Optimal. Leaf size=62 \[ \frac{2}{3} \left (x+\sqrt{x}+1\right )^{3/2}-\frac{1}{4} \left (2 \sqrt{x}+1\right ) \sqrt{x+\sqrt{x}+1}-\frac{3}{8} \sinh ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{3}}\right ) \]
[Out]
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Rubi [A] time = 0.0524468, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ \frac{2}{3} \left (x+\sqrt{x}+1\right )^{3/2}-\frac{1}{4} \left (2 \sqrt{x}+1\right ) \sqrt{x+\sqrt{x}+1}-\frac{3}{8} \sinh ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 + Sqrt[x] + x],x]
[Out]
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Rubi in Sympy [A] time = 2.05082, size = 63, normalized size = 1.02 \[ - \frac{\left (2 \sqrt{x} + 1\right ) \sqrt{\sqrt{x} + x + 1}}{4} + \frac{2 \left (\sqrt{x} + x + 1\right )^{\frac{3}{2}}}{3} - \frac{3 \operatorname{atanh}{\left (\frac{2 \sqrt{x} + 1}{2 \sqrt{\sqrt{x} + x + 1}} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+x+x**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0303024, size = 49, normalized size = 0.79 \[ \frac{1}{12} \sqrt{x+\sqrt{x}+1} \left (8 x+2 \sqrt{x}+5\right )-\frac{3}{8} \sinh ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 + Sqrt[x] + x],x]
[Out]
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Maple [A] time = 0.008, size = 42, normalized size = 0.7 \[{\frac{2}{3} \left ( 1+x+\sqrt{x} \right ) ^{{\frac{3}{2}}}}-{\frac{1}{4} \left ( 1+2\,\sqrt{x} \right ) \sqrt{1+x+\sqrt{x}}}-{\frac{3}{8}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( \sqrt{x}+{\frac{1}{2}} \right ) } \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+x+x^(1/2))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x + \sqrt{x} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + sqrt(x) + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.555843, size = 69, normalized size = 1.11 \[ \frac{1}{12} \,{\left (8 \, x + 2 \, \sqrt{x} + 5\right )} \sqrt{x + \sqrt{x} + 1} + \frac{3}{16} \, \log \left (4 \, \sqrt{x + \sqrt{x} + 1}{\left (2 \, \sqrt{x} + 1\right )} - 8 \, x - 8 \, \sqrt{x} - 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + sqrt(x) + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\sqrt{x} + x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+x+x**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.287548, size = 61, normalized size = 0.98 \[ \frac{1}{12} \,{\left (2 \, \sqrt{x}{\left (4 \, \sqrt{x} + 1\right )} + 5\right )} \sqrt{x + \sqrt{x} + 1} + \frac{3}{8} \,{\rm ln}\left (2 \, \sqrt{x + \sqrt{x} + 1} - 2 \, \sqrt{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + sqrt(x) + 1),x, algorithm="giac")
[Out]