\(\int \frac {1}{\sqrt {1+\sqrt {x}}} \, dx\) [230]

Optimal result

Integrand size = 11, antiderivative size = 29 \[ \int \frac {1}{\sqrt {1+\sqrt {x}}} \, dx=-4 \sqrt {1+\sqrt {x}}+\frac {4}{3} \left (1+\sqrt {x}\right )^{3/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {196, 45} \[ \int \frac {1}{\sqrt {1+\sqrt {x}}} \, dx=\frac {4}{3} \left (\sqrt {x}+1\right )^{3/2}-4 \sqrt {\sqrt {x}+1} \]

[In]

Int[1/Sqrt[1 + Sqrt[x]],x]

[Out]

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

Rule 45

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])

Rule 196

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]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x}{\sqrt {1+x}} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\sqrt {1+\sqrt {x}}} \, dx=\frac {4}{3} \left (-2+\sqrt {x}\right ) \sqrt {1+\sqrt {x}} \]

[In]

Integrate[1/Sqrt[1 + Sqrt[x]],x]

[Out]

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

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69

method	result	size
derivativedivides	\(\frac {4 \left (\sqrt {x}+1\right )^{\frac {3}{2}}}{3}-4 \sqrt {\sqrt {x}+1}\)	\(20\)
default	\(\frac {4 \left (\sqrt {x}+1\right )^{\frac {3}{2}}}{3}-4 \sqrt {\sqrt {x}+1}\)	\(20\)
meijerg	\(\frac {\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-4 \sqrt {x}+8\right ) \sqrt {\sqrt {x}+1}}{3}}{\sqrt {\pi }}\)	\(31\)

[In]

int(1/(x^(1/2)+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.48 \[ \int \frac {1}{\sqrt {1+\sqrt {x}}} \, dx=\frac {4}{3} \, \sqrt {\sqrt {x} + 1} {\left (\sqrt {x} - 2\right )} \]

[In]

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

[Out]

4/3*sqrt(sqrt(x) + 1)*(sqrt(x) - 2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (24) = 48\).

Time = 0.50 (sec) , antiderivative size = 117, normalized size of antiderivative = 4.03 \[ \int \frac {1}{\sqrt {1+\sqrt {x}}} \, dx=- \frac {4 x^{\frac {5}{2}} \sqrt {\sqrt {x} + 1}}{3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {8 x^{\frac {5}{2}}}{3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {4 x^{3} \sqrt {\sqrt {x} + 1}}{3 x^{\frac {5}{2}} + 3 x^{2}} - \frac {8 x^{2} \sqrt {\sqrt {x} + 1}}{3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {8 x^{2}}{3 x^{\frac {5}{2}} + 3 x^{2}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\sqrt {1+\sqrt {x}}} \, dx=\frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} - 4 \, \sqrt {\sqrt {x} + 1} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\sqrt {1+\sqrt {x}}} \, dx=\frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} - 4 \, \sqrt {\sqrt {x} + 1} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.41 \[ \int \frac {1}{\sqrt {1+\sqrt {x}}} \, dx=x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},2;\ 3;\ -\sqrt {x}\right ) \]

[In]

int(1/(x^(1/2) + 1)^(1/2),x)

[Out]

x*hypergeom([1/2, 2], 3, -x^(1/2))