Integrand size = 21, antiderivative size = 198 \[ \int \sqrt {d+\sqrt {c+\sqrt {b+a x}}} \, dx=\frac {8 \left (35 b-28 c^2+36 c d^2-16 d^4+35 a x\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{315 a}-\frac {64 \left (2 c d-d^3\right ) \sqrt {c+\sqrt {b+a x}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{315 a}+\sqrt {b+a x} \left (\frac {8 \left (7 c-6 d^2\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{315 a}+\frac {8 d \sqrt {c+\sqrt {b+a x}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{63 a}\right ) \] Output:
8/315*(-16*d^4+36*c*d^2+35*a*x-28*c^2+35*b)*(d+(c+(a*x+b)^(1/2))^(1/2))^(1 /2)/a-64/315*(-d^3+2*c*d)*(c+(a*x+b)^(1/2))^(1/2)*(d+(c+(a*x+b)^(1/2))^(1/ 2))^(1/2)/a+(a*x+b)^(1/2)*(8/315*(-6*d^2+7*c)*(d+(c+(a*x+b)^(1/2))^(1/2))^ (1/2)/a+8/63*d*(c+(a*x+b)^(1/2))^(1/2)*(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2)/a )
Time = 0.16 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.72 \[ \int \sqrt {d+\sqrt {c+\sqrt {b+a x}}} \, dx=\frac {8 d \sqrt {c+\sqrt {b+a x}} \left (-16 c+8 d^2+5 \sqrt {b+a x}\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{315 a}-\frac {8 \left (28 c^2-36 c d^2+16 d^4-7 c \sqrt {b+a x}+6 d^2 \sqrt {b+a x}-35 (b+a x)\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{315 a} \] Input:
Integrate[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]],x]
Output:
(8*d*Sqrt[c + Sqrt[b + a*x]]*(-16*c + 8*d^2 + 5*Sqrt[b + a*x])*Sqrt[d + Sq rt[c + Sqrt[b + a*x]]])/(315*a) - (8*(28*c^2 - 36*c*d^2 + 16*d^4 - 7*c*Sqr t[b + a*x] + 6*d^2*Sqrt[b + a*x] - 35*(b + a*x))*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/(315*a)
Time = 0.34 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.49, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {7267, 896, 25, 1732, 522, 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.
\(\displaystyle \int \sqrt {\sqrt {\sqrt {a x+b}+c}+d} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle \frac {2 \int \sqrt {b+a x} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}d\sqrt {b+a x}}{a}\) |
\(\Big \downarrow \) 896 |
\(\displaystyle \frac {2 \int \sqrt {b+a x} \sqrt {d+\sqrt [4]{b+a x}}d\left (c+\sqrt {b+a x}\right )}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {2 \int -\sqrt {b+a x} \sqrt {d+\sqrt [4]{b+a x}}d\left (c+\sqrt {b+a x}\right )}{a}\) |
\(\Big \downarrow \) 1732 |
\(\displaystyle -\frac {4 \int (-b+c-a x) \sqrt [4]{b+a x} \sqrt {d+\sqrt [4]{b+a x}}d\sqrt [4]{b+a x}}{a}\) |
\(\Big \downarrow \) 522 |
\(\displaystyle -\frac {4 \int \left (-\left (d+\sqrt [4]{b+a x}\right )^{7/2}+3 d \left (d+\sqrt [4]{b+a x}\right )^{5/2}+\left (c-3 d^2\right ) \left (d+\sqrt [4]{b+a x}\right )^{3/2}+\left (d^3-c d\right ) \sqrt {d+\sqrt [4]{b+a x}}\right )d\sqrt [4]{b+a x}}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 \left (\frac {2}{5} \left (c-3 d^2\right ) \left (\sqrt [4]{a x+b}+d\right )^{5/2}-\frac {2}{3} d \left (c-d^2\right ) \left (\sqrt [4]{a x+b}+d\right )^{3/2}-\frac {2}{9} \left (\sqrt [4]{a x+b}+d\right )^{9/2}+\frac {6}{7} d \left (\sqrt [4]{a x+b}+d\right )^{7/2}\right )}{a}\) |
Input:
Int[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]],x]
Output:
(-4*((-2*d*(c - d^2)*(d + (b + a*x)^(1/4))^(3/2))/3 + (2*(c - 3*d^2)*(d + (b + a*x)^(1/4))^(5/2))/5 + (6*d*(d + (b + a*x)^(1/4))^(7/2))/7 - (2*(d + (b + a*x)^(1/4))^(9/2))/9))/a
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
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_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symb ol] :> With[{g = Denominator[n]}, Simp[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]
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 = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 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 | \(\frac {\frac {8 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {9}{2}}}{9}-\frac {24 d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {7}{2}}}{7}+\frac {8 \left (3 d^{2}-c \right ) \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}}}{5}+\frac {8 \left (-d^{2}+c \right ) d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}}{a}\) | \(93\) |
Input:
int((d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x,method=_RETURNVERBOSE)
Output:
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))
Time = 0.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.47 \[ \int \sqrt {d+\sqrt {c+\sqrt {b+a x}}} \, dx=-\frac {8 \, {\left (16 \, d^{4} - 36 \, c d^{2} + 28 \, c^{2} - 35 \, a x + {\left (6 \, d^{2} - 7 \, c\right )} \sqrt {a x + b} - {\left (8 \, d^{3} - 16 \, c d + 5 \, \sqrt {a x + b} d\right )} \sqrt {c + \sqrt {a x + b}} - 35 \, b\right )} \sqrt {d + \sqrt {c + \sqrt {a x + b}}}}{315 \, a} \] Input:
