Integrand size = 40, antiderivative size = 218 \[ \int \frac {(e+f x)^2}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=-\frac {18 e^2+6 \left (1-\sqrt {1-6 b}\right ) e f+\left (1-\sqrt {1-6 b}-3 b\right ) f^2}{162 \sqrt {1-6 b} \left (1-\sqrt {1-6 b}-6 x\right )}-\frac {\left (18 e^2-\left (2-2 \sqrt {1-6 b}-15 b\right ) f^2+6 e \left (f+2 \sqrt {1-6 b} f\right )\right ) \log \left (1-\sqrt {1-6 b}-6 x\right )}{486 (1-6 b)}+\frac {\left (36 e^2+\left (5+4 \sqrt {1-6 b}-24 b\right ) f^2+12 e \left (f+2 \sqrt {1-6 b} f\right )\right ) \log \left (1+2 \sqrt {1-6 b}-6 x\right )}{972 (1-6 b)} \] Output:
-1/162*(18*e^2+6*(1-(1-6*b)^(1/2))*e*f+(1-(1-6*b)^(1/2)-3*b)*f^2)/(1-6*b)^ (1/2)/(1-(1-6*b)^(1/2)-6*x)-(18*e^2-(2-2*(1-6*b)^(1/2)-15*b)*f^2+6*e*(f+2* (1-6*b)^(1/2)*f))*ln(1-(1-6*b)^(1/2)-6*x)/(486-2916*b)+(36*e^2+(5+4*(1-6*b )^(1/2)-24*b)*f^2+12*e*(f+2*(1-6*b)^(1/2)*f))*ln(1+2*(1-6*b)^(1/2)-6*x)/(9 72-5832*b)
Leaf count is larger than twice the leaf count of optimal. \(460\) vs. \(2(218)=436\).
Time = 1.11 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.11 \[ \int \frac {(e+f x)^2}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\frac {\frac {6 \left (12 e f \left (b \left (\sqrt {1-6 b}-6 x\right )+x-\sqrt {1-6 b} x\right )-6 e^2 \left (-1-\sqrt {1-6 b}+6 b+6 \sqrt {1-6 b} x\right )+f^2 \left (6 b^2-2 \left (-1+\sqrt {1-6 b}\right ) x+b \left (-1+\sqrt {1-6 b}+6 \left (-2+\sqrt {1-6 b}\right ) x\right )\right )\right )}{(-1+6 b) (b+2 x (-1+3 x))}+\frac {2 \left (-36 \sqrt {1-6 b} e^2-12 \left (2+\sqrt {1-6 b}-12 b\right ) e f+\left (-4-5 \sqrt {1-6 b}+24 \left (1+\sqrt {1-6 b}\right ) b\right ) f^2\right ) \arctan \left (\frac {-1+6 x}{2 \sqrt {-1+6 b}}\right )}{(-1+6 b)^{3/2}}-\frac {4 \left (18 \sqrt {1-6 b} e^2+6 \left (2+\sqrt {1-6 b}-12 b\right ) e f+\left (2-2 \sqrt {1-6 b}+3 \left (-4+5 \sqrt {1-6 b}\right ) b\right ) f^2\right ) \arctan \left (\frac {-1+6 x}{\sqrt {-1+6 b}}\right )}{(-1+6 b)^{3/2}}-\frac {\left (36 e^2+\left (5+4 \sqrt {1-6 b}-24 b\right ) f^2+12 e \left (f+2 \sqrt {1-6 b} f\right )\right ) \log \left (1-8 b+4 x-12 x^2\right )}{-1+6 b}+\frac {2 \left (18 e^2+\left (-2+2 \sqrt {1-6 b}+15 b\right ) f^2+6 e \left (f+2 \sqrt {1-6 b} f\right )\right ) \log \left (b-2 x+6 x^2\right )}{-1+6 b}}{1944} \] Input:
Integrate[(e + f*x)^2/(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x ^3),x]
Output:
((6*(12*e*f*(b*(Sqrt[1 - 6*b] - 6*x) + x - Sqrt[1 - 6*b]*x) - 6*e^2*(-1 - Sqrt[1 - 6*b] + 6*b + 6*Sqrt[1 - 6*b]*x) + f^2*(6*b^2 - 2*(-1 + Sqrt[1 - 6 *b])*x + b*(-1 + Sqrt[1 - 6*b] + 6*(-2 + Sqrt[1 - 6*b])*x))))/((-1 + 6*b)* (b + 2*x*(-1 + 3*x))) + (2*(-36*Sqrt[1 - 6*b]*e^2 - 12*(2 + Sqrt[1 - 6*b] - 12*b)*e*f + (-4 - 5*Sqrt[1 - 6*b] + 24*(1 + Sqrt[1 - 6*b])*b)*f^2)*ArcTa n[(-1 + 6*x)/(2*Sqrt[-1 + 6*b])])/(-1 + 6*b)^(3/2) - (4*(18*Sqrt[1 - 6*b]* e^2 + 6*(2 + Sqrt[1 - 6*b] - 12*b)*e*f + (2 - 2*Sqrt[1 - 6*b] + 3*(-4 + 5* Sqrt[1 - 6*b])*b)*f^2)*ArcTan[(-1 + 6*x)/Sqrt[-1 + 6*b]])/(-1 + 6*b)^(3/2) - ((36*e^2 + (5 + 4*Sqrt[1 - 6*b] - 24*b)*f^2 + 12*e*(f + 2*Sqrt[1 - 6*b] *f))*Log[1 - 8*b + 4*x - 12*x^2])/(-1 + 6*b) + (2*(18*e^2 + (-2 + 2*Sqrt[1 - 6*b] + 15*b)*f^2 + 6*e*(f + 2*Sqrt[1 - 6*b]*f))*Log[b - 2*x + 6*x^2])/( -1 + 6*b))/1944
