Optimal. Leaf size=28 \[ -1-e^5+e^x+e^{x+\frac {e^3}{5+x+2 x^2}} \]
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Rubi [A] time = 0.90, antiderivative size = 22, normalized size of antiderivative = 0.79, number of steps used = 4, number of rules used = 3, integrand size = 106, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {6688, 2194, 6706} \begin {gather*} e^{\frac {e^3}{2 x^2+x+5}+x}+e^x \end {gather*}
Antiderivative was successfully verified.
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Rule 2194
Rule 6688
Rule 6706
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^x+\frac {e^{x+\frac {e^3}{5+x+2 x^2}} \left (25-e^3+2 \left (5-2 e^3\right ) x+21 x^2+4 x^3+4 x^4\right )}{\left (5+x+2 x^2\right )^2}\right ) \, dx\\ &=\int e^x \, dx+\int \frac {e^{x+\frac {e^3}{5+x+2 x^2}} \left (25-e^3+2 \left (5-2 e^3\right ) x+21 x^2+4 x^3+4 x^4\right )}{\left (5+x+2 x^2\right )^2} \, dx\\ &=e^x+e^{x+\frac {e^3}{5+x+2 x^2}}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.81, size = 22, normalized size = 0.79 \begin {gather*} e^x+e^{x+\frac {e^3}{5+x+2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 29, normalized size = 1.04 \begin {gather*} e^{x} + e^{\left (\frac {2 \, x^{3} + x^{2} + 5 \, x + e^{3}}{2 \, x^{2} + x + 5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.53, size = 53, normalized size = 1.89 \begin {gather*} {\left (e^{\left (x + 3\right )} + e^{\left (\frac {10 \, x^{3} - 2 \, x^{2} e^{3} + 5 \, x^{2} - x e^{3} + 25 \, x}{5 \, {\left (2 \, x^{2} + x + 5\right )}} + \frac {1}{5} \, e^{3} + 3\right )}\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 30, normalized size = 1.07
method | result | size |
risch | \({\mathrm e}^{x}+{\mathrm e}^{\frac {{\mathrm e}^{3}+2 x^{3}+x^{2}+5 x}{2 x^{2}+x +5}}\) | \(30\) |
norman | \(\frac {{\mathrm e}^{x} x +{\mathrm e}^{\frac {{\mathrm e}^{3}+2 x^{3}+x^{2}+5 x}{2 x^{2}+x +5}} x +2 \,{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{\frac {{\mathrm e}^{3}+2 x^{3}+x^{2}+5 x}{2 x^{2}+x +5}} x^{2}+5 \,{\mathrm e}^{x}+5 \,{\mathrm e}^{\frac {{\mathrm e}^{3}+2 x^{3}+x^{2}+5 x}{2 x^{2}+x +5}}}{2 x^{2}+x +5}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 19, normalized size = 0.68 \begin {gather*} e^{\left (x + \frac {e^{3}}{2 \, x^{2} + x + 5}\right )} + e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.40, size = 63, normalized size = 2.25 \begin {gather*} {\mathrm {e}}^x+{\mathrm {e}}^{\frac {x^2}{2\,x^2+x+5}}\,{\mathrm {e}}^{\frac {2\,x^3}{2\,x^2+x+5}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^3}{2\,x^2+x+5}}\,{\mathrm {e}}^{\frac {5\,x}{2\,x^2+x+5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 27, normalized size = 0.96 \begin {gather*} e^{x} + e^{\frac {2 x^{3} + x^{2} + 5 x + e^{3}}{2 x^{2} + x + 5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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