Integrand size = 40, antiderivative size = 211 \[ \int \frac {1}{(e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )} \, dx=-\frac {2}{3 \sqrt {1-6 b} \left (6 e+f-\sqrt {1-6 b} f\right ) \left (1-\sqrt {1-6 b}-6 x\right )}-\frac {2 \left (6 e+f-4 \sqrt {1-6 b} f\right ) \log \left (1-\sqrt {1-6 b}-6 x\right )}{9 (1-6 b) \left (6 e+f-\sqrt {1-6 b} f\right )^2}+\frac {2 \log \left (1+2 \sqrt {1-6 b}-6 x\right )}{9 (1-6 b) \left (6 e+f+2 \sqrt {1-6 b} f\right )}-\frac {2 f^2 \log (e+f x)}{\left (6 e+f-\sqrt {1-6 b} f\right )^2 \left (6 e+f+2 \sqrt {1-6 b} f\right )} \] Output:
-2/3/(1-6*b)^(1/2)/(6*e+f-(1-6*b)^(1/2)*f)/(1-(1-6*b)^(1/2)-6*x)-2/9*(6*e+ f-4*(1-6*b)^(1/2)*f)*ln(1-(1-6*b)^(1/2)-6*x)/(1-6*b)/(6*e+f-(1-6*b)^(1/2)* f)^2+2/9*ln(1+2*(1-6*b)^(1/2)-6*x)/(1-6*b)/(6*e+f+2*(1-6*b)^(1/2)*f)-2*f^2 *ln(f*x+e)/(6*e+f-(1-6*b)^(1/2)*f)^2/(6*e+f+2*(1-6*b)^(1/2)*f)
Leaf count is larger than twice the leaf count of optimal. \(621\) vs. \(2(211)=422\).
Time = 4.53 (sec) , antiderivative size = 621, normalized size of antiderivative = 2.94 \[ \int \frac {1}{(e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )} \, dx=\frac {1}{54} \left (-\frac {6 b f \left (2+\sqrt {1-6 b}-6 x\right )+2 \left (1+\sqrt {1-6 b}\right ) f (-1+3 x)+6 e \left (-1-\sqrt {1-6 b}+6 b+6 \sqrt {1-6 b} x\right )}{(-1+6 b) \left (6 e^2+2 e f+b f^2\right ) (b+2 x (-1+3 x))}-\frac {4 \left (6 \sqrt {1-6 b} e+\left (-2+\sqrt {1-6 b}+12 b\right ) f\right ) \arctan \left (\frac {-1+6 x}{2 \sqrt {-1+6 b}}\right )}{(-1+6 b)^{3/2} \left (12 e^2+4 e f+(-1+8 b) f^2\right )}+\frac {2 \left (-36 \sqrt {1-6 b} e^3-6 \left (-2+3 \sqrt {1-6 b}+12 b\right ) e^2 f+2 \left (2 \left (1+\sqrt {1-6 b}\right )-3 \left (4+7 \sqrt {1-6 b}\right ) b\right ) e f^2+\left (1+\sqrt {1-6 b}-\left (10+7 \sqrt {1-6 b}\right ) b+24 b^2\right ) f^3\right ) \arctan \left (\frac {-1+6 x}{\sqrt {-1+6 b}}\right )}{(-1+6 b)^{3/2} \left (6 e^2+2 e f+b f^2\right )^2}-\frac {2 f^2 \left (108 e^3+54 e^2 f+54 b e f^2+\left (-1-\sqrt {1-6 b}+\left (9+6 \sqrt {1-6 b}\right ) b\right ) f^3\right ) \log (e+f x)}{\left (6 e^2+2 e f+b f^2\right )^2 \left (12 e^2+4 e f+(-1+8 b) f^2\right )}-\frac {2 \left (6 e+f-2 \sqrt {1-6 b} f\right ) \log \left (1-8 b+4 x-12 x^2\right )}{(-1+6 b) \left (12 e^2+4 e f+(-1+8 b) f^2\right )}+\frac {\left (36 e^3-6 \left (-3+2 \sqrt {1-6 b}\right ) e^2 f-2 \left (2+2 \sqrt {1-6 b}-21 b\right ) e f^2+\left (-1-\sqrt {1-6 b}+\left (7+4 \sqrt {1-6 b}\right ) b\right ) f^3\right ) \log \left (b-2 x+6 x^2\right )}{(-1+6 b) \left (6 e^2+2 e f+b f^2\right )^2}\right ) \] Input:
