Integrand size = 11, antiderivative size = 40 \[ \int e^{\sqrt {5+3 x}} \, dx=-\frac {2}{3} e^{\sqrt {5+3 x}}+\frac {2}{3} e^{\sqrt {5+3 x}} \sqrt {5+3 x} \]
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Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2238, 2207, 2225} \[ \int e^{\sqrt {5+3 x}} \, dx=\frac {2}{3} e^{\sqrt {3 x+5}} \sqrt {3 x+5}-\frac {2}{3} e^{\sqrt {3 x+5}} \]
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Rule 2207
Rule 2225
Rule 2238
Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} \text {Subst}\left (\int e^x x \, dx,x,\sqrt {5+3 x}\right ) \\ & = \frac {2}{3} e^{\sqrt {5+3 x}} \sqrt {5+3 x}-\frac {2}{3} \text {Subst}\left (\int e^x \, dx,x,\sqrt {5+3 x}\right ) \\ & = -\frac {2}{3} e^{\sqrt {5+3 x}}+\frac {2}{3} e^{\sqrt {5+3 x}} \sqrt {5+3 x} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.65 \[ \int e^{\sqrt {5+3 x}} \, dx=\frac {2}{3} e^{\sqrt {5+3 x}} \left (-1+\sqrt {5+3 x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(-\frac {2 \,{\mathrm e}^{\sqrt {5+3 x}}}{3}+\frac {2 \,{\mathrm e}^{\sqrt {5+3 x}} \sqrt {5+3 x}}{3}\) | \(29\) |
default | \(-\frac {2 \,{\mathrm e}^{\sqrt {5+3 x}}}{3}+\frac {2 \,{\mathrm e}^{\sqrt {5+3 x}} \sqrt {5+3 x}}{3}\) | \(29\) |
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Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.48 \[ \int e^{\sqrt {5+3 x}} \, dx=\frac {2}{3} \, {\left (\sqrt {3 \, x + 5} - 1\right )} e^{\left (\sqrt {3 \, x + 5}\right )} \]
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Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int e^{\sqrt {5+3 x}} \, dx=\frac {2 \sqrt {3 x + 5} e^{\sqrt {3 x + 5}}}{3} - \frac {2 e^{\sqrt {3 x + 5}}}{3} \]
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Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.48 \[ \int e^{\sqrt {5+3 x}} \, dx=\frac {2}{3} \, {\left (\sqrt {3 \, x + 5} - 1\right )} e^{\left (\sqrt {3 \, x + 5}\right )} \]
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Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.48 \[ \int e^{\sqrt {5+3 x}} \, dx=\frac {2}{3} \, {\left (\sqrt {3 \, x + 5} - 1\right )} e^{\left (\sqrt {3 \, x + 5}\right )} \]
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Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.48 \[ \int e^{\sqrt {5+3 x}} \, dx=\frac {2\,{\mathrm {e}}^{\sqrt {3\,x+5}}\,\left (\sqrt {3\,x+5}-1\right )}{3} \]
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