Optimal. Leaf size=96 \[ \frac {2^{-1+2 x} \sqrt {a-2^{-x} b}}{a \log (2)}+\frac {3\ 2^{-2+x} b \sqrt {a-2^{-x} b}}{a^2 \log (2)}+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a-2^{-x} b}}{\sqrt {a}}\right )}{4 a^{5/2} \log (2)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2280, 44, 65,
214} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 65
Rule 214
Rule 2280
Rubi steps
\begin {align*} \int \frac {2^{2 x}}{\sqrt {a-2^{-x} b}} \, dx &=-\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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 115, normalized size = 1.20 \begin {gather*} \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 \sqrt {2^x a-b} b^2 \tanh ^{-1}\left (\frac {2^{x/2} \sqrt {a}}{\sqrt {2^x a-b}}\right )\right )}{a^{5/2} \sqrt {a-2^{-x} b} \log (2)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {2^{2 x}}{\sqrt {a -b 2^{-x}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 130, normalized size = 1.35 \begin {gather*} -\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 174, normalized size = 1.81 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2^{2 x}}{\sqrt {a - 2^{- x} b}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 5.71, size = 96, normalized size = 1.00 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {2^{2\,x}}{\sqrt {a-\frac {b}{2^x}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________