Integrand size = 34, antiderivative size = 16 \[ \int \frac {752+256 e^3}{2209+256 e^6+e^3 (1504-2560 x)-7520 x+6400 x^2} \, dx=-1+\frac {x}{\frac {47}{16}+e^3-5 x} \]
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Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.44, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {12, 2006, 27, 32} \[ \int \frac {752+256 e^3}{2209+256 e^6+e^3 (1504-2560 x)-7520 x+6400 x^2} \, dx=\frac {47+16 e^3}{5 \left (-80 x+16 e^3+47\right )} \]
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Rule 12
Rule 27
Rule 32
Rule 2006
Rubi steps \begin{align*} \text {integral}& = \left (16 \left (47+16 e^3\right )\right ) \int \frac {1}{2209+256 e^6+e^3 (1504-2560 x)-7520 x+6400 x^2} \, dx \\ & = \left (16 \left (47+16 e^3\right )\right ) \int \frac {1}{\left (47+16 e^3\right )^2-160 \left (47+16 e^3\right ) x+6400 x^2} \, dx \\ & = \left (16 \left (47+16 e^3\right )\right ) \int \frac {1}{\left (47+16 e^3-80 x\right )^2} \, dx \\ & = \frac {47+16 e^3}{5 \left (47+16 e^3-80 x\right )} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.44 \[ \int \frac {752+256 e^3}{2209+256 e^6+e^3 (1504-2560 x)-7520 x+6400 x^2} \, dx=\frac {47+16 e^3}{5 \left (47+16 e^3-80 x\right )} \]
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Time = 0.63 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19
method | result | size |
norman | \(\frac {\frac {47}{5}+\frac {16 \,{\mathrm e}^{3}}{5}}{16 \,{\mathrm e}^{3}-80 x +47}\) | \(19\) |
gosper | \(\frac {16 \,{\mathrm e}^{3}+47}{80 \,{\mathrm e}^{3}-400 x +235}\) | \(20\) |
parallelrisch | \(\frac {256 \,{\mathrm e}^{3}+752}{1280 \,{\mathrm e}^{3}-6400 x +3760}\) | \(20\) |
risch | \(\frac {{\mathrm e}^{3}}{\frac {235}{16}-25 x +5 \,{\mathrm e}^{3}}+\frac {47}{80 \left (\frac {47}{16}-5 x +{\mathrm e}^{3}\right )}\) | \(26\) |
meijerg | \(-\frac {47 x}{5 \left (-\frac {{\mathrm e}^{3}}{5}-\frac {47}{80}\right ) \left (16 \,{\mathrm e}^{3}+47\right ) \left (1-\frac {80 x}{16 \,{\mathrm e}^{3}+47}\right )}-\frac {16 \,{\mathrm e}^{3} x}{5 \left (-\frac {{\mathrm e}^{3}}{5}-\frac {47}{80}\right ) \left (16 \,{\mathrm e}^{3}+47\right ) \left (1-\frac {80 x}{16 \,{\mathrm e}^{3}+47}\right )}\) | \(72\) |
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Time = 0.35 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {752+256 e^3}{2209+256 e^6+e^3 (1504-2560 x)-7520 x+6400 x^2} \, dx=-\frac {16 \, e^{3} + 47}{5 \, {\left (80 \, x - 16 \, e^{3} - 47\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {752+256 e^3}{2209+256 e^6+e^3 (1504-2560 x)-7520 x+6400 x^2} \, dx=- \frac {752 + 256 e^{3}}{6400 x - 1280 e^{3} - 3760} \]
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Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {752+256 e^3}{2209+256 e^6+e^3 (1504-2560 x)-7520 x+6400 x^2} \, dx=-\frac {16 \, e^{3} + 47}{5 \, {\left (80 \, x - 16 \, e^{3} - 47\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {752+256 e^3}{2209+256 e^6+e^3 (1504-2560 x)-7520 x+6400 x^2} \, dx=-\frac {16 \, e^{3} + 47}{5 \, {\left (80 \, x - 16 \, e^{3} - 47\right )}} \]
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Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {752+256 e^3}{2209+256 e^6+e^3 (1504-2560 x)-7520 x+6400 x^2} \, dx=\frac {\frac {16\,{\mathrm {e}}^3}{5}+\frac {47}{5}}{16\,{\mathrm {e}}^3-80\,x+47} \]
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