Integrand size = 43, antiderivative size = 194 \[ \int \frac {A+B x+C x^2}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=-\frac {18 A+3 \left (1-\sqrt {1-6 b}\right ) B+\left (1-\sqrt {1-6 b}-3 b\right ) C}{162 \sqrt {1-6 b} \left (1-\sqrt {1-6 b}-6 x\right )}-\frac {\left (18 A+\left (3+6 \sqrt {1-6 b}\right ) B-\left (2-2 \sqrt {1-6 b}-15 b\right ) C\right ) \log \left (1-\sqrt {1-6 b}-6 x\right )}{486 (1-6 b)}+\frac {\left (36 A+6 B+5 C-24 b C+4 \sqrt {1-6 b} (3 B+C)\right ) \log \left (1+2 \sqrt {1-6 b}-6 x\right )}{972 (1-6 b)} \] Output:
-1/162*(18*A+3*(1-(1-6*b)^(1/2))*B+(1-(1-6*b)^(1/2)-3*b)*C)/(1-6*b)^(1/2)/ (1-(1-6*b)^(1/2)-6*x)-(18*A+(3+6*(1-6*b)^(1/2))*B-(2-2*(1-6*b)^(1/2)-15*b) *C)*ln(1-(1-6*b)^(1/2)-6*x)/(486-2916*b)+(36*A+6*B+5*C-24*C*b+4*(1-6*b)^(1 /2)*(3*B+C))*ln(1+2*(1-6*b)^(1/2)-6*x)/(972-5832*b)
Leaf count is larger than twice the leaf count of optimal. \(423\) vs. \(2(194)=388\).
Time = 1.07 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.18 \[ \int \frac {A+B x+C x^2}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\frac {\frac {6 \left (6 b^2 C+6 b B \left (\sqrt {1-6 b}-6 x\right )-2 \left (-1+\sqrt {1-6 b}\right ) (3 B+C) x+b C \left (-1+\sqrt {1-6 b}+6 \left (-2+\sqrt {1-6 b}\right ) x\right )-6 A \left (-1-\sqrt {1-6 b}+6 b+6 \sqrt {1-6 b} x\right )\right )}{(-1+6 b) (b+2 x (-1+3 x))}+\frac {2 \left (-36 A \sqrt {1-6 b}-6 \left (2+\sqrt {1-6 b}-12 b\right ) B+\left (-4-5 \sqrt {1-6 b}+24 \left (1+\sqrt {1-6 b}\right ) b\right ) C\right ) \arctan \left (\frac {-1+6 x}{2 \sqrt {-1+6 b}}\right )}{(-1+6 b)^{3/2}}-\frac {4 \left (18 A \sqrt {1-6 b}+3 \left (2+\sqrt {1-6 b}-12 b\right ) B+\left (2-2 \sqrt {1-6 b}+3 \left (-4+5 \sqrt {1-6 b}\right ) b\right ) C\right ) \arctan \left (\frac {-1+6 x}{\sqrt {-1+6 b}}\right )}{(-1+6 b)^{3/2}}-\frac {\left (36 A+6 \left (1+2 \sqrt {1-6 b}\right ) B+\left (5+4 \sqrt {1-6 b}-24 b\right ) C\right ) \log \left (1-8 b+4 x-12 x^2\right )}{-1+6 b}+\frac {2 \left (18 A+\left (3+6 \sqrt {1-6 b}\right ) B+\left (-2+2 \sqrt {1-6 b}+15 b\right ) C\right ) \log \left (b-2 x+6 x^2\right )}{-1+6 b}}{1944} \] Input:
Integrate[(A + B*x + C*x^2)/(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3),x]
Output:
((6*(6*b^2*C + 6*b*B*(Sqrt[1 - 6*b] - 6*x) - 2*(-1 + Sqrt[1 - 6*b])*(3*B + C)*x + b*C*(-1 + Sqrt[1 - 6*b] + 6*(-2 + Sqrt[1 - 6*b])*x) - 6*A*(-1 - Sq rt[1 - 6*b] + 6*b + 6*Sqrt[1 - 6*b]*x)))/((-1 + 6*b)*(b + 2*x*(-1 + 3*x))) + (2*(-36*A*Sqrt[1 - 6*b] - 6*(2 + Sqrt[1 - 6*b] - 12*b)*B + (-4 - 5*Sqrt [1 - 6*b] + 24*(1 + Sqrt[1 - 6*b])*b)*C)*ArcTan[(-1 + 6*x)/(2*Sqrt[-1 + 6* b])])/(-1 + 6*b)^(3/2) - (4*(18*A*Sqrt[1 - 6*b] + 3*(2 + Sqrt[1 - 6*b] - 1 2*b)*B + (2 - 2*Sqrt[1 - 6*b] + 3*(-4 + 5*Sqrt[1 - 6*b])*b)*C)*ArcTan[(-1 + 6*x)/Sqrt[-1 + 6*b]])/(-1 + 6*b)^(3/2) - ((36*A + 6*(1 + 2*Sqrt[1 - 6*b] )*B + (5 + 4*Sqrt[1 - 6*b] - 24*b)*C)*Log[1 - 8*b + 4*x - 12*x^2])/(-1 + 6 *b) + (2*(18*A + (3 + 6*Sqrt[1 - 6*b])*B + (-2 + 2*Sqrt[1 - 6*b] + 15*b)*C )*Log[b - 2*x + 6*x^2])/(-1 + 6*b))/1944
Time = 1.23 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {2525, 27, 2488, 27, 27, 86, 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.
