3.94 \(\int \sqrt{e^{a+b x}} \, dx\)

Optimal. Leaf size=16 \[ \frac{2 \sqrt{e^{a+b x}}}{b} \]

[Out]

(2*Sqrt[E^(a + b*x)])/b

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Rubi [A]  time = 0.0125971, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2 \sqrt{e^{a+b x}}}{b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[E^(a + b*x)],x]

[Out]

(2*Sqrt[E^(a + b*x)])/b

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Rubi in Sympy [A]  time = 1.19301, size = 12, normalized size = 0.75 \[ \frac{2 \sqrt{e^{a + b x}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(exp(a + b*x))/b

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Mathematica [A]  time = 0.00296304, size = 16, normalized size = 1. \[ \frac{2 \sqrt{e^{a+b x}}}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[E^(a + b*x)],x]

[Out]

(2*Sqrt[E^(a + b*x)])/b

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Maple [A]  time = 0.003, size = 14, normalized size = 0.9 \[ 2\,{\frac{\sqrt{{{\rm e}^{bx+a}}}}{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(b*x+a)^(1/2),x)

[Out]

2*exp(b*x+a)^(1/2)/b

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Maxima [A]  time = 0.787282, size = 19, normalized size = 1.19 \[ \frac{2 \, e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(1/2*b*x + 1/2*a),x, algorithm="maxima")

[Out]

2*e^(1/2*b*x + 1/2*a)/b

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Fricas [A]  time = 0.248416, size = 19, normalized size = 1.19 \[ \frac{2 \, e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(1/2*b*x + 1/2*a),x, algorithm="fricas")

[Out]

2*e^(1/2*b*x + 1/2*a)/b

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Sympy [A]  time = 0.1371, size = 14, normalized size = 0.88 \[ \begin{cases} \frac{2 \sqrt{e^{a + b x}}}{b} & \text{for}\: b \neq 0 \\x & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(b*x+a)**(1/2),x)

[Out]

Piecewise((2*sqrt(exp(a + b*x))/b, Ne(b, 0)), (x, True))

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GIAC/XCAS [A]  time = 0.237854, size = 19, normalized size = 1.19 \[ \frac{2 \, e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(1/2*b*x + 1/2*a),x, algorithm="giac")

[Out]

2*e^(1/2*b*x + 1/2*a)/b