\(\int \frac {(e+f x)^2}{(1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3)^2} \, dx\) [145]

Optimal result

Integrand size = 40, antiderivative size = 404 \[ \int \frac {(e+f x)^2}{\left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^2} \, dx=\frac {18 e^2+6 e f+f^2-3 b f^2-\sqrt {1-6 b} f (6 e+f)}{729 (1-6 b) \left (1-\sqrt {1-6 b}-6 x\right )^3}-\frac {36 e^2+6 \left (2+\sqrt {1-6 b}\right ) e f-\left (1-\sqrt {1-6 b}-12 b\right ) f^2}{1458 (1-6 b)^{3/2} \left (1-\sqrt {1-6 b}-6 x\right )^2}+\frac {(6 e+f) \left (6 e+f+2 \sqrt {1-6 b} f\right )}{1458 (1-6 b)^2 \left (1-\sqrt {1-6 b}-6 x\right )}+\frac {36 e^2+12 e f+5 f^2-24 b f^2+\sqrt {1-6 b} f (24 e+4 f)}{4374 (1-6 b)^2 \left (1+2 \sqrt {1-6 b}-6 x\right )}+\frac {\left (72 e^2+6 \left (4+5 \sqrt {1-6 b}\right ) e f+\left (4+5 \sqrt {1-6 b}-12 b\right ) f^2\right ) \log \left (1-\sqrt {1-6 b}-6 x\right )}{6561 (1-6 b)^{5/2}}-\frac {\left (72 e^2+6 \left (4+5 \sqrt {1-6 b}\right ) e f+\left (4+5 \sqrt {1-6 b}-12 b\right ) f^2\right ) \log \left (1+2 \sqrt {1-6 b}-6 x\right )}{6561 (1-6 b)^{5/2}} \] Output:

1/729*(18*e^2+6*e*f+f^2-3*b*f^2-(1-6*b)^(1/2)*f*(6*e+f))/(1-6*b)/(1-(1-6*b 
)^(1/2)-6*x)^3-1/1458*(36*e^2+6*(2+(1-6*b)^(1/2))*e*f-(1-(1-6*b)^(1/2)-12* 
b)*f^2)/(1-6*b)^(3/2)/(1-(1-6*b)^(1/2)-6*x)^2+1/1458*(6*e+f)*(6*e+f+2*(1-6 
*b)^(1/2)*f)/(1-6*b)^2/(1-(1-6*b)^(1/2)-6*x)+1/4374*(36*e^2+12*e*f+5*f^2-2 
4*b*f^2+(1-6*b)^(1/2)*f*(24*e+4*f))/(1-6*b)^2/(1+2*(1-6*b)^(1/2)-6*x)+1/65 
61*(72*e^2+6*(4+5*(1-6*b)^(1/2))*e*f+(4+5*(1-6*b)^(1/2)-12*b)*f^2)*ln(1-(1 
-6*b)^(1/2)-6*x)/(1-6*b)^(5/2)-1/6561*(72*e^2+6*(4+5*(1-6*b)^(1/2))*e*f+(4 
+5*(1-6*b)^(1/2)-12*b)*f^2)*ln(1+2*(1-6*b)^(1/2)-6*x)/(1-6*b)^(5/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(914\) vs. \(2(404)=808\).

