Optimal. Leaf size=104 \[ \frac {1}{3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}-\frac {1}{12} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}-\frac {1}{8} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\frac {1}{8} \cosh ^{-1}\left (\sqrt {x}\right ) \]
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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 = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {280, 323, 330, 52} \begin {gather*} \frac {1}{3} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}-\frac {1}{12} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}-\frac {1}{8} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\frac {1}{8} \cosh ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 280
Rule 323
Rule 330
Rubi steps
\begin {align*} \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2} \, dx &=\frac {1}{3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}-\frac {1}{6} \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {1}{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 {1}{8} \int \frac {\sqrt {x}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {1}{8} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {1}{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 {1}{16} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}} \, dx\\ &=-\frac {1}{8} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {1}{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 {1}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{8} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {1}{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 {1}{8} \cosh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 99, normalized size = 0.95 \begin {gather*} \frac {\sqrt {\sqrt {x}+1} \sqrt {x} \left (8 x^{5/2}-2 x^{3/2}-8 x^2+2 x-3 \sqrt {x}+3\right )+6 \sqrt {1-\sqrt {x}} \sin ^{-1}\left (\frac {\sqrt {1-\sqrt {x}}}{\sqrt {2}}\right )}{24 \sqrt {\sqrt {x}-1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.30, size = 176, normalized size = 1.69 \begin {gather*} \frac {\left (\frac {3 \left (\sqrt {x}-1\right )^5}{\left (\sqrt {x}+1\right )^5}+\frac {47 \left (\sqrt {x}-1\right )^4}{\left (\sqrt {x}+1\right )^4}+\frac {78 \left (\sqrt {x}-1\right )^3}{\left (\sqrt {x}+1\right )^3}+\frac {78 \left (\sqrt {x}-1\right )^2}{\left (\sqrt {x}+1\right )^2}+\frac {47 \left (\sqrt {x}-1\right )}{\sqrt {x}+1}+3\right ) \sqrt {\sqrt {x}-1}}{12 \left (\frac {\sqrt {x}-1}{\sqrt {x}+1}-1\right )^6 \sqrt {\sqrt {x}+1}}-\frac {1}{4} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x}-1}}{\sqrt {\sqrt {x}+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 57, normalized size = 0.55 \begin {gather*} \frac {1}{24} \, {\left (8 \, x^{2} - 2 \, x - 3\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {1}{16} \, \log \left (2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 127, normalized size = 1.22 \begin {gather*} \frac {1}{120} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, \sqrt {x} - 26\right )} {\left (\sqrt {x} + 1\right )} + 321\right )} {\left (\sqrt {x} + 1\right )} - 451\right )} {\left (\sqrt {x} + 1\right )} + 745\right )} {\left (\sqrt {x} + 1\right )} - 405\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {1}{60} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, \sqrt {x} - 17\right )} {\left (\sqrt {x} + 1\right )} + 133\right )} {\left (\sqrt {x} + 1\right )} - 295\right )} {\left (\sqrt {x} + 1\right )} + 195\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {1}{4} \, \log \left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 65, normalized size = 0.62 \begin {gather*} -\frac {\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (-8 \sqrt {x -1}\, x^{\frac {5}{2}}+2 \sqrt {x -1}\, x^{\frac {3}{2}}+3 \ln \left (\sqrt {x}+\sqrt {x -1}\right )+3 \sqrt {x -1}\, \sqrt {x}\right )}{24 \sqrt {x -1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 47, normalized size = 0.45 \begin {gather*} \frac {1}{3} \, {\left (x - 1\right )}^{\frac {3}{2}} x^{\frac {3}{2}} + \frac {1}{4} \, {\left (x - 1\right )}^{\frac {3}{2}} \sqrt {x} + \frac {1}{8} \, \sqrt {x - 1} \sqrt {x} - \frac {1}{8} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 31.39, size = 632, normalized size = 6.08 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {\sqrt {\sqrt {x}-1}-\mathrm {i}}{\sqrt {\sqrt {x}+1}-1}\right )}{2}-\frac {\frac {35\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^3}{6\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^3}+\frac {757\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^5}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^5}+\frac {7339\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^7}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^7}+\frac {41929\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^9}{3\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^9}+\frac {25661\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{11}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{11}}+\frac {25661\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{13}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{13}}+\frac {41929\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{15}}{3\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{15}}+\frac {7339\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{17}}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{17}}+\frac {757\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{19}}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{19}}+\frac {35\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{21}}{6\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{21}}-\frac {{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{23}}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^{23}}-\frac {\sqrt {\sqrt {x}-1}-\mathrm {i}}{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}} \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 x^{\frac {3}{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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