Integrate[1/((e + f*x)*(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108* x^3)),x]
Output:
(-((6*b*f*(2 + Sqrt[1 - 6*b] - 6*x) + 2*(1 + Sqrt[1 - 6*b])*f*(-1 + 3*x) + 6*e*(-1 - Sqrt[1 - 6*b] + 6*b + 6*Sqrt[1 - 6*b]*x))/((-1 + 6*b)*(6*e^2 + 2*e*f + b*f^2)*(b + 2*x*(-1 + 3*x)))) - (4*(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)*(12*e^2 + 4*e*f + (-1 + 8*b)*f^2)) + (2*(-36*Sqrt[1 - 6*b]*e^3 - 6*(-2 + 3*Sqrt[1 - 6*b] + 12*b)*e^2*f + 2*(2*(1 + Sqrt[1 - 6*b]) - 3*(4 + 7*Sqr t[1 - 6*b])*b)*e*f^2 + (1 + Sqrt[1 - 6*b] - (10 + 7*Sqrt[1 - 6*b])*b + 24* b^2)*f^3)*ArcTan[(-1 + 6*x)/Sqrt[-1 + 6*b]])/((-1 + 6*b)^(3/2)*(6*e^2 + 2* e*f + b*f^2)^2) - (2*f^2*(108*e^3 + 54*e^2*f + 54*b*e*f^2 + (-1 - Sqrt[1 - 6*b] + (9 + 6*Sqrt[1 - 6*b])*b)*f^3)*Log[e + f*x])/((6*e^2 + 2*e*f + b*f^ 2)^2*(12*e^2 + 4*e*f + (-1 + 8*b)*f^2)) - (2*(6*e + f - 2*Sqrt[1 - 6*b]*f) *Log[1 - 8*b + 4*x - 12*x^2])/((-1 + 6*b)*(12*e^2 + 4*e*f + (-1 + 8*b)*f^2 )) + ((36*e^3 - 6*(-3 + 2*Sqrt[1 - 6*b])*e^2*f - 2*(2 + 2*Sqrt[1 - 6*b] - 21*b)*e*f^2 + (-1 - Sqrt[1 - 6*b] + (7 + 4*Sqrt[1 - 6*b])*b)*f^3)*Log[b - 2*x + 6*x^2])/((-1 + 6*b)*(6*e^2 + 2*e*f + b*f^2)^2))/54
Time = 1.17 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.07, 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, 93, 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.
| \(\displaystyle \int \frac {1}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right ) (e+f x)} \, dx\) |
| \(\Big \downarrow \) 2488 |
| \(\displaystyle 99179645184 (1-6 b)^3 \int -\frac {1}{49589822592 \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \left (\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)-6 (1-6 b) x\right ) (e+f x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 (1-6 b)^3 \int \frac {1}{(1-6 b) \left (-6 x+2 \sqrt {1-6 b}+1\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 (e+f x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 (1-6 b)^2 \int \frac {1}{\left (-6 x+2 \sqrt {1-6 b}+1\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 (e+f x)}dx\) |
| \(\Big \downarrow \) 93 |
