Integrand size = 20, antiderivative size = 189 \[ \int \frac {d+e x}{a+b x^2+c x^4} \, dx=\frac {\sqrt {2} \sqrt {c} d \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} d \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {e \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
-e*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2)+d*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)-d*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 = 0.17 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.03 \[ \int \frac {d+e x}{a+b x^2+c x^4} \, dx=\frac {\frac {2 \sqrt {2} \sqrt {c} d \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}}}-\frac {2 \sqrt {2} \sqrt {c} d \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}}}+e \left (\log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )-\log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )\right )}{2 \sqrt {b^2-4 a c}} \]
((2*Sqrt[2]*Sqrt[c]*d*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c ]]])/Sqrt[b - Sqrt[b^2 - 4*a*c]] - (2*Sqrt[2]*Sqrt[c]*d*ArcTan[(Sqrt[2]*Sq rt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/Sqrt[b + Sqrt[b^2 - 4*a*c]] + e*(Lo g[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2] - Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2] ))/(2*Sqrt[b^2 - 4*a*c])
Time = 0.39 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2202, 27, 1406, 218, 1432, 1083, 219}
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.
\(\displaystyle \int \frac {d+e x}{a+b x^2+c x^4} \, dx\) |
\(\Big \downarrow \) 2202 |
\(\displaystyle \int \frac {d}{c x^4+b x^2+a}dx+\int \frac {e x}{c x^4+b x^2+a}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle d \int \frac {1}{c x^4+b x^2+a}dx+e \int \frac {x}{c x^4+b x^2+a}dx\) |
\(\Big \downarrow \) 1406 |
\(\displaystyle d \left (\frac {c \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx}{\sqrt {b^2-4 a c}}-\frac {c \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{\sqrt {b^2-4 a c}}\right )+e \int \frac {x}{c x^4+b x^2+a}dx\) |
\(\Big \downarrow \) 218 |
\(\displaystyle e \int \frac {x}{c x^4+b x^2+a}dx+d \left (\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}\right )\) |
\(\Big \downarrow \) 1432 |
\(\displaystyle \frac {1}{2} e \int \frac {1}{c x^4+b x^2+a}dx^2+d \left (\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle d \left (\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}\right )-e \int \frac {1}{-x^4+b^2-4 a c}d\left (2 c x^2+b\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle d \left (\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}\right )-\frac {e \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}\) |
d*((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]])) - (e*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]]) /Sqrt[b^2 - 4*a*c]
3.1.20.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
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_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.23
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\textit {\_R} e +d \right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}\right )}{2}\) | \(43\) |
default | \(4 c \left (\frac {\sqrt {-4 a c +b^{2}}\, \left (\frac {e \ln \left (2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b \right )}{4 c}+\frac {d \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 a c -2 b^{2}}-\frac {\sqrt {-4 a c +b^{2}}\, \left (\frac {e \ln \left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right )}{4 c}-\frac {d \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 a c -2 b^{2}}\right )\) | \(200\) |
Result contains complex when optimal does not.
Time = 3.16 (sec) , antiderivative size = 398481, normalized size of antiderivative = 2108.37 \[ \int \frac {d+e x}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {d+e x}{a+b x^2+c x^4} \, dx=\text {Timed out} \]
\[ \int \frac {d+e x}{a+b x^2+c x^4} \, dx=\int { \frac {e x + d}{c x^{4} + b x^{2} + a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1342 vs. \(2 (149) = 298\).
Time = 1.40 (sec) , antiderivative size = 1342, normalized size of antiderivative = 7.10 \[ \int \frac {d+e x}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
1/4*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 - 8*sqrt(2)*sqrt(b*c + sq rt(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)*sqrt(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 + 2*b^3*c^2 - 4*sqrt(2)*sqrt(b*c + sqrt (b^2 - 4*a*c)*c)*a*c^3 - 32*a^2*c^3 - 8*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c )*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b *c + sqrt(b^2 - 4*a*c)*c)*a*b*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + s qrt(b^2 - 4*a*c)*c)*b^2*c - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^2 + 2*(b^2 - 4*a*c)*b^2*c - 8*(b^2 - 4*a*c)*a*c^2 - 2*(b^2 - 4*a*c)*b*c^2)*d*arctan(2*sqrt(1/2)*x/sqrt((b + sqrt(b^2 - 4*a*c))/c))/( (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)*abs(c)) + 1/4*(sqrt(2)*sqrt(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 - sqr t(b^2 - 4*a*c)*c)*b^3*c + 2*b^4*c + 16*sqrt(2)*sqrt(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 + 2*b^3*c^2 - 4*sq rt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*c^3 + 32*a^2*c^3 - 8*a*b*c^3 - sqr t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3 + 4*sqrt(2)*sqr t(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c + 2*sqrt(2)*sqrt(b...
