3.715 \(\int \sqrt {2-\sqrt {4+\sqrt {-9+5 x}}} \, dx\)

Optimal. Leaf size=82 \[ \frac {8}{45} \left (2-\sqrt {\sqrt {5 x-9}+4}\right )^{9/2}-\frac {48}{35} \left (2-\sqrt {\sqrt {5 x-9}+4}\right )^{7/2}+\frac {64}{25} \left (2-\sqrt {\sqrt {5 x-9}+4}\right )^{5/2} \]

[Out]

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)

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Rubi [A]  time = 0.08, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {371, 1398, 772} \[ \frac {8}{45} \left (2-\sqrt {\sqrt {5 x-9}+4}\right )^{9/2}-\frac {48}{35} \left (2-\sqrt {\sqrt {5 x-9}+4}\right )^{7/2}+\frac {64}{25} \left (2-\sqrt {\sqrt {5 x-9}+4}\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 - Sqrt[4 + Sqrt[-9 + 5*x]]],x]

[Out]

(64*(2 - Sqrt[4 + Sqrt[-9 + 5*x]])^(5/2))/25 - (48*(2 - Sqrt[4 + Sqrt[-9 + 5*x]])^(7/2))/35 + (8*(2 - Sqrt[4 +
Sqrt[-9 + 5*x]])^(9/2))/45

Rule 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(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]

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rule 1398

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p
, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rubi steps

\begin {align*} \int \sqrt {2-\sqrt {4+\sqrt {-9+5 x}}} \, dx &=\frac {2}{5} \operatorname {Subst}\left (\int x \sqrt {2-\sqrt {4+x}} \, dx,x,\sqrt {-9+5 x}\right )\\ &=\frac {2}{5} \operatorname {Subst}\left (\int \sqrt {2-\sqrt {x}} (-4+x) \, dx,x,4+\sqrt {-9+5 x}\right )\\ &=\frac {4}{5} \operatorname {Subst}\left (\int \sqrt {2-x} x \left (-4+x^2\right ) \, dx,x,\sqrt {4+\sqrt {-9+5 x}}\right )\\ &=\frac {4}{5} \operatorname {Subst}\left (\int \left (-8 (2-x)^{3/2}+6 (2-x)^{5/2}-(2-x)^{7/2}\right ) \, dx,x,\sqrt {4+\sqrt {-9+5 x}}\right )\\ &=\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}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 57, normalized size = 0.70 \[ \frac {8 \left (2-\sqrt {\sqrt {5 x-9}+4}\right )^{5/2} \left (35 \sqrt {5 x-9}+130 \sqrt {\sqrt {5 x-9}+4}+244\right )}{1575} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 - Sqrt[4 + Sqrt[-9 + 5*x]]],x]

[Out]

(8*(2 - Sqrt[4 + Sqrt[-9 + 5*x]])^(5/2)*(244 + 35*Sqrt[-9 + 5*x] + 130*Sqrt[4 + Sqrt[-9 + 5*x]]))/1575

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fricas [A]  time = 0.47, size = 57, normalized size = 0.70 \[ -\frac {8}{1575} \, {\left (2 \, {\left (5 \, \sqrt {5 \, x - 9} - 32\right )} \sqrt {\sqrt {5 \, x - 9} + 4} - 175 \, x - 4 \, \sqrt {5 \, x - 9} + 443\right )} \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-(4+(-9+5*x)^(1/2))^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-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)