| \(\displaystyle \int \frac {A+B x+C x^2}{54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1} \, dx\) |
| \(\Big \downarrow \) 2525 |
| \(\displaystyle \frac {1}{324} \int \frac {54 (6 A-b C+2 (3 B+C) x)}{108 x^3-54 x^2+54 b x-(1-6 b)^{3/2}-9 b+1}dx+\frac {1}{324} C \log \left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \frac {6 A-b C+2 (3 B+C) x}{108 x^3-54 x^2+54 b x-(1-6 b)^{3/2}-9 b+1}dx+\frac {1}{324} C \log \left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )\) |
| \(\Big \downarrow \) 2488 |
| \(\displaystyle 16529940864 (1-6 b)^3 \int -\frac {6 A-b C+2 (3 B+C) x}{49589822592 \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \left (\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)-6 (1-6 b) x\right )}dx+\frac {1}{324} C \log \left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{324} C \log \left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )-\frac {1}{3} (1-6 b)^3 \int \frac {6 A-b C+2 (3 B+C) x}{(1-6 b) \left (-6 x+2 \sqrt {1-6 b}+1\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{324} C \log \left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )-\frac {1}{3} (1-6 b)^2 \int \frac {6 A-b C+2 (3 B+C) x}{\left (-6 x+2 \sqrt {1-6 b}+1\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {1}{324} C \log \left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )-\frac {1}{3} (1-6 b)^2 \int \left (\frac {18 A+3 \left (1-\sqrt {1-6 b}\right ) B+\left (-3 b-\sqrt {1-6 b}+1\right ) C}{9 (1-6 b)^{5/2} \left (-6 x-\sqrt {1-6 b}+1\right )^2}+\frac {-18 A-3 \left (2 \sqrt {1-6 b}+1\right ) B-\left (-3 b+2 \sqrt {1-6 b}+1\right ) C}{27 (1-6 b)^3 \left (-6 x-\sqrt {1-6 b}+1\right )}+\frac {18 A+3 \left (2 \sqrt {1-6 b}+1\right ) B+\left (-3 b+2 \sqrt {1-6 b}+1\right ) C}{27 (1-6 b)^3 \left (-6 x+2 \sqrt {1-6 b}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{324} C \log \left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )-\frac {1}{3} (1-6 b)^2 \left (\frac {18 A+3 \left (1-\sqrt {1-6 b}\right ) B+\left (-3 b-\sqrt {1-6 b}+1\right ) C}{54 (1-6 b)^{5/2} \left (-\sqrt {1-6 b}-6 x+1\right )}+\frac {\log \left (-\sqrt {1-6 b}-6 x+1\right ) \left (18 A+2 \sqrt {1-6 b} (3 B+C)-3 b C+3 B+C\right )}{162 (1-6 b)^3}-\frac {\log \left (2 \sqrt {1-6 b}-6 x+1\right ) \left (18 A+2 \sqrt {1-6 b} (3 B+C)-3 b C+3 B+C\right )}{162 (1-6 b)^3}\right )\) |
Input:
Int[(A + B*x + C*x^2)/(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x ^3),x]
Output:
-1/3*((1 - 6*b)^2*((18*A + 3*(1 - Sqrt[1 - 6*b])*B + (1 - Sqrt[1 - 6*b] - 3*b)*C)/(54*(1 - 6*b)^(5/2)*(1 - Sqrt[1 - 6*b] - 6*x)) + ((18*A + 3*B + C - 3*b*C + 2*Sqrt[1 - 6*b]*(3*B + C))*Log[1 - Sqrt[1 - 6*b] - 6*x])/(162*(1 - 6*b)^3) - ((18*A + 3*B + C - 3*b*C + 2*Sqrt[1 - 6*b]*(3*B + C))*Log[1 + 2*Sqrt[1 - 6*b] - 6*x])/(162*(1 - 6*b)^3))) + (C*Log[1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3])/324
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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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]
Int[(Pm_)/(Qn_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Si mp[Coeff[Pm, x, m]*(Log[Qn]/(n*Coeff[Qn, x, n])), x] + Simp[1/(n*Coeff[Qn, x, n]) Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x], x ]/Qn, x], x] /; EqQ[m, n - 1]] /; PolyQ[Pm, x] && PolyQ[Qn, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.56 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.36
