\(\int \frac {e+f x}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx\) [141]

Optimal result

Integrand size = 38, antiderivative size = 139 \[ \int \frac {e+f x}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=-\frac {6 e+f-\sqrt {1-6 b} f}{54 \sqrt {1-6 b} \left (1-\sqrt {1-6 b}-6 x\right )}-\frac {\left (6 e+f+2 \sqrt {1-6 b} f\right ) \log \left (1-\sqrt {1-6 b}-6 x\right )}{162 (1-6 b)}+\frac {\left (6 e+f+2 \sqrt {1-6 b} f\right ) \log \left (1+2 \sqrt {1-6 b}-6 x\right )}{162 (1-6 b)} \] Output:

-1/54*(6*e+f-(1-6*b)^(1/2)*f)/(1-6*b)^(1/2)/(1-(1-6*b)^(1/2)-6*x)-(6*e+f+2 
*(1-6*b)^(1/2)*f)*ln(1-(1-6*b)^(1/2)-6*x)/(162-972*b)+(6*e+f+2*(1-6*b)^(1/ 
2)*f)*ln(1+2*(1-6*b)^(1/2)-6*x)/(162-972*b)
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.99 \[ \int \frac {e+f x}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\frac {1}{324} \left (\frac {6 \left (e \left (1+\sqrt {1-6 b}-6 b-6 \sqrt {1-6 b} x\right )+f \left (b \left (\sqrt {1-6 b}-6 x\right )+x-\sqrt {1-6 b} x\right )\right )}{(-1+6 b) (b+2 x (-1+3 x))}-\frac {2 \left (6 \sqrt {1-6 b} e+\left (2+\sqrt {1-6 b}-12 b\right ) f\right ) \arctan \left (\frac {-1+6 x}{2 \sqrt {-1+6 b}}\right )}{(-1+6 b)^{3/2}}-\frac {2 \left (6 \sqrt {1-6 b} e+\left (2+\sqrt {1-6 b}-12 b\right ) f\right ) \arctan \left (\frac {-1+6 x}{\sqrt {-1+6 b}}\right )}{(-1+6 b)^{3/2}}-\frac {\left (6 e+f+2 \sqrt {1-6 b} f\right ) \log \left (1-8 b+4 x-12 x^2\right )}{-1+6 b}+\frac {\left (6 e+f+2 \sqrt {1-6 b} f\right ) \log \left (b-2 x+6 x^2\right )}{-1+6 b}\right ) \] Input:

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

Output:

((6*(e*(1 + Sqrt[1 - 6*b] - 6*b - 6*Sqrt[1 - 6*b]*x) + f*(b*(Sqrt[1 - 6*b] 
 - 6*x) + x - Sqrt[1 - 6*b]*x)))/((-1 + 6*b)*(b + 2*x*(-1 + 3*x))) - (2*(6 
*Sqrt[1 - 6*b]*e + (2 + Sqrt[1 - 6*b] - 12*b)*f)*ArcTan[(-1 + 6*x)/(2*Sqrt 
[-1 + 6*b])])/(-1 + 6*b)^(3/2) - (2*(6*Sqrt[1 - 6*b]*e + (2 + Sqrt[1 - 6*b 
] - 12*b)*f)*ArcTan[(-1 + 6*x)/Sqrt[-1 + 6*b]])/(-1 + 6*b)^(3/2) - ((6*e + 
 f + 2*Sqrt[1 - 6*b]*f)*Log[1 - 8*b + 4*x - 12*x^2])/(-1 + 6*b) + ((6*e + 
f + 2*Sqrt[1 - 6*b]*f)*Log[b - 2*x + 6*x^2])/(-1 + 6*b))/324
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2488, 27, 27, 86, 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.

