\(\int \frac {a+b x+c x^2}{d+e x^2+f x^4} \, dx\) [337]

Optimal result

Integrand size = 25, antiderivative size = 209 \[ \int \frac {a+b x+c x^2}{d+e x^2+f x^4} \, dx=\frac {\left (c-\frac {c e-2 a f}{\sqrt {e^2-4 d f}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {e-\sqrt {e^2-4 d f}}}\right )}{\sqrt {2} \sqrt {f} \sqrt {e-\sqrt {e^2-4 d f}}}+\frac {\left (c+\frac {c e-2 a f}{\sqrt {e^2-4 d f}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {e+\sqrt {e^2-4 d f}}}\right )}{\sqrt {2} \sqrt {f} \sqrt {e+\sqrt {e^2-4 d f}}}-\frac {b \text {arctanh}\left (\frac {e+2 f x^2}{\sqrt {e^2-4 d f}}\right )}{\sqrt {e^2-4 d f}} \]

[Out]

-b*arctanh((2*f*x^2+e)/(-4*d*f+e^2)^(1/2))/(-4*d*f+e^2)^(1/2)+1/2*arctan(x*2^(1/2)*f^(1/2)/(e-(-4*d*f+e^2)^(1/
2))^(1/2))*(c+(2*a*f-c*e)/(-4*d*f+e^2)^(1/2))*2^(1/2)/f^(1/2)/(e-(-4*d*f+e^2)^(1/2))^(1/2)+1/2*arctan(x*2^(1/2
)*f^(1/2)/(e+(-4*d*f+e^2)^(1/2))^(1/2))*(c+(-2*a*f+c*e)/(-4*d*f+e^2)^(1/2))*2^(1/2)/f^(1/2)/(e+(-4*d*f+e^2)^(1
/2))^(1/2)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1687, 1180, 211, 12, 1121, 632, 212} \[ \int \frac {a+b x+c x^2}{d+e x^2+f x^4} \, dx=\frac {\left (c-\frac {c e-2 a f}{\sqrt {e^2-4 d f}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {e-\sqrt {e^2-4 d f}}}\right )}{\sqrt {2} \sqrt {f} \sqrt {e-\sqrt {e^2-4 d f}}}+\frac {\left (\frac {c e-2 a f}{\sqrt {e^2-4 d f}}+c\right ) \arctan \left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {\sqrt {e^2-4 d f}+e}}\right )}{\sqrt {2} \sqrt {f} \sqrt {\sqrt {e^2-4 d f}+e}}-\frac {b \text {arctanh}\left (\frac {e+2 f x^2}{\sqrt {e^2-4 d f}}\right )}{\sqrt {e^2-4 d f}} \]

[In]

Int[(a + b*x + c*x^2)/(d + e*x^2 + f*x^4),x]

[Out]

((c - (c*e - 2*a*f)/Sqrt[e^2 - 4*d*f])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[e - Sqrt[e^2 - 4*d*f]]])/(Sqrt[2]*Sqrt[
f]*Sqrt[e - Sqrt[e^2 - 4*d*f]]) + ((c + (c*e - 2*a*f)/Sqrt[e^2 - 4*d*f])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[e + S
qrt[e^2 - 4*d*f]]])/(Sqrt[2]*Sqrt[f]*Sqrt[e + Sqrt[e^2 - 4*d*f]]) - (b*ArcTanh[(e + 2*f*x^2)/Sqrt[e^2 - 4*d*f]
])/Sqrt[e^2 - 4*d*f]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1121

