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 = 0.0065371, 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, Rules used = {2194} \[ \frac{2 \sqrt{e^{a+b x}}}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[E^(a + b*x)],x]

[Out]

(2*Sqrt[E^(a + b*x)])/b

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int \sqrt{e^{a+b x}} \, dx &=\frac{2 \sqrt{e^{a+b x}}}{b}\\ \end{align*}

Mathematica [A]  time = 0.0050288, 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 = 0.001, size = 14, normalized size = 0.9 \begin{align*} 2\,{\frac{\sqrt{{{\rm e}^{bx+a}}}}{b}} \end{align*}

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 = 1.04315, size = 19, normalized size = 1.19 \begin{align*} \frac{2 \, e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

2*e^(1/2*b*x + 1/2*a)/b

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Fricas [A]  time = 1.46732, size = 34, normalized size = 2.12 \begin{align*} \frac{2 \, e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2*e^(1/2*b*x + 1/2*a)/b

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Sympy [A]  time = 0.082823, size = 14, normalized size = 0.88 \begin{align*} \begin{cases} \frac{2 \sqrt{e^{a + b x}}}{b} & \text{for}\: b \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}

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 [A]  time = 1.24524, size = 19, normalized size = 1.19 \begin{align*} \frac{2 \, e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2*e^(1/2*b*x + 1/2*a)/b