\(\int \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx\) [152]

Optimal result

Integrand size = 34, antiderivative size = 187 \[ \int \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=-\frac {\sqrt {1-6 b} \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right ) \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}}{3 \sqrt {2} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right )}+\frac {\sqrt {1-6 b} \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right )^2 \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}}{15 \sqrt {2} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right )} \] Output:

-1/6*(1-6*b)^(1/2)*(2+(1-6*x)/(1-6*b)^(1/2))*(-2*(1-6*b)^(3/2)+3*(1-6*b)*( 
1-6*x)-(1-6*x)^3)^(1/2)*2^(1/2)/(1-(1-6*x)/(1-6*b)^(1/2))+1/30*(1-6*b)^(1/ 
2)*(2+(1-6*x)/(1-6*b)^(1/2))^2*(-2*(1-6*b)^(3/2)+3*(1-6*b)*(1-6*x)-(1-6*x) 
^3)^(1/2)*2^(1/2)/(1-(1-6*x)/(1-6*b)^(1/2))
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.

Time = 2.85 (sec) , antiderivative size = 1461, normalized size of antiderivative = 7.81 \[ \int \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx =\text {Too large to display} \] Input:

Integrate[Sqrt[1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3],x]
 

Output:

(((-1 + 6*x)*(1 - Sqrt[1 - 6*b] - 54*x^2 + 108*x^3 + b*(-9 + 6*Sqrt[1 - 6* 
b] + 54*x)))/3 - 6*(1 - 6*b)*Sqrt[(-x + Root[1 - Sqrt[1 - 6*b] - 9*b + 6*S 
qrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 1])/(Root[1 - Sqrt[1 - 6 
*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 1] - Root 
[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^ 
3 & , 3])]*Sqrt[-(((x - Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 
 54*b*#1 - 54*#1^2 + 108*#1^3 & , 2])*(x - Root[1 - Sqrt[1 - 6*b] - 9*b + 
6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 3]))/(Root[1 - Sqrt[1 
 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 2] - 
Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108 
*#1^3 & , 3])^2)]*(Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b 
*#1 - 54*#1^2 + 108*#1^3 & , 2] - Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 
- 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 3])*(EllipticF[ArcSin[Sqrt[(-x 
 + Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 
108*#1^3 & , 3])/(-Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b 
*#1 - 54*#1^2 + 108*#1^3 & , 2] + Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 
- 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 3])]], (Root[1 - Sqrt[1 - 6*b] 
 - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 2] - Root[1 
- Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & 
 , 3])/(Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54...
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2480, 27, 53, 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.

\(\displaystyle \int \sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} \, dx\)

\(\Big \downarrow \) 2480

\(\displaystyle -\frac {\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} \int -157464 \sqrt {2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}dx}{157464 \sqrt {2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} \int \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}dx}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} \int \left (-3 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (1-6 b)^{3/2}-\left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2}\right )dx}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} \left (-\frac {\left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{5/2}}{15 (1-6 b)}-\frac {1}{3} \sqrt {1-6 b} \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2}\right )}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\)

Input:

Int[Sqrt[1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3],x]
 

Output:

(Sqrt[1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3]*(-1/3*(Sqrt[1 
 - 6*b]*(-((1 + 2*Sqrt[1 - 6*b])*(1 - 6*b)) + 6*(1 - 6*b)*x)^(3/2)) - (-(( 
1 + 2*Sqrt[1 - 6*b])*(1 - 6*b)) + 6*(1 - 6*b)*x)^(5/2)/(15*(1 - 6*b))))/(( 
(1 - Sqrt[1 - 6*b])*(1 - 6*b) - 6*(1 - 6*b)*x)*Sqrt[-((1 + 2*Sqrt[1 - 6*b] 
)*(1 - 6*b)) + 6*(1 - 6*b)*x])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2480

Int[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] 
, c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Simp[Px^p/((c^3 - 4*b*c*d + 9* 
a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3*b*d)*x)^(2*p))   Int 
[(c^3 - 4*b*c*d + 9*a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3* 
b*d)*x)^(2*p), x], x] /; EqQ[b^2*c^2 - 4*a*c^3 - 4*b^3*d + 18*a*b*c*d - 27* 
a^2*d^2, 0] && NeQ[c^2 - 3*b*d, 0]] /; FreeQ[p, x] && PolyQ[Px, x, 3] &&  ! 
IntegerQ[p]
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.40

method	result	size
gosper	\(-\frac {\left (1+2 \sqrt {1-6 b}-6 x \right ) \left (6 x +3 \sqrt {1-6 b}-1\right ) \sqrt {1-\left (1-6 b \right )^{\frac {3}{2}}-9 b +54 b x -54 x^{2}+108 x^{3}}}{15 \left (-1+6 x +\sqrt {1-6 b}\right )}\)	\(75\)
risch	\(\frac {\left (6 x \sqrt {1-6 b}+36 x^{2}-\sqrt {1-6 b}+36 b -12 x -5\right ) \left (-1-2 \sqrt {1-6 b}+6 x \right ) \left (-1+6 x +\sqrt {1-6 b}\right )}{30 \sqrt {108 x^{3}+54 b x +6 \sqrt {1-6 b}\, b -54 x^{2}-9 b -\sqrt {1-6 b}+1}}\)	\(101\)

