Integrand size = 27, antiderivative size = 23 \[ \int \frac {800+e^5 (20-200 x)+1000 x+e^{10} (-10+12 x)}{e^{10}} \, dx=x^2+5 \left (1-x+\frac {10 \left (\frac {4}{5}+x\right )}{e^5}\right )^2 \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {12} \[ \int \frac {800+e^5 (20-200 x)+1000 x+e^{10} (-10+12 x)}{e^{10}} \, dx=\frac {500 x^2}{e^{10}}-\frac {(1-10 x)^2}{e^5}+\frac {1}{6} (5-6 x)^2+\frac {800 x}{e^{10}} \]
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Rule 12
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (800+e^5 (20-200 x)+1000 x+e^{10} (-10+12 x)\right ) \, dx}{e^{10}} \\ & = -\frac {(1-10 x)^2}{e^5}+\frac {1}{6} (5-6 x)^2+\frac {800 x}{e^{10}}+\frac {500 x^2}{e^{10}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {800+e^5 (20-200 x)+1000 x+e^{10} (-10+12 x)}{e^{10}} \, dx=2 \left (-5 x+\frac {400 x}{e^{10}}+\frac {10 x}{e^5}+3 x^2+\frac {250 x^2}{e^{10}}-\frac {50 x^2}{e^5}\right ) \]
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Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48
method | result | size |
risch | \(6 x^{2}-10 x -100 x^{2} {\mathrm e}^{-5}+20 x \,{\mathrm e}^{-5}+500 \,{\mathrm e}^{-10} x^{2}+800 \,{\mathrm e}^{-10} x\) | \(34\) |
gosper | \(2 x \left (3 x \,{\mathrm e}^{10}-5 \,{\mathrm e}^{10}-50 x \,{\mathrm e}^{5}+10 \,{\mathrm e}^{5}+250 x +400\right ) {\mathrm e}^{-10}\) | \(35\) |
default | \({\mathrm e}^{-10} \left (6 x^{2} {\mathrm e}^{10}-10 x \,{\mathrm e}^{10}-100 x^{2} {\mathrm e}^{5}+20 x \,{\mathrm e}^{5}+500 x^{2}+800 x \right )\) | \(43\) |
parallelrisch | \({\mathrm e}^{-10} \left (6 x^{2} {\mathrm e}^{10}-10 x \,{\mathrm e}^{10}-100 x^{2} {\mathrm e}^{5}+20 x \,{\mathrm e}^{5}+500 x^{2}+800 x \right )\) | \(43\) |
norman | \(\left (-10 \left ({\mathrm e}^{10}-2 \,{\mathrm e}^{5}-80\right ) {\mathrm e}^{-5} x +2 \left (3 \,{\mathrm e}^{10}-50 \,{\mathrm e}^{5}+250\right ) {\mathrm e}^{-5} x^{2}\right ) {\mathrm e}^{-5}\) | \(45\) |
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Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65 \[ \int \frac {800+e^5 (20-200 x)+1000 x+e^{10} (-10+12 x)}{e^{10}} \, dx=2 \, {\left (250 \, x^{2} + {\left (3 \, x^{2} - 5 \, x\right )} e^{10} - 10 \, {\left (5 \, x^{2} - x\right )} e^{5} + 400 \, x\right )} e^{\left (-10\right )} \]
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Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {800+e^5 (20-200 x)+1000 x+e^{10} (-10+12 x)}{e^{10}} \, dx=\frac {x^{2} \left (- 100 e^{5} + 500 + 6 e^{10}\right )}{e^{10}} + \frac {x \left (- 10 e^{10} + 800 + 20 e^{5}\right )}{e^{10}} \]
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Time = 0.18 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65 \[ \int \frac {800+e^5 (20-200 x)+1000 x+e^{10} (-10+12 x)}{e^{10}} \, dx=2 \, {\left (250 \, x^{2} + {\left (3 \, x^{2} - 5 \, x\right )} e^{10} - 10 \, {\left (5 \, x^{2} - x\right )} e^{5} + 400 \, x\right )} e^{\left (-10\right )} \]
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Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65 \[ \int \frac {800+e^5 (20-200 x)+1000 x+e^{10} (-10+12 x)}{e^{10}} \, dx=2 \, {\left (250 \, x^{2} + {\left (3 \, x^{2} - 5 \, x\right )} e^{10} - 10 \, {\left (5 \, x^{2} - x\right )} e^{5} + 400 \, x\right )} e^{\left (-10\right )} \]
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Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {800+e^5 (20-200 x)+1000 x+e^{10} (-10+12 x)}{e^{10}} \, dx=\frac {{\mathrm {e}}^{-10}\,\left (12\,{\mathrm {e}}^{10}-200\,{\mathrm {e}}^5+1000\right )\,x^2}{2}+{\mathrm {e}}^{-10}\,\left (20\,{\mathrm {e}}^5-10\,{\mathrm {e}}^{10}+800\right )\,x \]
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