Optimal. Leaf size=63 \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{3 x^{3/2}}+\frac{4 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{3 \sqrt{x}} \]
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Rubi [A] time = 0.0201345, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {272, 265} \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{3 x^{3/2}}+\frac{4 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{3 \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 272
Rule 265
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}} \, dx &=\frac{2 \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{3 x^{3/2}}+\frac{2}{3} \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}}}{3 x^{3/2}}+\frac{4 \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{3 \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.011744, size = 36, normalized size = 0.57 \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} (2 x+1)}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 25, normalized size = 0.4 \begin{align*}{\frac{4\,x+2}{3}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40682, size = 28, normalized size = 0.44 \begin{align*} \frac{4 \, \sqrt{x - 1}}{3 \, \sqrt{x}} + \frac{2 \, \sqrt{x - 1}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.00233, size = 99, normalized size = 1.57 \begin{align*} \frac{2 \,{\left ({\left (2 \, x + 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + 2 \, x^{2}\right )}}{3 \, x^{2}} \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{5}{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 = 1.06921, size = 65, normalized size = 1.03 \begin{align*} \frac{128 \,{\left (3 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4\right )}}{3 \,{\left ({\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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