Integrand size = 57, antiderivative size = 23 \[ \int \left (8+8 x+e^{2 x} \left (2 e^{10}+e^5 (-2-4 x)+2 x+2 x^2\right )+e^x \left (-4-12 x-4 x^2+e^5 (8+4 x)\right )\right ) \, dx=\left (2+2 x-e^x \left (1-\frac {e^5}{x}\right ) x\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(23)=46\).
Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.35, number of steps used = 21, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2227, 2225, 2207} \[ \int \left (8+8 x+e^{2 x} \left (2 e^{10}+e^5 (-2-4 x)+2 x+2 x^2\right )+e^x \left (-4-12 x-4 x^2+e^5 (8+4 x)\right )\right ) \, dx=-4 e^x x^2+e^{2 x} x^2+4 x^2-4 e^x x+8 x-4 e^{x+5}+e^{2 x+5}+e^{2 x+10}+4 e^{x+5} (x+2)-e^{2 x+5} (2 x+1) \]
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Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = 8 x+4 x^2+\int e^{2 x} \left (2 e^{10}+e^5 (-2-4 x)+2 x+2 x^2\right ) \, dx+\int e^x \left (-4-12 x-4 x^2+e^5 (8+4 x)\right ) \, dx \\ & = 8 x+4 x^2+\int \left (-4 e^x-12 e^x x-4 e^x x^2+4 e^{5+x} (2+x)\right ) \, dx+\int \left (2 e^{10+2 x}+2 e^{2 x} x+2 e^{2 x} x^2-2 e^{5+2 x} (1+2 x)\right ) \, dx \\ & = 8 x+4 x^2+2 \int e^{10+2 x} \, dx+2 \int e^{2 x} x \, dx+2 \int e^{2 x} x^2 \, dx-2 \int e^{5+2 x} (1+2 x) \, dx-4 \int e^x \, dx-4 \int e^x x^2 \, dx+4 \int e^{5+x} (2+x) \, dx-12 \int e^x x \, dx \\ & = -4 e^x+e^{10+2 x}+8 x-12 e^x x+e^{2 x} x+4 x^2-4 e^x x^2+e^{2 x} x^2+4 e^{5+x} (2+x)-e^{5+2 x} (1+2 x)+2 \int e^{5+2 x} \, dx-2 \int e^{2 x} x \, dx-4 \int e^{5+x} \, dx+8 \int e^x x \, dx+12 \int e^x \, dx-\int e^{2 x} \, dx \\ & = 8 e^x-\frac {e^{2 x}}{2}-4 e^{5+x}+e^{5+2 x}+e^{10+2 x}+8 x-4 e^x x+4 x^2-4 e^x x^2+e^{2 x} x^2+4 e^{5+x} (2+x)-e^{5+2 x} (1+2 x)-8 \int e^x \, dx+\int e^{2 x} \, dx \\ & = -4 e^{5+x}+e^{5+2 x}+e^{10+2 x}+8 x-4 e^x x+4 x^2-4 e^x x^2+e^{2 x} x^2+4 e^{5+x} (2+x)-e^{5+2 x} (1+2 x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(23)=46\).
Time = 0.35 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.22 \[ \int \left (8+8 x+e^{2 x} \left (2 e^{10}+e^5 (-2-4 x)+2 x+2 x^2\right )+e^x \left (-4-12 x-4 x^2+e^5 (8+4 x)\right )\right ) \, dx=8 x+4 x^2+e^x \left (4 e^5+4 \left (-1+e^5\right ) x-4 x^2\right )+e^{2 x} \left (e^{10}-2 e^5 x+x^2\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(21)=42\).
Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13
method | result | size |
risch | \(\left ({\mathrm e}^{10}-2 x \,{\mathrm e}^{5}+x^{2}\right ) {\mathrm e}^{2 x}+\left (4 x \,{\mathrm e}^{5}-4 x^{2}+4 \,{\mathrm e}^{5}-4 x \right ) {\mathrm e}^{x}+4 x^{2}+8 x\) | \(49\) |
norman | \({\mathrm e}^{2 x} {\mathrm e}^{10}+{\mathrm e}^{2 x} x^{2}+\left (-4+4 \,{\mathrm e}^{5}\right ) x \,{\mathrm e}^{x}+8 x +4 x^{2}+4 \,{\mathrm e}^{5} {\mathrm e}^{x}-4 \,{\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{2 x} {\mathrm e}^{5} x\) | \(59\) |
parallelrisch | \(-2 \,{\mathrm e}^{2 x} {\mathrm e}^{5} x +{\mathrm e}^{2 x} x^{2}+4 x \,{\mathrm e}^{5} {\mathrm e}^{x}-4 \,{\mathrm e}^{x} x^{2}+{\mathrm e}^{2 x} {\mathrm e}^{10}+4 \,{\mathrm e}^{5} {\mathrm e}^{x}-4 \,{\mathrm e}^{x} x +4 x^{2}+8 x\) | \(61\) |
default | \(8 x +{\mathrm e}^{2 x} x^{2}-{\mathrm e}^{5} {\mathrm e}^{2 x}+{\mathrm e}^{2 x} {\mathrm e}^{10}-4 \,{\mathrm e}^{5} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )-4 \,{\mathrm e}^{x} x -4 \,{\mathrm e}^{x} x^{2}+8 \,{\mathrm e}^{5} {\mathrm e}^{x}+4 \,{\mathrm e}^{5} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+4 x^{2}\) | \(84\) |
parts | \(8 x +{\mathrm e}^{2 x} x^{2}-{\mathrm e}^{5} {\mathrm e}^{2 x}+{\mathrm e}^{2 x} {\mathrm e}^{10}-4 \,{\mathrm e}^{5} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )-4 \,{\mathrm e}^{x} x -4 \,{\mathrm e}^{x} x^{2}+8 \,{\mathrm e}^{5} {\mathrm e}^{x}+4 \,{\mathrm e}^{5} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+4 x^{2}\) | \(84\) |
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Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78 \[ \int \left (8+8 x+e^{2 x} \left (2 e^{10}+e^5 (-2-4 x)+2 x+2 x^2\right )+e^x \left (-4-12 x-4 x^2+e^5 (8+4 x)\right )\right ) \, dx=4 \, x^{2} + {\left (x^{2} - 2 \, x e^{5} + e^{10}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{2} - {\left (x + 1\right )} e^{5} + x\right )} e^{x} + 8 \, x \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (17) = 34\).
Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \left (8+8 x+e^{2 x} \left (2 e^{10}+e^5 (-2-4 x)+2 x+2 x^2\right )+e^x \left (-4-12 x-4 x^2+e^5 (8+4 x)\right )\right ) \, dx=4 x^{2} + 8 x + \left (x^{2} - 2 x e^{5} + e^{10}\right ) e^{2 x} + \left (- 4 x^{2} - 4 x + 4 x e^{5} + 4 e^{5}\right ) e^{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \left (8+8 x+e^{2 x} \left (2 e^{10}+e^5 (-2-4 x)+2 x+2 x^2\right )+e^x \left (-4-12 x-4 x^2+e^5 (8+4 x)\right )\right ) \, dx=4 \, x^{2} + {\left (x^{2} - 2 \, x e^{5} + e^{10}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{2} - x {\left (e^{5} - 1\right )} - e^{5}\right )} e^{x} + 8 \, x \]
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Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (19) = 38\).
Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.17 \[ \int \left (8+8 x+e^{2 x} \left (2 e^{10}+e^5 (-2-4 x)+2 x+2 x^2\right )+e^x \left (-4-12 x-4 x^2+e^5 (8+4 x)\right )\right ) \, dx=x^{2} e^{\left (2 \, x\right )} + 4 \, x^{2} - 2 \, x e^{\left (2 \, x + 5\right )} + 4 \, {\left (x + 1\right )} e^{\left (x + 5\right )} - 4 \, {\left (x^{2} + x\right )} e^{x} + 8 \, x + e^{\left (2 \, x + 10\right )} \]
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Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39 \[ \int \left (8+8 x+e^{2 x} \left (2 e^{10}+e^5 (-2-4 x)+2 x+2 x^2\right )+e^x \left (-4-12 x-4 x^2+e^5 (8+4 x)\right )\right ) \, dx=8\,x+4\,{\mathrm {e}}^{x+5}+{\mathrm {e}}^{2\,x+10}-4\,x^2\,{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^{2\,x+5}+x^2\,{\mathrm {e}}^{2\,x}+4\,x^2+x\,{\mathrm {e}}^x\,\left (4\,{\mathrm {e}}^5-4\right ) \]
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