Time = 1.04 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.04, 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.
| \(\displaystyle \int \frac {(e+f x)^2}{54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} \, dx\) |
| \(\Big \downarrow \) 2488 |
| \(\displaystyle 99179645184 (1-6 b)^3 \int -\frac {(e+f x)^2}{49589822592 \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \left (\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)-6 (1-6 b) x\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 (1-6 b)^3 \int \frac {(e+f x)^2}{(1-6 b) \left (-6 x+2 \sqrt {1-6 b}+1\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 (1-6 b)^2 \int \frac {(e+f x)^2}{\left (-6 x+2 \sqrt {1-6 b}+1\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2}dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -2 (1-6 b)^2 \int \left (\frac {18 e^2+6 \left (1-\sqrt {1-6 b}\right ) f e+\left (-3 b-\sqrt {1-6 b}+1\right ) f^2}{54 (1-6 b)^{5/2} \left (-6 x-\sqrt {1-6 b}+1\right )^2}+\frac {-18 e^2-6 \left (2 \sqrt {1-6 b} f+f\right ) e+\left (-15 b-2 \sqrt {1-6 b}+2\right ) f^2}{162 (1-6 b)^3 \left (-6 x-\sqrt {1-6 b}+1\right )}+\frac {36 e^2+12 \left (2 \sqrt {1-6 b} f+f\right ) e+\left (-24 b+4 \sqrt {1-6 b}+5\right ) f^2}{324 (1-6 b)^3 \left (-6 x+2 \sqrt {1-6 b}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 (1-6 b)^2 \left (\frac {6 \left (1-\sqrt {1-6 b}\right ) e f+\left (-3 b-\sqrt {1-6 b}+1\right ) f^2+18 e^2}{324 (1-6 b)^{5/2} \left (-\sqrt {1-6 b}-6 x+1\right )}+\frac {\left (6 e \left (2 \sqrt {1-6 b} f+f\right )-\left (-15 b-2 \sqrt {1-6 b}+2\right ) f^2+18 e^2\right ) \log \left (-\sqrt {1-6 b}-6 x+1\right )}{972 (1-6 b)^3}-\frac {\left (12 e \left (2 \sqrt {1-6 b} f+f\right )+\left (-24 b+4 \sqrt {1-6 b}+5\right ) f^2+36 e^2\right ) \log \left (2 \sqrt {1-6 b}-6 x+1\right )}{1944 (1-6 b)^3}\right )\) |
Input:
Int[(e + f*x)^2/(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3),x]
Output:
-2*(1 - 6*b)^2*((18*e^2 + 6*(1 - Sqrt[1 - 6*b])*e*f + (1 - Sqrt[1 - 6*b] - 3*b)*f^2)/(324*(1 - 6*b)^(5/2)*(1 - Sqrt[1 - 6*b] - 6*x)) + ((18*e^2 - (2 - 2*Sqrt[1 - 6*b] - 15*b)*f^2 + 6*e*(f + 2*Sqrt[1 - 6*b]*f))*Log[1 - Sqrt [1 - 6*b] - 6*x])/(972*(1 - 6*b)^3) - ((36*e^2 + (5 + 4*Sqrt[1 - 6*b] - 24 *b)*f^2 + 12*e*(f + 2*Sqrt[1 - 6*b]*f))*Log[1 + 2*Sqrt[1 - 6*b] - 6*x])/(1 944*(1 - 6*b)^3))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]))
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 = 0.56 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.34