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-\left (1-6 b \right )^{\frac {3}{2}}+108 \textit {\_Z}^{3}+54 b \textit {\_Z} -54 \textit {\_Z}^{2}-9 b +1\right )}{\sum }\frac {\left (-C \,\textit {\_R}^{2}-B \textit {\_R} -A \right ) \ln \left (x -\textit {\_R} \right )}{-6 \textit {\_R}^{2}+2 \textit {\_R} -b}\right )}{54}\) | \(70\) |
| parallelrisch | \(\frac {6 C +18 B -36 b C -108 B b -6 B \sqrt {1-6 b}\, \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right )+6 B \sqrt {1-6 b}\, \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right )+36 B \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) x -36 B \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) x -8 C \sqrt {1-6 b}\, \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right )-C \sqrt {1-6 b}\, \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right )-24 C \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) x -54 C \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) b +72 B \sqrt {1-6 b}\, \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) x -72 B \sqrt {1-6 b}\, \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) x +24 C \sqrt {1-6 b}\, \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) x +30 C \sqrt {1-6 b}\, \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) b -24 C \sqrt {1-6 b}\, \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) x +24 C \sqrt {1-6 b}\, \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) b +180 C \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) x b +144 C \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) x b -18 B \sqrt {1-6 b}-6 C \sqrt {1-6 b}-108 A \sqrt {1-6 b}-30 C \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) x -36 A \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right )+36 A \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right )+6 B \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right )-6 B \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right )+8 C \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right )+C \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right )+18 C \sqrt {1-6 b}\, b -72 B \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) b +72 B \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) b +36 A \sqrt {1-6 b}\, \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right )-36 A \sqrt {1-6 b}\, \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right )+216 A \ln \left (-\frac {1}{6}+x +\frac {\sqrt {1-6 b}}{6}\right ) x -216 A \ln \left (x -\frac {1}{6}-\frac {\sqrt {1-6 b}}{3}\right ) x}{972 \left (-1+6 b \right ) \left (-1+6 x +\sqrt {1-6 b}\right )}\) | \(646\) |
Input:
int((C*x^2+B*x+A)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x,method=_RE TURNVERBOSE)
Output:
1/54*sum((-C*_R^2-B*_R-A)/(-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))
Time = 0.13 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.63 \[ \int \frac {A+B x+C x^2}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\frac {18 \, C b^{2} - 3 \, {\left (36 \, A + C\right )} b - 6 \, {\left (6 \, {\left (3 \, B + C\right )} b - 3 \, B - C\right )} x + 2 \, {\left (15 \, C b^{2} + 6 \, {\left (15 \, C b + 18 \, A + 3 \, B - 2 \, C\right )} x^{2} + {\left (18 \, A + 3 \, B - 2 \, C\right )} b - 2 \, {\left (15 \, C b + 18 \, A + 3 \, B - 2 \, C\right )} x + 2 \, {\left (6 \, {\left (3 \, B + C\right )} x^{2} + {\left (3 \, B + C\right )} b - 2 \, {\left (3 \, B + C\right )} x\right )} \sqrt {-6 \, b + 1}\right )} \log \left (6 \, x + \sqrt {-6 \, b + 1} - 1\right ) + {\left (24 \, C b^{2} + 6 \, {\left (24 \, C b - 36 \, A - 6 \, B - 5 \, C\right )} x^{2} - {\left (36 \, A + 6 \, B + 5 \, C\right )} b - 2 \, {\left (24 \, C b - 36 \, A - 6 \, B - 5 \, C\right )} x - 4 \, {\left (6 \, {\left (3 \, B + C\right )} x^{2} + {\left (3 \, B + C\right )} b - 2 \, {\left (3 \, B + C\right )} x\right )} \sqrt {-6 \, b + 1}\right )} \log \left (6 \, x - 2 \, \sqrt {-6 \, b + 1} - 1\right ) + 3 \, {\left ({\left (6 \, B + C\right )} b + 2 \, {\left (3 \, C b - 18 \, A - 3 \, B - C\right )} x + 6 \, A\right )} \sqrt {-6 \, b + 1} + 18 \, A}{972 \, {\left (6 \, {\left (6 \, b - 1\right )} x^{2} + 6 \, b^{2} - 2 \, {\left (6 \, b - 1\right )} x - b\right )}} \] Input:
integrate((C*x^2+B*x+A)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x, alg orithm="fricas")
Output:
1/972*(18*C*b^2 - 3*(36*A + C)*b - 6*(6*(3*B + C)*b - 3*B - C)*x + 2*(15*C *b^2 + 6*(15*C*b + 18*A + 3*B - 2*C)*x^2 + (18*A + 3*B - 2*C)*b - 2*(15*C* b + 18*A + 3*B - 2*C)*x + 2*(6*(3*B + C)*x^2 + (3*B + C)*b - 2*(3*B + C)*x )*sqrt(-6*b + 1))*log(6*x + sqrt(-6*b + 1) - 1) + (24*C*b^2 + 6*(24*C*b - 36*A - 6*B - 5*C)*x^2 - (36*A + 6*B + 5*C)*b - 2*(24*C*b - 36*A - 6*B - 5* C)*x - 4*(6*(3*B + C)*x^2 + (3*B + C)*b - 2*(3*B + C)*x)*sqrt(-6*b + 1))*l og(6*x - 2*sqrt(-6*b + 1) - 1) + 3*((6*B + C)*b + 2*(3*C*b - 18*A - 3*B - C)*x + 6*A)*sqrt(-6*b + 1) + 18*A)/(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. 247850 vs. \(2 (165) = 330\).
Time = 10.77 (sec) , antiderivative size = 247850, normalized size of antiderivative = 1277.58 \[ \int \frac {A+B x+C x^2}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\text {Too large to display} \] Input:
integrate((C*x**2+B*x+A)/(1-(1-6*b)**(3/2)-9*b+54*b*x-54*x**2+108*x**3),x)
Output:
(C/216 - sqrt((36*b**2 - 12*b + 1)*(-31104*A**2*b + 5184*A**2 - 10368*A*B* b + 1728*A*B - 5184*A*C*b**2 + 144*A*C + 20736*B**2*b**2 - 7776*B**2*b + 7 20*B**2 + 12960*B*C*b**2 - 4608*B*C*b + 408*B*C + 17280*C**2*b**3 - 6480*C **2*b**2 + 696*C**2*b + 81*C**2*(1 - 6*b)*(36*b**2 - 12*b + 1) - 16*C**2 + 16*sqrt(1 - 6*b)*(-1296*A*B*b + 216*A*B - 432*A*C*b + 72*A*C - 216*B**2*b + 36*B**2 - 108*B*C*b**2 - 72*B*C*b + 15*B*C - 36*C**2*b**2 + C**2)))/(19 44*sqrt(1 - 6*b)*(36*b**2 - 12*b + 1)))*log(-80621568*A**4*b/(-161243136*A **4*sqrt(1 - 6*b) + 1289945088*A**3*B*b - 107495424*A**3*B*sqrt(1 - 6*b) - 214990848*A**3*B - 53747712*A**3*C*b*sqrt(1 - 6*b) + 429981696*A**3*C*b - 8957952*A**3*C*sqrt(1 - 6*b) - 71663616*A**3*C + 644972544*A**2*B**2*b*sq rt(1 - 6*b) + 644972544*A**2*B**2*b - 134369280*A**2*B**2*sqrt(1 - 6*b) - 107495424*A**2*B**2 + 322486272*A**2*B*C*b**2 + 403107840*A**2*B*C*b*sqrt( 1 - 6*b) + 214990848*A**2*B*C*b - 76142592*A**2*B*C*sqrt(1 - 6*b) - 447897 60*A**2*B*C - 6718464*A**2*C**2*b**2*sqrt(1 - 6*b) + 107495424*A**2*C**2*b **2 + 69424128*A**2*C**2*b*sqrt(1 - 6*b) - 12130560*A**2*C**2*sqrt(1 - 6*b ) - 2985984*A**2*C**2 - 859963392*A*B**3*b**2 + 214990848*A*B**3*b*sqrt(1 - 6*b) + 394149888*A*B**3*b - 38817792*A*B**3*sqrt(1 - 6*b) - 41803776*A*B **3 + 107495424*A*B**2*C*b**2*sqrt(1 - 6*b) - 752467968*A*B**2*C*b**2 + 13 8848256*A*B**2*C*b*sqrt(1 - 6*b) + 322486272*A*B**2*C*b - 27620352*A*B**2* C*sqrt(1 - 6*b) - 32845824*A*B**2*C + 26873856*A*B*C**2*b**3 + 69424128...
