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

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

[Out]

(-2*x^(3/2))/3 + (2*(1 + x)^(3/2))/3

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Rubi [A]  time = 0.0135762, 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 \[ \frac{2}{3} (x+1)^{3/2}-\frac{2 x^{3/2}}{3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*x^(3/2))/3 + (2*(1 + x)^(3/2))/3

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Rubi in Sympy [A]  time = 1.42524, 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/(x**(1/2)+(1+x)**(1/2)),x)

[Out]

-2*x**(3/2)/3 + 2*(x + 1)**(3/2)/3

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Mathematica [A]  time = 0.0224673, size = 19, normalized size = 0.9 \[ \frac{2}{3} \left ((x+1)^{3/2}-x^{3/2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*(-x^(3/2) + (1 + x)^(3/2)))/3

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Maple [A]  time = 0.003, size = 14, normalized size = 0.7 \[ -{\frac{2}{3}{x}^{{\frac{3}{2}}}}+{\frac{2}{3} \left ( 1+x \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^(1/2)+(1+x)^(1/2)),x)

[Out]

-2/3*x^(3/2)+2/3*(1+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 = 0.308378, 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 = 1.41415, 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/(x**(1/2)+(1+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 = 0.292285, 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)