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 = 0.0410894, antiderivative size = 44, normalized size of antiderivative = 1., 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 2248
Rule 43
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*}
Mathematica [A] time = 0.0115383, 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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Maple [A] time = 0.01, size = 29, normalized size = 0.7 \begin{align*} -{\frac{-2\,{2}^{x}b+4\,a}{3\,{b}^{2}\ln \left ( 2 \right ) }\sqrt{a+{2}^{x}b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973952, size = 51, normalized size = 1.16 \begin{align*} \frac{2 \,{\left (2^{x} b + a\right )}^{\frac{3}{2}}}{3 \, b^{2} \log \left (2\right )} - \frac{2 \, \sqrt{2^{x} b + a} a}{b^{2} \log \left (2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56375, size = 65, normalized size = 1.48 \begin{align*} \frac{2 \, \sqrt{2^{x} b + a}{\left (2^{x} b - 2 \, a\right )}}{3 \, b^{2} \log \left (2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.955978, size = 58, normalized size = 1.32 \begin{align*} \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{2^{2 x}}{2 \sqrt{a} \log{\left (2 \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28933, size = 42, normalized size = 0.95 \begin{align*} \frac{2 \,{\left ({\left (2^{x} b + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{2^{x} b + a} a\right )}}{3 \, b^{2} \log \left (2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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