Input:

int((1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x,method=_RETURNVERB 
OSE)
 

Output:

-1/15*(1+2*(1-6*b)^(1/2)-6*x)*(6*x+3*(1-6*b)^(1/2)-1)*(1-(1-6*b)^(3/2)-9*b 
+54*b*x-54*x^2+108*x^3)^(1/2)/(-1+6*x+(1-6*b)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.42 \[ \int \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\frac {\sqrt {108 \, x^{3} + 54 \, b x - 54 \, x^{2} + {\left (6 \, b - 1\right )} \sqrt {-6 \, b + 1} - 9 \, b + 1} {\left (36 \, x^{3} + 2 \, {\left (21 \, b - 2\right )} x - 18 \, x^{2} + {\left (-6 \, b + 1\right )}^{\frac {3}{2}} - 7 \, b + 1\right )}}{15 \, {\left (6 \, x^{2} + b - 2 \, x\right )}} \] Input:

integrate((1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x, algorithm=" 
fricas")
 

Output:

1/15*sqrt(108*x^3 + 54*b*x - 54*x^2 + (6*b - 1)*sqrt(-6*b + 1) - 9*b + 1)* 
(36*x^3 + 2*(21*b - 2)*x - 18*x^2 + (-6*b + 1)^(3/2) - 7*b + 1)/(6*x^2 + b 
 - 2*x)
 

Sympy [F]

\[ \int \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\int \sqrt {54 b x - 9 b + 108 x^{3} - 54 x^{2} - \left (1 - 6 b\right )^{\frac {3}{2}} + 1}\, dx \] Input:

integrate((1-(1-6*b)**(3/2)-9*b+54*b*x-54*x**2+108*x**3)**(1/2),x)
 

Output:

Integral(sqrt(54*b*x - 9*b + 108*x**3 - 54*x**2 - (1 - 6*b)**(3/2) + 1), x 
)
 

Maxima [F]

\[ \int \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\int { \sqrt {108 \, x^{3} + 54 \, b x - 54 \, x^{2} - {\left (-6 \, b + 1\right )}^{\frac {3}{2}} - 9 \, b + 1} \,d x } \] Input:

integrate((1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x, algorithm=" 
maxima")
 

Output:

integrate(sqrt(108*x^3 + 54*b*x - 54*x^2 - (-6*b + 1)^(3/2) - 9*b + 1), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (153) = 306\).

Time = 0.12 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.92 \[ \int \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=-\frac {1}{45} \, \sqrt {\frac {1}{2}} {\left (5 \, {\left ({\left (6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1\right )}^{\frac {3}{2}} + 6 \, \sqrt {-6 \, b + 1} \sqrt {6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1} + 3 \, \sqrt {6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1}\right )} \sqrt {-6 \, b + 1} \mathrm {sgn}\left (6 \, x + \sqrt {-6 \, b + 1} - 1\right ) - 180 \, b \sqrt {6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1} \mathrm {sgn}\left (6 \, x + \sqrt {-6 \, b + 1} - 1\right ) - {\left (3 \, {\left (6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1\right )}^{\frac {5}{2}} + 10 \, {\left (6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1\right )}^{\frac {3}{2}} {\left (2 \, \sqrt {-6 \, b + 1} + 1\right )} - 15 \, {\left (24 \, b - 4 \, \sqrt {-6 \, b + 1} - 5\right )} \sqrt {6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1}\right )} \mathrm {sgn}\left (6 \, x + \sqrt {-6 \, b + 1} - 1\right ) + 10 \, {\left ({\left (6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1\right )}^{\frac {3}{2}} + 6 \, \sqrt {-6 \, b + 1} \sqrt {6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1} + 3 \, \sqrt {6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1}\right )} \mathrm {sgn}\left (6 \, x + \sqrt {-6 \, b + 1} - 1\right ) - 15 \, \sqrt {-6 \, b + 1} \sqrt {6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1} \mathrm {sgn}\left (6 \, x + \sqrt {-6 \, b + 1} - 1\right ) + 15 \, \sqrt {6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1} \mathrm {sgn}\left (6 \, x + \sqrt {-6 \, b + 1} - 1\right )\right )} \] Input:

integrate((1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x, algorithm=" 
giac")
 

Output:

