\(\int \frac {1}{(1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3)^{3/2}} \, dx\) [162]

Optimal result

Integrand size = 34, antiderivative size = 399 \[ \int \frac {1}{\left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^{3/2}} \, dx=-\frac {\sqrt {1-6 b} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right ) \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right )}{9 \sqrt {2} \left (-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3\right )^{3/2}}-\frac {5 \sqrt {1-6 b} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right )^2 \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right )}{54 \sqrt {2} \left (-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3\right )^{3/2}}+\frac {5 \sqrt {1-6 b} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right )^3 \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right )}{54 \sqrt {2} \left (-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3\right )^{3/2}}-\frac {5 \sqrt {1-6 b} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right )^3 \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {2+\frac {1-6 x}{\sqrt {1-6 b}}}}{\sqrt {3}}\right )}{54 \sqrt {6} \left (-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3\right )^{3/2}} \] Output:

-1/18*(1-6*b)^(1/2)*(1-(1-6*x)/(1-6*b)^(1/2))*(2+(1-6*x)/(1-6*b)^(1/2))*2^ 
(1/2)/(-2*(1-6*b)^(3/2)+3*(1-6*b)*(1-6*x)-(1-6*x)^3)^(3/2)-5/108*(1-6*b)^( 
1/2)*(1-(1-6*x)/(1-6*b)^(1/2))^2*(2+(1-6*x)/(1-6*b)^(1/2))*2^(1/2)/(-2*(1- 
6*b)^(3/2)+3*(1-6*b)*(1-6*x)-(1-6*x)^3)^(3/2)+5/108*(1-6*b)^(1/2)*(1-(1-6* 
x)/(1-6*b)^(1/2))^3*(2+(1-6*x)/(1-6*b)^(1/2))*2^(1/2)/(-2*(1-6*b)^(3/2)+3* 
(1-6*b)*(1-6*x)-(1-6*x)^3)^(3/2)-5/324*(1-6*b)^(1/2)*(1-(1-6*x)/(1-6*b)^(1 
/2))^3*(2+(1-6*x)/(1-6*b)^(1/2))^(3/2)*arctanh(1/3*(2+(1-6*x)/(1-6*b)^(1/2 
))^(1/2)*3^(1/2))*6^(1/2)/(-2*(1-6*b)^(3/2)+3*(1-6*b)*(1-6*x)-(1-6*x)^3)^( 
3/2)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {2480, 27, 52, 52, 61, 73, 217}

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.

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

\(\Big \downarrow \) 2480

\(\displaystyle -\frac {7808611824626688 \sqrt {2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \int -\frac {1}{7808611824626688 \sqrt {2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2}}dx}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 52

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (-\frac {5 \int \frac {1}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2}}dx}{12 (1-6 b)^{3/2}}-\frac {1}{36 (1-6 b)^{5/2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\right )}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (-\frac {5 \left (-\frac {\int \frac {1}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2}}dx}{2 (1-6 b)^{3/2}}-\frac {1}{18 (1-6 b)^{5/2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\right )}{12 (1-6 b)^{3/2}}-\frac {1}{36 (1-6 b)^{5/2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\right )}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (-\frac {5 \left (-\frac {\frac {1}{9 (1-6 b)^{5/2} \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}-\frac {\int \frac {1}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}dx}{3 (1-6 b)^{3/2}}}{2 (1-6 b)^{3/2}}-\frac {1}{18 (1-6 b)^{5/2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\right )}{12 (1-6 b)^{3/2}}-\frac {1}{36 (1-6 b)^{5/2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\right )}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (-\frac {5 \left (-\frac {\frac {1}{9 (1-6 b)^{5/2} \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}-\frac {\int \frac {1}{-3 (1-6 b)^{3/2}+\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)-6 x (1-6 b)}d\sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{9 (1-6 b)^{5/2}}}{2 (1-6 b)^{3/2}}-\frac {1}{18 (1-6 b)^{5/2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\right )}{12 (1-6 b)^{3/2}}-\frac {1}{36 (1-6 b)^{5/2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\right )}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (-\frac {5 \left (-\frac {\frac {\arctan \left (\frac {\sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{\sqrt {3} (1-6 b)^{3/4}}\right )}{9 \sqrt {3} (1-6 b)^{13/4}}+\frac {1}{9 (1-6 b)^{5/2} \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}}{2 (1-6 b)^{3/2}}-\frac {1}{18 (1-6 b)^{5/2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\right )}{12 (1-6 b)^{3/2}}-\frac {1}{36 (1-6 b)^{5/2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}\right )}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^{3/2}}\)

