Integrand size = 70, antiderivative size = 26 \[ \int \frac {1}{9} \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx=\frac {1}{9} x^2 \left (-11+3 x+e^x x-(4+x)^2\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(26)=52\).
Time = 0.18 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.77, number of steps used = 34, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {12, 1607, 2227, 2207, 2225} \[ \int \frac {1}{9} \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx=\frac {x^6}{9}-\frac {2 e^x x^5}{9}+\frac {10 x^5}{9}-\frac {10 e^x x^4}{9}+\frac {1}{9} e^{2 x} x^4+\frac {79 x^4}{9}-6 e^x x^3+30 x^3+81 x^2 \]
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Rule 12
Rule 1607
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} \int \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx \\ & = 81 x^2+30 x^3+\frac {79 x^4}{9}+\frac {10 x^5}{9}+\frac {x^6}{9}+\frac {1}{9} \int e^{2 x} \left (4 x^3+2 x^4\right ) \, dx+\frac {1}{9} \int e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right ) \, dx \\ & = 81 x^2+30 x^3+\frac {79 x^4}{9}+\frac {10 x^5}{9}+\frac {x^6}{9}+\frac {1}{9} \int e^{2 x} x^3 (4+2 x) \, dx+\frac {1}{9} \int \left (-162 e^x x^2-94 e^x x^3-20 e^x x^4-2 e^x x^5\right ) \, dx \\ & = 81 x^2+30 x^3+\frac {79 x^4}{9}+\frac {10 x^5}{9}+\frac {x^6}{9}+\frac {1}{9} \int \left (4 e^{2 x} x^3+2 e^{2 x} x^4\right ) \, dx-\frac {2}{9} \int e^x x^5 \, dx-\frac {20}{9} \int e^x x^4 \, dx-\frac {94}{9} \int e^x x^3 \, dx-18 \int e^x x^2 \, dx \\ & = 81 x^2-18 e^x x^2+30 x^3-\frac {94 e^x x^3}{9}+\frac {79 x^4}{9}-\frac {20 e^x x^4}{9}+\frac {10 x^5}{9}-\frac {2 e^x x^5}{9}+\frac {x^6}{9}+\frac {2}{9} \int e^{2 x} x^4 \, dx+\frac {4}{9} \int e^{2 x} x^3 \, dx+\frac {10}{9} \int e^x x^4 \, dx+\frac {80}{9} \int e^x x^3 \, dx+\frac {94}{3} \int e^x x^2 \, dx+36 \int e^x x \, dx \\ & = 36 e^x x+81 x^2+\frac {40 e^x x^2}{3}+30 x^3-\frac {14 e^x x^3}{9}+\frac {2}{9} e^{2 x} x^3+\frac {79 x^4}{9}-\frac {10 e^x x^4}{9}+\frac {1}{9} e^{2 x} x^4+\frac {10 x^5}{9}-\frac {2 e^x x^5}{9}+\frac {x^6}{9}-\frac {4}{9} \int e^{2 x} x^3 \, dx-\frac {2}{3} \int e^{2 x} x^2 \, dx-\frac {40}{9} \int e^x x^3 \, dx-\frac {80}{3} \int e^x x^2 \, dx-36 \int e^x \, dx-\frac {188}{3} \int e^x x \, dx \\ & = -36 e^x-\frac {80 e^x x}{3}+81 x^2-\frac {40 e^x x^2}{3}-\frac {1}{3} e^{2 x} x^2+30 x^3-6 e^x x^3+\frac {79 x^4}{9}-\frac {10 e^x x^4}{9}+\frac {1}{9} e^{2 x} x^4+\frac {10 x^5}{9}-\frac {2 e^x x^5}{9}+\frac {x^6}{9}+\frac {2}{3} \int e^{2 x} x \, dx+\frac {2}{3} \int e^{2 x} x^2 \, dx+\frac {40}{3} \int e^x x^2 \, dx+\frac {160}{3} \int e^x x \, dx+\frac {188 \int e^x \, dx}{3} \\ & = \frac {80 e^x}{3}+\frac {80 e^x x}{3}+\frac {1}{3} e^{2 x} x+81 x^2+30 x^3-6 e^x x^3+\frac {79 x^4}{9}-\frac {10 e^x x^4}{9}+\frac {1}{9} e^{2 x} x^4+\frac {10 x^5}{9}-\frac {2 e^x x^5}{9}+\frac {x^6}{9}-\frac {1}{3} \int e^{2 x} \, dx-\frac {2}{3} \int e^{2 x} x \, dx-\frac {80}{3} \int e^x x \, dx-\frac {160 \int e^x \, dx}{3} \\ & = -\frac {80 e^x}{3}-\frac {e^{2 x}}{6}+81 x^2+30 x^3-6 e^x x^3+\frac {79 x^4}{9}-\frac {10 e^x x^4}{9}+\frac {1}{9} e^{2 x} x^4+\frac {10 x^5}{9}-\frac {2 e^x x^5}{9}+\frac {x^6}{9}+\frac {1}{3} \int e^{2 x} \, dx+\frac {80 \int e^x \, dx}{3} \\ & = 81 x^2+30 x^3-6 e^x x^3+\frac {79 x^4}{9}-\frac {10 e^x x^4}{9}+\frac {1}{9} e^{2 x} x^4+\frac {10 x^5}{9}-\frac {2 e^x x^5}{9}+\frac {x^6}{9} \\ \end{align*}
Time = 2.71 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {1}{9} \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx=\frac {1}{9} x^2 \left (27-\left (-5+e^x\right ) x+x^2\right )^2 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(23)=46\).