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1687

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rubi steps \begin{align*} \text {integral}& = \int \frac {b x}{d+e x^2+f x^4} \, dx+\int \frac {a+c x^2}{d+e x^2+f x^4} \, dx \\ & = b \int \frac {x}{d+e x^2+f x^4} \, dx+\frac {1}{2} \left (c-\frac {c e-2 a f}{\sqrt {e^2-4 d f}}\right ) \int \frac {1}{\frac {e}{2}-\frac {1}{2} \sqrt {e^2-4 d f}+f x^2} \, dx+\frac {1}{2} \left (c+\frac {c e-2 a f}{\sqrt {e^2-4 d f}}\right ) \int \frac {1}{\frac {e}{2}+\frac {1}{2} \sqrt {e^2-4 d f}+f x^2} \, dx \\ & = \frac {\left (c-\frac {c e-2 a f}{\sqrt {e^2-4 d f}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {e-\sqrt {e^2-4 d f}}}\right )}{\sqrt {2} \sqrt {f} \sqrt {e-\sqrt {e^2-4 d f}}}+\frac {\left (c+\frac {c e-2 a f}{\sqrt {e^2-4 d f}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {e+\sqrt {e^2-4 d f}}}\right )}{\sqrt {2} \sqrt {f} \sqrt {e+\sqrt {e^2-4 d f}}}+\frac {1}{2} b \text {Subst}\left (\int \frac {1}{d+e x+f x^2} \, dx,x,x^2\right ) \\ & = \frac {\left (c-\frac {c e-2 a f}{\sqrt {e^2-4 d f}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {e-\sqrt {e^2-4 d f}}}\right )}{\sqrt {2} \sqrt {f} \sqrt {e-\sqrt {e^2-4 d f}}}+\frac {\left (c+\frac {c e-2 a f}{\sqrt {e^2-4 d f}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {e+\sqrt {e^2-4 d f}}}\right )}{\sqrt {2} \sqrt {f} \sqrt {e+\sqrt {e^2-4 d f}}}-b \text {Subst}\left (\int \frac {1}{e^2-4 d f-x^2} \, dx,x,e+2 f x^2\right ) \\ & = \frac {\left (c-\frac {c e-2 a f}{\sqrt {e^2-4 d f}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {e-\sqrt {e^2-4 d f}}}\right )}{\sqrt {2} \sqrt {f} \sqrt {e-\sqrt {e^2-4 d f}}}+\frac {\left (c+\frac {c e-2 a f}{\sqrt {e^2-4 d f}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {e+\sqrt {e^2-4 d f}}}\right )}{\sqrt {2} \sqrt {f} \sqrt {e+\sqrt {e^2-4 d f}}}-\frac {b \tanh ^{-1}\left (\frac {e+2 f x^2}{\sqrt {e^2-4 d f}}\right )}{\sqrt {e^2-4 d f}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.12 \[ \int \frac {a+b x+c x^2}{d+e x^2+f x^4} \, dx=\frac {\frac {\sqrt {2} \left (2 a f+c \left (-e+\sqrt {e^2-4 d f}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {e-\sqrt {e^2-4 d f}}}\right )}{\sqrt {f} \sqrt {e-\sqrt {e^2-4 d f}}}+\frac {\sqrt {2} \left (-2 a f+c \left (e+\sqrt {e^2-4 d f}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {e+\sqrt {e^2-4 d f}}}\right )}{\sqrt {f} \sqrt {e+\sqrt {e^2-4 d f}}}+b \log \left (-e+\sqrt {e^2-4 d f}-2 f x^2\right )-b \log \left (e+\sqrt {e^2-4 d f}+2 f x^2\right )}{2 \sqrt {e^2-4 d f}} \]

[In]

Integrate[(a + b*x + c*x^2)/(d + e*x^2 + f*x^4),x]

[Out]

((Sqrt[2]*(2*a*f + c*(-e + Sqrt[e^2 - 4*d*f]))*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[e - Sqrt[e^2 - 4*d*f]]])/(Sqrt[
f]*Sqrt[e - Sqrt[e^2 - 4*d*f]]) + (Sqrt[2]*(-2*a*f + c*(e + Sqrt[e^2 - 4*d*f]))*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqr
t[e + Sqrt[e^2 - 4*d*f]]])/(Sqrt[f]*Sqrt[e + Sqrt[e^2 - 4*d*f]]) + b*Log[-e + Sqrt[e^2 - 4*d*f] - 2*f*x^2] - b
*Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x^2])/(2*Sqrt[e^2 - 4*d*f])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.23

