3.8 \(\int \frac{1}{\left (\sqrt{-1+x}+\sqrt{x}\right )^2 \sqrt{-1+x}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{4 x^{3/2}}{3}+\frac{4}{3} (x-1)^{3/2}+2 \sqrt{x-1} \]

[Out]

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

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Rubi [A]  time = 0.127849, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{4 x^{3/2}}{3}+\frac{4}{3} (x-1)^{3/2}+2 \sqrt{x-1} \]

Antiderivative was successfully verified.

[In]  Int[1/((Sqrt[-1 + x] + Sqrt[x])^2*Sqrt[-1 + x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.97118, size = 26, normalized size = 0.87 \[ - \frac{4 x^{\frac{3}{2}}}{3} + \frac{4 \left (x - 1\right )^{\frac{3}{2}}}{3} + 2 \sqrt{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0229325, size = 29, normalized size = 0.97 \[ \frac{2}{3} \left (-2 x^{3/2}+2 \sqrt{x-1} x+\sqrt{x-1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((Sqrt[-1 + x] + Sqrt[x])^2*Sqrt[-1 + x]),x]

[Out]

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

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Maple [A]  time = 0.004, size = 21, normalized size = 0.7 \[{\frac{4}{3} \left ( -1+x \right ) ^{{\frac{3}{2}}}}-{\frac{4}{3}{x}^{{\frac{3}{2}}}}+2\,\sqrt{-1+x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-1+x)^(1/2)/((-1+x)^(1/2)+x^(1/2))^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x - 1}{\left (\sqrt{x - 1} + \sqrt{x}\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x - 1)*(sqrt(x - 1) + sqrt(x))^2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x - 1)*(sqrt(x - 1) + sqrt(x))^2), x)

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Fricas [A]  time = 0.207534, size = 24, normalized size = 0.8 \[ \frac{2}{3} \,{\left (2 \, x + 1\right )} \sqrt{x - 1} - \frac{4}{3} \, x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x - 1)*(sqrt(x - 1) + sqrt(x))^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.50173, size = 53, normalized size = 1.77 \[ - \frac{4 \sqrt{x}}{6 \sqrt{x} \sqrt{x - 1} + 6 x - 3} - \frac{2 \sqrt{x - 1}}{6 \sqrt{x} \sqrt{x - 1} + 6 x - 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*sqrt(x)/(6*sqrt(x)*sqrt(x - 1) + 6*x - 3) - 2*sqrt(x - 1)/(6*sqrt(x)*sqrt(x -
1) + 6*x - 3)

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GIAC/XCAS [A]  time = 0.205963, size = 27, normalized size = 0.9 \[ \frac{4}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} - \frac{4}{3} \, x^{\frac{3}{2}} + 2 \, \sqrt{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x - 1)*(sqrt(x - 1) + sqrt(x))^2),x, algorithm="giac")

[Out]

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