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

\(\Big \downarrow \) 2488

\(\displaystyle 99179645184 (1-6 b)^3 \int -\frac {e+f x}{49589822592 \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \left (\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)-6 (1-6 b) x\right )}dx\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 86

\(\displaystyle -2 (1-6 b)^2 \int \left (\frac {-6 e-2 \sqrt {1-6 b} f-f}{54 (6 b-1)^3 \left (-6 x+2 \sqrt {1-6 b}+1\right )}+\frac {-6 e-2 \sqrt {1-6 b} f-f}{54 (6 b-1)^3 \left (6 x+\sqrt {1-6 b}-1\right )}+\frac {6 e-\sqrt {1-6 b} f+f}{18 \sqrt {1-6 b} (6 b-1)^2 \left (6 x+\sqrt {1-6 b}-1\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 (1-6 b)^2 \left (\frac {-\sqrt {1-6 b} f+6 e+f}{108 (1-6 b)^{5/2} \left (-\sqrt {1-6 b}-6 x+1\right )}+\frac {\left (2 \sqrt {1-6 b} f+6 e+f\right ) \log \left (-\sqrt {1-6 b}-6 x+1\right )}{324 (1-6 b)^3}-\frac {\left (2 \sqrt {1-6 b} f+6 e+f\right ) \log \left (2 \sqrt {1-6 b}-6 x+1\right )}{324 (1-6 b)^3}\right )\)

Input:

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

Output:

-2*(1 - 6*b)^2*((6*e + f - Sqrt[1 - 6*b]*f)/(108*(1 - 6*b)^(5/2)*(1 - Sqrt 
[1 - 6*b] - 6*x)) + ((6*e + f + 2*Sqrt[1 - 6*b]*f)*Log[1 - Sqrt[1 - 6*b] - 
 6*x])/(324*(1 - 6*b)^3) - ((6*e + f + 2*Sqrt[1 - 6*b]*f)*Log[1 + 2*Sqrt[1 
 - 6*b] - 6*x])/(324*(1 - 6*b)^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 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 2009

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

rule 2488

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*( 
x_)^3)^(p_), x_Symbol] :> Simp[1/(4^p*(c^2 - 3*b*d)^(3*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 - 27*a^2*d^2, 0] & 
& ILtQ[p, 0]
 

Maple [C] (warning: unable to verify)

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

Time = 0.48 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.46

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-\left (1-6 b \right )^{\frac {3}{2}}+108 \textit {\_Z}^{3}+54 b \textit {\_Z} -54 \textit {\_Z}^{2}-9 b +1\right )}{\sum }\frac {\left (-\textit {\_R} f -e \right ) \ln \left (x -\textit {\_R} \right )}{-6 \textit {\_R}^{2}+2 \textit {\_R} -b}\right )}{54}\)	\(64\)
parallelrisch	\(\frac {3 f -12 \sqrt {1-6 b}\, \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) x f +12 \sqrt {1-6 b}\, \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) x f -18 b f -18 \sqrt {1-6 b}\, e -6 \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) e +\ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) f +6 \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) e -\ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) f -3 \sqrt {1-6 b}\, f +36 \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) x e +6 \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) x f -36 \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) x e -12 \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) f b +12 \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) f b +6 \sqrt {1-6 b}\, \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) e -\sqrt {1-6 b}\, \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) f -6 \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) x f -6 \sqrt {1-6 b}\, \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) e +\sqrt {1-6 b}\, \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) f}{162 \left (-1+6 b \right ) \left (-1+6 x +\sqrt {1-6 b}\right )}\)	\(356\)

Input:

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

Output:

1/54*sum((-_R*f-e)/(-6*_R^2+2*_R-b)*ln(x-_R),_R=RootOf(-(1-6*b)^(3/2)+108* 
_Z^3+54*b*_Z-54*_Z^2-9*b+1))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.44 \[ \int \frac {e+f x}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=-\frac {3 \, {\left (6 \, b - 1\right )} f x + 3 \, {\left (6 \, b - 1\right )} e - {\left (6 \, {\left (6 \, e + f\right )} x^{2} + 6 \, b e + b f - 2 \, {\left (6 \, e + f\right )} x + 2 \, {\left (6 \, f x^{2} + b f - 2 \, f x\right )} \sqrt {-6 \, b + 1}\right )} \log \left (6 \, x + \sqrt {-6 \, b + 1} - 1\right ) + {\left (6 \, {\left (6 \, e + f\right )} x^{2} + 6 \, b e + b f - 2 \, {\left (6 \, e + f\right )} x + 2 \, {\left (6 \, f x^{2} + b f - 2 \, f x\right )} \sqrt {-6 \, b + 1}\right )} \log \left (6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1\right ) - 3 \, {\left (b f - {\left (6 \, e + f\right )} x + e\right )} \sqrt {-6 \, b + 1}}{162 \, {\left (6 \, {\left (6 \, b - 1\right )} x^{2} + 6 \, b^{2} - 2 \, {\left (6 \, b - 1\right )} x - b\right )}} \] Input:

