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

Optimal. Leaf size=21 \[ \frac{2 x^{3/2}}{3}-\frac{2}{3} (x-1)^{3/2} \]

[Out]

(-2*(-1 + x)^(3/2))/3 + (2*x^(3/2))/3

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Rubi [A]  time = 0.0144696, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 x^{3/2}}{3}-\frac{2}{3} (x-1)^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(-1 + x)^(3/2))/3 + (2*x^(3/2))/3

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Rubi in Sympy [A]  time = 1.46981, size = 17, normalized size = 0.81 \[ \frac{2 x^{\frac{3}{2}}}{3} - \frac{2 \left (x - 1\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(3/2)/3 - 2*(x - 1)**(3/2)/3

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Mathematica [A]  time = 0.0234333, size = 19, normalized size = 0.9 \[ \frac{2}{3} \left (x^{3/2}-(x-1)^{3/2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*(-(-1 + x)^(3/2) + x^(3/2)))/3

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Maple [A]  time = 0.002, size = 14, normalized size = 0.7 \[ -{\frac{2}{3} \left ( -1+x \right ) ^{{\frac{3}{2}}}}+{\frac{2}{3}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((-1+x)^(1/2)+x^(1/2)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x - 1} + \sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x - 1) + sqrt(x)), x)

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Fricas [A]  time = 0.27261, size = 18, normalized size = 0.86 \[ -\frac{2}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} + \frac{2}{3} \, x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(x - 1)^(3/2) + 2/3*x^(3/2)

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Sympy [A]  time = 1.44117, size = 63, normalized size = 3. \[ \frac{2 \sqrt{x} \sqrt{x - 1}}{3 \sqrt{x} + 3 \sqrt{x - 1}} + \frac{4 x}{3 \sqrt{x} + 3 \sqrt{x - 1}} - \frac{2}{3 \sqrt{x} + 3 \sqrt{x - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.278179, size = 18, normalized size = 0.86 \[ -\frac{2}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} + \frac{2}{3} \, x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(x - 1)^(3/2) + 2/3*x^(3/2)