-1/45*sqrt(1/2)*(5*((6*x - 2*sqrt(-6*b + 1) - 1)^(3/2) + 6*sqrt(-6*b + 1)* 
sqrt(6*x - 2*sqrt(-6*b + 1) - 1) + 3*sqrt(6*x - 2*sqrt(-6*b + 1) - 1))*sqr 
t(-6*b + 1)*sgn(6*x + sqrt(-6*b + 1) - 1) - 180*b*sqrt(6*x - 2*sqrt(-6*b + 
 1) - 1)*sgn(6*x + sqrt(-6*b + 1) - 1) - (3*(6*x - 2*sqrt(-6*b + 1) - 1)^( 
5/2) + 10*(6*x - 2*sqrt(-6*b + 1) - 1)^(3/2)*(2*sqrt(-6*b + 1) + 1) - 15*( 
24*b - 4*sqrt(-6*b + 1) - 5)*sqrt(6*x - 2*sqrt(-6*b + 1) - 1))*sgn(6*x + s 
qrt(-6*b + 1) - 1) + 10*((6*x - 2*sqrt(-6*b + 1) - 1)^(3/2) + 6*sqrt(-6*b 
+ 1)*sqrt(6*x - 2*sqrt(-6*b + 1) - 1) + 3*sqrt(6*x - 2*sqrt(-6*b + 1) - 1) 
)*sgn(6*x + sqrt(-6*b + 1) - 1) - 15*sqrt(-6*b + 1)*sqrt(6*x - 2*sqrt(-6*b 
 + 1) - 1)*sgn(6*x + sqrt(-6*b + 1) - 1) + 15*sqrt(6*x - 2*sqrt(-6*b + 1) 
- 1)*sgn(6*x + sqrt(-6*b + 1) - 1))
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\int \sqrt {54\,b\,x-9\,b-{\left (1-6\,b\right )}^{3/2}-54\,x^2+108\,x^3+1} \,d x \] Input:

int((54*b*x - 9*b - (1 - 6*b)^(3/2) - 54*x^2 + 108*x^3 + 1)^(1/2),x)
 

Output:

int((54*b*x - 9*b - (1 - 6*b)^(3/2) - 54*x^2 + 108*x^3 + 1)^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx =\text {Too large to display} \] Input:

int((1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x)
 

Output:

( - 72*sqrt( - 6*b + 1)*int(sqrt(6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 
 54*b*x - 9*b + 108*x**3 - 54*x**2 + 1)/(8*b**2 + 60*b*x**2 - 20*b*x - b + 
 72*x**4 - 48*x**3 + 2*x**2 + 2*x),x)*b**2 + 6*sqrt( - 6*b + 1)*int(sqrt(6 
*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9*b + 108*x**3 - 54*x**2 
 + 1)/(8*b**2 + 60*b*x**2 - 20*b*x - b + 72*x**4 - 48*x**3 + 2*x**2 + 2*x) 
,x)*b + sqrt( - 6*b + 1)*int(sqrt(6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) 
+ 54*b*x - 9*b + 108*x**3 - 54*x**2 + 1)/(8*b**2 + 60*b*x**2 - 20*b*x - b 
+ 72*x**4 - 48*x**3 + 2*x**2 + 2*x),x) + 108*sqrt( - 6*b + 1)*int((sqrt(6* 
sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9*b + 108*x**3 - 54*x**2 
+ 1)*x)/(8*b**2 + 60*b*x**2 - 20*b*x - b + 72*x**4 - 48*x**3 + 2*x**2 + 2* 
x),x)*b - 18*sqrt( - 6*b + 1)*int((sqrt(6*sqrt( - 6*b + 1)*b - sqrt( - 6*b 
 + 1) + 54*b*x - 9*b + 108*x**3 - 54*x**2 + 1)*x)/(8*b**2 + 60*b*x**2 - 20 
*b*x - b + 72*x**4 - 48*x**3 + 2*x**2 + 2*x),x) - 12*sqrt(6*sqrt( - 6*b + 
1)*b - sqrt( - 6*b + 1) + 54*b*x - 9*b + 108*x**3 - 54*x**2 + 1)*b + 18*sq 
rt(6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9*b + 108*x**3 - 54* 
x**2 + 1)*x - sqrt(6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9*b 
+ 108*x**3 - 54*x**2 + 1) + 108*int(sqrt(6*sqrt( - 6*b + 1)*b - sqrt( - 6* 
b + 1) + 54*b*x - 9*b + 108*x**3 - 54*x**2 + 1)/(8*b**2 + 60*b*x**2 - 20*b 
*x - b + 72*x**4 - 48*x**3 + 2*x**2 + 2*x),x)*b**2 - 24*int(sqrt(6*sqrt( - 
 6*b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9*b + 108*x**3 - 54*x**2 + 1)...