3.1.5 \(\int \frac {A+B x}{(a+b x+c x^2)^2 (d+f x^2)} \, dx\)
Optimal. Leaf size=596 \[ -\frac {f^{3/2} \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {d}}\right ) \left (-A (c d-a f)^2+2 b B d (c d-a f)+A b^2 d f\right )}{\sqrt {d} \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2}-\frac {f \log \left (a+b x+c x^2\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right )}{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2}+\frac {f \log \left (d+f x^2\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right )}{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2}-\frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-4 A b^2 c f \left (3 a^2 f^2-3 a c d f+2 c^2 d^2\right )+b^3 B f \left (-a^2 f^2-4 a c d f+5 c^2 d^2\right )+2 b B c \left (3 a^3 f^3+3 a^2 c d f^2-7 a c^2 d^2 f+c^3 d^3\right )-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^5 B d f^2\right )}{\left (b^2-4 a c\right )^{3/2} \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2}+\frac {-(A b-a B) \left (-2 a c f+b^2 f+2 c^2 d\right )-c x \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )+A b c (a f+c d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2+b^2 d f\right )} \]
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Rubi [A] time = 1.77, antiderivative size = 596, normalized size of antiderivative = 1.00,
number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used =
{1018, 1074, 634, 618, 206, 628, 635, 205, 260} \begin {gather*} -\frac {f^{3/2} \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {d}}\right ) \left (-A (c d-a f)^2+2 b B d (c d-a f)+A b^2 d f\right )}{\sqrt {d} \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2}-\frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-4 A b^2 c f \left (3 a^2 f^2-3 a c d f+2 c^2 d^2\right )+b^3 B f \left (-a^2 f^2-4 a c d f+5 c^2 d^2\right )+2 b B c \left (3 a^2 c d f^2+3 a^3 f^3-7 a c^2 d^2 f+c^3 d^3\right )-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^5 B d f^2\right )}{\left (b^2-4 a c\right )^{3/2} \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2}-\frac {f \log \left (a+b x+c x^2\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right )}{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2}+\frac {f \log \left (d+f x^2\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right )}{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2}+\frac {-(A b-a B) \left (-2 a c f+b^2 f+2 c^2 d\right )-c x \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )+A b c (a f+c d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2+b^2 d f\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((a + b*x + c*x^2)^2*(d + f*x^2)),x]
[Out]
(A*b*c*(c*d + a*f) - (A*b - a*B)*(2*c^2*d + b^2*f - 2*a*c*f) - c*(A*b^2*f + 2*A*c*(c*d - a*f) - b*B*(c*d + a*f
))*x)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(a + b*x + c*x^2)) - (f^(3/2)*(A*b^2*d*f + 2*b*B*d*(c*d - a*f)
- A*(c*d - a*f)^2)*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/(Sqrt[d]*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2) - ((b^5*
B*d*f^2 - 2*A*b^4*f^2*(c*d - a*f) - 4*A*c^2*(c*d - 3*a*f)*(c*d - a*f)^2 + b^3*B*f*(5*c^2*d^2 - 4*a*c*d*f - a^2
*f^2) - 4*A*b^2*c*f*(2*c^2*d^2 - 3*a*c*d*f + 3*a^2*f^2) + 2*b*B*c*(c^3*d^3 - 7*a*c^2*d^2*f + 3*a^2*c*d*f^2 + 3
*a^3*f^3))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(3/2)*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*
f))^2) - (f*(2*A*b*f*(c*d - a*f) + B*(c^2*d^2 - 2*a*c*d*f - f*(b^2*d - a^2*f)))*Log[a + b*x + c*x^2])/(2*(c^2*
d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2) + (f*(2*A*b*f*(c*d - a*f) + B*(c^2*d^2 - 2*a*c*d*f - f*(b^2*d - a^2*f)
))*Log[d + f*x^2])/(2*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2)
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 260
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]
Rule 618
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 628
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 634
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]
Rule 635
Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && !NiceSqrtQ[-(a*c)]
Rule 1018
Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp
[((a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q + 1)*((g*c)*(-(b*(c*d + a*f))) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(
2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(2*a*f)) - h*(b*c*d + a*b*f))*x))/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)
*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + f
*x^2)^q*Simp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d)*(-(b*f)))*(p + 1) + (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*
(c*d - a*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f*((g*c)*(-(b*(c*d + a*f))) + (g*b - a*h)*(2*c^2*d + b^2*f - c*
(2*a*f)))*(p + q + 2) - (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(b*f*(p + 1)))*x - c*f*(b^2*(g*f
) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}
, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] && !( !IntegerQ[p] && ILtQ[q, -1
])
Rule 1074
Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (f_.)*(x_)^2)), x_Symbol]
:> With[{q = c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2}, Dist[1/q, Int[(A*c^2*d - a*c*C*d + A*b^2*f - a*b*B*f -
a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d + A*b*f - a*B*f)*x)/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[(c*C*d^2 +
b*B*d*f - A*c*d*f - a*C*d*f + a*A*f^2 - f*(B*c*d - b*C*d + A*b*f - a*B*f)*x)/(d + f*x^2), x], x] /; NeQ[q, 0]]
/; FreeQ[{a, b, c, d, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0]
Rubi steps
\begin {align*} \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+f x^2\right )} \, dx &=\frac {A b c (c d+a f)-(A b-a B) \left (2 c^2 d+b^2 f-2 a c f\right )-c \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d f+(c d-a f)^2\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {-\left ((b B-2 A c) \left (b^2 d f+(c d-a f)^2\right )\right )-a f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )+\left (b^2-4 a c\right ) f (B c d+A b f-a B f) x+c f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) x^2}{\left (a+b x+c x^2\right ) \left (d+f x^2\right )} \, dx}{\left (b^2-4 a c\right ) \left (b^2 d f+(c d-a f)^2\right )}\\ &=\frac {A b c (c d+a f)-(A b-a B) \left (2 c^2 d+b^2 f-2 a c f\right )-c \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d f+(c d-a f)^2\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {-a b \left (b^2-4 a c\right ) f^2 (B c d+A b f-a B f)-a c^2 d f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )+a^2 c f^2 \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )+c^2 