\[ \int \frac {A+B x+C x^2}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\int { \frac {C x^{2} + B x + A}{108 \, x^{3} + 54 \, b x - 54 \, x^{2} - {\left (-6 \, b + 1\right )}^{\frac {3}{2}} - 9 \, b + 1} \,d x } \] Input:
integrate((C*x^2+B*x+A)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x, alg orithm="maxima")
Output:
integrate((C*x^2 + B*x + A)/(108*x^3 + 54*b*x - 54*x^2 - (-6*b + 1)^(3/2) - 9*b + 1), x)
Exception generated. \[ \int \frac {A+B x+C x^2}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((C*x^2+B*x+A)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x, alg orithm="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 = 13.20 (sec) , antiderivative size = 1077, normalized size of antiderivative = 5.55 \[ \int \frac {A+B x+C x^2}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx =\text {Too large to display} \] Input:
int(-(A + B*x + C*x^2)/(9*b - 54*b*x + (1 - 6*b)^(3/2) + 54*x^2 - 108*x^3 - 1),x)
Output:
(B/324 + C/972 - ((1 - 6*b)^(3/2)*(A/1944 + B/11664 + C/34992 - (C*b)/1166 4))/(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 + 7 2*b*x*(-(6*b - 1)^3)^(1/2) + 3)*(24*C - 576*C*b - 3*C*(6*b - 1)^3 + 24*B*( 864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + 8*C*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + 5184*C*b^2 - 20736*C*b^3 + 31104*C* b^4 + 864*B*b^2*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + 288 *C*b^2*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + 18*C*b*(6*b - 1)^3 - 288*B*b*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) - 96 *C*b*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + 72*A*(1 - 6*b) ^(3/2)*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + 12*B*(1 - 6* b)^(3/2)*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + C*(1 - 6*b )^(3/2)*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + 6*C*b*(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 + 64 8*b^3 + 72*b*x*(-(6*b - 1)^3)^(1/2) - 3)*(576*C*b - 24*C + 3*C*(6*b - 1)^3 + 24*B*(864*b^2 - (6*b - 1)^3 - 144*b - 1728*b^3 + 8)^(1/2) + 8*C*(864...
Time = 0.16 (sec) , antiderivative size = 3061, normalized size of antiderivative = 15.78 \[ \int \frac {A+B x+C x^2}{1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3} \, dx =\text {Too large to display} \] Input:
int((C*x^2+B*x+A)/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3),x)
Output:
( - 72*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*a*b - 432*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*a*x**2 + 144*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*a*x - 60* sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*c - 12*s qrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2 - 360*sqr t(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*c*x**2 + 120*s qrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*c*x + 8*sqrt (6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*c - 72*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*x**2 + 24*sqrt(6*b - 1)*sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*x + 48*sqrt(6*b - 1) *sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*c*x**2 - 16*sqrt(6*b - 1)* sqrt( - 6*b + 1)*atan((6*x - 1)/sqrt(6*b - 1))*c*x + 144*sqrt(6*b - 1)*ata n((6*x - 1)/sqrt(6*b - 1))*b**3 + 48*sqrt(6*b - 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*c + 864*sqrt(6*b - 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*x**2 - 288*sqrt(6*b - 1)*atan((6*x - 1)/sqrt(6*b - 1))*b**2*x - 24*sqrt(6*b - 1 )*atan((6*x - 1)/sqrt(6*b - 1))*b**2 + 288*sqrt(6*b - 1)*atan((6*x - 1)/sq rt(6*b - 1))*b*c*x**2 - 96*sqrt(6*b - 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*c *x - 8*sqrt(6*b - 1)*atan((6*x - 1)/sqrt(6*b - 1))*b*c - 144*sqrt(6*b - 1) *atan((6*x - 1)/sqrt(6*b - 1))*b*x**2 + 48*sqrt(6*b - 1)*atan((6*x - 1)/sq rt(6*b - 1))*b*x - 48*sqrt(6*b - 1)*atan((6*x - 1)/sqrt(6*b - 1))*c*x**...