3.2251 \(\int \frac{1}{\left (1+\sqrt{x}\right )^3 \sqrt{x}} \, dx\)

Optimal. Leaf size=11 \[ -\frac{1}{\left (\sqrt{x}+1\right )^2} \]

[Out]

-(1 + Sqrt[x])^(-2)

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Rubi [A]  time = 0.0103994, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{1}{\left (\sqrt{x}+1\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 + Sqrt[x])^3*Sqrt[x]),x]

[Out]

-(1 + Sqrt[x])^(-2)

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Rubi in Sympy [A]  time = 1.68544, size = 10, normalized size = 0.91 \[ - \frac{1}{\left (\sqrt{x} + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(1/2)/(1+x**(1/2))**3,x)

[Out]

-1/(sqrt(x) + 1)**2

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Mathematica [A]  time = 0.00371788, size = 11, normalized size = 1. \[ -\frac{1}{\left (\sqrt{x}+1\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 + Sqrt[x])^3*Sqrt[x]),x]

[Out]

-(1 + Sqrt[x])^(-2)

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Maple [A]  time = 0.002, size = 10, normalized size = 0.9 \[ - \left ( 1+\sqrt{x} \right ) ^{-2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(1/2)/(1+x^(1/2))^3,x)

[Out]

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

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Maxima [A]  time = 1.44626, size = 12, normalized size = 1.09 \[ -\frac{1}{{\left (\sqrt{x} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x)*(sqrt(x) + 1)^3),x, algorithm="maxima")

[Out]

-1/(sqrt(x) + 1)^2

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Fricas [A]  time = 0.233711, size = 16, normalized size = 1.45 \[ -\frac{1}{x + 2 \, \sqrt{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x)*(sqrt(x) + 1)^3),x, algorithm="fricas")

[Out]

-1/(x + 2*sqrt(x) + 1)

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Sympy [A]  time = 1.70747, size = 12, normalized size = 1.09 \[ - \frac{1}{2 \sqrt{x} + x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(1/2)/(1+x**(1/2))**3,x)

[Out]

-1/(2*sqrt(x) + x + 1)

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GIAC/XCAS [A]  time = 0.24483, size = 12, normalized size = 1.09 \[ -\frac{1}{{\left (\sqrt{x} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x)*(sqrt(x) + 1)^3),x, algorithm="giac")

[Out]

-1/(sqrt(x) + 1)^2