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 = 0.03, antiderivative size = 94, normalized size of antiderivative = 1.00, 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 265
Rule 272
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.56, size = 44, normalized size = 0.47 \[ \frac {2 \, {\left (8 \, x^{4} + {\left (8 \, x^{3} + 4 \, x^{2} + 3 \, x - 15\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1}\right )}}{105 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 111, normalized size = 1.18 \[ \frac {4096 \, {\left (35 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{16} - 70 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{12} + 168 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{8} + 224 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 128\right )}}{105 \, {\left ({\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 4\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 33, normalized size = 0.35 \[ \frac {2 \sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (x -1\right ) \left (8 x^{2}+12 x +15\right )}{105 x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.55, size = 31, normalized size = 0.33 \[ \frac {16 \, {\left (x - 1\right )}^{\frac {3}{2}}}{105 \, x^{\frac {3}{2}}} + \frac {8 \, {\left (x - 1\right )}^{\frac {3}{2}}}{35 \, x^{\frac {5}{2}}} + \frac {2 \, {\left (x - 1\right )}^{\frac {3}{2}}}{7 \, x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.04, size = 55, normalized size = 0.59 \[ \frac {\sqrt {\sqrt {x}-1}\,\left (\frac {2\,x\,\sqrt {\sqrt {x}+1}}{35}-\frac {2\,\sqrt {\sqrt {x}+1}}{7}+\frac {8\,x^2\,\sqrt {\sqrt {x}+1}}{105}+\frac {16\,x^3\,\sqrt {\sqrt {x}+1}}{105}\right )}{x^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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