Optimal. Leaf size=26 \[ 4+5 e^{-\frac {16}{x}+x^2}+\left (1+e^x\right ) \left (\frac {2}{3}+x\right ) \]
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Rubi [A] time = 0.60, antiderivative size = 32, normalized size of antiderivative = 1.23, number of steps used = 6, number of rules used = 5, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {12, 6688, 2176, 2194, 6706} \begin {gather*} 5 e^{x^2-\frac {16}{x}}+x-e^x+\frac {1}{3} e^x (3 x+5) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rule 6688
Rule 6706
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {e^{-\frac {16-x^3}{x}} \left (240+3 e^{\frac {16-x^3}{x}} x^2+30 x^3+e^{x+\frac {16-x^3}{x}} \left (5 x^2+3 x^3\right )\right )}{x^2} \, dx\\ &=\frac {1}{3} \int \left (3+e^x (5+3 x)+\frac {30 e^{-\frac {16}{x}+x^2} \left (8+x^3\right )}{x^2}\right ) \, dx\\ &=x+\frac {1}{3} \int e^x (5+3 x) \, dx+10 \int \frac {e^{-\frac {16}{x}+x^2} \left (8+x^3\right )}{x^2} \, dx\\ &=5 e^{-\frac {16}{x}+x^2}+x+\frac {1}{3} e^x (5+3 x)-\int e^x \, dx\\ &=-e^x+5 e^{-\frac {16}{x}+x^2}+x+\frac {1}{3} e^x (5+3 x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 24, normalized size = 0.92 \begin {gather*} 5 e^{-\frac {16}{x}+x^2}+x+e^x \left (\frac {2}{3}+x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 38, normalized size = 1.46 \begin {gather*} \frac {1}{3} \, {\left ({\left (3 \, x + 2\right )} e^{\left (-\frac {x^{3} - x^{2} - 16}{x}\right )} + 15\right )} e^{\left (\frac {x^{3} - 16}{x}\right )} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 22, normalized size = 0.85 \begin {gather*} x e^{x} + x + \frac {2}{3} \, e^{x} + 5 \, e^{\left (\frac {x^{3} - 16}{x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 24, normalized size = 0.92
method | result | size |
risch | \(x +\frac {\left (3 x +2\right ) {\mathrm e}^{x}}{3}+5 \,{\mathrm e}^{\frac {x^{3}-16}{x}}\) | \(24\) |
norman | \(\frac {\left (x^{2} {\mathrm e}^{\frac {-x^{3}+16}{x}}+{\mathrm e}^{x} x^{2} {\mathrm e}^{\frac {-x^{3}+16}{x}}+5 x +\frac {2 \,{\mathrm e}^{x} x \,{\mathrm e}^{\frac {-x^{3}+16}{x}}}{3}\right ) {\mathrm e}^{-\frac {-x^{3}+16}{x}}}{x}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 24, normalized size = 0.92 \begin {gather*} {\left (x - 1\right )} e^{x} + x + 5 \, e^{\left (x^{2} - \frac {16}{x}\right )} + \frac {5}{3} \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.42, size = 22, normalized size = 0.85 \begin {gather*} x+5\,{\mathrm {e}}^{x^2-\frac {16}{x}}+\frac {2\,{\mathrm {e}}^x}{3}+x\,{\mathrm {e}}^x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 20, normalized size = 0.77 \begin {gather*} x + \frac {\left (3 x + 2\right ) e^{x}}{3} + 5 e^{- \frac {16 - x^{3}}{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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