Integrand size = 28, antiderivative size = 229 \[ \int \frac {A+B x+C x^2}{x \left (a+b x^2+c x^4\right )} \, dx=\frac {\sqrt {2} B \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} B \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {(A b-2 a C) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a \sqrt {b^2-4 a c}}+\frac {A \log (x)}{a}-\frac {A \log \left (a+b x^2+c x^4\right )}{4 a} \]
A*ln(x)/a-1/4*A*ln(c*x^4+b*x^2+a)/a+1/2*(A*b-2*C*a)*arctanh((2*c*x^2+b)/(- 4*a*c+b^2)^(1/2))/a/(-4*a*c+b^2)^(1/2)+B*arctan(x*2^(1/2)*c^(1/2)/(b-(-4*a *c+b^2)^(1/2))^(1/2))*2^(1/2)*c^(1/2)/(-4*a*c+b^2)^(1/2)/(b-(-4*a*c+b^2)^( 1/2))^(1/2)-B*arctan(x*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/ 2)*c^(1/2)/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 0.29 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.24 \[ \int \frac {A+B x+C x^2}{x \left (a+b x^2+c x^4\right )} \, dx=\frac {\sqrt {2} B \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} B \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {A \log (x)}{a}-\frac {\left (A \left (b+\sqrt {b^2-4 a c}\right )-2 a C\right ) \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{4 a \sqrt {b^2-4 a c}}-\frac {\left (A \left (-b+\sqrt {b^2-4 a c}\right )+2 a C\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{4 a \sqrt {b^2-4 a c}} \]
(Sqrt[2]*B*Sqrt[c]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]] )/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*B*Sqrt[c]*Arc Tan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*S qrt[b + Sqrt[b^2 - 4*a*c]]) + (A*Log[x])/a - ((A*(b + Sqrt[b^2 - 4*a*c]) - 2*a*C)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(4*a*Sqrt[b^2 - 4*a*c]) - ( (A*(-b + Sqrt[b^2 - 4*a*c]) + 2*a*C)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2]) /(4*a*Sqrt[b^2 - 4*a*c])
Time = 0.52 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2193, 27, 1406, 218, 1578, 1200, 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.
| \(\displaystyle \int \frac {A+B x+C x^2}{x \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2193 |
| \(\displaystyle \int \frac {C x^2+A}{x \left (c x^4+b x^2+a\right )}dx+\int \frac {B}{c x^4+b x^2+a}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {C x^2+A}{x \left (c x^4+b x^2+a\right )}dx+B \int \frac {1}{c x^4+b x^2+a}dx\) |
| \(\Big \downarrow \) 1406 |
| \(\displaystyle \int \frac {C x^2+A}{x \left (c x^4+b x^2+a\right )}dx+B \left (\frac {c \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx}{\sqrt {b^2-4 a c}}-\frac {c \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{\sqrt {b^2-4 a c}}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \int \frac {C x^2+A}{x \left (c x^4+b x^2+a\right )}dx+B \left (\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}\right )\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {C x^2+A}{x^2 \left (c x^4+b x^2+a\right )}dx^2+B \left (\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}\right )\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {A}{a x^2}+\frac {-A c x^2-A b+a C}{a \left (c x^4+b x^2+a\right )}\right )dx^2+B \left (\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {(A b-2 a C) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c}}-\frac {A \log \left (a+b x^2+c x^4\right )}{2 a}+\frac {A \log \left (x^2\right )}{a}\right )+B \left (\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}\right )\) |
B*((Sqrt[2]*Sqrt[c]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]] ])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*Sqrt[c]*ArcT an[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sq rt[b + Sqrt[b^2 - 4*a*c]])) + (((A*b - 2*a*C)*ArcTanh[(b + 2*c*x^2)/Sqrt[b ^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]) + (A*Log[x^2])/a - (A*Log[a + b*x^2 + c*x^4])/(2*a))/2
3.1.26.3.1 Defintions of rubi rules used
