Integrand size = 25, antiderivative size = 233 \[ \int \sqrt {2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}} \, dx=-\frac {16}{3} \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{3/2}+\frac {136}{5} \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{5/2}-\frac {480}{7} \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{7/2}+\frac {304}{3} \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{9/2}-\frac {760}{11} \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{11/2}+\frac {300}{13} \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{13/2}-\frac {56}{15} \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{15/2}+\frac {4}{17} \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{17/2} \] Output:
-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)
Time = 0.19 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.80 \[ \int \sqrt {2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}} \, dx=\frac {8 \sqrt {2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}} \left (8 \left (-15510-7428 \sqrt {3+\sqrt {-1+2 \sqrt {x}}}+211 \sqrt {-1+2 \sqrt {x}}+1700 \sqrt {3+\sqrt {-1+2 \sqrt {x}}} \sqrt {-1+2 \sqrt {x}}\right )+7 \left (-549-672 \sqrt {3+\sqrt {-1+2 \sqrt {x}}}-121 \sqrt {-1+2 \sqrt {x}}+286 \sqrt {3+\sqrt {-1+2 \sqrt {x}}} \sqrt {-1+2 \sqrt {x}}\right ) \sqrt {x}+30030 x\right )}{255255} \] Input:
Integrate[Sqrt[2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]],x]
Output:
(8*Sqrt[2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]]*(8*(-15510 - 7428*Sqrt[3 + Sqr t[-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*Sqrt[ 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
Time = 1.20 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {7267, 7267, 7267, 25, 2091, 2123, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sqrt {\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int \sqrt {\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2} \sqrt {x}d\sqrt {x}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \int \sqrt {\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2} \sqrt {2 \sqrt {x}-1} (x+1)d\sqrt {2 \sqrt {x}-1}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int -\sqrt {\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2} \sqrt {\sqrt {2 \sqrt {x}-1}+3} \left ((x-3)^2+1\right ) (3-x)d\sqrt {\sqrt {2 \sqrt {x}-1}+3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \sqrt {\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2} \sqrt {\sqrt {2 \sqrt {x}-1}+3} \left ((x-3)^2+1\right ) (3-x)d\sqrt {\sqrt {2 \sqrt {x}-1}+3}\) |
| \(\Big \downarrow \) 2091 |
| \(\displaystyle -2 \int \sqrt {\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2} \sqrt {\sqrt {2 \sqrt {x}-1}+3} (3-x) \left (x^2-6 x+10\right )d\sqrt {\sqrt {2 \sqrt {x}-1}+3}\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle -2 \int \left (-\left (\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2\right )^{15/2}+14 \left (\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2\right )^{13/2}-75 \left (\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2\right )^{11/2}+190 \left (\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2\right )^{9/2}-228 \left (\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2\right )^{7/2}+120 \left (\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2\right )^{5/2}-34 \left (\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2\right )^{3/2}+4 \sqrt {\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2}\right )d\sqrt {\sqrt {2 \sqrt {x}-1}+3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {2}{17} \left (\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2\right )^{17/2}-\frac {28}{15} \left (\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2\right )^{15/2}+\frac {150}{13} \left (\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2\right )^{13/2}-\frac {380}{11} \left (\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2\right )^{11/2}+\frac {152}{3} \left (\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2\right )^{9/2}-\frac {240}{7} \left (\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2\right )^{7/2}+\frac {68}{5} \left (\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2\right )^{5/2}-\frac {8}{3} \left (\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2\right )^{3/2}\right )\) |
