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