Integrand size = 23, antiderivative size = 19 \[ \int \frac {-1+4 e^6 x+4 e^{6+x} x}{e^5 x} \, dx=-5+4 e \left (e^x+x\right )-\frac {\log (4 x)}{e^5} \]
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Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {12, 14, 2225, 45} \[ \int \frac {-1+4 e^6 x+4 e^{6+x} x}{e^5 x} \, dx=4 e x+4 e^{x+1}-\frac {\log (x)}{e^5} \]
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Rule 12
Rule 14
Rule 45
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-1+4 e^6 x+4 e^{6+x} x}{x} \, dx}{e^5} \\ & = \frac {\int \left (4 e^{6+x}+\frac {-1+4 e^6 x}{x}\right ) \, dx}{e^5} \\ & = \frac {\int \frac {-1+4 e^6 x}{x} \, dx}{e^5}+\frac {4 \int e^{6+x} \, dx}{e^5} \\ & = 4 e^{1+x}+\frac {\int \left (4 e^6-\frac {1}{x}\right ) \, dx}{e^5} \\ & = 4 e^{1+x}+4 e x-\frac {\log (x)}{e^5} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-1+4 e^6 x+4 e^{6+x} x}{e^5 x} \, dx=4 e^{1+x}+4 e x-\frac {\log (x)}{e^5} \]
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Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00
method | result | size |
risch | \(4 x \,{\mathrm e}-{\mathrm e}^{-5} \ln \left (x \right )+4 \,{\mathrm e}^{1+x}\) | \(19\) |
norman | \(4 x \,{\mathrm e}+4 \,{\mathrm e} \,{\mathrm e}^{x}-{\mathrm e}^{-5} \ln \left (x \right )\) | \(21\) |
parts | \({\mathrm e}^{-5} \left (4 x \,{\mathrm e}^{6}-\ln \left (x \right )\right )+4 \,{\mathrm e} \,{\mathrm e}^{x}\) | \(23\) |
default | \({\mathrm e}^{-5} \left (-\ln \left (x \right )+4 x \,{\mathrm e} \,{\mathrm e}^{5}+4 \,{\mathrm e} \,{\mathrm e}^{5} {\mathrm e}^{x}\right )\) | \(26\) |
parallelrisch | \({\mathrm e}^{-5} \left (-\ln \left (x \right )+4 x \,{\mathrm e} \,{\mathrm e}^{5}+4 \,{\mathrm e} \,{\mathrm e}^{5} {\mathrm e}^{x}\right )\) | \(26\) |
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-1+4 e^6 x+4 e^{6+x} x}{e^5 x} \, dx={\left (4 \, x e^{6} + 4 \, e^{\left (x + 6\right )} - \log \left (x\right )\right )} e^{\left (-5\right )} \]
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Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {-1+4 e^6 x+4 e^{6+x} x}{e^5 x} \, dx=\frac {4 x e^{6} - \log {\left (x \right )}}{e^{5}} + 4 e e^{x} \]
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Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-1+4 e^6 x+4 e^{6+x} x}{e^5 x} \, dx={\left (4 \, x e^{6} + 4 \, e^{\left (x + 6\right )} - \log \left (x\right )\right )} e^{\left (-5\right )} \]
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Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-1+4 e^6 x+4 e^{6+x} x}{e^5 x} \, dx={\left (4 \, x e^{6} + 4 \, e^{\left (x + 6\right )} - \log \left (x\right )\right )} e^{\left (-5\right )} \]
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Time = 10.65 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {-1+4 e^6 x+4 e^{6+x} x}{e^5 x} \, dx=4\,x\,\mathrm {e}+4\,\mathrm {e}\,{\mathrm {e}}^x-{\mathrm {e}}^{-5}\,\ln \left (x\right ) \]
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