\(\int \frac {1}{(e+f x)^2 \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx\) [159]

Optimal result

Integrand size = 42, antiderivative size = 493 \[ \int \frac {1}{(e+f x)^2 \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx=\frac {\sqrt {2} (1-6 b) f \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right ) \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right )}{\left (6 e+f-\sqrt {1-6 b} f\right ) \left (6 e+f+2 \sqrt {1-6 b} f\right ) \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3} (e+f x)}-\frac {4 \sqrt {6} \sqrt {1-6 b} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right ) \sqrt {2+\frac {1-6 x}{\sqrt {1-6 b}}} \text {arctanh}\left (\frac {\sqrt {2+\frac {1-6 x}{\sqrt {1-6 b}}}}{\sqrt {3}}\right )}{\left (6 e+f-\sqrt {1-6 b} f\right )^2 \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}}+\frac {18 \sqrt {2} \sqrt [4]{1-6 b} \sqrt {f} \left (6 \sqrt {1-6 b} e+\left (1+\sqrt {1-6 b}-6 b\right ) f\right ) \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right ) \sqrt {2+\frac {1-6 x}{\sqrt {1-6 b}}} \text {arctanh}\left (\frac {\sqrt [4]{1-6 b} \sqrt {f} \sqrt {2+\frac {1-6 x}{\sqrt {1-6 b}}}}{\sqrt {6 e+f+2 \sqrt {1-6 b} f}}\right )}{\left (6 e+f-\sqrt {1-6 b} f\right )^2 \left (6 e+f+2 \sqrt {1-6 b} f\right )^{3/2} \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}} \] Output:

2^(1/2)*(1-6*b)*f*(1-(1-6*x)/(1-6*b)^(1/2))*(2+(1-6*x)/(1-6*b)^(1/2))/(6*e 
+f-(1-6*b)^(1/2)*f)/(6*e+f+2*(1-6*b)^(1/2)*f)/(-2*(1-6*b)^(3/2)+3*(1-6*b)* 
(1-6*x)-(1-6*x)^3)^(1/2)/(f*x+e)-4*6^(1/2)*(1-6*b)^(1/2)*(1-(1-6*x)/(1-6*b 
)^(1/2))*(2+(1-6*x)/(1-6*b)^(1/2))^(1/2)*arctanh(1/3*(2+(1-6*x)/(1-6*b)^(1 
/2))^(1/2)*3^(1/2))/(6*e+f-(1-6*b)^(1/2)*f)^2/(-2*(1-6*b)^(3/2)+3*(1-6*b)* 
(1-6*x)-(1-6*x)^3)^(1/2)+18*2^(1/2)*(1-6*b)^(1/4)*f^(1/2)*(6*(1-6*b)^(1/2) 
*e+(1+(1-6*b)^(1/2)-6*b)*f)*(1-(1-6*x)/(1-6*b)^(1/2))*(2+(1-6*x)/(1-6*b)^( 
1/2))^(1/2)*arctanh((1-6*b)^(1/4)*f^(1/2)*(2+(1-6*x)/(1-6*b)^(1/2))^(1/2)/ 
(6*e+f+2*(1-6*b)^(1/2)*f)^(1/2))/(6*e+f-(1-6*b)^(1/2)*f)^2/(6*e+f+2*(1-6*b 
)^(1/2)*f)^(3/2)/(-2*(1-6*b)^(3/2)+3*(1-6*b)*(1-6*x)-(1-6*x)^3)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 88.26 (sec) , antiderivative size = 451671, normalized size of antiderivative = 916.17 \[ \int \frac {1}{(e+f x)^2 \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2489, 27, 114, 27, 174, 73, 217, 218}

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.