Input:
Int[Sqrt[2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]],x]
Output:
2*((-8*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(3/2))/3 + (68*(2 + Sqrt[3 + S qrt[-1 + 2*Sqrt[x]]])^(5/2))/5 - (240*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]) ^(7/2))/7 + (152*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(9/2))/3 - (380*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(11/2))/11 + (150*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(13/2))/13 - (28*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(15/2) )/15 + (2*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(17/2))/17)
Int[(Px_)*(u_)^(p_.)*(z_)^(q_.), x_Symbol] :> Int[Px*ExpandToSum[z, x]^q*Ex pandToSum[u, x]^p, x] /; FreeQ[{p, q}, x] && PolyQ[Px, x] && BinomialQ[z, x ] && TrinomialQ[u, x] && !(BinomialMatchQ[z, x] && TrinomialMatchQ[u, x])
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])
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.95 (sec) , antiderivative size = 154, normalized size of antiderivative = 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 | \(-\frac {16 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {3}{2}}}{3}+\frac {136 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {5}{2}}}{5}-\frac {480 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {7}{2}}}{7}+\frac {304 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {9}{2}}}{3}-\frac {760 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {11}{2}}}{11}+\frac {300 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {13}{2}}}{13}-\frac {56 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {15}{2}}}{15}+\frac {4 \left (2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}\right )^{\frac {17}{2}}}{17}\) | \(154\) |
Input:
int((2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(1/2),x,method=_RETURNVERBOSE)
Output:
-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)
Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.36 \[ \int \sqrt {2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}} \, dx=-\frac {8}{255255} \, {\left ({\left (847 \, \sqrt {x} - 1688\right )} \sqrt {2 \, \sqrt {x} - 1} - 2 \, {\left ({\left (1001 \, \sqrt {x} + 6800\right )} \sqrt {2 \, \sqrt {x} - 1} - 2352 \, \sqrt {x} - 29712\right )} \sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} - 30030 \, x + 3843 \, \sqrt {x} + 124080\right )} \sqrt {\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2} \] Input:
integrate((2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(1/2),x, algorithm="fricas")
Output:
-8/255255*((847*sqrt(x) - 1688)*sqrt(2*sqrt(x) - 1) - 2*((1001*sqrt(x) + 6 800)*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)
Time = 0.77 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.87 \[ \int \sqrt {2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}} \, dx=\frac {4 \left (\sqrt {\sqrt {2 \sqrt {x} - 1} + 3} + 2\right )^{\frac {17}{2}}}{17} - \frac {56 \left (\sqrt {\sqrt {2 \sqrt {x} - 1} + 3} + 2\right )^{\frac {15}{2}}}{15} + \frac {300 \left (\sqrt {\sqrt {2 \sqrt {x} - 1} + 3} + 2\right )^{\frac {13}{2}}}{13} - \frac {760 \left (\sqrt {\sqrt {2 \sqrt {x} - 1} + 3} + 2\right )^{\frac {11}{2}}}{11} + \frac {304 \left (\sqrt {\sqrt {2 \sqrt {x} - 1} + 3} + 2\right )^{\frac {9}{2}}}{3} - \frac {480 \left (\sqrt {\sqrt {2 \sqrt {x} - 1} + 3} + 2\right )^{\frac {7}{2}}}{7} + \frac {136 \left (\sqrt {\sqrt {2 \sqrt {x} - 1} + 3} + 2\right )^{\frac {5}{2}}}{5} - \frac {16 \left (\sqrt {\sqrt {2 \sqrt {x} - 1} + 3} + 2\right )^{\frac {3}{2}}}{3} \] Input:
integrate((2+(3+(-1+2*x**(1/2))**(1/2))**(1/2))**(1/2),x)
Output:
4*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)**(17/2)/17 - 56*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)**(15/2)/15 + 300*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)**(13/ 2)/13 - 760*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)**(11/2)/11 + 304*(sqrt(sqr t(2*sqrt(x) - 1) + 3) + 2)**(9/2)/3 - 480*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)**(7/2)/7 + 136*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)**(5/2)/5 - 16*(sqrt (sqrt(2*sqrt(x) - 1) + 3) + 2)**(3/2)/3
