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 = 0.05, antiderivative size = 58, 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, 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 44
Rule 2248
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.41, size = 39, normalized size = 0.67 \[ \frac {2^{2 \, x} a^{2} - 2 \cdot 2^{x} a b + 2 \, b^{2} \log \left (2^{x} a + b\right )}{2 \, a^{3} \log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 48, normalized size = 0.83 \[ \frac {b^{2} \log \left ({\left | 2^{x} a + b \right |}\right )}{a^{3} \log \relax (2)} + \frac {2^{2 \, x} a \log \relax (2) - 2 \cdot 2^{x} b \log \relax (2)}{2 \, a^{2} \log \relax (2)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 54, normalized size = 0.93 \[ \frac {{\mathrm e}^{2 \ln \relax (2) x}}{2 \ln \relax (2) a}-\frac {b \,{\mathrm e}^{\ln \relax (2) x}}{\ln \relax (2) a^{2}}+\frac {b^{2} \ln \left (a \,{\mathrm e}^{\ln \relax (2) x}+b \right )}{\ln \relax (2) a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 59, normalized size = 1.02 \[ \frac {b^{2} x}{a^{3}} - \frac {{\left (2^{-x + 1} b - a\right )} 2^{2 \, x - 1}}{a^{2} \log \relax (2)} + \frac {b^{2} \log \left (a + \frac {b}{2^{x}}\right )}{a^{3} \log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.68, size = 47, normalized size = 0.81 \[ \frac {2^{2\,x}}{2\,a\,\ln \relax (2)}-\frac {2^x\,b}{a^2\,\ln \relax (2)}+\frac {b^2\,\ln \left (b+2^x\,a\right )}{a^3\,\ln \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 92, normalized size = 1.59 \[ \begin {cases} \frac {2^{2 x} a^{2} \log {\relax (2 )} - 2 \cdot 2^{x} a b \log {\relax (2 )}}{2 a^{3} \log {\relax (2 )}^{2}} & \text {for}\: 2 a^{3} \log {\relax (2 )}^{2} \neq 0 \\x \left (- \frac {b^{2}}{a^{3}} + \frac {a^{2} - a b + b^{2}}{a^{3}}\right ) & \text {otherwise} \end {cases} + \frac {b^{2} x}{a^{3}} + \frac {b^{2} \log {\left (\frac {a}{b} + 2^{- x} \right )}}{a^{3} \log {\relax (2 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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