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

Output:

-1/162*(3*(6*b - 1)*f*x + 3*(6*b - 1)*e - (6*(6*e + f)*x^2 + 6*b*e + b*f - 
 2*(6*e + f)*x + 2*(6*f*x^2 + b*f - 2*f*x)*sqrt(-6*b + 1))*log(6*x + sqrt( 
-6*b + 1) - 1) + (6*(6*e + f)*x^2 + 6*b*e + b*f - 2*(6*e + f)*x + 2*(6*f*x 
^2 + b*f - 2*f*x)*sqrt(-6*b + 1))*log(6*x - 2*sqrt(-6*b + 1) - 1) - 3*(b*f 
 - (6*e + f)*x + e)*sqrt(-6*b + 1))/(6*(6*b - 1)*x^2 + 6*b^2 - 2*(6*b - 1) 
*x - b)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3281 vs. \(2 (116) = 232\).

Time = 1.46 (sec) , antiderivative size = 3281, normalized size of antiderivative = 23.60 \[ \int \frac {e+f x}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-sqrt((-24*b*f**2 + 36*e**2 + 24*e*f*sqrt(1 - 6*b) + 12*e*f + 4*f**2*sqrt( 
1 - 6*b) + 5*f**2)/(36*b**2 - 12*b + 1))*log(144*b**2*f**2/(1728*b*e*f + 2 
88*b*f**2*sqrt(1 - 6*b) + 288*b*f**2 - 432*e**2*sqrt(1 - 6*b) - 144*e*f*sq 
rt(1 - 6*b) - 288*e*f - 60*f**2*sqrt(1 - 6*b) - 48*f**2) - 216*b*e**2/(172 
8*b*e*f + 288*b*f**2*sqrt(1 - 6*b) + 288*b*f**2 - 432*e**2*sqrt(1 - 6*b) - 
 144*e*f*sqrt(1 - 6*b) - 288*e*f - 60*f**2*sqrt(1 - 6*b) - 48*f**2) - 144* 
b*e*f*sqrt(1 - 6*b)/(1728*b*e*f + 288*b*f**2*sqrt(1 - 6*b) + 288*b*f**2 - 
432*e**2*sqrt(1 - 6*b) - 144*e*f*sqrt(1 - 6*b) - 288*e*f - 60*f**2*sqrt(1 
- 6*b) - 48*f**2) - 360*b*e*f/(1728*b*e*f + 288*b*f**2*sqrt(1 - 6*b) + 288 
*b*f**2 - 432*e**2*sqrt(1 - 6*b) - 144*e*f*sqrt(1 - 6*b) - 288*e*f - 60*f* 
*2*sqrt(1 - 6*b) - 48*f**2) - 72*b*f**2*sqrt(1 - 6*b)/(1728*b*e*f + 288*b* 
f**2*sqrt(1 - 6*b) + 288*b*f**2 - 432*e**2*sqrt(1 - 6*b) - 144*e*f*sqrt(1 
- 6*b) - 288*e*f - 60*f**2*sqrt(1 - 6*b) - 48*f**2) - 102*b*f**2/(1728*b*e 
*f + 288*b*f**2*sqrt(1 - 6*b) + 288*b*f**2 - 432*e**2*sqrt(1 - 6*b) - 144* 
e*f*sqrt(1 - 6*b) - 288*e*f - 60*f**2*sqrt(1 - 6*b) - 48*f**2) + 72*e**2*s 
qrt(1 - 6*b)/(1728*b*e*f + 288*b*f**2*sqrt(1 - 6*b) + 288*b*f**2 - 432*e** 
2*sqrt(1 - 6*b) - 144*e*f*sqrt(1 - 6*b) - 288*e*f - 60*f**2*sqrt(1 - 6*b) 
- 48*f**2) + 36*e**2/(1728*b*e*f + 288*b*f**2*sqrt(1 - 6*b) + 288*b*f**2 - 
 432*e**2*sqrt(1 - 6*b) - 144*e*f*sqrt(1 - 6*b) - 288*e*f - 60*f**2*sqrt(1 
 - 6*b) - 48*f**2) + 48*e*f*sqrt(1 - 6*b)/(1728*b*e*f + 288*b*f**2*sqrt...
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate((f*x+e)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),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%%%{%%%{-972,[1]%%%}+%%%{162,[0]%%%},[2]%%%}+%%%{%%{[%%%{-324, 
[1]%%%}+%
 

