Integrand size = 81, antiderivative size = 25 \[ \int e^{-10 \sqrt {\frac {2}{3}}} \left (4 e^6 x^3+10 e^3 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (10 x^4+2 x^5+e^3 \left (8 x^3+2 x^4\right )\right )\right ) \, dx=e^{-10 \sqrt {\frac {2}{3}}} x^4 \left (e^3+e^x+x\right )^2 \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(110\) vs. \(2(25)=50\).
Time = 0.24 (sec) , antiderivative size = 110, normalized size of antiderivative = 4.40, number of steps used = 38, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {12, 1607, 2227, 2207, 2225} \[ \int e^{-10 \sqrt {\frac {2}{3}}} \left (4 e^6 x^3+10 e^3 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (10 x^4+2 x^5+e^3 \left (8 x^3+2 x^4\right )\right )\right ) \, dx=e^{-10 \sqrt {\frac {2}{3}}} x^6+2 e^{x-10 \sqrt {\frac {2}{3}}} x^5+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+2 e^{x+\frac {1}{3} \left (9-10 \sqrt {6}\right )} x^4+e^{2 x-10 \sqrt {\frac {2}{3}}} x^4+e^{6-10 \sqrt {\frac {2}{3}}} x^4 \]
[In]
[Out]
Rule 12
Rule 1607
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = e^{-10 \sqrt {\frac {2}{3}}} \int \left (4 e^6 x^3+10 e^3 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (10 x^4+2 x^5+e^3 \left (8 x^3+2 x^4\right )\right )\right ) \, dx \\ & = e^{6-10 \sqrt {\frac {2}{3}}} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6+e^{-10 \sqrt {\frac {2}{3}}} \int e^{2 x} \left (4 x^3+2 x^4\right ) \, dx+e^{-10 \sqrt {\frac {2}{3}}} \int e^x \left (10 x^4+2 x^5+e^3 \left (8 x^3+2 x^4\right )\right ) \, dx \\ & = e^{6-10 \sqrt {\frac {2}{3}}} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6+e^{-10 \sqrt {\frac {2}{3}}} \int e^{2 x} x^3 (4+2 x) \, dx+e^{-10 \sqrt {\frac {2}{3}}} \int \left (10 e^x x^4+2 e^x x^5+2 e^{3+x} x^3 (4+x)\right ) \, dx \\ & = e^{6-10 \sqrt {\frac {2}{3}}} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6+e^{-10 \sqrt {\frac {2}{3}}} \int \left (4 e^{2 x} x^3+2 e^{2 x} x^4\right ) \, dx+\left (2 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x^5 \, dx+\left (2 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} x^3 (4+x) \, dx+\left (10 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x^4 \, dx \\ & = e^{6-10 \sqrt {\frac {2}{3}}} x^4+10 e^{-10 \sqrt {\frac {2}{3}}+x} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+2 e^{-10 \sqrt {\frac {2}{3}}+x} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6+\left (2 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} x^4 \, dx+\left (2 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int \left (4 e^{3+x} x^3+e^{3+x} x^4\right ) \, dx+\left (4 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} x^3 \, dx-\left (10 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x^4 \, dx-\left (40 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x^3 \, dx \\ & = -40 e^{-10 \sqrt {\frac {2}{3}}+x} x^3+2 e^{-10 \sqrt {\frac {2}{3}}+2 x} x^3+e^{6-10 \sqrt {\frac {2}{3}}} x^4+e^{-10 \sqrt {\frac {2}{3}}+2 x} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+2 e^{-10 \sqrt {\frac {2}{3}}+x} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6+\left (2 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} x^4 \, dx-\left (4 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} x^3 \, dx-\left (6 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} x^2 \, dx+\left (8 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} x^3 \, dx+\left (40 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x^3 \, dx+\left (120 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x^2 \, dx \\ & = 120 e^{-10 \sqrt {\frac {2}{3}}+x} x^2-3 e^{-10 \sqrt {\frac {2}{3}}+2 x} x^2+8 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x} x^3+e^{6-10 \sqrt {\frac {2}{3}}} x^4+2 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x} x^4+e^{-10 \sqrt {\frac {2}{3}}+2 x} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+2 