Integrand size = 23, antiderivative size = 82 \[ \int \sqrt {2-\sqrt {4+\sqrt {-9+5 x}}} \, dx=\frac {64}{25} \left (2-\sqrt {4+\sqrt {-9+5 x}}\right )^{5/2}-\frac {48}{35} \left (2-\sqrt {4+\sqrt {-9+5 x}}\right )^{7/2}+\frac {8}{45} \left (2-\sqrt {4+\sqrt {-9+5 x}}\right )^{9/2} \]
64/25*(2-(4+(-9+5*x)^(1/2))^(1/2))^(5/2)-48/35*(2-(4+(-9+5*x)^(1/2))^(1/2) )^(7/2)+8/45*(2-(4+(-9+5*x)^(1/2))^(1/2))^(9/2)
Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05 \[ \int \sqrt {2-\sqrt {4+\sqrt {-9+5 x}}} \, dx=-\frac {8 \sqrt {2-\sqrt {4+\sqrt {-9+5 x}}} \left (443-175 x-4 \sqrt {-9+5 x}-64 \sqrt {4+\sqrt {-9+5 x}}+10 \sqrt {-9+5 x} \sqrt {4+\sqrt {-9+5 x}}\right )}{1575} \]
(-8*Sqrt[2 - Sqrt[4 + Sqrt[-9 + 5*x]]]*(443 - 175*x - 4*Sqrt[-9 + 5*x] - 6 4*Sqrt[4 + Sqrt[-9 + 5*x]] + 10*Sqrt[-9 + 5*x]*Sqrt[4 + Sqrt[-9 + 5*x]]))/ 1575
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {7267, 896, 25, 1388, 900, 86, 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.
| \(\displaystyle \int \sqrt {2-\sqrt {\sqrt {5 x-9}+4}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {2}{5} \int \sqrt {5 x-9} \sqrt {2-\sqrt {\sqrt {5 x-9}+4}}d\sqrt {5 x-9}\) |
| \(\Big \downarrow \) 896 |
| \(\displaystyle \frac {2}{5} \int \sqrt {5 x-9} \sqrt {2-\sqrt [4]{5 x-9}}d\left (\sqrt {5 x-9}+4\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2}{5} \int -\sqrt {5 x-9} \sqrt {2-\sqrt [4]{5 x-9}}d\left (\sqrt {5 x-9}+4\right )\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {2}{5} \int \left (2-\sqrt [4]{5 x-9}\right )^{3/2} \left (\sqrt [4]{5 x-9}+2\right )d\left (\sqrt {5 x-9}+4\right )\) |
| \(\Big \downarrow \) 900 |
| \(\displaystyle -\frac {4}{5} \int \sqrt [4]{5 x-9} \left (-\sqrt {5 x-9}-2\right )^{3/2} \left (\sqrt {5 x-9}+6\right )d\sqrt [4]{5 x-9}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle -\frac {4}{5} \int \left (\left (-\sqrt {5 x-9}-2\right )^{7/2}-6 \left (-\sqrt {5 x-9}-2\right )^{5/2}+8 \left (-\sqrt {5 x-9}-2\right )^{3/2}\right )d\sqrt [4]{5 x-9}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4}{5} \left (-\frac {2}{9} \left (-\sqrt {5 x-9}-2\right )^{9/2}+\frac {12}{7} \left (-\sqrt {5 x-9}-2\right )^{7/2}-\frac {16}{5} \left (-\sqrt {5 x-9}-2\right )^{5/2}\right )\) |
(-4*((-16*(-2 - Sqrt[-9 + 5*x])^(5/2))/5 + (12*(-2 - Sqrt[-9 + 5*x])^(7/2) )/7 - (2*(-2 - Sqrt[-9 + 5*x])^(9/2))/9))/5
3.8.15.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1) Subst[Int[Si mplifyIntegrand[(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]