d \left (-\left ((b B-2 A c) \left (b^2 d f+(c d-a f)^2\right )\right )-a f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )\right )+b^2 f \left (-\left ((b B-2 A c) \left (b^2 d f+(c d-a f)^2\right )\right )-a f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )\right )-a c f \left (-\left ((b B-2 A c) \left (b^2 d f+(c d-a f)^2\right )\right )-a f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )\right )+c \left (c \left (b^2-4 a c\right ) d f (B c d+A b f-a B f)-a \left (b^2-4 a c\right ) f^2 (B c d+A b f-a B f)-b c d f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )+b f \left (-\left ((b B-2 A c) \left (b^2 d f+(c d-a f)^2\right )\right )-a f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )\right )\right ) x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}-\frac {\int \frac {b \left (b^2-4 a c\right ) d f^2 (B c d+A b f-a B f)+c^2 d^2 f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )-a c d f^2 \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )-c d f \left (-\left ((b B-2 A c) \left (b^2 d f+(c d-a f)^2\right )\right )-a f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )\right )+a f^2 \left (-\left ((b B-2 A c) \left (b^2 d f+(c d-a f)^2\right )\right )-a f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )\right )-f \left (c \left (b^2-4 a c\right ) d f (B c d+A b f-a B f)-a \left (b^2-4 a c\right ) f^2 (B c d+A b f-a B f)-b c d f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )+b f \left (-\left ((b B-2 A c) \left (b^2 d f+(c d-a f)^2\right )\right )-a f \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right )\right )\right ) x}{d+f x^2} \, dx}{\left (b^2-4 a c\right ) \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}\\ &=\frac {A b c (c d+a f)-(A b-a B) \left (2 c^2 d+b^2 f-2 a c f\right )-c \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d f+(c d-a f)^2\right ) \left (a+b x+c x^2\right )}-\frac {\left (f^2 \left (A b^2 d f+2 b B d (c d-a f)-A (c d-a f)^2\right )\right ) \int \frac {1}{d+f x^2} \, dx}{\left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}+\frac {\left (b^5 B d f^2-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^3 B f \left (5 c^2 d^2-4 a c d f-a^2 f^2\right )-4 A b^2 c f \left (2 c^2 d^2-3 a c d f+3 a^2 f^2\right )+2 b B c \left (c^3 d^3-7 a c^2 d^2 f+3 a^2 c d f^2+3 a^3 f^3\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right ) \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}-\frac {\left (f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c d f-f \left (b^2 d-a^2 f\right )\right )\right )\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}+\frac {\left (f^2 \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c d f-f \left (b^2 d-a^2 f\right )\right )\right )\right ) \int \frac {x}{d+f x^2} \, dx}{\left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}\\ &=\frac {A b c (c d+a f)-(A b-a B) \left (2 c^2 d+b^2 f-2 a c f\right )-c \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d f+(c d-a f)^2\right ) \left (a+b x+c x^2\right )}-\frac {f^{3/2} \left (A b^2 d f+2 b B d (c d-a f)-A (c d-a f)^2\right ) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {d}}\right )}{\sqrt {d} \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}-\frac {f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c d f-f \left (b^2 d-a^2 f\right )\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}+\frac {f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c d f-f \left (b^2 d-a^2 f\right )\right )\right ) \log \left (d+f x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}-\frac {\left (b^5 B d f^2-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^3 B f \left (5 c^2 d^2-4 a c d f-a^2 f^2\right )-4 A b^2 c f \left (2 c^2 d^2-3 a c d f+3 a^2 f^2\right )+2 b B c \left (c^3 d^3-7 a c^2 d^2 f+3 a^2 c d f^2+3 a^3 f^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right ) \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}\\ &=\frac {A b c (c d+a f)-(A b-a B) \left (2 c^2 d+b^2 f-2 a c f\right )-c \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d f+(c d-a f)^2\right ) \left (a+b x+c x^2\right )}-\frac {f^{3/2} \left (A b^2 d f+2 b B d (c d-a f)-A (c d-a f)^2\right ) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {d}}\right )}{\sqrt {d} \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}-\frac {\left (b^5 B d f^2-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^3 B f \left (5 c^2 d^2-4 a c d f-a^2 f^2\right )-4 A b^2 c f \left (2 c^2 d^2-3 a c d f+3 a^2 f^2\right )+2 b B c \left (c^3 d^3-7 a c^2 d^2 f+3 a^2 c d f^2+3 a^3 f^3\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}-\frac {f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c d f-f \left (b^2 d-a^2 f\right )\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}+\frac {f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c d f-f \left (b^2 d-a^2 f\right )\right )\right ) \log \left (d+f x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}\\ \end {align*}
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Mathematica [A] time = 1.84, size = 523, normalized size = 0.88 \begin {gather*} \frac {f \log \left (d+f x^2\right ) \left (B \left (f \left (a^2 f-b^2 d\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right )+f \log (a+x (b+c x)) \left (B \left (f \left (b^2 d-a^2 f\right )+2 a c d f-c^2 d^2\right )+2 A b f (a f-c d)\right )-\frac {2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right ) \left (B \left (2 a^2 c f-a \left (b^2 f+b c f x+2 c^2 d\right )-b c^2 d x\right )+A \left (b c (c d-3 a f)+2 c^2 x (c d-a f)+b^3 f+b^2 c f x\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac {2 \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right ) \left (-4 A b^2 c f \left (3 a^2 f^2-3 a c d f+2 c^2 d^2\right )-b^3 B f \left (a^2 f^2+4 a c d f-5 c^2 d^2\right )+2 b B c \left (3 a^3 f^3+3 a^2 c d f^2-7 a c^2 d^2 f+c^3 d^3\right )+2 A b^4 f^2 (a f-c d)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^5 B d f^2\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac {2 f^{3/2} \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {d}}\right ) \left (A (c d-a f)^2+2 b B d (a f-c d)-A b^2 d f\right )}{\sqrt {d}}}{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((a + b*x + c*x^2)^2*(d + f*x^2)),x]
[Out]
((-2*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))*(A*(b^3*f + b*c*(c*d - 3*a*f) + b^2*c*f*x + 2*c^2*(c*d - a*f)*x
) + B*(2*a^2*c*f - b*c^2*d*x - a*(2*c^2*d + b^2*f + b*c*f*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*f^(3/2)
*(-(A*b^2*d*f) + A*(c*d - a*f)^2 + 2*b*B*d*(-(c*d) + a*f))*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/Sqrt[d] - (2*(b^5*B*d*