Time = 6.91 (sec) , antiderivative size = 914, normalized size of antiderivative = 2.26 \[ \int \frac {(e+f x)^2}{\left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^2} \, dx=\frac {6 e^2+6 \sqrt {1-6 b} e^2+12 b e f+b f^2-\sqrt {1-6 b} b f^2-36 e^2 x-12 e f x+12 \sqrt {1-6 b} e f x-2 f^2 x+2 \sqrt {1-6 b} f^2 x+6 b f^2 x}{13122 \left (b-2 x+6 x^2\right )^3}+\frac {18 e^2-18 \sqrt {1-6 b} e^2+30 e f+6 \sqrt {1-6 b} e f-144 b e f+2 f^2-2 \sqrt {1-6 b} f^2-9 b f^2+21 \sqrt {1-6 b} b f^2-108 e^2 x-36 e f x-72 \sqrt {1-6 b} e f x+12 f^2 x-12 \sqrt {1-6 b} f^2 x-90 b f^2 x}{26244 (-1+6 b) \left (b-2 x+6 x^2\right )^2}+\frac {18 e^2+9 e f+6 \sqrt {1-6 b} e f-18 b e f+f^2+2 \sqrt {1-6 b} f^2-3 b f^2-6 \sqrt {1-6 b} b f^2-108 e^2 x-36 e f x-36 \sqrt {1-6 b} e f x-3 f^2 x-6 \sqrt {1-6 b} f^2 x}{4374 (-1+6 b)^2 \left (b-2 x+6 x^2\right )}+\frac {12 e^2-24 \sqrt {1-6 b} e^2-12 e f+96 b e f-f^2-2 \sqrt {1-6 b} f^2+8 b f^2+16 \sqrt {1-6 b} b f^2-72 e^2 x-24 e f x-48 \sqrt {1-6 b} e f x-10 f^2 x-8 \sqrt {1-6 b} f^2 x+48 b f^2 x}{4374 (-1+6 b)^2 \left (-1+8 b-4 x+12 x^2\right )}+\frac {\left (-72 e^2-24 e f-30 \sqrt {1-6 b} e f-4 f^2-5 \sqrt {1-6 b} f^2+12 b f^2\right ) \arctan \left (\frac {-1+6 x}{2 \sqrt {-1+6 b}}\right )}{6561 (-1+6 b)^{5/2}}+\frac {\left (-72 e^2-24 e f-30 \sqrt {1-6 b} e f-4 f^2-5 \sqrt {1-6 b} f^2+12 b f^2\right ) \arctan \left (\frac {-1+6 x}{\sqrt {-1+6 b}}\right )}{6561 (-1+6 b)^{5/2}}+\frac {\left (72 \sqrt {1-6 b} e^2+30 e f+24 \sqrt {1-6 b} e f-180 b e f+5 f^2+4 \sqrt {1-6 b} f^2-30 b f^2-12 \sqrt {1-6 b} b f^2\right ) \log \left (1-8 b+4 x-12 x^2\right )}{13122 (-1+6 b)^3}+\frac {\left (-72 \sqrt {1-6 b} e^2-30 e f-24 \sqrt {1-6 b} e f+180 b e f-5 f^2-4 \sqrt {1-6 b} f^2+30 b f^2+12 \sqrt {1-6 b} b f^2\right ) \log \left (b-2 x+6 x^2\right )}{13122 (-1+6 b)^3} \] Input:

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

Output:

