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

Optimal. Leaf size=46 \[ \frac{2 \left (a-b 2^x\right )^{3/2}}{3 b^2 \log (2)}-\frac{2 a \sqrt{a-b 2^x}}{b^2 \log (2)} \]

[Out]

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

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Rubi [A]  time = 0.078743, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2 \left (a-b 2^x\right )^{3/2}}{3 b^2 \log (2)}-\frac{2 a \sqrt{a-b 2^x}}{b^2 \log (2)} \]

Antiderivative was successfully verified.

[In]  Int[4^x/Sqrt[a - 2^x*b],x]

[Out]

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

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Rubi in Sympy [A]  time = 10.0612, size = 39, normalized size = 0.85 \[ - \frac{2 a \sqrt{- 2^{x} b + a}}{b^{2} \log{\left (2 \right )}} + \frac{2 \left (- 2^{x} b + a\right )^{\frac{3}{2}}}{3 b^{2} \log{\left (2 \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(4**x/(a-2**x*b)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.10092, size = 57, normalized size = 1.24 \[ \frac{2 \left (2 a^2 \left (\sqrt{1-\frac{b 2^x}{a}}-1\right )+a b 2^x+b^2 4^x\right )}{b^2 \log (8) \sqrt{a-b 2^x}} \]

Antiderivative was successfully verified.

[In]  Integrate[4^x/Sqrt[a - 2^x*b],x]

[Out]

(2*(2^x*a*b + 4^x*b^2 + 2*a^2*(-1 + Sqrt[1 - (2^x*b)/a])))/(b^2*Sqrt[a - 2^x*b]*
Log[8])

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Maple [A]  time = 0.024, size = 29, normalized size = 0.6 \[ -{\frac{2\,{2}^{x}b+4\,a}{3\,{b}^{2}\ln \left ( 2 \right ) }\sqrt{a-{2}^{x}b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(4^x/(a-2^x*b)^(1/2),x)

[Out]

-2/3*(2^x*b+2*a)/b^2*(a-2^x*b)^(1/2)/ln(2)

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Maxima [A]  time = 0.902818, size = 96, normalized size = 2.09 \[ \frac{2^{2 \, x + 1}}{3 \, \sqrt{-2^{x} b + a} \log \left (2\right )} + \frac{2^{x + 1} a}{3 \, \sqrt{-2^{x} b + a} b \log \left (2\right )} - \frac{4 \, a^{2}}{3 \, \sqrt{-2^{x} b + a} b^{2} \log \left (2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*2^(2*x + 1)/(sqrt(-2^x*b + a)*log(2)) + 1/3*2^(x + 1)*a/(sqrt(-2^x*b + a)*b*
log(2)) - 4/3*a^2/(sqrt(-2^x*b + a)*b^2*log(2))

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Fricas [A]  time = 0.243144, size = 38, normalized size = 0.83 \[ -\frac{2 \,{\left (2^{x} b + 2 \, a\right )} \sqrt{-2^{x} b + a}}{3 \, b^{2} \log \left (2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(2^x*b + 2*a)*sqrt(-2^x*b + a)/(b^2*log(2))

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Sympy [A]  time = 1.62732, size = 58, normalized size = 1.26 \[ \begin{cases} - \frac{2 \cdot 2^{x} \sqrt{- 2^{x} b + a}}{3 b \log{\left (2 \right )}} - \frac{4 a \sqrt{- 2^{x} b + a}}{3 b^{2} \log{\left (2 \right )}} & \text{for}\: b \neq 0 \\\frac{4^{x}}{2 \sqrt{a} \log{\left (2 \right )}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*2**x*sqrt(-2**x*b + a)/(3*b*log(2)) - 4*a*sqrt(-2**x*b + a)/(3*b**
2*log(2)), Ne(b, 0)), (4**x/(2*sqrt(a)*log(2)), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(4^x/sqrt(-2^x*b + a), x)