Integrand size = 48, antiderivative size = 23 \[ \int e^{-20 x} \left (-7812500-2812500 e^{5 x}+e^{20 x} (-21+2 x)+e^{15 x} (-15300+1500 x)+e^{10 x} (-351250+12500 x)\right ) \, dx=-2-x+\left (-1-\left (3+25 e^{-5 x}\right )^2+x\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(23)=46\).
Time = 0.17 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.57, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6874, 2225, 2207} \[ \int e^{-20 x} \left (-7812500-2812500 e^{5 x}+e^{20 x} (-21+2 x)+e^{15 x} (-15300+1500 x)+e^{10 x} (-351250+12500 x)\right ) \, dx=x^2-21 x+390625 e^{-20 x}+187500 e^{-15 x}-125 e^{-10 x}-60 e^{-5 x}+125 e^{-10 x} (281-10 x)+60 e^{-5 x} (51-5 x) \]
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Rule 2207
Rule 2225
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-21-7812500 e^{-20 x}-2812500 e^{-15 x}+2 x+300 e^{-5 x} (-51+5 x)+1250 e^{-10 x} (-281+10 x)\right ) \, dx \\ & = -21 x+x^2+300 \int e^{-5 x} (-51+5 x) \, dx+1250 \int e^{-10 x} (-281+10 x) \, dx-2812500 \int e^{-15 x} \, dx-7812500 \int e^{-20 x} \, dx \\ & = 390625 e^{-20 x}+187500 e^{-15 x}+125 e^{-10 x} (281-10 x)+60 e^{-5 x} (51-5 x)-21 x+x^2+300 \int e^{-5 x} \, dx+1250 \int e^{-10 x} \, dx \\ & = 390625 e^{-20 x}+187500 e^{-15 x}-125 e^{-10 x}-60 e^{-5 x}+125 e^{-10 x} (281-10 x)+60 e^{-5 x} (51-5 x)-21 x+x^2 \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.96 \[ \int e^{-20 x} \left (-7812500-2812500 e^{5 x}+e^{20 x} (-21+2 x)+e^{15 x} (-15300+1500 x)+e^{10 x} (-351250+12500 x)\right ) \, dx=390625 e^{-20 x}+187500 e^{-15 x}+300 e^{-5 x} (10-x)+1250 e^{-10 x} (28-x)-21 x+x^2 \]
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Time = 1.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74
method | result | size |
risch | \(x^{2}-21 x +\left (3000-300 x \right ) {\mathrm e}^{-5 x}+\left (35000-1250 x \right ) {\mathrm e}^{-10 x}+187500 \,{\mathrm e}^{-15 x}+390625 \,{\mathrm e}^{-20 x}\) | \(40\) |
derivativedivides | \(-300 \,{\mathrm e}^{-5 x} x -1250 \,{\mathrm e}^{-10 x} x +x^{2}+3000 \,{\mathrm e}^{-5 x}+35000 \,{\mathrm e}^{-10 x}+187500 \,{\mathrm e}^{-15 x}+390625 \,{\mathrm e}^{-20 x}-21 x\) | \(58\) |
default | \(-300 \,{\mathrm e}^{-5 x} x -1250 \,{\mathrm e}^{-10 x} x +x^{2}+3000 \,{\mathrm e}^{-5 x}+35000 \,{\mathrm e}^{-10 x}+187500 \,{\mathrm e}^{-15 x}+390625 \,{\mathrm e}^{-20 x}-21 x\) | \(58\) |
parts | \(-300 \,{\mathrm e}^{-5 x} x -1250 \,{\mathrm e}^{-10 x} x +x^{2}+3000 \,{\mathrm e}^{-5 x}+35000 \,{\mathrm e}^{-10 x}+187500 \,{\mathrm e}^{-15 x}+390625 \,{\mathrm e}^{-20 x}-21 x\) | \(58\) |
norman | \(\left (390625+x^{2} {\mathrm e}^{20 x}+35000 \,{\mathrm e}^{10 x}+3000 \,{\mathrm e}^{15 x}-1250 \,{\mathrm e}^{10 x} x -300 \,{\mathrm e}^{15 x} x -21 x \,{\mathrm e}^{20 x}+187500 \,{\mathrm e}^{5 x}\right ) {\mathrm e}^{-20 x}\) | \(69\) |
