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

Optimal. Leaf size=29 \[ \frac{4}{3} \left (\sqrt{x}+1\right )^{3/2}-4 \sqrt{\sqrt{x}+1} \]

[Out]

-4*Sqrt[1 + Sqrt[x]] + (4*(1 + Sqrt[x])^(3/2))/3

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Rubi [A]  time = 0.0073292, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac{4}{3} \left (\sqrt{x}+1\right )^{3/2}-4 \sqrt{\sqrt{x}+1} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 + Sqrt[x]],x]

[Out]

-4*Sqrt[1 + Sqrt[x]] + (4*(1 + Sqrt[x])^(3/2))/3

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}{\sqrt{1+\sqrt{x}}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+x}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{1}{\sqrt{1+x}}+\sqrt{1+x}\right ) \, dx,x,\sqrt{x}\right )\\ &=-4 \sqrt{1+\sqrt{x}}+\frac{4}{3} \left (1+\sqrt{x}\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.00741, size = 22, normalized size = 0.76 \[ \frac{4}{3} \left (\sqrt{x}-2\right ) \sqrt{\sqrt{x}+1} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + Sqrt[x]],x]

[Out]

(4*(-2 + Sqrt[x])*Sqrt[1 + Sqrt[x]])/3

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Maple [A]  time = 0.005, size = 20, normalized size = 0.7 \begin{align*}{\frac{4}{3} \left ( \sqrt{x}+1 \right ) ^{{\frac{3}{2}}}}-4\,\sqrt{\sqrt{x}+1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(x^(1/2)+1)^(3/2)-4*(x^(1/2)+1)^(1/2)

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Maxima [A]  time = 0.928268, size = 26, normalized size = 0.9 \begin{align*} \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} - 4 \, \sqrt{\sqrt{x} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(sqrt(x) + 1)^(3/2) - 4*sqrt(sqrt(x) + 1)

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Fricas [A]  time = 2.05516, size = 50, normalized size = 1.72 \begin{align*} \frac{4}{3} \, \sqrt{\sqrt{x} + 1}{\left (\sqrt{x} - 2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*sqrt(sqrt(x) + 1)*(sqrt(x) - 2)

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Sympy [B]  time = 0.760373, size = 117, normalized size = 4.03 \begin{align*} - \frac{4 x^{\frac{5}{2}} \sqrt{\sqrt{x} + 1}}{3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{8 x^{\frac{5}{2}}}{3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{4 x^{3} \sqrt{\sqrt{x} + 1}}{3 x^{\frac{5}{2}} + 3 x^{2}} - \frac{8 x^{2} \sqrt{\sqrt{x} + 1}}{3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{8 x^{2}}{3 x^{\frac{5}{2}} + 3 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*x**(5/2)*sqrt(sqrt(x) + 1)/(3*x**(5/2) + 3*x**2) + 8*x**(5/2)/(3*x**(5/2) + 3*x**2) + 4*x**3*sqrt(sqrt(x) +
1)/(3*x**(5/2) + 3*x**2) - 8*x**2*sqrt(sqrt(x) + 1)/(3*x**(5/2) + 3*x**2) + 8*x**2/(3*x**(5/2) + 3*x**2)

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Giac [A]  time = 1.06352, size = 26, normalized size = 0.9 \begin{align*} \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} - 4 \, \sqrt{\sqrt{x} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(sqrt(x) + 1)^(3/2) - 4*sqrt(sqrt(x) + 1)