Optimal. Leaf size=27 \[ 2 e^{e^{\frac {x^2}{5+x}}-x} \left (1+\frac {10}{e^4}+x\right ) \]
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Rubi [F]
time = 3.94, 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 {e^{-4+e^{\frac {x^2}{5+x}}-x} \left (-500-200 x-20 x^2+e^4 \left (-50 x-20 x^2-2 x^3\right )+e^{\frac {x^2}{5+x}} \left (200 x+20 x^2+e^4 \left (20 x+22 x^2+2 x^3\right )\right )\right )}{25+10 x+x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-4+e^{\frac {x^2}{5+x}}-x} \left (-500-200 x-20 x^2+e^4 \left (-50 x-20 x^2-2 x^3\right )+e^{\frac {x^2}{5+x}} \left (200 x+20 x^2+e^4 \left (20 x+22 x^2+2 x^3\right )\right )\right )}{(5+x)^2} \, dx\\ &=\int \left (-2 e^{-4+e^{\frac {x^2}{5+x}}-x} \left (10+e^4 x\right )+\frac {2 e^{-4+e^{\frac {x^2}{5+x}}-x+\frac {x^2}{5+x}} x (10+x) \left (10+e^4+e^4 x\right )}{(5+x)^2}\right ) \, dx\\ &=-\left (2 \int e^{-4+e^{\frac {x^2}{5+x}}-x} \left (10+e^4 x\right ) \, dx\right )+2 \int \frac {e^{-4+e^{\frac {x^2}{5+x}}-x+\frac {x^2}{5+x}} x (10+x) \left (10+e^4+e^4 x\right )}{(5+x)^2} \, dx\\ &=-\left (2 \int \left (10 e^{-4+e^{\frac {x^2}{5+x}}-x}+e^{e^{\frac {x^2}{5+x}}-x} x\right ) \, dx\right )+2 \int \left (10 e^{-4+e^{\frac {x^2}{5+x}}-x+\frac {x^2}{5+x}} \left (1+\frac {e^4}{10}\right )+e^{e^{\frac {x^2}{5+x}}-x+\frac {x^2}{5+x}} x+\frac {50 e^{-4+e^{\frac {x^2}{5+x}}-x+\frac {x^2}{5+x}} \left (-5+2 e^4\right )}{(5+x)^2}-\frac {25 e^{e^{\frac {x^2}{5+x}}-x+\frac {x^2}{5+x}}}{5+x}\right ) \, dx\\ &=-\left (2 \int e^{e^{\frac {x^2}{5+x}}-x} x \, dx\right )+2 \int e^{e^{\frac {x^2}{5+x}}-x+\frac {x^2}{5+x}} x \, dx-20 \int e^{-4+e^{\frac {x^2}{5+x}}-x} \, dx-50 \int \frac {e^{e^{\frac {x^2}{5+x}}-x+\frac {x^2}{5+x}}}{5+x} \, dx-\left (100 \left (5-2 e^4\right )\right ) \int \frac {e^{-4+e^{\frac {x^2}{5+x}}-x+\frac {x^2}{5+x}}}{(5+x)^2} \, dx+\left (2 \left (10+e^4\right )\right ) \int e^{-4+e^{\frac {x^2}{5+x}}-x+\frac {x^2}{5+x}} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 1.10, size = 29, normalized size = 1.07 \begin {gather*} e^{-4+e^{\frac {x^2}{5+x}}-x} \left (20+2 e^4 (1+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.13, size = 29, normalized size = 1.07
method | result | size |
risch | \(\left (2 x \,{\mathrm e}^{4}+2 \,{\mathrm e}^{4}+20\right ) {\mathrm e}^{-4+{\mathrm e}^{\frac {x^{2}}{5+x}}-x}\) | \(29\) |
norman | \(\frac {\left (2 x^{2}+10 \left (10+{\mathrm e}^{4}\right ) {\mathrm e}^{-4}+4 \left (5+3 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-4} x \right ) {\mathrm e}^{{\mathrm e}^{\frac {x^{2}}{5+x}}-x}}{5+x}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 27, normalized size = 1.00 \begin {gather*} 2 \, {\left (x e^{4} + e^{4} + 10\right )} e^{\left (-x + e^{\left (x + \frac {25}{x + 5} - 5\right )} - 4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 26, normalized size = 0.96 \begin {gather*} 2 \, {\left ({\left (x + 1\right )} e^{4} + 10\right )} e^{\left (-x + e^{\left (\frac {x^{2}}{x + 5}\right )} - 4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 9.03, size = 27, normalized size = 1.00 \begin {gather*} \frac {\left (2 x e^{4} + 20 + 2 e^{4}\right ) e^{- x + e^{\frac {x^{2}}{x + 5}}}}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.89, size = 29, normalized size = 1.07 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^{\frac {x^2}{x+5}}-x}\,\left (2\,x+{\mathrm {e}}^{-4}\,\left (2\,{\mathrm {e}}^4+20\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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