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> With[{g = Denominator[n]}, Simp[g Subst[Int[x^(g - 1)*(a + b*x^(g*n) )^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[n]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
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 = 0.36 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.72
| method | result | size |
| derivativedivides | \(\frac {64 \left (2-\sqrt {4+\sqrt {-9+5 x}}\right )^{\frac {5}{2}}}{25}-\frac {48 \left (2-\sqrt {4+\sqrt {-9+5 x}}\right )^{\frac {7}{2}}}{35}+\frac {8 \left (2-\sqrt {4+\sqrt {-9+5 x}}\right )^{\frac {9}{2}}}{45}\) | \(59\) |
| default | \(\frac {64 \left (2-\sqrt {4+\sqrt {-9+5 x}}\right )^{\frac {5}{2}}}{25}-\frac {48 \left (2-\sqrt {4+\sqrt {-9+5 x}}\right )^{\frac {7}{2}}}{35}+\frac {8 \left (2-\sqrt {4+\sqrt {-9+5 x}}\right )^{\frac {9}{2}}}{45}\) | \(59\) |
64/25*(2-(4+(-9+5*x)^(1/2))^(1/2))^(5/2)-48/35*(2-(4+(-9+5*x)^(1/2))^(1/2) )^(7/2)+8/45*(2-(4+(-9+5*x)^(1/2))^(1/2))^(9/2)
Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.70 \[ \int \sqrt {2-\sqrt {4+\sqrt {-9+5 x}}} \, dx=-\frac {8}{1575} \, {\left (2 \, {\left (5 \, \sqrt {5 \, x - 9} - 32\right )} \sqrt {\sqrt {5 \, x - 9} + 4} - 175 \, x - 4 \, \sqrt {5 \, x - 9} + 443\right )} \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2} \]
-8/1575*(2*(5*sqrt(5*x - 9) - 32)*sqrt(sqrt(5*x - 9) + 4) - 175*x - 4*sqrt (5*x - 9) + 443)*sqrt(-sqrt(sqrt(5*x - 9) + 4) + 2)
Time = 0.65 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.79 \[ \int \sqrt {2-\sqrt {4+\sqrt {-9+5 x}}} \, dx=\frac {8 \left (2 - \sqrt {\sqrt {5 x - 9} + 4}\right )^{\frac {9}{2}}}{45} - \frac {48 \left (2 - \sqrt {\sqrt {5 x - 9} + 4}\right )^{\frac {7}{2}}}{35} + \frac {64 \left (2 - \sqrt {\sqrt {5 x - 9} + 4}\right )^{\frac {5}{2}}}{25} \]
8*(2 - sqrt(sqrt(5*x - 9) + 4))**(9/2)/45 - 48*(2 - sqrt(sqrt(5*x - 9) + 4 ))**(7/2)/35 + 64*(2 - sqrt(sqrt(5*x - 9) + 4))**(5/2)/25
Time = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.71 \[ \int \sqrt {2-\sqrt {4+\sqrt {-9+5 x}}} \, dx=\frac {8}{45} \, {\left (-\sqrt {\sqrt {5 \, x - 9} + 4} + 2\right )}^{\frac {9}{2}} - \frac {48}{35} \, {\left (-\sqrt {\sqrt {5 \, x - 9} + 4} + 2\right )}^{\frac {7}{2}} + \frac {64}{25} \, {\left (-\sqrt {\sqrt {5 \, x - 9} + 4} + 2\right )}^{\frac {5}{2}} \]
8/45*(-sqrt(sqrt(5*x - 9) + 4) + 2)^(9/2) - 48/35*(-sqrt(sqrt(5*x - 9) + 4 ) + 2)^(7/2) + 64/25*(-sqrt(sqrt(5*x - 9) + 4) + 2)^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (58) = 116\).
