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 = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}
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^{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*}
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Mathematica [A] time = 0.02, size = 36, normalized size = 0.57 \begin {gather*} \frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2} (2 x+3)}{15 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 10.09, size = 836, normalized size = 13.27 \begin {gather*} \frac {\left (\frac {\left (\sqrt {\sqrt {x}-1}-1\right )^2}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^2}+1\right ) \left (\frac {45 \left (\sqrt {\sqrt {x}-1}-1\right )^{16}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{16}}+\frac {300 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^{15}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{15}}+\frac {2960 \left (\sqrt {\sqrt {x}-1}-1\right )^{14}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{14}}+\frac {6620 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^{13}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{13}}+\frac {34252 \left (\sqrt {\sqrt {x}-1}-1\right )^{12}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{12}}+\frac {46980 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^{11}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{11}}+\frac {152688 \left (\sqrt {\sqrt {x}-1}-1\right )^{10}}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{10}}+\frac {129140 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^9}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^9}+\frac {254222 \left (\sqrt {\sqrt {x}-1}-1\right )^8}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^8}+\frac {129140 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^7}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^7}+\frac {152688 \left (\sqrt {\sqrt {x}-1}-1\right )^6}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^6}+\frac {46980 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^5}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^5}+\frac {34252 \left (\sqrt {\sqrt {x}-1}-1\right )^4}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^4}+\frac {6620 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )^3}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^3}+\frac {2960 \left (\sqrt {\sqrt {x}-1}-1\right )^2}{\left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^2}+\frac {300 \sqrt {3} \left (\sqrt {\sqrt {x}-1}-1\right )}{\sqrt {3}-\sqrt {\sqrt {x}+1}}+45\right ) \left (\frac {1}{122880}-\frac {\sqrt {\sqrt {x}-1}}{122880}\right ) \left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right )^{20}}{\left (\sqrt {\sqrt {x}+1}-\sqrt {3}\right ) \left (-2 x-2 \sqrt {\sqrt {x}-1} \sqrt {x}+\sqrt {3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}+2 \sqrt {3} \sqrt {\sqrt {x}+1} \sqrt {x}-3 \sqrt {x}\right )^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 37, normalized size = 0.59 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 90, normalized size = 1.43 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 28, normalized size = 0.44 \begin {gather*} \frac {2 \sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (x -1\right ) \left (2 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 = 1.31, size = 21, normalized size = 0.33 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.07, size = 43, normalized size = 0.68 \begin {gather*} \frac {\sqrt {\sqrt {x}-1}\,\left (\frac {2\,x\,\sqrt {\sqrt {x}+1}}{15}-\frac {2\,\sqrt {\sqrt {x}+1}}{5}+\frac {4\,x^2\,\sqrt {\sqrt {x}+1}}{15}\right )}{x^{5/2}} \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 {\sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}{x^{\frac {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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