Optimal. Leaf size=95 \[ -\frac{1}{2} \sqrt{x} \left (-x-\sqrt{x}+1\right )^{3/2}+\frac{5}{12} \left (-x-\sqrt{x}+1\right )^{3/2}+\frac{9}{32} \left (2 \sqrt{x}+1\right ) \sqrt{-x-\sqrt{x}+1}+\frac{45}{64} \sin ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{5}}\right ) \]
[Out]
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Rubi [A] time = 0.111151, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{1}{2} \sqrt{x} \left (-x-\sqrt{x}+1\right )^{3/2}+\frac{5}{12} \left (-x-\sqrt{x}+1\right )^{3/2}+\frac{9}{32} \left (2 \sqrt{x}+1\right ) \sqrt{-x-\sqrt{x}+1}+\frac{45}{64} \sin ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{5}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - Sqrt[x] - x]*Sqrt[x],x]
[Out]
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Rubi in Sympy [A] time = 6.77057, size = 87, normalized size = 0.92 \[ - \frac{\sqrt{x} \left (- \sqrt{x} - x + 1\right )^{\frac{3}{2}}}{2} + \frac{9 \left (2 \sqrt{x} + 1\right ) \sqrt{- \sqrt{x} - x + 1}}{32} + \frac{5 \left (- \sqrt{x} - x + 1\right )^{\frac{3}{2}}}{12} + \frac{45 \operatorname{atan}{\left (- \frac{- 2 \sqrt{x} - 1}{2 \sqrt{- \sqrt{x} - x + 1}} \right )}}{64} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)*(1-x-x**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0483693, 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.
[In] Integrate[Sqrt[1 - Sqrt[x] - x]*Sqrt[x],x]
[Out]
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Maple [A] time = 0.005, size = 67, normalized size = 0.7 \[ -{\frac{1}{2} \left ( 1-x-\sqrt{x} \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{5}{12} \left ( 1-x-\sqrt{x} \right ) ^{{\frac{3}{2}}}}-{\frac{9}{32} \left ( -2\,\sqrt{x}-1 \right ) \sqrt{1-x-\sqrt{x}}}+{\frac{45}{64}\arcsin \left ({\frac{2\,\sqrt{5}}{5} \left ( \sqrt{x}+{\frac{1}{2}} \right ) } \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)*(1-x-x^(1/2))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x} \sqrt{-x - \sqrt{x} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)*sqrt(-x - sqrt(x) + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.22884, size = 89, normalized size = 0.94 \[ \frac{1}{96} \,{\left (2 \,{\left (24 \, x - 17\right )} \sqrt{x} + 8 \, x + 67\right )} \sqrt{-x - \sqrt{x} + 1} + \frac{45}{128} \, \arctan \left (\frac{8 \, x + 8 \, \sqrt{x} - 3}{4 \, \sqrt{-x - \sqrt{x} + 1}{\left (2 \, \sqrt{x} + 1\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)*sqrt(-x - sqrt(x) + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x} \sqrt{- \sqrt{x} - x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)*(1-x-x**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.268708, size = 69, normalized size = 0.73 \[ \frac{1}{96} \,{\left (2 \,{\left (4 \, \sqrt{x}{\left (6 \, \sqrt{x} + 1\right )} - 17\right )} \sqrt{x} + 67\right )} \sqrt{-x - \sqrt{x} + 1} + \frac{45}{64} \, \arcsin \left (\frac{1}{5} \, \sqrt{5}{\left (2 \, \sqrt{x} + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)*sqrt(-x - sqrt(x) + 1),x, algorithm="giac")
[Out]