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 = 0.150186, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b 2^{-x}}}{\sqrt{a}}\right )}{4 a^{5/2} \log (2)}-\frac{3 b 2^{x-2} \sqrt{a+b 2^{-x}}}{a^2 \log (2)}+\frac{2^{2 x-1} \sqrt{a+b 2^{-x}}}{a \log (2)} \]
Antiderivative was successfully verified.
[In] Int[2^(2*x)/Sqrt[a + b/2^x],x]
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Rubi in Sympy [A] time = 12.5274, size = 78, normalized size = 0.84 \[ \frac{2^{2 x} \sqrt{a + 2^{- x} b}}{2 a \log{\left (2 \right )}} - \frac{3 \cdot 2^{x} b \sqrt{a + 2^{- x} b}}{4 a^{2} \log{\left (2 \right )}} + \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + 2^{- x} b}}{\sqrt{a}} \right )}}{4 a^{\frac{5}{2}} \log{\left (2 \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(2**(2*x)/(a+b/(2**x))**(1/2),x)
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Mathematica [A] time = 0.0324053, size = 114, normalized size = 1.23 \[ \frac{2^{-\frac{x}{2}-2} \left (\sqrt{a} 2^{x/2} \left (a^2 2^{2 x+1}-a b 2^x-3 b^2\right )+3 b^2 \sqrt{a 2^x+b} \log \left (\sqrt{a} \sqrt{a 2^x+b}+a 2^{x/2}\right )\right )}{a^{5/2} \log (2) \sqrt{a+b 2^{-x}}} \]
Antiderivative was successfully verified.
[In] Integrate[2^(2*x)/Sqrt[a + b/2^x],x]
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Maple [F] time = 0.023, size = 0, normalized size = 0. \[ \int{{2}^{2\,x}{\frac{1}{\sqrt{a+{\frac{b}{{2}^{x}}}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(2^(2*x)/(a+b/(2^x))^(1/2),x)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(2^(2*x)/sqrt(a + b/2^x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.25099, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} \log \left (2 \cdot 2^{x} a \sqrt{\frac{2^{x} a + b}{2^{x}}} +{\left (2 \cdot 2^{x} a + b\right )} \sqrt{a}\right ) + 2 \,{\left (2 \cdot 2^{2 \, x} a - 3 \cdot 2^{x} b\right )} \sqrt{a} \sqrt{\frac{2^{x} a + b}{2^{x}}}}{8 \, a^{\frac{5}{2}} \log \left (2\right )}, -\frac{3 \, b^{2} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{2^{x} a + b}{2^{x}}}}\right ) -{\left (2 \cdot 2^{2 \, x} a - 3 \cdot 2^{x} b\right )} \sqrt{-a} \sqrt{\frac{2^{x} a + b}{2^{x}}}}{4 \, \sqrt{-a} a^{2} \log \left (2\right )}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(2^(2*x)/sqrt(a + b/2^x),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{2^{2 x}}{\sqrt{a + 2^{- x} b}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(2**(2*x)/(a+b/(2**x))**(1/2),x)
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GIAC/XCAS [A] time = 0.297582, size = 127, normalized size = 1.37 \[ \frac{2 \, \sqrt{2^{2 \, x} a + 2^{x} b}{\left (\frac{2 \cdot 2^{x}}{a} - \frac{3 \, b}{a^{2}}\right )} - \frac{3 \, b^{2}{\rm ln}\left ({\left | -2 \,{\left (2^{x} \sqrt{a} - \sqrt{2^{2 \, x} a + 2^{x} b}\right )} \sqrt{a} - b \right |}\right )}{a^{\frac{5}{2}}} + \frac{3 \, b^{2}{\rm ln}\left ({\left | b \right |}\right )}{a^{\frac{5}{2}}}}{8 \,{\rm ln}\left (2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(2^(2*x)/sqrt(a + b/2^x),x, algorithm="giac")
[Out]