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{4}+e \,\textit {\_Z}^{2}+d \right )}{\sum }\frac {\left (c \,\textit {\_R}^{2}+b \textit {\_R} +a \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} f +\textit {\_R} e}\right )}{2}\)	\(48\)
default	\(4 f \left (-\frac {\sqrt {-4 d f +e^{2}}\, \left (\frac {b \ln \left (-2 f \,x^{2}+\sqrt {-4 d f +e^{2}}-e \right )}{2}+\frac {\left (-\sqrt {-4 d f +e^{2}}\, c -2 a f +e c \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {f x \sqrt {2}}{\sqrt {\left (\sqrt {-4 d f +e^{2}}-e \right ) f}}\right )}{2 \sqrt {\left (\sqrt {-4 d f +e^{2}}-e \right ) f}}\right )}{4 f \left (4 d f -e^{2}\right )}-\frac {\sqrt {-4 d f +e^{2}}\, \left (-\frac {b \ln \left (2 f \,x^{2}+\sqrt {-4 d f +e^{2}}+e \right )}{2}+\frac {\left (\sqrt {-4 d f +e^{2}}\, c -2 a f +e c \right ) \sqrt {2}\, \arctan \left (\frac {f x \sqrt {2}}{\sqrt {\left (e +\sqrt {-4 d f +e^{2}}\right ) f}}\right )}{2 \sqrt {\left (e +\sqrt {-4 d f +e^{2}}\right ) f}}\right )}{4 f \left (4 d f -e^{2}\right )}\right )\)	\(240\)

[In]

int((c*x^2+b*x+a)/(f*x^4+e*x^2+d),x,method=_RETURNVERBOSE)

[Out]

1/2*sum((_R^2*c+_R*b+a)/(2*_R^3*f+_R*e)*ln(x-_R),_R=RootOf(_Z^4*f+_Z^2*e+d))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 29.02 (sec) , antiderivative size = 578003, normalized size of antiderivative = 2765.56 \[ \int \frac {a+b x+c x^2}{d+e x^2+f x^4} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+b*x+a)/(f*x^4+e*x^2+d),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b x+c x^2}{d+e x^2+f x^4} \, dx=\text {Timed out} \]

[In]

integrate((c*x**2+b*x+a)/(f*x**4+e*x**2+d),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {a+b x+c x^2}{d+e x^2+f x^4} \, dx=\int { \frac {c x^{2} + b x + a}{f x^{4} + e x^{2} + d} \,d x } \]

[In]

integrate((c*x^2+b*x+a)/(f*x^4+e*x^2+d),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)/(f*x^4 + e*x^2 + d), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1714 vs. \(2 (171) = 342\).

Time = 1.12 (sec) , antiderivative size = 1714, normalized size of antiderivative = 8.20 \[ \int \frac {a+b x+c x^2}{d+e x^2+f x^4} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+b*x+a)/(f*x^4+e*x^2+d),x, algorithm="giac")

[Out]

