Integrand size = 17, antiderivative size = 57 \[ \int \sqrt {-a+b e^{c+d x}} \, dx=\frac {2 \sqrt {-a+b e^{c+d x}}}{d}-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {-a+b e^{c+d x}}}{\sqrt {a}}\right )}{d} \]
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Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2320, 52, 65, 211} \[ \int \sqrt {-a+b e^{c+d x}} \, dx=\frac {2 \sqrt {b e^{c+d x}-a}}{d}-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {b e^{c+d x}-a}}{\sqrt {a}}\right )}{d} \]
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Rule 52
Rule 65
Rule 211
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sqrt {-a+b x}}{x} \, dx,x,e^{c+d x}\right )}{d} \\ & = \frac {2 \sqrt {-a+b e^{c+d x}}}{d}-\frac {a \text {Subst}\left (\int \frac {1}{x \sqrt {-a+b x}} \, dx,x,e^{c+d x}\right )}{d} \\ & = \frac {2 \sqrt {-a+b e^{c+d x}}}{d}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b e^{c+d x}}\right )}{b d} \\ & = \frac {2 \sqrt {-a+b e^{c+d x}}}{d}-\frac {2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {-a+b e^{c+d x}}}{\sqrt {a}}\right )}{d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \sqrt {-a+b e^{c+d x}} \, dx=\frac {2 \left (\sqrt {-a+b e^{c+d x}}-\sqrt {a} \arctan \left (\frac {\sqrt {-a+b e^{c+d x}}}{\sqrt {a}}\right )\right )}{d} \]
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Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {2 \sqrt {-a +b \,{\mathrm e}^{d x +c}}-2 \sqrt {a}\, \arctan \left (\frac {\sqrt {-a +b \,{\mathrm e}^{d x +c}}}{\sqrt {a}}\right )}{d}\) | \(46\) |
default | \(\frac {2 \sqrt {-a +b \,{\mathrm e}^{d x +c}}-2 \sqrt {a}\, \arctan \left (\frac {\sqrt {-a +b \,{\mathrm e}^{d x +c}}}{\sqrt {a}}\right )}{d}\) | \(46\) |
risch | \(-\frac {2 \left (a -b \,{\mathrm e}^{d x +c}\right )}{d \sqrt {-a +b \,{\mathrm e}^{d x +c}}}-\frac {2 \arctan \left (\frac {\sqrt {-a +b \,{\mathrm e}^{d x +c}}}{\sqrt {a}}\right ) \sqrt {a}}{d}\) | \(59\) |
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Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.05 \[ \int \sqrt {-a+b e^{c+d x}} \, dx=\left [\frac {\sqrt {-a} \log \left ({\left (b e^{\left (d x + c\right )} - 2 \, \sqrt {b e^{\left (d x + c\right )} - a} \sqrt {-a} - 2 \, a\right )} e^{\left (-d x - c\right )}\right ) + 2 \, \sqrt {b e^{\left (d x + c\right )} - a}}{d}, -\frac {2 \, {\left (\sqrt {a} \arctan \left (\frac {\sqrt {b e^{\left (d x + c\right )} - a}}{\sqrt {a}}\right ) - \sqrt {b e^{\left (d x + c\right )} - a}\right )}}{d}\right ] \]
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Time = 0.83 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.32 \[ \int \sqrt {-a+b e^{c+d x}} \, dx=\begin {cases} \frac {\begin {cases} - 2 \sqrt {a} \operatorname {atan}{\left (\frac {\sqrt {- a + b e^{c} e^{d x}}}{\sqrt {a}} \right )} + 2 \sqrt {- a + b e^{c} e^{d x}} & \text {for}\: b e^{c} \neq 0 \\\sqrt {- a} \log {\left (e^{d x} \right )} & \text {otherwise} \end {cases}}{d} & \text {for}\: d \neq 0 \\x \sqrt {- a + b e^{c}} & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.82 \[ \int \sqrt {-a+b e^{c+d x}} \, dx=-\frac {2 \, \sqrt {a} \arctan \left (\frac {\sqrt {b e^{\left (d x + c\right )} - a}}{\sqrt {a}}\right )}{d} + \frac {2 \, \sqrt {b e^{\left (d x + c\right )} - a}}{d} \]
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Time = 0.31 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.79 \[ \int \sqrt {-a+b e^{c+d x}} \, dx=-\frac {2 \, {\left (\sqrt {a} \arctan \left (\frac {\sqrt {b e^{\left (d x + c\right )} - a}}{\sqrt {a}}\right ) - \sqrt {b e^{\left (d x + c\right )} - a}\right )}}{d} \]
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Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.82 \[ \int \sqrt {-a+b e^{c+d x}} \, dx=\frac {2\,\sqrt {b\,{\mathrm {e}}^{c+d\,x}-a}}{d}-\frac {2\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-a}}{\sqrt {a}}\right )}{d} \]
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