Optimal. Leaf size=70 \[ -\frac {2}{3} \left (-x-\sqrt {x}+1\right )^{3/2}-\frac {1}{4} \left (2 \sqrt {x}+1\right ) \sqrt {-x-\sqrt {x}+1}-\frac {5}{8} \sin ^{-1}\left (\frac {2 \sqrt {x}+1}{\sqrt {5}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1341, 640, 612, 619, 216} \[ -\frac {2}{3} \left (-x-\sqrt {x}+1\right )^{3/2}-\frac {1}{4} \left (2 \sqrt {x}+1\right ) \sqrt {-x-\sqrt {x}+1}-\frac {5}{8} \sin ^{-1}\left (\frac {2 \sqrt {x}+1}{\sqrt {5}}\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 612
Rule 619
Rule 640
Rule 1341
Rubi steps
\begin {align*} \int \sqrt {1-\sqrt {x}-x} \, dx &=2 \operatorname {Subst}\left (\int x \sqrt {1-x-x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}-\operatorname {Subst}\left (\int \sqrt {1-x-x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}-\frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}-\frac {5}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x-x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}-\frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}+\frac {1}{8} \sqrt {5} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{5}}} \, dx,x,-1-2 \sqrt {x}\right )\\ &=-\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}-\frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}-\frac {5}{8} \sin ^{-1}\left (\frac {1+2 \sqrt {x}}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.76 \[ \frac {1}{12} \sqrt {-x-\sqrt {x}+1} \left (8 x+2 \sqrt {x}-11\right )+\frac {5}{8} \sin ^{-1}\left (\frac {-2 \sqrt {x}-1}{\sqrt {5}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.35, size = 84, normalized size = 1.20 \[ \frac {1}{12} \, {\left (8 \, x + 2 \, \sqrt {x} - 11\right )} \sqrt {-x - \sqrt {x} + 1} + \frac {5}{16} \, \arctan \left (-\frac {{\left (8 \, x^{2} - {\left (16 \, x^{2} - 38 \, x + 11\right )} \sqrt {x} - 9 \, x + 3\right )} \sqrt {-x - \sqrt {x} + 1}}{4 \, {\left (4 \, x^{3} - 13 \, x^{2} + 7 \, x - 1\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 44, normalized size = 0.63 \[ \frac {1}{12} \, {\left (2 \, \sqrt {x} {\left (4 \, \sqrt {x} + 1\right )} - 11\right )} \sqrt {-x - \sqrt {x} + 1} - \frac {5}{8} \, \arcsin \left (\frac {1}{5} \, \sqrt {5} {\left (2 \, \sqrt {x} + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 50, normalized size = 0.71 \[ -\frac {5 \arcsin \left (\frac {2 \sqrt {5}\, \left (\sqrt {x}+\frac {1}{2}\right )}{5}\right )}{8}-\frac {2 \left (-x -\sqrt {x}+1\right )^{\frac {3}{2}}}{3}+\frac {\left (-2 \sqrt {x}-1\right ) \sqrt {-x -\sqrt {x}+1}}{4} \]
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 \[ \int \sqrt {1-\sqrt {x}-x} \,d x \]
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 \[ \int \sqrt {- \sqrt {x} - x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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