Optimal. Leaf size=63 \[ \frac{4 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{15 x^{3/2}}+\frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{5 x^{5/2}} \]
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Rubi [A] time = 0.0203429, 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} \[ \frac{4 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{15 x^{3/2}}+\frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{5 x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 272
Rule 265
Rubi steps
\begin{align*} \int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{x^{7/2}} \, dx &=\frac{2 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{5 x^{5/2}}+\frac{2}{5} \int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{x^{5/2}} \, dx\\ &=\frac{2 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{5 x^{5/2}}+\frac{4 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{15 x^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0139119, size = 36, normalized size = 0.57 \[ \frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2} (2 x+3)}{15 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 28, normalized size = 0.4 \begin{align*}{\frac{ \left ( -2+2\,x \right ) \left ( 2\,x+3 \right ) }{15}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43155, size = 28, normalized size = 0.44 \begin{align*} \frac{4 \,{\left (x - 1\right )}^{\frac{3}{2}}}{15 \, x^{\frac{3}{2}}} + \frac{2 \,{\left (x - 1\right )}^{\frac{3}{2}}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.994124, size = 108, normalized size = 1.71 \begin{align*} \frac{2 \,{\left (2 \, x^{3} +{\left (2 \, x^{2} + x - 3\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1}\right )}}{15 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11736, size = 122, normalized size = 1.94 \begin{align*} \frac{128 \,{\left (15 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{12} - 20 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{8} + 80 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 64\right )}}{15 \,{\left ({\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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