\(\int \frac {\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}}{(e+f x)^2} \, dx\) [154]

Optimal result

Integrand size = 42, antiderivative size = 435 \[ \int \frac {\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}}{(e+f x)^2} \, dx=\frac {9 \sqrt {2} \left (6 e+f+\sqrt {1-6 b} f\right ) \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}}{\sqrt {1-6 b} f^2 \left (6 e+f+2 \sqrt {1-6 b} f\right ) \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right )}+\frac {\left (6 e+f-\sqrt {1-6 b} f\right ) \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right ) \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}}{\sqrt {2} f \left (6 e+f+2 \sqrt {1-6 b} f\right ) \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right ) (e+f x)}-\frac {9 \sqrt {2} \left (6 e+f+\sqrt {1-6 b} f\right ) \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3} \text {arctanh}\left (\frac {\sqrt [4]{1-6 b} \sqrt {f} \sqrt {2+\frac {1-6 x}{\sqrt {1-6 b}}}}{\sqrt {6 e+f+2 \sqrt {1-6 b} f}}\right )}{(1-6 b)^{3/4} f^{5/2} \sqrt {6 e+2 \left (\frac {1}{2}+\sqrt {1-6 b}\right ) f} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right ) \sqrt {2+\frac {1-6 x}{\sqrt {1-6 b}}}} \] Output:

9*2^(1/2)*(6*e+f+(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)/(1-6*b)^(1/2)/f^2/(6*e+f+2*(1-6*b)^(1/2)*f)/(1-(1-6*x)/(1-6*b 
)^(1/2))+1/2*(6*e+f-(1-6*b)^(1/2)*f)*(2+(1-6*x)/(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^(1/2)/f/(6*e+f+2*(1-6*b)^(1/2) 
*f)/(1-(1-6*x)/(1-6*b)^(1/2))/(f*x+e)-9*2^(1/2)*(6*e+f+(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)*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))/(1 
-6*b)^(3/4)/f^(5/2)/(6*e+2*(1/2+(1-6*b)^(1/2))*f)^(1/2)/(1-(1-6*x)/(1-6*b) 
^(1/2))/(2+(1-6*x)/(1-6*b)^(1/2))^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

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

Time = 16.25 (sec) , antiderivative size = 5628, normalized size of antiderivative = 12.94 \[ \int \frac {\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}}{(e+f x)^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.86, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2489, 27, 87, 60, 73, 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.

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

\(\Big \downarrow \) 2489

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[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 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 1309, normalized size of antiderivative = 3.01 \[ \int \frac {\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}}{(e+f x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[(3*sqrt(3)*(6*f*x^3 + 2*(3*e - f)*x^2 + b*e + (b*f - 2*e)*x)*sqrt(-((6*b 
- 1)*sqrt(-6*b + 1)*f^3 + 54*b*e*f^2 + (9*b - 1)*f^3 + 108*e^3 + 54*e^2*f) 
/((8*b - 1)*f^3 + 12*e^2*f + 4*e*f^2))*log((9*b^2*f^3 + 162*b*e^3 + 108*b* 
e^2*f + 9*(3*b^2 + 2*b)*e*f^2 - 162*(b*f^3 + 6*e^2*f + 2*e*f^2)*x^3 + 54*( 
(3*b + 4)*e*f^2 + 2*b*f^3 + 18*e^3 + 18*e^2*f)*x^2 + sqrt(3)*(3*(2*b - 1)* 
e*f^2 - b*f^3 - 6*e^2*f + ((12*b - 1)*f^3 + 36*e^2*f + 12*e*f^2)*x - (2*b* 
f^3 + 6*e^2*f + 3*e*f^2 - (6*e*f^2 + f^3)*x)*sqrt(-6*b + 1))*sqrt(108*x^3 
+ 54*b*x - 54*x^2 + (6*b - 1)*sqrt(-6*b + 1) - 9*b + 1)*sqrt(-((6*b - 1)*s 
qrt(-6*b + 1)*f^3 + 54*b*e*f^2 + (9*b - 1)*f^3 + 108*e^3 + 54*e^2*f)/((8*b 
 - 1)*f^3 + 12*e^2*f + 4*e*f^2)) - 9*(6*(3*b + 4)*e^2*f + 4*(3*b + 1)*e*f^ 
2 + (3*b^2 + 2*b)*f^3 + 36*e^3)*x + 18*(b^2*f^3 + 6*b*e^2*f + 2*b*e*f^2 + 
6*(b*f^3 + 6*e^2*f + 2*e*f^2)*x^2 - 2*(b*f^3 + 6*e^2*f + 2*e*f^2)*x)*sqrt( 
-6*b + 1))/(6*f*x^3 + 2*(3*e - f)*x^2 + b*e + (b*f - 2*e)*x)) + (12*f*x^2 
- b*f + (18*e - f)*x - 3*(f*x + e)*sqrt(-6*b + 1) - 3*e)*sqrt(108*x^3 + 54 
*b*x - 54*x^2 + (6*b - 1)*sqrt(-6*b + 1) - 9*b + 1))/(6*f^3*x^3 + b*e*f^2 
+ 2*(3*e*f^2 - f^3)*x^2 + (b*f^3 - 2*e*f^2)*x), (6*sqrt(3)*(6*f*x^3 + 2*(3 
*e - f)*x^2 + b*e + (b*f - 2*e)*x)*sqrt(((6*b - 1)*sqrt(-6*b + 1)*f^3 + 54 
*b*e*f^2 + (9*b - 1)*f^3 + 108*e^3 + 54*e^2*f)/((8*b - 1)*f^3 + 12*e^2*f + 
 4*e*f^2))*arctan(1/9*sqrt(3)*(2*(12*b - 1)*e^2 + (10*b - 1)*e*f + (8*b^2 
- b)*f^2 + 2*((12*b - 1)*f^2 + 36*e^2 + 12*e*f)*x^2 - (6*(2*b + 1)*e*f ...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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