(6*e^2 + 6*Sqrt[1 - 6*b]*e^2 + 12*b*e*f + b*f^2 - Sqrt[1 - 6*b]*b*f^2 - 36 
*e^2*x - 12*e*f*x + 12*Sqrt[1 - 6*b]*e*f*x - 2*f^2*x + 2*Sqrt[1 - 6*b]*f^2 
*x + 6*b*f^2*x)/(13122*(b - 2*x + 6*x^2)^3) + (18*e^2 - 18*Sqrt[1 - 6*b]*e 
^2 + 30*e*f + 6*Sqrt[1 - 6*b]*e*f - 144*b*e*f + 2*f^2 - 2*Sqrt[1 - 6*b]*f^ 
2 - 9*b*f^2 + 21*Sqrt[1 - 6*b]*b*f^2 - 108*e^2*x - 36*e*f*x - 72*Sqrt[1 - 
6*b]*e*f*x + 12*f^2*x - 12*Sqrt[1 - 6*b]*f^2*x - 90*b*f^2*x)/(26244*(-1 + 
6*b)*(b - 2*x + 6*x^2)^2) + (18*e^2 + 9*e*f + 6*Sqrt[1 - 6*b]*e*f - 18*b*e 
*f + f^2 + 2*Sqrt[1 - 6*b]*f^2 - 3*b*f^2 - 6*Sqrt[1 - 6*b]*b*f^2 - 108*e^2 
*x - 36*e*f*x - 36*Sqrt[1 - 6*b]*e*f*x - 3*f^2*x - 6*Sqrt[1 - 6*b]*f^2*x)/ 
(4374*(-1 + 6*b)^2*(b - 2*x + 6*x^2)) + (12*e^2 - 24*Sqrt[1 - 6*b]*e^2 - 1 
2*e*f + 96*b*e*f - f^2 - 2*Sqrt[1 - 6*b]*f^2 + 8*b*f^2 + 16*Sqrt[1 - 6*b]* 
b*f^2 - 72*e^2*x - 24*e*f*x - 48*Sqrt[1 - 6*b]*e*f*x - 10*f^2*x - 8*Sqrt[1 
 - 6*b]*f^2*x + 48*b*f^2*x)/(4374*(-1 + 6*b)^2*(-1 + 8*b - 4*x + 12*x^2)) 
+ ((-72*e^2 - 24*e*f - 30*Sqrt[1 - 6*b]*e*f - 4*f^2 - 5*Sqrt[1 - 6*b]*f^2 
+ 12*b*f^2)*ArcTan[(-1 + 6*x)/(2*Sqrt[-1 + 6*b])])/(6561*(-1 + 6*b)^(5/2)) 
 + ((-72*e^2 - 24*e*f - 30*Sqrt[1 - 6*b]*e*f - 4*f^2 - 5*Sqrt[1 - 6*b]*f^2 
 + 12*b*f^2)*ArcTan[(-1 + 6*x)/Sqrt[-1 + 6*b]])/(6561*(-1 + 6*b)^(5/2)) + 
((72*Sqrt[1 - 6*b]*e^2 + 30*e*f + 24*Sqrt[1 - 6*b]*e*f - 180*b*e*f + 5*f^2 
 + 4*Sqrt[1 - 6*b]*f^2 - 30*b*f^2 - 12*Sqrt[1 - 6*b]*b*f^2)*Log[1 - 8*b + 
4*x - 12*x^2])/(13122*(-1 + 6*b)^3) + ((-72*Sqrt[1 - 6*b]*e^2 - 30*e*f ...
 

Rubi [A] (verified)

Time = 2.03 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2488, 27, 27, 99, 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.

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

\(\Big \downarrow \) 2488

\(\displaystyle 9836602018824134393856 (1-6 b)^6 \int \frac {(e+f x)^2}{2459150504706033598464 \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^4 \left (\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)-6 (1-6 b) x\right )^2}dx\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 99