e^{-10 \sqrt {\frac {2}{3}}+x} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6+\left (6 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} x \, dx+\left (6 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} x^2 \, dx-\left (8 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} x^3 \, dx-\left (24 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} x^2 \, dx-\left (120 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x^2 \, dx-\left (240 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x \, dx \\ & = -240 e^{-10 \sqrt {\frac {2}{3}}+x} x+3 e^{-10 \sqrt {\frac {2}{3}}+2 x} x-24 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x} x^2+e^{6-10 \sqrt {\frac {2}{3}}} x^4+2 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x} x^4+e^{-10 \sqrt {\frac {2}{3}}+2 x} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+2 e^{-10 \sqrt {\frac {2}{3}}+x} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6-\left (3 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} \, dx-\left (6 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} x \, dx+\left (24 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} x^2 \, dx+\left (48 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} x \, dx+\left (240 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x \, dx+\left (240 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x x \, dx \\ & = 240 e^{-10 \sqrt {\frac {2}{3}}+x}-\frac {3}{2} e^{-10 \sqrt {\frac {2}{3}}+2 x}+48 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x} x+e^{6-10 \sqrt {\frac {2}{3}}} x^4+2 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x} x^4+e^{-10 \sqrt {\frac {2}{3}}+2 x} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+2 e^{-10 \sqrt {\frac {2}{3}}+x} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6+\left (3 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{2 x} \, dx-\left (48 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} \, dx-\left (48 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} x \, dx-\left (240 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^x \, dx \\ & = -48 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x}+e^{6-10 \sqrt {\frac {2}{3}}} x^4+2 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x} x^4+e^{-10 \sqrt {\frac {2}{3}}+2 x} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+2 e^{-10 \sqrt {\frac {2}{3}}+x} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6+\left (48 e^{-10 \sqrt {\frac {2}{3}}}\right ) \int e^{3+x} \, dx \\ & = e^{6-10 \sqrt {\frac {2}{3}}} x^4+2 e^{\frac {1}{3} \left (9-10 \sqrt {6}\right )+x} x^4+e^{-10 \sqrt {\frac {2}{3}}+2 x} x^4+2 e^{3-10 \sqrt {\frac {2}{3}}} x^5+2 e^{-10 \sqrt {\frac {2}{3}}+x} x^5+e^{-10 \sqrt {\frac {2}{3}}} x^6 \\ \end{align*}
Time = 2.33 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int e^{-10 \sqrt {\frac {2}{3}}} \left (4 e^6 x^3+10 e^3 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (10 x^4+2 x^5+e^3 \left (8 x^3+2 x^4\right )\right )\right ) \, dx=e^{-10 \sqrt {\frac {2}{3}}} x^4 \left (e^3+e^x+x\right )^2 \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(23)=46\).
Time = 1.97 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24
method | result | size |
parallelrisch | \({\mathrm e}^{-\frac {10 \sqrt {6}}{3}} \left (2 \,{\mathrm e}^{x} {\mathrm e}^{3} x^{4}+{\mathrm e}^{6} x^{4}+2 x^{5} {\mathrm e}^{3}+x^{6}+{\mathrm e}^{2 x} x^{4}+2 x^{5} {\mathrm e}^{x}\right )\) | \(56\) |
risch | \(x^{4} {\mathrm e}^{-\frac {10 \sqrt {3}\, \sqrt {2}}{3}+6}+2 x^{5} {\mathrm e}^{-\frac {10 \sqrt {3}\, \sqrt {2}}{3}+3}+{\mathrm e}^{-\frac {10 \sqrt {3}\, \sqrt {2}}{3}} x^{6}+x^{4} {\mathrm e}^{-\frac {10 \sqrt {3}\, \sqrt {2}}{3}+2 x}+\left (2 x^{4} {\mathrm e}^{3}+2 x^{5}\right ) {\mathrm e}^{-\frac {10 \sqrt {3}\, \sqrt {2}}{3}+x}\) | \(88\) |