| \(\displaystyle -2 (1-6 b)^2 \int \left (\frac {f^3}{(1-6 b)^2 \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )^2 \left (6 e+2 \sqrt {1-6 b} f+f\right ) (e+f x)}-\frac {2}{3 (6 b-1)^3 \left (6 e+2 \sqrt {1-6 b} f+f\right ) \left (-6 x+2 \sqrt {1-6 b}+1\right )}-\frac {2 \left (6 e-4 \sqrt {1-6 b} f+f\right )}{3 (6 b-1)^3 \left (6 e-\sqrt {1-6 b} f+f\right )^2 \left (6 x+\sqrt {1-6 b}-1\right )}+\frac {2}{(1-6 b)^{5/2} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right ) \left (-6 x-\sqrt {1-6 b}+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 (1-6 b)^2 \left (\frac {f^2 \log (e+f x)}{(1-6 b)^2 \left (-\sqrt {1-6 b} f+6 e+f\right )^2 \left (2 \sqrt {1-6 b} f+6 e+f\right )}+\frac {1}{3 (1-6 b)^{5/2} \left (-\sqrt {1-6 b}-6 x+1\right ) \left (-\sqrt {1-6 b} f+6 e+f\right )}+\frac {\left (-4 \sqrt {1-6 b} f+6 e+f\right ) \log \left (-\sqrt {1-6 b}-6 x+1\right )}{9 (1-6 b)^3 \left (-\sqrt {1-6 b} f+6 e+f\right )^2}-\frac {\log \left (2 \sqrt {1-6 b}-6 x+1\right )}{9 (1-6 b)^3 \left (2 \sqrt {1-6 b} f+6 e+f\right )}\right )\) |
Input:
Int[1/((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*(1/(3*(1 - 6*b)^(5/2)*(6*e + f - Sqrt[1 - 6*b]*f)*(1 - Sqrt [1 - 6*b] - 6*x)) + ((6*e + f - 4*Sqrt[1 - 6*b]*f)*Log[1 - Sqrt[1 - 6*b] - 6*x])/(9*(1 - 6*b)^3*(6*e + f - Sqrt[1 - 6*b]*f)^2) - Log[1 + 2*Sqrt[1 - 6*b] - 6*x]/(9*(1 - 6*b)^3*(6*e + f + 2*Sqrt[1 - 6*b]*f)) + (f^2*Log[e + f *x])/((1 - 6*b)^2*(6*e + f - Sqrt[1 - 6*b]*f)^2*(6*e + f + 2*Sqrt[1 - 6*b] *f)))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Int[ExpandIntegrand[(e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.30 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.89
| method | result | size |
| default | \(-\frac {f^{2} \ln \left (f x +e \right )}{\left (1-6 b \right )^{\frac {3}{2}} f^{3}+54 b e \,f^{2}+9 b \,f^{3}+108 e^{3}+54 e^{2} f -f^{3}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (-\left (1-6 b \right )^{\frac {3}{2}}+108 \textit {\_Z}^{3}+54 b \textit {\_Z} -54 \textit {\_Z}^{2}-9 b +1\right )}{\sum }\frac {\left (-2 \textit {\_R}^{2} f^{2}+2 \textit {\_R} e f +\textit {\_R} \,f^{2}-b \,f^{2}-2 e^{2}-e f \right ) \ln \left (x -\textit {\_R} \right )}{-6 \textit {\_R}^{2}+2 \textit {\_R} -b}}{\left (1-6 b \right )^{\frac {3}{2}} f^{3}+54 b e \,f^{2}+9 b \,f^{3}+108 e^{3}+54 e^{2} f -f^{3}}\) | \(187\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(8887\) |
Input:
int(1/(f*x+e)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x,method=_RETURN VERBOSE)
Output:
-f^2/((1-6*b)^(3/2)*f^3+54*b*e*f^2+9*b*f^3+108*e^3+54*e^2*f-f^3)*ln(f*x+e) +sum((-2*_R^2*f^2+2*_R*e*f+_R*f^2-b*f^2-2*e^2-e*f)/(-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))/((1-6*b)^(3/2 )*f^3+54*b*e*f^2+9*b*f^3+108*e^3+54*e^2*f-f^3)
Leaf count of result is larger than twice the leaf count of optimal. 1738 vs. \(2 (188) = 376\).
