Optimal. Leaf size=74 \[ \frac{2}{3} \sqrt{x+\sqrt{x}} x+\frac{1}{6} \sqrt{x+\sqrt{x}} \sqrt{x}-\frac{\sqrt{x+\sqrt{x}}}{4}+\frac{1}{4} \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+\sqrt{x}}}\right ) \]
[Out]
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Rubi [A] time = 0.0846573, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546 \[ \frac{2}{3} \sqrt{x+\sqrt{x}} x+\frac{1}{6} \sqrt{x+\sqrt{x}} \sqrt{x}-\frac{\sqrt{x+\sqrt{x}}}{4}+\frac{1}{4} \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+\sqrt{x}}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[Sqrt[x] + x],x]
[Out]
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Rubi in Sympy [A] time = 4.9406, size = 61, normalized size = 0.82 \[ \frac{\sqrt{x} \sqrt{\sqrt{x} + x}}{6} + \frac{2 x \sqrt{\sqrt{x} + x}}{3} - \frac{\sqrt{\sqrt{x} + x}}{4} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{x}}{\sqrt{\sqrt{x} + x}} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x+x**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0247638, size = 55, normalized size = 0.74 \[ \frac{1}{12} \sqrt{x+\sqrt{x}} \left (8 x+2 \sqrt{x}-3\right )+\frac{1}{8} \log \left (2 \sqrt{x}+2 \sqrt{x+\sqrt{x}}+1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[Sqrt[x] + x],x]
[Out]
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Maple [A] time = 0.006, size = 42, normalized size = 0.6 \[{\frac{2}{3} \left ( x+\sqrt{x} \right ) ^{{\frac{3}{2}}}}-{\frac{1}{4} \left ( 1+2\,\sqrt{x} \right ) \sqrt{x+\sqrt{x}}}+{\frac{1}{8}\ln \left ({\frac{1}{2}}+\sqrt{x}+\sqrt{x+\sqrt{x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x+x^(1/2))^(1/2),x)
[Out]
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Maxima [A] time = 0.761675, size = 147, normalized size = 1.99 \[ -\frac{\frac{3 \,{\left (\sqrt{x} + 1\right )}^{\frac{5}{2}}}{x^{\frac{5}{4}}} - \frac{8 \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}}}{x^{\frac{3}{4}}} - \frac{3 \, \sqrt{\sqrt{x} + 1}}{x^{\frac{1}{4}}}}{12 \,{\left (\frac{{\left (\sqrt{x} + 1\right )}^{3}}{x^{\frac{3}{2}}} - \frac{3 \,{\left (\sqrt{x} + 1\right )}^{2}}{x} + \frac{3 \,{\left (\sqrt{x} + 1\right )}}{\sqrt{x}} - 1\right )}} + \frac{1}{8} \, \log \left (\frac{\sqrt{\sqrt{x} + 1}}{x^{\frac{1}{4}}} + 1\right ) - \frac{1}{8} \, \log \left (\frac{\sqrt{\sqrt{x} + 1}}{x^{\frac{1}{4}}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.541969, size = 66, normalized size = 0.89 \[ \frac{1}{12} \,{\left (8 \, x + 2 \, \sqrt{x} - 3\right )} \sqrt{x + \sqrt{x}} + \frac{1}{16} \, \log \left (4 \, \sqrt{x + \sqrt{x}}{\left (2 \, \sqrt{x} + 1\right )} + 8 \, x + 8 \, \sqrt{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + sqrt(x)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\sqrt{x} + x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x+x**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + sqrt(x)),x, algorithm="giac")
[Out]