3.25.48 \(\int \sqrt {d+\sqrt {c+\sqrt {b+a x}}} \, dx\)

Optimal. Leaf size=198 \[ \frac {8 \left (35 a x+35 b-28 c^2+36 c d^2-16 d^4\right ) \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{315 a}-\frac {64 \left (2 c d-d^3\right ) \sqrt {\sqrt {a x+b}+c} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{315 a}+\sqrt {a x+b} \left (\frac {8 \left (7 c-6 d^2\right ) \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{315 a}+\frac {8 d \sqrt {\sqrt {a x+b}+c} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{63 a}\right ) \]

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Rubi [A]  time = 0.15, antiderivative size = 129, normalized size of antiderivative = 0.65, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {371, 1398, 772} \begin {gather*} -\frac {8 \left (c-3 d^2\right ) \left (\sqrt {\sqrt {a x+b}+c}+d\right )^{5/2}}{5 a}+\frac {8 d \left (c-d^2\right ) \left (\sqrt {\sqrt {a x+b}+c}+d\right )^{3/2}}{3 a}+\frac {8 \left (\sqrt {\sqrt {a x+b}+c}+d\right )^{9/2}}{9 a}-\frac {24 d \left (\sqrt {\sqrt {a x+b}+c}+d\right )^{7/2}}{7 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]],x]

[Out]

(8*d*(c - d^2)*(d + Sqrt[c + Sqrt[b + a*x]])^(3/2))/(3*a) - (8*(c - 3*d^2)*(d + Sqrt[c + Sqrt[b + a*x]])^(5/2)
)/(5*a) - (24*d*(d + Sqrt[c + Sqrt[b + a*x]])^(7/2))/(7*a) + (8*(d + Sqrt[c + Sqrt[b + a*x]])^(9/2))/(9*a)

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 {d+\sqrt {c+\sqrt {b+a x}}} \, dx &=\frac {2 \operatorname {Subst}\left (\int x \sqrt {d+\sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right )}{a}\\ &=\frac {2 \operatorname {Subst}\left (\int \sqrt {d+\sqrt {x}} (-c+x) \, dx,x,c+\sqrt {b+a x}\right )}{a}\\ &=\frac {4 \operatorname {Subst}\left (\int x \sqrt {d+x} \left (-c+x^2\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {4 \operatorname {Subst}\left (\int \left (-d \left (-c+d^2\right ) \sqrt {d+x}+\left (-c+3 d^2\right ) (d+x)^{3/2}-3 d (d+x)^{5/2}+(d+x)^{7/2}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {8 d \left (c-d^2\right ) \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}}{3 a}-\frac {8 \left (c-3 d^2\right ) \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}}{5 a}-\frac {24 d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{7/2}}{7 a}+\frac {8 \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{9/2}}{9 a}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 114, normalized size = 0.58 \begin {gather*} \frac {8 \left (\sqrt {\sqrt {a x+b}+c}+d\right )^{3/2} \left (24 d^2 \sqrt {\sqrt {a x+b}+c}-28 c \sqrt {\sqrt {a x+b}+c}+35 \sqrt {a x+b} \sqrt {\sqrt {a x+b}+c}-30 d \sqrt {a x+b}+12 c d-16 d^3\right )}{315 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]],x]

[Out]

(8*(d + Sqrt[c + Sqrt[b + a*x]])^(3/2)*(12*c*d - 16*d^3 - 30*d*Sqrt[b + a*x] - 28*c*Sqrt[c + Sqrt[b + a*x]] +
24*d^2*Sqrt[c + Sqrt[b + a*x]] + 35*Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]))/(315*a)

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IntegrateAlgebraic [A]  time = 0.12, size = 144, normalized size = 0.73 \begin {gather*} -\frac {8 \sqrt {c+\sqrt {b+a x}} \left (16 c d-8 d^3-5 d \sqrt {b+a x}\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{315 a}-\frac {8 \left (28 c^2-36 c d^2+16 d^4-7 c \sqrt {b+a x}+6 d^2 \sqrt {b+a x}-35 (b+a x)\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{315 a} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]],x]

[Out]

(-8*Sqrt[c + Sqrt[b + a*x]]*(16*c*d - 8*d^3 - 5*d*Sqrt[b + a*x])*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/(315*a) -
(8*(28*c^2 - 36*c*d^2 + 16*d^4 - 7*c*Sqrt[b + a*x] + 6*d^2*Sqrt[b + a*x] - 35*(b + a*x))*Sqrt[d + Sqrt[c + Sqr
t[b + a*x]]])/(315*a)

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fricas [A]  time = 0.75, size = 94, normalized size = 0.47 \begin {gather*} -\frac {8 \, {\left (16 \, d^{4} - 36 \, c d^{2} + 28 \, c^{2} - 35 \, a x + {\left (6 \, d^{2} - 7 \, c\right )} \sqrt {a x + b} - {\left (8 \, d^{3} - 16 \, c d + 5 \, \sqrt {a x + b} d\right )} \sqrt {c + \sqrt {a x + b}} - 35 \, b\right )} \sqrt {d + \sqrt {c + \sqrt {a x + b}}}}{315 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-8/315*(16*d^4 - 36*c*d^2 + 28*c^2 - 35*a*x + (6*d^2 - 7*c)*sqrt(a*x + b) - (8*d^3 - 16*c*d + 5*sqrt(a*x + b)*
d)*sqrt(c + sqrt(a*x + b)) - 35*b)*sqrt(d + sqrt(c + sqrt(a*x + b)))/a

