Integrand size = 69, antiderivative size = 15 \[ \int e^{3 x} \left (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 \left (2+18 x+18 x^2\right )+e \left (8 x+24 x^2+12 x^3\right )\right ) \, dx=e^{3 x} \left (x+(e+x)^2\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(109\) vs. \(2(15)=30\).
Time = 0.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 7.27, number of steps used = 38, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2227, 2225, 2207} \[ \int e^{3 x} \left (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 \left (2+18 x+18 x^2\right )+e \left (8 x+24 x^2+12 x^3\right )\right ) \, dx=e^{3 x} x^4+2 e^{3 x} x^3+4 e^{3 x+1} x^3+e^{3 x} x^2+4 e^{3 x+1} x^2+6 e^{3 x+2} x^2+2 e^{3 x+2} x-\frac {4}{3} e^{3 x+3}+e^{3 x+4}+\frac {4}{3} e^{3 x+3} (3 x+1) \]
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Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \int \left (3 e^{4+3 x}+2 e^{3 x} x+9 e^{3 x} x^2+10 e^{3 x} x^3+3 e^{3 x} x^4+4 e^{3+3 x} (1+3 x)+4 e^{1+3 x} x \left (2+6 x+3 x^2\right )+2 e^{2+3 x} \left (1+9 x+9 x^2\right )\right ) \, dx \\ & = 2 \int e^{3 x} x \, dx+2 \int e^{2+3 x} \left (1+9 x+9 x^2\right ) \, dx+3 \int e^{4+3 x} \, dx+3 \int e^{3 x} x^4 \, dx+4 \int e^{3+3 x} (1+3 x) \, dx+4 \int e^{1+3 x} x \left (2+6 x+3 x^2\right ) \, dx+9 \int e^{3 x} x^2 \, dx+10 \int e^{3 x} x^3 \, dx \\ & = e^{4+3 x}+\frac {2}{3} e^{3 x} x+3 e^{3 x} x^2+\frac {10}{3} e^{3 x} x^3+e^{3 x} x^4+\frac {4}{3} e^{3+3 x} (1+3 x)-\frac {2}{3} \int e^{3 x} \, dx+2 \int \left (e^{2+3 x}+9 e^{2+3 x} x+9 e^{2+3 x} x^2\right ) \, dx-4 \int e^{3+3 x} \, dx-4 \int e^{3 x} x^3 \, dx+4 \int \left (2 e^{1+3 x} x+6 e^{1+3 x} x^2+3 e^{1+3 x} x^3\right ) \, dx-6 \int e^{3 x} x \, dx-10 \int e^{3 x} x^2 \, dx \\ & = -\frac {2 e^{3 x}}{9}-\frac {4}{3} e^{3+3 x}+e^{4+3 x}-\frac {4}{3} e^{3 x} x-\frac {1}{3} e^{3 x} x^2+2 e^{3 x} x^3+e^{3 x} x^4+\frac {4}{3} e^{3+3 x} (1+3 x)+2 \int e^{3 x} \, dx+2 \int e^{2+3 x} \, dx+4 \int e^{3 x} x^2 \, dx+\frac {20}{3} \int e^{3 x} x \, dx+8 \int e^{1+3 x} x \, dx+12 \int e^{1+3 x} x^3 \, dx+18 \int e^{2+3 x} x \, dx+18 \int e^{2+3 x} x^2 \, dx+24 \int e^{1+3 x} x^2 \, dx \\ & = \frac {4 e^{3 x}}{9}+\frac {2}{3} e^{2+3 x}-\frac {4}{3} e^{3+3 x}+e^{4+3 x}+\frac {8}{9} e^{3 x} x+\frac {8}{3} e^{1+3 x} x+6 e^{2+3 x} x+e^{3 x} x^2+8 e^{1+3 x} x^2+6 e^{2+3 x} x^2+2 e^{3 x} x^3+4 e^{1+3 x} x^3+e^{3 x} x^4+\frac {4}{3} e^{3+3 x} (1+3 x)-\frac {20}{9} \int e^{3 x} \, dx-\frac {8}{3} \int e^{1+3 x} \, dx-\frac {8}{3} \int e^{3 x} x \, dx-6 \int e^{2+3 x} \, dx-12 \int e^{2+3 x} x \, dx-12 \int e^{1+3 x} x^2 \, dx-16 \int e^{1+3 x} x \, dx \\ & = -\frac {8 e^{3 x}}{27}-\frac {8}{9} e^{1+3 x}-\frac {4}{3} e^{2+3 x}-\frac {4}{3} e^{3+3 x}+e^{4+3 x}-\frac {8}{3} e^{1+3 x} x+2 e^{2+3 