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*Sqrt[a + b/2^(2*x)])/(a*Log[2])

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Rubi [A]  time = 0.0479708, antiderivative size = 24, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.0312534, 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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Maple [A]  time = 0.029, size = 40, normalized size = 1.7 \begin{align*}{\frac{a \left ({2}^{x} \right ) ^{2}+b}{a{2}^{x}\ln \left ( 2 \right ) }{\frac{1}{\sqrt{{\frac{a \left ({2}^{x} \right ) ^{2}+b}{ \left ({2}^{x} \right ) ^{2}}}}}}} \end{align*}

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 = 1.65125, size = 26, normalized size = 1.08 \begin{align*} \frac{\sqrt{2^{2 \, x} a + b}}{a \log \left (2\right )} \end{align*}

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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Fricas [A]  time = 1.52985, size = 62, normalized size = 2.58 \begin{align*} \frac{2^{x} \sqrt{\frac{2^{2 \, x} a + b}{2^{2 \, x}}}}{a \log \left (2\right )} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2^{x}}{\sqrt{a + 4^{- x} b}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2^{x}}{\sqrt{a + \frac{b}{4^{x}}}}\,{d x} \end{align*}

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)