Optimal. Leaf size=18 \[ e^{2 x}+x+\frac {-28+x^2}{1+x} \]
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Rubi [A] time = 0.10, antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {27, 6742, 2194, 683} \begin {gather*} 2 x+e^{2 x}-\frac {27}{x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 683
Rule 2194
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {29+4 x+2 x^2+e^{2 x} \left (2+4 x+2 x^2\right )}{(1+x)^2} \, dx\\ &=\int \left (2 e^{2 x}+\frac {29+4 x+2 x^2}{(1+x)^2}\right ) \, dx\\ &=2 \int e^{2 x} \, dx+\int \frac {29+4 x+2 x^2}{(1+x)^2} \, dx\\ &=e^{2 x}+\int \left (2+\frac {27}{(1+x)^2}\right ) \, dx\\ &=e^{2 x}+2 x-\frac {27}{1+x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 16, normalized size = 0.89 \begin {gather*} e^{2 x}+2 x-\frac {27}{1+x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 24, normalized size = 1.33 \begin {gather*} \frac {2 \, x^{2} + {\left (x + 1\right )} e^{\left (2 \, x\right )} + 2 \, x - 27}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 26, normalized size = 1.44 \begin {gather*} \frac {2 \, x^{2} + x e^{\left (2 \, x\right )} + 2 \, x + e^{\left (2 \, x\right )} - 27}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.57, size = 16, normalized size = 0.89
method | result | size |
risch | \(2 x -\frac {27}{x +1}+{\mathrm e}^{2 x}\) | \(16\) |
derivativedivides | \(2 x -\frac {54}{2 x +2}+{\mathrm e}^{2 x}\) | \(18\) |
default | \(2 x -\frac {54}{2 x +2}+{\mathrm e}^{2 x}\) | \(18\) |
norman | \(\frac {x \,{\mathrm e}^{2 x}+2 x^{2}-29+{\mathrm e}^{2 x}}{x +1}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 2 \, x + \frac {{\left (x^{2} + 2 \, x\right )} e^{\left (2 \, x\right )}}{x^{2} + 2 \, x + 1} - \frac {2 \, e^{\left (-2\right )} E_{2}\left (-2 \, x - 2\right )}{x + 1} - \frac {27}{x + 1} - 2 \, \int \frac {e^{\left (2 \, x\right )}}{x^{3} + 3 \, x^{2} + 3 \, x + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 15, normalized size = 0.83 \begin {gather*} 2\,x+{\mathrm {e}}^{2\,x}-\frac {27}{x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 12, normalized size = 0.67 \begin {gather*} 2 x + e^{2 x} - \frac {27}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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