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

Optimal. Leaf size=44 \[ \frac {2 \left (a+b 2^x\right )^{3/2}}{3 b^2 \log (2)}-\frac {2 a \sqrt {a+b 2^x}}{b^2 \log (2)} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2248, 43} \[ \frac {2 \left (a+b 2^x\right )^{3/2}}{3 b^2 \log (2)}-\frac {2 a \sqrt {a+b 2^x}}{b^2 \log (2)} \]

Antiderivative was successfully verified.

[In]

Int[2^(2*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])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2248

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[(g*h*Log[G])/(d*e*Log[F])]}, Dist[(Denominator[m]*G^(f*h - (c*g*h)/d))/(d*e*Log[F]), Subst
[Int[x^(Numerator[m] - 1)*(a + b*x^Denominator[m])^p, x], x, F^((e*(c + d*x))/Denominator[m])], x] /; LeQ[m, -
1] || GeQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

Rubi steps

\begin {align*} \int \frac {2^{2 x}}{\sqrt {a+2^x b}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b x}} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b}\right ) \, dx,x,2^x\right )}{\log (2)}\\ &=-\frac {2 a \sqrt {a+2^x b}}{b^2 \log (2)}+\frac {2 \left (a+2^x b\right )^{3/2}}{3 b^2 \log (2)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 0.66 \[ \frac {2 \left (b 2^x-2 a\right ) \sqrt {a+b 2^x}}{b^2 \log (8)} \]

Antiderivative was successfully verified.

[In]

Integrate[2^(2*x)/Sqrt[a + 2^x*b],x]

[Out]

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

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fricas [A]  time = 0.45, size = 27, normalized size = 0.61 \[ \frac {2 \, \sqrt {2^{x} b + a} {\left (2^{x} b - 2 \, a\right )}}{3 \, b^{2} \log \relax (2)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2^(2*x)/(a+2^x*b)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.28, size = 31, normalized size = 0.70 \[ \frac {2 \, {\left ({\left (2^{x} b + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {2^{x} b + a} a\right )}}{3 \, b^{2} \log \relax (2)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2^(2*x)/(a+2^x*b)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 29, normalized size = 0.66 \[ -\frac {2 \left (-b 2^{x}+2 a \right ) \sqrt {b 2^{x}+a}}{3 \ln \relax (2) b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2^(2*x)/(b*2^x+a)^(1/2),x)

[Out]

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

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maxima [A]  time = 0.83, size = 38, normalized size = 0.86 \[ \frac {2 \, {\left (2^{x} b + a\right )}^{\frac {3}{2}}}{3 \, b^{2} \log \relax (2)} - \frac {2 \, \sqrt {2^{x} b + a} a}{b^{2} \log \relax (2)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2^(2*x)/(a+2^x*b)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.62, size = 28, normalized size = 0.64 \[ -\frac {2\,\sqrt {a+2^x\,b}\,\left (2\,a-2^x\,b\right )}{3\,b^2\,\ln \relax (2)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2^(2*x)/(a + 2^x*b)^(1/2),x)

[Out]

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

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sympy [A]  time = 0.97, size = 58, normalized size = 1.32 \[ \begin {cases} \frac {2 \cdot 2^{x} \sqrt {2^{x} b + a}}{3 b \log {\relax (2 )}} - \frac {4 a \sqrt {2^{x} b + a}}{3 b^{2} \log {\relax (2 )}} & \text {for}\: b \neq 0 \\\frac {2^{2 x}}{2 \sqrt {a} \log {\relax (2 )}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2**(2*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)), (2**(2*x)/(
2*sqrt(a)*log(2)), True))

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