3.2253 \(\int \frac {1}{(1+\sqrt {x})^3 \sqrt {x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {261} \[ -\frac {1}{\left (\sqrt {x}+1\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 + Sqrt[x])^3*Sqrt[x]),x]

[Out]

-(1 + Sqrt[x])^(-2)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (1+\sqrt {x}\right )^3 \sqrt {x}} \, dx &=-\frac {1}{\left (1+\sqrt {x}\right )^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \[ -\frac {1}{\left (\sqrt {x}+1\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1 + Sqrt[x])^(-2)

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fricas [B]  time = 0.86, size = 20, normalized size = 1.82 \[ -\frac {x - 2 \, \sqrt {x} + 1}{x^{2} - 2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(1+x^(1/2))^3,x, algorithm="fricas")

[Out]

-(x - 2*sqrt(x) + 1)/(x^2 - 2*x + 1)

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giac [A]  time = 0.15, size = 9, normalized size = 0.82 \[ -\frac {1}{{\left (\sqrt {x} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(1+x^(1/2))^3,x, algorithm="giac")

[Out]

-1/(sqrt(x) + 1)^2

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maple [A]  time = 0.00, 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]

int(1/x^(1/2)/(x^(1/2)+1)^3,x)

[Out]

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

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maxima [A]  time = 0.85, size = 9, normalized size = 0.82 \[ -\frac {1}{{\left (\sqrt {x} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(1+x^(1/2))^3,x, algorithm="maxima")

[Out]

-1/(sqrt(x) + 1)^2

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mupad [B]  time = 0.03, size = 9, normalized size = 0.82 \[ -\frac {1}{{\left (\sqrt {x}+1\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(x^(1/2) + 1)^3),x)

[Out]

-1/(x^(1/2) + 1)^2

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sympy [A]  time = 0.44, 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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