Integrand size = 17, antiderivative size = 93 \[ \int \frac {4^x}{\sqrt {a+2^{-x} b}} \, dx=\frac {2^{-1+2 x} \sqrt {a+2^{-x} b}}{a \log (2)}-\frac {3\ 2^{-2+x} b \sqrt {a+2^{-x} b}}{a^2 \log (2)}+\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+2^{-x} b}}{\sqrt {a}}\right )}{4 a^{5/2} \log (2)} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2280, 44, 65, 214} \[ \int \frac {4^x}{\sqrt {a+2^{-x} b}} \, dx=\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+b 2^{-x}}}{\sqrt {a}}\right )}{4 a^{5/2} \log (2)}-\frac {3 b 2^{x-2} \sqrt {a+b 2^{-x}}}{a^2 \log (2)}+\frac {2^{2 x-1} \sqrt {a+b 2^{-x}}}{a \log (2)} \]
[In]
[Out]
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 = 0.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.19 \[ \int \frac {4^x}{\sqrt {a+2^{-x} b}} \, dx=\frac {2^{-2-\frac {x}{2}} \left (2^{x/2} \sqrt {a} \left (2^{1+2 x} a^2-2^x a b-3 b^2\right )+3 b^2 \sqrt {2^x a+b} \text {arctanh}\left (\frac {2^{x/2} \sqrt {a}}{\sqrt {2^x a+b}}\right )\right )}{a^{5/2} \sqrt {a+2^{-x} b} \log (2)} \]
[In]
[Out]
\[\int \frac {4^{x}}{\sqrt {a +b 2^{-x}}}d x\]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.78 \[ \int \frac {4^x}{\sqrt {a+2^{-x} b}} \, dx=\left [\frac {3 \, \sqrt {a} b^{2} \log \left (2 \cdot 2^{x} a + 2 \cdot 2^{x} \sqrt {a} \sqrt {\frac {2^{x} a + b}{2^{x}}} + b\right ) + 2 \, {\left (2 \cdot 2^{2 \, x} a^{2} - 3 \cdot 2^{x} a b\right )} \sqrt {\frac {2^{x} a + b}{2^{x}}}}{8 \, a^{3} \log \left (2\right )}, -\frac {3 \, \sqrt {-a} b^{2} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {2^{x} a + b}{2^{x}}}}{a}\right ) - {\left (2 \cdot 2^{2 \, x} a^{2} - 3 \cdot 2^{x} a b\right )} \sqrt {\frac {2^{x} a + b}{2^{x}}}}{4 \, a^{3} \log \left (2\right )}\right ] \]
[In]
[Out]
\[ \int \frac {4^x}{\sqrt {a+2^{-x} b}} \, dx=\int \frac {4^{x}}{\sqrt {a + 2^{- x} b}}\, dx \]
[In]
[Out]
\[ \int \frac {4^x}{\sqrt {a+2^{-x} b}} \, dx=\int { \frac {4^{x}}{\sqrt {a + \frac {b}{2^{x}}}} \,d x } \]
[In]
[Out]
\[ \int \frac {4^x}{\sqrt {a+2^{-x} b}} \, dx=\int { \frac {4^{x}}{\sqrt {a + \frac {b}{2^{x}}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {4^x}{\sqrt {a+2^{-x} b}} \, dx=\int \frac {4^x}{\sqrt {a+\frac {b}{2^x}}} \,d x \]
[In]
[Out]