Optimal. Leaf size=104 \[ -\frac {1}{8} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {1}{12} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}+\frac {1}{3} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}-\frac {1}{8} \cosh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A]
time = 0.03, 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 = {286, 329, 336,
54} \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 54
Rule 286
Rule 329
Rule 336
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} \text {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 = 1.15, size = 87, normalized size = 0.84 \begin {gather*} \frac {1}{24} \left (\sqrt {\frac {-1+\sqrt {x}}{1+\sqrt {x}}} \sqrt {x} \left (-3-3 \sqrt {x}-2 x-2 x^{3/2}+8 x^2+8 x^{5/2}\right )-6 \tanh ^{-1}\left (\sqrt {\frac {-1+\sqrt {x}}{1+\sqrt {x}}}\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.33, size = 65, normalized size = 0.62
method | result | size |
derivativedivides | \(-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {\sqrt {x}+1}\, \left (-8 x^{\frac {5}{2}} \sqrt {x -1}+2 x^{\frac {3}{2}} \sqrt {x -1}+3 \sqrt {x}\, \sqrt {x -1}+3 \ln \left (\sqrt {x}+\sqrt {x -1}\right )\right )}{24 \sqrt {x -1}}\) | \(65\) |
default | \(-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {\sqrt {x}+1}\, \left (-8 x^{\frac {5}{2}} \sqrt {x -1}+2 x^{\frac {3}{2}} \sqrt {x -1}+3 \sqrt {x}\, \sqrt {x -1}+3 \ln \left (\sqrt {x}+\sqrt {x -1}\right )\right )}{24 \sqrt {x -1}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, 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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Fricas [A]
time = 1.27, 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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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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Giac [A]
time = 2.84, 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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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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