Integrand size = 19, antiderivative size = 93 \[ \int \frac {2^{2 x}}{\sqrt {a+2^{-x} b}} \, dx=\frac {2^{-1+2 x} \sqrt {a+2^{-x} b}}{a \log (2)}-\frac {3\ 2^{-2+x} b \sqrt {a+2^{-x} b}}{a^2 \log (2)}+\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+2^{-x} b}}{\sqrt {a}}\right )}{4 a^{5/2} \log (2)} \]
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Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2280, 44, 65, 214} \[ \int \frac {2^{2 x}}{\sqrt {a+2^{-x} b}} \, dx=\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+b 2^{-x}}}{\sqrt {a}}\right )}{4 a^{5/2} \log (2)}-\frac {3 b 2^{x-2} \sqrt {a+b 2^{-x}}}{a^2 \log (2)}+\frac {2^{2 x-1} \sqrt {a+b 2^{-x}}}{a \log (2)} \]
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Rule 44
Rule 65
Rule 214
Rule 2280
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {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) \text {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 ) \text {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) \text {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*}
Time = 0.00 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.19 \[ \int \frac {2^{2 x}}{\sqrt {a+2^{-x} b}} \, dx=\frac {2^{-2-\frac {x}{2}} \left (2^{x/2} \sqrt {a} \left (2^{1+2 x} a^2-2^x a b-3 b^2\right )+3 b^2 \sqrt {2^x a+b} \text {arctanh}\left (\frac {2^{x/2} \sqrt {a}}{\sqrt {2^x a+b}}\right )\right )}{a^{5/2} \sqrt {a+2^{-x} b} \log (2)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(181\) vs. \(2(81)=162\).
Time = 0.33 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.96
method | result | size |
derivativedivides | \(-\frac {\sqrt {\left (a 2^{x}+b \right ) 2^{-x}}\, 2^{x} \left (-4 a^{\frac {5}{2}} \sqrt {a 2^{2 x}+2^{x} b}\, 2^{x}-4 \ln \left (\frac {2 \sqrt {\left (a 2^{x}+b \right ) 2^{x}}\, \sqrt {a}+2 a 2^{x}+b}{2 \sqrt {a}}\right ) a \,b^{2}+8 \sqrt {\left (a 2^{x}+b \right ) 2^{x}}\, a^{\frac {3}{2}} b -2 a^{\frac {3}{2}} \sqrt {a 2^{2 x}+2^{x} b}\, b +b^{2} \ln \left (\frac {2 \sqrt {a 2^{2 x}+2^{x} b}\, \sqrt {a}+2 a 2^{x}+b}{2 \sqrt {a}}\right ) a \right )}{8 \ln \left (2\right ) \sqrt {\left (a 2^{x}+b \right ) 2^{x}}\, a^{\frac {7}{2}}}\) | \(182\) |
default | \(-\frac {\sqrt {\left (a 2^{x}+b \right ) 2^{-x}}\, 2^{x} \left (-4 a^{\frac {5}{2}} \sqrt {a 2^{2 x}+2^{x} b}\, 2^{x}-4 \ln \left (\frac {2 \sqrt {\left (a 2^{x}+b \right ) 2^{x}}\, \sqrt {a}+2 a 2^{x}+b}{2 \sqrt {a}}\right ) a \,b^{2}+8 \sqrt {\left (a 2^{x}+b \right ) 2^{x}}\, a^{\frac {3}{2}} b -2 a^{\frac {3}{2}} \sqrt {a 2^{2 x}+2^{x} b}\, b +b^{2} \ln \left (\frac {2 \sqrt {a 2^{2 x}+2^{x} b}\, \sqrt {a}+2 a 2^{x}+b}{2 \sqrt {a}}\right ) a \right )}{8 \ln \left (2\right ) \sqrt {\left (a 2^{x}+b \right ) 2^{x}}\, a^{\frac {7}{2}}}\) | \(182\) |
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none
Time = 0.32 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.78 \[ \int \frac {2^{2 x}}{\sqrt {a+2^{-x} b}} \, dx=\left [\frac {3 \, \sqrt {a} b^{2} \log \left (2 \cdot 2^{x} a + 2 \cdot 2^{x} \sqrt {a} \sqrt {\frac {2^{x} a + b}{2^{x}}} + b\right ) + 2 \, {\left (2 \cdot 2^{2 \, x} a^{2} - 3 \cdot 2^{x} a b\right )} \sqrt {\frac {2^{x} a + b}{2^{x}}}}{8 \, a^{3} \log \left (2\right )}, -\frac {3 \, \sqrt {-a} b^{2} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {2^{x} a + b}{2^{x}}}}{a}\right ) - {\left (2 \cdot 2^{2 \, x} a^{2} - 3 \cdot 2^{x} a b\right )} \sqrt {\frac {2^{x} a + b}{2^{x}}}}{4 \, a^{3} \log \left (2\right )}\right ] \]
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\[ \int \frac {2^{2 x}}{\sqrt {a+2^{-x} b}} \, dx=\int \frac {2^{2 x}}{\sqrt {a + 2^{- x} b}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.33 \[ \int \frac {2^{2 x}}{\sqrt {a+2^{-x} b}} \, dx=-\frac {3 \, b^{2} \log \left (\frac {\sqrt {a + \frac {b}{2^{x}}} - \sqrt {a}}{\sqrt {a + \frac {b}{2^{x}}} + \sqrt {a}}\right )}{8 \, a^{\frac {5}{2}} \log \left (2\right )} - \frac {3 \, {\left (a + \frac {b}{2^{x}}\right )}^{\frac {3}{2}} b^{2} - 5 \, \sqrt {a + \frac {b}{2^{x}}} a b^{2}}{4 \, {\left ({\left (a + \frac {b}{2^{x}}\right )}^{2} a^{2} - 2 \, {\left (a + \frac {b}{2^{x}}\right )} a^{3} + a^{4}\right )} \log \left (2\right )} \]
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Time = 0.46 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.99 \[ \int \frac {2^{2 x}}{\sqrt {a+2^{-x} b}} \, dx=\frac {2 \, \sqrt {2^{2 \, x} a + 2^{x} b} {\left (\frac {2 \cdot 2^{x}}{a} - \frac {3 \, b}{a^{2}}\right )} - \frac {3 \, b^{2} \log \left ({\left | 2 \, {\left (2^{x} \sqrt {a} - \sqrt {2^{2 \, x} a + 2^{x} b}\right )} \sqrt {a} + b \right |}\right )}{a^{\frac {5}{2}}} + \frac {3 \, b^{2} \log \left ({\left | b \right |}\right )}{a^{\frac {5}{2}}}}{8 \, \log \left (2\right )} \]
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Timed out. \[ \int \frac {2^{2 x}}{\sqrt {a+2^{-x} b}} \, dx=\int \frac {2^{2\,x}}{\sqrt {a+\frac {b}{2^x}}} \,d x \]
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