3.493 \(\int \frac {2^x}{\sqrt {a+4^{-x} b}} \, dx\)

Optimal. Leaf size=24 \[ \frac {2^x \sqrt {a+b 2^{-2 x}}}{a \log (2)} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2249, 191} \[ \frac {2^x \sqrt {a+b 2^{-2 x}}}{a \log (2)} \]

Antiderivative was successfully verified.

[In]

Int[2^x/Sqrt[a + b/4^x],x]

[Out]

(2^x*Sqrt[a + b/2^(2*x)])/(a*Log[2])

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 2249

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[(d*e*Log[F])/(g*h*Log[G])]}, Dist[Denominator[m]/(g*h*Log[G]), Subst[Int[x^(Denominator[m]
- 1)*(a + b*F^(c*e - (d*e*f)/g)*x^Numerator[m])^p, x], x, G^((h*(f + g*x))/Denominator[m])], x] /; LtQ[m, -1]
|| GtQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

Rubi steps

\begin {align*} \int \frac {2^x}{\sqrt {a+4^{-x} b}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b}{x^2}}} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac {2^x \sqrt {a+2^{-2 x} b}}{a \log (2)}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 35, normalized size = 1.46 \[ \frac {2^{-x} \left (a 2^{2 x}+b\right )}{a \log (2) \sqrt {a+b 2^{-2 x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[2^x/Sqrt[a + b/4^x],x]

[Out]

(2^(2*x)*a + b)/(2^x*a*Sqrt[a + b/2^(2*x)]*Log[2])

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fricas [A]  time = 0.44, size = 30, normalized size = 1.25 \[ \frac {2^{x} \sqrt {\frac {2^{2 \, x} a + b}{2^{2 \, x}}}}{a \log \relax (2)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2^x/(a+b/(4^x))^(1/2),x, algorithm="fricas")

[Out]

2^x*sqrt((2^(2*x)*a + b)/2^(2*x))/(a*log(2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2^{x}}{\sqrt {a + \frac {b}{4^{x}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2^x/(a+b/(4^x))^(1/2),x, algorithm="giac")

[Out]

integrate(2^x/sqrt(a + b/4^x), x)

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maple [A]  time = 0.05, size = 40, normalized size = 1.67 \[ \frac {\left (a 2^{2 x}+b \right ) 2^{-x}}{\sqrt {\left (a 2^{2 x}+b \right ) 2^{-2 x}}\, \ln \relax (2) a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2^x/(a+b/(4^x))^(1/2),x)

[Out]

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

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maxima [A]  time = 2.49, size = 19, normalized size = 0.79 \[ \frac {\sqrt {2^{2 \, x} a + b}}{a \log \relax (2)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2^x/(a+b/(4^x))^(1/2),x, algorithm="maxima")

[Out]

sqrt(2^(2*x)*a + b)/(a*log(2))

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mupad [B]  time = 3.51, size = 24, normalized size = 1.00 \[ \frac {2^x\,\sqrt {a+\frac {b}{2^{2\,x}}}}{a\,\ln \relax (2)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2^x/(a + b/4^x)^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2^{x}}{\sqrt {a + 4^{- x} b}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2**x/(a+b/(4**x))**(1/2),x)

[Out]

Integral(2**x/sqrt(a + 4**(-x)*b), x)

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