Time = 0.41 (sec) , antiderivative size = 474, normalized size of antiderivative = 5.78 \[ \int \sqrt {2-\sqrt {4+\sqrt {-9+5 x}}} \, dx=-\frac {8}{1575} \, {\left ({\left (35 \, {\left (\sqrt {\sqrt {5 \, x - 9} + 4} - 2\right )}^{4} \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2} + 360 \, {\left (\sqrt {\sqrt {5 \, x - 9} + 4} - 2\right )}^{3} \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2} + 1512 \, {\left (\sqrt {\sqrt {5 \, x - 9} + 4} - 2\right )}^{2} \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2} - 3360 \, {\left (-\sqrt {\sqrt {5 \, x - 9} + 4} + 2\right )}^{\frac {3}{2}} + 5040 \, \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2}\right )} \mathrm {sgn}\left (-4 \, {\left (\sqrt {5 \, x - 9} + 4\right )}^{2} + 32 \, \sqrt {5 \, x - 9} + 79\right ) - 18 \, {\left (5 \, {\left (\sqrt {\sqrt {5 \, x - 9} + 4} - 2\right )}^{3} \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2} + 42 \, {\left (\sqrt {\sqrt {5 \, x - 9} + 4} - 2\right )}^{2} \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2} - 140 \, {\left (-\sqrt {\sqrt {5 \, x - 9} + 4} + 2\right )}^{\frac {3}{2}} + 280 \, \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2}\right )} \mathrm {sgn}\left (-4 \, {\left (\sqrt {5 \, x - 9} + 4\right )}^{2} + 32 \, \sqrt {5 \, x - 9} + 79\right ) - 84 \, {\left (3 \, {\left (\sqrt {\sqrt {5 \, x - 9} + 4} - 2\right )}^{2} \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2} - 20 \, {\left (-\sqrt {\sqrt {5 \, x - 9} + 4} + 2\right )}^{\frac {3}{2}} + 60 \, \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2}\right )} \mathrm {sgn}\left (-4 \, {\left (\sqrt {5 \, x - 9} + 4\right )}^{2} + 32 \, \sqrt {5 \, x - 9} + 79\right ) - 840 \, {\left ({\left (-\sqrt {\sqrt {5 \, x - 9} + 4} + 2\right )}^{\frac {3}{2}} - 6 \, \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2}\right )} \mathrm {sgn}\left (-4 \, {\left (\sqrt {5 \, x - 9} + 4\right )}^{2} + 32 \, \sqrt {5 \, x - 9} + 79\right )\right )} \mathrm {sgn}\left (20 \, x - 51\right ) \]
-8/1575*((35*(sqrt(sqrt(5*x - 9) + 4) - 2)^4*sqrt(-sqrt(sqrt(5*x - 9) + 4) + 2) + 360*(sqrt(sqrt(5*x - 9) + 4) - 2)^3*sqrt(-sqrt(sqrt(5*x - 9) + 4) + 2) + 1512*(sqrt(sqrt(5*x - 9) + 4) - 2)^2*sqrt(-sqrt(sqrt(5*x - 9) + 4) + 2) - 3360*(-sqrt(sqrt(5*x - 9) + 4) + 2)^(3/2) + 5040*sqrt(-sqrt(sqrt(5* x - 9) + 4) + 2))*sgn(-4*(sqrt(5*x - 9) + 4)^2 + 32*sqrt(5*x - 9) + 79) - 18*(5*(sqrt(sqrt(5*x - 9) + 4) - 2)^3*sqrt(-sqrt(sqrt(5*x - 9) + 4) + 2) + 42*(sqrt(sqrt(5*x - 9) + 4) - 2)^2*sqrt(-sqrt(sqrt(5*x - 9) + 4) + 2) - 1 40*(-sqrt(sqrt(5*x - 9) + 4) + 2)^(3/2) + 280*sqrt(-sqrt(sqrt(5*x - 9) + 4 ) + 2))*sgn(-4*(sqrt(5*x - 9) + 4)^2 + 32*sqrt(5*x - 9) + 79) - 84*(3*(sqr t(sqrt(5*x - 9) + 4) - 2)^2*sqrt(-sqrt(sqrt(5*x - 9) + 4) + 2) - 20*(-sqrt (sqrt(5*x - 9) + 4) + 2)^(3/2) + 60*sqrt(-sqrt(sqrt(5*x - 9) + 4) + 2))*sg n(-4*(sqrt(5*x - 9) + 4)^2 + 32*sqrt(5*x - 9) + 79) - 840*((-sqrt(sqrt(5*x - 9) + 4) + 2)^(3/2) - 6*sqrt(-sqrt(sqrt(5*x - 9) + 4) + 2))*sgn(-4*(sqrt (5*x - 9) + 4)^2 + 32*sqrt(5*x - 9) + 79))*sgn(20*x - 51)
Timed out. \[ \int \sqrt {2-\sqrt {4+\sqrt {-9+5 x}}} \, dx=\int \sqrt {2-\sqrt {\sqrt {5\,x-9}+4}} \,d x \]