Integrand size = 25, antiderivative size = 21 \[ \int \left (8+e^{\frac {5-4 x}{x}} \left (5 x^2-4 x^3\right )\right ) \, dx=-5+8 x-e^{-5+\frac {5+x}{x}} x^4 \]
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Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1607, 2326} \[ \int \left (8+e^{\frac {5-4 x}{x}} \left (5 x^2-4 x^3\right )\right ) \, dx=8 x-\frac {5 e^{\frac {5-4 x}{x}} x^2}{\frac {5-4 x}{x^2}+\frac {4}{x}} \]
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Rule 1607
Rule 2326
Rubi steps \begin{align*} \text {integral}& = 8 x+\int e^{\frac {5-4 x}{x}} \left (5 x^2-4 x^3\right ) \, dx \\ & = 8 x+\int e^{\frac {5-4 x}{x}} (5-4 x) x^2 \, dx \\ & = 8 x-\frac {5 e^{\frac {5-4 x}{x}} x^2}{\frac {5-4 x}{x^2}+\frac {4}{x}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \left (8+e^{\frac {5-4 x}{x}} \left (5 x^2-4 x^3\right )\right ) \, dx=8 x-e^{-4+\frac {5}{x}} x^4 \]
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Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(8 x -{\mathrm e}^{\frac {5}{x}-4} x^{4}\) | \(18\) |
default | \(8 x -{\mathrm e}^{\frac {5}{x}-4} x^{4}\) | \(18\) |
parts | \(8 x -{\mathrm e}^{\frac {5}{x}-4} x^{4}\) | \(18\) |
norman | \(8 x -x^{4} {\mathrm e}^{\frac {-4 x +5}{x}}\) | \(20\) |
risch | \(8 x -x^{4} {\mathrm e}^{-\frac {-5+4 x}{x}}\) | \(21\) |
parallelrisch | \(8 x -x^{4} {\mathrm e}^{-\frac {-5+4 x}{x}}\) | \(21\) |
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none
Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \left (8+e^{\frac {5-4 x}{x}} \left (5 x^2-4 x^3\right )\right ) \, dx=-x^{4} e^{\left (-\frac {4 \, x - 5}{x}\right )} + 8 \, x \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \left (8+e^{\frac {5-4 x}{x}} \left (5 x^2-4 x^3\right )\right ) \, dx=- x^{4} e^{\frac {5 - 4 x}{x}} + 8 x \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \left (8+e^{\frac {5-4 x}{x}} \left (5 x^2-4 x^3\right )\right ) \, dx=-625 \, e^{\left (-4\right )} \Gamma \left (-3, -\frac {5}{x}\right ) - 2500 \, e^{\left (-4\right )} \Gamma \left (-4, -\frac {5}{x}\right ) + 8 \, x \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.14 \[ \int \left (8+e^{\frac {5-4 x}{x}} \left (5 x^2-4 x^3\right )\right ) \, dx=8 \, x - \frac {625 \, e^{\left (-\frac {4 \, x - 5}{x}\right )}}{\frac {{\left (4 \, x - 5\right )}^{4}}{x^{4}} - \frac {16 \, {\left (4 \, x - 5\right )}^{3}}{x^{3}} + \frac {96 \, {\left (4 \, x - 5\right )}^{2}}{x^{2}} - \frac {256 \, {\left (4 \, x - 5\right )}}{x} + 256} \]
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Time = 8.94 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \left (8+e^{\frac {5-4 x}{x}} \left (5 x^2-4 x^3\right )\right ) \, dx=8\,x-x^4\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{5/x} \]
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