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-\left (1-6 b \right )^{\frac {3}{2}}+108 \textit {\_Z}^{3}+54 b \textit {\_Z} -54 \textit {\_Z}^{2}-9 b +1\right )}{\sum }\frac {\left (-\textit {\_R}^{2} f^{2}-2 \textit {\_R} e f -e^{2}\right ) \ln \left (x -\textit {\_R} \right )}{-6 \textit {\_R}^{2}+2 \textit {\_R} -b}\right )}{54}\) | \(75\) |
| parallelrisch | \(\frac {6 f^{2}+24 \sqrt {1-6 b}\, \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) x \,f^{2}+30 \sqrt {1-6 b}\, \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) b \,f^{2}+144 \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) e f b -24 \sqrt {1-6 b}\, \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) x \,f^{2}+24 \sqrt {1-6 b}\, \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) b \,f^{2}+180 \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) x b \,f^{2}+144 \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) x b \,f^{2}-12 \sqrt {1-6 b}\, \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) e f +12 \sqrt {1-6 b}\, \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) e f +72 \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) x e f -72 \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) x e f -144 \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) e f b +144 \sqrt {1-6 b}\, \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) x e f -144 \sqrt {1-6 b}\, \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) x e f -6 \sqrt {1-6 b}\, f^{2}-12 \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) e f +36 \sqrt {1-6 b}\, \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) e^{2}-8 \sqrt {1-6 b}\, \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) f^{2}-36 \sqrt {1-6 b}\, \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) e^{2}-\sqrt {1-6 b}\, \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) f^{2}+216 \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) x \,e^{2}-108 \sqrt {1-6 b}\, e^{2}-36 b \,f^{2}+36 e f -216 b e f -36 \sqrt {1-6 b}\, e f -36 \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) e^{2}+8 \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) f^{2}+36 \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) e^{2}+\ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) f^{2}-24 \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) x \,f^{2}-54 \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) b \,f^{2}-216 \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) x \,e^{2}-30 \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) x \,f^{2}+12 \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) e f +18 \sqrt {1-6 b}\, b \,f^{2}}{972 \left (-1+6 b \right ) \left (-1+6 x +\sqrt {1-6 b}\right )}\) | \(707\) |
Input:
int((f*x+e)^2/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x,method=_RETURN VERBOSE)
Output:
1/54*sum((-_R^2*f^2-2*_R*e*f-e^2)/(-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))
Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (185) = 370\).