f^2 - 4*A*c^2*(c*d - 3*a*f)*(c*d - a*f)^2 + 2*A*b^4*f^2*(-(c*d) + a*f) - b^3*B*f*(-5*c^2*d^2 + 4*a*c*d*f + a^2
*f^2) - 4*A*b^2*c*f*(2*c^2*d^2 - 3*a*c*d*f + 3*a^2*f^2) + 2*b*B*c*(c^3*d^3 - 7*a*c^2*d^2*f + 3*a^2*c*d*f^2 + 3
*a^3*f^3))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + f*(2*A*b*f*(c*d - a*f) + B*(c^2*d^2
- 2*a*c*d*f + f*(-(b^2*d) + a^2*f)))*Log[d + f*x^2] + f*(2*A*b*f*(-(c*d) + a*f) + B*(-(c^2*d^2) + 2*a*c*d*f +
f*(b^2*d - a^2*f)))*Log[a + x*(b + c*x)])/(2*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2)
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+f x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(A + B*x)/((a + b*x + c*x^2)^2*(d + f*x^2)),x]
[Out]
IntegrateAlgebraic[(A + B*x)/((a + b*x + c*x^2)^2*(d + f*x^2)), x]
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(c*x^2+b*x+a)^2/(f*x^2+d),x, algorithm="fricas")
[Out]
Timed out
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giac [B] time = 0.21, size = 1313, normalized size = 2.20
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(c*x^2+b*x+a)^2/(f*x^2+d),x, algorithm="giac")
[Out]
-1/2*(B*c^2*d^2*f - B*b^2*d*f^2 - 2*B*a*c*d*f^2 + 2*A*b*c*d*f^2 + B*a^2*f^3 - 2*A*a*b*f^3)*log(c*x^2 + b*x + a
)/(c^4*d^4 + 2*b^2*c^2*d^3*f - 4*a*c^3*d^3*f + b^4*d^2*f^2 - 4*a*b^2*c*d^2*f^2 + 6*a^2*c^2*d^2*f^2 + 2*a^2*b^2
*d*f^3 - 4*a^3*c*d*f^3 + a^4*f^4) + 1/2*(B*c^2*d^2*f - B*b^2*d*f^2 - 2*B*a*c*d*f^2 + 2*A*b*c*d*f^2 + B*a^2*f^3
- 2*A*a*b*f^3)*log(f*x^2 + d)/(c^4*d^4 + 2*b^2*c^2*d^3*f - 4*a*c^3*d^3*f + b^4*d^2*f^2 - 4*a*b^2*c*d^2*f^2 +
6*a^2*c^2*d^2*f^2 + 2*a^2*b^2*d*f^3 - 4*a^3*c*d*f^3 + a^4*f^4) - (2*B*b*c*d^2*f^2 - A*c^2*d^2*f^2 - 2*B*a*b*d*
f^3 + A*b^2*d*f^3 + 2*A*a*c*d*f^3 - A*a^2*f^4)*arctan(f*x/sqrt(d*f))/((c^4*d^4 + 2*b^2*c^2*d^3*f - 4*a*c^3*d^3
*f + b^4*d^2*f^2 - 4*a*b^2*c*d^2*f^2 + 6*a^2*c^2*d^2*f^2 + 2*a^2*b^2*d*f^3 - 4*a^3*c*d*f^3 + a^4*f^4)*sqrt(d*f
)) + (2*B*b*c^4*d^3 - 4*A*c^5*d^3 + 5*B*b^3*c^2*d^2*f - 14*B*a*b*c^3*d^2*f - 8*A*b^2*c^3*d^2*f + 20*A*a*c^4*d^
2*f + B*b^5*d*f^2 - 4*B*a*b^3*c*d*f^2 - 2*A*b^4*c*d*f^2 + 6*B*a^2*b*c^2*d*f^2 + 12*A*a*b^2*c^2*d*f^2 - 28*A*a^
2*c^3*d*f^2 - B*a^2*b^3*f^3 + 2*A*a*b^4*f^3 + 6*B*a^3*b*c*f^3 - 12*A*a^2*b^2*c*f^3 + 12*A*a^3*c^2*f^3)*arctan(
(2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^4*d^4 - 4*a*c^5*d^4 + 2*b^4*c^2*d^3*f - 12*a*b^2*c^3*d^3*f + 16*a^2*c^
4*d^3*f + b^6*d^2*f^2 - 8*a*b^4*c*d^2*f^2 + 22*a^2*b^2*c^2*d^2*f^2 - 24*a^3*c^3*d^2*f^2 + 2*a^2*b^4*d*f^3 - 12
*a^3*b^2*c*d*f^3 + 16*a^4*c^2*d*f^3 + a^4*b^2*f^4 - 4*a^5*c*f^4)*sqrt(-b^2 + 4*a*c)) + (2*B*a*c^4*d^3 - A*b*c^
4*d^3 + 3*B*a*b^2*c^2*d^2*f - 2*A*b^3*c^2*d^2*f - 6*B*a^2*c^3*d^2*f + 5*A*a*b*c^3*d^2*f + B*a*b^4*d*f^2 - A*b^
5*d*f^2 - 4*B*a^2*b^2*c*d*f^2 + 5*A*a*b^3*c*d*f^2 + 6*B*a^3*c^2*d*f^2 - 7*A*a^2*b*c^2*d*f^2 + B*a^3*b^2*f^3 -
A*a^2*b^3*f^3 - 2*B*a^4*c*f^3 + 3*A*a^3*b*c*f^3 + (B*b*c^4*d^3 - 2*A*c^5*d^3 + B*b^3*c^2*d^2*f - B*a*b*c^3*d^2
*f - 3*A*b^2*c^3*d^2*f + 6*A*a*c^4*d^2*f + B*a*b^3*c*d*f^2 - A*b^4*c*d*f^2 - B*a^2*b*c^2*d*f^2 + 4*A*a*b^2*c^2
*d*f^2 - 6*A*a^2*c^3*d*f^2 + B*a^3*b*c*f^3 - A*a^2*b^2*c*f^3 + 2*A*a^3*c^2*f^3)*x)/((c^2*d^2 + b^2*d*f - 2*a*c
*d*f + a^2*f^2)^2*(c*x^2 + b*x + a)*(b^2 - 4*a*c))
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maple [B] time = 0.06, size = 9311, normalized size = 15.62 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/(c*x^2+b*x+a)^2/(f*x^2+d),x)
[Out]
result too large to display
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(c*x^2+b*x+a)^2/(f*x^2+d),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?
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mupad [B] time = 7.53, size = 23006, normalized size = 38.60
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x)/((d + f*x^2)*(a + b*x + c*x^2)^2),x)
[Out]
((A*b^3*f + A*b*c^2*d - 2*B*a*c^2*d - B*a*b^2*f + 2*B*a^2*c*f - 3*A*a*b*c*f)/((4*a*c - b^2)*(a^2*f^2 + c^2*d^2
+ b^2*d*f - 2*a*c*d*f)) - (x*(2*A*a*c^2*f - 2*A*c^3*d + B*b*c^2*d - A*b^2*c*f + B*a*b*c*f))/((4*a*c - b^2)*(a
^2*f^2 + c^2*d^2 + b^2*d*f - 2*a*c*d*f)))/(a + b*x + c*x^2) + symsum(log((x*(4*A^3*b^3*c^4*f^6 + 16*B^3*a^3*c^
4*f^6 - 3*B^3*a^2*b^2*c^3*f^6 + B^3*b^2*c^5*d^2*f^4 - 16*A^3*a*b*c^5*f^6 + 20*A^2*B*a^2*c^5*f^6 - 3*A^2*B*b^4*
c^3*f^6 + 4*A^2*B*c^7*d^2*f^4 - 16*B^3*a^2*c^5*d*f^5 + 6*B^3*a*b^2*c^4*d*f^5 - 24*A^2*B*a*c^6*d*f^5 + 6*A*B^2*
a*b^3*c^3*f^6 - 28*A*B^2*a^2*b*c^4*f^6 + 8*A^2*B*a*b^2*c^4*f^6 - 4*A*B^2*b*c^6*d^2*f^4 - 6*A*B^2*b^3*c^4*d*f^5
+ 8*A^2*B*b^2*c^5*d*f^5 + 16*A*B^2*a*b*c^5*d*f^5))/(16*a^2*c^6*d^4 + a^4*b^4*f^4 + b^4*c^4*d^4 + 16*a^6*c^2*f
^4 + b^8*d^2*f^2 - 8*a*b^2*c^5*d^4 - 8*a^5*b^2*c*f^4 + 2*a^2*b^6*d*f^3 - 64*a^3*c^5*d^3*f - 64*a^5*c^3*d*f^3 +
2*b^6*c^2*d^3*f + 96*a^4*c^4*d^2*f^2 + 54*a^2*b^4*c^2*d^2*f^2 - 112*a^3*b^2*c^3*d^2*f^2 - 20*a*b^4*c^3*d^3*f
- 12*a*b^6*c*d^2*f^2 - 20*a^3*b^4*c*d*f^3 + 64*a^2*b^2*c^4*d^3*f + 64*a^4*b^2*c^2*d*f^3) - root(2560*a^3*b^2*c
^9*d^8*f*z^4 - 1152*a^2*b^4*c^8*d^8*f*z^4 + 384*a^5*b^8*c*d^3*f^6*z^4 + 384*a*b^8*c^5*d^7*f^2*z^4 + 288*a^3*b^
10*c*d^4*f^5*z^4 + 288*a*b^10*c^3*d^6*f^3*z^4 + 224*a^7*b^6*c*d^2*f^7*z^4 - 192*a^10*b^2*c^2*d*f^8*z^4 + 224*a
*b^6*c^7*d^8*f*z^4 + 80*a*b^12*c*d^5*f^4*z^4 + 48*a^9*b^4*c*d*f^8*z^4 - 33920*a^6*b^2*c^6*d^5*f^4*z^4 + 27936*
a^5*b^4*c^5*d^5*f^4*z^4 + 26112*a^7*b^2*c^5*d^4*f^5*z^4 + 26112*a^5*b^2*c^7*d^6*f^3*z^4 - 20352*a^6*b^4*c^4*d^
4*f^5*z^4 - 20352*a^4*b^4*c^6*d^6*f^3*z^4 - 13080*a^4*b^6*c^4*d^5*f^4*z^4 - 11520*a^8*b^2*c^4*d^3*f^6*z^4 - 11
520*a^4*b^2*c^8*d^7*f^2*z^4 + 8736*a^5*b^6*c^3*d^4*f^5*z^4 + 8736*a^3*b^6*c^5*d^6*f^3*z^4 + 7488*a^7*b^4*c^3*d
^3*f^6*z^4 + 7488*a^3*b^4*c^7*d^7*f^2*z^4 + 3840*a^3*b^8*c^3*d^5*f^4*z^4 + 2560*a^9*b^2*c^3*d^2*f^7*z^4 - 2416
*a^6*b^6*c^2*d^3*f^6*z^4 - 2416*a^2*b^6*c^6*d^7*f^2*z^4 - 2160*a^4*b^8*c^2*d^4*f^5*z^4 - 2160*a^2*b^8*c^4*d^6*
f^3*z^4 - 1152*a^8*b^4*c^2*d^2*f^7*z^4 - 720*a^2*b^10*c^2*d^5*f^4*z^4 - 16*b^8*c^6*d^8*f*z^4 - 2048*a^4*c^10*d
^8*f*z^4 + 256*a^11*c^3*d*f^8*z^4 - 4*a^8*b^6*d*f^8*z^4 + 48*a*b^4*c^9*d^9*z^4 - 24*b^10*c^4*d^7*f^2*z^4 - 16*
b^12*c^2*d^6*f^3*z^4 + 17920*a^7*c^7*d^5*f^4*z^4 - 14336*a^8*c^6*d^4*f^5*z^4 - 14336*a^6*c^8*d^6*f^3*z^4 + 716
8*a^9*c^5*d^3*f^6*z^4 + 7168*a^5*c^9*d^7*f^2*z^4 - 2048*a^10*c^4*d^2*f^7*z^4 - 24*a^4*b^10*d^3*f^6*z^4 - 16*a^
6*b^8*d^2*f^7*z^4 - 16*a^2*b^12*d^4*f^5*z^4 - 192*a^2*b^2*c^10*d^9*z^4 - 4*b^14*d^5*f^4*z^4 - 4*b^6*c^8*d^9*z^
4 + 256*a^3*c^11*d^9*z^4 + 912*A*B*a^6*b*c^3*d*f^6*z^2 + 192*A*B*a^4*b^5*c*d*f^6*z^2 + 920*A*B*a^4*b^3*c^3*d^2
*f^5*z^2 - 480*A*B*a^2*b^5*c^3*d^3*f^4*z^2 - 336*A*B*a^2*b^3*c^5*d^4*f^3*z^2 - 272*A*B*a^3*b^3*c^4*d^3*f^4*z^2
+ 240*A*B*a^3*b^5*c^2*d^2*f^5*z^2 + 192*A*B*a*b*c^8*d^6*f*z^2 - 2496*A*B*a^5*b*c^4*d^2*f^5*z^2 + 1872*A*B*a^4
