Integrand size = 23, antiderivative size = 34 \[ \int \frac {1}{375} e^{-x} \left (-25+e^x (50-2 x)+25 x\right ) \, dx=\frac {1}{5} \left (2+e^2+x+\frac {1}{3} \left (4-x-e^{-x} x-\frac {x^2}{25}\right )\right ) \]
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Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {12, 6874, 2225, 2207} \[ \int \frac {1}{375} e^{-x} \left (-25+e^x (50-2 x)+25 x\right ) \, dx=-\frac {1}{375} (25-x)^2-\frac {e^{-x} x}{15} \]
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Rule 12
Rule 2207
Rule 2225
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{375} \int e^{-x} \left (-25+e^x (50-2 x)+25 x\right ) \, dx \\ & = \frac {1}{375} \int \left (-25 e^{-x}-2 (-25+x)+25 e^{-x} x\right ) \, dx \\ & = -\frac {1}{375} (25-x)^2-\frac {1}{15} \int e^{-x} \, dx+\frac {1}{15} \int e^{-x} x \, dx \\ & = \frac {e^{-x}}{15}-\frac {1}{375} (25-x)^2-\frac {e^{-x} x}{15}+\frac {1}{15} \int e^{-x} \, dx \\ & = -\frac {1}{375} (25-x)^2-\frac {e^{-x} x}{15} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.68 \[ \int \frac {1}{375} e^{-x} \left (-25+e^x (50-2 x)+25 x\right ) \, dx=\frac {2 x}{15}-\frac {e^{-x} x}{15}-\frac {x^2}{375} \]
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Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.50
method | result | size |
default | \(-\frac {x^{2}}{375}+\frac {2 x}{15}-\frac {x \,{\mathrm e}^{-x}}{15}\) | \(17\) |
risch | \(-\frac {x^{2}}{375}+\frac {2 x}{15}-\frac {x \,{\mathrm e}^{-x}}{15}\) | \(17\) |
parts | \(-\frac {x^{2}}{375}+\frac {2 x}{15}-\frac {x \,{\mathrm e}^{-x}}{15}\) | \(17\) |
norman | \(\left (-\frac {x}{15}+\frac {2 \,{\mathrm e}^{x} x}{15}-\frac {{\mathrm e}^{x} x^{2}}{375}\right ) {\mathrm e}^{-x}\) | \(22\) |
parallelrisch | \(\frac {\left (-{\mathrm e}^{x} x^{2}+50 \,{\mathrm e}^{x} x -25 x \right ) {\mathrm e}^{-x}}{375}\) | \(23\) |
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Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.59 \[ \int \frac {1}{375} e^{-x} \left (-25+e^x (50-2 x)+25 x\right ) \, dx=-\frac {1}{375} \, {\left ({\left (x^{2} - 50 \, x\right )} e^{x} + 25 \, x\right )} e^{\left (-x\right )} \]
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Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.44 \[ \int \frac {1}{375} e^{-x} \left (-25+e^x (50-2 x)+25 x\right ) \, dx=- \frac {x^{2}}{375} + \frac {2 x}{15} - \frac {x e^{- x}}{15} \]
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Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71 \[ \int \frac {1}{375} e^{-x} \left (-25+e^x (50-2 x)+25 x\right ) \, dx=-\frac {1}{375} \, x^{2} - \frac {1}{15} \, {\left (x + 1\right )} e^{\left (-x\right )} + \frac {2}{15} \, x + \frac {1}{15} \, e^{\left (-x\right )} \]
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Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.47 \[ \int \frac {1}{375} e^{-x} \left (-25+e^x (50-2 x)+25 x\right ) \, dx=-\frac {1}{375} \, x^{2} - \frac {1}{15} \, x e^{\left (-x\right )} + \frac {2}{15} \, x \]
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Time = 12.64 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.35 \[ \int \frac {1}{375} e^{-x} \left (-25+e^x (50-2 x)+25 x\right ) \, dx=-\frac {x\,\left (x+25\,{\mathrm {e}}^{-x}-50\right )}{375} \]
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