Optimal. Leaf size=25 \[ \frac {1}{2} e^{\frac {1}{2} \left (-9+x-\frac {4}{x+x^2}\right )} x^2 \]
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Rubi [F]
time = 0.74, 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^{\frac {-4-9 x-8 x^2+x^3}{2 x+2 x^2}} \left (4+12 x+9 x^2+6 x^3+x^4\right )}{4+8 x+4 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^{\frac {-4-9 x-8 x^2+x^3}{2 x+2 x^2}} \left (4+12 x+9 x^2+6 x^3+x^4\right )}{4 (1+x)^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{\frac {-4-9 x-8 x^2+x^3}{2 x+2 x^2}} \left (4+12 x+9 x^2+6 x^3+x^4\right )}{(1+x)^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}} \left (4+12 x+9 x^2+6 x^3+x^4\right )}{(1+x)^2} \, dx\\ &=\frac {1}{4} \int \left (4 e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}} x+e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}} x^2-\frac {4 e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}}}{(1+x)^2}+\frac {8 e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}}}{1+x}\right ) \, dx\\ &=\frac {1}{4} \int e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}} x^2 \, dx+2 \int \frac {e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}}}{1+x} \, dx+\int e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}} x \, dx-\int \frac {e^{\frac {-4-9 x-8 x^2+x^3}{x (2+2 x)}}}{(1+x)^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.18, size = 30, normalized size = 1.20 \begin {gather*} \frac {1}{2} e^{-\frac {9}{2}-\frac {2}{x}+\frac {x}{2}+\frac {2}{1+x}} x^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.72, size = 30, normalized size = 1.20
method | result | size |
gosper | \(\frac {x^{2} {\mathrm e}^{\frac {x^{3}-8 x^{2}-9 x -4}{2 \left (x +1\right ) x}}}{2}\) | \(30\) |
risch | \(\frac {x^{2} {\mathrm e}^{\frac {x^{3}-8 x^{2}-9 x -4}{2 \left (x +1\right ) x}}}{2}\) | \(30\) |
norman | \(\frac {\frac {x^{2} {\mathrm e}^{\frac {x^{3}-8 x^{2}-9 x -4}{2 x^{2}+2 x}}}{2}+\frac {x^{3} {\mathrm e}^{\frac {x^{3}-8 x^{2}-9 x -4}{2 x^{2}+2 x}}}{2}}{x +1}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 23, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, x^{2} e^{\left (\frac {1}{2} \, x + \frac {2}{x + 1} - \frac {2}{x} - \frac {9}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.48, size = 28, normalized size = 1.12 \begin {gather*} \frac {1}{2} \, x^{2} e^{\left (\frac {x^{3} - 8 \, x^{2} - 9 \, x - 4}{2 \, {\left (x^{2} + x\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 27, normalized size = 1.08 \begin {gather*} \frac {x^{2} e^{\frac {x^{3} - 8 x^{2} - 9 x - 4}{2 x^{2} + 2 x}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 28, normalized size = 1.12 \begin {gather*} \frac {1}{2} \, x^{2} e^{\left (\frac {x^{3} - 8 \, x^{2} - 9 \, x - 4}{2 \, {\left (x^{2} + x\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.36, size = 46, normalized size = 1.84 \begin {gather*} \frac {x^2\,{\mathrm {e}}^{\frac {x^2}{2\,x+2}}\,{\mathrm {e}}^{-\frac {9}{2\,x+2}}\,{\mathrm {e}}^{-\frac {4\,x}{x+1}}\,{\mathrm {e}}^{-\frac {2}{x^2+x}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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