\(\int \sqrt {e^{a+b x}} \, dx\) [94]

Optimal result

Integrand size = 11, antiderivative size = 16 \[ \int \sqrt {e^{a+b x}} \, dx=\frac {2 \sqrt {e^{a+b x}}}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2225} \[ \int \sqrt {e^{a+b x}} \, dx=\frac {2 \sqrt {e^{a+b x}}}{b} \]

[In]

Int[Sqrt[E^(a + b*x)],x]

[Out]

(2*Sqrt[E^(a + b*x)])/b

Rule 2225

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*} \text {integral}& = \frac {2 \sqrt {e^{a+b x}}}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {e^{a+b x}} \, dx=\frac {2 \sqrt {e^{a+b x}}}{b} \]

[In]

Integrate[Sqrt[E^(a + b*x)],x]

[Out]

(2*Sqrt[E^(a + b*x)])/b

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88

method	result	size
gosper	\(\frac {2 \sqrt {{\mathrm e}^{b x +a}}}{b}\)	\(14\)
derivativedivides	\(\frac {2 \sqrt {{\mathrm e}^{b x +a}}}{b}\)	\(14\)
default	\(\frac {2 \sqrt {{\mathrm e}^{b x +a}}}{b}\)	\(14\)
risch	\(\frac {2 \sqrt {{\mathrm e}^{b x +a}}}{b}\)	\(14\)
parallelrisch	\(\frac {2 \sqrt {{\mathrm e}^{b x +a}}}{b}\)	\(14\)
meijerg	\(-\frac {2 \sqrt {{\mathrm e}^{b x +a}}\, {\mathrm e}^{-\frac {a}{2}-\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}} \left (1-{\mathrm e}^{\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}}\right )}{b}\)	\(40\)

[In]

int(exp(b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \sqrt {e^{a+b x}} \, dx=\frac {2 \, e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{b} \]

[In]

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

[Out]

2*e^(1/2*b*x + 1/2*a)/b

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \sqrt {e^{a+b x}} \, dx=\begin {cases} \frac {2 \sqrt {e^{a + b x}}}{b} & \text {for}\: b \neq 0 \\x & \text {otherwise} \end {cases} \]

[In]

integrate(exp(b*x+a)**(1/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \sqrt {e^{a+b x}} \, dx=\frac {2 \, e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{b} \]

[In]

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

[Out]

2*e^(1/2*b*x + 1/2*a)/b

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \sqrt {e^{a+b x}} \, dx=\frac {2 \, e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{b} \]

[In]

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

[Out]

2*e^(1/2*b*x + 1/2*a)/b

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \sqrt {e^{a+b x}} \, dx=\frac {2\,\sqrt {{\mathrm {e}}^{a+b\,x}}}{b} \]

[In]

int(exp(a + b*x)^(1/2),x)

[Out]

(2*exp(a + b*x)^(1/2))/b