Optimal. Leaf size=57 \[ \frac {2 \sqrt {-a+b e^{c+d x}}}{d}-\frac {2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {-a+b e^{c+d x}}}{\sqrt {a}}\right )}{d} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2320, 52, 65,
211} \begin {gather*} \frac {2 \sqrt {b e^{c+d x}-a}}{d}-\frac {2 \sqrt {a} \text {ArcTan}\left (\frac {\sqrt {b e^{c+d x}-a}}{\sqrt {a}}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 211
Rule 2320
Rubi steps
\begin {align*} \int \sqrt {-a+b e^{c+d x}} \, dx &=\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*}
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Mathematica [A]
time = 0.04, size = 54, normalized size = 0.95 \begin {gather*} \frac {2 \left (\sqrt {-a+b e^{c+d x}}-\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {-a+b e^{c+d x}}}{\sqrt {a}}\right )\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 46, normalized size = 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 | \(-\frac {2 \left (a -b \,{\mathrm e}^{d x +c}\right )}{d \sqrt {-a +b \,{\mathrm e}^{d x +c}}}-\frac {2 \arctan \left (\frac {\sqrt {-a +b \,{\mathrm e}^{d x +c}}}{\sqrt {a}}\right ) \sqrt {a}}{d}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 47, normalized size = 0.82 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 117, normalized size = 2.05 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- a + b e^{c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.29, size = 45, normalized size = 0.79 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.60, size = 47, normalized size = 0.82 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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