Time = 0.10 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.78 \[ \int \frac {(e+f x)^2}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=-\frac {18 \, {\left (6 \, b - 1\right )} e^{2} - 3 \, {\left (6 \, b^{2} - b\right )} f^{2} + 6 \, {\left (6 \, {\left (6 \, b - 1\right )} e f + {\left (6 \, b - 1\right )} f^{2}\right )} x - 2 \, {\left (18 \, b e^{2} + 6 \, b e f + {\left (15 \, b^{2} - 2 \, b\right )} f^{2} + 6 \, {\left ({\left (15 \, b - 2\right )} f^{2} + 18 \, e^{2} + 6 \, e f\right )} x^{2} - 2 \, {\left ({\left (15 \, b - 2\right )} f^{2} + 18 \, e^{2} + 6 \, e f\right )} x + 2 \, {\left (6 \, b e f + b f^{2} + 6 \, {\left (6 \, e f + f^{2}\right )} x^{2} - 2 \, {\left (6 \, e f + f^{2}\right )} x\right )} \sqrt {-6 \, b + 1}\right )} \log \left (6 \, x + \sqrt {-6 \, b + 1} - 1\right ) + {\left (36 \, b e^{2} + 12 \, b e f - {\left (24 \, b^{2} - 5 \, b\right )} f^{2} - 6 \, {\left ({\left (24 \, b - 5\right )} f^{2} - 36 \, e^{2} - 12 \, e f\right )} x^{2} + 2 \, {\left ({\left (24 \, b - 5\right )} f^{2} - 36 \, e^{2} - 12 \, e f\right )} x + 4 \, {\left (6 \, b e f + b f^{2} + 6 \, {\left (6 \, e f + f^{2}\right )} x^{2} - 2 \, {\left (6 \, e f + f^{2}\right )} x\right )} \sqrt {-6 \, b + 1}\right )} \log \left (6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1\right ) - 3 \, {\left (12 \, b e f + b f^{2} + 6 \, e^{2} + 2 \, {\left ({\left (3 \, b - 1\right )} f^{2} - 18 \, e^{2} - 6 \, e f\right )} x\right )} \sqrt {-6 \, b + 1}}{972 \, {\left (6 \, {\left (6 \, b - 1\right )} x^{2} + 6 \, b^{2} - 2 \, {\left (6 \, b - 1\right )} x - b\right )}} \] Input:
integrate((f*x+e)^2/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x, algorit hm="fricas")
Output:
-1/972*(18*(6*b - 1)*e^2 - 3*(6*b^2 - b)*f^2 + 6*(6*(6*b - 1)*e*f + (6*b - 1)*f^2)*x - 2*(18*b*e^2 + 6*b*e*f + (15*b^2 - 2*b)*f^2 + 6*((15*b - 2)*f^ 2 + 18*e^2 + 6*e*f)*x^2 - 2*((15*b - 2)*f^2 + 18*e^2 + 6*e*f)*x + 2*(6*b*e *f + b*f^2 + 6*(6*e*f + f^2)*x^2 - 2*(6*e*f + f^2)*x)*sqrt(-6*b + 1))*log( 6*x + sqrt(-6*b + 1) - 1) + (36*b*e^2 + 12*b*e*f - (24*b^2 - 5*b)*f^2 - 6* ((24*b - 5)*f^2 - 36*e^2 - 12*e*f)*x^2 + 2*((24*b - 5)*f^2 - 36*e^2 - 12*e *f)*x + 4*(6*b*e*f + b*f^2 + 6*(6*e*f + f^2)*x^2 - 2*(6*e*f + f^2)*x)*sqrt (-6*b + 1))*log(6*x - 2*sqrt(-6*b + 1) - 1) - 3*(12*b*e*f + b*f^2 + 6*e^2 + 2*((3*b - 1)*f^2 - 18*e^2 - 6*e*f)*x)*sqrt(-6*b + 1))/(6*(6*b - 1)*x^2 + 6*b^2 - 2*(6*b - 1)*x - b)
Leaf count of result is larger than twice the leaf count of optimal. 95302 vs. \(2 (187) = 374\).
