3.716 \(\int \frac {1}{\sqrt {2+\sqrt {1+\sqrt {x}}}} \, dx\)

Optimal. Leaf size=83 \[ \frac {8}{7} \left (\sqrt {\sqrt {x}+1}+2\right )^{7/2}-\frac {48}{5} \left (\sqrt {\sqrt {x}+1}+2\right )^{5/2}+\frac {88}{3} \left (\sqrt {\sqrt {x}+1}+2\right )^{3/2}-48 \sqrt {\sqrt {\sqrt {x}+1}+2} \]

[Out]

88/3*(2+(1+x^(1/2))^(1/2))^(3/2)-48/5*(2+(1+x^(1/2))^(1/2))^(5/2)+8/7*(2+(1+x^(1/2))^(1/2))^(7/2)-48*(2+(1+x^(
1/2))^(1/2))^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {371, 1398, 772} \[ \frac {8}{7} \left (\sqrt {\sqrt {x}+1}+2\right )^{7/2}-\frac {48}{5} \left (\sqrt {\sqrt {x}+1}+2\right )^{5/2}+\frac {88}{3} \left (\sqrt {\sqrt {x}+1}+2\right )^{3/2}-48 \sqrt {\sqrt {\sqrt {x}+1}+2} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + Sqrt[1 + Sqrt[x]]],x]

[Out]

-48*Sqrt[2 + Sqrt[1 + Sqrt[x]]] + (88*(2 + Sqrt[1 + Sqrt[x]])^(3/2))/3 - (48*(2 + Sqrt[1 + Sqrt[x]])^(5/2))/5
+ (8*(2 + Sqrt[1 + Sqrt[x]])^(7/2))/7

Rule 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rule 1398

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p
, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {2+\sqrt {1+\sqrt {x}}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x}{\sqrt {2+\sqrt {1+x}}} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {-1+x}{\sqrt {2+\sqrt {x}}} \, dx,x,1+\sqrt {x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {x \left (-1+x^2\right )}{\sqrt {2+x}} \, dx,x,\sqrt {1+\sqrt {x}}\right )\\ &=4 \operatorname {Subst}\left (\int \left (-\frac {6}{\sqrt {2+x}}+11 \sqrt {2+x}-6 (2+x)^{3/2}+(2+x)^{5/2}\right ) \, dx,x,\sqrt {1+\sqrt {x}}\right )\\ &=-48 \sqrt {2+\sqrt {1+\sqrt {x}}}+\frac {88}{3} \left (2+\sqrt {1+\sqrt {x}}\right )^{3/2}-\frac {48}{5} \left (2+\sqrt {1+\sqrt {x}}\right )^{5/2}+\frac {8}{7} \left (2+\sqrt {1+\sqrt {x}}\right )^{7/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 58, normalized size = 0.70 \[ \frac {8}{105} \sqrt {\sqrt {\sqrt {x}+1}+2} \left (3 \sqrt {x} \left (5 \sqrt {\sqrt {x}+1}-12\right )+76 \sqrt {\sqrt {x}+1}-280\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[2 + Sqrt[1 + Sqrt[x]]],x]

[Out]

(8*Sqrt[2 + Sqrt[1 + Sqrt[x]]]*(-280 + 76*Sqrt[1 + Sqrt[x]] + 3*(-12 + 5*Sqrt[1 + Sqrt[x]])*Sqrt[x]))/105

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fricas [A]  time = 0.45, size = 35, normalized size = 0.42 \[ \frac {8}{105} \, {\left ({\left (15 \, \sqrt {x} + 76\right )} \sqrt {\sqrt {x} + 1} - 36 \, \sqrt {x} - 280\right )} \sqrt {\sqrt {\sqrt {x} + 1} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/105*((15*sqrt(x) + 76)*sqrt(sqrt(x) + 1) - 36*sqrt(x) - 280)*sqrt(sqrt(sqrt(x) + 1) + 2)

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giac [A]  time = 3.18, size = 82, normalized size = 0.99 \[ \frac {8 \, {\left (15 \, {\left (\sqrt {\sqrt {x} + 1} + 2\right )}^{\frac {7}{2}} - 126 \, {\left (\sqrt {\sqrt {x} + 1} + 2\right )}^{\frac {5}{2}} + 385 \, {\left (\sqrt {\sqrt {x} + 1} + 2\right )}^{\frac {3}{2}} - 630 \, \sqrt {\sqrt {\sqrt {x} + 1} + 2}\right )}}{105 \, \mathrm {sgn}\left (4 \, {\left (\sqrt {x} + 1\right )}^{2} - 8 \, \sqrt {x} - 7\right ) \mathrm {sgn}\left (4 \, x - 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/105*(15*(sqrt(sqrt(x) + 1) + 2)^(7/2) - 126*(sqrt(sqrt(x) + 1) + 2)^(5/2) + 385*(sqrt(sqrt(x) + 1) + 2)^(3/2
) - 630*sqrt(sqrt(sqrt(x) + 1) + 2))/(sgn(4*(sqrt(x) + 1)^2 - 8*sqrt(x) - 7)*sgn(4*x - 3))

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maple [A]  time = 0.01, size = 54, normalized size = 0.65 \[ \frac {88 \left (2+\sqrt {\sqrt {x}+1}\right )^{\frac {3}{2}}}{3}-\frac {48 \left (2+\sqrt {\sqrt {x}+1}\right )^{\frac {5}{2}}}{5}+\frac {8 \left (2+\sqrt {\sqrt {x}+1}\right )^{\frac {7}{2}}}{7}-48 \sqrt {2+\sqrt {\sqrt {x}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

88/3*(2+(x^(1/2)+1)^(1/2))^(3/2)-48/5*(2+(x^(1/2)+1)^(1/2))^(5/2)+8/7*(2+(x^(1/2)+1)^(1/2))^(7/2)-48*(2+(x^(1/
2)+1)^(1/2))^(1/2)

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maxima [A]  time = 0.88, size = 53, normalized size = 0.64 \[ \frac {8}{7} \, {\left (\sqrt {\sqrt {x} + 1} + 2\right )}^{\frac {7}{2}} - \frac {48}{5} \, {\left (\sqrt {\sqrt {x} + 1} + 2\right )}^{\frac {5}{2}} + \frac {88}{3} \, {\left (\sqrt {\sqrt {x} + 1} + 2\right )}^{\frac {3}{2}} - 48 \, \sqrt {\sqrt {\sqrt {x} + 1} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/7*(sqrt(sqrt(x) + 1) + 2)^(7/2) - 48/5*(sqrt(sqrt(x) + 1) + 2)^(5/2) + 88/3*(sqrt(sqrt(x) + 1) + 2)^(3/2) -
48*sqrt(sqrt(sqrt(x) + 1) + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {\sqrt {\sqrt {x}+1}+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\sqrt {\sqrt {x} + 1} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(sqrt(sqrt(x) + 1) + 2), x)

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