3.684 \(\int \sqrt{-a+b e^{c+d x}} \, dx\)

Optimal. Leaf size=57 \[ \frac{2 \sqrt{b e^{c+d x}-a}}{d}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b e^{c+d x}-a}}{\sqrt{a}}\right )}{d} \]

[Out]

(2*Sqrt[-a + b*E^(c + d*x)])/d - (2*Sqrt[a]*ArcTan[Sqrt[-a + b*E^(c + d*x)]/Sqrt
[a]])/d

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Rubi [A]  time = 0.0758488, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{2 \sqrt{b e^{c+d x}-a}}{d}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b e^{c+d x}-a}}{\sqrt{a}}\right )}{d} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-a + b*E^(c + d*x)],x]

[Out]

(2*Sqrt[-a + b*E^(c + d*x)])/d - (2*Sqrt[a]*ArcTan[Sqrt[-a + b*E^(c + d*x)]/Sqrt
[a]])/d

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Rubi in Sympy [A]  time = 9.89564, size = 75, normalized size = 1.32 \[ - \frac{2 \sqrt{a} e^{- c - d x} e^{c + d x} \operatorname{atan}{\left (\frac{\sqrt{- a + b e^{c + d x}}}{\sqrt{a}} \right )}}{d} + \frac{2 \sqrt{- a + b e^{c + d x}} e^{- c - d x} e^{c + d x}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-a+b*exp(d*x+c))**(1/2),x)

[Out]

-2*sqrt(a)*exp(-c - d*x)*exp(c + d*x)*atan(sqrt(-a + b*exp(c + d*x))/sqrt(a))/d
+ 2*sqrt(-a + b*exp(c + d*x))*exp(-c - d*x)*exp(c + d*x)/d

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Mathematica [A]  time = 0.0385192, size = 54, normalized size = 0.95 \[ \frac{2 \left (\sqrt{b e^{c+d x}-a}-\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b e^{c+d x}-a}}{\sqrt{a}}\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-a + b*E^(c + d*x)],x]

[Out]

(2*(Sqrt[-a + b*E^(c + d*x)] - Sqrt[a]*ArcTan[Sqrt[-a + b*E^(c + d*x)]/Sqrt[a]])
)/d

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Maple [A]  time = 0.006, size = 48, normalized size = 0.8 \[ -2\,{\frac{\sqrt{a}}{d}\arctan \left ({\frac{\sqrt{-a+b{{\rm e}^{dx+c}}}}{\sqrt{a}}} \right ) }+2\,{\frac{\sqrt{-a+b{{\rm e}^{dx+c}}}}{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-a+b*exp(d*x+c))^(1/2),x)

[Out]

-2*arctan((-a+b*exp(d*x+c))^(1/2)/a^(1/2))*a^(1/2)/d+2*(-a+b*exp(d*x+c))^(1/2)/d

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*e^(d*x + c) - a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.26014, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{-a} \log \left ({\left (b e^{\left (d x + c\right )} - 2 \, \sqrt{b e^{\left (d x + c\right )} - a} \sqrt{-a} - 2 \, a\right )} e^{\left (-d x - c\right )}\right ) + 2 \, \sqrt{b e^{\left (d x + c\right )} - a}}{d}, -\frac{2 \,{\left (\sqrt{a} \arctan \left (\frac{\sqrt{b e^{\left (d x + c\right )} - a}}{\sqrt{a}}\right ) - \sqrt{b e^{\left (d x + c\right )} - a}\right )}}{d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*e^(d*x + c) - a),x, algorithm="fricas")

[Out]

[(sqrt(-a)*log((b*e^(d*x + c) - 2*sqrt(b*e^(d*x + c) - a)*sqrt(-a) - 2*a)*e^(-d*
x - c)) + 2*sqrt(b*e^(d*x + c) - a))/d, -2*(sqrt(a)*arctan(sqrt(b*e^(d*x + c) -
a)/sqrt(a)) - sqrt(b*e^(d*x + c) - a))/d]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- a + b e^{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-a+b*exp(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(-a + b*exp(c + d*x)), x)

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GIAC/XCAS [A]  time = 0.317329, size = 61, normalized size = 1.07 \[ -\frac{2 \,{\left (\sqrt{a} \arctan \left (\frac{\sqrt{b e^{\left (d x + c\right )} - a}}{\sqrt{a}}\right ) - \sqrt{b e^{\left (d x + c\right )} - a}\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*e^(d*x + c) - a),x, algorithm="giac")

[Out]

-2*(sqrt(a)*arctan(sqrt(b*e^(d*x + c) - a)/sqrt(a)) - sqrt(b*e^(d*x + c) - a))/d