*b*c^5*d^3*f^4*z^2 - 744*A*B*a^5*b^3*c^2*d*f^6*z^2 - 720*A*B*a^2*b*c^7*d^5*f^2*z^2 + 504*A*B*a*b^3*c^6*d^5*f^2
*z^2 + 256*A*B*a^3*b*c^6*d^4*f^3*z^2 + 168*A*B*a*b^7*c^2*d^3*f^4*z^2 - 144*A*B*a^2*b^7*c*d^2*f^5*z^2 + 144*A*B
*a*b^5*c^4*d^4*f^3*z^2 - 56*B^2*a*b^2*c^7*d^6*f*z^2 - 36*B^2*a^5*b^4*c*d*f^6*z^2 - 16*B^2*a*b^8*c*d^3*f^4*z^2
- 164*A^2*a^3*b^6*c*d*f^6*z^2 - 16*A^2*a*b^8*c*d^2*f^5*z^2 - 96*A*B*b^5*c^5*d^5*f^2*z^2 - 24*A*B*b^7*c^3*d^4*f
^3*z^2 - 580*B^2*a^4*b^2*c^4*d^3*f^4*z^2 + 536*B^2*a^3*b^4*c^3*d^3*f^4*z^2 - 348*B^2*a^4*b^4*c^2*d^2*f^5*z^2 +
316*B^2*a^2*b^2*c^6*d^5*f^2*z^2 + 200*B^2*a^5*b^2*c^3*d^2*f^5*z^2 - 120*B^2*a^2*b^4*c^4*d^4*f^3*z^2 - 66*B^2*
a^2*b^6*c^2*d^3*f^4*z^2 - 16*B^2*a^3*b^2*c^5*d^4*f^3*z^2 + 1952*A^2*a^4*b^2*c^4*d^2*f^5*z^2 - 1792*A^2*a^3*b^2
*c^5*d^3*f^4*z^2 - 1272*A^2*a^3*b^4*c^3*d^2*f^5*z^2 + 976*A^2*a^2*b^2*c^6*d^4*f^3*z^2 + 960*A^2*a^2*b^4*c^4*d^
3*f^4*z^2 + 282*A^2*a^2*b^6*c^2*d^2*f^5*z^2 - 72*A*B*b^3*c^7*d^6*f*z^2 - 16*A*B*b^9*c*d^3*f^4*z^2 - 16*A*B*a^3
*b^7*d*f^6*z^2 + 16*A*B*a*b^9*d^2*f^5*z^2 - 180*B^2*a*b^4*c^5*d^5*f^2*z^2 + 132*B^2*a^6*b^2*c^2*d*f^6*z^2 + 10
8*B^2*a^3*b^6*c*d^2*f^5*z^2 + 20*B^2*a*b^6*c^3*d^4*f^3*z^2 - 736*A^2*a^5*b^2*c^3*d*f^6*z^2 + 624*A^2*a^4*b^4*c
^2*d*f^6*z^2 - 416*A^2*a*b^2*c^7*d^5*f^2*z^2 - 276*A^2*a*b^4*c^5*d^4*f^3*z^2 - 196*A^2*a*b^6*c^3*d^3*f^4*z^2 +
31*B^2*b^6*c^4*d^5*f^2*z^2 + 2*B^2*b^8*c^2*d^4*f^3*z^2 - 768*B^2*a^5*c^5*d^3*f^4*z^2 + 512*B^2*a^6*c^4*d^2*f^
5*z^2 + 512*B^2*a^4*c^6*d^4*f^3*z^2 - 128*B^2*a^3*c^7*d^5*f^2*z^2 + 80*A^2*b^4*c^6*d^5*f^2*z^2 + 31*A^2*b^6*c^
4*d^4*f^3*z^2 + 14*A^2*b^8*c^2*d^3*f^4*z^2 - 1152*A^2*a^3*c^7*d^4*f^3*z^2 + 1008*A^2*a^4*c^6*d^3*f^4*z^2 + 624
*A^2*a^2*c^8*d^5*f^2*z^2 - 288*A^2*a^5*c^5*d^2*f^5*z^2 - 10*B^2*a^2*b^8*d^2*f^5*z^2 - 48*A^2*a^6*b^2*c^2*f^7*z
^2 - 16*A*B*b*c^9*d^7*z^2 + 20*B^2*b^4*c^6*d^6*f*z^2 - 128*B^2*a^7*c^3*d*f^6*z^2 + 64*A^2*b^2*c^8*d^6*f*z^2 -
112*A^2*a^6*c^4*d*f^6*z^2 + 3*B^2*a^4*b^6*d*f^6*z^2 + 14*A^2*a^2*b^8*d*f^6*z^2 + 12*A^2*a^5*b^4*c*f^7*z^2 - 16
0*A^2*a*c^9*d^6*f*z^2 + 3*B^2*b^10*d^3*f^4*z^2 - A^2*b^10*d^2*f^5*z^2 + 64*A^2*a^7*c^3*f^7*z^2 + 4*B^2*b^2*c^8
*d^7*z^2 - A^2*a^4*b^6*f^7*z^2 + 16*A^2*c^10*d^7*z^2 - 160*A*B^2*a*b*c^6*d^4*f^2*z + 112*A*B^2*a^4*b*c^3*d*f^5
*z - 24*A*B^2*a^2*b^5*c*d*f^5*z + 480*A^2*B*a^2*b^2*c^4*d^2*f^4*z - 176*A*B^2*a^2*b^3*c^3*d^2*f^4*z - 10*A^2*B
*a*b^6*c*d*f^5*z + 384*A*B^2*a^2*b*c^5*d^3*f^3*z - 352*A*B^2*a^3*b*c^4*d^2*f^4*z - 288*A^2*B*a*b^2*c^5*d^3*f^3
*z - 160*A^2*B*a^3*b^2*c^3*d*f^5*z - 148*A^2*B*a*b^4*c^3*d^2*f^4*z + 112*A*B^2*a*b^3*c^4*d^3*f^3*z + 72*A^2*B*
a^2*b^4*c^2*d*f^5*z + 72*A*B^2*a*b^5*c^2*d^2*f^4*z + 48*A*B^2*a^3*b^3*c^2*d*f^5*z + 48*B^3*a*b^2*c^5*d^4*f^2*z
- 36*B^3*a^4*b^2*c^2*d*f^5*z - 4*B^3*a*b^4*c^3*d^3*f^3*z - 480*A^3*a^2*b*c^5*d^2*f^4*z - 160*A^3*a^2*b^3*c^3*
d*f^5*z + 128*A^3*a*b^3*c^4*d^2*f^4*z + 112*A^2*B*b^4*c^4*d^3*f^3*z - 64*A*B^2*b^5*c^3*d^3*f^3*z + 16*A^2*B*b^
2*c^6*d^4*f^2*z + 16*A*B^2*b^3*c^5*d^4*f^2*z - A^2*B*b^6*c^2*d^2*f^4*z + 448*A^2*B*a^3*c^5*d^2*f^4*z - 352*A^2
*B*a^2*c^6*d^3*f^3*z - 48*A^2*B*a^4*b^2*c^2*f^6*z + 12*B^3*a^3*b^4*c*d*f^5*z - 10*B^3*a*b^6*c*d^2*f^4*z + 416*
A^3*a^3*b*c^4*d*f^5*z + 224*A^3*a*b*c^6*d^3*f^3*z + 24*A^3*a*b^5*c^2*d*f^5*z - 2*A*B^2*b^7*c*d^2*f^4*z - 272*A
^2*B*a^4*c^4*d*f^5*z + 128*A^2*B*a*c^7*d^4*f^2*z + 12*A^2*B*a^3*b^4*c*f^6*z - 120*B^3*a^2*b^2*c^4*d^3*f^3*z +
112*B^3*a^3*b^2*c^3*d^2*f^4*z + 16*A*B^2*b*c^7*d^5*f*z + 2*A*B^2*a*b^7*d*f^5*z - 2*A^3*b^7*c*d*f^5*z - 16*A^2*
B*c^8*d^5*f*z + 11*B^3*b^6*c^2*d^3*f^3*z - 8*B^3*b^4*c^4*d^4*f^2*z - 64*A^3*b^3*c^5*d^3*f^3*z + 96*A^3*a^3*b^3
*c^2*f^6*z - 4*B^3*b^2*c^6*d^5*f*z - 32*A^3*b*c^7*d^4*f^2*z - B^3*a^2*b^6*d*f^5*z - 128*A^3*a^4*b*c^3*f^6*z -
24*A^3*a^2*b^5*c*f^6*z + 64*A^2*B*a^5*c^3*f^6*z - A^2*B*a^2*b^6*f^6*z + A^2*B*b^8*d*f^5*z + 2*A^3*a*b^7*f^6*z
+ B^3*b^8*d^2*f^4*z + 32*A^3*B*a*b*c^4*d*f^4 - 18*A^2*B^2*a*b^2*c^3*d*f^4 + 32*A*B^3*a*b*c^4*d^2*f^3 - 28*A*B^
3*a^2*b*c^3*d*f^4 + 6*A*B^3*a*b^3*c^2*d*f^4 - 10*A^3*B*b^3*c^3*d*f^4 - 4*A^3*B*b*c^5*d^2*f^3 - 4*A*B^3*b*c^5*d
^3*f^2 - 28*A^3*B*a^2*b*c^3*f^5 + 6*A^3*B*a*b^3*c^2*f^5 + 9*A^2*B^2*b^2*c^4*d^2*f^3 - 3*A^2*B^2*a^2*b^2*c^2*f^
5 - 10*B^4*a*b^2*c^3*d^2*f^3 - 3*B^4*a^2*b^2*c^2*d*f^4 - 10*A*B^3*b^3*c^3*d^2*f^3 + 3*A^2*B^2*b^4*c^2*d*f^4 +
36*A^2*B^2*a^2*c^4*d*f^4 - 24*A^2*B^2*a*c^5*d^2*f^3 + 4*A^2*B^2*c^6*d^3*f^2 + 16*A^2*B^2*a^3*c^3*f^5 + 16*B^4*
a^3*c^3*d*f^4 + 8*A^4*b^2*c^4*d*f^4 - 8*A^4*a*b^2*c^3*f^5 - 24*A^4*a*c^5*d*f^4 + 3*B^4*b^4*c^2*d^2*f^3 + 4*A^4
*c^6*d^2*f^3 + 36*A^4*a^2*c^4*f^5 + B^4*b^2*c^4*d^3*f^2, z, k)*(root(2560*a^3*b^2*c^9*d^8*f*z^4 - 1152*a^2*b^4
*c^8*d^8*f*z^4 + 384*a^5*b^8*c*d^3*f^6*z^4 + 384*a*b^8*c^5*d^7*f^2*z^4 + 288*a^3*b^10*c*d^4*f^5*z^4 + 288*a*b^
10*c^3*d^6*f^3*z^4 + 224*a^7*b^6*c*d^2*f^7*z^4 - 192*a^10*b^2*c^2*d*f^8*z^4 + 224*a*b^6*c^7*d^8*f*z^4 + 80*a*b
^12*c*d^5*f^4*z^4 + 48*a^9*b^4*c*d*f^8*z^4 - 33920*a^6*b^2*c^6*d^5*f^4*z^4 + 27936*a^5*b^4*c^5*d^5*f^4*z^4 + 2
6112*a^7*b^2*c^5*d^4*f^5*z^4 + 26112*a^5*b^2*c^7*d^6*f^3*z^4 - 20352*a^6*b^4*c^4*d^4*f^5*z^4 - 20352*a^4*b^4*c
^6*d^6*f^3*z^4 - 13080*a^4*b^6*c^4*d^5*f^4*z^4 - 11520*a^8*b^2*c^4*d^3*f^6*z^4 - 11520*a^4*b^2*c^8*d^7*f^2*z^4
+ 8736*a^5*b^6*c^3*d^4*f^5*z^4 + 8736*a^3*b^6*c^5*d^6*f^3*z^4 + 7488*a^7*b^4*c^3*d^3*f^6*z^4 + 7488*a^3*b^4*c
^7*d^7*f^2*z^4 + 3840*a^3*b^8*c^3*d^5*f^4*z^4 + 2560*a^9*b^2*c^3*d^2*f^7*z^4 - 2416*a^6*b^6*c^2*d^3*f^6*z^4 -
2416*a^2*b^6*c^6*d^7*f^2*z^4 - 2160*a^4*b^8*c^2*d^4*f^5*z^4 - 2160*a^2*b^8*c^4*d^6*f^3*z^4 - 1152*a^8*b^4*c^2*
d^2*f^7*z^4 - 720*a^2*b^10*c^2*d^5*f^4*z^4 - 16*b^8*c^6*d^8*f*z^4 - 2048*a^4*c^10*d^8*f*z^4 + 256*a^11*c^3*d*f
^8*z^4 - 4*a^8*b^6*d*f^8*z^4 + 48*a*b^4*c^9*d^9*z^4 - 24*b^10*c^4*d^7*f^2*z^4 - 16*b^12*c^2*d^6*f^3*z^4 + 1792
0*a^7*c^7*d^5*f^4*z^4 - 14336*a^8*c^6*d^4*f^5*z^4 - 14336*a^6*c^8*d^6*f^3*z^4 + 7168*a^9*c^5*d^3*f^6*z^4 + 716
8*a^5*c^9*d^7*f^2*z^4 - 2048*a^10*c^4*d^2*f^7*z^4 - 24*a^4*b^10*d^3*f^6*z^4 - 16*a^6*b^8*d^2*f^7*z^4 - 16*a^2*
b^12*d^4*f^5*z^4 - 192*a^2*b^2*c^10*d^9*z^4 - 4*b^14*d^5*f^4*z^4 - 4*b^6*c^8*d^9*z^4 + 256*a^3*c^11*d^9*z^4 +
912*A*B*a^6*b*c^3*d*f^6*z^2 + 192*A*B*a^4*b^5*c*d*f^6*z^2 + 920*A*B*a^4*b^3*c^3*d^2*f^5*z^2 - 480*A*B*a^2*b^5*
c^3*d^3*f^4*z^2 - 336*A*B*a^2*b^3*c^5*d^4*f^3*z^2 - 272*A*B*a^3*b^3*c^4*d^3*f^4*z^2 + 240*A*B*a^3*b^5*c^2*d^2*
f^5*z^2 + 192*A*B*a*b*c^8*d^6*f*z^2 - 2496*A*B*a^5*b*c^4*d^2*f^5*z^2 + 1872*A*B*a^4*b*c^5*d^3*f^4*z^2 - 744*A*
B*a^5*b^3*c^2*d*f^6*z^2 - 720*A*B*a^2*b*c^7*d^5*f^2*z^2 + 504*A*B*a*b^3*c^6*d^5*f^2*z^2 + 256*A*B*a^3*b*c^6*d^
4*f^3*z^2 + 168*A*B*a*b^7*c^2*d^3*f^4*z^2 - 144*A*B*a^2*b^7*c*d^2*f^5*z^2 + 144*A*B*a*b^5*c^4*d^4*f^3*z^2 - 56
