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]

-16/3*(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)

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Rubi [A]  time = 0.38, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, 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*}

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Mathematica [A]  time = 0.13, 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

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fricas [A]  time = 0.46, size = 85, normalized size = 0.36 \[ -\frac {8}{255255} \, {\left ({\left (847 \, \sqrt {x} - 1688\right )} \sqrt {2 \, \sqrt {x} - 1} - 2 \, {\left ({\left (1001 \, \sqrt {x} + 6800\right )} \sqrt {2 \, \sqrt {x} - 1} - 2352 \, \sqrt {x} - 29712\right )} \sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} - 30030 \, x + 3843 \, \sqrt {x} + 124080\right )} \sqrt {\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2} \]

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)

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giac [A]  time = 51.25, size = 271, normalized size = 1.16 \[ \frac {4}{255255} \, {\left (15015 \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {17}{2}} - 238238 \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {15}{2}} + 1472625 \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {13}{2}} - 4408950 \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {11}{2}} + 6466460 \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {9}{2}} - 4375800 \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {7}{2}} + 1735734 \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {5}{2}} - 340340 \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {3}{2}}\right )} \mathrm {sgn}\left (131072 \, x^{23} + 6029312 \, x^{22} + 131596288 \, x^{21} + 1823539200 \, x^{20} + 18092523520 \, x^{19} + 137313009664 \, x^{18} + 830934196224 \, x^{17} + 4121209913344 \, x^{16} + 17059018985472 \, x^{15} + 59571270234112 \, x^{14} + 176317166240256 \, x^{13} + 442104199109632 \, x^{12} + 934792487842816 \, x^{11} + 1653389259996160 \, x^{10} + 2419262240692992 \, x^{9} + 2886578907966976 \, x^{8} + 2756595188687360 \, x^{7} + 2055315711024768 \, x^{6} + 1156127428771360 \, x^{5} + 466803251648192 \, x^{4} + 125285938081152 \, x^{3} + 19649836876032 \, x^{2} + 1399854182400 \, x + 14929920000\right ) \]

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]

4/255255*(15015*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(17/2) - 238238*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(15/2)
+ 1472625*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(13/2) - 4408950*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(11/2) + 6
466460*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(9/2) - 4375800*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(7/2) + 1735734
*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(5/2) - 340340*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(3/2))*sgn(131072*x^23
+ 6029312*x^22 + 131596288*x^21 + 1823539200*x^20 + 18092523520*x^19 + 137313009664*x^18 + 830934196224*x^17
+ 4121209913344*x^16 + 17059018985472*x^15 + 59571270234112*x^14 + 176317166240256*x^13 + 442104199109632*x^12
+ 934792487842816*x^11 + 1653389259996160*x^10 + 2419262240692992*x^9 + 2886578907966976*x^8 + 27565951886873
60*x^7 + 2055315711024768*x^6 + 1156127428771360*x^5 + 466803251648192*x^4 + 125285938081152*x^3 + 19649836876
032*x^2 + 1399854182400*x + 14929920000)

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maple [A]  time = 0.02, size = 154, normalized size = 0.66 \[ -\frac {16 \left (2+\sqrt {3+\sqrt {2 \sqrt {x}-1}}\right )^{\frac {3}{2}}}{3}+\frac {136 \left (2+\sqrt {3+\sqrt {2 \sqrt {x}-1}}\right )^{\frac {5}{2}}}{5}-\frac {480 \left (2+\sqrt {3+\sqrt {2 \sqrt {x}-1}}\right )^{\frac {7}{2}}}{7}+\frac {304 \left (2+\sqrt {3+\sqrt {2 \sqrt {x}-1}}\right )^{\frac {9}{2}}}{3}-\frac {760 \left (2+\sqrt {3+\sqrt {2 \sqrt {x}-1}}\right )^{\frac {11}{2}}}{11}+\frac {300 \left (2+\sqrt {3+\sqrt {2 \sqrt {x}-1}}\right )^{\frac {13}{2}}}{13}-\frac {56 \left (2+\sqrt {3+\sqrt {2 \sqrt {x}-1}}\right )^{\frac {15}{2}}}{15}+\frac {4 \left (2+\sqrt {3+\sqrt {2 \sqrt {x}-1}}\right )^{\frac {17}{2}}}{17} \]

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)

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maxima [A]  time = 0.93, size = 153, normalized size = 0.66 \[ \frac {4}{17} \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {17}{2}} - \frac {56}{15} \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {15}{2}} + \frac {300}{13} \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {13}{2}} - \frac {760}{11} \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {11}{2}} + \frac {304}{3} \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {9}{2}} - \frac {480}{7} \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {7}{2}} + \frac {136}{5} \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {5}{2}} - \frac {16}{3} \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {3}{2}} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {\sqrt {\sqrt {2\,\sqrt {x}-1}+3}+2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sqrt {\sqrt {2 \sqrt {x} - 1} + 3} + 2}\, dx \]

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)

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