\(\int \frac {\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}}{e+f x} \, dx\) [153]

Optimal result

Integrand size = 42, antiderivative size = 365 \[ \int \frac {\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}}{e+f x} \, dx=\frac {\sqrt {2} \left (f-\frac {6 e+f}{\sqrt {1-6 b}}\right ) \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}}{f^2 \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right )}-\frac {\sqrt {2} \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right ) \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3}}{3 f \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right )}+\frac {\sqrt {2} \left (6 e+f-\sqrt {1-6 b} f\right ) \sqrt {6 e+f+2 \sqrt {1-6 b} f} \sqrt {-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3} \text {arctanh}\left (\frac {\sqrt [4]{1-6 b} \sqrt {f} \sqrt {2+\frac {1-6 x}{\sqrt {1-6 b}}}}{\sqrt {6 e+f+2 \sqrt {1-6 b} f}}\right )}{(1-6 b)^{3/4} f^{5/2} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right ) \sqrt {2+\frac {1-6 x}{\sqrt {1-6 b}}}} \] Output:

2^(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)/f^2/(1-(1-6*x)/(1-6*b)^(1/2))-1/3*2^(1/2)*(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)/f/(1-(1-6*x)/( 
1-6*b)^(1/2))+2^(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)*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)/(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 = 16.69 (sec) , antiderivative size = 13435, normalized size of antiderivative = 36.81 \[ \int \frac {\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}}{e+f x} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 313, 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, 90, 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.

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

\(\Big \downarrow \) 2489

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

(Sqrt[1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3]*((-2*(-((1 + 
2*Sqrt[1 - 6*b])*(1 - 6*b)) + 6*(1 - 6*b)*x)^(3/2))/(3*f) + ((1 - 6*b)*(6* 
e + (1 - 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]*ArcT 
an[(Sqrt[f]*Sqrt[-((1 + 2*Sqrt[1 - 6*b])*(1 - 6*b)) + 6*(1 - 6*b)*x])/(Sqr 
t[1 - 6*b]*Sqrt[6*e + f + 2*Sqrt[1 - 6*b]*f])])/f^(3/2)))/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 90

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

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

Input:

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

Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1033, normalized size of antiderivative = 2.83 \[ \int \frac {\sqrt {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}}{e+f x} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/3*(3*(6*x^2 + b - 2*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)/f)*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 - 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 - ((3*b - 1)*f^2 - 3 
*e*f + 3*(6*e*f + f^2)*x + (3*f^2*x - 3*e*f - f^2)*sqrt(-6*b + 1))*sqrt(10 
8*x^3 + 54*b*x - 54*x^2 + (6*b - 1)*sqrt(-6*b + 1) - 9*b + 1)*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)/ 
f) + 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 + (2*b + 1)*f - 2*(18*e + 5*f)* 
x + sqrt(-6*b + 1)*(6*e + f) + 6*e)*sqrt(108*x^3 + 54*b*x - 54*x^2 + (6*b 
- 1)*sqrt(-6*b + 1) - 9*b + 1))/(6*f^2*x^2 + b*f^2 - 2*f^2*x), -1/3*(6*(6* 
x^2 + b - 2*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)/f)*arctan(1/27*(18*(6*e + f)*x^2 + 3*(12*b - 1 
)*e + (9*b - 1)*f - 3*((6*b + 1)*f + 12*e)*x + (18*f*x^2 + (6*b - 1)*f + 3 
*(6*e - f)*x - 3*e)*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)*sqrt(-6*b + 1)*f^3 - 54*b* 
e*f^2 - (9*b - 1)*f^3 - 108*e^3 - 54*e^2*f)/f)/(72*(b*f^2 + 6*e^2 + 2*e*f) 
*x^4 - 48*(b*f^2 + 6*e^2 + 2*e*f)*x^3 + 6*(8*b^2 - b)*e^2 + 2*(8*b^2 - ...
 

Sympy [F]

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

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

Maxima [F]

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

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

Output:

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

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

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

Reduce [F]

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

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