default | \({\mathrm e}^{-\frac {10 \sqrt {6}}{3}} \left (2 x^{5} {\mathrm e}^{x}+8 \,{\mathrm e}^{3} \left ({\mathrm e}^{x} x^{3}-3 \,{\mathrm e}^{x} x^{2}+6 \,{\mathrm e}^{x} x -6 \,{\mathrm e}^{x}\right )+2 \,{\mathrm e}^{3} \left ({\mathrm e}^{x} x^{4}-4 \,{\mathrm e}^{x} x^{3}+12 \,{\mathrm e}^{x} x^{2}-24 \,{\mathrm e}^{x} x +24 \,{\mathrm e}^{x}\right )+{\mathrm e}^{2 x} x^{4}+x^{6}+{\mathrm e}^{6} x^{4}+2 x^{5} {\mathrm e}^{3}\right )\) | \(108\) |
norman | \(\left ({\mathrm e}^{-\frac {5 \sqrt {6}}{3}} x^{6}+{\mathrm e}^{-\frac {5 \sqrt {6}}{3}} x^{4} {\mathrm e}^{2 x}+{\mathrm e}^{-\frac {5 \sqrt {6}}{3}} {\mathrm e}^{6} x^{4}+2 \,{\mathrm e}^{-\frac {5 \sqrt {6}}{3}} x^{5} {\mathrm e}^{x}+2 \,{\mathrm e}^{-\frac {5 \sqrt {6}}{3}} {\mathrm e}^{3} x^{5}+2 \,{\mathrm e}^{-\frac {5 \sqrt {6}}{3}} {\mathrm e}^{3} x^{4} {\mathrm e}^{x}\right ) {\mathrm e}^{-\frac {5 \sqrt {6}}{3}}\) | \(117\) |
parts | \(2 \,{\mathrm e}^{-\frac {10 \sqrt {6}}{3}} \left (\frac {x^{6}}{2}+x^{5} {\mathrm e}^{3}+\frac {{\mathrm e}^{6} x^{4}}{2}\right )+{\mathrm e}^{-\frac {10 \sqrt {6}}{3}} {\mathrm e}^{2 x} x^{4}+2 \,{\mathrm e}^{-\frac {10 \sqrt {6}}{3}} \left (x^{5} {\mathrm e}^{x}+{\mathrm e}^{3} \left ({\mathrm e}^{x} x^{4}-4 \,{\mathrm e}^{x} x^{3}+12 \,{\mathrm e}^{x} x^{2}-24 \,{\mathrm e}^{x} x +24 \,{\mathrm e}^{x}\right )+4 \,{\mathrm e}^{3} \left ({\mathrm e}^{x} x^{3}-3 \,{\mathrm e}^{x} x^{2}+6 \,{\mathrm e}^{x} x -6 \,{\mathrm e}^{x}\right )\right )\) | \(135\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int e^{-10 \sqrt {\frac {2}{3}}} \left (4 e^6 x^3+10 e^3 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (10 x^4+2 x^5+e^3 \left (8 x^3+2 x^4\right )\right )\right ) \, dx={\left (x^{6} + 2 \, x^{5} e^{3} + x^{4} e^{6} + x^{4} e^{\left (2 \, x\right )} + 2 \, {\left (x^{5} + x^{4} e^{3}\right )} e^{x}\right )} e^{\left (-\frac {10}{3} \, \sqrt {3} \sqrt {2}\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (22) = 44\).
Time = 0.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.56 \[ \int e^{-10 \sqrt {\frac {2}{3}}} \left (4 e^6 x^3+10 e^3 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (10 x^4+2 x^5+e^3 \left (8 x^3+2 x^4\right )\right )\right ) \, dx=\frac {x^{6}}{e^{\frac {10 \sqrt {6}}{3}}} + \frac {2 x^{5} e^{3}}{e^{\frac {10 \sqrt {6}}{3}}} + \frac {x^{4} e^{6}}{e^{\frac {10 \sqrt {6}}{3}}} + \frac {x^{4} e^{\frac {10 \sqrt {6}}{3}} e^{2 x} + \left (2 x^{5} e^{\frac {10 \sqrt {6}}{3}} + 2 x^{4} e^{3} e^{\frac {10 \sqrt {6}}{3}}\right ) e^{x}}{e^{\frac {20 \sqrt {6}}{3}}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (21) = 42\).
Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int e^{-10 \sqrt {\frac {2}{3}}} \left (4 e^6 x^3+10 e^3 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (10 x^4+2 x^5+e^3 \left (8 x^3+2 x^4\right )\right )\right ) \, dx={\left (x^{6} + 2 \, x^{5} e^{3} + x^{4} e^{6} + x^{4} e^{\left (2 \, x\right )} + 2 \, {\left (x^{5} + x^{4} e^{3}\right )} e^{x}\right )} e^{\left (-\frac {10}{3} \, \sqrt {3} \sqrt {2}\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (21) = 42\).
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04 \[ \int e^{-10 \sqrt {\frac {2}{3}}} \left (4 e^6 x^3+10 e^3 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (10 x^4+2 x^5+e^3 \left (8 x^3+2 x^4\right )\right )\right ) \, dx={\left (x^{6} + 2 \, x^{5} e^{3} + 2 \, x^{5} e^{x} + x^{4} e^{6} + x^{4} e^{\left (2 \, x\right )} + 2 \, x^{4} e^{\left (x + 3\right )}\right )} e^{\left (-\frac {10}{3} \, \sqrt {3} \sqrt {2}\right )} \]
[In]
[Out]
Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int e^{-10 \sqrt {\frac {2}{3}}} \left (4 e^6 x^3+10 e^3 x^4+6 x^5+e^{2 x} \left (4 x^3+2 x^4\right )+e^x \left (10 x^4+2 x^5+e^3 \left (8 x^3+2 x^4\right )\right )\right ) \, dx=x^4\,{\mathrm {e}}^{-\frac {10\,\sqrt {6}}{3}}\,{\left (x+{\mathrm {e}}^3+{\mathrm {e}}^x\right )}^2 \]
[In]
[Out]