Optimal. Leaf size=31 \[ \frac {4}{-e^3-x+\log \left (5 \left (1+(2+x)^2\right ) \left (2-\log \left (\frac {25}{3}\right )\right )\right )} \]
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Rubi [A]
time = 0.19, antiderivative size = 30, normalized size of antiderivative = 0.97, number of steps
used = 4, number of rules used = 4, integrand size = 151, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {27, 12, 6820,
6818} \begin {gather*} -\frac {4}{-\log \left (5 \left (x^2+4 x+5\right ) \left (2-\log \left (\frac {25}{3}\right )\right )\right )+x+e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 6818
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 (1+x)^2}{5 x^2+4 x^3+x^4+e^6 \left (5+4 x+x^2\right )+e^3 \left (10 x+8 x^2+2 x^3\right )+\left (-10 x-8 x^2-2 x^3+e^3 \left (-10-8 x-2 x^2\right )\right ) \log \left (50+40 x+10 x^2-\left (25+20 x+5 x^2\right ) \log \left (\frac {25}{3}\right )\right )+\left (5+4 x+x^2\right ) \log ^2\left (50+40 x+10 x^2-\left (25+20 x+5 x^2\right ) \log \left (\frac {25}{3}\right )\right )} \, dx\\ &=4 \int \frac {(1+x)^2}{5 x^2+4 x^3+x^4+e^6 \left (5+4 x+x^2\right )+e^3 \left (10 x+8 x^2+2 x^3\right )+\left (-10 x-8 x^2-2 x^3+e^3 \left (-10-8 x-2 x^2\right )\right ) \log \left (50+40 x+10 x^2-\left (25+20 x+5 x^2\right ) \log \left (\frac {25}{3}\right )\right )+\left (5+4 x+x^2\right ) \log ^2\left (50+40 x+10 x^2-\left (25+20 x+5 x^2\right ) \log \left (\frac {25}{3}\right )\right )} \, dx\\ &=4 \int \frac {(1+x)^2}{\left (5+4 x+x^2\right ) \left (e^3+x-\log \left (-5 \left (5+4 x+x^2\right ) \left (-2+\log \left (\frac {25}{3}\right )\right )\right )\right )^2} \, dx\\ &=-\frac {4}{e^3+x-\log \left (5 \left (5+4 x+x^2\right ) \left (2-\log \left (\frac {25}{3}\right )\right )\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.90 \begin {gather*} -\frac {4}{e^3+x-\log \left (-5 \left (5+4 x+x^2\right ) \left (-2+\log \left (\frac {25}{3}\right )\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.50, size = 35, normalized size = 1.13
method | result | size |
norman | \(-\frac {4}{{\mathrm e}^{3}+x -\ln \left (\left (5 x^{2}+20 x +25\right ) \ln \left (\frac {3}{25}\right )+10 x^{2}+40 x +50\right )}\) | \(35\) |
risch | \(-\frac {4}{{\mathrm e}^{3}+x -\ln \left (\left (5 x^{2}+20 x +25\right ) \left (\ln \left (3\right )-2 \ln \left (5\right )\right )+10 x^{2}+40 x +50\right )}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.50, size = 37, normalized size = 1.19 \begin {gather*} \frac {4}{i \, \pi - x - e^{3} + \log \left (5\right ) + \log \left (x^{2} + 4 \, x + 5\right ) + \log \left (2 \, \log \left (5\right ) - \log \left (3\right ) - 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 33, normalized size = 1.06 \begin {gather*} -\frac {4}{x + e^{3} - \log \left (10 \, x^{2} + 5 \, {\left (x^{2} + 4 \, x + 5\right )} \log \left (\frac {3}{25}\right ) + 40 \, x + 50\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 32, normalized size = 1.03 \begin {gather*} \frac {4}{- x + \log {\left (10 x^{2} + 40 x + \left (5 x^{2} + 20 x + 25\right ) \log {\left (\frac {3}{25} \right )} + 50 \right )} - e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 40, normalized size = 1.29 \begin {gather*} -\frac {4}{x + e^{3} - \log \left (5\right ) - \log \left (x^{2} \log \left (\frac {3}{25}\right ) + 2 \, x^{2} + 4 \, x \log \left (\frac {3}{25}\right ) + 8 \, x + 5 \, \log \left (\frac {3}{25}\right ) + 10\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.11, size = 34, normalized size = 1.10 \begin {gather*} -\frac {4}{x-\ln \left (40\,x+\ln \left (\frac {3}{25}\right )\,\left (5\,x^2+20\,x+25\right )+10\,x^2+50\right )+{\mathrm {e}}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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