Time = 7.55 (sec) , antiderivative size = 95302, normalized size of antiderivative = 437.17 \[ \int \frac {(e+f x)^2}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)**2/(1-(1-6*b)**(3/2)-9*b+54*b*x-54*x**2+108*x**3),x)
Output:
(f**2/216 - sqrt((36*b**2 - 12*b + 1)*(17280*b**3*f**4 + 77760*b**2*e**2*f **2 + 25920*b**2*e*f**3 - 6480*b**2*f**4 - 31104*b*e**4 - 20736*b*e**3*f - 31104*b*e**2*f**2 - 9216*b*e*f**3 + 696*b*f**4 + 5184*e**4 + 3456*e**3*f + 3024*e**2*f**2 + 816*e*f**3 + 81*f**4*(1 - 6*b)*(36*b**2 - 12*b + 1) - 1 6*f**4 + 16*f*sqrt(1 - 6*b)*(-216*b**2*e*f**2 - 36*b**2*f**3 - 2592*b*e**3 - 1296*b*e**2*f - 144*b*e*f**2 + 432*e**3 + 216*e**2*f + 30*e*f**2 + f**3 )))/(1944*sqrt(1 - 6*b)*(36*b**2 - 12*b + 1)))*log(-108864*b**5*f**8/(1492 992*b**4*e*f**7 - 7776*b**4*f**8*sqrt(1 - 6*b) + 248832*b**4*f**8 - 519561 216*b**3*e**3*f**5 + 17542656*b**3*e**2*f**6*sqrt(1 - 6*b) - 259780608*b** 3*e**2*f**6 + 5847552*b**3*e*f**7*sqrt(1 - 6*b) - 44292096*b**3*e*f**7 + 4 92480*b**3*f**8*sqrt(1 - 6*b) - 2571264*b**3*f**8 - 6234734592*b**2*e**5*f **3 - 723354624*b**2*e**4*f**4*sqrt(1 - 6*b) - 5195612160*b**2*e**4*f**4 - 482236416*b**2*e**3*f**5*sqrt(1 - 6*b) - 1472090112*b**2*e**3*f**5 - 1293 30432*b**2*e**2*f**6*sqrt(1 - 6*b) - 158754816*b**2*e**2*f**6 - 16319232*b **2*e*f**7*sqrt(1 - 6*b) - 2156544*b**2*e*f**7 - 803088*b**2*f**8*sqrt(1 - 6*b) + 442368*b**2*f**8 + 2579890176*b*e**7*f + 2526142464*b*e**6*f**2*sq rt(1 - 6*b) + 3009871872*b*e**6*f**2 + 2526142464*b*e**5*f**3*sqrt(1 - 6*b ) + 3583180800*b*e**5*f**3 + 1293677568*b*e**4*f**4*sqrt(1 - 6*b) + 214990 8480*b*e**4*f**4 + 394647552*b*e**3*f**5*sqrt(1 - 6*b) + 603666432*b*e**3* f**5 + 70886016*b*e**2*f**6*sqrt(1 - 6*b) + 81533952*b*e**2*f**6 + 6901...
