Optimal. Leaf size=94 \[ \frac {8 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{15 x^{3/2}}+\frac {2 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{5 x^{5/2}}+\frac {16 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{15 \sqrt {x}} \]
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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} \begin {gather*} \frac {8 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{15 x^{3/2}}+\frac {2 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{5 x^{5/2}}+\frac {16 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{15 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 265
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}} \, dx &=\frac {2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{5 x^{5/2}}+\frac {4}{5} \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}}}{5 x^{5/2}}+\frac {8 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{15 x^{3/2}}+\frac {8}{15} \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}}}{5 x^{5/2}}+\frac {8 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{15 x^{3/2}}+\frac {16 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{15 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 0.44 \begin {gather*} \frac {2 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \left (8 x^2+4 x+3\right )}{15 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.37, size = 127, normalized size = 1.35 \begin {gather*} \frac {4 \left (\frac {15 \left (\sqrt {x}-1\right )^4}{\left (\sqrt {x}+1\right )^4}+\frac {20 \left (\sqrt {x}-1\right )^3}{\left (\sqrt {x}+1\right )^3}+\frac {58 \left (\sqrt {x}-1\right )^2}{\left (\sqrt {x}+1\right )^2}+\frac {20 \left (\sqrt {x}-1\right )}{\sqrt {x}+1}+15\right ) \sqrt {\sqrt {x}-1}}{15 \left (\frac {\sqrt {x}-1}{\sqrt {x}+1}+1\right )^5 \sqrt {\sqrt {x}+1}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 39, normalized size = 0.41 \begin {gather*} \frac {2 \, {\left (8 \, x^{3} + {\left (8 \, x^{2} + 4 \, x + 3\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1}\right )}}{15 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 69, normalized size = 0.73 \begin {gather*} \frac {4096 \, {\left (5 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{8} + 10 \, {\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 8\right )}}{15 \, {\left ({\left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right )}^{4} + 4\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 30, normalized size = 0.32 \begin {gather*} \frac {2 \sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (8 x^{2}+4 x +3\right )}{15 x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 31, normalized size = 0.33 \begin {gather*} \frac {16 \, \sqrt {x - 1}}{15 \, \sqrt {x}} + \frac {8 \, \sqrt {x - 1}}{15 \, x^{\frac {3}{2}}} + \frac {2 \, \sqrt {x - 1}}{5 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.66, size = 43, normalized size = 0.46 \begin {gather*} \frac {\sqrt {\sqrt {x}-1}\,\left (\frac {8\,x}{15}+\frac {16\,x^2}{15}+\frac {2\,\sqrt {x}}{5}+\frac {8\,x^{3/2}}{15}+\frac {16\,x^{5/2}}{15}+\frac {2}{5}\right )}{x^{5/2}\,\sqrt {\sqrt {x}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{\frac {7}{2}} \sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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