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

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 = 0.0743935, 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 \[ \frac{2^x \sqrt{a+b 2^{-2 x}}}{a \log (2)} \]

Antiderivative was successfully verified.

[In]  Int[2^x/Sqrt[a + b/2^(2*x)],x]

[Out]

(2^x*Sqrt[a + b/2^(2*x)])/(a*Log[2])

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Rubi in Sympy [A]  time = 6.14553, size = 19, normalized size = 0.79 \[ \frac{2^{x} \sqrt{a + 2^{- 2 x} b}}{a \log{\left (2 \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(2**x/(a+b/(2**(2*x)))**(1/2),x)

[Out]

2**x*sqrt(a + 2**(-2*x)*b)/(a*log(2))

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Mathematica [A]  time = 0.00811669, 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/2^(2*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 = 0.03, size = 40, normalized size = 1.7 \[{\frac{a \left ({2}^{x} \right ) ^{2}+b}{a{2}^{x}\ln \left ( 2 \right ) }{\frac{1}{\sqrt{{\frac{a \left ({2}^{x} \right ) ^{2}+b}{ \left ({2}^{x} \right ) ^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(2^x/(a+b/(2^(2*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 = 0.789313, size = 32, normalized size = 1.33 \[ \frac{2^{x} \sqrt{a + \frac{b}{2^{2 \, x}}}}{a \log \left (2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2^x/sqrt(a + b/2^(2*x)),x, algorithm="maxima")

[Out]

2^x*sqrt(a + b/2^(2*x))/(a*log(2))

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Fricas [A]  time = 0.266132, size = 41, normalized size = 1.71 \[ \frac{2^{x} \sqrt{\frac{2^{2 \, x} a + b}{2^{2 \, x}}}}{a \log \left (2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2^x/sqrt(a + b/2^(2*x)),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. \[ \int \frac{2^{x}}{\sqrt{a + 2^{- 2 x} b}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2**x/(a+b/(2**(2*x)))**(1/2),x)

[Out]

Integral(2**x/sqrt(a + 2**(-2*x)*b), x)

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GIAC/XCAS [A]  time = 0.219793, size = 39, normalized size = 1.62 \[ \frac{\frac{\sqrt{2^{2 \, x} a + b}}{a} - \frac{\sqrt{b}}{a}}{{\rm ln}\left (2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2^x/sqrt(a + b/2^(2*x)),x, algorithm="giac")

[Out]

(sqrt(2^(2*x)*a + b)/a - sqrt(b)/a)/ln(2)