Integrand size = 45, antiderivative size = 28 \[ \int \frac {e^{\frac {45-9 x}{x^2}} (270-27 x)+3 e^5 x^3-6 e^{7-2 x} x^3}{e^5 x^3} \, dx=3 \left (-4-e^{-5+\frac {9 (5-x)}{x^2}}+e^{2-2 x}+x\right ) \]
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Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 14, 2225, 6820, 6838} \[ \int \frac {e^{\frac {45-9 x}{x^2}} (270-27 x)+3 e^5 x^3-6 e^{7-2 x} x^3}{e^5 x^3} \, dx=-3 e^{\frac {9 (5-x)}{x^2}-5}+3 x+3 e^{2-2 x} \]
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Rule 12
Rule 14
Rule 2225
Rule 6820
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {e^{\frac {45-9 x}{x^2}} (270-27 x)+3 e^5 x^3-6 e^{7-2 x} x^3}{x^3} \, dx}{e^5} \\ & = \frac {\int \left (-6 e^{7-2 x}+\frac {3 e^{-9/x} \left (90 e^{\frac {45}{x^2}}-9 e^{\frac {45}{x^2}} x+e^{5+\frac {9}{x}} x^3\right )}{x^3}\right ) \, dx}{e^5} \\ & = \frac {3 \int \frac {e^{-9/x} \left (90 e^{\frac {45}{x^2}}-9 e^{\frac {45}{x^2}} x+e^{5+\frac {9}{x}} x^3\right )}{x^3} \, dx}{e^5}-\frac {6 \int e^{7-2 x} \, dx}{e^5} \\ & = 3 e^{2-2 x}+\frac {3 \int \left (e^5-\frac {9 e^{-\frac {9 (-5+x)}{x^2}} (-10+x)}{x^3}\right ) \, dx}{e^5} \\ & = 3 e^{2-2 x}+3 x-\frac {27 \int \frac {e^{-\frac {9 (-5+x)}{x^2}} (-10+x)}{x^3} \, dx}{e^5} \\ & = -3 e^{-5+\frac {9 (5-x)}{x^2}}+3 e^{2-2 x}+3 x \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {45-9 x}{x^2}} (270-27 x)+3 e^5 x^3-6 e^{7-2 x} x^3}{e^5 x^3} \, dx=-3 e^{-5+\frac {45}{x^2}-\frac {9}{x}}+3 e^{2-2 x}+3 x \]
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Time = 0.70 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04
method | result | size |
parts | \(3 x +3 \,{\mathrm e}^{2-2 x}-3 \,{\mathrm e}^{-5} {\mathrm e}^{\frac {-9 x +45}{x^{2}}}\) | \(29\) |
risch | \(3 x +3 \,{\mathrm e}^{2-2 x}-3 \,{\mathrm e}^{-\frac {5 x^{2}+9 x -45}{x^{2}}}\) | \(31\) |
parallelrisch | \({\mathrm e}^{-5} \left (3 x \,{\mathrm e}^{5}+3 \,{\mathrm e}^{5} {\mathrm e}^{2-2 x}-3 \,{\mathrm e}^{-\frac {9 \left (-5+x \right )}{x^{2}}}\right )\) | \(33\) |
default | \({\mathrm e}^{-5} \left (-3 \,{\mathrm e}^{-\frac {9}{x}+\frac {45}{x^{2}}}+3 \,{\mathrm e}^{5} {\mathrm e}^{2} {\mathrm e}^{-2 x}+3 x \,{\mathrm e}^{5}\right )\) | \(36\) |
norman | \(\frac {3 x^{3}+3 x^{2} {\mathrm e}^{2-2 x}-3 x^{2} {\mathrm e}^{-5} {\mathrm e}^{\frac {-9 x +45}{x^{2}}}}{x^{2}}\) | \(41\) |
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Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {45-9 x}{x^2}} (270-27 x)+3 e^5 x^3-6 e^{7-2 x} x^3}{e^5 x^3} \, dx=3 \, {\left (x e^{5} + e^{\left (-2 \, x + 7\right )} - e^{\left (-\frac {9 \, {\left (x - 5\right )}}{x^{2}}\right )}\right )} e^{\left (-5\right )} \]
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Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {45-9 x}{x^2}} (270-27 x)+3 e^5 x^3-6 e^{7-2 x} x^3}{e^5 x^3} \, dx=3 x - \frac {3 e^{\frac {45 - 9 x}{x^{2}}}}{e^{5}} + 3 e^{2 - 2 x} \]
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Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {45-9 x}{x^2}} (270-27 x)+3 e^5 x^3-6 e^{7-2 x} x^3}{e^5 x^3} \, dx=3 \, {\left (x e^{5} - {\left (e^{\left (2 \, x + \frac {45}{x^{2}}\right )} - e^{\left (\frac {9}{x} + 7\right )}\right )} e^{\left (-2 \, x - \frac {9}{x}\right )}\right )} e^{\left (-5\right )} \]
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Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {45-9 x}{x^2}} (270-27 x)+3 e^5 x^3-6 e^{7-2 x} x^3}{e^5 x^3} \, dx=3 \, {\left (x e^{5} + e^{\left (-2 \, x + 7\right )} - e^{\left (-\frac {9}{x} + \frac {45}{x^{2}}\right )}\right )} e^{\left (-5\right )} \]
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Time = 14.45 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {45-9 x}{x^2}} (270-27 x)+3 e^5 x^3-6 e^{7-2 x} x^3}{e^5 x^3} \, dx=3\,x+3\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^2-3\,{\mathrm {e}}^{-5}\,{\mathrm {e}}^{-\frac {9}{x}}\,{\mathrm {e}}^{\frac {45}{x^2}} \]
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