Time = 8.36 (sec) , antiderivative size = 1308, normalized size of antiderivative = 6.92 \[ \int \frac {d+e x}{a+b x^2+c x^4} \, dx=\sum _{k=1}^4\ln \left (c^2\,\left (d\,e^2+e^3\,x+{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^2\,b^2\,d\,4-{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^3\,b^3\,x\,8-{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^2\,a\,c\,d\,16+\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )\,b\,e^2\,x\,2-\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )\,c\,d^2\,x\,4-{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^2\,b^2\,e\,x\,4+\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )\,b\,d\,e\,4+{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^3\,a\,b\,c\,x\,32+{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^2\,a\,c\,e\,x\,16\right )\right )\,\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right ) \]
symsum(log(c^2*(d*e^2 + e^3*x + 4*root(128*a^2*b^2*c*z^4 - 256*a^3*c^2*z^4 - 16*a*b^4*z^4 + 16*a*b*c*d^2*z^2 - 32*a^2*c*e^2*z^2 + 8*a*b^2*e^2*z^2 - 4*b^3*d^2*z^2 + 16*a*c*d^2*e*z - 4*b^2*d^2*e*z - b*d^2*e^2 - c*d^4 - a*e^4 , z, k)^2*b^2*d - 8*root(128*a^2*b^2*c*z^4 - 256*a^3*c^2*z^4 - 16*a*b^4*z^ 4 + 16*a*b*c*d^2*z^2 - 32*a^2*c*e^2*z^2 + 8*a*b^2*e^2*z^2 - 4*b^3*d^2*z^2 + 16*a*c*d^2*e*z - 4*b^2*d^2*e*z - b*d^2*e^2 - c*d^4 - a*e^4, z, k)^3*b^3* x - 16*root(128*a^2*b^2*c*z^4 - 256*a^3*c^2*z^4 - 16*a*b^4*z^4 + 16*a*b*c* d^2*z^2 - 32*a^2*c*e^2*z^2 + 8*a*b^2*e^2*z^2 - 4*b^3*d^2*z^2 + 16*a*c*d^2* e*z - 4*b^2*d^2*e*z - b*d^2*e^2 - c*d^4 - a*e^4, z, k)^2*a*c*d + 2*root(12 8*a^2*b^2*c*z^4 - 256*a^3*c^2*z^4 - 16*a*b^4*z^4 + 16*a*b*c*d^2*z^2 - 32*a ^2*c*e^2*z^2 + 8*a*b^2*e^2*z^2 - 4*b^3*d^2*z^2 + 16*a*c*d^2*e*z - 4*b^2*d^ 2*e*z - b*d^2*e^2 - c*d^4 - a*e^4, z, k)*b*e^2*x - 4*root(128*a^2*b^2*c*z^ 4 - 256*a^3*c^2*z^4 - 16*a*b^4*z^4 + 16*a*b*c*d^2*z^2 - 32*a^2*c*e^2*z^2 + 8*a*b^2*e^2*z^2 - 4*b^3*d^2*z^2 + 16*a*c*d^2*e*z - 4*b^2*d^2*e*z - b*d^2* e^2 - c*d^4 - a*e^4, z, k)*c*d^2*x - 4*root(128*a^2*b^2*c*z^4 - 256*a^3*c^ 2*z^4 - 16*a*b^4*z^4 + 16*a*b*c*d^2*z^2 - 32*a^2*c*e^2*z^2 + 8*a*b^2*e^2*z ^2 - 4*b^3*d^2*z^2 + 16*a*c*d^2*e*z - 4*b^2*d^2*e*z - b*d^2*e^2 - c*d^4 - a*e^4, z, k)^2*b^2*e*x + 4*root(128*a^2*b^2*c*z^4 - 256*a^3*c^2*z^4 - 16*a *b^4*z^4 + 16*a*b*c*d^2*z^2 - 32*a^2*c*e^2*z^2 + 8*a*b^2*e^2*z^2 - 4*b^3*d ^2*z^2 + 16*a*c*d^2*e*z - 4*b^2*d^2*e*z - b*d^2*e^2 - c*d^4 - a*e^4, z,...