Integrand size = 38, antiderivative size = 24 \[ \int \frac {2+e^x \left (-25 e^4-2 x\right )+50 e^4 x+4 x^2}{25 e^4+2 x} \, dx=x^2+\log \left (\frac {1}{4} e^{-e^x} \left (25+\frac {2 x}{e^4}\right )\right ) \]
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Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {6874, 2225, 712} \[ \int \frac {2+e^x \left (-25 e^4-2 x\right )+50 e^4 x+4 x^2}{25 e^4+2 x} \, dx=x^2-e^x+\log \left (2 x+25 e^4\right ) \]
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Rule 712
Rule 2225
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-e^x+\frac {2 \left (1+25 e^4 x+2 x^2\right )}{25 e^4+2 x}\right ) \, dx \\ & = 2 \int \frac {1+25 e^4 x+2 x^2}{25 e^4+2 x} \, dx-\int e^x \, dx \\ & = -e^x+2 \int \left (x+\frac {1}{25 e^4+2 x}\right ) \, dx \\ & = -e^x+x^2+\log \left (25 e^4+2 x\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {2+e^x \left (-25 e^4-2 x\right )+50 e^4 x+4 x^2}{25 e^4+2 x} \, dx=-e^x+x^2+\log \left (25 e^4+2 x\right ) \]
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Time = 0.35 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67
method | result | size |
parallelrisch | \(x^{2}-{\mathrm e}^{x}+\ln \left (x +\frac {25 \,{\mathrm e}^{4}}{2}\right )\) | \(16\) |
norman | \(-{\mathrm e}^{x}+x^{2}+\ln \left (25 \,{\mathrm e}^{4}+2 x \right )\) | \(18\) |
risch | \(-{\mathrm e}^{x}+x^{2}+\ln \left (25 \,{\mathrm e}^{4}+2 x \right )\) | \(18\) |
parts | \(-{\mathrm e}^{x}+x^{2}+\ln \left (25 \,{\mathrm e}^{4}+2 x \right )\) | \(18\) |
default | \(-{\mathrm e}^{x}+x^{2}+\ln \left (25 \,{\mathrm e}^{4}+2 x \right )\) | \(46\) |
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Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {2+e^x \left (-25 e^4-2 x\right )+50 e^4 x+4 x^2}{25 e^4+2 x} \, dx=x^{2} - e^{x} + \log \left (2 \, x + 25 \, e^{4}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {2+e^x \left (-25 e^4-2 x\right )+50 e^4 x+4 x^2}{25 e^4+2 x} \, dx=x^{2} - e^{x} + \log {\left (2 x + 25 e^{4} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (19) = 38\).
Time = 0.23 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.33 \[ \int \frac {2+e^x \left (-25 e^4-2 x\right )+50 e^4 x+4 x^2}{25 e^4+2 x} \, dx=x^{2} - \frac {25}{2} \, {\left (25 \, e^{4} \log \left (2 \, x + 25 \, e^{4}\right ) - 2 \, x\right )} e^{4} - 25 \, x e^{4} + \frac {625}{2} \, e^{8} \log \left (2 \, x + 25 \, e^{4}\right ) - e^{x} + \log \left (2 \, x + 25 \, e^{4}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {2+e^x \left (-25 e^4-2 x\right )+50 e^4 x+4 x^2}{25 e^4+2 x} \, dx=x^{2} - e^{x} + \log \left (2 \, x + 25 \, e^{4}\right ) \]
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Time = 12.65 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {2+e^x \left (-25 e^4-2 x\right )+50 e^4 x+4 x^2}{25 e^4+2 x} \, dx=\ln \left (x+\frac {25\,{\mathrm {e}}^4}{2}\right )-{\mathrm {e}}^x+x^2 \]
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