3.718 \(\int \sqrt{2+\sqrt{3+\sqrt{-1+2 \sqrt{x}}}} \, dx\)

Optimal. Leaf size=233 \[ \frac{4}{17} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{17/2}-\frac{56}{15} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{15/2}+\frac{300}{13} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{13/2}-\frac{760}{11} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{11/2}+\frac{304}{3} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{9/2}-\frac{480}{7} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{7/2}+\frac{136}{5} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{5/2}-\frac{16}{3} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{3/2} \]

[Out]

(-16*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(3/2))/3 + (136*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(5/2))/5 - (480
*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(7/2))/7 + (304*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(9/2))/3 - (760*(2
+ Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(11/2))/11 + (300*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(13/2))/13 - (56*(2 +
Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(15/2))/15 + (4*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(17/2))/17

________________________________________________________________________________________

Rubi [A]  time = 0.38133, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {1620} \[ \frac{4}{17} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{17/2}-\frac{56}{15} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{15/2}+\frac{300}{13} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{13/2}-\frac{760}{11} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{11/2}+\frac{304}{3} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{9/2}-\frac{480}{7} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{7/2}+\frac{136}{5} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{5/2}-\frac{16}{3} \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]],x]

[Out]

(-16*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(3/2))/3 + (136*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(5/2))/5 - (480
*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(7/2))/7 + (304*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(9/2))/3 - (760*(2
+ Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(11/2))/11 + (300*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(13/2))/13 - (56*(2 +
Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(15/2))/15 + (4*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(17/2))/17

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int \sqrt{2+\sqrt{3+\sqrt{-1+2 \sqrt{x}}}} \, dx &=2 \operatorname{Subst}\left (\int x \sqrt{2+\sqrt{3+\sqrt{-1+2 x}}} \, dx,x,\sqrt{x}\right )\\ &=\operatorname{Subst}\left (\int x \left (1+x^2\right ) \sqrt{2+\sqrt{3+x}} \, dx,x,\sqrt{-1+2 \sqrt{x}}\right )\\ &=2 \operatorname{Subst}\left (\int x \sqrt{2+x} \left (-3+x^2\right ) \left (1+\left (-3+x^2\right )^2\right ) \, dx,x,\sqrt{3+\sqrt{-1+2 \sqrt{x}}}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-4 \sqrt{2+x}+34 (2+x)^{3/2}-120 (2+x)^{5/2}+228 (2+x)^{7/2}-190 (2+x)^{9/2}+75 (2+x)^{11/2}-14 (2+x)^{13/2}+(2+x)^{15/2}\right ) \, dx,x,\sqrt{3+\sqrt{-1+2 \sqrt{x}}}\right )\\ &=-\frac{16}{3} \left (2+\sqrt{3+\sqrt{-1+2 \sqrt{x}}}\right )^{3/2}+\frac{136}{5} \left (2+\sqrt{3+\sqrt{-1+2 \sqrt{x}}}\right )^{5/2}-\frac{480}{7} \left (2+\sqrt{3+\sqrt{-1+2 \sqrt{x}}}\right )^{7/2}+\frac{304}{3} \left (2+\sqrt{3+\sqrt{-1+2 \sqrt{x}}}\right )^{9/2}-\frac{760}{11} \left (2+\sqrt{3+\sqrt{-1+2 \sqrt{x}}}\right )^{11/2}+\frac{300}{13} \left (2+\sqrt{3+\sqrt{-1+2 \sqrt{x}}}\right )^{13/2}-\frac{56}{15} \left (2+\sqrt{3+\sqrt{-1+2 \sqrt{x}}}\right )^{15/2}+\frac{4}{17} \left (2+\sqrt{3+\sqrt{-1+2 \sqrt{x}}}\right )^{17/2}\\ \end{align*}

Mathematica [A]  time = 0.130451, size = 183, normalized size = 0.79 \[ \frac{8 \left (\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right )^{3/2} \left (7 \sqrt{x} \left (2145 \sqrt{2 \sqrt{x}-1} \sqrt{\sqrt{2 \sqrt{x}-1}+3}+1452 \sqrt{\sqrt{2 \sqrt{x}-1}+3}-4004 \sqrt{2 \sqrt{x}-1}-3576\right )+4 \left (3843 \sqrt{2 \sqrt{x}-1} \sqrt{\sqrt{2 \sqrt{x}-1}+3}-2535 \sqrt{\sqrt{2 \sqrt{x}-1}+3}-4286 \sqrt{2 \sqrt{x}-1}-9786\right )\right )}{255255} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]],x]

[Out]