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_)^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]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c , 0] && PosQ[b^2 - 4*a*c]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_S ymbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[Pq, x, 2*k]*x^(2*k), {k, 0, q/2 + 1}]*(d*x)^m*(a + b*x^2 + c*x^4)^p, x] + Simp[1/d Int[Sum[Coe ff[Pq, x, 2*k + 1]*x^(2*k), {k, 0, (q + 1)/2}]*(d*x)^(m + 1)*(a + b*x^2 + c *x^4)^p, x], x]] /; FreeQ[{a, b, c, d, m, p}, x] && PolyQ[Pq, x] && !PolyQ [Pq, x^2]
Time = 0.10 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.10
| method | result | size |
| default | \(\frac {A \ln \left (x \right )}{a}+\frac {4 c \left (\frac {\sqrt {-4 a c +b^{2}}\, \left (\frac {\left (A \sqrt {-4 a c +b^{2}}-A b +2 C a \right ) \ln \left (2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b \right )}{4 c}+\frac {B a \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a c -4 b^{2}}+\frac {\sqrt {-4 a c +b^{2}}\, \left (-\frac {\left (-A \sqrt {-4 a c +b^{2}}-A b +2 C a \right ) \ln \left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right )}{4 c}+\frac {B a \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a c -4 b^{2}}\right )}{a}\) | \(251\) |
| risch | \(\frac {A \ln \left (x \right )}{a}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16 a^{4} c^{2}-8 a^{3} b^{2} c +a^{2} b^{4}\right ) \textit {\_Z}^{4}+\left (32 A \,a^{3} c^{2}-16 A \,a^{2} b^{2} c +2 A a \,b^{4}\right ) \textit {\_Z}^{3}+\left (24 a^{2} c^{2} A^{2}-10 a \,b^{2} c \,A^{2}+b^{4} A^{2}-8 A C \,a^{2} b c +2 A C a \,b^{3}-4 B^{2} a^{2} b c +B^{2} a \,b^{3}+8 C^{2} a^{3} c -2 C^{2} a^{2} b^{2}\right ) \textit {\_Z}^{2}+\left (8 A^{3} a \,c^{2}-2 A^{3} b^{2} c -8 A^{2} C a b c +2 A^{2} C \,b^{3}+8 A \,C^{2} a^{2} c -2 A \,C^{2} a \,b^{2}-8 B^{2} C \,a^{2} c +2 B^{2} C a \,b^{2}\right ) \textit {\_Z} +A^{4} c^{2}-2 A^{3} C b c +A^{2} B^{2} b c +2 A^{2} C^{2} a c +A^{2} C^{2} b^{2}-4 A \,B^{2} C a c -2 A \,C^{3} a b +B^{4} a c +B^{2} C^{2} a b +C^{4} a^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (40 a^{3} c^{2}-22 a^{2} b^{2} c +3 b^{4} a \right ) \textit {\_R}^{4}+\left (60 A \,a^{2} c^{2}-27 A a \,b^{2} c +3 A \,b^{4}-4 C \,a^{2} b c +C a \,b^{3}\right ) \textit {\_R}^{3}+\left (30 A^{2} a \,c^{2}-8 A^{2} b^{2} c -14 A C a b c +4 A C \,b^{3}-7 B^{2} a b c +2 B^{2} b^{3}+18 C^{2} a^{2} c -5 C^{2} a \,b^{2}\right ) \textit {\_R}^{2}+\left (5 A^{3} c^{2}-6 A^{2} C b c -A \,B^{2} b c +13 A \,C^{2} a c -A \,C^{2} b^{2}-17 B^{2} C a c +4 B^{2} C \,b^{2}-C^{3} a b \right ) \textit {\_R} +2 A^{2} C^{2} c -6 A \,B^{2} C c -2 A \,C^{3} b +2 B^{4} c +2 B^{2} C^{2} b +2 C^{4} a \right ) x +\left (4 a^{2} b B c -B a \,b^{3}\right ) \textit {\_R}^{3}+\left (-6 A B a b c +2 A B \,b^{3}+4 B C \,a^{2} c -2 B C a \,b^{2}\right ) \textit {\_R}^{2}+\left (-4 A^{2} B b c -6 A B C a c +4 A B C \,b^{2}-a c \,B^{3}-B \,C^{2} a b \right ) \textit {\_R} -4 A^{2} B C c +2 A \,B^{3} c +2 A B \,C^{2} b \right )\right )}{2}\) | \(706\) |
A*ln(x)/a+4/a*c*((-4*a*c+b^2)^(1/2)/(16*a*c-4*b^2)*(1/4*(A*(-4*a*c+b^2)^(1 /2)-A*b+2*C*a)/c*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)+B*a*2^(1/2)/((b+(-4*a*c+ b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))) +(-4*a*c+b^2)^(1/2)/(16*a*c-4*b^2)*(-1/4*(-A*(-4*a*c+b^2)^(1/2)-A*b+2*C*a) /c*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)+B*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))* c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))))
Timed out. \[ \int \frac {A+B x+C x^2}{x \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B x+C x^2}{x \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x+C x^2}{x \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (c x^{4} + b x^{2} + a\right )} x} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 2339 vs. \(2 (186) = 372\).
