Optimal. Leaf size=29 \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{\sqrt{x}} \]
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Rubi [A] time = 0.0098801, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {265} \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 265
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}} \, dx &=\frac{2 \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{\sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0091006, size = 29, normalized size = 1. \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 20, normalized size = 0.7 \begin{align*} 2\,{\frac{\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}}{\sqrt{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42919, size = 14, normalized size = 0.48 \begin{align*} \frac{2 \, \sqrt{x - 1}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.01802, size = 74, normalized size = 2.55 \begin{align*} \frac{2 \,{\left (\sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + x\right )}}{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 \frac{1}{x^{\frac{3}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.094, size = 34, normalized size = 1.17 \begin{align*} \frac{16}{{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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