Optimal. Leaf size=68 \[ \frac {2}{3} \left (x+\sqrt {x-1}\right )^{3/2}-\frac {1}{4} \left (2 \sqrt {x-1}+1\right ) \sqrt {x+\sqrt {x-1}}-\frac {3}{8} \sinh ^{-1}\left (\frac {2 \sqrt {x-1}+1}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {640, 612, 619, 215} \[ \frac {2}{3} \left (x+\sqrt {x-1}\right )^{3/2}-\frac {1}{4} \left (2 \sqrt {x-1}+1\right ) \sqrt {x+\sqrt {x-1}}-\frac {3}{8} \sinh ^{-1}\left (\frac {2 \sqrt {x-1}+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 215
Rule 612
Rule 619
Rule 640
Rubi steps
\begin {align*} \int \sqrt {\sqrt {-1+x}+x} \, dx &=2 \operatorname {Subst}\left (\int x \sqrt {1+x+x^2} \, dx,x,\sqrt {-1+x}\right )\\ &=\frac {2}{3} \left (\sqrt {-1+x}+x\right )^{3/2}-\operatorname {Subst}\left (\int \sqrt {1+x+x^2} \, dx,x,\sqrt {-1+x}\right )\\ &=-\frac {1}{4} \left (1+2 \sqrt {-1+x}\right ) \sqrt {\sqrt {-1+x}+x}+\frac {2}{3} \left (\sqrt {-1+x}+x\right )^{3/2}-\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x+x^2}} \, dx,x,\sqrt {-1+x}\right )\\ &=-\frac {1}{4} \left (1+2 \sqrt {-1+x}\right ) \sqrt {\sqrt {-1+x}+x}+\frac {2}{3} \left (\sqrt {-1+x}+x\right )^{3/2}-\frac {1}{8} \sqrt {3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 \sqrt {-1+x}\right )\\ &=-\frac {1}{4} \left (1+2 \sqrt {-1+x}\right ) \sqrt {\sqrt {-1+x}+x}+\frac {2}{3} \left (\sqrt {-1+x}+x\right )^{3/2}-\frac {3}{8} \sinh ^{-1}\left (\frac {1+2 \sqrt {-1+x}}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 54, normalized size = 0.79 \[ \frac {1}{24} \left (2 \sqrt {x+\sqrt {x-1}} \left (8 x+2 \sqrt {x-1}-3\right )-9 \sinh ^{-1}\left (\frac {2 \sqrt {x-1}+1}{\sqrt {3}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 59, normalized size = 0.87 \[ \frac {1}{12} \, {\left (8 \, x + 2 \, \sqrt {x - 1} - 3\right )} \sqrt {x + \sqrt {x - 1}} + \frac {3}{16} \, \log \left (-4 \, \sqrt {x + \sqrt {x - 1}} {\left (2 \, \sqrt {x - 1} + 1\right )} + 8 \, x + 8 \, \sqrt {x - 1} - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 53, normalized size = 0.78 \[ \frac {1}{12} \, {\left (2 \, \sqrt {x - 1} {\left (4 \, \sqrt {x - 1} + 1\right )} + 5\right )} \sqrt {x + \sqrt {x - 1}} + \frac {3}{8} \, \log \left (2 \, \sqrt {x + \sqrt {x - 1}} - 2 \, \sqrt {x - 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 0.71 \[ -\frac {3 \arcsinh \left (\frac {2 \sqrt {3}\, \left (\sqrt {x -1}+\frac {1}{2}\right )}{3}\right )}{8}+\frac {2 \left (x +\sqrt {x -1}\right )^{\frac {3}{2}}}{3}-\frac {\left (1+2 \sqrt {x -1}\right ) \sqrt {x +\sqrt {x -1}}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x + \sqrt {x - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {x+\sqrt {x-1}} \,d x \]
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 \[ \int \sqrt {x + \sqrt {x - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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