Time = 0.33 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15
method | result | size |
risch | \(\frac {{\mathrm e}^{2 x} x^{4}}{9}+\frac {\left (-2 x^{5}-10 x^{4}-54 x^{3}\right ) {\mathrm e}^{x}}{9}+\frac {x^{6}}{9}+\frac {10 x^{5}}{9}+\frac {79 x^{4}}{9}+30 x^{3}+81 x^{2}\) | \(56\) |
default | \(\frac {{\mathrm e}^{2 x} x^{4}}{9}-\frac {2 x^{5} {\mathrm e}^{x}}{9}-\frac {10 \,{\mathrm e}^{x} x^{4}}{9}-6 \,{\mathrm e}^{x} x^{3}+\frac {x^{6}}{9}+\frac {10 x^{5}}{9}+\frac {79 x^{4}}{9}+30 x^{3}+81 x^{2}\) | \(57\) |
norman | \(\frac {{\mathrm e}^{2 x} x^{4}}{9}-\frac {2 x^{5} {\mathrm e}^{x}}{9}-\frac {10 \,{\mathrm e}^{x} x^{4}}{9}-6 \,{\mathrm e}^{x} x^{3}+\frac {x^{6}}{9}+\frac {10 x^{5}}{9}+\frac {79 x^{4}}{9}+30 x^{3}+81 x^{2}\) | \(57\) |
parallelrisch | \(\frac {{\mathrm e}^{2 x} x^{4}}{9}-\frac {2 x^{5} {\mathrm e}^{x}}{9}-\frac {10 \,{\mathrm e}^{x} x^{4}}{9}-6 \,{\mathrm e}^{x} x^{3}+\frac {x^{6}}{9}+\frac {10 x^{5}}{9}+\frac {79 x^{4}}{9}+30 x^{3}+81 x^{2}\) | \(57\) |
parts | \(\frac {{\mathrm e}^{2 x} x^{4}}{9}-\frac {2 x^{5} {\mathrm e}^{x}}{9}-\frac {10 \,{\mathrm e}^{x} x^{4}}{9}-6 \,{\mathrm e}^{x} x^{3}+\frac {x^{6}}{9}+\frac {10 x^{5}}{9}+\frac {79 x^{4}}{9}+30 x^{3}+81 x^{2}\) | \(57\) |
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).
Time = 0.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {1}{9} \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx=\frac {1}{9} \, x^{6} + \frac {10}{9} \, x^{5} + \frac {1}{9} \, x^{4} e^{\left (2 \, x\right )} + \frac {79}{9} \, x^{4} + 30 \, x^{3} + 81 \, x^{2} - \frac {2}{9} \, {\left (x^{5} + 5 \, x^{4} + 27 \, x^{3}\right )} e^{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {1}{9} \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx=\frac {x^{6}}{9} + \frac {10 x^{5}}{9} + \frac {x^{4} e^{2 x}}{9} + \frac {79 x^{4}}{9} + 30 x^{3} + 81 x^{2} + \frac {\left (- 18 x^{5} - 90 x^{4} - 486 x^{3}\right ) e^{x}}{81} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).
Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {1}{9} \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx=\frac {1}{9} \, x^{6} + \frac {10}{9} \, x^{5} + \frac {1}{9} \, x^{4} e^{\left (2 \, x\right )} + \frac {79}{9} \, x^{4} + 30 \, x^{3} + 81 \, x^{2} - \frac {2}{9} \, {\left (x^{5} + 5 \, x^{4} + 27 \, x^{3}\right )} e^{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).
Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {1}{9} \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx=\frac {1}{9} \, x^{6} + \frac {10}{9} \, x^{5} + \frac {1}{9} \, x^{4} e^{\left (2 \, x\right )} + \frac {79}{9} \, x^{4} + 30 \, x^{3} + 81 \, x^{2} - \frac {2}{9} \, {\left (x^{5} + 5 \, x^{4} + 27 \, x^{3}\right )} e^{x} \]
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Time = 9.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {1}{9} \left (1458 x+810 x^2+316 x^3+50 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (-162 x^2-94 x^3-20 x^4-2 x^5\right )\right ) \, dx=\frac {x^2\,{\left (5\,x-x\,{\mathrm {e}}^x+x^2+27\right )}^2}{9} \]
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