\(\displaystyle 4 (1-6 b)^4 \int \left (\frac {(6 e+f) \left (6 e+2 \sqrt {1-6 b} f+f\right )}{972 (6 b-1)^6 \left (6 x+\sqrt {1-6 b}-1\right )^2}+\frac {-72 e^2-6 \left (5 \sqrt {1-6 b}+4\right ) f e-\left (-12 b+5 \sqrt {1-6 b}+4\right ) f^2}{4374 (1-6 b)^{13/2} \left (-6 x-\sqrt {1-6 b}+1\right )}+\frac {72 e^2+6 \left (5 \sqrt {1-6 b}+4\right ) f e+\left (-12 b+5 \sqrt {1-6 b}+4\right ) f^2}{4374 (1-6 b)^{13/2} \left (-6 x+2 \sqrt {1-6 b}+1\right )}+\frac {36 e^2+12 \left (2 \sqrt {1-6 b} f+f\right ) e+\left (-24 b+4 \sqrt {1-6 b}+5\right ) f^2}{2916 (1-6 b)^6 \left (-6 x+2 \sqrt {1-6 b}+1\right )^2}+\frac {-36 e^2-6 \left (\sqrt {1-6 b}+2\right ) f e+\left (-12 b-\sqrt {1-6 b}+1\right ) f^2}{486 (1-6 b)^{11/2} \left (-6 x-\sqrt {1-6 b}+1\right )^3}+\frac {18 e^2+6 \left (1-\sqrt {1-6 b}\right ) f e+\left (-3 b-\sqrt {1-6 b}+1\right ) f^2}{162 (1-6 b)^5 \left (-6 x-\sqrt {1-6 b}+1\right )^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 4 (1-6 b)^4 \left (\frac {12 e \left (2 \sqrt {1-6 b} f+f\right )+\left (-24 b+4 \sqrt {1-6 b}+5\right ) f^2+36 e^2}{17496 (1-6 b)^6 \left (2 \sqrt {1-6 b}-6 x+1\right )}-\frac {6 \left (\sqrt {1-6 b}+2\right ) e f-\left (-12 b-\sqrt {1-6 b}+1\right ) f^2+36 e^2}{5832 (1-6 b)^{11/2} \left (-\sqrt {1-6 b}-6 x+1\right )^2}+\frac {6 \left (1-\sqrt {1-6 b}\right ) e f+\left (-3 b-\sqrt {1-6 b}+1\right ) f^2+18 e^2}{2916 (1-6 b)^5 \left (-\sqrt {1-6 b}-6 x+1\right )^3}+\frac {\left (6 \left (5 \sqrt {1-6 b}+4\right ) e f+\left (-12 b+5 \sqrt {1-6 b}+4\right ) f^2+72 e^2\right ) \log \left (-\sqrt {1-6 b}-6 x+1\right )}{26244 (1-6 b)^{13/2}}-\frac {\left (6 \left (5 \sqrt {1-6 b}+4\right ) e f+\left (-12 b+5 \sqrt {1-6 b}+4\right ) f^2+72 e^2\right ) \log \left (2 \sqrt {1-6 b}-6 x+1\right )}{26244 (1-6 b)^{13/2}}+\frac {(6 e+f) \left (2 \sqrt {1-6 b} f+6 e+f\right )}{5832 (1-6 b)^6 \left (-\sqrt {1-6 b}-6 x+1\right )}\right )\)

