3.3061 \(\int \sqrt{\sqrt{\frac{1}{x}}+\frac{1}{x}} \, dx\)

Optimal. Leaf size=26 \[ \frac{4 \left (\sqrt{\frac{1}{x}}+\frac{1}{x}\right )^{3/2}}{3 \left (\frac{1}{x}\right )^{3/2}} \]

[Out]

(4*(Sqrt[x^(-1)] + x^(-1))^(3/2))/(3*(x^(-1))^(3/2))

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Rubi [A]  time = 0.0429081, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{4 \left (\sqrt{\frac{1}{x}}+\frac{1}{x}\right )^{3/2}}{3 \left (\frac{1}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[Sqrt[x^(-1)] + x^(-1)],x]

[Out]

(4*(Sqrt[x^(-1)] + x^(-1))^(3/2))/(3*(x^(-1))^(3/2))

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Rubi in Sympy [A]  time = 5.27735, size = 22, normalized size = 0.85 \[ \frac{4 \left (\sqrt{\frac{1}{x}} + \frac{1}{x}\right )^{\frac{3}{2}}}{3 \left (\frac{1}{x}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1/x+(1/x)**(1/2))**(1/2),x)

[Out]

4*(sqrt(1/x) + 1/x)**(3/2)/(3*(1/x)**(3/2))

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Mathematica [A]  time = 0.0444018, size = 0, normalized size = 0. \[ \int \sqrt{\sqrt{\frac{1}{x}}+\frac{1}{x}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[Sqrt[x^(-1)] + x^(-1)],x]

[Out]

Integrate[Sqrt[Sqrt[x^(-1)] + x^(-1)], x]

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Maple [A]  time = 0.029, size = 32, normalized size = 1.2 \[{\frac{4}{3}\sqrt{{\frac{1}{x} \left ( \sqrt{{x}^{-1}}x+1 \right ) }} \left ( \sqrt{{x}^{-1}}x+1 \right ){\frac{1}{\sqrt{{x}^{-1}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1/x+(1/x)^(1/2))^(1/2),x)

[Out]

4/3*(((1/x)^(1/2)*x+1)/x)^(1/2)*((1/x)^(1/2)*x+1)/(1/x)^(1/2)

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Maxima [A]  time = 1.37492, size = 12, normalized size = 0.46 \[ \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(1/sqrt(x) + 1/x),x, algorithm="maxima")

[Out]

4/3*(sqrt(x) + 1)^(3/2)

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Fricas [A]  time = 0.243532, size = 28, normalized size = 1.08 \[ \frac{4 \,{\left (x^{\frac{3}{2}} + x\right )} \sqrt{\frac{x + \sqrt{x}}{x^{\frac{3}{2}}}}}{3 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(1/sqrt(x) + 1/x),x, algorithm="fricas")

[Out]

4/3*(x^(3/2) + x)*sqrt((x + sqrt(x))/x^(3/2))/sqrt(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\sqrt{\frac{1}{x}} + \frac{1}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1/x+(1/x)**(1/2))**(1/2),x)

[Out]

Integral(sqrt(sqrt(1/x) + 1/x), x)

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GIAC/XCAS [A]  time = 0.213938, size = 15, normalized size = 0.58 \[ \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} - \frac{4}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(1/sqrt(x) + 1/x),x, algorithm="giac")

[Out]

4/3*(sqrt(x) + 1)^(3/2) - 4/3