\(\int \frac {1}{(e+f x) \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx\) [158]

Optimal result

Integrand size = 42, antiderivative size = 329 \[ \int \frac {1}{(e+f x) \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx=\frac {2 \sqrt {\frac {2}{3}} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right ) \sqrt {2+\frac {1-6 x}{\sqrt {1-6 b}}} \text {arctanh}\left (\frac {\sqrt {2+\frac {1-6 x}{\sqrt {1-6 b}}}}{\sqrt {3}}\right )}{\left (f-\frac {6 e+f}{\sqrt {1-6 b}}\right ) \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}}+\frac {2 \sqrt {2} (1-6 b)^{3/4} \sqrt {f} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right ) \sqrt {2+\frac {1-6 x}{\sqrt {1-6 b}}} \text {arctanh}\left (\frac {\sqrt [4]{1-6 b} \sqrt {f} \sqrt {2+\frac {1-6 x}{\sqrt {1-6 b}}}}{\sqrt {6 e+f+2 \sqrt {1-6 b} f}}\right )}{\left (6 e+f-\sqrt {1-6 b} f\right ) \sqrt {6 e+f+2 \sqrt {1-6 b} f} \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}} \] Output:

2/3*6^(1/2)*(1-(1-6*x)/(1-6*b)^(1/2))*(2+(1-6*x)/(1-6*b)^(1/2))^(1/2)*arct 
anh(1/3*(2+(1-6*x)/(1-6*b)^(1/2))^(1/2)*3^(1/2))/(f-(6*e+f)/(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*2^(1/2)*(1-6*b)^(3 
/4)*f^(1/2)*(1-(1-6*x)/(1-6*b)^(1/2))*(2+(1-6*x)/(1-6*b)^(1/2))^(1/2)*arct 
anh((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)/(6*e+f+2*(1-6*b)^(1/2)*f)^(1/2)/( 
-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 = 20.36 (sec) , antiderivative size = 1352, normalized size of antiderivative = 4.11 \[ \int \frac {1}{(e+f x) \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx =\text {Too large to display} \] Input:

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

Output:

(16*(-1 + 6*b)^3*(5832*e^6 + 5832*(1 + Sqrt[1 - 6*b])*e^5*f + 4860*(1 + Sq 
rt[1 - 6*b] - 3*b)*e^4*f^2 - 1080*(-2*(1 + Sqrt[1 - 6*b]) + 3*(3 + Sqrt[1 
- 6*b])*b)*e^3*f^3 - 270*(-2*(1 + Sqrt[1 - 6*b]) + 6*(2 + Sqrt[1 - 6*b])*b 
 - 9*b^2)*e^2*f^4 + 18*(4*(1 + Sqrt[1 - 6*b]) - 6*(5 + 3*Sqrt[1 - 6*b])*b 
+ 9*(5 + Sqrt[1 - 6*b])*b^2)*e*f^5 + (4*(1 + Sqrt[1 - 6*b]) - 12*(3 + 2*Sq 
rt[1 - 6*b])*b + 27*(3 + Sqrt[1 - 6*b])*b^2 - 27*b^3)*f^6)*EllipticPi[(f*( 
-Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 10 
8*#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]))/(e + f*Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqr 
t[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 3]), ArcSin[Sqrt[(-x + Roo 
t[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*#1^2 + 1 
08*#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 & , 1])/(Root[1 - Sqrt...
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2489, 27, 97, 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.

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

\(\Big \downarrow \) 2489

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 97

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 218

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

Input:

Int[1/((e + f*x)*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]*((-2*ArcTan[Sqrt[-((1 + 2*Sqrt[1 - 6*b])*( 
1 - 6*b)) + 6*(1 - 6*b)*x]/(Sqrt[3]*(1 - 6*b)^(3/4))])/(Sqrt[3]*(1 - 6*b)^ 
(7/4)*(6*e + f - Sqrt[1 - 6*b]*f)) + (2*Sqrt[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)*Sqr 
t[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 97

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

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]

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

Input:

int(1/(f*x+e)/(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)/(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. 920 vs. \(2 (273) = 546\).

Time = 0.50 (sec) , antiderivative size = 3961, normalized size of antiderivative = 12.04 \[ \int \frac {1}{(e+f x) \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

integrate(1/(f*x+e)/(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)*sqrt(54*b*x + 6*b*sqrt(1 - 6*b) - 9*b + 108*x**3 - 5 
4*x**2 - sqrt(1 - 6*b) + 1)), x)
 

Maxima [F]

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

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

Giac [F(-2)]

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

integrate(1/(f*x+e)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(1/2),x, a 
lgorithm="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. \[ \int \frac {1}{(e+f x) \sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}} \, dx=\text {Hanged} \] Input:

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

Output:

\text{Hanged}
 

Reduce [F]

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

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