Input:

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

Output:

4*(1 - 6*b)^4*((18*e^2 + 6*(1 - Sqrt[1 - 6*b])*e*f + (1 - Sqrt[1 - 6*b] - 
3*b)*f^2)/(2916*(1 - 6*b)^5*(1 - Sqrt[1 - 6*b] - 6*x)^3) - (36*e^2 + 6*(2 
+ Sqrt[1 - 6*b])*e*f - (1 - Sqrt[1 - 6*b] - 12*b)*f^2)/(5832*(1 - 6*b)^(11 
/2)*(1 - Sqrt[1 - 6*b] - 6*x)^2) + ((6*e + f)*(6*e + f + 2*Sqrt[1 - 6*b]*f 
))/(5832*(1 - 6*b)^6*(1 - Sqrt[1 - 6*b] - 6*x)) + (36*e^2 + (5 + 4*Sqrt[1 
- 6*b] - 24*b)*f^2 + 12*e*(f + 2*Sqrt[1 - 6*b]*f))/(17496*(1 - 6*b)^6*(1 + 
 2*Sqrt[1 - 6*b] - 6*x)) + ((72*e^2 + 6*(4 + 5*Sqrt[1 - 6*b])*e*f + (4 + 5 
*Sqrt[1 - 6*b] - 12*b)*f^2)*Log[1 - Sqrt[1 - 6*b] - 6*x])/(26244*(1 - 6*b) 
^(13/2)) - ((72*e^2 + 6*(4 + 5*Sqrt[1 - 6*b])*e*f + (4 + 5*Sqrt[1 - 6*b] - 
 12*b)*f^2)*Log[1 + 2*Sqrt[1 - 6*b] - 6*x])/(26244*(1 - 6*b)^(13/2)))
 

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 99

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

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 [A] (verified)

Time = 2.58 (sec) , antiderivative size = 703, normalized size of antiderivative = 1.74

