Integrand size = 171, antiderivative size = 29 \[ \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx=e^{e^{\frac {1}{2} x \left (x-81 \left (-3+e^{\frac {x}{3+2 x}}\right )^2 x\right )}} \]
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\[ \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx=\int \frac {\exp \left (\exp \left (\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right )+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right ) \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\exp \left (\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right )+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right ) \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{(3+2 x)^2} \, dx \\ & = \int \left (-728 \exp \left (\exp \left (\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right )+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right ) x-\frac {81 \exp \left (\exp \left (\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right )+\frac {2 x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right ) x (3+x) (3+4 x)}{(3+2 x)^2}+\frac {243 \exp \left (\exp \left (\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right )+\frac {x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right ) x \left (18+27 x+8 x^2\right )}{(3+2 x)^2}\right ) \, dx \\ & = -\left (81 \int \frac {\exp \left (\exp \left (\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right )+\frac {2 x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right ) x (3+x) (3+4 x)}{(3+2 x)^2} \, dx\right )+243 \int \frac {\exp \left (\exp \left (\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right )+\frac {x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right ) x \left (18+27 x+8 x^2\right )}{(3+2 x)^2} \, dx-728 \int \exp \left (\exp \left (\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right )+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )\right ) x \, dx \\ & = -\left (81 \int \left (\frac {3}{4} e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {2 x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {2 x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} x+\frac {27 e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {2 x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}}{4 (3+2 x)^2}-\frac {9 e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {2 x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}}{2 (3+2 x)}\right ) \, dx\right )+243 \int \left (\frac {3}{4} e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+2 e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} x+\frac {27 e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}}{4 (3+2 x)^2}-\frac {9 e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}}{2 (3+2 x)}\right ) \, dx-728 \int e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} x \, dx \\ & = -\left (\frac {243}{4} \int e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {2 x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \, dx\right )-81 \int e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {2 x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} x \, dx+\frac {729}{4} \int e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \, dx+\frac {729}{2} \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {2 x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}}{3+2 x} \, dx+486 \int e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} x \, dx-\frac {2187}{4} \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {2 x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}}{(3+2 x)^2} \, dx-728 \int e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} x \, dx-\frac {2187}{2} \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}}{3+2 x} \, dx+\frac {6561}{4} \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {x}{3+2 x}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}}{(3+2 x)^2} \, dx \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx=e^{e^{-\frac {1}{2} \left (728-486 e^{\frac {x}{3+2 x}}+81 e^{\frac {2 x}{3+2 x}}\right ) x^2}} \]
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Time = 3.80 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21
method | result | size |
risch | \({\mathrm e}^{{\mathrm e}^{\frac {x^{2} \left (-81 \,{\mathrm e}^{\frac {2 x}{3+2 x}}+486 \,{\mathrm e}^{\frac {x}{3+2 x}}-728\right )}{2}}}\) | \(35\) |
parallelrisch | \({\mathrm e}^{{\mathrm e}^{-\frac {x^{2} \left (81 \,{\mathrm e}^{\frac {2 x}{3+2 x}}-486 \,{\mathrm e}^{\frac {x}{3+2 x}}+728\right )}{2}}}\) | \(36\) |
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Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx=e^{\left (e^{\left (-\frac {81}{2} \, x^{2} e^{\left (\frac {2 \, x}{2 \, x + 3}\right )} + 243 \, x^{2} e^{\left (\frac {x}{2 \, x + 3}\right )} - 364 \, x^{2}\right )}\right )} \]
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Time = 2.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx=e^{e^{- \frac {81 x^{2} e^{\frac {2 x}{2 x + 3}}}{2} + 243 x^{2} e^{\frac {x}{2 x + 3}} - 364 x^{2}}} \]
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Time = 0.55 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx=e^{\left (e^{\left (243 \, x^{2} e^{\left (-\frac {3}{2 \, {\left (2 \, x + 3\right )}} + \frac {1}{2}\right )} - \frac {81}{2} \, x^{2} e^{\left (-\frac {3}{2 \, x + 3} + 1\right )} - 364 \, x^{2}\right )}\right )} \]
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\[ \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx=\int { -\frac {{\left (2912 \, x^{3} + 8736 \, x^{2} + 81 \, {\left (4 \, x^{3} + 15 \, x^{2} + 9 \, x\right )} e^{\left (\frac {2 \, x}{2 \, x + 3}\right )} - 243 \, {\left (8 \, x^{3} + 27 \, x^{2} + 18 \, x\right )} e^{\left (\frac {x}{2 \, x + 3}\right )} + 6552 \, x\right )} e^{\left (-\frac {81}{2} \, x^{2} e^{\left (\frac {2 \, x}{2 \, x + 3}\right )} + 243 \, x^{2} e^{\left (\frac {x}{2 \, x + 3}\right )} - 364 \, x^{2} + e^{\left (-\frac {81}{2} \, x^{2} e^{\left (\frac {2 \, x}{2 \, x + 3}\right )} + 243 \, x^{2} e^{\left (\frac {x}{2 \, x + 3}\right )} - 364 \, x^{2}\right )}\right )}}{4 \, x^{2} + 12 \, x + 9} \,d x } \]
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Time = 11.68 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx={\mathrm {e}}^{{\mathrm {e}}^{-\frac {81\,x^2\,{\mathrm {e}}^{\frac {2\,x}{2\,x+3}}}{2}}\,{\mathrm {e}}^{243\,x^2\,{\mathrm {e}}^{\frac {x}{2\,x+3}}}\,{\mathrm {e}}^{-364\,x^2}} \]
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