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giac [B]  time = 2.66, size = 444, normalized size = 2.24 \begin {gather*} \frac {8 \, {\left (35 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {9}{2}} - 180 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {7}{2}} d + 378 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {d + \sqrt {c + \sqrt {a x + b}}} d^{4} - 126 \, c {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} + 420 \, c {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} d - 630 \, c \sqrt {d + \sqrt {c + \sqrt {a x + b}}} d^{2} + 315 \, c^{2} \sqrt {d + \sqrt {c + \sqrt {a x + b}}} + 21 \, {\left (3 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} - 10 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} d + 15 \, \sqrt {d + \sqrt {c + \sqrt {a x + b}}} d^{2} - 15 \, c \sqrt {d + \sqrt {c + \sqrt {a x + b}}}\right )} c + 3 \, {\left (15 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right ) - 63 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} d \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right ) + 105 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} d^{2} \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right ) - 105 \, \sqrt {d + \sqrt {c + \sqrt {a x + b}}} d^{3} \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right ) - 35 \, c {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right ) + 105 \, c \sqrt {d + \sqrt {c + \sqrt {a x + b}}} d \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right )\right )} d\right )}}{315 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="giac")

[Out]

8/315*(35*(d + sqrt(c + sqrt(a*x + b)))^(9/2) - 180*(d + sqrt(c + sqrt(a*x + b)))^(7/2)*d + 378*(d + sqrt(c +
sqrt(a*x + b)))^(5/2)*d^2 - 420*(d + sqrt(c + sqrt(a*x + b)))^(3/2)*d^3 + 315*sqrt(d + sqrt(c + sqrt(a*x + b))
)*d^4 - 126*c*(d + sqrt(c + sqrt(a*x + b)))^(5/2) + 420*c*(d + sqrt(c + sqrt(a*x + b)))^(3/2)*d - 630*c*sqrt(d
+ sqrt(c + sqrt(a*x + b)))*d^2 + 315*c^2*sqrt(d + sqrt(c + sqrt(a*x + b))) + 21*(3*(d + sqrt(c + sqrt(a*x + b
)))^(5/2) - 10*(d + sqrt(c + sqrt(a*x + b)))^(3/2)*d + 15*sqrt(d + sqrt(c + sqrt(a*x + b)))*d^2 - 15*c*sqrt(d
+ sqrt(c + sqrt(a*x + b))))*c + 3*(15*(d + sqrt(c + sqrt(a*x + b)))^(7/2)*sgn(sqrt(c + sqrt(a*x + b))) - 63*(d
+ sqrt(c + sqrt(a*x + b)))^(5/2)*d*sgn(sqrt(c + sqrt(a*x + b))) + 105*(d + sqrt(c + sqrt(a*x + b)))^(3/2)*d^2
*sgn(sqrt(c + sqrt(a*x + b))) - 105*sqrt(d + sqrt(c + sqrt(a*x + b)))*d^3*sgn(sqrt(c + sqrt(a*x + b))) - 35*c*
(d + sqrt(c + sqrt(a*x + b)))^(3/2)*sgn(sqrt(c + sqrt(a*x + b))) + 105*c*sqrt(d + sqrt(c + sqrt(a*x + b)))*d*s
gn(sqrt(c + sqrt(a*x + b))))*d)/a

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maple [A]  time = 0.19, size = 93, normalized size = 0.47

method	result	size
derivativedivides	\(\frac {\frac {8 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {9}{2}}}{9}-\frac {24 d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {7}{2}}}{7}+\frac {8 \left (3 d^{2}-c \right ) \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}}}{5}-\frac {8 \left (d^{2}-c \right ) d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}}{a}\)	\(93\)
default	\(\frac {\frac {8 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {9}{2}}}{9}-\frac {24 d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {7}{2}}}{7}+\frac {8 \left (3 d^{2}-c \right ) \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}}}{5}-\frac {8 \left (d^{2}-c \right ) d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}}{a}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/a*(4/9*(d+(c+(a*x+b)^(1/2))^(1/2))^(9/2)-12/7*d*(d+(c+(a*x+b)^(1/2))^(1/2))^(7/2)+4/5*(3*d^2-c)*(d+(c+(a*x+b
)^(1/2))^(1/2))^(5/2)-4/3*(d^2-c)*d*(d+(c+(a*x+b)^(1/2))^(1/2))^(3/2))

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maxima [A]  time = 0.34, size = 92, normalized size = 0.46 \begin {gather*} \frac {8 \, {\left (35 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {9}{2}} - 135 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {7}{2}} d + 63 \, {\left (3 \, d^{2} - c\right )} {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} - 105 \, {\left (d^{3} - c d\right )} {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}}\right )}}{315 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

8/315*(35*(d + sqrt(c + sqrt(a*x + b)))^(9/2) - 135*(d + sqrt(c + sqrt(a*x + b)))^(7/2)*d + 63*(3*d^2 - c)*(d
+ sqrt(c + sqrt(a*x + b)))^(5/2) - 105*(d^3 - c*d)*(d + sqrt(c + sqrt(a*x + b)))^(3/2))/a

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {d+\sqrt {c+\sqrt {b+a\,x}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + (c + (b + a*x)^(1/2))^(1/2))^(1/2),x)

[Out]

int((d + (c + (b + a*x)^(1/2))^(1/2))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {d + \sqrt {c + \sqrt {a x + b}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+(c+(a*x+b)**(1/2))**(1/2))**(1/2),x)

[Out]

Integral(sqrt(d + sqrt(c + sqrt(a*x + b))), x)

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