Optimal. Leaf size=24 \[ \frac {2^x \sqrt {a+b 2^{-2 x}}}{a \log (2)} \]
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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.
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Rule 191
Rule 2249
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.
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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.
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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.
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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.
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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.
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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.
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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.
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