Time = 0.03 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.66 \[ \int \sqrt {2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}} \, dx=\frac {4}{17} \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {17}{2}} - \frac {56}{15} \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {15}{2}} + \frac {300}{13} \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {13}{2}} - \frac {760}{11} \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {11}{2}} + \frac {304}{3} \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {9}{2}} - \frac {480}{7} \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {7}{2}} + \frac {136}{5} \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {5}{2}} - \frac {16}{3} \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {3}{2}} \] Input:
integrate((2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(1/2),x, algorithm="maxima")
Output:
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*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(13/ 2) - 760/11*(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)^(11/2) + 304/3*(sqrt(sqrt( 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)
Time = 3.52 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.16 \[ \int \sqrt {2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}} \, dx=\frac {4}{255255} \, {\left (15015 \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {17}{2}} - 238238 \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {15}{2}} + 1472625 \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {13}{2}} - 4408950 \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {11}{2}} + 6466460 \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {9}{2}} - 4375800 \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {7}{2}} + 1735734 \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {5}{2}} - 340340 \, {\left (\sqrt {\sqrt {2 \, \sqrt {x} - 1} + 3} + 2\right )}^{\frac {3}{2}}\right )} \mathrm {sgn}\left (8192 \, x^{23} + 376832 \, x^{22} + 8224768 \, x^{21} + 113971200 \, x^{20} + 1130782720 \, x^{19} + 8582063104 \, x^{18} + 51933387264 \, x^{17} + 257575619584 \, x^{16} + 1066188686592 \, x^{15} + 3723204389632 \, x^{14} + 11019822890016 \, x^{13} + 27631512444352 \, x^{12} + 58424530490176 \, x^{11} + 103336828749760 \, x^{10} + 151203890043312 \, x^{9} + 180411181747936 \, x^{8} + 172287199292960 \, x^{7} + 128457231939048 \, x^{6} + 72257964298210 \, x^{5} + 29175203228012 \, x^{4} + 7830371130072 \, x^{3} + 1228114804752 \, x^{2} + 87490886400 \, x + 933120000\right ) \] Input:
integrate((2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(1/2),x, algorithm="giac")
Output:
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) + 646 6460*(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 + 376 832*x^22 + 8224768*x^21 + 113971200*x^20 + 1130782720*x^19 + 8582063104*x^ 18 + 51933387264*x^17 + 257575619584*x^16 + 1066188686592*x^15 + 372320438 9632*x^14 + 11019822890016*x^13 + 27631512444352*x^12 + 58424530490176*x^1 1 + 103336828749760*x^10 + 151203890043312*x^9 + 180411181747936*x^8 + 172 287199292960*x^7 + 128457231939048*x^6 + 72257964298210*x^5 + 291752032280 12*x^4 + 7830371130072*x^3 + 1228114804752*x^2 + 87490886400*x + 933120000 )
Timed out. \[ \int \sqrt {2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}} \, dx=\int \sqrt {\sqrt {\sqrt {2\,\sqrt {x}-1}+3}+2} \,d x \] Input:
int((((2*x^(1/2) - 1)^(1/2) + 3)^(1/2) + 2)^(1/2),x)
Output:
int((((2*x^(1/2) - 1)^(1/2) + 3)^(1/2) + 2)^(1/2), x)
Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.47 \[ \int \sqrt {2+\sqrt {3+\sqrt {-1+2 \sqrt {x}}}} \, dx=\frac {8 \sqrt {\sqrt {\sqrt {2 \sqrt {x}-1}+3}+2}\, \left (2002 \sqrt {x}\, \sqrt {2 \sqrt {x}-1}\, \sqrt {\sqrt {2 \sqrt {x}-1}+3}+13600 \sqrt {2 \sqrt {x}-1}\, \sqrt {\sqrt {2 \sqrt {x}-1}+3}-4704 \sqrt {x}\, \sqrt {\sqrt {2 \sqrt {x}-1}+3}-59424 \sqrt {\sqrt {2 \sqrt {x}-1}+3}-847 \sqrt {x}\, \sqrt {2 \sqrt {x}-1}+1688 \sqrt {2 \sqrt {x}-1}-3843 \sqrt {x}+30030 x -124080\right )}{255255} \] Input:
int((2+(3+(-1+2*x^(1/2))^(1/2))^(1/2))^(1/2),x)
Output:
(8*sqrt(sqrt(sqrt(2*sqrt(x) - 1) + 3) + 2)*(2002*sqrt(x)*sqrt(2*sqrt(x) - 1)*sqrt(sqrt(2*sqrt(x) - 1) + 3) + 13600*sqrt(2*sqrt(x) - 1)*sqrt(sqrt(2*s qrt(x) - 1) + 3) - 4704*sqrt(x)*sqrt(sqrt(2*sqrt(x) - 1) + 3) - 59424*sqrt (sqrt(2*sqrt(x) - 1) + 3) - 847*sqrt(x)*sqrt(2*sqrt(x) - 1) + 1688*sqrt(2* sqrt(x) - 1) - 3843*sqrt(x) + 30030*x - 124080))/255255