Input:

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

Output:

(((1 - Sqrt[1 - 6*b])*(1 - 6*b) - 6*(1 - 6*b)*x)^3*(-((1 + 2*Sqrt[1 - 6*b] 
)*(1 - 6*b)) + 6*(1 - 6*b)*x)^(3/2)*(-1/36*1/((1 - 6*b)^(5/2)*((1 - Sqrt[1 
 - 6*b])*(1 - 6*b) - 6*(1 - 6*b)*x)^2*Sqrt[-((1 + 2*Sqrt[1 - 6*b])*(1 - 6* 
b)) + 6*(1 - 6*b)*x]) - (5*(-1/18*1/((1 - 6*b)^(5/2)*((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]) - (1/(9*(1 - 6*b)^(5/2)*Sqrt[-((1 + 2*Sqrt[1 - 6*b])*(1 - 6*b)) 
 + 6*(1 - 6*b)*x]) + ArcTan[Sqrt[-((1 + 2*Sqrt[1 - 6*b])*(1 - 6*b)) + 6*(1 
 - 6*b)*x]/(Sqrt[3]*(1 - 6*b)^(3/4))]/(9*Sqrt[3]*(1 - 6*b)^(13/4)))/(2*(1 
- 6*b)^(3/2))))/(12*(1 - 6*b)^(3/2))))/(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x 
 - 54*x^2 + 108*x^3)^(3/2)
 

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 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

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 [F]

\[\int \frac {1}{\left (1-\left (1-6 b \right )^{\frac {3}{2}}-9 b +54 b x -54 x^{2}+108 x^{3}\right )^{\frac {3}{2}}}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 607, normalized size of antiderivative = 1.52 \[ \int \frac {1}{\left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2916*(270*sqrt(1/6)*(2592*x^8 + 1512*(2*b + 1)*x^6 - 3456*x^7 - 168*(18* 
b + 1)*x^5 + 20*(54*b^2 + 45*b - 2)*x^4 + 8*b^4 - 8*(90*b^2 + 5*b - 1)*x^3 
 - b^3 + 6*(26*b^3 + 17*b^2 - 2*b)*x^2 - 2*(26*b^3 - 3*b^2)*x)*(-6*b + 1)^ 
(1/4)*arctan(1/3*sqrt(1/6)*sqrt(108*x^3 + 54*b*x - 54*x^2 + (6*b - 1)*sqrt 
(-6*b + 1) - 9*b + 1)*(36*x^2 + sqrt(-6*b + 1)*(6*x - 1) + 12*b - 12*x - 1 
)*(-6*b + 1)^(1/4)/(72*x^4 + 2*(30*b + 1)*x^2 - 48*x^3 + 8*b^2 - 2*(10*b - 
 1)*x - b)) - (288*(36*b^2 - 12*b + 1)*x^3 - 720*b^3 - 144*(36*b^2 - 12*b 
+ 1)*x^2 + 312*b^2 + 4*(1080*b^3 - 324*b^2 + 18*b + 1)*x + (19440*x^6 + 10 
8*(156*b + 49)*x^4 - 19440*x^5 - 72*(156*b - 1)*x^3 - 216*b^3 + 36*(99*b^2 
 + 45*b - 4)*x^2 + 207*b^2 - 4*(297*b^2 - 21*b - 1)*x - 38*b + 2)*sqrt(-6* 
b + 1) - 44*b + 2)*sqrt(108*x^3 + 54*b*x - 54*x^2 + (6*b - 1)*sqrt(-6*b + 
1) - 9*b + 1))/(2592*(36*b^2 - 12*b + 1)*x^8 - 3456*(36*b^2 - 12*b + 1)*x^ 
7 + 1512*(72*b^3 + 12*b^2 - 10*b + 1)*x^6 + 288*b^6 - 168*(648*b^3 - 180*b 
^2 + 6*b + 1)*x^5 - 132*b^5 + 20*(1944*b^4 + 972*b^3 - 558*b^2 + 69*b - 2) 
*x^4 + 20*b^4 - 8*(3240*b^4 - 900*b^3 - 6*b^2 + 17*b - 1)*x^3 - b^3 + 6*(9 
36*b^5 + 300*b^4 - 250*b^3 + 41*b^2 - 2*b)*x^2 - 2*(936*b^5 - 420*b^4 + 62 
*b^3 - 3*b^2)*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{%%{[17794363111902506549001421627856069105221941539635 
2,0]:[1,0
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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