Optimal. Leaf size=125 \[ \frac {32 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{315 x^{3/2}}+\frac {16 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{105 x^{5/2}}+\frac {4 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{21 x^{7/2}}+\frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{9 x^{9/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {272, 265} \[ \frac {32 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{315 x^{3/2}}+\frac {16 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{105 x^{5/2}}+\frac {4 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{21 x^{7/2}}+\frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{9 x^{9/2}} \]
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^{11/2}} \, dx &=\frac {2 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{9 x^{9/2}}+\frac {2}{3} \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}}{9 x^{9/2}}+\frac {4 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{21 x^{7/2}}+\frac {8}{21} \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}}{9 x^{9/2}}+\frac {4 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{21 x^{7/2}}+\frac {16 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{105 x^{5/2}}+\frac {16}{105} \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}}{9 x^{9/2}}+\frac {4 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{21 x^{7/2}}+\frac {16 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{105 x^{5/2}}+\frac {32 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{315 x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 0.37 \[ \frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2} \left (16 x^3+24 x^2+30 x+35\right )}{315 x^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 49, normalized size = 0.39 \[ \frac {2 \, {\left (16 \, x^{5} + {\left (16 \, x^{4} + 8 \, x^{3} + 6 \, x^{2} + 5 \, x - 35\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1}\right )}}{315 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 132, normalized size = 1.06 \[ \frac {16384 \, {\left (315 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{20} - 756 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{16} + 1344 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{12} + 2304 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{8} + 2304 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 1024\right )}}{315 \, {\left ({\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 4\right )}^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 38, normalized size = 0.30 \[ \frac {2 \sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (x -1\right ) \left (16 x^{3}+24 x^{2}+30 x +35\right )}{315 x^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 41, normalized size = 0.33 \[ \frac {32 \, {\left (x - 1\right )}^{\frac {3}{2}}}{315 \, x^{\frac {3}{2}}} + \frac {16 \, {\left (x - 1\right )}^{\frac {3}{2}}}{105 \, x^{\frac {5}{2}}} + \frac {4 \, {\left (x - 1\right )}^{\frac {3}{2}}}{21 \, x^{\frac {7}{2}}} + \frac {2 \, {\left (x - 1\right )}^{\frac {3}{2}}}{9 \, x^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.02, size = 67, normalized size = 0.54 \[ \frac {\sqrt {\sqrt {x}-1}\,\left (\frac {2\,x\,\sqrt {\sqrt {x}+1}}{63}-\frac {2\,\sqrt {\sqrt {x}+1}}{9}+\frac {4\,x^2\,\sqrt {\sqrt {x}+1}}{105}+\frac {16\,x^3\,\sqrt {\sqrt {x}+1}}{315}+\frac {32\,x^4\,\sqrt {\sqrt {x}+1}}{315}\right )}{x^{9/2}} \]
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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