Optimal. Leaf size=24 \[ \frac {2^x \sqrt {a+2^{-2 x} b}}{a \log (2)} \]
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Rubi [A]
time = 0.03, 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 = {2281, 197}
\begin {gather*} \frac {2^x \sqrt {a+b 2^{-2 x}}}{a \log (2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 2281
Rubi steps
\begin {align*} \int \frac {2^x}{\sqrt {a+4^{-x} b}} \, dx &=\frac {\text {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 = 0.05, size = 35, normalized size = 1.46 \begin {gather*} \frac {2^{-x} \left (2^{2 x} a+b\right )}{a \sqrt {a+2^{-2 x} b} \log (2)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 40, normalized size = 1.67
method | result | size |
risch | \(\frac {\left (a 2^{2 x}+b \right ) 2^{-x}}{\sqrt {\left (a 2^{2 x}+b \right ) 2^{-2 x}}\, a \ln \left (2\right )}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 19, normalized size = 0.79 \begin {gather*} \frac {\sqrt {2^{2 \, x} a + b}}{a \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 30, normalized size = 1.25 \begin {gather*} \frac {2^{x} \sqrt {\frac {2^{2 \, x} a + b}{2^{2 \, x}}}}{a \log \left (2\right )} \end {gather*}
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 \begin {gather*} \int \frac {2^{x}}{\sqrt {a + 4^{- x} b}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \frac {2^x\,\sqrt {a+\frac {b}{2^{2\,x}}}}{a\,\ln \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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