Mupad [B] (verification not implemented)

Time = 12.26 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.27 \[ \int \frac {e+f x}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\frac {\frac {f}{324}-\frac {{\left (1-6\,b\right )}^{3/2}\,\left (\frac {e}{1944}+\frac {f}{11664}\right )}{b^2-\frac {b}{3}+\frac {1}{36}}}{\frac {{\left (1-6\,b\right )}^{3/2}}{36\,\left (b-\frac {1}{6}\right )}-x+\frac {1}{6}}+\frac {\mathrm {atan}\left (\frac {-b\,12{}\mathrm {i}-x\,12{}\mathrm {i}+b\,x\,72{}\mathrm {i}+{\left (1-6\,b\right )}^{3/2}\,1{}\mathrm {i}+2{}\mathrm {i}}{\sqrt {864\,b^2-{\left (6\,b-1\right )}^3-144\,b-1728\,b^3+8}}\right )\,\left (2\,f-24\,b\,f+6\,e\,{\left (1-6\,b\right )}^{3/2}+f\,{\left (1-6\,b\right )}^{3/2}+72\,b^2\,f\right )\,1{}\mathrm {i}}{27\,\left (6\,b-1\right )\,\sqrt {864\,b^2-{\left (6\,b-1\right )}^3-144\,b-1728\,b^3+8}} \] Input:

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

Output:

(f/324 - ((1 - 6*b)^(3/2)*(e/1944 + f/11664))/(b^2 - b/3 + 1/36))/((1 - 6* 
b)^(3/2)/(36*(b - 1/6)) - x + 1/6) + (atan((b*x*72i - x*12i - b*12i + (1 - 
 6*b)^(3/2)*1i + 2i)/(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2)) 
*(2*f - 24*b*f + 6*e*(1 - 6*b)^(3/2) + f*(1 - 6*b)^(3/2) + 72*b^2*f)*1i)/( 
27*(6*b - 1)*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1625, normalized size of antiderivative = 11.69 \[ \int \frac {e+f x}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 12*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*e - 
2*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*f - 72*sq 
rt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*e*x**2 + 24*sqr 
t(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*e*x - 12*sqrt(6* 
b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*f*x**2 + 4*sqrt(6*b 
- 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*f*x + 24*sqrt(6*b - 1) 
*atan((6*x - 1)/sqrt(6*b - 1))*b**2*f + 144*sqrt(6*b - 1)*atan((6*x - 1)/s 
qrt(6*b - 1))*b*f*x**2 - 48*sqrt(6*b - 1)*atan((6*x - 1)/sqrt(6*b - 1))*b* 
f*x - 4*sqrt(6*b - 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*f - 24*sqrt(6*b - 1) 
*atan((6*x - 1)/sqrt(6*b - 1))*f*x**2 + 8*sqrt(6*b - 1)*atan((6*x - 1)/sqr 
t(6*b - 1))*f*x - 12*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/(2*sqrt 
(6*b - 1)))*b*e - 2*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/(2*sqrt( 
6*b - 1)))*b*f - 72*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/(2*sqrt( 
6*b - 1)))*e*x**2 + 24*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/(2*sq 
rt(6*b - 1)))*e*x - 12*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/(2*sq 
rt(6*b - 1)))*f*x**2 + 4*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/(2* 
sqrt(6*b - 1)))*f*x + 24*sqrt(6*b - 1)*atan((6*x - 1)/(2*sqrt(6*b - 1)))*b 
**2*f + 144*sqrt(6*b - 1)*atan((6*x - 1)/(2*sqrt(6*b - 1)))*b*f*x**2 - 48* 
sqrt(6*b - 1)*atan((6*x - 1)/(2*sqrt(6*b - 1)))*b*f*x - 4*sqrt(6*b - 1)*at 
an((6*x - 1)/(2*sqrt(6*b - 1)))*b*f - 24*sqrt(6*b - 1)*atan((6*x - 1)/(...