Optimal. Leaf size=93 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b 2^{-x}}}{\sqrt{a}}\right )}{4 a^{5/2} \log (2)}-\frac{3 b 2^{x-2} \sqrt{a+b 2^{-x}}}{a^2 \log (2)}+\frac{2^{2 x-1} \sqrt{a+b 2^{-x}}}{a \log (2)} \]
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Rubi [A] time = 0.0743125, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2248, 51, 63, 208} \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b 2^{-x}}}{\sqrt{a}}\right )}{4 a^{5/2} \log (2)}-\frac{3 b 2^{x-2} \sqrt{a+b 2^{-x}}}{a^2 \log (2)}+\frac{2^{2 x-1} \sqrt{a+b 2^{-x}}}{a \log (2)} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{4^x}{\sqrt{a+2^{-x} b}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,2^{-x}\right )}{\log (2)}\\ &=\frac{2^{-1+2 x} \sqrt{a+2^{-x} b}}{a \log (2)}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,2^{-x}\right )}{4 a \log (2)}\\ &=\frac{2^{-1+2 x} \sqrt{a+2^{-x} b}}{a \log (2)}-\frac{3\ 2^{-2+x} b \sqrt{a+2^{-x} b}}{a^2 \log (2)}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,2^{-x}\right )}{8 a^2 \log (2)}\\ &=\frac{2^{-1+2 x} \sqrt{a+2^{-x} b}}{a \log (2)}-\frac{3\ 2^{-2+x} b \sqrt{a+2^{-x} b}}{a^2 \log (2)}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+2^{-x} b}\right )}{4 a^2 \log (2)}\\ &=\frac{2^{-1+2 x} \sqrt{a+2^{-x} b}}{a \log (2)}-\frac{3\ 2^{-2+x} b \sqrt{a+2^{-x} b}}{a^2 \log (2)}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+2^{-x} b}}{\sqrt{a}}\right )}{4 a^{5/2} \log (2)}\\ \end{align*}
Mathematica [A] time = 0.0861008, size = 111, normalized size = 1.19 \[ \frac{2^{-\frac{x}{2}-2} \left (\sqrt{a} 2^{x/2} \left (a^2 2^{2 x+1}-a b 2^x-3 b^2\right )+3 b^2 \sqrt{a 2^x+b} \tanh ^{-1}\left (\frac{\sqrt{a} 2^{x/2}}{\sqrt{a 2^x+b}}\right )\right )}{a^{5/2} \log (2) \sqrt{a+b 2^{-x}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{{4}^{x}{\frac{1}{\sqrt{a+{\frac{b}{{2}^{x}}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{4^{x}}{\sqrt{a + \frac{b}{2^{x}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54, size = 373, normalized size = 4.01 \begin{align*} \left [\frac{3 \, \sqrt{a} b^{2} \log \left (2 \cdot 2^{x} a + 2 \cdot 2^{x} \sqrt{a} \sqrt{\frac{2^{x} a + b}{2^{x}}} + b\right ) + 2 \,{\left (2 \cdot 2^{2 \, x} a^{2} - 3 \cdot 2^{x} a b\right )} \sqrt{\frac{2^{x} a + b}{2^{x}}}}{8 \, a^{3} \log \left (2\right )}, -\frac{3 \, \sqrt{-a} b^{2} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{2^{x} a + b}{2^{x}}}}{a}\right ) -{\left (2 \cdot 2^{2 \, x} a^{2} - 3 \cdot 2^{x} a b\right )} \sqrt{\frac{2^{x} a + b}{2^{x}}}}{4 \, a^{3} \log \left (2\right )}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{4^{x}}{\sqrt{a + 2^{- x} b}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{4^{x}}{\sqrt{a + \frac{b}{2^{x}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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