*B^2*a*b^2*c^7*d^6*f*z^2 - 36*B^2*a^5*b^4*c*d*f^6*z^2 - 16*B^2*a*b^8*c*d^3*f^4*z^2 - 164*A^2*a^3*b^6*c*d*f^6*z
^2 - 16*A^2*a*b^8*c*d^2*f^5*z^2 - 96*A*B*b^5*c^5*d^5*f^2*z^2 - 24*A*B*b^7*c^3*d^4*f^3*z^2 - 580*B^2*a^4*b^2*c^
4*d^3*f^4*z^2 + 536*B^2*a^3*b^4*c^3*d^3*f^4*z^2 - 348*B^2*a^4*b^4*c^2*d^2*f^5*z^2 + 316*B^2*a^2*b^2*c^6*d^5*f^
2*z^2 + 200*B^2*a^5*b^2*c^3*d^2*f^5*z^2 - 120*B^2*a^2*b^4*c^4*d^4*f^3*z^2 - 66*B^2*a^2*b^6*c^2*d^3*f^4*z^2 - 1
6*B^2*a^3*b^2*c^5*d^4*f^3*z^2 + 1952*A^2*a^4*b^2*c^4*d^2*f^5*z^2 - 1792*A^2*a^3*b^2*c^5*d^3*f^4*z^2 - 1272*A^2
*a^3*b^4*c^3*d^2*f^5*z^2 + 976*A^2*a^2*b^2*c^6*d^4*f^3*z^2 + 960*A^2*a^2*b^4*c^4*d^3*f^4*z^2 + 282*A^2*a^2*b^6
*c^2*d^2*f^5*z^2 - 72*A*B*b^3*c^7*d^6*f*z^2 - 16*A*B*b^9*c*d^3*f^4*z^2 - 16*A*B*a^3*b^7*d*f^6*z^2 + 16*A*B*a*b
^9*d^2*f^5*z^2 - 180*B^2*a*b^4*c^5*d^5*f^2*z^2 + 132*B^2*a^6*b^2*c^2*d*f^6*z^2 + 108*B^2*a^3*b^6*c*d^2*f^5*z^2
+ 20*B^2*a*b^6*c^3*d^4*f^3*z^2 - 736*A^2*a^5*b^2*c^3*d*f^6*z^2 + 624*A^2*a^4*b^4*c^2*d*f^6*z^2 - 416*A^2*a*b^
2*c^7*d^5*f^2*z^2 - 276*A^2*a*b^4*c^5*d^4*f^3*z^2 - 196*A^2*a*b^6*c^3*d^3*f^4*z^2 + 31*B^2*b^6*c^4*d^5*f^2*z^2
+ 2*B^2*b^8*c^2*d^4*f^3*z^2 - 768*B^2*a^5*c^5*d^3*f^4*z^2 + 512*B^2*a^6*c^4*d^2*f^5*z^2 + 512*B^2*a^4*c^6*d^4
*f^3*z^2 - 128*B^2*a^3*c^7*d^5*f^2*z^2 + 80*A^2*b^4*c^6*d^5*f^2*z^2 + 31*A^2*b^6*c^4*d^4*f^3*z^2 + 14*A^2*b^8*
c^2*d^3*f^4*z^2 - 1152*A^2*a^3*c^7*d^4*f^3*z^2 + 1008*A^2*a^4*c^6*d^3*f^4*z^2 + 624*A^2*a^2*c^8*d^5*f^2*z^2 -
288*A^2*a^5*c^5*d^2*f^5*z^2 - 10*B^2*a^2*b^8*d^2*f^5*z^2 - 48*A^2*a^6*b^2*c^2*f^7*z^2 - 16*A*B*b*c^9*d^7*z^2 +
20*B^2*b^4*c^6*d^6*f*z^2 - 128*B^2*a^7*c^3*d*f^6*z^2 + 64*A^2*b^2*c^8*d^6*f*z^2 - 112*A^2*a^6*c^4*d*f^6*z^2 +
3*B^2*a^4*b^6*d*f^6*z^2 + 14*A^2*a^2*b^8*d*f^6*z^2 + 12*A^2*a^5*b^4*c*f^7*z^2 - 160*A^2*a*c^9*d^6*f*z^2 + 3*B
^2*b^10*d^3*f^4*z^2 - A^2*b^10*d^2*f^5*z^2 + 64*A^2*a^7*c^3*f^7*z^2 + 4*B^2*b^2*c^8*d^7*z^2 - A^2*a^4*b^6*f^7*
z^2 + 16*A^2*c^10*d^7*z^2 - 160*A*B^2*a*b*c^6*d^4*f^2*z + 112*A*B^2*a^4*b*c^3*d*f^5*z - 24*A*B^2*a^2*b^5*c*d*f
^5*z + 480*A^2*B*a^2*b^2*c^4*d^2*f^4*z - 176*A*B^2*a^2*b^3*c^3*d^2*f^4*z - 10*A^2*B*a*b^6*c*d*f^5*z + 384*A*B^
2*a^2*b*c^5*d^3*f^3*z - 352*A*B^2*a^3*b*c^4*d^2*f^4*z - 288*A^2*B*a*b^2*c^5*d^3*f^3*z - 160*A^2*B*a^3*b^2*c^3*
d*f^5*z - 148*A^2*B*a*b^4*c^3*d^2*f^4*z + 112*A*B^2*a*b^3*c^4*d^3*f^3*z + 72*A^2*B*a^2*b^4*c^2*d*f^5*z + 72*A*
B^2*a*b^5*c^2*d^2*f^4*z + 48*A*B^2*a^3*b^3*c^2*d*f^5*z + 48*B^3*a*b^2*c^5*d^4*f^2*z - 36*B^3*a^4*b^2*c^2*d*f^5
*z - 4*B^3*a*b^4*c^3*d^3*f^3*z - 480*A^3*a^2*b*c^5*d^2*f^4*z - 160*A^3*a^2*b^3*c^3*d*f^5*z + 128*A^3*a*b^3*c^4
*d^2*f^4*z + 112*A^2*B*b^4*c^4*d^3*f^3*z - 64*A*B^2*b^5*c^3*d^3*f^3*z + 16*A^2*B*b^2*c^6*d^4*f^2*z + 16*A*B^2*
b^3*c^5*d^4*f^2*z - A^2*B*b^6*c^2*d^2*f^4*z + 448*A^2*B*a^3*c^5*d^2*f^4*z - 352*A^2*B*a^2*c^6*d^3*f^3*z - 48*A
^2*B*a^4*b^2*c^2*f^6*z + 12*B^3*a^3*b^4*c*d*f^5*z - 10*B^3*a*b^6*c*d^2*f^4*z + 416*A^3*a^3*b*c^4*d*f^5*z + 224
*A^3*a*b*c^6*d^3*f^3*z + 24*A^3*a*b^5*c^2*d*f^5*z - 2*A*B^2*b^7*c*d^2*f^4*z - 272*A^2*B*a^4*c^4*d*f^5*z + 128*
A^2*B*a*c^7*d^4*f^2*z + 12*A^2*B*a^3*b^4*c*f^6*z - 120*B^3*a^2*b^2*c^4*d^3*f^3*z + 112*B^3*a^3*b^2*c^3*d^2*f^4
*z + 16*A*B^2*b*c^7*d^5*f*z + 2*A*B^2*a*b^7*d*f^5*z - 2*A^3*b^7*c*d*f^5*z - 16*A^2*B*c^8*d^5*f*z + 11*B^3*b^6*
c^2*d^3*f^3*z - 8*B^3*b^4*c^4*d^4*f^2*z - 64*A^3*b^3*c^5*d^3*f^3*z + 96*A^3*a^3*b^3*c^2*f^6*z - 4*B^3*b^2*c^6*
d^5*f*z - 32*A^3*b*c^7*d^4*f^2*z - B^3*a^2*b^6*d*f^5*z - 128*A^3*a^4*b*c^3*f^6*z - 24*A^3*a^2*b^5*c*f^6*z + 64
*A^2*B*a^5*c^3*f^6*z - A^2*B*a^2*b^6*f^6*z + A^2*B*b^8*d*f^5*z + 2*A^3*a*b^7*f^6*z + B^3*b^8*d^2*f^4*z + 32*A^
3*B*a*b*c^4*d*f^4 - 18*A^2*B^2*a*b^2*c^3*d*f^4 + 32*A*B^3*a*b*c^4*d^2*f^3 - 28*A*B^3*a^2*b*c^3*d*f^4 + 6*A*B^3
*a*b^3*c^2*d*f^4 - 10*A^3*B*b^3*c^3*d*f^4 - 4*A^3*B*b*c^5*d^2*f^3 - 4*A*B^3*b*c^5*d^3*f^2 - 28*A^3*B*a^2*b*c^3
*f^5 + 6*A^3*B*a*b^3*c^2*f^5 + 9*A^2*B^2*b^2*c^4*d^2*f^3 - 3*A^2*B^2*a^2*b^2*c^2*f^5 - 10*B^4*a*b^2*c^3*d^2*f^
3 - 3*B^4*a^2*b^2*c^2*d*f^4 - 10*A*B^3*b^3*c^3*d^2*f^3 + 3*A^2*B^2*b^4*c^2*d*f^4 + 36*A^2*B^2*a^2*c^4*d*f^4 -
24*A^2*B^2*a*c^5*d^2*f^3 + 4*A^2*B^2*c^6*d^3*f^2 + 16*A^2*B^2*a^3*c^3*f^5 + 16*B^4*a^3*c^3*d*f^4 + 8*A^4*b^2*c
^4*d*f^4 - 8*A^4*a*b^2*c^3*f^5 - 24*A^4*a*c^5*d*f^4 + 3*B^4*b^4*c^2*d^2*f^3 + 4*A^4*c^6*d^2*f^3 + 36*A^4*a^2*c
^4*f^5 + B^4*b^2*c^4*d^3*f^2, z, k)*(root(2560*a^3*b^2*c^9*d^8*f*z^4 - 1152*a^2*b^4*c^8*d^8*f*z^4 + 384*a^5*b^
8*c*d^3*f^6*z^4 + 384*a*b^8*c^5*d^7*f^2*z^4 + 288*a^3*b^10*c*d^4*f^5*z^4 + 288*a*b^10*c^3*d^6*f^3*z^4 + 224*a^
7*b^6*c*d^2*f^7*z^4 - 192*a^10*b^2*c^2*d*f^8*z^4 + 224*a*b^6*c^7*d^8*f*z^4 + 80*a*b^12*c*d^5*f^4*z^4 + 48*a^9*
b^4*c*d*f^8*z^4 - 33920*a^6*b^2*c^6*d^5*f^4*z^4 + 27936*a^5*b^4*c^5*d^5*f^4*z^4 + 26112*a^7*b^2*c^5*d^4*f^5*z^
4 + 26112*a^5*b^2*c^7*d^6*f^3*z^4 - 20352*a^6*b^4*c^4*d^4*f^5*z^4 - 20352*a^4*b^4*c^6*d^6*f^3*z^4 - 13080*a^4*
b^6*c^4*d^5*f^4*z^4 - 11520*a^8*b^2*c^4*d^3*f^6*z^4 - 11520*a^4*b^2*c^8*d^7*f^2*z^4 + 8736*a^5*b^6*c^3*d^4*f^5
*z^4 + 8736*a^3*b^6*c^5*d^6*f^3*z^4 + 7488*a^7*b^4*c^3*d^3*f^6*z^4 + 7488*a^3*b^4*c^7*d^7*f^2*z^4 + 3840*a^3*b
^8*c^3*d^5*f^4*z^4 + 2560*a^9*b^2*c^3*d^2*f^7*z^4 - 2416*a^6*b^6*c^2*d^3*f^6*z^4 - 2416*a^2*b^6*c^6*d^7*f^2*z^
4 - 2160*a^4*b^8*c^2*d^4*f^5*z^4 - 2160*a^2*b^8*c^4*d^6*f^3*z^4 - 1152*a^8*b^4*c^2*d^2*f^7*z^4 - 720*a^2*b^10*
c^2*d^5*f^4*z^4 - 16*b^8*c^6*d^8*f*z^4 - 2048*a^4*c^10*d^8*f*z^4 + 256*a^11*c^3*d*f^8*z^4 - 4*a^8*b^6*d*f^8*z^
4 + 48*a*b^4*c^9*d^9*z^4 - 24*b^10*c^4*d^7*f^2*z^4 - 16*b^12*c^2*d^6*f^3*z^4 + 17920*a^7*c^7*d^5*f^4*z^4 - 143
36*a^8*c^6*d^4*f^5*z^4 - 14336*a^6*c^8*d^6*f^3*z^4 + 7168*a^9*c^5*d^3*f^6*z^4 + 7168*a^5*c^9*d^7*f^2*z^4 - 204
8*a^10*c^4*d^2*f^7*z^4 - 24*a^4*b^10*d^3*f^6*z^4 - 16*a^6*b^8*d^2*f^7*z^4 - 16*a^2*b^12*d^4*f^5*z^4 - 192*a^2*
b^2*c^10*d^9*z^4 - 4*b^14*d^5*f^4*z^4 - 4*b^6*c^8*d^9*z^4 + 256*a^3*c^11*d^9*z^4 + 912*A*B*a^6*b*c^3*d*f^6*z^2
+ 192*A*B*a^4*b^5*c*d*f^6*z^2 + 920*A*B*a^4*b^3*c^3*d^2*f^5*z^2 - 480*A*B*a^2*b^5*c^3*d^3*f^4*z^2 - 336*A*B*a
^2*b^3*c^5*d^4*f^3*z^2 - 272*A*B*a^3*b^3*c^4*d^3*f^4*z^2 + 240*A*B*a^3*b^5*c^2*d^2*f^5*z^2 + 192*A*B*a*b*c^8*d
^6*f*z^2 - 2496*A*B*a^5*b*c^4*d^2*f^5*z^2 + 1872*A*B*a^4*b*c^5*d^3*f^4*z^2 - 744*A*B*a^5*b^3*c^2*d*f^6*z^2 - 7
20*A*B*a^2*b*c^7*d^5*f^2*z^2 + 504*A*B*a*b^3*c^6*d^5*f^2*z^2 + 256*A*B*a^3*b*c^6*d^4*f^3*z^2 + 168*A*B*a*b^7*c
^2*d^3*f^4*z^2 - 144*A*B*a^2*b^7*c*d^2*f^5*z^2 + 144*A*B*a*b^5*c^4*d^4*f^3*z^2 - 56*B^2*a*b^2*c^7*d^6*f*z^2 -
36*B^2*a^5*b^4*c*d*f^6*z^2 - 16*B^2*a*b^8*c*d^3*f^4*z^2 - 164*A^2*a^3*b^6*c*d*f^6*z^2 - 16*A^2*a*b^8*c*d^2*f^5
*z^2 - 96*A*B*b^5*c^5*d^5*f^2*z^2 - 24*A*B*b^7*c^3*d^4*f^3*z^2 - 580*B^2*a^4*b^2*c^4*d^3*f^4*z^2 + 536*B^2*a^3
*b^4*c^3*d^3*f^4*z^2 - 348*B^2*a^4*b^4*c^2*d^2*f^5*z^2 + 316*B^2*a^2*b^2*c^6*d^5*f^2*z^2 + 200*B^2*a^5*b^2*c^3
