3.501 \(\int \frac{4^x}{\sqrt{a+2^{-x} b}} \, dx\)

Optimal. Leaf size=93 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b 2^{-x}}}{\sqrt{a}}\right )}{4 a^{5/2} \log (2)}-\frac{3 b 2^{x-2} \sqrt{a+b 2^{-x}}}{a^2 \log (2)}+\frac{2^{2 x-1} \sqrt{a+b 2^{-x}}}{a \log (2)} \]

[Out]

(2^(-1 + 2*x)*Sqrt[a + b/2^x])/(a*Log[2]) - (3*2^(-2 + x)*b*Sqrt[a + b/2^x])/(a^
2*Log[2]) + (3*b^2*ArcTanh[Sqrt[a + b/2^x]/Sqrt[a]])/(4*a^(5/2)*Log[2])

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Rubi [A]  time = 0.152391, 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 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b 2^{-x}}}{\sqrt{a}}\right )}{4 a^{5/2} \log (2)}-\frac{3 b 2^{x-2} \sqrt{a+b 2^{-x}}}{a^2 \log (2)}+\frac{2^{2 x-1} \sqrt{a+b 2^{-x}}}{a \log (2)} \]

Antiderivative was successfully verified.

[In]  Int[4^x/Sqrt[a + b/2^x],x]

[Out]

(2^(-1 + 2*x)*Sqrt[a + b/2^x])/(a*Log[2]) - (3*2^(-2 + x)*b*Sqrt[a + b/2^x])/(a^
2*Log[2]) + (3*b^2*ArcTanh[Sqrt[a + b/2^x]/Sqrt[a]])/(4*a^(5/2)*Log[2])

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Rubi in Sympy [A]  time = 12.5381, 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(4**x/(a+b/(2**x))**(1/2),x)

[Out]

2**(2*x)*sqrt(a + 2**(-x)*b)/(2*a*log(2)) - 3*2**x*b*sqrt(a + 2**(-x)*b)/(4*a**2
*log(2)) + 3*b**2*atanh(sqrt(a + 2**(-x)*b)/sqrt(a))/(4*a**(5/2)*log(2))

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Mathematica [A]  time = 0.133565, 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[4^x/Sqrt[a + b/2^x],x]

[Out]

(2^(-2 - x/2)*(2^(x/2)*Sqrt[a]*(2^(1 + 2*x)*a^2 - 2^x*a*b - 3*b^2) + 3*b^2*Sqrt[
2^x*a + b]*Log[2^(x/2)*a + Sqrt[a]*Sqrt[2^x*a + b]]))/(a^(5/2)*Sqrt[a + b/2^x]*L
og[2])

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{{4}^{x}{\frac{1}{\sqrt{a+{\frac{b}{{2}^{x}}}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(4^x/(a+b/(2^x))^(1/2),x)

[Out]

int(4^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(4^x/sqrt(a + b/2^x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.26341, 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(4^x/sqrt(a + b/2^x),x, algorithm="fricas")

[Out]

[1/8*(3*b^2*log(2*2^x*a*sqrt((2^x*a + b)/2^x) + (2*2^x*a + b)*sqrt(a)) + 2*(2*2^
(2*x)*a - 3*2^x*b)*sqrt(a)*sqrt((2^x*a + b)/2^x))/(a^(5/2)*log(2)), -1/4*(3*b^2*
arctan(a/(sqrt(-a)*sqrt((2^x*a + b)/2^x))) - (2*2^(2*x)*a - 3*2^x*b)*sqrt(-a)*sq
rt((2^x*a + b)/2^x))/(sqrt(-a)*a^2*log(2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{4^{x}}{\sqrt{a + 2^{- x} b}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(4**x/(a+b/(2**x))**(1/2),x)

[Out]

Integral(4**x/sqrt(a + 2**(-x)*b), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{4^{x}}{\sqrt{a + \frac{b}{2^{x}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(4^x/sqrt(a + b/2^x),x, algorithm="giac")

[Out]

integrate(4^x/sqrt(a + b/2^x), x)