3.2250 \(\int \frac{\sqrt{x}}{1+\sqrt{x}} \, dx\)

Optimal. Leaf size=19 \[ x-2 \sqrt{x}+2 \log \left (\sqrt{x}+1\right ) \]

[Out]

-2*Sqrt[x] + x + 2*Log[1 + Sqrt[x]]

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Rubi [A]  time = 0.0107918, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ x-2 \sqrt{x}+2 \log \left (\sqrt{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(1 + Sqrt[x]),x]

[Out]

-2*Sqrt[x] + x + 2*Log[1 + Sqrt[x]]

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0075446, size = 19, normalized size = 1. \[ x-2 \sqrt{x}+2 \log \left (\sqrt{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(1 + Sqrt[x]),x]

[Out]

-2*Sqrt[x] + x + 2*Log[1 + Sqrt[x]]

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Maple [A]  time = 0.003, size = 16, normalized size = 0.8 \begin{align*} x+2\,\ln \left ( \sqrt{x}+1 \right ) -2\,\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.961382, size = 30, normalized size = 1.58 \begin{align*}{\left (\sqrt{x} + 1\right )}^{2} - 4 \, \sqrt{x} + 2 \, \log \left (\sqrt{x} + 1\right ) - 4 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(x) + 1)^2 - 4*sqrt(x) + 2*log(sqrt(x) + 1) - 4

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Fricas [A]  time = 1.36597, size = 49, normalized size = 2.58 \begin{align*} x - 2 \, \sqrt{x} + 2 \, \log \left (\sqrt{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*sqrt(x) + 2*log(sqrt(x) + 1)

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Sympy [A]  time = 0.12822, size = 17, normalized size = 0.89 \begin{align*} - 2 \sqrt{x} + x + 2 \log{\left (\sqrt{x} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1/2)/(1+x**(1/2)),x)

[Out]

-2*sqrt(x) + x + 2*log(sqrt(x) + 1)

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Giac [A]  time = 1.0978, size = 20, normalized size = 1.05 \begin{align*} x - 2 \, \sqrt{x} + 2 \, \log \left (\sqrt{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*sqrt(x) + 2*log(sqrt(x) + 1)