Optimal. Leaf size=58 \[ \frac{b^2 x}{a^3}+\frac{b^2 \log \left (a+b 2^{-x}\right )}{a^3 \log (2)}-\frac{b 2^x}{a^2 \log (2)}+\frac{2^{2 x-1}}{a \log (2)} \]
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Rubi [A] time = 0.0508679, antiderivative size = 58, 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, 44} \[ \frac{b^2 x}{a^3}+\frac{b^2 \log \left (a+b 2^{-x}\right )}{a^3 \log (2)}-\frac{b 2^x}{a^2 \log (2)}+\frac{2^{2 x-1}}{a \log (2)} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 44
Rubi steps
\begin{align*} \int \frac{2^{2 x}}{a+2^{-x} b} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)} \, dx,x,2^{-x}\right )}{\log (2)}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^3}-\frac{b}{a^2 x^2}+\frac{b^2}{a^3 x}-\frac{b^3}{a^3 (a+b x)}\right ) \, dx,x,2^{-x}\right )}{\log (2)}\\ &=\frac{b^2 x}{a^3}+\frac{2^{-1+2 x}}{a \log (2)}-\frac{2^x b}{a^2 \log (2)}+\frac{b^2 \log \left (a+2^{-x} b\right )}{a^3 \log (2)}\\ \end{align*}
Mathematica [A] time = 0.0176555, size = 36, normalized size = 0.62 \[ \frac{2 b^2 \log \left (a 2^x+b\right )+a 2^x \left (a 2^x-2 b\right )}{a^3 \log (4)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 54, normalized size = 0.9 \begin{align*}{\frac{ \left ({{\rm e}^{x\ln \left ( 2 \right ) }} \right ) ^{2}}{2\,a\ln \left ( 2 \right ) }}-{\frac{{{\rm e}^{x\ln \left ( 2 \right ) }}b}{\ln \left ( 2 \right ){a}^{2}}}+{\frac{{b}^{2}\ln \left ( a{{\rm e}^{x\ln \left ( 2 \right ) }}+b \right ) }{{a}^{3}\ln \left ( 2 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992382, size = 80, normalized size = 1.38 \begin{align*} \frac{b^{2} x}{a^{3}} - \frac{{\left (2^{-x + 1} b - a\right )} 2^{2 \, x - 1}}{a^{2} \log \left (2\right )} + \frac{b^{2} \log \left (a + \frac{b}{2^{x}}\right )}{a^{3} \log \left (2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57711, size = 90, normalized size = 1.55 \begin{align*} \frac{2^{2 \, x} a^{2} - 2 \cdot 2^{x} a b + 2 \, b^{2} \log \left (2^{x} a + b\right )}{2 \, a^{3} \log \left (2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.192232, size = 68, normalized size = 1.17 \begin{align*} \begin{cases} \frac{2^{2 x} a^{2} \log{\left (2 \right )} - 2 \cdot 2^{x} a b \log{\left (2 \right )}}{2 a^{3} \log{\left (2 \right )}^{2}} & \text{for}\: 2 a^{3} \log{\left (2 \right )}^{2} \neq 0 \\\frac{x \left (a - b\right )}{a^{2}} & \text{otherwise} \end{cases} + \frac{b^{2} \log{\left (2^{x} + \frac{b}{a} \right )}}{a^{3} \log{\left (2 \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21713, size = 65, normalized size = 1.12 \begin{align*} \frac{b^{2} \log \left ({\left | 2^{x} a + b \right |}\right )}{a^{3} \log \left (2\right )} + \frac{2^{2 \, x} a \log \left (2\right ) - 2 \cdot 2^{x} b \log \left (2\right )}{2 \, a^{2} \log \left (2\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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