*d^2*f^5*z^2 - 120*B^2*a^2*b^4*c^4*d^4*f^3*z^2 - 66*B^2*a^2*b^6*c^2*d^3*f^4*z^2 - 16*B^2*a^3*b^2*c^5*d^4*f^3*z
^2 + 1952*A^2*a^4*b^2*c^4*d^2*f^5*z^2 - 1792*A^2*a^3*b^2*c^5*d^3*f^4*z^2 - 1272*A^2*a^3*b^4*c^3*d^2*f^5*z^2 +
976*A^2*a^2*b^2*c^6*d^4*f^3*z^2 + 960*A^2*a^2*b^4*c^4*d^3*f^4*z^2 + 282*A^2*a^2*b^6*c^2*d^2*f^5*z^2 - 72*A*B*b
^3*c^7*d^6*f*z^2 - 16*A*B*b^9*c*d^3*f^4*z^2 - 16*A*B*a^3*b^7*d*f^6*z^2 + 16*A*B*a*b^9*d^2*f^5*z^2 - 180*B^2*a*
b^4*c^5*d^5*f^2*z^2 + 132*B^2*a^6*b^2*c^2*d*f^6*z^2 + 108*B^2*a^3*b^6*c*d^2*f^5*z^2 + 20*B^2*a*b^6*c^3*d^4*f^3
*z^2 - 736*A^2*a^5*b^2*c^3*d*f^6*z^2 + 624*A^2*a^4*b^4*c^2*d*f^6*z^2 - 416*A^2*a*b^2*c^7*d^5*f^2*z^2 - 276*A^2
*a*b^4*c^5*d^4*f^3*z^2 - 196*A^2*a*b^6*c^3*d^3*f^4*z^2 + 31*B^2*b^6*c^4*d^5*f^2*z^2 + 2*B^2*b^8*c^2*d^4*f^3*z^
2 - 768*B^2*a^5*c^5*d^3*f^4*z^2 + 512*B^2*a^6*c^4*d^2*f^5*z^2 + 512*B^2*a^4*c^6*d^4*f^3*z^2 - 128*B^2*a^3*c^7*
d^5*f^2*z^2 + 80*A^2*b^4*c^6*d^5*f^2*z^2 + 31*A^2*b^6*c^4*d^4*f^3*z^2 + 14*A^2*b^8*c^2*d^3*f^4*z^2 - 1152*A^2*
a^3*c^7*d^4*f^3*z^2 + 1008*A^2*a^4*c^6*d^3*f^4*z^2 + 624*A^2*a^2*c^8*d^5*f^2*z^2 - 288*A^2*a^5*c^5*d^2*f^5*z^2
- 10*B^2*a^2*b^8*d^2*f^5*z^2 - 48*A^2*a^6*b^2*c^2*f^7*z^2 - 16*A*B*b*c^9*d^7*z^2 + 20*B^2*b^4*c^6*d^6*f*z^2 -
128*B^2*a^7*c^3*d*f^6*z^2 + 64*A^2*b^2*c^8*d^6*f*z^2 - 112*A^2*a^6*c^4*d*f^6*z^2 + 3*B^2*a^4*b^6*d*f^6*z^2 +
14*A^2*a^2*b^8*d*f^6*z^2 + 12*A^2*a^5*b^4*c*f^7*z^2 - 160*A^2*a*c^9*d^6*f*z^2 + 3*B^2*b^10*d^3*f^4*z^2 - A^2*b
^10*d^2*f^5*z^2 + 64*A^2*a^7*c^3*f^7*z^2 + 4*B^2*b^2*c^8*d^7*z^2 - A^2*a^4*b^6*f^7*z^2 + 16*A^2*c^10*d^7*z^2 -
160*A*B^2*a*b*c^6*d^4*f^2*z + 112*A*B^2*a^4*b*c^3*d*f^5*z - 24*A*B^2*a^2*b^5*c*d*f^5*z + 480*A^2*B*a^2*b^2*c^
4*d^2*f^4*z - 176*A*B^2*a^2*b^3*c^3*d^2*f^4*z - 10*A^2*B*a*b^6*c*d*f^5*z + 384*A*B^2*a^2*b*c^5*d^3*f^3*z - 352
*A*B^2*a^3*b*c^4*d^2*f^4*z - 288*A^2*B*a*b^2*c^5*d^3*f^3*z - 160*A^2*B*a^3*b^2*c^3*d*f^5*z - 148*A^2*B*a*b^4*c
^3*d^2*f^4*z + 112*A*B^2*a*b^3*c^4*d^3*f^3*z + 72*A^2*B*a^2*b^4*c^2*d*f^5*z + 72*A*B^2*a*b^5*c^2*d^2*f^4*z + 4
8*A*B^2*a^3*b^3*c^2*d*f^5*z + 48*B^3*a*b^2*c^5*d^4*f^2*z - 36*B^3*a^4*b^2*c^2*d*f^5*z - 4*B^3*a*b^4*c^3*d^3*f^
3*z - 480*A^3*a^2*b*c^5*d^2*f^4*z - 160*A^3*a^2*b^3*c^3*d*f^5*z + 128*A^3*a*b^3*c^4*d^2*f^4*z + 112*A^2*B*b^4*
c^4*d^3*f^3*z - 64*A*B^2*b^5*c^3*d^3*f^3*z + 16*A^2*B*b^2*c^6*d^4*f^2*z + 16*A*B^2*b^3*c^5*d^4*f^2*z - A^2*B*b
^6*c^2*d^2*f^4*z + 448*A^2*B*a^3*c^5*d^2*f^4*z - 352*A^2*B*a^2*c^6*d^3*f^3*z - 48*A^2*B*a^4*b^2*c^2*f^6*z + 12
*B^3*a^3*b^4*c*d*f^5*z - 10*B^3*a*b^6*c*d^2*f^4*z + 416*A^3*a^3*b*c^4*d*f^5*z + 224*A^3*a*b*c^6*d^3*f^3*z + 24
*A^3*a*b^5*c^2*d*f^5*z - 2*A*B^2*b^7*c*d^2*f^4*z - 272*A^2*B*a^4*c^4*d*f^5*z + 128*A^2*B*a*c^7*d^4*f^2*z + 12*
A^2*B*a^3*b^4*c*f^6*z - 120*B^3*a^2*b^2*c^4*d^3*f^3*z + 112*B^3*a^3*b^2*c^3*d^2*f^4*z + 16*A*B^2*b*c^7*d^5*f*z
+ 2*A*B^2*a*b^7*d*f^5*z - 2*A^3*b^7*c*d*f^5*z - 16*A^2*B*c^8*d^5*f*z + 11*B^3*b^6*c^2*d^3*f^3*z - 8*B^3*b^4*c
^4*d^4*f^2*z - 64*A^3*b^3*c^5*d^3*f^3*z + 96*A^3*a^3*b^3*c^2*f^6*z - 4*B^3*b^2*c^6*d^5*f*z - 32*A^3*b*c^7*d^4*
f^2*z - B^3*a^2*b^6*d*f^5*z - 128*A^3*a^4*b*c^3*f^6*z - 24*A^3*a^2*b^5*c*f^6*z + 64*A^2*B*a^5*c^3*f^6*z - A^2*
B*a^2*b^6*f^6*z + A^2*B*b^8*d*f^5*z + 2*A^3*a*b^7*f^6*z + B^3*b^8*d^2*f^4*z + 32*A^3*B*a*b*c^4*d*f^4 - 18*A^2*
B^2*a*b^2*c^3*d*f^4 + 32*A*B^3*a*b*c^4*d^2*f^3 - 28*A*B^3*a^2*b*c^3*d*f^4 + 6*A*B^3*a*b^3*c^2*d*f^4 - 10*A^3*B
*b^3*c^3*d*f^4 - 4*A^3*B*b*c^5*d^2*f^3 - 4*A*B^3*b*c^5*d^3*f^2 - 28*A^3*B*a^2*b*c^3*f^5 + 6*A^3*B*a*b^3*c^2*f^
5 + 9*A^2*B^2*b^2*c^4*d^2*f^3 - 3*A^2*B^2*a^2*b^2*c^2*f^5 - 10*B^4*a*b^2*c^3*d^2*f^3 - 3*B^4*a^2*b^2*c^2*d*f^4
- 10*A*B^3*b^3*c^3*d^2*f^3 + 3*A^2*B^2*b^4*c^2*d*f^4 + 36*A^2*B^2*a^2*c^4*d*f^4 - 24*A^2*B^2*a*c^5*d^2*f^3 +
4*A^2*B^2*c^6*d^3*f^2 + 16*A^2*B^2*a^3*c^3*f^5 + 16*B^4*a^3*c^3*d*f^4 + 8*A^4*b^2*c^4*d*f^4 - 8*A^4*a*b^2*c^3*
f^5 - 24*A^4*a*c^5*d*f^4 + 3*B^4*b^4*c^2*d^2*f^3 + 4*A^4*c^6*d^2*f^3 + 36*A^4*a^2*c^4*f^5 + B^4*b^2*c^4*d^3*f^
2, z, k)*((4*b^5*c^8*d^7*f^2 + 4*b^7*c^6*d^6*f^3 - 4*b^9*c^4*d^5*f^4 - 4*b^11*c^2*d^4*f^5 - 612*a^2*b^5*c^6*d^
5*f^4 - 712*a^2*b^7*c^4*d^4*f^5 - 132*a^2*b^9*c^2*d^3*f^6 + 1696*a^3*b^3*c^7*d^5*f^4 + 2736*a^3*b^5*c^5*d^4*f^
5 + 896*a^3*b^7*c^3*d^3*f^6 - 5120*a^4*b^3*c^6*d^4*f^5 - 3140*a^4*b^5*c^4*d^3*f^6 - 220*a^4*b^7*c^2*d^2*f^7 +
5664*a^5*b^3*c^5*d^3*f^6 + 1128*a^5*b^5*c^3*d^2*f^7 - 2560*a^6*b^3*c^4*d^2*f^7 + 8*a*b^11*c*d^3*f^6 + 8*a^5*b^
7*c*d*f^8 - 448*a^8*b*c^4*d*f^8 - 32*a*b^3*c^9*d^7*f^2 - 24*a*b^5*c^7*d^6*f^3 + 88*a*b^7*c^5*d^5*f^4 + 88*a*b^
9*c^3*d^4*f^5 + 64*a^2*b*c^10*d^7*f^2 + 128*a^3*b*c^9*d^6*f^3 + 16*a^3*b^9*c*d^2*f^7 - 1600*a^4*b*c^8*d^5*f^4
+ 3840*a^5*b*c^7*d^4*f^5 - 4160*a^6*b*c^6*d^3*f^6 - 92*a^6*b^5*c^2*d*f^8 + 2176*a^7*b*c^5*d^2*f^7 + 352*a^7*b^
3*c^3*d*f^8)/(16*a^2*c^6*d^4 + a^4*b^4*f^4 + b^4*c^4*d^4 + 16*a^6*c^2*f^4 + b^8*d^2*f^2 - 8*a*b^2*c^5*d^4 - 8*
a^5*b^2*c*f^4 + 2*a^2*b^6*d*f^3 - 64*a^3*c^5*d^3*f - 64*a^5*c^3*d*f^3 + 2*b^6*c^2*d^3*f + 96*a^4*c^4*d^2*f^2 +
54*a^2*b^4*c^2*d^2*f^2 - 112*a^3*b^2*c^3*d^2*f^2 - 20*a*b^4*c^3*d^3*f - 12*a*b^6*c*d^2*f^2 - 20*a^3*b^4*c*d*f
^3 + 64*a^2*b^2*c^4*d^3*f + 64*a^4*b^2*c^2*d*f^3) + (x*(128*a^9*c^4*f^9 - 2*a^6*b^6*c*f^9 - 640*a^8*c^5*d*f^8
+ 6*b^12*c*d^3*f^6 + 24*a^7*b^4*c^2*f^9 - 96*a^8*b^2*c^3*f^9 + 128*a^2*c^11*d^7*f^2 - 640*a^3*c^10*d^6*f^3 + 1
152*a^4*c^9*d^5*f^4 - 640*a^5*c^8*d^4*f^5 - 640*a^6*c^7*d^3*f^6 + 1152*a^7*c^6*d^2*f^7 + 8*b^4*c^9*d^7*f^2 + 2
2*b^6*c^7*d^6*f^3 + 26*b^8*c^5*d^5*f^4 + 18*b^10*c^3*d^4*f^5 + 672*a^2*b^2*c^9*d^6*f^3 + 1224*a^2*b^4*c^7*d^5*
f^4 + 1202*a^2*b^6*c^5*d^4*f^5 + 564*a^2*b^8*c^3*d^3*f^6 - 2048*a^3*b^2*c^8*d^5*f^4 - 2744*a^3*b^4*c^6*d^4*f^5
- 1736*a^3*b^6*c^4*d^3*f^6 - 128*a^3*b^8*c^2*d^2*f^7 + 2656*a^4*b^2*c^7*d^4*f^5 + 2648*a^4*b^4*c^5*d^3*f^6 +
570*a^4*b^6*c^3*d^2*f^7 - 1344*a^5*b^2*c^6*d^3*f^6 - 904*a^5*b^4*c^4*d^2*f^7 - 160*a^6*b^2*c^5*d^2*f^7 + 2*a^4
*b^8*c*d*f^8 - 64*a*b^2*c^10*d^7*f^2 - 216*a*b^4*c^8*d^6*f^3 - 300*a*b^6*c^6*d^5*f^4 - 240*a*b^8*c^4*d^4*f^5 -
92*a*b^10*c^2*d^3*f^6 + 10*a^2*b^10*c*d^2*f^7 - 12*a^5*b^6*c^2*d*f^8 - 40*a^6*b^4*c^3*d*f^8 + 384*a^7*b^2*c^4
*d*f^8))/(16*a^2*c^6*d^4 + a^4*b^4*f^4 + b^4*c^4*d^4 + 16*a^6*c^2*f^4 + b^8*d^2*f^2 - 8*a*b^2*c^5*d^4 - 8*a^5*
b^2*c*f^4 + 2*a^2*b^6*d*f^3 - 64*a^3*c^5*d^3*f - 64*a^5*c^3*d*f^3 + 2*b^6*c^2*d^3*f + 96*a^4*c^4*d^2*f^2 + 54*
a^2*b^4*c^2*d^2*f^2 - 112*a^3*b^2*c^3*d^2*f^2 - 20*a*b^4*c^3*d^3*f - 12*a*b^6*c*d^2*f^2 - 20*a^3*b^4*c*d*f^3 +
64*a^2*b^2*c^4*d^3*f + 64*a^4*b^2*c^2*d*f^3)) + (A*a^4*b^6*c*f^8 - 64*A*a^7*c^4*f^8 - 32*A*a*c^10*d^6*f^2 + 3
52*A*a^6*c^5*d*f^7 + A*b^10*c*d^2*f^6 - 12*A*a^5*b^4*c^2*f^8 + 48*A*a^6*b^2*c^3*f^8 + 224*A*a^2*c^9*d^5*f^3 -
