Optimal. Leaf size=30 \[ 2+e^{-x^2+\frac {2+5 x^2}{4 x}}+x+4 x^2 \]
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Rubi [A] time = 0.20, antiderivative size = 33, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 3, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {12, 14, 6706} \begin {gather*} e^{\frac {-4 x^3+5 x^2+2}{4 x}}+\frac {1}{16} (8 x+1)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 6706
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {4 x^2+32 x^3+e^{\frac {2+5 x^2-4 x^3}{4 x}} \left (-2+5 x^2-8 x^3\right )}{x^2} \, dx\\ &=\frac {1}{4} \int \left (4 (1+8 x)+\frac {e^{\frac {2+5 x^2-4 x^3}{4 x}} \left (-2+5 x^2-8 x^3\right )}{x^2}\right ) \, dx\\ &=\frac {1}{16} (1+8 x)^2+\frac {1}{4} \int \frac {e^{\frac {2+5 x^2-4 x^3}{4 x}} \left (-2+5 x^2-8 x^3\right )}{x^2} \, dx\\ &=e^{\frac {2+5 x^2-4 x^3}{4 x}}+\frac {1}{16} (1+8 x)^2\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 27, normalized size = 0.90 \begin {gather*} e^{\frac {1}{2 x}+\frac {5 x}{4}-x^2}+x+4 x^2 \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 25, normalized size = 0.83 \begin {gather*} 4 \, x^{2} + x + e^{\left (-\frac {4 \, x^{3} - 5 \, x^{2} - 2}{4 \, x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 25, normalized size = 0.83 \begin {gather*} 4 \, x^{2} + x + e^{\left (-\frac {4 \, x^{3} - 5 \, x^{2} - 2}{4 \, x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 26, normalized size = 0.87
method | result | size |
risch | \(4 x^{2}+x +{\mathrm e}^{-\frac {4 x^{3}-5 x^{2}-2}{4 x}}\) | \(26\) |
norman | \(\frac {x^{2}+x \,{\mathrm e}^{\frac {-4 x^{3}+5 x^{2}+2}{4 x}}+4 x^{3}}{x}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 22, normalized size = 0.73 \begin {gather*} 4 \, x^{2} + x + e^{\left (-x^{2} + \frac {5}{4} \, x + \frac {1}{2 \, x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 22, normalized size = 0.73 \begin {gather*} x+{\mathrm {e}}^{\frac {5\,x}{4}+\frac {1}{2\,x}-x^2}+4\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 22, normalized size = 0.73 \begin {gather*} 4 x^{2} + x + e^{\frac {- x^{3} + \frac {5 x^{2}}{4} + \frac {1}{2}}{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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