Time = 1.43 (sec) , antiderivative size = 2339, normalized size of antiderivative = 10.21 \[ \int \frac {A+B x+C x^2}{x \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
-1/4*A*log(abs(c*x^4 + b*x^2 + a))/a + A*log(abs(x))/a + 1/4*((sqrt(2)*sqr t(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)* c)*a*b^2*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c - 2*b^4*c + 1 6*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^2 + 8*sqrt(2)*sqrt(b*c + s qrt(b^2 - 4*a*c)*c)*a*b*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2* c^2 + 16*a*b^2*c^2 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^3 - 32* a^2*c^3 + 2*(b^2 - 4*a*c)*b^2*c - 8*(b^2 - 4*a*c)*a*c^2)*B*abs(c) - (2*b^3 *c^3 - 8*a*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)* c)*b^3*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b *c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^3 - 2*(b^ 2 - 4*a*c)*b*c^3)*B)*arctan(2*sqrt(1/2)*x/sqrt((a^2*b*c + sqrt(a^4*b^2*c^2 - 4*a^5*c^3))/(a^2*c^2)))/((a*b^4 - 8*a^2*b^2*c - 2*a*b^3*c + 16*a^3*c^2 + 8*a^2*b*c^2 + a*b^2*c^2 - 4*a^2*c^3)*c^2) + 1/4*((sqrt(2)*sqrt(b*c - sqr t(b^2 - 4*a*c)*c)*b^4 - 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c + 2*b^4*c + 16*sqrt(2)*s qrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*c^2 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4 *a*c)*c)*a*b*c^2 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c^2 - 16*a* b^2*c^2 - 4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*c^3 + 32*a^2*c^3 - 2 *(b^2 - 4*a*c)*b^2*c + 8*(b^2 - 4*a*c)*a*c^2)*B*abs(c) - (2*b^3*c^3 - 8...
Time = 8.37 (sec) , antiderivative size = 2258, normalized size of antiderivative = 9.86 \[ \int \frac {A+B x+C x^2}{x \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
symsum(log(x*(B^4*c^3 + C^4*a*c^2 + A^2*C^2*c^3 - 3*A*B^2*C*c^3 - A*C^3*b* c^2 + B^2*C^2*b*c^2) - root(128*a^3*b^2*c*z^4 - 256*a^4*c^2*z^4 - 16*a^2*b ^4*z^4 + 128*A*a^2*b^2*c*z^3 - 256*A*a^3*c^2*z^3 - 16*A*a*b^4*z^3 + 32*A*C *a^2*b*c*z^2 - 8*A*C*a*b^3*z^2 + 16*B^2*a^2*b*c*z^2 + 40*A^2*a*b^2*c*z^2 - 32*C^2*a^3*c*z^2 - 4*B^2*a*b^3*z^2 + 8*C^2*a^2*b^2*z^2 - 96*A^2*a^2*c^2*z ^2 - 4*A^2*b^4*z^2 + 16*A^2*C*a*b*c*z + 16*B^2*C*a^2*c*z - 16*A*C^2*a^2*c* z - 4*B^2*C*a*b^2*z + 4*A*C^2*a*b^2*z + 4*A^3*b^2*c*z - 16*A^3*a*c^2*z - 4 *A^2*C*b^3*z + 4*A*B^2*C*a*c - 2*A^2*C^2*a*c + 2*A^3*C*b*c + 2*A*C^3*a*b - B^2*C^2*a*b - A^2*B^2*b*c - B^4*a*c - A^2*C^2*b^2 - C^4*a^2 - A^4*c^2, z, k)*(x*(A*B^2*b*c^3 - 5*A^3*c^4 - 13*A*C^2*a*c^3 + 6*A^2*C*b*c^3 + 17*B^2* C*a*c^3 + C^3*a*b*c^2 + A*C^2*b^2*c^2 - 4*B^2*C*b^2*c^2) - root(128*a^3*b^ 2*c*z^4 - 256*a^4*c^2*z^4 - 16*a^2*b^4*z^4 + 128*A*a^2*b^2*c*z^3 - 256*A*a ^3*c^2*z^3 - 16*A*a*b^4*z^3 + 32*A*C*a^2*b*c*z^2 - 8*A*C*a*b^3*z^2 + 16*B^ 2*a^2*b*c*z^2 + 40*A^2*a*b^2*c*z^2 - 32*C^2*a^3*c*z^2 - 4*B^2*a*b^3*z^2 + 8*C^2*a^2*b^2*z^2 - 96*A^2*a^2*c^2*z^2 - 4*A^2*b^4*z^2 + 16*A^2*C*a*b*c*z + 16*B^2*C*a^2*c*z - 16*A*C^2*a^2*c*z - 4*B^2*C*a*b^2*z + 4*A*C^2*a*b^2*z + 4*A^3*b^2*c*z - 16*A^3*a*c^2*z - 4*A^2*C*b^3*z + 4*A*B^2*C*a*c - 2*A^2*C ^2*a*c + 2*A^3*C*b*c + 2*A*C^3*a*b - B^2*C^2*a*b - A^2*B^2*b*c - B^4*a*c - A^2*C^2*b^2 - C^4*a^2 - A^4*c^2, z, k)*(x*(60*A^2*a*c^4 - 16*A^2*b^2*c^3 + 4*B^2*b^3*c^2 + 36*C^2*a^2*c^3 + 8*A*C*b^3*c^2 - 14*B^2*a*b*c^3 - 10*...