method	result	size
default	\(\frac {-\frac {\left (30 \sqrt {1-6 b}\, e f +5 \sqrt {1-6 b}\, f^{2}-12 b \,f^{2}+72 e^{2}+24 e f +4 f^{2}\right ) x^{3}}{13122 \left (36 b^{2}-12 b +1\right )}-\frac {\left (30 \sqrt {1-6 b}\, e f +5 \sqrt {1-6 b}\, f^{2}-12 b \,f^{2}+72 e^{2}+24 e f +4 f^{2}\right ) \left (\sqrt {1-6 b}-2\right ) x^{2}}{52488 \left (36 b^{2}-12 b +1\right )}-\frac {\left (6 b +1-2 \sqrt {1-6 b}\right ) \left (864 b^{2} e^{2}-72 e^{2}-f^{2}+360 \sqrt {1-6 b}\, b e f +360 \sqrt {1-6 b}\, b^{2} e f +864 b \,e^{2}+288 b^{2} e f +276 \sqrt {1-6 b}\, b^{2} f^{2}+504 b^{3} f^{2}+42 b \,f^{2}-204 b^{2} f^{2}-24 e f +\sqrt {1-6 b}\, f^{2}+288 b e f -30 \sqrt {1-6 b}\, e f -12 \sqrt {1-6 b}\, b \,f^{2}\right ) x}{314928 \left (12 b^{2}+12 b -1\right ) \left (36 b^{2}-12 b +1\right )}-\frac {\left (102 \sqrt {1-6 b}\, b -11 \sqrt {1-6 b}-36 b +2\right ) \left (2448 \sqrt {1-6 b}\, b^{3} e f +3972 \sqrt {1-6 b}\, b^{3} f^{2}+12240 b^{4} f^{2}+3060 \sqrt {1-6 b}\, b^{2} e f -1164 \sqrt {1-6 b}\, b^{2} f^{2}+124848 b^{3} e^{2}+24120 b^{3} e f -6600 b^{3} f^{2}-804 \sqrt {1-6 b}\, b e f +127 \sqrt {1-6 b}\, b \,f^{2}-45144 b^{2} e^{2}-8244 b^{2} e f +1464 b^{2} f^{2}+51 \sqrt {1-6 b}\, e f -5 \sqrt {1-6 b}\, f^{2}+5652 b \,e^{2}+1074 b e f -142 b \,f^{2}-234 e^{2}-51 e f +5 f^{2}\right )}{2834352 \left (6936 b^{3}-2508 b^{2}+314 b -13\right ) \left (36 b^{2}-12 b +1\right )}}{\left (x^{3}+\frac {\sqrt {1-6 b}\, b}{18}+\frac {b x}{2}-\frac {x^{2}}{2}-\frac {\sqrt {1-6 b}}{108}-\frac {b}{12}+\frac {1}{108}\right ) \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right )}+\frac {8 \left (30 \sqrt {1-6 b}\, e f +5 \sqrt {1-6 b}\, f^{2}-12 b \,f^{2}+72 e^{2}+24 e f +4 f^{2}\right ) \arctan \left (\frac {-72 x +6 \sqrt {1-6 b}+12}{18 \sqrt {-1+6 b}}\right )}{81 \left (11664 b^{2}-3888 b +324\right ) \sqrt {-1+6 b}}\)	\(703\)
parallelrisch	\(\text {Expression too large to display}\)	\(4720\)

Input:

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

Output:

(-1/13122*(30*(1-6*b)^(1/2)*e*f+5*(1-6*b)^(1/2)*f^2-12*b*f^2+72*e^2+24*e*f 
+4*f^2)/(36*b^2-12*b+1)*x^3-1/52488*(30*(1-6*b)^(1/2)*e*f+5*(1-6*b)^(1/2)* 
f^2-12*b*f^2+72*e^2+24*e*f+4*f^2)/(36*b^2-12*b+1)*((1-6*b)^(1/2)-2)*x^2-1/ 
314928*(6*b+1-2*(1-6*b)^(1/2))*(864*b^2*e^2-72*e^2-f^2+360*(1-6*b)^(1/2)*b 
*e*f+360*(1-6*b)^(1/2)*b^2*e*f+864*b*e^2+288*b^2*e*f+276*(1-6*b)^(1/2)*b^2 
*f^2+504*b^3*f^2+42*b*f^2-204*b^2*f^2-24*e*f+(1-6*b)^(1/2)*f^2+288*b*e*f-3 
0*(1-6*b)^(1/2)*e*f-12*(1-6*b)^(1/2)*b*f^2)/(12*b^2+12*b-1)/(36*b^2-12*b+1 
)*x-1/2834352*(102*(1-6*b)^(1/2)*b-11*(1-6*b)^(1/2)-36*b+2)*(2448*(1-6*b)^ 
(1/2)*b^3*e*f+3972*(1-6*b)^(1/2)*b^3*f^2+12240*b^4*f^2+3060*(1-6*b)^(1/2)* 
b^2*e*f-1164*(1-6*b)^(1/2)*b^2*f^2+124848*b^3*e^2+24120*b^3*e*f-6600*b^3*f 
^2-804*(1-6*b)^(1/2)*b*e*f+127*(1-6*b)^(1/2)*b*f^2-45144*b^2*e^2-8244*b^2* 
e*f+1464*b^2*f^2+51*(1-6*b)^(1/2)*e*f-5*(1-6*b)^(1/2)*f^2+5652*b*e^2+1074* 
b*e*f-142*b*f^2-234*e^2-51*e*f+5*f^2)/(6936*b^3-2508*b^2+314*b-13)/(36*b^2 
-12*b+1))/(x^3+1/18*(1-6*b)^(1/2)*b+1/2*b*x-1/2*x^2-1/108*(1-6*b)^(1/2)-1/ 
12*b+1/108)/(-1/6+x+1/6*(1-6*b)^(1/2))+8/81*(30*(1-6*b)^(1/2)*e*f+5*(1-6*b 
)^(1/2)*f^2-12*b*f^2+72*e^2+24*e*f+4*f^2)/(11664*b^2-3888*b+324)/(-1+6*b)^ 
(1/2)*arctan(1/18*(-72*x+6*(1-6*b)^(1/2)+12)/(-1+6*b)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2724 vs. \(2 (347) = 694\).

