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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \[ \frac {2 \sqrt {e^{a+b x}}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[E^(a + b*x)],x]

[Out]

(2*Sqrt[E^(a + b*x)])/b

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fricas [A]  time = 0.42, size = 14, normalized size = 0.88 \[ \frac {2 \, e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^(1/2*b*x + 1/2*a)/b

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giac [A]  time = 0.32, size = 14, normalized size = 0.88 \[ \frac {2 \, e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^(1/2*b*x + 1/2*a)/b

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maple [A]  time = 0.00, size = 14, normalized size = 0.88 \[ \frac {2 \sqrt {{\mathrm e}^{b x +a}}}{b} \]

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 = 0.87, size = 14, normalized size = 0.88 \[ \frac {2 \, e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^(1/2*b*x + 1/2*a)/b

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mupad [B]  time = 3.39, size = 13, normalized size = 0.81 \[ \frac {2\,\sqrt {{\mathrm {e}}^{a+b\,x}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a + b*x)^(1/2),x)

[Out]

(2*exp(a + b*x)^(1/2))/b

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sympy [A]  time = 0.09, 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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