parallelrisch | \(\left (390625+x^{2} {\mathrm e}^{20 x}+35000 \,{\mathrm e}^{10 x}+3000 \,{\mathrm e}^{15 x}-1250 \,{\mathrm e}^{10 x} x -300 \,{\mathrm e}^{15 x} x -21 x \,{\mathrm e}^{20 x}+187500 \,{\mathrm e}^{5 x}\right ) {\mathrm e}^{-20 x}\) | \(69\) |
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Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87 \[ \int e^{-20 x} \left (-7812500-2812500 e^{5 x}+e^{20 x} (-21+2 x)+e^{15 x} (-15300+1500 x)+e^{10 x} (-351250+12500 x)\right ) \, dx={\left ({\left (x^{2} - 21 \, x\right )} e^{\left (20 \, x\right )} - 300 \, {\left (x - 10\right )} e^{\left (15 \, x\right )} - 1250 \, {\left (x - 28\right )} e^{\left (10 \, x\right )} + 187500 \, e^{\left (5 \, x\right )} + 390625\right )} e^{\left (-20 \, x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int e^{-20 x} \left (-7812500-2812500 e^{5 x}+e^{20 x} (-21+2 x)+e^{15 x} (-15300+1500 x)+e^{10 x} (-351250+12500 x)\right ) \, dx=x^{2} - 21 x + \left (3000 - 300 x\right ) e^{- 5 x} + \left (35000 - 1250 x\right ) e^{- 10 x} + 187500 e^{- 15 x} + 390625 e^{- 20 x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).
Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.30 \[ \int e^{-20 x} \left (-7812500-2812500 e^{5 x}+e^{20 x} (-21+2 x)+e^{15 x} (-15300+1500 x)+e^{10 x} (-351250+12500 x)\right ) \, dx=x^{2} - 60 \, {\left (5 \, x + 1\right )} e^{\left (-5 \, x\right )} - 125 \, {\left (10 \, x + 1\right )} e^{\left (-10 \, x\right )} - 21 \, x + 3060 \, e^{\left (-5 \, x\right )} + 35125 \, e^{\left (-10 \, x\right )} + 187500 \, e^{\left (-15 \, x\right )} + 390625 \, e^{\left (-20 \, x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (22) = 44\).
Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.96 \[ \int e^{-20 x} \left (-7812500-2812500 e^{5 x}+e^{20 x} (-21+2 x)+e^{15 x} (-15300+1500 x)+e^{10 x} (-351250+12500 x)\right ) \, dx=x^{2} - 300 \, x e^{\left (-5 \, x\right )} - 1250 \, x e^{\left (-10 \, x\right )} - 21 \, x + 3000 \, e^{\left (-5 \, x\right )} + 35000 \, e^{\left (-10 \, x\right )} + 187500 \, e^{\left (-15 \, x\right )} + 390625 \, e^{\left (-20 \, x\right )} \]
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Time = 13.48 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78 \[ \int e^{-20 x} \left (-7812500-2812500 e^{5 x}+e^{20 x} (-21+2 x)+e^{15 x} (-15300+1500 x)+e^{10 x} (-351250+12500 x)\right ) \, dx=187500\,{\mathrm {e}}^{-15\,x}-21\,x+390625\,{\mathrm {e}}^{-20\,x}-{\mathrm {e}}^{-5\,x}\,\left (300\,x-3000\right )-{\mathrm {e}}^{-10\,x}\,\left (1250\,x-35000\right )+x^2 \]
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