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

\(\Big \downarrow \) 2489

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 114

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (\frac {\int \frac {9 (1-6 b)^2 \left (4 e+\sqrt {1-6 b} f+f-2 f x\right )}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}dx}{(1-6 b)^2 \left (-\sqrt {1-6 b} f+6 e+f\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right )}+\frac {f \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{(1-6 b)^2 \left (-\sqrt {1-6 b} f+6 e+f\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right ) (e+f x)}\right )}{\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (\frac {9 \int \frac {4 e+\sqrt {1-6 b} f+f-2 f x}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}dx}{\left (-\sqrt {1-6 b} f+6 e+f\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right )}+\frac {f \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{(1-6 b)^2 \left (-\sqrt {1-6 b} f+6 e+f\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right ) (e+f x)}\right )}{\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1}}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (\frac {9 \left (\frac {4 \left (2 \sqrt {1-6 b} f+6 e+f\right ) \int \frac {1}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}dx}{-\sqrt {1-6 b} f+6 e+f}+\frac {f \left (\sqrt {1-6 b} f+6 e+f\right ) \int \frac {1}{\sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}dx}{(1-6 b) \left (-\sqrt {1-6 b} f+6 e+f\right )}\right )}{\left (-\sqrt {1-6 b} f+6 e+f\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right )}+\frac {f \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{(1-6 b)^2 \left (-\sqrt {1-6 b} f+6 e+f\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right ) (e+f x)}\right )}{\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (\frac {9 \left (\frac {4 \left (2 \sqrt {1-6 b} f+6 e+f\right ) \int \frac {1}{-3 (1-6 b)^{3/2}+\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)-6 x (1-6 b)}d\sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{3 (1-6 b) \left (-\sqrt {1-6 b} f+6 e+f\right )}+\frac {f \left (\sqrt {1-6 b} f+6 e+f\right ) \int \frac {1}{\frac {1}{6} \left (6 e+2 \sqrt {1-6 b} f+f\right )+\frac {f \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )}{6 (1-6 b)}}d\sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{3 (1-6 b)^2 \left (-\sqrt {1-6 b} f+6 e+f\right )}\right )}{\left (-\sqrt {1-6 b} f+6 e+f\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right )}+\frac {f \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{(1-6 b)^2 \left (-\sqrt {1-6 b} f+6 e+f\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right ) (e+f x)}\right )}{\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (\frac {9 \left (\frac {f \left (\sqrt {1-6 b} f+6 e+f\right ) \int \frac {1}{\frac {1}{6} \left (6 e+2 \sqrt {1-6 b} f+f\right )+\frac {f \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )}{6 (1-6 b)}}d\sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{3 (1-6 b)^2 \left (-\sqrt {1-6 b} f+6 e+f\right )}-\frac {4 \arctan \left (\frac {\sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{\sqrt {3} (1-6 b)^{3/4}}\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right )}{3 \sqrt {3} (1-6 b)^{7/4} \left (-\sqrt {1-6 b} f+6 e+f\right )}\right )}{\left (-\sqrt {1-6 b} f+6 e+f\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right )}+\frac {f \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{(1-6 b)^2 \left (-\sqrt {1-6 b} f+6 e+f\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right ) (e+f x)}\right )}{\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (\frac {9 \left (\frac {2 \sqrt {f} \left (\sqrt {1-6 b} f+6 e+f\right ) \arctan \left (\frac {\sqrt {f} \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{\sqrt {1-6 b} \sqrt {2 \sqrt {1-6 b} f+6 e+f}}\right )}{(1-6 b)^{3/2} \left (-\sqrt {1-6 b} f+6 e+f\right ) \sqrt {2 \sqrt {1-6 b} f+6 e+f}}-\frac {4 \arctan \left (\frac {\sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{\sqrt {3} (1-6 b)^{3/4}}\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right )}{3 \sqrt {3} (1-6 b)^{7/4} \left (-\sqrt {1-6 b} f+6 e+f\right )}\right )}{\left (-\sqrt {1-6 b} f+6 e+f\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right )}+\frac {f \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{(1-6 b)^2 \left (-\sqrt {1-6 b} f+6 e+f\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right ) (e+f x)}\right )}{\sqrt {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1}}\)

Input:

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

Output:

(((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]*((f*Sqrt[-((1 + 2*Sqrt[1 - 6*b])*(1 - 6*b) 
) + 6*(1 - 6*b)*x])/((1 - 6*b)^2*(6*e + f - Sqrt[1 - 6*b]*f)*(6*e + f + 2* 
Sqrt[1 - 6*b]*f)*(e + f*x)) + (9*((-4*(6*e + f + 2*Sqrt[1 - 6*b]*f)*ArcTan 
[Sqrt[-((1 + 2*Sqrt[1 - 6*b])*(1 - 6*b)) + 6*(1 - 6*b)*x]/(Sqrt[3]*(1 - 6* 
b)^(3/4))])/(3*Sqrt[3]*(1 - 6*b)^(7/4)*(6*e + f - Sqrt[1 - 6*b]*f)) + (2*S 
qrt[f]*(6*e + f + Sqrt[1 - 6*b]*f)*ArcTan[(Sqrt[f]*Sqrt[-((1 + 2*Sqrt[1 - 
6*b])*(1 - 6*b)) + 6*(1 - 6*b)*x])/(Sqrt[1 - 6*b]*Sqrt[6*e + f + 2*Sqrt[1 
- 6*b]*f])])/((1 - 6*b)^(3/2)*(6*e + f - Sqrt[1 - 6*b]*f)*Sqrt[6*e + f + 2 
*Sqrt[1 - 6*b]*f])))/((6*e + f - Sqrt[1 - 6*b]*f)*(6*e + f + 2*Sqrt[1 - 6* 
b]*f))))/Sqrt[1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3]
 

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 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 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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 218

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

rule 2489

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*( 
x_)^3)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2 + d*x^3)^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[(e + f*x)^m*(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] /; FreeQ[{a, b, c, d, e, f, m, p}, x] 
 && NeQ[c^2 - 3*b*d, 0] && EqQ[b^2*c^2 - 4*a*c^3 - 4*b^3*d + 18*a*b*c*d - 2 
7*a^2*d^2, 0] &&  !IntegerQ[p]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3641 vs. \(2 (421) = 842\).

Time = 4.51 (sec) , antiderivative size = 15167, normalized size of antiderivative = 30.76 \[ \int \frac {1}{(e+f x)^2 \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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