3.8.18 \(\int \sqrt {2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}} \, dx\) [718]

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

[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 = 0.26, 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 = {1634} \begin {gather*} \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} \end {gather*}

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 1634

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 \text {Subst}\left (\int x \sqrt {2+\sqrt {3+\sqrt {-1+2 x}}} \, dx,x,\sqrt {x}\right )\\ &=\text {Subst}\left (\int x \left (1+x^2\right ) \sqrt {2+\sqrt {3+x}} \, dx,x,\sqrt {-1+2 \sqrt {x}}\right )\\ &=2 \text {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 \text {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 = 0.16, size = 186, normalized size = 0.80 \begin {gather*} \frac {8 \sqrt {2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}} \left (8 \left (-15510-7428 \sqrt {3+\sqrt {-1+2 \sqrt {x}}}+211 \sqrt {-1+2 \sqrt {x}}+1700 \sqrt {3+\sqrt {-1+2 \sqrt {x}}} \sqrt {-1+2 \sqrt {x}}\right )+7 \left (-549-672 \sqrt {3+\sqrt {-1+2 \sqrt {x}}}-121 \sqrt {-1+2 \sqrt {x}}+286 \sqrt {3+\sqrt {-1+2 \sqrt {x}}} \sqrt {-1+2 \sqrt {x}}\right ) \sqrt {x}+30030 x\right )}{255255} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*Sqrt[2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]]*(8*(-15510 - 7428*Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]] + 211*Sqrt[-1 + 2
*Sqrt[x]] + 1700*Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]*Sqrt[-1 + 2*Sqrt[x]]) + 7*(-549 - 672*Sqrt[3 + Sqrt[-1 + 2*Sqr
t[x]]] - 121*Sqrt[-1 + 2*Sqrt[x]] + 286*Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]*Sqrt[-1 + 2*Sqrt[x]])*Sqrt[x] + 30030*x
))/255255

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Maple [A]
time = 0.42, size = 154, normalized size = 0.66

method	result	size
derivativedivides	\(-\frac {16 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {3}{2}}}{3}+\frac {136 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {5}{2}}}{5}-\frac {480 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {7}{2}}}{7}+\frac {304 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {9}{2}}}{3}-\frac {760 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {11}{2}}}{11}+\frac {300 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {13}{2}}}{13}-\frac {56 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {15}{2}}}{15}+\frac {4 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {17}{2}}}{17}\)	\(154\)
default	\(-\frac {16 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {3}{2}}}{3}+\frac {136 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {5}{2}}}{5}-\frac {480 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {7}{2}}}{7}+\frac {304 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {9}{2}}}{3}-\frac {760 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {11}{2}}}{11}+\frac {300 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {13}{2}}}{13}-\frac {56 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {15}{2}}}{15}+\frac {4 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {17}{2}}}{17}\)	\(154\)

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,method=_RETURNVERBOSE)

[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 = 0.28, size = 153, normalized size = 0.66 \begin {gather*} \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 {gather*}

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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Fricas [A]
time = 0.37, size = 85, normalized size = 0.36 \begin {gather*} -\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 {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\sqrt {\sqrt {2 \sqrt {x} - 1} + 3} + 2}\, dx \end {gather*}

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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Giac [A]
time = 6.11, size = 271, normalized size = 1.16 \begin {gather*} \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 (8192 \, x^{23} + 376832 \, x^{22} + 8224768 \, x^{21} + 113971200 \, x^{20} + 1130782720 \, x^{19} + 8582063104 \, x^{18} + 51933387264 \, x^{17} + 257575619584 \, x^{16} + 1066188686592 \, x^{15} + 3723204389632 \, x^{14} + 11019822890016 \, x^{13} + 27631512444352 \, x^{12} + 58424530490176 \, x^{11} + 103336828749760 \, x^{10} + 151203890043312 \, x^{9} + 180411181747936 \, x^{8} + 172287199292960 \, x^{7} + 128457231939048 \, x^{6} + 72257964298210 \, x^{5} + 29175203228012 \, x^{4} + 7830371130072 \, x^{3} + 1228114804752 \, x^{2} + 87490886400 \, x + 933120000\right ) \end {gather*}

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(8192*x^23 +
376832*x^22 + 8224768*x^21 + 113971200*x^20 + 1130782720*x^19 + 8582063104*x^18 + 51933387264*x^17 + 25757561
9584*x^16 + 1066188686592*x^15 + 3723204389632*x^14 + 11019822890016*x^13 + 27631512444352*x^12 + 584245304901
76*x^11 + 103336828749760*x^10 + 151203890043312*x^9 + 180411181747936*x^8 + 172287199292960*x^7 + 12845723193
9048*x^6 + 72257964298210*x^5 + 29175203228012*x^4 + 7830371130072*x^3 + 1228114804752*x^2 + 87490886400*x + 9
33120000)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {\sqrt {\sqrt {2\,\sqrt {x}-1}+3}+2} \,d x \end {gather*}

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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