-1/2*(e^2*f^2 - 4*d*f^3 - 2*e*f^3 + f^4)*sqrt(e^2 - 4*d*f)*b*log(x^2 + 1/2*(e - sqrt(e^2 - 4*d*f))/f)/((e^4 -
8*d*e^2*f - 2*e^3*f + 16*d^2*f^2 + 8*d*e*f^2 + e^2*f^2 - 4*d*f^3)*f^2) + 1/4*((sqrt(2)*sqrt(e*f + sqrt(e^2 - 4
*d*f)*f)*e^4 - 8*sqrt(2)*sqrt(e*f + sqrt(e^2 - 4*d*f)*f)*d*e^2*f - 2*sqrt(2)*sqrt(e*f + sqrt(e^2 - 4*d*f)*f)*e
^3*f - 2*e^4*f + 16*sqrt(2)*sqrt(e*f + sqrt(e^2 - 4*d*f)*f)*d^2*f^2 + 8*sqrt(2)*sqrt(e*f + sqrt(e^2 - 4*d*f)*f
)*d*e*f^2 + sqrt(2)*sqrt(e*f + sqrt(e^2 - 4*d*f)*f)*e^2*f^2 + 16*d*e^2*f^2 + 2*e^3*f^2 - 4*sqrt(2)*sqrt(e*f +
sqrt(e^2 - 4*d*f)*f)*d*f^3 - 32*d^2*f^3 - 8*d*e*f^3 - sqrt(2)*sqrt(e^2 - 4*d*f)*sqrt(e*f + sqrt(e^2 - 4*d*f)*f
)*e^3 + 4*sqrt(2)*sqrt(e^2 - 4*d*f)*sqrt(e*f + sqrt(e^2 - 4*d*f)*f)*d*e*f + 2*sqrt(2)*sqrt(e^2 - 4*d*f)*sqrt(e
*f + sqrt(e^2 - 4*d*f)*f)*e^2*f - sqrt(2)*sqrt(e^2 - 4*d*f)*sqrt(e*f + sqrt(e^2 - 4*d*f)*f)*e*f^2 + 2*(e^2 - 4
*d*f)*e^2*f - 8*(e^2 - 4*d*f)*d*f^2 - 2*(e^2 - 4*d*f)*e*f^2)*a - 2*(2*d*e^2*f^2 - 8*d^2*f^3 - sqrt(2)*sqrt(e^2
- 4*d*f)*sqrt(e*f + sqrt(e^2 - 4*d*f)*f)*d*e^2 + 4*sqrt(2)*sqrt(e^2 - 4*d*f)*sqrt(e*f + sqrt(e^2 - 4*d*f)*f)*
d^2*f + 2*sqrt(2)*sqrt(e^2 - 4*d*f)*sqrt(e*f + sqrt(e^2 - 4*d*f)*f)*d*e*f - sqrt(2)*sqrt(e^2 - 4*d*f)*sqrt(e*f
+ sqrt(e^2 - 4*d*f)*f)*d*f^2 - 2*(e^2 - 4*d*f)*d*f^2)*c)*arctan(2*sqrt(1/2)*x/sqrt((e + sqrt(e^2 - 4*d*f))/f)
)/((d*e^4 - 8*d^2*e^2*f - 2*d*e^3*f + 16*d^3*f^2 + 8*d^2*e*f^2 + d*e^2*f^2 - 4*d^2*f^3)*abs(f)) + 1/4*((sqrt(2
)*sqrt(e*f - sqrt(e^2 - 4*d*f)*f)*e^4 - 8*sqrt(2)*sqrt(e*f - sqrt(e^2 - 4*d*f)*f)*d*e^2*f - 2*sqrt(2)*sqrt(e*f
- sqrt(e^2 - 4*d*f)*f)*e^3*f + 2*e^4*f + 16*sqrt(2)*sqrt(e*f - sqrt(e^2 - 4*d*f)*f)*d^2*f^2 + 8*sqrt(2)*sqrt(
e*f - sqrt(e^2 - 4*d*f)*f)*d*e*f^2 + sqrt(2)*sqrt(e*f - sqrt(e^2 - 4*d*f)*f)*e^2*f^2 - 16*d*e^2*f^2 + 2*e^3*f^
2 - 4*sqrt(2)*sqrt(e*f - sqrt(e^2 - 4*d*f)*f)*d*f^3 + 32*d^2*f^3 - 8*d*e*f^3 - sqrt(2)*sqrt(e^2 - 4*d*f)*sqrt(
e*f - sqrt(e^2 - 4*d*f)*f)*e^3 + 4*sqrt(2)*sqrt(e^2 - 4*d*f)*sqrt(e*f - sqrt(e^2 - 4*d*f)*f)*d*e*f + 2*sqrt(2)
*sqrt(e^2 - 4*d*f)*sqrt(e*f - sqrt(e^2 - 4*d*f)*f)*e^2*f - sqrt(2)*sqrt(e^2 - 4*d*f)*sqrt(e*f - sqrt(e^2 - 4*d
*f)*f)*e*f^2 - 2*(e^2 - 4*d*f)*e^2*f + 8*(e^2 - 4*d*f)*d*f^2 - 2*(e^2 - 4*d*f)*e*f^2)*a - 2*(2*d*e^2*f^2 - 8*d
^2*f^3 - sqrt(2)*sqrt(e^2 - 4*d*f)*sqrt(e*f - sqrt(e^2 - 4*d*f)*f)*d*e^2 + 4*sqrt(2)*sqrt(e^2 - 4*d*f)*sqrt(e*
f - sqrt(e^2 - 4*d*f)*f)*d^2*f + 2*sqrt(2)*sqrt(e^2 - 4*d*f)*sqrt(e*f - sqrt(e^2 - 4*d*f)*f)*d*e*f - sqrt(2)*s
qrt(e^2 - 4*d*f)*sqrt(e*f - sqrt(e^2 - 4*d*f)*f)*d*f^2 - 2*(e^2 - 4*d*f)*d*f^2)*c)*arctan(2*sqrt(1/2)*x/sqrt((
e - sqrt(e^2 - 4*d*f))/f))/((d*e^4 - 8*d^2*e^2*f - 2*d*e^3*f + 16*d^3*f^2 + 8*d^2*e*f^2 + d*e^2*f^2 - 4*d^2*f^
3)*abs(f)) + 1/4*(e^5*f - 8*d*e^3*f^2 - 2*e^4*f^2 + 16*d^2*e*f^3 + 8*d*e^2*f^3 + e^3*f^3 - 4*d*e*f^4 + (e^4*f
- 6*d*e^2*f^2 - 2*e^3*f^2 + 8*d^2*f^3 + 4*d*e*f^3 + e^2*f^3 - 2*d*f^4)*sqrt(e^2 - 4*d*f))*b*log(x^2 + 1/2*(e +
sqrt(e^2 - 4*d*f))/f)/((d*e^4 - 8*d^2*e^2*f - 2*d*e^3*f + 16*d^3*f^2 + 8*d^2*e*f^2 + d*e^2*f^2 - 4*d^2*f^3)*f
^2)