Time = 6.91 (sec) , antiderivative size = 1738, normalized size of antiderivative = 8.24 \[ \int \frac {1}{(e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/(f*x+e)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x, algorit hm="fricas")
Output:
-1/27*(216*(6*b - 1)*e^5 + 216*(6*b - 1)*e^4*f + 18*(60*b^2 + 8*b - 3)*e^3 *f^2 + 4*(180*b^2 - 36*b + 1)*e^2*f^3 + (144*b^3 + 78*b^2 - 29*b + 2)*e*f^ 4 + (48*b^3 - 14*b^2 + b)*f^5 - 3*(72*(6*b - 1)*e^4*f + 48*(6*b - 1)*e^3*f ^2 + 2*(180*b^2 - 24*b - 1)*e^2*f^3 + 2*(60*b^2 - 16*b + 1)*e*f^4 + (48*b^ 3 - 14*b^2 + b)*f^5)*x + (108*(6*b^2 - b)*e^3*f^2 + 54*(6*b^2 - b)*e^2*f^3 + 54*(6*b^3 - b^2)*e*f^4 + (54*b^3 - 15*b^2 + b)*f^5 + 6*(108*(6*b - 1)*e ^3*f^2 + 54*(6*b - 1)*e^2*f^3 + 54*(6*b^2 - b)*e*f^4 + (54*b^2 - 15*b + 1) *f^5)*x^2 - 2*(108*(6*b - 1)*e^3*f^2 + 54*(6*b - 1)*e^2*f^3 + 54*(6*b^2 - b)*e*f^4 + (54*b^2 - 15*b + 1)*f^5)*x)*log(f*x + e) - (432*b*e^5 + 360*b*e ^4*f + 12*(66*b^2 - b)*e^3*f^2 + 2*(198*b^2 - 23*b)*e^2*f^3 + 2*(168*b^3 - 23*b^2)*e*f^4 + (56*b^3 - 15*b^2 + b)*f^5 + 6*(12*(66*b - 1)*e^3*f^2 + 2* (198*b - 23)*e^2*f^3 + 2*(168*b^2 - 23*b)*e*f^4 + (56*b^2 - 15*b + 1)*f^5 + 432*e^5 + 360*e^4*f)*x^2 - 2*(12*(66*b - 1)*e^3*f^2 + 2*(198*b - 23)*e^2 *f^3 + 2*(168*b^2 - 23*b)*e*f^4 + (56*b^2 - 15*b + 1)*f^5 + 432*e^5 + 360* e^4*f)*x - (16*b^2*e*f^4 + 144*b*e^4*f + 96*b*e^3*f^2 + 16*(3*b^2 + b)*e^2 *f^3 - (32*b^3 - 12*b^2 + b)*f^5 + 6*(16*(3*b + 1)*e^2*f^3 + 16*b*e*f^4 - (32*b^2 - 12*b + 1)*f^5 + 144*e^4*f + 96*e^3*f^2)*x^2 - 2*(16*(3*b + 1)*e^ 2*f^3 + 16*b*e*f^4 - (32*b^2 - 12*b + 1)*f^5 + 144*e^4*f + 96*e^3*f^2)*x)* sqrt(-6*b + 1))*log(6*x + sqrt(-6*b + 1) - 1) + 2*(b^3*f^5 + 216*b*e^5 + 1 80*b*e^4*f + 24*(3*b^2 + 2*b)*e^3*f^2 + 4*(9*b^2 + b)*e^2*f^3 + 2*(3*b^...
Timed out. \[ \int \frac {1}{(e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(f*x+e)/(1-(1-6*b)**(3/2)-9*b+54*b*x-54*x**2+108*x**3),x)
Output:
Timed out
\[ \int \frac {1}{(e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )} \, dx=\int { \frac {1}{{\left (108 \, x^{3} + 54 \, b x - 54 \, x^{2} - {\left (-6 \, b + 1\right )}^{\frac {3}{2}} - 9 \, b + 1\right )} {\left (f x + e\right )}} \,d x } \] Input:
integrate(1/(f*x+e)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x, algorit hm="maxima")
Output:
-f^2*log(f*x + e)/(54*b*e*f^2 + ((-6*b + 1)^(3/2) + 9*b - 1)*f^3 + 108*e^3 + 54*e^2*f) + 54*integrate((2*f^2*x^2 + b*f^2 + 2*e^2 + e*f - (2*e*f + f^ 2)*x)/(108*x^3 + 54*b*x - 54*x^2 - (-6*b + 1)^(3/2) - 9*b + 1), x)/(54*b*e *f^2 + ((-6*b + 1)^(3/2) + 9*b - 1)*f^3 + 108*e^3 + 54*e^2*f)
Exception generated. \[ \int \frac {1}{(e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(f*x+e)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x, algorit hm="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]%%%}+%
Time = 48.64 (sec) , antiderivative size = 12009, normalized size of antiderivative = 56.91 \[ \int \frac {1}{(e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )} \, dx=\text {Too large to display} \] Input:
int(-1/((e + f*x)*(9*b - 54*b*x + (1 - 6*b)^(3/2) + 54*x^2 - 108*x^3 - 1)) ,x)
Output:
