Optimal. Leaf size=95 \[ -\frac {1}{2} \sqrt {x} \left (-x-\sqrt {x}+1\right )^{3/2}+\frac {5}{12} \left (-x-\sqrt {x}+1\right )^{3/2}+\frac {9}{32} \left (2 \sqrt {x}+1\right ) \sqrt {-x-\sqrt {x}+1}+\frac {45}{64} \sin ^{-1}\left (\frac {2 \sqrt {x}+1}{\sqrt {5}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1357, 742, 640, 612, 619, 216} \[ -\frac {1}{2} \sqrt {x} \left (-x-\sqrt {x}+1\right )^{3/2}+\frac {5}{12} \left (-x-\sqrt {x}+1\right )^{3/2}+\frac {9}{32} \left (2 \sqrt {x}+1\right ) \sqrt {-x-\sqrt {x}+1}+\frac {45}{64} \sin ^{-1}\left (\frac {2 \sqrt {x}+1}{\sqrt {5}}\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 612
Rule 619
Rule 640
Rule 742
Rule 1357
Rubi steps
\begin {align*} \int \sqrt {1-\sqrt {x}-x} \sqrt {x} \, dx &=2 \operatorname {Subst}\left (\int x^2 \sqrt {1-x-x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{2} \left (1-\sqrt {x}-x\right )^{3/2} \sqrt {x}-\frac {1}{2} \operatorname {Subst}\left (\int \left (-1+\frac {5 x}{2}\right ) \sqrt {1-x-x^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {5}{12} \left (1-\sqrt {x}-x\right )^{3/2}-\frac {1}{2} \left (1-\sqrt {x}-x\right )^{3/2} \sqrt {x}+\frac {9}{8} \operatorname {Subst}\left (\int \sqrt {1-x-x^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {9}{32} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}+\frac {5}{12} \left (1-\sqrt {x}-x\right )^{3/2}-\frac {1}{2} \left (1-\sqrt {x}-x\right )^{3/2} \sqrt {x}+\frac {45}{64} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x-x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {9}{32} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}+\frac {5}{12} \left (1-\sqrt {x}-x\right )^{3/2}-\frac {1}{2} \left (1-\sqrt {x}-x\right )^{3/2} \sqrt {x}-\frac {1}{64} \left (9 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{5}}} \, dx,x,-1-2 \sqrt {x}\right )\\ &=\frac {9}{32} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}+\frac {5}{12} \left (1-\sqrt {x}-x\right )^{3/2}-\frac {1}{2} \left (1-\sqrt {x}-x\right )^{3/2} \sqrt {x}+\frac {45}{64} \sin ^{-1}\left (\frac {1+2 \sqrt {x}}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 60, normalized size = 0.63 \[ \frac {1}{96} \sqrt {-x-\sqrt {x}+1} \left (48 x^{3/2}+8 x-34 \sqrt {x}+67\right )-\frac {45}{64} \sin ^{-1}\left (\frac {-2 \sqrt {x}-1}{\sqrt {5}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 2.32, size = 89, normalized size = 0.94 \[ \frac {1}{96} \, {\left (2 \, {\left (24 \, x - 17\right )} \sqrt {x} + 8 \, x + 67\right )} \sqrt {-x - \sqrt {x} + 1} - \frac {45}{128} \, \arctan \left (-\frac {{\left (8 \, x^{2} - {\left (16 \, x^{2} - 38 \, x + 11\right )} \sqrt {x} - 9 \, x + 3\right )} \sqrt {-x - \sqrt {x} + 1}}{4 \, {\left (4 \, x^{3} - 13 \, x^{2} + 7 \, x - 1\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 51, normalized size = 0.54 \[ \frac {1}{96} \, {\left (2 \, {\left (4 \, \sqrt {x} {\left (6 \, \sqrt {x} + 1\right )} - 17\right )} \sqrt {x} + 67\right )} \sqrt {-x - \sqrt {x} + 1} + \frac {45}{64} \, \arcsin \left (\frac {1}{5} \, \sqrt {5} {\left (2 \, \sqrt {x} + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 67, normalized size = 0.71 \[ \frac {45 \arcsin \left (\frac {2 \sqrt {5}\, \left (\sqrt {x}+\frac {1}{2}\right )}{5}\right )}{64}-\frac {\left (-x -\sqrt {x}+1\right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {5 \left (-x -\sqrt {x}+1\right )^{\frac {3}{2}}}{12}-\frac {9 \left (-2 \sqrt {x}-1\right ) \sqrt {-x -\sqrt {x}+1}}{32} \]
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 \[ \int \sqrt {x} \sqrt {-x - \sqrt {x} + 1}\,{d x} \]
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 \[ \int \sqrt {x}\,\sqrt {1-\sqrt {x}-x} \,d x \]
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 \[ \int \sqrt {x} \sqrt {- \sqrt {x} - x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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