3.536 \(\int \sqrt{\sqrt{x}+x} \, dx\)

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]

-Sqrt[Sqrt[x] + x]/4 + (Sqrt[x]*Sqrt[Sqrt[x] + x])/6 + (2*x*Sqrt[Sqrt[x] + x])/3
+ ArcTanh[Sqrt[x]/Sqrt[Sqrt[x] + x]]/4

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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]

-Sqrt[Sqrt[x] + x]/4 + (Sqrt[x]*Sqrt[Sqrt[x] + x])/6 + (2*x*Sqrt[Sqrt[x] + x])/3
+ ArcTanh[Sqrt[x]/Sqrt[Sqrt[x] + x]]/4

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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]

sqrt(x)*sqrt(sqrt(x) + x)/6 + 2*x*sqrt(sqrt(x) + x)/3 - sqrt(sqrt(x) + x)/4 + at
anh(sqrt(x)/sqrt(sqrt(x) + x))/4

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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]

(Sqrt[Sqrt[x] + x]*(-3 + 2*Sqrt[x] + 8*x))/12 + Log[1 + 2*Sqrt[x] + 2*Sqrt[Sqrt[
x] + x]]/8

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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]

2/3*(x+x^(1/2))^(3/2)-1/4*(1+2*x^(1/2))*(x+x^(1/2))^(1/2)+1/8*ln(1/2+x^(1/2)+(x+
x^(1/2))^(1/2))

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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]

-1/12*(3*(sqrt(x) + 1)^(5/2)/x^(5/4) - 8*(sqrt(x) + 1)^(3/2)/x^(3/4) - 3*sqrt(sq
rt(x) + 1)/x^(1/4))/((sqrt(x) + 1)^3/x^(3/2) - 3*(sqrt(x) + 1)^2/x + 3*(sqrt(x)
+ 1)/sqrt(x) - 1) + 1/8*log(sqrt(sqrt(x) + 1)/x^(1/4) + 1) - 1/8*log(sqrt(sqrt(x
) + 1)/x^(1/4) - 1)

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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]

1/12*(8*x + 2*sqrt(x) - 3)*sqrt(x + sqrt(x)) + 1/16*log(4*sqrt(x + sqrt(x))*(2*s
qrt(x) + 1) + 8*x + 8*sqrt(x) + 1)

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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]

Integral(sqrt(sqrt(x) + x), x)

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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]

Timed out