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giac [B]  time = 7.17, size = 474, normalized size = 5.78 \[ -\frac {8}{1575} \, {\left ({\left (35 \, {\left (\sqrt {\sqrt {5 \, x - 9} + 4} - 2\right )}^{4} \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2} + 360 \, {\left (\sqrt {\sqrt {5 \, x - 9} + 4} - 2\right )}^{3} \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2} + 1512 \, {\left (\sqrt {\sqrt {5 \, x - 9} + 4} - 2\right )}^{2} \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2} - 3360 \, {\left (-\sqrt {\sqrt {5 \, x - 9} + 4} + 2\right )}^{\frac {3}{2}} + 5040 \, \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2}\right )} \mathrm {sgn}\left (-4 \, {\left (\sqrt {5 \, x - 9} + 4\right )}^{2} + 32 \, \sqrt {5 \, x - 9} + 79\right ) - 18 \, {\left (5 \, {\left (\sqrt {\sqrt {5 \, x - 9} + 4} - 2\right )}^{3} \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2} + 42 \, {\left (\sqrt {\sqrt {5 \, x - 9} + 4} - 2\right )}^{2} \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2} - 140 \, {\left (-\sqrt {\sqrt {5 \, x - 9} + 4} + 2\right )}^{\frac {3}{2}} + 280 \, \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2}\right )} \mathrm {sgn}\left (-4 \, {\left (\sqrt {5 \, x - 9} + 4\right )}^{2} + 32 \, \sqrt {5 \, x - 9} + 79\right ) - 84 \, {\left (3 \, {\left (\sqrt {\sqrt {5 \, x - 9} + 4} - 2\right )}^{2} \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2} - 20 \, {\left (-\sqrt {\sqrt {5 \, x - 9} + 4} + 2\right )}^{\frac {3}{2}} + 60 \, \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2}\right )} \mathrm {sgn}\left (-4 \, {\left (\sqrt {5 \, x - 9} + 4\right )}^{2} + 32 \, \sqrt {5 \, x - 9} + 79\right ) - 840 \, {\left ({\left (-\sqrt {\sqrt {5 \, x - 9} + 4} + 2\right )}^{\frac {3}{2}} - 6 \, \sqrt {-\sqrt {\sqrt {5 \, x - 9} + 4} + 2}\right )} \mathrm {sgn}\left (-4 \, {\left (\sqrt {5 \, x - 9} + 4\right )}^{2} + 32 \, \sqrt {5 \, x - 9} + 79\right )\right )} \mathrm {sgn}\left (20 \, x - 51\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-(4+(-9+5*x)^(1/2))^(1/2))^(1/2),x, algorithm="giac")

[Out]

-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) - 140*(-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*(sqrt(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))*sgn(-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)

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maple [A]  time = 0.01, size = 59, normalized size = 0.72 \[ \frac {64 \left (2-\sqrt {4+\sqrt {5 x -9}}\right )^{\frac {5}{2}}}{25}-\frac {48 \left (2-\sqrt {4+\sqrt {5 x -9}}\right )^{\frac {7}{2}}}{35}+\frac {8 \left (2-\sqrt {4+\sqrt {5 x -9}}\right )^{\frac {9}{2}}}{45} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2-(4+(-9+5*x)^(1/2))^(1/2))^(1/2),x)

[Out]

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)

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maxima [A]  time = 0.87, size = 58, normalized size = 0.71 \[ \frac {8}{45} \, {\left (-\sqrt {\sqrt {5 \, x - 9} + 4} + 2\right )}^{\frac {9}{2}} - \frac {48}{35} \, {\left (-\sqrt {\sqrt {5 \, x - 9} + 4} + 2\right )}^{\frac {7}{2}} + \frac {64}{25} \, {\left (-\sqrt {\sqrt {5 \, x - 9} + 4} + 2\right )}^{\frac {5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-(4+(-9+5*x)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {2-\sqrt {\sqrt {5\,x-9}+4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2 - ((5*x - 9)^(1/2) + 4)^(1/2))^(1/2),x)

[Out]

int((2 - ((5*x - 9)^(1/2) + 4)^(1/2))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {2 - \sqrt {\sqrt {5 x - 9} + 4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-(4+(-9+5*x)**(1/2))**(1/2))**(1/2),x)

[Out]

Integral(sqrt(2 - sqrt(sqrt(5*x - 9) + 4)), x)

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