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

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]

-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

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Rubi [A]  time = 0.0588152, antiderivative size = 83, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.0598087, size = 58, normalized size = 0.7 \[ \frac{8}{105} \sqrt{\sqrt{\sqrt{x}+1}+2} \left (3 \sqrt{x} \left (5 \sqrt{\sqrt{x}+1}-12\right )+76 \sqrt{\sqrt{x}+1}-280\right ) \]

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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Maple [A]  time = 0.01, size = 54, normalized size = 0.7 \begin{align*}{\frac{88}{3} \left ( 2+\sqrt{1+\sqrt{x}} \right ) ^{{\frac{3}{2}}}}-{\frac{48}{5} \left ( 2+\sqrt{1+\sqrt{x}} \right ) ^{{\frac{5}{2}}}}+{\frac{8}{7} \left ( 2+\sqrt{1+\sqrt{x}} \right ) ^{{\frac{7}{2}}}}-48\,\sqrt{2+\sqrt{1+\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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Maxima [A]  time = 1.0017, size = 72, normalized size = 0.87 \begin{align*} \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} \end{align*}

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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Fricas [A]  time = 1.50503, size = 124, normalized size = 1.49 \begin{align*} \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} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\sqrt{\sqrt{x} + 1} + 2}}\, dx \end{align*}

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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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]

Exception raised: TypeError