(8*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(3/2)*(4*(-9786 - 2535*Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]] - 4286*Sqrt[-1 +
2*Sqrt[x]] + 3843*Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]*Sqrt[-1 + 2*Sqrt[x]]) + 7*(-3576 + 1452*Sqrt[3 + Sqrt[-1 + 2
*Sqrt[x]]] - 4004*Sqrt[-1 + 2*Sqrt[x]] + 2145*Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]*Sqrt[-1 + 2*Sqrt[x]])*Sqrt[x]))/2
55255

________________________________________________________________________________________

Maple [A]  time = 0.017, size = 154, normalized size = 0.7 \begin{align*} -{\frac{16}{3} \left ( 2+\sqrt{3+\sqrt{-1+2\,\sqrt{x}}} \right ) ^{{\frac{3}{2}}}}+{\frac{136}{5} \left ( 2+\sqrt{3+\sqrt{-1+2\,\sqrt{x}}} \right ) ^{{\frac{5}{2}}}}-{\frac{480}{7} \left ( 2+\sqrt{3+\sqrt{-1+2\,\sqrt{x}}} \right ) ^{{\frac{7}{2}}}}+{\frac{304}{3} \left ( 2+\sqrt{3+\sqrt{-1+2\,\sqrt{x}}} \right ) ^{{\frac{9}{2}}}}-{\frac{760}{11} \left ( 2+\sqrt{3+\sqrt{-1+2\,\sqrt{x}}} \right ) ^{{\frac{11}{2}}}}+{\frac{300}{13} \left ( 2+\sqrt{3+\sqrt{-1+2\,\sqrt{x}}} \right ) ^{{\frac{13}{2}}}}-{\frac{56}{15} \left ( 2+\sqrt{3+\sqrt{-1+2\,\sqrt{x}}} \right ) ^{{\frac{15}{2}}}}+{\frac{4}{17} \left ( 2+\sqrt{3+\sqrt{-1+2\,\sqrt{x}}} \right ) ^{{\frac{17}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(1/2),x)

[Out]

-16/3*(2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(3/2)+136/5*(2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(5/2)-480/7*(2+(3+(-1+
2*x^(1/2))^(1/2))^(1/2))^(7/2)+304/3*(2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(9/2)-760/11*(2+(3+(-1+2*x^(1/2))^(1/2
))^(1/2))^(11/2)+300/13*(2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(13/2)-56/15*(2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(15
/2)+4/17*(2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(17/2)

________________________________________________________________________________________

Maxima [A]  time = 1.01166, size = 207, normalized size = 0.89 \begin{align*} \frac{4}{17} \,{\left (\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2\right )}^{\frac{17}{2}} - \frac{56}{15} \,{\left (\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2\right )}^{\frac{15}{2}} + \frac{300}{13} \,{\left (\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2\right )}^{\frac{13}{2}} - \frac{760}{11} \,{\left (\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2\right )}^{\frac{11}{2}} + \frac{304}{3} \,{\left (\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2\right )}^{\frac{9}{2}} - \frac{480}{7} \,{\left (\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2\right )}^{\frac{7}{2}} + \frac{136}{5} \,{\left (\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2\right )}^{\frac{5}{2}} - \frac{16}{3} \,{\left (\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

4/17*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(17/2) - 56/15*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(15/2) + 300/13*(s
qrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(13/2) - 760/11*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(11/2) + 304/3*(sqrt(sqr
t(2*sqrt(x) - 1) + 3) + 2)^(9/2) - 480/7*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(7/2) + 136/5*(sqrt(sqrt(2*sqrt(x
) - 1) + 3) + 2)^(5/2) - 16/3*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 1.49924, size = 309, normalized size = 1.33 \begin{align*} -\frac{8}{255255} \,{\left ({\left (847 \, \sqrt{x} - 1688\right )} \sqrt{2 \, \sqrt{x} - 1} - 2 \,{\left ({\left (1001 \, \sqrt{x} + 6800\right )} \sqrt{2 \, \sqrt{x} - 1} - 2352 \, \sqrt{x} - 29712\right )} \sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} - 30030 \, x + 3843 \, \sqrt{x} + 124080\right )} \sqrt{\sqrt{\sqrt{2 \, \sqrt{x} - 1} + 3} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-8/255255*((847*sqrt(x) - 1688)*sqrt(2*sqrt(x) - 1) - 2*((1001*sqrt(x) + 6800)*sqrt(2*sqrt(x) - 1) - 2352*sqrt
(x) - 29712)*sqrt(sqrt(2*sqrt(x) - 1) + 3) - 30030*x + 3843*sqrt(x) + 124080)*sqrt(sqrt(sqrt(2*sqrt(x) - 1) +
3) + 2)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sqrt{\sqrt{2 \sqrt{x} - 1} + 3} + 2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+(3+(-1+2*x**(1/2))**(1/2))**(1/2))**(1/2),x)

[Out]

Integral(sqrt(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2), x)

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(1/2),x, algorithm="giac")

[Out]

Timed out