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

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 = 0.0055358, 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, Rules used = {2106, 30, 32} \[ \frac{2}{3} (x+1)^{3/2}-\frac{2 x^{3/2}}{3} \]

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

Rule 2106

Int[(u_.)/((d_.)*(x_)^(n_.) + (c_.)*Sqrt[(a_.) + (b_.)*(x_)^(p_.)]), x_Symbol] :> -Dist[b/(a*d), Int[u*x^n, x]
, x] + Dist[1/(a*c), Int[u*Sqrt[a + b*x^(2*n)], x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 2*n] && EqQ[b*c^
2 - d^2, 0]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0158995, size = 21, normalized size = 1. \[ \frac{2}{3} (x+1)^{3/2}-\frac{2 x^{3/2}}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x^(3/2))/3 + (2*(1 + x)^(3/2))/3

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Maple [A]  time = 0.002, size = 14, normalized size = 0.7 \begin{align*} -{\frac{2}{3}{x}^{{\frac{3}{2}}}}+{\frac{2}{3} \left ( 1+x \right ) ^{{\frac{3}{2}}}} \end{align*}

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. \begin{align*} \int \frac{1}{\sqrt{x + 1} + \sqrt{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x + 1) + sqrt(x)), x)

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Fricas [A]  time = 1.84066, size = 45, normalized size = 2.14 \begin{align*} \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - \frac{2}{3} \, x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2) - 2/3*x^(3/2)

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Sympy [B]  time = 0.351887, size = 63, normalized size = 3. \begin{align*} \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}} \end{align*}

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 [A]  time = 1.05009, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - \frac{2}{3} \, x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2) - 2/3*x^(3/2)