Time = 0.16 (sec) , antiderivative size = 2724, normalized size of antiderivative = 6.74 \[ \int \frac {(e+f x)^2}{\left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/26244*(20736*(18*(6*b - 1)*e^2 + 6*(6*b - 1)*e*f - (18*b^2 - 9*b + 1)*f 
^2)*x^7 - 1296*(336*(6*b - 1)*e^2 + 6*(60*b^2 + 92*b - 17)*e*f - (276*b^2 
- 148*b + 17)*f^2)*x^6 + 648*(48*(72*b^2 + 18*b - 5)*e^2 + 12*(156*b^2 + 4 
*b - 5)*e*f + (72*b^3 - 156*b^2 + 102*b - 13)*f^2)*x^5 - 24*(360*(216*b^2 
- 30*b - 1)*e^2 - 3*(3888*b^3 - 13284*b^2 + 2424*b - 53)*e*f - (324*b^3 - 
1152*b^2 - 63*b + 41)*f^2)*x^4 + 12*(6*(12312*b^3 + 2484*b^2 - 1014*b + 43 
)*e^2 + 6*(1512*b^3 + 2724*b^2 - 754*b + 43)*e*f + (11016*b^4 - 6804*b^3 + 
 2766*b^2 - 419*b + 16)*f^2)*x^3 - 6*(5256*b^4 - 2820*b^3 + 582*b^2 - 55*b 
 + 2)*e^2 + 6*(288*b^5 - 804*b^4 + 264*b^3 - 29*b^2 + b)*e*f - 3*(96*b^5 + 
 20*b^4 - 12*b^3 + b^2)*f^2 - 18*(2*(12312*b^3 - 3276*b^2 + 234*b - 5)*e^2 
 - 2*(7128*b^4 - 7560*b^3 + 1482*b^2 - 52*b - 3)*e*f + (1296*b^4 - 492*b^3 
 + 184*b^2 - 35*b + 2)*f^2)*x^2 + 6*(2*(15768*b^4 - 4356*b^3 + 270*b^2 + 9 
*b - 1)*e^2 - 2*(1872*b^4 - 2868*b^3 + 852*b^2 - 89*b + 3)*e*f + (576*b^5 
- 144*b^4 + 164*b^3 - 44*b^2 + 3*b)*f^2)*x - 4*(12960*(6*(6*b - 1)*e*f + ( 
6*b - 1)*f^2)*x^8 - 17280*(6*(6*b - 1)*e*f + (6*b - 1)*f^2)*x^7 + 7560*(6* 
(12*b^2 + 4*b - 1)*e*f + (12*b^2 + 4*b - 1)*f^2)*x^6 - 840*(6*(108*b^2 - 1 
2*b - 1)*e*f + (108*b^2 - 12*b - 1)*f^2)*x^5 + 100*(6*(324*b^3 + 216*b^2 - 
 57*b + 2)*e*f + (324*b^3 + 216*b^2 - 57*b + 2)*f^2)*x^4 - 40*(6*(540*b^3 
- 60*b^2 - 11*b + 1)*e*f + (540*b^3 - 60*b^2 - 11*b + 1)*f^2)*x^3 + 30*(48 
*b^5 - 14*b^4 + b^3)*e*f + 5*(48*b^5 - 14*b^4 + b^3)*f^2 + 30*(6*(156*b...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/324*(18*(72*b^2 - (-6*b + 1)^(3/2) - 18*b + 1)*e^2 + 12*(3*(2*(-6*b + 1) 
^(3/2) - 5)*b + 54*b^2 - (-6*b + 1)^(3/2) + 1)*e*f - ((6*b - 1)^3 - 15*((- 
6*b + 1)^(3/2) - 1)*b - 54*b^2 + 2*(-6*b + 1)^(3/2) - 1)*f^2 + 36*(18*(6*b 
 - 1)*e^2 + 6*((-6*b + 1)^(3/2) + 6*b - 1)*e*f - (18*b^2 - (-6*b + 1)^(3/2 
) - 9*b + 1)*f^2)*x^2 + 6*(18*((-6*b + 1)^(3/2) - 12*b + 2)*e^2 - 6*(36*b^ 
2 + (-6*b + 1)^(3/2) - 1)*e*f - (3*((-6*b + 1)^(3/2) + 2)*b + (-6*b + 1)^( 
3/2) - 1)*f^2)*x)/((-6*b + 1)^(9/2) + 108*(2*(-6*b + 1)^(3/2) - 11)*b^3 + 
1944*b^4 + 108*((6*b - 1)^3 - 216*b^3 + 108*b^2 - 18*b + 1)*x^3 + (6*b - 1 
)^3 - 54*(2*(-6*b + 1)^(3/2) - 5)*b^2 - 54*((6*b - 1)^3 - 216*b^3 + 108*b^ 
2 - 18*b + 1)*x^2 - 9*((6*b - 1)^3 - 2*(-6*b + 1)^(3/2) + 3)*b - 54*(216*b 
^4 - 108*b^3 - ((6*b - 1)^3 + 1)*b + 18*b^2)*x - (-6*b + 1)^(3/2) + 1) - 1 
/54*integrate(-(18*(2*(-6*b + 1)^(3/2) - 6*b + 1)*e^2 - 6*(72*b^2 - (-6*b 
+ 1)^(3/2) - 18*b + 1)*e*f - (3*(2*(-6*b + 1)^(3/2) - 5)*b + 54*b^2 - (-6* 
b + 1)^(3/2) + 1)*f^2 + 6*(18*(6*b - 1)*e^2 + 6*((-6*b + 1)^(3/2) + 6*b - 
1)*e*f - (18*b^2 - (-6*b + 1)^(3/2) - 9*b + 1)*f^2)*x)/(108*x^3 + 54*b*x - 
 54*x^2 - (-6*b + 1)^(3/2) - 9*b + 1), x)/((6*b - 1)^3 - 216*b^3 + 108*b^2 
 - 18*b + 1)
 