(log((x*(6*e*f - 108*b*e^2 - 9*b*f^2 + f^2*(1 - 6*b)^(3/2) + 18*e^2 + f^2 + 18*b^2*f^2 - 36*b*e*f + 6*e*f*(1 - 6*b)^(3/2)))/(1458*(6*b - 1)^2*(2*e*f + b*f^2 + 6*e^2)^2) - (216*b*e^3 - 12*b*f^3 + 4*e*f^2 - 12*e^2*f + f^3*(1 - 6*b)^(3/2) - 36*e^3 + f^3 + 36*b^2*f^3 - 3*b*f^3*(1 - 6*b)^(3/2) + 4*e* f^2*(1 - 6*b)^(3/2) + 6*e^2*f*(1 - 6*b)^(3/2) + 180*b^2*e*f^2 - 54*b*e*f^2 + 72*b*e^2*f)/(2916*f*(6*b - 1)^2*(2*e*f + b*f^2 + 6*e^2)^2) + ((54*e*f^2 - 216*b*f^3 - 9*f^3*(-(6*b - 1)^3)^(1/2) + 8*f^3*(1 - 6*b)^(3/2) + f^3*(1 - 6*b)^(9/2) + 9*f^3 + 1944*b^2*f^3 - 7776*b^3*f^3 + 11664*b^4*f^3 - 144* b*f^3*(1 - 6*b)^(3/2) - 828*b^2*f^3*(-(6*b - 1)^3)^(1/2) + 1512*b^3*f^3*(- (6*b - 1)^3)^(1/2) + 11664*b^2*e*f^2 - 46656*b^3*e*f^2 + 69984*b^4*e*f^2 + 864*b^2*f^3*(1 - 6*b)^(3/2) - 1728*b^3*f^3*(1 - 6*b)^(3/2) + 150*b*f^3*(- (6*b - 1)^3)^(1/2) - 24*e*f^2*(-(6*b - 1)^3)^(1/2) - 72*e^2*f*(-(6*b - 1)^ 3)^(1/2) + 432*e^3*(1 - 6*b)^(3/2)*(-(6*b - 1)^3)^(1/2) - 9*f^3*(1 - 6*b)^ (3/2)*(-(6*b - 1)^3)^(1/2) - 1296*b*e*f^2 + 288*b*e*f^2*(-(6*b - 1)^3)^(1/ 2) + 864*b*e^2*f*(-(6*b - 1)^3)^(1/2) + 66*b*f^3*(1 - 6*b)^(3/2)*(-(6*b - 1)^3)^(1/2) - 30*e*f^2*(1 - 6*b)^(3/2)*(-(6*b - 1)^3)^(1/2) + 216*e^2*f*(1 - 6*b)^(3/2)*(-(6*b - 1)^3)^(1/2) - 864*b^2*e*f^2*(-(6*b - 1)^3)^(1/2) - 2592*b^2*e^2*f*(-(6*b - 1)^3)^(1/2) + 396*b*e*f^2*(1 - 6*b)^(3/2)*(-(6*b - 1)^3)^(1/2))*(15552*e^5*(-(6*b - 1)^3)^(1/2) + 684*b*f^5 + 396*e*f^4 - 77 76*e^4*f - 972*f^5*x + 46656*e^5*(1 - 6*b)^(3/2) - 5*f^5*(1 - 6*b)^(3/2...
Time = 0.18 (sec) , antiderivative size = 11241, normalized size of antiderivative = 53.27 \[ \int \frac {1}{(e+f x) \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )} \, dx =\text {Too large to display} \] Input:
int(1/(f*x+e)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x)
Output:
( - 672*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**3* e*f**4 - 112*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))* b**3*f**5 - 1584*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*e**3*f**2 - 792*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqr t(6*b - 1))*b**2*e**2*f**3 - 4032*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*e*f**4*x**2 + 1344*sqrt(6*b - 1)*sqrt( - 6*b + 1 )*atan((6*x - 1)/sqrt(6*b - 1))*b**2*e*f**4*x + 92*sqrt(6*b - 1)*sqrt( - 6 *b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*e*f**4 - 672*sqrt(6*b - 1)*sqrt ( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*f**5*x**2 + 224*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*f**5*x + 30*sqrt(6 *b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*f**5 - 864*sqr t(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*e**5 - 720*sqr t(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*e**4*f - 9504* sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*e**3*f**2*x **2 + 3168*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b* e**3*f**2*x + 24*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*e**3*f**2 - 4752*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt( 6*b - 1))*b*e**2*f**3*x**2 + 1584*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*e**2*f**3*x + 92*sqrt(6*b - 1)*sqrt( - 6*b + 1)*ata n((6*x - 1)/sqrt(6*b - 1))*b*e**2*f**3 + 552*sqrt(6*b - 1)*sqrt( - 6*b ...