Optimal. Leaf size=94 \[ \frac{16 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{105 x^{3/2}}+\frac{8 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{35 x^{5/2}}+\frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{7 x^{7/2}} \]
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Rubi [A] time = 0.0312135, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {272, 265} \[ \frac{16 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{105 x^{3/2}}+\frac{8 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{35 x^{5/2}}+\frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{7 x^{7/2}} \]
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^{9/2}} \, dx &=\frac{2 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{7 x^{7/2}}+\frac{4}{7} \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}}{7 x^{7/2}}+\frac{8 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{35 x^{5/2}}+\frac{8}{35} \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}}{7 x^{7/2}}+\frac{8 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{35 x^{5/2}}+\frac{16 \left (-1+\sqrt{x}\right )^{3/2} \left (1+\sqrt{x}\right )^{3/2}}{105 x^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0162812, size = 41, normalized size = 0.44 \[ \frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2} \left (8 x^2+12 x+15\right )}{105 x^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 33, normalized size = 0.4 \begin{align*}{\frac{ \left ( -2+2\,x \right ) \left ( 8\,{x}^{2}+12\,x+15 \right ) }{105}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}{x}^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40464, size = 42, normalized size = 0.45 \begin{align*} \frac{16 \,{\left (x - 1\right )}^{\frac{3}{2}}}{105 \, x^{\frac{3}{2}}} + \frac{8 \,{\left (x - 1\right )}^{\frac{3}{2}}}{35 \, x^{\frac{5}{2}}} + \frac{2 \,{\left (x - 1\right )}^{\frac{3}{2}}}{7 \, x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.971025, size = 124, normalized size = 1.32 \begin{align*} \frac{2 \,{\left (8 \, x^{4} +{\left (8 \, x^{3} + 4 \, x^{2} + 3 \, x - 15\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1}\right )}}{105 \, x^{4}} \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 [A] time = 1.12485, size = 150, normalized size = 1.6 \begin{align*} \frac{4096 \,{\left (35 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{16} - 70 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{12} + 168 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{8} + 224 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 128\right )}}{105 \,{\left ({\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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