640*A*a^3*c^8*d^4*f^4 + 960*A*a^4*c^7*d^3*f^5 - 800*A*a^5*c^6*d^2*f^6 + 8*A*b^2*c^9*d^6*f^2 + 16*A*b^4*c^7*d^5
*f^3 + A*b^6*c^5*d^4*f^4 - 6*A*b^8*c^3*d^3*f^5 - 4*B*b^3*c^8*d^6*f^2 - 12*B*b^5*c^6*d^5*f^3 - 4*B*b^7*c^4*d^4*
f^4 + 4*B*b^9*c^2*d^3*f^5 - 120*A*a*b^2*c^8*d^5*f^3 - 60*A*a*b^4*c^6*d^4*f^4 + 36*A*a*b^6*c^4*d^3*f^5 - 8*A*a*
b^8*c^2*d^2*f^6 - 20*A*a^3*b^6*c^2*d*f^7 + 80*A*a^4*b^4*c^3*d*f^7 - 216*A*a^5*b^2*c^4*d*f^7 + 92*B*a*b^3*c^7*d
^5*f^3 + 72*B*a*b^5*c^5*d^4*f^4 - 20*B*a*b^7*c^3*d^3*f^5 - 176*B*a^2*b*c^8*d^5*f^3 + 544*B*a^3*b*c^7*d^4*f^4 -
736*B*a^4*b*c^6*d^3*f^5 - 4*B*a^4*b^5*c^2*d*f^7 + 464*B*a^5*b*c^5*d^2*f^6 + 44*B*a^5*b^3*c^3*d*f^7 + 384*A*a^
2*b^2*c^7*d^4*f^4 + 32*A*a^2*b^4*c^5*d^3*f^5 + 14*A*a^2*b^6*c^3*d^2*f^6 - 560*A*a^3*b^2*c^6*d^3*f^5 - 56*A*a^3
*b^4*c^4*d^2*f^6 + 456*A*a^4*b^2*c^5*d^2*f^6 - 360*B*a^2*b^3*c^6*d^4*f^4 - 64*B*a^2*b^5*c^4*d^3*f^5 + 504*B*a^
3*b^3*c^5*d^3*f^5 + 40*B*a^3*b^5*c^3*d^2*f^6 - 276*B*a^4*b^3*c^4*d^2*f^6 + 2*A*a^2*b^8*c*d*f^7 + 16*B*a*b*c^9*
d^6*f^2 - 112*B*a^6*b*c^4*d*f^7)/(16*a^2*c^6*d^4 + a^4*b^4*f^4 + b^4*c^4*d^4 + 16*a^6*c^2*f^4 + b^8*d^2*f^2 -
8*a*b^2*c^5*d^4 - 8*a^5*b^2*c*f^4 + 2*a^2*b^6*d*f^3 - 64*a^3*c^5*d^3*f - 64*a^5*c^3*d*f^3 + 2*b^6*c^2*d^3*f +
96*a^4*c^4*d^2*f^2 + 54*a^2*b^4*c^2*d^2*f^2 - 112*a^3*b^2*c^3*d^2*f^2 - 20*a*b^4*c^3*d^3*f - 12*a*b^6*c*d^2*f^
2 - 20*a^3*b^4*c*d*f^3 + 64*a^2*b^2*c^4*d^3*f + 64*a^4*b^2*c^2*d*f^3) + (x*(64*B*a^7*c^4*f^8 + 4*A*a^3*b^7*c*f
^8 - 256*A*a^6*b*c^4*f^8 - B*a^4*b^6*c*f^8 - 320*B*a^6*c^5*d*f^7 + 3*B*b^10*c*d^2*f^6 - 48*A*a^4*b^5*c^2*f^8 +
192*A*a^5*b^3*c^3*f^8 + 12*B*a^5*b^4*c^2*f^8 - 48*B*a^6*b^2*c^3*f^8 - 16*A*b^3*c^8*d^5*f^3 - 48*A*b^5*c^6*d^4
*f^4 - 36*A*b^7*c^4*d^3*f^5 - 4*A*b^9*c^2*d^2*f^6 - 64*B*a^2*c^9*d^5*f^3 + 320*B*a^3*c^8*d^4*f^4 - 640*B*a^4*c
^7*d^3*f^5 + 640*B*a^5*c^6*d^2*f^6 + 4*B*b^4*c^7*d^5*f^3 + 23*B*b^6*c^5*d^4*f^4 + 22*B*b^8*c^3*d^3*f^5 + 320*A
*a*b^3*c^7*d^4*f^4 + 352*A*a*b^5*c^5*d^3*f^5 + 76*A*a*b^7*c^3*d^2*f^6 - 512*A*a^2*b*c^8*d^4*f^4 - 60*A*a^2*b^7
*c^2*d*f^7 + 1408*A*a^3*b*c^7*d^3*f^5 + 352*A*a^3*b^5*c^3*d*f^7 - 1792*A*a^4*b*c^6*d^2*f^6 - 976*A*a^4*b^3*c^4
*d*f^7 - 132*B*a*b^4*c^6*d^4*f^4 - 196*B*a*b^6*c^4*d^3*f^5 - 40*B*a*b^8*c^2*d^2*f^6 - 20*B*a^3*b^6*c^2*d*f^7 +
52*B*a^4*b^4*c^3*d*f^7 + 64*B*a^5*b^2*c^4*d*f^7 + 4*A*a*b^9*c*d*f^7 - 1184*A*a^2*b^3*c^6*d^3*f^5 - 544*A*a^2*
b^5*c^4*d^2*f^6 + 1664*A*a^3*b^3*c^5*d^2*f^6 + 80*B*a^2*b^2*c^7*d^4*f^4 + 520*B*a^2*b^4*c^5*d^3*f^5 + 210*B*a^
2*b^6*c^3*d^2*f^6 - 192*B*a^3*b^2*c^6*d^3*f^5 - 456*B*a^3*b^4*c^4*d^2*f^6 + 96*B*a^4*b^2*c^5*d^2*f^6 + 64*A*a*
b*c^9*d^5*f^3 + 1088*A*a^5*b*c^5*d*f^7 + 2*B*a^2*b^8*c*d*f^7))/(16*a^2*c^6*d^4 + a^4*b^4*f^4 + b^4*c^4*d^4 + 1
6*a^6*c^2*f^4 + b^8*d^2*f^2 - 8*a*b^2*c^5*d^4 - 8*a^5*b^2*c*f^4 + 2*a^2*b^6*d*f^3 - 64*a^3*c^5*d^3*f - 64*a^5*
c^3*d*f^3 + 2*b^6*c^2*d^3*f + 96*a^4*c^4*d^2*f^2 + 54*a^2*b^4*c^2*d^2*f^2 - 112*a^3*b^2*c^3*d^2*f^2 - 20*a*b^4
*c^3*d^3*f - 12*a*b^6*c*d^2*f^2 - 20*a^3*b^4*c*d*f^3 + 64*a^2*b^2*c^4*d^3*f + 64*a^4*b^2*c^2*d*f^3)) + (13*A^2
*a^2*b^5*c^2*f^7 - 56*A^2*a^3*b^3*c^3*f^7 + 16*A^2*b^3*c^6*d^3*f^4 + A^2*b^5*c^4*d^2*f^5 + 8*B^2*b^3*c^6*d^4*f
^3 + 9*B^2*b^5*c^4*d^3*f^4 - 64*A*B*a^5*c^4*f^7 - A^2*a*b^7*c*f^7 + 80*A^2*a^4*b*c^4*f^7 + 16*A^2*b*c^8*d^4*f^
3 + 2*A^2*b^7*c^2*d*f^6 - 48*A^2*a*b*c^7*d^3*f^4 - 22*A^2*a*b^5*c^3*d*f^6 - 48*A^2*a^3*b*c^5*d*f^6 - 16*B^2*a*
b*c^7*d^4*f^3 - 64*B^2*a^4*b*c^4*d*f^6 - A*B*b^8*c*d*f^6 - 8*A^2*a*b^3*c^5*d^2*f^5 + 64*A^2*a^2*b^3*c^4*d*f^6
- 56*B^2*a*b^3*c^5*d^3*f^4 + 2*B^2*a*b^5*c^3*d^2*f^5 + 96*B^2*a^2*b*c^6*d^3*f^4 - 11*B^2*a^2*b^5*c^2*d*f^6 - 1
6*B^2*a^3*b*c^5*d^2*f^5 + 40*B^2*a^3*b^3*c^3*d*f^6 + A*B*a^2*b^6*c*f^7 + 32*A*B*a*c^8*d^4*f^3 + 32*A*B*a^4*c^5
*d*f^6 + B^2*a*b^7*c*d*f^6 - 8*B^2*a^2*b^3*c^4*d^2*f^5 - 12*A*B*a^3*b^4*c^2*f^7 + 48*A*B*a^4*b^2*c^3*f^7 - 160
*A*B*a^2*c^7*d^3*f^4 + 160*A*B*a^3*c^6*d^2*f^5 - 24*A*B*b^2*c^7*d^4*f^3 - 24*A*B*b^4*c^5*d^3*f^4 + A*B*b^6*c^3
*d^2*f^5 + 120*A*B*a*b^2*c^6*d^3*f^4 - 4*A*B*a*b^4*c^4*d^2*f^5 - 24*A*B*a^2*b^4*c^3*d*f^6 + 8*A*B*a^3*b^2*c^4*
d*f^6 - 24*A*B*a^2*b^2*c^5*d^2*f^5 + 10*A*B*a*b^6*c^2*d*f^6)/(16*a^2*c^6*d^4 + a^4*b^4*f^4 + b^4*c^4*d^4 + 16*
a^6*c^2*f^4 + b^8*d^2*f^2 - 8*a*b^2*c^5*d^4 - 8*a^5*b^2*c*f^4 + 2*a^2*b^6*d*f^3 - 64*a^3*c^5*d^3*f - 64*a^5*c^
3*d*f^3 + 2*b^6*c^2*d^3*f + 96*a^4*c^4*d^2*f^2 + 54*a^2*b^4*c^2*d^2*f^2 - 112*a^3*b^2*c^3*d^2*f^2 - 20*a*b^4*c
^3*d^3*f - 12*a*b^6*c*d^2*f^2 - 20*a^3*b^4*c*d*f^3 + 64*a^2*b^2*c^4*d^3*f + 64*a^4*b^2*c^2*d*f^3) + (x*(104*A^
2*a^4*c^5*f^7 - 32*B^2*a^5*c^4*f^7 + 8*A^2*c^9*d^4*f^3 + A^2*b^8*c*f^7 + 50*A^2*a^2*b^4*c^3*f^7 - 96*A^2*a^3*b
^2*c^4*f^7 - 12*B^2*a^3*b^4*c^2*f^7 + 42*B^2*a^4*b^2*c^3*f^7 + 208*A^2*a^2*c^7*d^2*f^5 + 8*A^2*b^2*c^7*d^3*f^4
+ 18*A^2*b^4*c^5*d^2*f^5 - 32*B^2*a^2*c^7*d^3*f^4 + 32*B^2*a^3*c^6*d^2*f^5 + 2*B^2*b^2*c^7*d^4*f^3 - 6*B^2*b^
4*c^5*d^3*f^4 + 9*B^2*b^6*c^3*d^2*f^5 - 12*A^2*a*b^6*c^2*f^7 + B^2*a^2*b^6*c*f^7 - 64*A^2*a*c^8*d^3*f^4 - 256*
A^2*a^3*c^6*d*f^6 + 2*A^2*b^6*c^3*d*f^6 + 32*B^2*a^4*c^5*d*f^6 - 36*A^2*a*b^4*c^4*d*f^6 - 2*B^2*a*b^6*c^2*d*f^
6 - 2*A*B*a*b^7*c*f^7 - 144*A^2*a*b^2*c^6*d^2*f^5 + 168*A^2*a^2*b^2*c^5*d*f^6 + 24*B^2*a*b^2*c^6*d^3*f^4 - 64*
B^2*a*b^4*c^4*d^2*f^5 + 26*B^2*a^2*b^4*c^3*d*f^6 - 88*B^2*a^3*b^2*c^4*d*f^6 + 72*A*B*a^4*b*c^4*f^7 - 8*A*B*b*c
^8*d^4*f^3 + 2*A*B*b^7*c^2*d*f^6 + 84*B^2*a^2*b^2*c^5*d^2*f^5 + 24*A*B*a^2*b^5*c^2*f^7 - 84*A*B*a^3*b^3*c^3*f^
7 + 4*A*B*b^3*c^6*d^3*f^4 - 20*A*B*b^5*c^4*d^2*f^5 + 148*A*B*a*b^3*c^5*d^2*f^5 - 192*A*B*a^2*b*c^6*d^2*f^5 - 4
*A*B*a^2*b^3*c^4*d*f^6 + 16*A*B*a*b*c^7*d^3*f^4 - 12*A*B*a*b^5*c^3*d*f^6 + 112*A*B*a^3*b*c^5*d*f^6))/(16*a^2*c
^6*d^4 + a^4*b^4*f^4 + b^4*c^4*d^4 + 16*a^6*c^2*f^4 + b^8*d^2*f^2 - 8*a*b^2*c^5*d^4 - 8*a^5*b^2*c*f^4 + 2*a^2*
b^6*d*f^3 - 64*a^3*c^5*d^3*f - 64*a^5*c^3*d*f^3 + 2*b^6*c^2*d^3*f + 96*a^4*c^4*d^2*f^2 + 54*a^2*b^4*c^2*d^2*f^
2 - 112*a^3*b^2*c^3*d^2*f^2 - 20*a*b^4*c^3*d^3*f - 12*a*b^6*c*d^2*f^2 - 20*a^3*b^4*c*d*f^3 + 64*a^2*b^2*c^4*d^
3*f + 64*a^4*b^2*c^2*d*f^3)) - (16*A^3*a*c^6*d*f^5 - 4*A^3*c^7*d^2*f^4 - B^3*b^3*c^4*d^2*f^4 - 12*A^3*a^2*c^5*
f^6 - 16*A*B^2*a^3*c^4*f^6 + 2*A^3*a*b^2*c^4*f^6 - 6*A^3*b^2*c^5*d*f^5 + 4*B^3*a*b*c^5*d^2*f^4 + 3*B^3*a*b^3*c
^3*d*f^5 - 12*B^3*a^2*b*c^4*d*f^5 + 3*A*B^2*a^2*b^2*c^3*f^6 + A*B^2*b^2*c^5*d^2*f^4 - 3*A^2*B*a*b^3*c^3*f^6 +
16*A^2*B*a^2*b*c^4*f^6 - 8*A*B^2*a*c^6*d^2*f^4 + 24*A*B^2*a^2*c^5*d*f^5 - 3*A*B^2*b^4*c^3*d*f^5 + 4*A^2*B*b*c^
6*d^2*f^4 + 9*A^2*B*b^3*c^4*d*f^5 + 4*A*B^2*a*b^2*c^4*d*f^5 - 28*A^2*B*a*b*c^5*d*f^5)/(16*a^2*c^6*d^4 + a^4*b^
4*f^4 + b^4*c^4*d^4 + 16*a^6*c^2*f^4 + b^8*d^2*f^2 - 8*a*b^2*c^5*d^4 - 8*a^5*b^2*c*f^4 + 2*a^2*b^6*d*f^3 - 64*
a^3*c^5*d^3*f - 64*a^5*c^3*d*f^3 + 2*b^6*c^2*d^3*f + 96*a^4*c^4*d^2*f^2 + 54*a^2*b^4*c^2*d^2*f^2 - 112*a^3*b^2
