Optimal. Leaf size=44 \[ \frac{2 \left (a+b 2^x\right )^{3/2}}{3 b^2 \log (2)}-\frac{2 a \sqrt{a+b 2^x}}{b^2 \log (2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.040342, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2248, 43} \[ \frac{2 \left (a+b 2^x\right )^{3/2}}{3 b^2 \log (2)}-\frac{2 a \sqrt{a+b 2^x}}{b^2 \log (2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2248
Rule 43
Rubi steps
\begin{align*} \int \frac{4^x}{\sqrt{a+2^x b}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b x}} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b}\right ) \, dx,x,2^x\right )}{\log (2)}\\ &=-\frac{2 a \sqrt{a+2^x b}}{b^2 \log (2)}+\frac{2 \left (a+2^x b\right )^{3/2}}{3 b^2 \log (2)}\\ \end{align*}
Mathematica [A] time = 0.0215607, size = 29, normalized size = 0.66 \[ \frac{2 \left (b 2^x-2 a\right ) \sqrt{a+b 2^x}}{b^2 \log (8)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.015, size = 29, normalized size = 0.7 \begin{align*} -{\frac{-2\,{2}^{x}b+4\,a}{3\,{b}^{2}\ln \left ( 2 \right ) }\sqrt{a+{2}^{x}b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.46217, size = 92, normalized size = 2.09 \begin{align*} \frac{2^{2 \, x + 1}}{3 \, \sqrt{2^{x} b + a} \log \left (2\right )} - \frac{2^{x + 1} a}{3 \, \sqrt{2^{x} b + a} b \log \left (2\right )} - \frac{4 \, a^{2}}{3 \, \sqrt{2^{x} b + a} b^{2} \log \left (2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.55118, size = 65, normalized size = 1.48 \begin{align*} \frac{2 \, \sqrt{2^{x} b + a}{\left (2^{x} b - 2 \, a\right )}}{3 \, b^{2} \log \left (2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.969486, size = 56, normalized size = 1.27 \begin{align*} \begin{cases} \frac{2 \cdot 2^{x} \sqrt{2^{x} b + a}}{3 b \log{\left (2 \right )}} - \frac{4 a \sqrt{2^{x} b + a}}{3 b^{2} \log{\left (2 \right )}} & \text{for}\: b \neq 0 \\\frac{4^{x}}{2 \sqrt{a} \log{\left (2 \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{4^{x}}{\sqrt{2^{x} b + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]