Optimal. Leaf size=57 \[ \frac {2 \sqrt {b e^{c+d x}-a}}{d}-\frac {2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b e^{c+d x}-a}}{\sqrt {a}}\right )}{d} \]
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Rubi [A] time = 0.04, 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 = {2282, 50, 63, 205} \[ \frac {2 \sqrt {b e^{c+d x}-a}}{d}-\frac {2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b e^{c+d x}-a}}{\sqrt {a}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 205
Rule 2282
Rubi steps
\begin {align*} \int \sqrt {-a+b e^{c+d x}} \, dx &=\frac {\operatorname {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 \operatorname {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) \operatorname {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 = 0.02, size = 55, normalized size = 0.96 \[ \frac {2 \sqrt {b e^{c+d x}-a}-2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b e^{c+d x}-a}}{\sqrt {a}}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 117, normalized size = 2.05 \[ \left [\frac {\sqrt {-a} \log \left ({\left (b e^{\left (d x + c\right )} - 2 \, \sqrt {b e^{\left (d x + c\right )} - a} \sqrt {-a} - 2 \, a\right )} e^{\left (-d x - c\right )}\right ) + 2 \, \sqrt {b e^{\left (d x + c\right )} - a}}{d}, -\frac {2 \, {\left (\sqrt {a} \arctan \left (\frac {\sqrt {b e^{\left (d x + c\right )} - a}}{\sqrt {a}}\right ) - \sqrt {b e^{\left (d x + c\right )} - a}\right )}}{d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 45, normalized size = 0.79 \[ -\frac {2 \, {\left (\sqrt {a} \arctan \left (\frac {\sqrt {b e^{\left (d x + c\right )} - a}}{\sqrt {a}}\right ) - \sqrt {b e^{\left (d x + c\right )} - a}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 48, normalized size = 0.84 \[ -\frac {2 \sqrt {a}\, \arctan \left (\frac {\sqrt {b \,{\mathrm e}^{d x +c}-a}}{\sqrt {a}}\right )}{d}+\frac {2 \sqrt {b \,{\mathrm e}^{d x +c}-a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.19, size = 47, normalized size = 0.82 \[ -\frac {2 \, \sqrt {a} \arctan \left (\frac {\sqrt {b e^{\left (d x + c\right )} - a}}{\sqrt {a}}\right )}{d} + \frac {2 \, \sqrt {b e^{\left (d x + c\right )} - a}}{d} \]
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 \[ \frac {2\,\sqrt {b\,{\mathrm {e}}^{c+d\,x}-a}}{d}-\frac {2\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-a}}{\sqrt {a}}\right )}{d} \]
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 \[ \int \sqrt {- a + b e^{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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