Optimal. Leaf size=104 \[ \frac {1}{3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}+\frac {5}{12} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}+\frac {5}{8} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}+\frac {5}{8} \cosh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {323, 330, 52} \[ \frac {1}{3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}+\frac {5}{12} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}+\frac {5}{8} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}+\frac {5}{8} \cosh ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 52
Rule 323
Rule 330
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx &=\frac {1}{3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {5}{6} \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=\frac {5}{12} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}+\frac {1}{3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {5}{8} \int \frac {\sqrt {x}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=\frac {5}{8} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+\frac {5}{12} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}+\frac {1}{3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {5}{16} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}} \, dx\\ &=\frac {5}{8} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+\frac {5}{12} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}+\frac {1}{3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {5}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,\sqrt {x}\right )\\ &=\frac {5}{8} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+\frac {5}{12} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}+\frac {1}{3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {5}{8} \cosh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 67, normalized size = 0.64 \[ \frac {1}{24} \left (\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x} \left (8 x^2+10 x+15\right )+30 \tanh ^{-1}\left (\sqrt {\frac {\sqrt {x}-1}{\sqrt {x}+1}}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.57, size = 57, normalized size = 0.55 \[ \frac {1}{24} \, {\left (8 \, x^{2} + 10 \, x + 15\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - \frac {5}{16} \, \log \left (2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 76, normalized size = 0.73 \[ \frac {1}{24} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (\sqrt {x} + 1\right )} {\left (\sqrt {x} - 4\right )} + 45\right )} {\left (\sqrt {x} + 1\right )} - 55\right )} {\left (\sqrt {x} + 1\right )} + 85\right )} {\left (\sqrt {x} + 1\right )} - 33\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - \frac {5}{4} \, \log \left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 65, normalized size = 0.62 \[ \frac {\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (8 \sqrt {x -1}\, x^{\frac {5}{2}}+10 \sqrt {x -1}\, x^{\frac {3}{2}}+15 \ln \left (\sqrt {x}+\sqrt {x -1}\right )+15 \sqrt {x -1}\, \sqrt {x}\right )}{24 \sqrt {x -1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 47, normalized size = 0.45 \[ \frac {1}{3} \, \sqrt {x - 1} x^{\frac {5}{2}} + \frac {5}{12} \, \sqrt {x - 1} x^{\frac {3}{2}} + \frac {5}{8} \, \sqrt {x - 1} \sqrt {x} + \frac {5}{8} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 27.09, size = 632, normalized size = 6.08 \[ \frac {5\,\mathrm {atanh}\left (\frac {\sqrt {\sqrt {x}-1}-\mathrm {i}}{\sqrt {\sqrt {x}+1}-1}\right )}{2}-\frac {-\frac {175\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^3}{6\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^3}+\frac {311\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^5}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^5}+\frac {8361\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^7}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^7}+\frac {42259\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^9}{3\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^9}+\frac {25295\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{11}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{11}}+\frac {25295\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{13}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{13}}+\frac {42259\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{15}}{3\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{15}}+\frac {8361\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{17}}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{17}}+\frac {311\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{19}}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{19}}-\frac {175\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{21}}{6\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{21}}+\frac {5\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{23}}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{23}}+\frac {5\,\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}{2\,\left (\sqrt {\sqrt {x}+1}-1\right )}}{1+\frac {66\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^4}-\frac {220\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^6}+\frac {495\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^8}-\frac {792\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{10}}+\frac {924\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{12}}-\frac {792\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{14}}+\frac {495\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{16}}-\frac {220\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{18}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{18}}+\frac {66\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{20}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{20}}-\frac {12\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{22}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{22}}+\frac {{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{24}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{24}}-\frac {12\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^2}} \]
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 \[ \int \frac {x^{\frac {5}{2}}}{\sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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