Optimal. Leaf size=26 \[ \frac{4 \left (\sqrt{\frac{1}{x}}+\frac{1}{x}\right )^{3/2}}{3 \left (\frac{1}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.0259805, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1966, 2014} \[ \frac{4 \left (\sqrt{\frac{1}{x}}+\frac{1}{x}\right )^{3/2}}{3 \left (\frac{1}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1966
Rule 2014
Rubi steps
\begin{align*} \int \sqrt{\sqrt{\frac{1}{x}}+\frac{1}{x}} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{\sqrt{x}+x}}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4 \left (\sqrt{\frac{1}{x}}+\frac{1}{x}\right )^{3/2}}{3 \left (\frac{1}{x}\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0175556, size = 26, normalized size = 1. \[ \frac{4 \left (\sqrt{\frac{1}{x}}+\frac{1}{x}\right )^{3/2}}{3 \left (\frac{1}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.104, size = 32, normalized size = 1.2 \begin{align*}{\frac{4}{3}\sqrt{{\frac{1}{x} \left ( \sqrt{{x}^{-1}}x+1 \right ) }} \left ( \sqrt{{x}^{-1}}x+1 \right ){\frac{1}{\sqrt{{x}^{-1}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{1}{\sqrt{x}} + \frac{1}{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.44551, size = 55, normalized size = 2.12 \begin{align*} \frac{4}{3} \,{\left (x + \sqrt{x}\right )} \sqrt{\frac{\sqrt{x} + 1}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sqrt{\frac{1}{x}} + \frac{1}{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09235, size = 15, normalized size = 0.58 \begin{align*} \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} - \frac{4}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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