integrate((d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="fricas")
Output:
-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)*sqr t(d + sqrt(c + sqrt(a*x + b)))/a
Time = 0.59 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.61 \[ \int \sqrt {d+\sqrt {c+\sqrt {b+a x}}} \, dx=\begin {cases} \frac {2 \left (- \frac {12 d \left (d + \sqrt {c + \sqrt {a x + b}}\right )^{\frac {7}{2}}}{7} + \frac {4 \left (- c + 3 d^{2}\right ) \left (d + \sqrt {c + \sqrt {a x + b}}\right )^{\frac {5}{2}}}{5} + \frac {4 \left (d + \sqrt {c + \sqrt {a x + b}}\right )^{\frac {9}{2}}}{9} + \frac {4 \left (d + \sqrt {c + \sqrt {a x + b}}\right )^{\frac {3}{2}} \left (c d - d^{3}\right )}{3}\right )}{a} & \text {for}\: a \neq 0 \\x \sqrt {d + \sqrt {\sqrt {b} + c}} & \text {otherwise} \end {cases} \] Input:
integrate((d+(c+(a*x+b)**(1/2))**(1/2))**(1/2),x)
Output:
Piecewise((2*(-12*d*(d + sqrt(c + sqrt(a*x + b)))**(7/2)/7 + 4*(-c + 3*d** 2)*(d + sqrt(c + sqrt(a*x + b)))**(5/2)/5 + 4*(d + sqrt(c + sqrt(a*x + b)) )**(9/2)/9 + 4*(d + sqrt(c + sqrt(a*x + b)))**(3/2)*(c*d - d**3)/3)/a, Ne( a, 0)), (x*sqrt(d + sqrt(sqrt(b) + c)), True))
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.46 \[ \int \sqrt {d+\sqrt {c+\sqrt {b+a x}}} \, dx=\frac {8 \, {\left (35 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {9}{2}} - 135 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {7}{2}} d + 63 \, {\left (3 \, d^{2} - c\right )} {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} - 105 \, {\left (d^{3} - c d\right )} {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}}\right )}}{315 \, a} \] Input:
integrate((d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (155) = 310\).
Time = 5.15 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.24 \[ \int \sqrt {d+\sqrt {c+\sqrt {b+a x}}} \, dx=\frac {8 \, {\left (35 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {9}{2}} - 180 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {7}{2}} d + 378 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {d + \sqrt {c + \sqrt {a x + b}}} d^{4} - 126 \, c {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} + 420 \, c {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} d - 630 \, c \sqrt {d + \sqrt {c + \sqrt {a x + b}}} d^{2} + 315 \, c^{2} \sqrt {d + \sqrt {c + \sqrt {a x + b}}} + 21 \, {\left (3 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} - 10 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} d + 15 \, \sqrt {d + \sqrt {c + \sqrt {a x + b}}} d^{2} - 15 \, c \sqrt {d + \sqrt {c + \sqrt {a x + b}}}\right )} c + 3 \, {\left (15 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right ) - 63 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} d \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right ) + 105 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} d^{2} \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right ) - 105 \, \sqrt {d + \sqrt {c + \sqrt {a x + b}}} d^{3} \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right ) - 35 \, c {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right ) + 105 \, c \sqrt {d + \sqrt {c + \sqrt {a x + b}}} d \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right )\right )} d\right )}}{315 \, a} \] Input:
integrate((d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="giac")
Output:
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 + s qrt(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 + sq rt(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 + sqr t(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 + sqr t(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*sgn(sqrt(c + sqrt(a*x + b))))*d)/a
Timed out. \[ \int \sqrt {d+\sqrt {c+\sqrt {b+a x}}} \, dx=\int \sqrt {d+\sqrt {c+\sqrt {b+a\,x}}} \,d x \] Input:
int((d + (c + (b + a*x)^(1/2))^(1/2))^(1/2),x)
Output:
int((d + (c + (b + a*x)^(1/2))^(1/2))^(1/2), x)
Time = 0.17 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.54 \[ \int \sqrt {d+\sqrt {c+\sqrt {b+a x}}} \, dx=\frac {8 \sqrt {\sqrt {\sqrt {a x +b}+c}+d}\, \left (5 \sqrt {a x +b}\, \sqrt {\sqrt {a x +b}+c}\, d -16 \sqrt {\sqrt {a x +b}+c}\, c d +8 \sqrt {\sqrt {a x +b}+c}\, d^{3}+7 \sqrt {a x +b}\, c -6 \sqrt {a x +b}\, d^{2}+35 a x +35 b -28 c^{2}+36 c \,d^{2}-16 d^{4}\right )}{315 a} \] Input:
int((d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x)
Output:
(8*sqrt(sqrt(sqrt(a*x + b) + c) + d)*(5*sqrt(a*x + b)*sqrt(sqrt(a*x + b) + c)*d - 16*sqrt(sqrt(a*x + b) + c)*c*d + 8*sqrt(sqrt(a*x + b) + c)*d**3 + 7*sqrt(a*x + b)*c - 6*sqrt(a*x + b)*d**2 + 35*a*x + 35*b - 28*c**2 + 36*c* d**2 - 16*d**4))/(315*a)