Optimal. Leaf size=93 \[ \frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b 2^{-x}}}{\sqrt {a}}\right )}{4 a^{5/2} \log (2)}-\frac {3 b 2^{x-2} \sqrt {a+b 2^{-x}}}{a^2 \log (2)}+\frac {2^{2 x-1} \sqrt {a+b 2^{-x}}}{a \log (2)} \]
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Rubi [A] time = 0.07, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2248, 51, 63, 208} \[ \frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b 2^{-x}}}{\sqrt {a}}\right )}{4 a^{5/2} \log (2)}-\frac {3 b 2^{x-2} \sqrt {a+b 2^{-x}}}{a^2 \log (2)}+\frac {2^{2 x-1} \sqrt {a+b 2^{-x}}}{a \log (2)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 2248
Rubi steps
\begin {align*} \int \frac {2^{2 x}}{\sqrt {a+2^{-x} b}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,2^{-x}\right )}{\log (2)}\\ &=\frac {2^{-1+2 x} \sqrt {a+2^{-x} b}}{a \log (2)}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,2^{-x}\right )}{4 a \log (2)}\\ &=\frac {2^{-1+2 x} \sqrt {a+2^{-x} b}}{a \log (2)}-\frac {3\ 2^{-2+x} b \sqrt {a+2^{-x} b}}{a^2 \log (2)}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,2^{-x}\right )}{8 a^2 \log (2)}\\ &=\frac {2^{-1+2 x} \sqrt {a+2^{-x} b}}{a \log (2)}-\frac {3\ 2^{-2+x} b \sqrt {a+2^{-x} b}}{a^2 \log (2)}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+2^{-x} b}\right )}{4 a^2 \log (2)}\\ &=\frac {2^{-1+2 x} \sqrt {a+2^{-x} b}}{a \log (2)}-\frac {3\ 2^{-2+x} b \sqrt {a+2^{-x} b}}{a^2 \log (2)}+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+2^{-x} b}}{\sqrt {a}}\right )}{4 a^{5/2} \log (2)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 111, normalized size = 1.19 \[ \frac {2^{-\frac {x}{2}-2} \left (\sqrt {a} 2^{x/2} \left (a^2 2^{2 x+1}-a b 2^x-3 b^2\right )+3 b^2 \sqrt {a 2^x+b} \tanh ^{-1}\left (\frac {\sqrt {a} 2^{x/2}}{\sqrt {a 2^x+b}}\right )\right )}{a^{5/2} \log (2) \sqrt {a+b 2^{-x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 166, normalized size = 1.78 \[ \left [\frac {3 \, \sqrt {a} b^{2} \log \left (2 \cdot 2^{x} a + 2 \cdot 2^{x} \sqrt {a} \sqrt {\frac {2^{x} a + b}{2^{x}}} + b\right ) + 2 \, {\left (2 \cdot 2^{2 \, x} a^{2} - 3 \cdot 2^{x} a b\right )} \sqrt {\frac {2^{x} a + b}{2^{x}}}}{8 \, a^{3} \log \relax (2)}, -\frac {3 \, \sqrt {-a} b^{2} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {2^{x} a + b}{2^{x}}}}{a}\right ) - {\left (2 \cdot 2^{2 \, x} a^{2} - 3 \cdot 2^{x} a b\right )} \sqrt {\frac {2^{x} a + b}{2^{x}}}}{4 \, a^{3} \log \relax (2)}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 94, normalized size = 1.01 \[ \frac {2 \, \sqrt {2^{2 \, x} a + 2^{x} b} {\left (\frac {2 \cdot 2^{x}}{a} - \frac {3 \, b}{a^{2}}\right )} - \frac {3 \, b^{2} \log \left ({\left | -2 \, {\left (2^{x} \sqrt {a} - \sqrt {2^{2 \, x} a + 2^{x} b}\right )} \sqrt {a} - b \right |}\right )}{a^{\frac {5}{2}}} + \frac {3 \, b^{2} \log \left ({\left | b \right |}\right )}{a^{\frac {5}{2}}}}{8 \, \log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {2^{2 x}}{\sqrt {b 2^{-x}+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.04, size = 124, normalized size = 1.33 \[ -\frac {3 \, b^{2} \log \left (\frac {\sqrt {a + \frac {b}{2^{x}}} - \sqrt {a}}{\sqrt {a + \frac {b}{2^{x}}} + \sqrt {a}}\right )}{8 \, a^{\frac {5}{2}} \log \relax (2)} - \frac {3 \, {\left (a + \frac {b}{2^{x}}\right )}^{\frac {3}{2}} b^{2} - 5 \, \sqrt {a + \frac {b}{2^{x}}} a b^{2}}{4 \, {\left ({\left (a + \frac {b}{2^{x}}\right )}^{2} a^{2} - 2 \, {\left (a + \frac {b}{2^{x}}\right )} a^{3} + a^{4}\right )} \log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {2^{2\,x}}{\sqrt {a+\frac {b}{2^x}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2^{2 x}}{\sqrt {a + 2^{- x} b}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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