Optimal. Leaf size=30 \[ \frac {x+x^2 \left (3+\frac {4-x}{\left (\frac {2 (3+x)}{5}+\log (x)\right )^2}\right )}{x} \]
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Rubi [F]
time = 0.46, 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 {248+398 x+216 x^2+24 x^3+\left (2120+830 x+180 x^2\right ) \log (x)+(1350+450 x) \log ^2(x)+375 \log ^3(x)}{216+216 x+72 x^2+8 x^3+\left (540+360 x+60 x^2\right ) \log (x)+(450+150 x) \log ^2(x)+125 \log ^3(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 {248+398 x+216 x^2+24 x^3+10 \left (212+83 x+18 x^2\right ) \log (x)+450 (3+x) \log ^2(x)+375 \log ^3(x)}{(6+2 x+5 \log (x))^3} \, dx\\ &=\int \left (3+\frac {50 \left (-20-3 x+2 x^2\right )}{(6+2 x+5 \log (x))^3}-\frac {50 (-2+x)}{(6+2 x+5 \log (x))^2}\right ) \, dx\\ &=3 x+50 \int \frac {-20-3 x+2 x^2}{(6+2 x+5 \log (x))^3} \, dx-50 \int \frac {-2+x}{(6+2 x+5 \log (x))^2} \, dx\\ &=3 x+50 \int \left (-\frac {20}{(6+2 x+5 \log (x))^3}-\frac {3 x}{(6+2 x+5 \log (x))^3}+\frac {2 x^2}{(6+2 x+5 \log (x))^3}\right ) \, dx-50 \int \left (-\frac {2}{(6+2 x+5 \log (x))^2}+\frac {x}{(6+2 x+5 \log (x))^2}\right ) \, dx\\ &=3 x-50 \int \frac {x}{(6+2 x+5 \log (x))^2} \, dx+100 \int \frac {x^2}{(6+2 x+5 \log (x))^3} \, dx+100 \int \frac {1}{(6+2 x+5 \log (x))^2} \, dx-150 \int \frac {x}{(6+2 x+5 \log (x))^3} \, dx-1000 \int \frac {1}{(6+2 x+5 \log (x))^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.13, size = 21, normalized size = 0.70 \begin {gather*} 3 x-\frac {25 (-4+x) x}{(6+2 x+5 \log (x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 52, normalized size = 1.73
method | result | size |
risch | \(3 x -\frac {25 \left (x -4\right ) x}{\left (5 \ln \left (x \right )+2 x +6\right )^{2}}\) | \(22\) |
default | \(\frac {-540 \ln \left (x \right )-225 \ln \left (x \right )^{2}-8 x +11 x^{2}+12 x^{3}+75 x \ln \left (x \right )^{2}+60 x^{2} \ln \left (x \right )-324}{\left (5 \ln \left (x \right )+2 x +6\right )^{2}}\) | \(52\) |
norman | \(\frac {-540 \ln \left (x \right )-225 \ln \left (x \right )^{2}-8 x +11 x^{2}+12 x^{3}+75 x \ln \left (x \right )^{2}+60 x^{2} \ln \left (x \right )-324}{\left (5 \ln \left (x \right )+2 x +6\right )^{2}}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 58, normalized size = 1.93 \begin {gather*} \frac {12 \, x^{3} + 75 \, x \log \left (x\right )^{2} + 47 \, x^{2} + 60 \, {\left (x^{2} + 3 \, x\right )} \log \left (x\right ) + 208 \, x}{4 \, x^{2} + 20 \, {\left (x + 3\right )} \log \left (x\right ) + 25 \, \log \left (x\right )^{2} + 24 \, x + 36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 58, normalized size = 1.93 \begin {gather*} \frac {12 \, x^{3} + 75 \, x \log \left (x\right )^{2} + 47 \, x^{2} + 60 \, {\left (x^{2} + 3 \, x\right )} \log \left (x\right ) + 208 \, x}{4 \, x^{2} + 20 \, {\left (x + 3\right )} \log \left (x\right ) + 25 \, \log \left (x\right )^{2} + 24 \, x + 36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 41, normalized size = 1.37 \begin {gather*} 3 x + \frac {- x^{2} + 4 x}{\frac {4 x^{2}}{25} + \frac {24 x}{25} + \left (\frac {4 x}{5} + \frac {12}{5}\right ) \log {\left (x \right )} + \log {\left (x \right )}^{2} + \frac {36}{25}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs.
\(2 (31) = 62\).
time = 0.42, size = 66, normalized size = 2.20 \begin {gather*} 3 \, x - \frac {25 \, {\left (2 \, x^{3} - 3 \, x^{2} - 20 \, x\right )}}{8 \, x^{3} + 40 \, x^{2} \log \left (x\right ) + 50 \, x \log \left (x\right )^{2} + 68 \, x^{2} + 220 \, x \log \left (x\right ) + 125 \, \log \left (x\right )^{2} + 192 \, x + 300 \, \log \left (x\right ) + 180} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.14, size = 38, normalized size = 1.27 \begin {gather*} \frac {x\,\left (12\,x^2+60\,x\,\ln \left (x\right )+47\,x+75\,{\ln \left (x\right )}^2+180\,\ln \left (x\right )+208\right )}{{\left (2\,x+5\,\ln \left (x\right )+6\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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