Optimal. Leaf size=20 \[ -e^{2+\frac {8}{1+x}}-\log \left (x^5\right ) \]
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Rubi [A] time = 0.47, antiderivative size = 18, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {1594, 27, 6742, 2230, 2209} \begin {gather*} -e^{\frac {8}{x+1}+2}-5 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 1594
Rule 2209
Rule 2230
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-5-10 x+8 e^{\frac {10+2 x}{1+x}} x-5 x^2}{x \left (1+2 x+x^2\right )} \, dx\\ &=\int \frac {-5-10 x+8 e^{\frac {10+2 x}{1+x}} x-5 x^2}{x (1+x)^2} \, dx\\ &=\int \left (-\frac {5}{x}+\frac {8 e^{\frac {2 (5+x)}{1+x}}}{(1+x)^2}\right ) \, dx\\ &=-5 \log (x)+8 \int \frac {e^{\frac {2 (5+x)}{1+x}}}{(1+x)^2} \, dx\\ &=-5 \log (x)+8 \int \frac {e^{2+\frac {8}{1+x}}}{(1+x)^2} \, dx\\ &=-e^{2+\frac {8}{1+x}}-5 \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.13, size = 18, normalized size = 0.90 \begin {gather*} -e^{2+\frac {8}{1+x}}-5 \log (x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 18, normalized size = 0.90 \begin {gather*} -e^{\left (\frac {2 \, {\left (x + 5\right )}}{x + 1}\right )} - 5 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 44, normalized size = 2.20 \begin {gather*} -e^{\left (\frac {2 \, {\left (x + 5\right )}}{x + 1}\right )} + 5 \, \log \left (\frac {2 \, {\left (x + 5\right )}}{x + 1} - 2\right ) - 5 \, \log \left (\frac {2 \, {\left (x + 5\right )}}{x + 1} - 10\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 19, normalized size = 0.95
method | result | size |
risch | \(-5 \ln \relax (x )-{\mathrm e}^{\frac {2 x +10}{x +1}}\) | \(19\) |
derivativedivides | \(-5 \ln \left (-8+\frac {8}{x +1}\right )+5 \ln \left (\frac {8}{x +1}\right )-{\mathrm e}^{2+\frac {8}{x +1}}\) | \(36\) |
default | \(-5 \ln \left (-8+\frac {8}{x +1}\right )+5 \ln \left (\frac {8}{x +1}\right )-{\mathrm e}^{2+\frac {8}{x +1}}\) | \(36\) |
norman | \(\frac {-x \,{\mathrm e}^{\frac {2 x +10}{x +1}}-{\mathrm e}^{\frac {2 x +10}{x +1}}}{x +1}-5 \ln \relax (x )\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 17, normalized size = 0.85 \begin {gather*} -e^{\left (\frac {8}{x + 1} + 2\right )} - 5 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 24, normalized size = 1.20 \begin {gather*} -5\,\ln \relax (x)-{\mathrm {e}}^{\frac {2\,x}{x+1}}\,{\mathrm {e}}^{\frac {10}{x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 15, normalized size = 0.75 \begin {gather*} - e^{\frac {2 x + 10}{x + 1}} - 5 \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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