Optimal. Leaf size=135 \[ \frac {1}{4} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{7/2}-\frac {1}{24} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}-\frac {5}{96} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}-\frac {5}{64} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\frac {5}{64} \cosh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 6, 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}{4} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{7/2}-\frac {1}{24} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}-\frac {5}{96} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}-\frac {5}{64} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\frac {5}{64} \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^{5/2} \, dx &=\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {1}{8} \int \frac {x^{5/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5}{48} \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {5}{96} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5}{64} \int \frac {\sqrt {x}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {5}{64} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{96} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5}{128} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}} \, dx\\ &=-\frac {5}{64} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{96} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5}{64} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {5}{64} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{96} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5}{64} \cosh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 111, normalized size = 0.82 \begin {gather*} \frac {\sqrt {\sqrt {x}+1} \sqrt {x} \left (48 x^{7/2}-8 x^{5/2}-10 x^{3/2}-48 x^3+8 x^2+10 x-15 \sqrt {x}+15\right )+30 \sqrt {1-\sqrt {x}} \sin ^{-1}\left (\frac {\sqrt {1-\sqrt {x}}}{\sqrt {2}}\right )}{192 \sqrt {\sqrt {x}-1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.44, size = 216, normalized size = 1.60 \begin {gather*} \frac {\left (\frac {15 \left (\sqrt {x}-1\right )^7}{\left (\sqrt {x}+1\right )^7}+\frac {397 \left (\sqrt {x}-1\right )^6}{\left (\sqrt {x}+1\right )^6}+\frac {895 \left (\sqrt {x}-1\right )^5}{\left (\sqrt {x}+1\right )^5}+\frac {1765 \left (\sqrt {x}-1\right )^4}{\left (\sqrt {x}+1\right )^4}+\frac {1765 \left (\sqrt {x}-1\right )^3}{\left (\sqrt {x}+1\right )^3}+\frac {895 \left (\sqrt {x}-1\right )^2}{\left (\sqrt {x}+1\right )^2}+\frac {397 \left (\sqrt {x}-1\right )}{\sqrt {x}+1}+15\right ) \sqrt {\sqrt {x}-1}}{96 \left (\frac {\sqrt {x}-1}{\sqrt {x}+1}-1\right )^8 \sqrt {\sqrt {x}+1}}-\frac {5}{32} \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 = 0.39, size = 62, normalized size = 0.46 \begin {gather*} \frac {1}{192} \, {\left (48 \, x^{3} - 8 \, x^{2} - 10 \, x - 15\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {5}{128} \, \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 = 0.29, size = 162, normalized size = 1.20 \begin {gather*} \frac {1}{6720} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, {\left (7 \, \sqrt {x} - 50\right )} {\left (\sqrt {x} + 1\right )} + 1219\right )} {\left (\sqrt {x} + 1\right )} - 12463\right )} {\left (\sqrt {x} + 1\right )} + 64233\right )} {\left (\sqrt {x} + 1\right )} - 53963\right )} {\left (\sqrt {x} + 1\right )} + 59465\right )} {\left (\sqrt {x} + 1\right )} - 23205\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {1}{840} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, \sqrt {x} - 37\right )} {\left (\sqrt {x} + 1\right )} + 661\right )} {\left (\sqrt {x} + 1\right )} - 4551\right )} {\left (\sqrt {x} + 1\right )} + 4781\right )} {\left (\sqrt {x} + 1\right )} - 6335\right )} {\left (\sqrt {x} + 1\right )} + 2835\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {5}{32} \, \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 = 75, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (-48 \sqrt {x -1}\, x^{\frac {7}{2}}+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 )}{192 \sqrt {x -1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 57, normalized size = 0.42 \begin {gather*} \frac {1}{4} \, {\left (x - 1\right )}^{\frac {3}{2}} x^{\frac {5}{2}} + \frac {5}{24} \, {\left (x - 1\right )}^{\frac {3}{2}} x^{\frac {3}{2}} + \frac {5}{32} \, {\left (x - 1\right )}^{\frac {3}{2}} \sqrt {x} + \frac {5}{64} \, \sqrt {x - 1} \sqrt {x} - \frac {5}{64} \, \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 = 52.03, size = 831, normalized size = 6.16
result too large to display
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 {5}{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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