Giac [F(-2)]

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

integrate((f*x+e)^2/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^2,x, algor 
ithm="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%%%{%%%{17006112,[3]%%%}+%%%{-8503056,[2]%%%}+%%%{1417176,[1]% 
%%}+%%%{-
 

Mupad [B] (verification not implemented)

Time = 1.95 (sec) , antiderivative size = 645, normalized size of antiderivative = 1.60 \[ \int \frac {(e+f x)^2}{\left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(x^2*(((e*f)/104976 + (b*f^2)/314928 + e^2/13122 + f^2/629856 + (5*b*e*f)/ 
52488)/(b^2 - b/3 + 1/36) - ((1 - 6*b)^(3/2)*((e*f)/314928 + (b*f^2)/94478 
4 - e^2/157464 + f^2/1889568))/(b/12 - b^2/2 + b^3 - 1/216)) - x^3*(((e*f) 
/19683 - (b*f^2)/39366 + e^2/6561 + f^2/118098)/(b^2 - b/3 + 1/36) - ((1 - 
 6*b)^(3/2)*((5*e*f)/472392 + (5*f^2)/2834352))/(b/12 - b^2/2 + b^3 - 1/21 
6)) - x*(((b*e^2)/26244 - (e*f)/314928 - (b*f^2)/314928 + e^2/157464 + f^2 
/3779136 + (7*b^2*f^2)/314928 + (7*b*e*f)/157464)/(b^2 - b/3 + 1/36) + ((1 
 - 6*b)^(3/2)*((e*f)/3779136 - (b*f^2)/1259712 + e^2/472392 + f^2/22674816 
 - (5*b*e*f)/1889568))/(b/12 - b^2/2 + b^3 - 1/216)) + ((b*e^2)/157464 - ( 
17*e*f)/34012224 - (29*b*f^2)/34012224 - e^2/2834352 + (5*f^2)/102036672 + 
 (23*b^2*f^2)/5668704 + (23*b*e*f)/5668704 + (b^2*e*f)/472392)/(b^2 - b/3 
+ 1/36) + ((1 - 6*b)^(3/2)*((17*b*e^2)/5668704 - (17*e*f)/204073344 - (b*f 
^2)/8503056 - (11*e^2)/34012224 + (5*f^2)/612220032 + (5*b^2*f^2)/17006112 
 + (19*b*e*f)/34012224))/(b/12 - b^2/2 + b^3 - 1/216))/((7*b)/216 - x*(b/6 
 + ((1 - 6*b)^(3/2)*((5*b)/216 - 1/648))/(b - 1/6) - 1/108) + x^2*(b/2 + ( 
1 - 6*b)^(3/2)/(72*(b - 1/6)) + 1/12) - x^3*((1 - 6*b)^(3/2)/(36*(b - 1/6) 
) + 2/3) - b^2/18 + x^4 + ((1 - 6*b)^(3/2)*((5*b)/1296 - 1/1944))/(b - 1/6 
) - 1/324) + (atan((b*x*72i - x*12i - b*12i + (1 - 6*b)^(3/2)*1i + 2i)/(86 
4*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2))*(24*e*f - 432*b*e^2 - 3 
6*b*f^2 + 5*f^2*(1 - 6*b)^(3/2) + 72*e^2 + 4*f^2 + 72*b^2*f^2 - 144*b*e...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 15132, normalized size of antiderivative = 37.46 \[ \int \frac {(e+f x)^2}{\left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 1920*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**4 
*e*f - 320*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b* 
*4*f**2 - 37440*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1 
))*b**3*e*f*x**2 + 12480*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqr 
t(6*b - 1))*b**3*e*f*x + 240*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1) 
/sqrt(6*b - 1))*b**3*e*f - 6240*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 
 1)/sqrt(6*b - 1))*b**3*f**2*x**2 + 2080*sqrt(6*b - 1)*sqrt( - 6*b + 1)*at 
an((6*x - 1)/sqrt(6*b - 1))*b**3*f**2*x + 40*sqrt(6*b - 1)*sqrt( - 6*b + 1 
)*atan((6*x - 1)/sqrt(6*b - 1))*b**3*f**2 - 259200*sqrt(6*b - 1)*sqrt( - 6 
*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*e*f*x**4 + 172800*sqrt(6*b - 1) 
*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*e*f*x**3 - 24480*sqrt 
(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*e*f*x**2 - 1 
440*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*e*f* 
x - 43200*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b** 
2*f**2*x**4 + 28800*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b 
 - 1))*b**2*f**2*x**3 - 4080*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1) 
/sqrt(6*b - 1))*b**2*f**2*x**2 - 240*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan(( 
6*x - 1)/sqrt(6*b - 1))*b**2*f**2*x - 725760*sqrt(6*b - 1)*sqrt( - 6*b + 1 
)*atan((6*x - 1)/sqrt(6*b - 1))*b*e*f*x**6 + 725760*sqrt(6*b - 1)*sqrt( - 
6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*e*f*x**5 - 216000*sqrt(6*b - 1...