Optimal. Leaf size=22 \[ e^{x \left (x+\frac {8}{-3-\frac {9}{x}+(3+x)^2}\right )} \]
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Rubi [F]
time = 2.68, 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 {\exp \left (\frac {-x^2+6 x^3+6 x^4+x^5}{-9+6 x+6 x^2+x^3}\right ) \left (18 x-168 x^2-144 x^3+100 x^4+96 x^5+24 x^6+2 x^7\right )}{81-108 x-72 x^2+54 x^3+48 x^4+12 x^5+x^6} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x \left (9-84 x-72 x^2+50 x^3+48 x^4+12 x^5+x^6\right )}{\left (9-6 x-6 x^2-x^3\right )^2} \, dx\\ &=2 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x \left (9-84 x-72 x^2+50 x^3+48 x^4+12 x^5+x^6\right )}{\left (9-6 x-6 x^2-x^3\right )^2} \, dx\\ &=2 \int \left (\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x+\frac {12 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{(3+x)^2}+\frac {12 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) (-5+6 x)}{\left (-3+3 x+x^2\right )^2}-\frac {16 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{-3+3 x+x^2}\right ) \, dx\\ &=2 \int \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x \, dx+24 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{(3+x)^2} \, dx+24 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) (-5+6 x)}{\left (-3+3 x+x^2\right )^2} \, dx-32 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{-3+3 x+x^2} \, dx\\ &=2 \int \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x \, dx+24 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{(3+x)^2} \, dx+24 \int \left (-\frac {5 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{\left (-3+3 x+x^2\right )^2}+\frac {6 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x}{\left (-3+3 x+x^2\right )^2}\right ) \, dx-32 \int \left (-\frac {2 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{\sqrt {21} \left (-3+\sqrt {21}-2 x\right )}-\frac {2 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{\sqrt {21} \left (3+\sqrt {21}+2 x\right )}\right ) \, dx\\ &=2 \int \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x \, dx+24 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{(3+x)^2} \, dx-120 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{\left (-3+3 x+x^2\right )^2} \, dx+144 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x}{\left (-3+3 x+x^2\right )^2} \, dx+\frac {64 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{-3+\sqrt {21}-2 x} \, dx}{\sqrt {21}}+\frac {64 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{3+\sqrt {21}+2 x} \, dx}{\sqrt {21}}\\ &=2 \int \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x \, dx+24 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{(3+x)^2} \, dx-120 \int \left (\frac {4 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{21 \left (-3+\sqrt {21}-2 x\right )^2}+\frac {4 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{21 \sqrt {21} \left (-3+\sqrt {21}-2 x\right )}+\frac {4 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{21 \left (3+\sqrt {21}+2 x\right )^2}+\frac {4 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{21 \sqrt {21} \left (3+\sqrt {21}+2 x\right )}\right ) \, dx+144 \int \left (\frac {2 \left (-3+\sqrt {21}\right ) \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{21 \left (-3+\sqrt {21}-2 x\right )^2}-\frac {2 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{7 \sqrt {21} \left (-3+\sqrt {21}-2 x\right )}+\frac {2 \left (-3-\sqrt {21}\right ) \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{21 \left (3+\sqrt {21}+2 x\right )^2}-\frac {2 \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{7 \sqrt {21} \left (3+\sqrt {21}+2 x\right )}\right ) \, dx+\frac {64 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{-3+\sqrt {21}-2 x} \, dx}{\sqrt {21}}+\frac {64 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{3+\sqrt {21}+2 x} \, dx}{\sqrt {21}}\\ &=2 \int \exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right ) x \, dx-\frac {160}{7} \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{\left (-3+\sqrt {21}-2 x\right )^2} \, dx-\frac {160}{7} \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{\left (3+\sqrt {21}+2 x\right )^2} \, dx+24 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{(3+x)^2} \, dx-\frac {1}{7} \left (96 \sqrt {\frac {3}{7}}\right ) \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{-3+\sqrt {21}-2 x} \, dx-\frac {1}{7} \left (96 \sqrt {\frac {3}{7}}\right ) \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{3+\sqrt {21}+2 x} \, dx-\frac {160 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{-3+\sqrt {21}-2 x} \, dx}{7 \sqrt {21}}-\frac {160 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{3+\sqrt {21}+2 x} \, dx}{7 \sqrt {21}}+\frac {64 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{-3+\sqrt {21}-2 x} \, dx}{\sqrt {21}}+\frac {64 \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{3+\sqrt {21}+2 x} \, dx}{\sqrt {21}}-\frac {1}{7} \left (96 \left (3-\sqrt {21}\right )\right ) \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{\left (-3+\sqrt {21}-2 x\right )^2} \, dx-\frac {1}{7} \left (96 \left (3+\sqrt {21}\right )\right ) \int \frac {\exp \left (\frac {x^2 \left (-1+6 x+6 x^2+x^3\right )}{-9+6 x+6 x^2+x^3}\right )}{\left (3+\sqrt {21}+2 x\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.54, size = 26, normalized size = 1.18 \begin {gather*} e^{x^2+\frac {8 x^2}{-9+6 x+6 x^2+x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 34, normalized size = 1.55
method | result | size |
gosper | \({\mathrm e}^{\frac {x^{2} \left (x^{3}+6 x^{2}+6 x -1\right )}{x^{3}+6 x^{2}+6 x -9}}\) | \(34\) |
risch | \({\mathrm e}^{\frac {x^{2} \left (x^{3}+6 x^{2}+6 x -1\right )}{\left (3+x \right ) \left (x^{2}+3 x -3\right )}}\) | \(34\) |
norman | \(\frac {x^{3} {\mathrm e}^{\frac {x^{5}+6 x^{4}+6 x^{3}-x^{2}}{x^{3}+6 x^{2}+6 x -9}}+6 x \,{\mathrm e}^{\frac {x^{5}+6 x^{4}+6 x^{3}-x^{2}}{x^{3}+6 x^{2}+6 x -9}}+6 x^{2} {\mathrm e}^{\frac {x^{5}+6 x^{4}+6 x^{3}-x^{2}}{x^{3}+6 x^{2}+6 x -9}}-9 \,{\mathrm e}^{\frac {x^{5}+6 x^{4}+6 x^{3}-x^{2}}{x^{3}+6 x^{2}+6 x -9}}}{x^{3}+6 x^{2}+6 x -9}\) | \(176\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.61, size = 37, normalized size = 1.68 \begin {gather*} e^{\left (x^{2} + \frac {32 \, x}{x^{2} + 3 \, x - 3} - \frac {24}{x^{2} + 3 \, x - 3} - \frac {24}{x + 3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 36, normalized size = 1.64 \begin {gather*} e^{\left (\frac {x^{5} + 6 \, x^{4} + 6 \, x^{3} - x^{2}}{x^{3} + 6 \, x^{2} + 6 \, x - 9}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (15) = 30\).
time = 0.12, size = 31, normalized size = 1.41 \begin {gather*} e^{\frac {x^{5} + 6 x^{4} + 6 x^{3} - x^{2}}{x^{3} + 6 x^{2} + 6 x - 9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs.
\(2 (21) = 42\).
time = 0.40, size = 81, normalized size = 3.68 \begin {gather*} e^{\left (\frac {x^{5}}{x^{3} + 6 \, x^{2} + 6 \, x - 9} + \frac {6 \, x^{4}}{x^{3} + 6 \, x^{2} + 6 \, x - 9} + \frac {6 \, x^{3}}{x^{3} + 6 \, x^{2} + 6 \, x - 9} - \frac {x^{2}}{x^{3} + 6 \, x^{2} + 6 \, x - 9}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.39, size = 84, normalized size = 3.82 \begin {gather*} {\mathrm {e}}^{-\frac {x^2}{x^3+6\,x^2+6\,x-9}}\,{\mathrm {e}}^{\frac {x^5}{x^3+6\,x^2+6\,x-9}}\,{\mathrm {e}}^{\frac {6\,x^3}{x^3+6\,x^2+6\,x-9}}\,{\mathrm {e}}^{\frac {6\,x^4}{x^3+6\,x^2+6\,x-9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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