Optimal. Leaf size=44 \[ \frac {2 \left (a+b 2^x\right )^{3/2}}{3 b^2 \log (2)}-\frac {2 a \sqrt {a+b 2^x}}{b^2 \log (2)} \]
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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.
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Rule 43
Rule 2248
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.
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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.
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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.
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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.
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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.
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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.
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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.
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