*c^3*d^2*f^2 - 20*a*b^4*c^3*d^3*f - 12*a*b^6*c*d^2*f^2 - 20*a^3*b^4*c*d*f^3 + 64*a^2*b^2*c^4*d^3*f + 64*a^4*b^
2*c^2*d*f^3))*root(2560*a^3*b^2*c^9*d^8*f*z^4 - 1152*a^2*b^4*c^8*d^8*f*z^4 + 384*a^5*b^8*c*d^3*f^6*z^4 + 384*a
*b^8*c^5*d^7*f^2*z^4 + 288*a^3*b^10*c*d^4*f^5*z^4 + 288*a*b^10*c^3*d^6*f^3*z^4 + 224*a^7*b^6*c*d^2*f^7*z^4 - 1
92*a^10*b^2*c^2*d*f^8*z^4 + 224*a*b^6*c^7*d^8*f*z^4 + 80*a*b^12*c*d^5*f^4*z^4 + 48*a^9*b^4*c*d*f^8*z^4 - 33920
*a^6*b^2*c^6*d^5*f^4*z^4 + 27936*a^5*b^4*c^5*d^5*f^4*z^4 + 26112*a^7*b^2*c^5*d^4*f^5*z^4 + 26112*a^5*b^2*c^7*d
^6*f^3*z^4 - 20352*a^6*b^4*c^4*d^4*f^5*z^4 - 20352*a^4*b^4*c^6*d^6*f^3*z^4 - 13080*a^4*b^6*c^4*d^5*f^4*z^4 - 1
1520*a^8*b^2*c^4*d^3*f^6*z^4 - 11520*a^4*b^2*c^8*d^7*f^2*z^4 + 8736*a^5*b^6*c^3*d^4*f^5*z^4 + 8736*a^3*b^6*c^5
*d^6*f^3*z^4 + 7488*a^7*b^4*c^3*d^3*f^6*z^4 + 7488*a^3*b^4*c^7*d^7*f^2*z^4 + 3840*a^3*b^8*c^3*d^5*f^4*z^4 + 25
60*a^9*b^2*c^3*d^2*f^7*z^4 - 2416*a^6*b^6*c^2*d^3*f^6*z^4 - 2416*a^2*b^6*c^6*d^7*f^2*z^4 - 2160*a^4*b^8*c^2*d^
4*f^5*z^4 - 2160*a^2*b^8*c^4*d^6*f^3*z^4 - 1152*a^8*b^4*c^2*d^2*f^7*z^4 - 720*a^2*b^10*c^2*d^5*f^4*z^4 - 16*b^
8*c^6*d^8*f*z^4 - 2048*a^4*c^10*d^8*f*z^4 + 256*a^11*c^3*d*f^8*z^4 - 4*a^8*b^6*d*f^8*z^4 + 48*a*b^4*c^9*d^9*z^
4 - 24*b^10*c^4*d^7*f^2*z^4 - 16*b^12*c^2*d^6*f^3*z^4 + 17920*a^7*c^7*d^5*f^4*z^4 - 14336*a^8*c^6*d^4*f^5*z^4
- 14336*a^6*c^8*d^6*f^3*z^4 + 7168*a^9*c^5*d^3*f^6*z^4 + 7168*a^5*c^9*d^7*f^2*z^4 - 2048*a^10*c^4*d^2*f^7*z^4
- 24*a^4*b^10*d^3*f^6*z^4 - 16*a^6*b^8*d^2*f^7*z^4 - 16*a^2*b^12*d^4*f^5*z^4 - 192*a^2*b^2*c^10*d^9*z^4 - 4*b^
14*d^5*f^4*z^4 - 4*b^6*c^8*d^9*z^4 + 256*a^3*c^11*d^9*z^4 + 912*A*B*a^6*b*c^3*d*f^6*z^2 + 192*A*B*a^4*b^5*c*d*
f^6*z^2 + 920*A*B*a^4*b^3*c^3*d^2*f^5*z^2 - 480*A*B*a^2*b^5*c^3*d^3*f^4*z^2 - 336*A*B*a^2*b^3*c^5*d^4*f^3*z^2
- 272*A*B*a^3*b^3*c^4*d^3*f^4*z^2 + 240*A*B*a^3*b^5*c^2*d^2*f^5*z^2 + 192*A*B*a*b*c^8*d^6*f*z^2 - 2496*A*B*a^5
*b*c^4*d^2*f^5*z^2 + 1872*A*B*a^4*b*c^5*d^3*f^4*z^2 - 744*A*B*a^5*b^3*c^2*d*f^6*z^2 - 720*A*B*a^2*b*c^7*d^5*f^
2*z^2 + 504*A*B*a*b^3*c^6*d^5*f^2*z^2 + 256*A*B*a^3*b*c^6*d^4*f^3*z^2 + 168*A*B*a*b^7*c^2*d^3*f^4*z^2 - 144*A*
B*a^2*b^7*c*d^2*f^5*z^2 + 144*A*B*a*b^5*c^4*d^4*f^3*z^2 - 56*B^2*a*b^2*c^7*d^6*f*z^2 - 36*B^2*a^5*b^4*c*d*f^6*
z^2 - 16*B^2*a*b^8*c*d^3*f^4*z^2 - 164*A^2*a^3*b^6*c*d*f^6*z^2 - 16*A^2*a*b^8*c*d^2*f^5*z^2 - 96*A*B*b^5*c^5*d
^5*f^2*z^2 - 24*A*B*b^7*c^3*d^4*f^3*z^2 - 580*B^2*a^4*b^2*c^4*d^3*f^4*z^2 + 536*B^2*a^3*b^4*c^3*d^3*f^4*z^2 -
348*B^2*a^4*b^4*c^2*d^2*f^5*z^2 + 316*B^2*a^2*b^2*c^6*d^5*f^2*z^2 + 200*B^2*a^5*b^2*c^3*d^2*f^5*z^2 - 120*B^2*
a^2*b^4*c^4*d^4*f^3*z^2 - 66*B^2*a^2*b^6*c^2*d^3*f^4*z^2 - 16*B^2*a^3*b^2*c^5*d^4*f^3*z^2 + 1952*A^2*a^4*b^2*c
^4*d^2*f^5*z^2 - 1792*A^2*a^3*b^2*c^5*d^3*f^4*z^2 - 1272*A^2*a^3*b^4*c^3*d^2*f^5*z^2 + 976*A^2*a^2*b^2*c^6*d^4
*f^3*z^2 + 960*A^2*a^2*b^4*c^4*d^3*f^4*z^2 + 282*A^2*a^2*b^6*c^2*d^2*f^5*z^2 - 72*A*B*b^3*c^7*d^6*f*z^2 - 16*A
*B*b^9*c*d^3*f^4*z^2 - 16*A*B*a^3*b^7*d*f^6*z^2 + 16*A*B*a*b^9*d^2*f^5*z^2 - 180*B^2*a*b^4*c^5*d^5*f^2*z^2 + 1
32*B^2*a^6*b^2*c^2*d*f^6*z^2 + 108*B^2*a^3*b^6*c*d^2*f^5*z^2 + 20*B^2*a*b^6*c^3*d^4*f^3*z^2 - 736*A^2*a^5*b^2*
c^3*d*f^6*z^2 + 624*A^2*a^4*b^4*c^2*d*f^6*z^2 - 416*A^2*a*b^2*c^7*d^5*f^2*z^2 - 276*A^2*a*b^4*c^5*d^4*f^3*z^2
- 196*A^2*a*b^6*c^3*d^3*f^4*z^2 + 31*B^2*b^6*c^4*d^5*f^2*z^2 + 2*B^2*b^8*c^2*d^4*f^3*z^2 - 768*B^2*a^5*c^5*d^3
*f^4*z^2 + 512*B^2*a^6*c^4*d^2*f^5*z^2 + 512*B^2*a^4*c^6*d^4*f^3*z^2 - 128*B^2*a^3*c^7*d^5*f^2*z^2 + 80*A^2*b^
4*c^6*d^5*f^2*z^2 + 31*A^2*b^6*c^4*d^4*f^3*z^2 + 14*A^2*b^8*c^2*d^3*f^4*z^2 - 1152*A^2*a^3*c^7*d^4*f^3*z^2 + 1
008*A^2*a^4*c^6*d^3*f^4*z^2 + 624*A^2*a^2*c^8*d^5*f^2*z^2 - 288*A^2*a^5*c^5*d^2*f^5*z^2 - 10*B^2*a^2*b^8*d^2*f
^5*z^2 - 48*A^2*a^6*b^2*c^2*f^7*z^2 - 16*A*B*b*c^9*d^7*z^2 + 20*B^2*b^4*c^6*d^6*f*z^2 - 128*B^2*a^7*c^3*d*f^6*
z^2 + 64*A^2*b^2*c^8*d^6*f*z^2 - 112*A^2*a^6*c^4*d*f^6*z^2 + 3*B^2*a^4*b^6*d*f^6*z^2 + 14*A^2*a^2*b^8*d*f^6*z^
2 + 12*A^2*a^5*b^4*c*f^7*z^2 - 160*A^2*a*c^9*d^6*f*z^2 + 3*B^2*b^10*d^3*f^4*z^2 - A^2*b^10*d^2*f^5*z^2 + 64*A^
2*a^7*c^3*f^7*z^2 + 4*B^2*b^2*c^8*d^7*z^2 - A^2*a^4*b^6*f^7*z^2 + 16*A^2*c^10*d^7*z^2 - 160*A*B^2*a*b*c^6*d^4*
f^2*z + 112*A*B^2*a^4*b*c^3*d*f^5*z - 24*A*B^2*a^2*b^5*c*d*f^5*z + 480*A^2*B*a^2*b^2*c^4*d^2*f^4*z - 176*A*B^2
*a^2*b^3*c^3*d^2*f^4*z - 10*A^2*B*a*b^6*c*d*f^5*z + 384*A*B^2*a^2*b*c^5*d^3*f^3*z - 352*A*B^2*a^3*b*c^4*d^2*f^
4*z - 288*A^2*B*a*b^2*c^5*d^3*f^3*z - 160*A^2*B*a^3*b^2*c^3*d*f^5*z - 148*A^2*B*a*b^4*c^3*d^2*f^4*z + 112*A*B^
2*a*b^3*c^4*d^3*f^3*z + 72*A^2*B*a^2*b^4*c^2*d*f^5*z + 72*A*B^2*a*b^5*c^2*d^2*f^4*z + 48*A*B^2*a^3*b^3*c^2*d*f
^5*z + 48*B^3*a*b^2*c^5*d^4*f^2*z - 36*B^3*a^4*b^2*c^2*d*f^5*z - 4*B^3*a*b^4*c^3*d^3*f^3*z - 480*A^3*a^2*b*c^5
*d^2*f^4*z - 160*A^3*a^2*b^3*c^3*d*f^5*z + 128*A^3*a*b^3*c^4*d^2*f^4*z + 112*A^2*B*b^4*c^4*d^3*f^3*z - 64*A*B^
2*b^5*c^3*d^3*f^3*z + 16*A^2*B*b^2*c^6*d^4*f^2*z + 16*A*B^2*b^3*c^5*d^4*f^2*z - A^2*B*b^6*c^2*d^2*f^4*z + 448*
A^2*B*a^3*c^5*d^2*f^4*z - 352*A^2*B*a^2*c^6*d^3*f^3*z - 48*A^2*B*a^4*b^2*c^2*f^6*z + 12*B^3*a^3*b^4*c*d*f^5*z
- 10*B^3*a*b^6*c*d^2*f^4*z + 416*A^3*a^3*b*c^4*d*f^5*z + 224*A^3*a*b*c^6*d^3*f^3*z + 24*A^3*a*b^5*c^2*d*f^5*z
- 2*A*B^2*b^7*c*d^2*f^4*z - 272*A^2*B*a^4*c^4*d*f^5*z + 128*A^2*B*a*c^7*d^4*f^2*z + 12*A^2*B*a^3*b^4*c*f^6*z -
120*B^3*a^2*b^2*c^4*d^3*f^3*z + 112*B^3*a^3*b^2*c^3*d^2*f^4*z + 16*A*B^2*b*c^7*d^5*f*z + 2*A*B^2*a*b^7*d*f^5*
z - 2*A^3*b^7*c*d*f^5*z - 16*A^2*B*c^8*d^5*f*z + 11*B^3*b^6*c^2*d^3*f^3*z - 8*B^3*b^4*c^4*d^4*f^2*z - 64*A^3*b
^3*c^5*d^3*f^3*z + 96*A^3*a^3*b^3*c^2*f^6*z - 4*B^3*b^2*c^6*d^5*f*z - 32*A^3*b*c^7*d^4*f^2*z - B^3*a^2*b^6*d*f
^5*z - 128*A^3*a^4*b*c^3*f^6*z - 24*A^3*a^2*b^5*c*f^6*z + 64*A^2*B*a^5*c^3*f^6*z - A^2*B*a^2*b^6*f^6*z + A^2*B
*b^8*d*f^5*z + 2*A^3*a*b^7*f^6*z + B^3*b^8*d^2*f^4*z + 32*A^3*B*a*b*c^4*d*f^4 - 18*A^2*B^2*a*b^2*c^3*d*f^4 + 3
2*A*B^3*a*b*c^4*d^2*f^3 - 28*A*B^3*a^2*b*c^3*d*f^4 + 6*A*B^3*a*b^3*c^2*d*f^4 - 10*A^3*B*b^3*c^3*d*f^4 - 4*A^3*
B*b*c^5*d^2*f^3 - 4*A*B^3*b*c^5*d^3*f^2 - 28*A^3*B*a^2*b*c^3*f^5 + 6*A^3*B*a*b^3*c^2*f^5 + 9*A^2*B^2*b^2*c^4*d
^2*f^3 - 3*A^2*B^2*a^2*b^2*c^2*f^5 - 10*B^4*a*b^2*c^3*d^2*f^3 - 3*B^4*a^2*b^2*c^2*d*f^4 - 10*A*B^3*b^3*c^3*d^2
*f^3 + 3*A^2*B^2*b^4*c^2*d*f^4 + 36*A^2*B^2*a^2*c^4*d*f^4 - 24*A^2*B^2*a*c^5*d^2*f^3 + 4*A^2*B^2*c^6*d^3*f^2 +
16*A^2*B^2*a^3*c^3*f^5 + 16*B^4*a^3*c^3*d*f^4 + 8*A^4*b^2*c^4*d*f^4 - 8*A^4*a*b^2*c^3*f^5 - 24*A^4*a*c^5*d*f^
4 + 3*B^4*b^4*c^2*d^2*f^3 + 4*A^4*c^6*d^2*f^3 + 36*A^4*a^2*c^4*f^5 + B^4*b^2*c^4*d^3*f^2, z, k), k, 1, 4)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(c*x**2+b*x+a)**2/(f*x**2+d),x)
[Out]
Timed out
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