### 3.222 $$\int \frac{1}{1+\sqrt{x}} \, dx$$

Optimal. Leaf size=18 $2 \sqrt{x}-2 \log \left (\sqrt{x}+1\right )$

[Out]

2*Sqrt[x] - 2*Log[1 + Sqrt[x]]

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Rubi [A]  time = 0.0059158, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {190, 43} $2 \sqrt{x}-2 \log \left (\sqrt{x}+1\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(1 + Sqrt[x])^(-1),x]

[Out]

2*Sqrt[x] - 2*Log[1 + Sqrt[x]]

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[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{1}{1+\sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x}{1+x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (1+\frac{1}{-1-x}\right ) \, dx,x,\sqrt{x}\right )\\ &=2 \sqrt{x}-2 \log \left (1+\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0066791, size = 18, normalized size = 1. $2 \sqrt{x}-2 \log \left (\sqrt{x}+1\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(1 + Sqrt[x])^(-1),x]

[Out]

2*Sqrt[x] - 2*Log[1 + Sqrt[x]]

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Maple [A]  time = 0.003, size = 27, normalized size = 1.5 \begin{align*} 2\,\sqrt{x}+\ln \left ( \sqrt{x}-1 \right ) -\ln \left ( \sqrt{x}+1 \right ) -\ln \left ( -1+x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.928509, size = 20, normalized size = 1.11 \begin{align*} 2 \, \sqrt{x} - 2 \, \log \left (\sqrt{x} + 1\right ) + 2 \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x) - 2*log(sqrt(x) + 1) + 2

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Fricas [A]  time = 1.79365, size = 43, normalized size = 2.39 \begin{align*} 2 \, \sqrt{x} - 2 \, \log \left (\sqrt{x} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x) - 2*log(sqrt(x) + 1)

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Sympy [A]  time = 0.109201, size = 15, normalized size = 0.83 \begin{align*} 2 \sqrt{x} - 2 \log{\left (\sqrt{x} + 1 \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x) - 2*log(sqrt(x) + 1)

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Giac [A]  time = 1.04502, size = 19, normalized size = 1.06 \begin{align*} 2 \, \sqrt{x} - 2 \, \log \left (\sqrt{x} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x) - 2*log(sqrt(x) + 1)