Integrand size = 54, antiderivative size = 33 \[ \int \frac {e^{-\frac {4 x^2}{25}} \left (-25 x^2-32 x^3+8 x^4+e^{\frac {4 x^2}{25}} \left (200+450 x^2-675 x^4\right )\right )}{225 x^2} \, dx=\frac {(-4+x) \left (2-e^{-\frac {4 x^2}{25}} x\right )}{9 x}-x \left (-2+x^2\right ) \]
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Time = 0.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 6820, 2258, 2236, 2240, 2243} \[ \int \frac {e^{-\frac {4 x^2}{25}} \left (-25 x^2-32 x^3+8 x^4+e^{\frac {4 x^2}{25}} \left (200+450 x^2-675 x^4\right )\right )}{225 x^2} \, dx=-x^3-\frac {1}{9} e^{-\frac {4 x^2}{25}} x+\frac {4}{9} e^{-\frac {4 x^2}{25}}+2 x-\frac {8}{9 x} \]
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Rule 12
Rule 2236
Rule 2240
Rule 2243
Rule 2258
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \frac {1}{225} \int \frac {e^{-\frac {4 x^2}{25}} \left (-25 x^2-32 x^3+8 x^4+e^{\frac {4 x^2}{25}} \left (200+450 x^2-675 x^4\right )\right )}{x^2} \, dx \\ & = \frac {1}{225} \int \left (25 \left (18+\frac {8}{x^2}-27 x^2\right )+e^{-\frac {4 x^2}{25}} \left (-25-32 x+8 x^2\right )\right ) \, dx \\ & = \frac {1}{225} \int e^{-\frac {4 x^2}{25}} \left (-25-32 x+8 x^2\right ) \, dx+\frac {1}{9} \int \left (18+\frac {8}{x^2}-27 x^2\right ) \, dx \\ & = -\frac {8}{9 x}+2 x-x^3+\frac {1}{225} \int \left (-25 e^{-\frac {4 x^2}{25}}-32 e^{-\frac {4 x^2}{25}} x+8 e^{-\frac {4 x^2}{25}} x^2\right ) \, dx \\ & = -\frac {8}{9 x}+2 x-x^3+\frac {8}{225} \int e^{-\frac {4 x^2}{25}} x^2 \, dx-\frac {1}{9} \int e^{-\frac {4 x^2}{25}} \, dx-\frac {32}{225} \int e^{-\frac {4 x^2}{25}} x \, dx \\ & = \frac {4}{9} e^{-\frac {4 x^2}{25}}-\frac {8}{9 x}+2 x-\frac {1}{9} e^{-\frac {4 x^2}{25}} x-x^3-\frac {5}{36} \sqrt {\pi } \text {erf}\left (\frac {2 x}{5}\right )+\frac {1}{9} \int e^{-\frac {4 x^2}{25}} \, dx \\ & = \frac {4}{9} e^{-\frac {4 x^2}{25}}-\frac {8}{9 x}+2 x-\frac {1}{9} e^{-\frac {4 x^2}{25}} x-x^3 \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30 \[ \int \frac {e^{-\frac {4 x^2}{25}} \left (-25 x^2-32 x^3+8 x^4+e^{\frac {4 x^2}{25}} \left (200+450 x^2-675 x^4\right )\right )}{225 x^2} \, dx=\frac {4}{9} e^{-\frac {4 x^2}{25}}-\frac {8}{9 x}+2 x-\frac {1}{9} e^{-\frac {4 x^2}{25}} x-x^3 \]
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Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85
method | result | size |
risch | \(2 x -\frac {8}{9 x}-x^{3}+\frac {\left (-25 x +100\right ) {\mathrm e}^{-\frac {4 x^{2}}{25}}}{225}\) | \(28\) |
default | \(2 x -\frac {8}{9 x}-x^{3}+\frac {4 \,{\mathrm e}^{-\frac {4 x^{2}}{25}}}{9}-\frac {x \,{\mathrm e}^{-\frac {4 x^{2}}{25}}}{9}\) | \(34\) |
parts | \(2 x -\frac {8}{9 x}-x^{3}+\frac {4 \,{\mathrm e}^{-\frac {4 x^{2}}{25}}}{9}-\frac {x \,{\mathrm e}^{-\frac {4 x^{2}}{25}}}{9}\) | \(34\) |
norman | \(\frac {\left (\frac {4 x}{9}-\frac {x^{2}}{9}+2 x^{2} {\mathrm e}^{\frac {4 x^{2}}{25}}-{\mathrm e}^{\frac {4 x^{2}}{25}} x^{4}-\frac {8 \,{\mathrm e}^{\frac {4 x^{2}}{25}}}{9}\right ) {\mathrm e}^{-\frac {4 x^{2}}{25}}}{x}\) | \(52\) |
parallelrisch | \(\frac {\left (-5625 \,{\mathrm e}^{\frac {4 x^{2}}{25}} x^{4}+11250 x^{2} {\mathrm e}^{\frac {4 x^{2}}{25}}-625 x^{2}+2500 x -5000 \,{\mathrm e}^{\frac {4 x^{2}}{25}}\right ) {\mathrm e}^{-\frac {4 x^{2}}{25}}}{5625 x}\) | \(53\) |
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Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-\frac {4 x^2}{25}} \left (-25 x^2-32 x^3+8 x^4+e^{\frac {4 x^2}{25}} \left (200+450 x^2-675 x^4\right )\right )}{225 x^2} \, dx=-\frac {{\left (x^{2} + {\left (9 \, x^{4} - 18 \, x^{2} + 8\right )} e^{\left (\frac {4}{25} \, x^{2}\right )} - 4 \, x\right )} e^{\left (-\frac {4}{25} \, x^{2}\right )}}{9 \, x} \]
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Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \frac {e^{-\frac {4 x^2}{25}} \left (-25 x^2-32 x^3+8 x^4+e^{\frac {4 x^2}{25}} \left (200+450 x^2-675 x^4\right )\right )}{225 x^2} \, dx=- x^{3} + 2 x + \frac {\left (4 - x\right ) e^{- \frac {4 x^{2}}{25}}}{9} - \frac {8}{9 x} \]
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Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {e^{-\frac {4 x^2}{25}} \left (-25 x^2-32 x^3+8 x^4+e^{\frac {4 x^2}{25}} \left (200+450 x^2-675 x^4\right )\right )}{225 x^2} \, dx=-x^{3} - \frac {1}{9} \, x e^{\left (-\frac {4}{25} \, x^{2}\right )} + 2 \, x - \frac {8}{9 \, x} + \frac {4}{9} \, e^{\left (-\frac {4}{25} \, x^{2}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {e^{-\frac {4 x^2}{25}} \left (-25 x^2-32 x^3+8 x^4+e^{\frac {4 x^2}{25}} \left (200+450 x^2-675 x^4\right )\right )}{225 x^2} \, dx=-\frac {9 \, x^{4} + x^{2} e^{\left (-\frac {4}{25} \, x^{2}\right )} - 18 \, x^{2} - 4 \, x e^{\left (-\frac {4}{25} \, x^{2}\right )} + 8}{9 \, x} \]
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Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-\frac {4 x^2}{25}} \left (-25 x^2-32 x^3+8 x^4+e^{\frac {4 x^2}{25}} \left (200+450 x^2-675 x^4\right )\right )}{225 x^2} \, dx=\frac {4\,{\mathrm {e}}^{-\frac {4\,x^2}{25}}}{9}-x\,\left (\frac {{\mathrm {e}}^{-\frac {4\,x^2}{25}}}{9}-2\right )-\frac {8}{9\,x}-x^3 \]
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