x} x+e^{3 x} x^2+4 e^{1+3 x} x^2+6 e^{2+3 x} x^2+2 e^{3 x} x^3+4 e^{1+3 x} x^3+e^{3 x} x^4+\frac {4}{3} e^{3+3 x} (1+3 x)+\frac {8}{9} \int e^{3 x} \, dx+4 \int e^{2+3 x} \, dx+\frac {16}{3} \int e^{1+3 x} \, dx+8 \int e^{1+3 x} x \, dx \\ & = \frac {8}{9} e^{1+3 x}-\frac {4}{3} e^{3+3 x}+e^{4+3 x}+2 e^{2+3 x} x+e^{3 x} x^2+4 e^{1+3 x} x^2+6 e^{2+3 x} x^2+2 e^{3 x} x^3+4 e^{1+3 x} x^3+e^{3 x} x^4+\frac {4}{3} e^{3+3 x} (1+3 x)-\frac {8}{3} \int e^{1+3 x} \, dx \\ & = -\frac {4}{3} e^{3+3 x}+e^{4+3 x}+2 e^{2+3 x} x+e^{3 x} x^2+4 e^{1+3 x} x^2+6 e^{2+3 x} x^2+2 e^{3 x} x^3+4 e^{1+3 x} x^3+e^{3 x} x^4+\frac {4}{3} e^{3+3 x} (1+3 x) \\ \end{align*}
Time = 5.67 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33 \[ \int e^{3 x} \left (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 \left (2+18 x+18 x^2\right )+e \left (8 x+24 x^2+12 x^3\right )\right ) \, dx=e^{3 x} \left (e^2+x+2 e x+x^2\right )^2 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(15)=30\).
Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 3.40
method | result | size |
risch | \(\left ({\mathrm e}^{4}+4 x \,{\mathrm e}^{3}+6 x^{2} {\mathrm e}^{2}+4 x^{3} {\mathrm e}+x^{4}+2 \,{\mathrm e}^{2} x +4 x^{2} {\mathrm e}+2 x^{3}+x^{2}\right ) {\mathrm e}^{3 x}\) | \(51\) |
gosper | \(\left ({\mathrm e}^{4}+4 x \,{\mathrm e}^{3}+6 x^{2} {\mathrm e}^{2}+4 x^{3} {\mathrm e}+x^{4}+2 \,{\mathrm e}^{2} x +4 x^{2} {\mathrm e}+2 x^{3}+x^{2}\right ) {\mathrm e}^{3 x}\) | \(59\) |
norman | \({\mathrm e}^{3 x} x^{4}+{\mathrm e}^{3 x} {\mathrm e}^{4}+\left (4 \,{\mathrm e}^{3}+2 \,{\mathrm e}^{2}\right ) x \,{\mathrm e}^{3 x}+\left (4 \,{\mathrm e}+2\right ) x^{3} {\mathrm e}^{3 x}+\left (6 \,{\mathrm e}^{2}+4 \,{\mathrm e}+1\right ) x^{2} {\mathrm e}^{3 x}\) | \(72\) |
parallelrisch | \({\mathrm e}^{3 x} {\mathrm e}^{4}+4 x \,{\mathrm e}^{3} {\mathrm e}^{3 x}+6 \,{\mathrm e}^{2} x^{2} {\mathrm e}^{3 x}+4 \,{\mathrm e} x^{3} {\mathrm e}^{3 x}+{\mathrm e}^{3 x} x^{4}+2 \,{\mathrm e}^{2} x \,{\mathrm e}^{3 x}+4 x^{2} {\mathrm e} \,{\mathrm e}^{3 x}+2 x^{3} {\mathrm e}^{3 x}+x^{2} {\mathrm e}^{3 x}\) | \(93\) |
meijerg | \(-\frac {\left (-4 \,{\mathrm e}^{3}-6 \,{\mathrm e}^{2}-\frac {8 \,{\mathrm e}}{3}-\frac {2}{3}\right ) \left (1-\frac {\left (-6 x +2\right ) {\mathrm e}^{3 x}}{2}\right )}{3}-\frac {\left (2 \,{\mathrm e}^{2}+\frac {8 \,{\mathrm e}}{3}+1\right ) \left (2-\frac {\left (27 x^{2}-18 x +6\right ) {\mathrm e}^{3 x}}{3}\right )}{3}-\frac {\left (-\frac {4 \,{\mathrm e}}{9}-\frac {10}{27}\right ) \left (6-\frac {\left (-108 x^{3}+108 x^{2}-72 x +24\right ) {\mathrm e}^{3 x}}{4}\right )}{3}-\frac {8}{27}+\frac {\left (405 x^{4}-540 x^{3}+540 x^{2}-360 x +120\right ) {\mathrm e}^{3 x}}{405}-{\mathrm e}^{4} \left (1-{\mathrm e}^{3 x}\right )-\frac {4 \,{\mathrm e}^{3} \left (1-{\mathrm e}^{3 x}\right )}{3}-\frac {2 \,{\mathrm e}^{2} \left (1-{\mathrm e}^{3 x}\right )}{3}\) | \(155\) |
