dx+\int \frac {x \left (x+50 x^2+2500 x^4+2 \log (x)+100 x^2 \log (x)\right )}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2} \, dx\\ &=x-2 \int \frac {x}{-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)} \, dx+\int \left (\frac {x^2}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2}+\frac {50 x^3}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2}+\frac {2500 x^5}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2}+\frac {2 x \log (x)}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2}+\frac {100 x^3 \log (x)}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2}\right ) \, dx\\ &=x+2 \int \frac {x \log (x)}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2} \, dx-2 \int \frac {x}{-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)} \, dx+50 \int \frac {x^3}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2} \, dx+100 \int \frac {x^3 \log (x)}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2} \, dx+2500 \int \frac {x^5}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x^2}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 28, normalized size = 1.27 \begin {gather*} x-\frac {x^2}{-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 44, normalized size = 2.00 \begin {gather*} \frac {625 \, x^{5} + 50 \, x^{3} \log \relax (x) + x \log \relax (x)^{2} - 5 \, x}{625 \, x^{4} + 50 \, x^{2} \log \relax (x) + \log \relax (x)^{2} + x - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 28, normalized size = 1.27 \begin {gather*} x - \frac {x^{2}}{625 \, x^{4} + 50 \, x^{2} \log \relax (x) + \log \relax (x)^{2} + x - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 29, normalized size = 1.32
method | result | size |
risch | \(x -\frac {x^{2}}{625 x^{4}+50 x^{2} \ln \relax (x )+\ln \relax (x )^{2}+x -5}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 44, normalized size = 2.00 \begin {gather*} \frac {625 \, x^{5} + 50 \, x^{3} \log \relax (x) + x \log \relax (x)^{2} - 5 \, x}{625 \, x^{4} + 50 \, x^{2} \log \relax (x) + \log \relax (x)^{2} + x - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.98, size = 28, normalized size = 1.27 \begin {gather*} x-\frac {x^2}{625\,x^4+50\,x^2\,\ln \relax (x)+x+{\ln \relax (x)}^2-5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 26, normalized size = 1.18 \begin {gather*} - \frac {x^{2}}{625 x^{4} + 50 x^{2} \log {\relax (x )} + x + \log {\relax (x )}^{2} - 5} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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