\[ \int \frac {(e+f x)^2}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\int { \frac {{\left (f x + e\right )}^{2}}{108 \, x^{3} + 54 \, b x - 54 \, x^{2} - {\left (-6 \, b + 1\right )}^{\frac {3}{2}} - 9 \, b + 1} \,d x } \] Input:
integrate((f*x+e)^2/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x, algorit hm="maxima")
Output:
integrate((f*x + e)^2/(108*x^3 + 54*b*x - 54*x^2 - (-6*b + 1)^(3/2) - 9*b + 1), x)
Exception generated. \[ \int \frac {(e+f x)^2}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((f*x+e)^2/(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 = 13.07 (sec) , antiderivative size = 1147, normalized size of antiderivative = 5.26 \[ \int \frac {(e+f x)^2}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx =\text {Too large to display} \] Input:
int(-(e + f*x)^2/(9*b - 54*b*x + (1 - 6*b)^(3/2) + 54*x^2 - 108*x^3 - 1),x )
Output:
((e*f)/162 + f^2/972 - ((1 - 6*b)^(3/2)*((e*f)/5832 - (b*f^2)/11664 + e^2/ 1944 + f^2/34992))/(b^2 - b/3 + 1/36))/((1 - 6*b)^(3/2)/(36*(b - 1/6)) - x + 1/6) + (log((1 - 6*b)^(3/2)*(-(6*b - 1)^3)^(1/2) - 12*b*(-(6*b - 1)^3)^ (1/2) - 12*x*(-(6*b - 1)^3)^(1/2) - 54*b + 2*(-(6*b - 1)^3)^(1/2) + 324*b^ 2 - 648*b^3 + 72*b*x*(-(6*b - 1)^3)^(1/2) + 3)*(24*f^2 - 3*f^2*(6*b - 1)^3 - 576*b*f^2 + 8*f^2*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + 5184*b^2*f^2 - 20736*b^3*f^2 + 31104*b^4*f^2 + 18*b*f^2*(6*b - 1)^3 - 96 *b*f^2*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + 72*e^2*(1 - 6*b)^(3/2)*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + f^2*(1 - 6*b)^(3/2)*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + 288*b^2 *f^2*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + 48*e*f*(864*b^ 2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) - 576*b*e*f*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + 24*e*f*(1 - 6*b)^(3/2)*(864*b^2 - ( 6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + 1728*b^2*e*f*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + 6*b*f^2*(1 - 6*b)^(3/2)*(864*b^2 - (6 *b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2)))/(648*(6*b*(6*b - 1)^3 - 192*b - (6*b - 1)^3 + 1728*b^2 - 6912*b^3 + 10368*b^4 + 8)) - (log(54*b - 12*b*(-( 6*b - 1)^3)^(1/2) - 12*x*(-(6*b - 1)^3)^(1/2) + (1 - 6*b)^(3/2)*(-(6*b - 1 )^3)^(1/2) + 2*(-(6*b - 1)^3)^(1/2) - 324*b^2 + 648*b^3 + 72*b*x*(-(6*b - 1)^3)^(1/2) - 3)*(576*b*f^2 + 3*f^2*(6*b - 1)^3 - 24*f^2 + 8*f^2*(864*b...
Time = 0.17 (sec) , antiderivative size = 3275, normalized size of antiderivative = 15.02 \[ \int \frac {(e+f x)^2}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx =\text {Too large to display} \] Input:
int((f*x+e)^2/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x)
Output:
( - 60*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*f **2 - 72*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*e* *2 - 24*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*e*f - 360*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*f**2 *x**2 + 120*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b *f**2*x + 8*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b *f**2 - 432*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*e **2*x**2 + 144*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1) )*e**2*x - 144*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1) )*e*f*x**2 + 48*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1 ))*e*f*x + 48*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1)) *f**2*x**2 - 16*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1 ))*f**2*x + 288*sqrt(6*b - 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*e*f + 48* sqrt(6*b - 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*f**2 + 1728*sqrt(6*b - 1) *atan((6*x - 1)/sqrt(6*b - 1))*b*e*f*x**2 - 576*sqrt(6*b - 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*e*f*x - 48*sqrt(6*b - 1)*atan((6*x - 1)/sqrt(6*b - 1)) *b*e*f + 288*sqrt(6*b - 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*f**2*x**2 - 96* sqrt(6*b - 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*f**2*x - 8*sqrt(6*b - 1)*ata n((6*x - 1)/sqrt(6*b - 1))*b*f**2 - 288*sqrt(6*b - 1)*atan((6*x - 1)/sqrt( 6*b - 1))*e*f*x**2 + 96*sqrt(6*b - 1)*atan((6*x - 1)/sqrt(6*b - 1))*e*f...