parts | \({\mathrm e}^{3 x} {\mathrm e}^{4}+4 x \,{\mathrm e}^{3} {\mathrm e}^{3 x}+6 \,{\mathrm e}^{2} x^{2} {\mathrm e}^{3 x}+4 \,{\mathrm e} x^{3} {\mathrm e}^{3 x}+{\mathrm e}^{3 x} x^{4}+6 \,{\mathrm e}^{2} x \,{\mathrm e}^{3 x}+8 x^{2} {\mathrm e} \,{\mathrm e}^{3 x}+2 x^{3} {\mathrm e}^{3 x}-\frac {4 \,{\mathrm e}^{3 x} {\mathrm e}^{2}}{3}+\frac {8 \,{\mathrm e} \,{\mathrm e}^{3 x} x}{3}+x^{2} {\mathrm e}^{3 x}-\frac {8 \,{\mathrm e} \,{\mathrm e}^{3 x}}{9}-\frac {16 \,{\mathrm e} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )}{9}-\frac {4 \,{\mathrm e} \left (9 x^{2} {\mathrm e}^{3 x}-6 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x}\right )}{9}-\frac {4 \,{\mathrm e}^{2} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )}{3}\) | \(185\) |
derivativedivides | \({\mathrm e}^{3 x} x^{4}+2 x^{3} {\mathrm e}^{3 x}+x^{2} {\mathrm e}^{3 x}+\frac {2 \,{\mathrm e}^{3 x} {\mathrm e}^{2}}{3}+\frac {4 \,{\mathrm e}^{3 x} {\mathrm e}^{3}}{3}+{\mathrm e}^{3 x} {\mathrm e}^{4}+\frac {8 \,{\mathrm e} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )}{9}+\frac {8 \,{\mathrm e} \left (9 x^{2} {\mathrm e}^{3 x}-6 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x}\right )}{9}+\frac {4 \,{\mathrm e} \left (27 x^{3} {\mathrm e}^{3 x}-27 x^{2} {\mathrm e}^{3 x}+18 x \,{\mathrm e}^{3 x}-6 \,{\mathrm e}^{3 x}\right )}{27}+2 \,{\mathrm e}^{2} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )+\frac {2 \,{\mathrm e}^{2} \left (9 x^{2} {\mathrm e}^{3 x}-6 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x}\right )}{3}+\frac {4 \,{\mathrm e}^{3} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )}{3}\) | \(206\) |
default | \({\mathrm e}^{3 x} x^{4}+2 x^{3} {\mathrm e}^{3 x}+x^{2} {\mathrm e}^{3 x}+\frac {2 \,{\mathrm e}^{3 x} {\mathrm e}^{2}}{3}+\frac {4 \,{\mathrm e}^{3 x} {\mathrm e}^{3}}{3}+{\mathrm e}^{3 x} {\mathrm e}^{4}+\frac {8 \,{\mathrm e} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )}{9}+\frac {8 \,{\mathrm e} \left (9 x^{2} {\mathrm e}^{3 x}-6 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x}\right )}{9}+\frac {4 \,{\mathrm e} \left (27 x^{3} {\mathrm e}^{3 x}-27 x^{2} {\mathrm e}^{3 x}+18 x \,{\mathrm e}^{3 x}-6 \,{\mathrm e}^{3 x}\right )}{27}+2 \,{\mathrm e}^{2} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )+\frac {2 \,{\mathrm e}^{2} \left (9 x^{2} {\mathrm e}^{3 x}-6 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x}\right )}{3}+\frac {4 \,{\mathrm e}^{3} \left (3 x \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x}\right )}{3}\) | \(206\) |
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (15) = 30\).
Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 3.07 \[ \int e^{3 x} \left (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 \left (2+18 x+18 x^2\right )+e \left (8 x+24 x^2+12 x^3\right )\right ) \, dx={\left (x^{4} + 2 \, x^{3} + x^{2} + 4 \, x e^{3} + 2 \, {\left (3 \, x^{2} + x\right )} e^{2} + 4 \, {\left (x^{3} + x^{2}\right )} e + e^{4}\right )} e^{\left (3 \, x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (14) = 28\).
Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.87 \[ \int e^{3 x} \left (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 \left (2+18 x+18 x^2\right )+e \left (8 x+24 x^2+12 x^3\right )\right ) \, dx=\left (x^{4} + 2 x^{3} + 4 e x^{3} + x^{2} + 4 e x^{2} + 6 x^{2} e^{2} + 2 x e^{2} + 4 x e^{3} + e^{4}\right ) e^{3 x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (15) = 30\).
Time = 0.20 (sec) , antiderivative size = 221, normalized size of antiderivative = 14.73 \[ \int e^{3 x} \left (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 \left (2+18 x+18 x^2\right )+e \left (8 x+24 x^2+12 x^3\right )\right ) \, dx=\frac {1}{27} \, {\left (27 \, x^{4} - 36 \, x^{3} + 36 \, x^{2} - 24 \, x + 8\right )} e^{\left (3 \, x\right )} + \frac {4}{9} \, {\left (9 \, x^{3} e - 9 \, x^{2} e + 6 \, x e - 2 \, e\right )} e^{\left (3 \, x\right )} + \frac {10}{27} \, {\left (9 \, x^{3} - 9 \, x^{2} + 6 \, x - 2\right )} e^{\left (3 \, x\right )} + \frac {2}{3} \, {\left (9 \, x^{2} e^{2} - 6 \, x e^{2} + 2 \, e^{2}\right )} e^{\left (3 \, x\right )} + \frac {8}{9} \, {\left (9 \, x^{2} e - 6 \, x e + 2 \, e\right )} e^{\left (3 \, x\right )} + \frac {1}{3} \, {\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} + \frac {4}{3} \, {\left (3 \, x e^{3} - e^{3}\right )} e^{\left (3 \, x\right )} + 2 \, {\left (3 \, x e^{2} - e^{2}\right )} e^{\left (3 \, x\right )} + \frac {8}{9} \, {\left (3 \, x e - e\right )} e^{\left (3 \, x\right )} + \frac {2}{9} \, {\left (3 \, x - 1\right )} e^{\left (3 \, x\right )} + e^{\left (3 \, x + 4\right )} + \frac {4}{3} \, e^{\left (3 \, x + 3\right )} + \frac {2}{3} \, e^{\left (3 \, x + 2\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (15) = 30\).
Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 4.20 \[ \int e^{3 x} \left (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 \left (2+18 x+18 x^2\right )+e \left (8 x+24 x^2+12 x^3\right )\right ) \, dx={\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{\left (3 \, x\right )} + 4 \, x e^{\left (3 \, x + 3\right )} + 2 \, {\left (3 \, x^{2} + x\right )} e^{\left (3 \, x + 2\right )} + 4 \, {\left (x^{3} + x^{2}\right )} e^{\left (3 \, x + 1\right )} + e^{\left (3 \, x + 4\right )} \]
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Time = 10.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.27 \[ \int e^{3 x} \left (3 e^4+2 x+9 x^2+10 x^3+3 x^4+e^3 (4+12 x)+e^2 \left (2+18 x+18 x^2\right )+e \left (8 x+24 x^2+12 x^3\right )\right ) \, dx={\mathrm {e}}^{3\,x}\,{\left (x+{\mathrm {e}}^2+2\,x\,\mathrm {e}+x^2\right )}^2 \]
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