Mupad [B] (verification not implemented)

Time = 10.02 (sec) , antiderivative size = 3942, normalized size of antiderivative = 18.86 \[ \int \frac {a+b x+c x^2}{d+e x^2+f x^4} \, dx=\text {Too large to display} \]

[In]

int((a + b*x + c*x^2)/(d + e*x^2 + f*x^4),x)

[Out]

symsum(log(a*b^2*f^2 - a^2*c*f^2 + b^3*f^2*x - c^3*d*f - 8*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3
*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z
^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e
^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*
f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)^3*e^3*f^2*x + a*c^2*e*f - 16*root(16*d*e^4*f*z^4 - 128*d^
2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2
*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*
b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b
^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)^2*a*d*f^3 - 4*root(16*d*e^4*f*z^4
- 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2
*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z
+ 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3
*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)*a^2*f^3*x + 4*root(16*d*e^
4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2
- 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*
d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f -
2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)^2*a*e^2*f^2 + 16*r
oot(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d
*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2
+ 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*
a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)^2*b*d*f
^3*x + 2*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2
- 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*
e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^
2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z,
k)*b^2*e*f^2*x + 4*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d
^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^
2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f
+ 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^
4*f^2, z, k)*c^2*d*f^2*x - 2*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2
- 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^
2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*
b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^
4*d^2 + a^4*f^2, z, k)*c^2*e^2*f*x + 32*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d
*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2
*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f
^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c
^2*e^2 + c^4*d^2 + a^4*f^2, z, k)^3*d*e*f^3*x - 4*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4
- 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b
^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16
*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d
*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)^2*b*e^2*f^2*x + 4*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*
d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^
2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*
d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2
*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)*a*b*e*f^2 - 8*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^
4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*
c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z -
4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e +
a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)*b*c*d*f^2 - 2*a*b*c*f^2*x + b*c^2*e*f*x + 4*ro
ot(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8*b^2*d*
e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*f*z^2 +
16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*f - 2*a
^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k)*a*c*e*f^
2*x)*root(16*d*e^4*f*z^4 - 128*d^2*e^2*f^2*z^4 + 256*d^3*f^3*z^4 - 16*a*c*d*e^2*f*z^2 - 16*c^2*d^2*e*f*z^2 - 8
*b^2*d*e^2*f*z^2 - 16*a^2*d*e*f^2*z^2 + 64*a*c*d^2*f^2*z^2 + 32*b^2*d^2*f^2*z^2 + 4*c^2*d*e^3*z^2 + 4*a^2*e^3*
f*z^2 + 16*b*c^2*d^2*f*z + 4*a^2*b*e^2*f*z - 4*b*c^2*d*e^2*z - 16*a^2*b*d*f^2*z - 4*a*b^2*c*d*f + 2*a^2*c^2*d*
f - 2*a^3*c*e*f - 2*a*c^3*d*e + b^2*c^2*d*e + a^2*b^2*e*f + b^4*d*f + a^2*c^2*e^2 + c^4*d^2 + a^4*f^2, z, k),
k, 1, 4)