Optimal. Leaf size=18 \[ -5+\frac {4}{3+5 e^{-x}+2 x} \]
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Rubi [F] time = 0.93, 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 {20 e^x-8 e^{2 x}}{25+e^x (30+20 x)+e^{2 x} \left (9+12 x+4 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 e^x \left (5-2 e^x\right )}{\left (5+e^x (3+2 x)\right )^2} \, dx\\ &=4 \int \frac {e^x \left (5-2 e^x\right )}{\left (5+e^x (3+2 x)\right )^2} \, dx\\ &=4 \int \left (\frac {5 e^x (5+2 x)}{(3+2 x) \left (5+3 e^x+2 e^x x\right )^2}-\frac {2 e^x}{(3+2 x) \left (5+3 e^x+2 e^x x\right )}\right ) \, dx\\ &=-\left (8 \int \frac {e^x}{(3+2 x) \left (5+3 e^x+2 e^x x\right )} \, dx\right )+20 \int \frac {e^x (5+2 x)}{(3+2 x) \left (5+3 e^x+2 e^x x\right )^2} \, dx\\ &=-\left (8 \int \frac {e^x}{(3+2 x) \left (5+3 e^x+2 e^x x\right )} \, dx\right )+20 \int \left (\frac {e^x}{\left (5+3 e^x+2 e^x x\right )^2}+\frac {2 e^x}{(3+2 x) \left (5+3 e^x+2 e^x x\right )^2}\right ) \, dx\\ &=-\left (8 \int \frac {e^x}{(3+2 x) \left (5+3 e^x+2 e^x x\right )} \, dx\right )+20 \int \frac {e^x}{\left (5+3 e^x+2 e^x x\right )^2} \, dx+40 \int \frac {e^x}{(3+2 x) \left (5+3 e^x+2 e^x x\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.19, size = 18, normalized size = 1.00 \begin {gather*} \frac {4 e^x}{5+e^x (3+2 x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 16, normalized size = 0.89 \begin {gather*} \frac {4 \, e^{x}}{{\left (2 \, x + 3\right )} e^{x} + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 17, normalized size = 0.94 \begin {gather*} \frac {4 \, e^{x}}{2 \, x e^{x} + 3 \, e^{x} + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 18, normalized size = 1.00
method | result | size |
norman | \(\frac {4 \,{\mathrm e}^{x}}{2 \,{\mathrm e}^{x} x +3 \,{\mathrm e}^{x}+5}\) | \(18\) |
risch | \(\frac {2}{x +\frac {3}{2}}-\frac {20}{\left (2 x +3\right ) \left (2 \,{\mathrm e}^{x} x +3 \,{\mathrm e}^{x}+5\right )}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 16, normalized size = 0.89 \begin {gather*} \frac {4 \, e^{x}}{{\left (2 \, x + 3\right )} e^{x} + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.70, size = 31, normalized size = 1.72 \begin {gather*} \frac {4}{2\,x+3}-\frac {20}{\left (2\,x+3\right )\,\left ({\mathrm {e}}^x\,\left (2\,x+3\right )+5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.17, size = 26, normalized size = 1.44 \begin {gather*} - \frac {20}{10 x + \left (4 x^{2} + 12 x + 9\right ) e^{x} + 15} + \frac {8}{4 x + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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