3.1.16 \(\int \frac {A+B x}{(a+b x+c x^2)^2 (d+e x+f x^2)} \, dx\) [16]

Optimal. Leaf size=1075 \[ -\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )+c \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (a+b x+c x^2\right )}-\frac {\left (b^5 (B d-A e) f^2-2 b^4 f \left (B c d e-A \left (c e^2-c d f+a f^2\right )\right )-4 c^2 \left (A \left (c^3 d^3-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )+a c^2 d \left (3 e^2-5 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )-4 b^2 c \left (B c^2 d^2 e+A f \left (2 c^2 d^2+3 a^2 f^2+3 a c \left (e^2-d f\right )\right )\right )+2 b c \left (B \left (c^3 d^3+3 a^3 f^3+a c^2 d \left (e^2-7 d f\right )+3 a^2 c f \left (e^2+d f\right )\right )+A c e \left (3 c^2 d^2+3 a^2 f^2+a c \left (3 e^2+2 d f\right )\right )\right )-b^3 \left (A c e \left (c e^2-2 c d f-4 a f^2\right )+B \left (4 a c d f^2+a^2 f^3-c^2 d \left (e^2+5 d f\right )\right )\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+f \left (b^2 d-a b e+a^2 f\right )-c \left (b d e-a \left (e^2-2 d f\right )\right )\right )^2}+\frac {\left (B \left (c^2 d e \left (e^2-3 d f\right )-2 c d f \left (b e^2-2 b d f-a e f\right )+f^2 \left (b^2 d e-4 a b d f+a^2 e f\right )\right )-A \left (c^2 \left (e^4-4 d e^2 f+2 d^2 f^2\right )-f^2 \left (2 a b e f-2 a^2 f^2-b^2 \left (e^2-2 d f\right )\right )+2 c f \left (a f \left (e^2-2 d f\right )-b \left (e^3-3 d e f\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {e+2 f x}{\sqrt {e^2-4 d f}}\right )}{\sqrt {e^2-4 d f} \left (c^2 d^2+f \left (b^2 d-a b e+a^2 f\right )-c \left (b d e-a \left (e^2-2 d f\right )\right )\right )^2}+\frac {\left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c^2 d^2+f \left (b^2 d-a b e+a^2 f\right )-c \left (b d e-a \left (e^2-2 d f\right )\right )\right )^2}-\frac {\left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right ) \log \left (d+e x+f x^2\right )}{2 \left (c^2 d^2+f \left (b^2 d-a b e+a^2 f\right )-c \left (b d e-a \left (e^2-2 d f\right )\right )\right )^2} \]

[Out]

(-A*c*(2*a*c*e-b*(a*f+c*d))-(A*b-B*a)*(2*c^2*d+b^2*f-c*(2*a*f+b*e))-c*(A*b^2*f+2*c*(-A*a*f+A*c*d+B*a*e)-b*(A*c
*e+B*a*f+B*c*d))*x)/(-4*a*c+b^2)/((-a*f+c*d)^2-(-a*e+b*d)*(-b*f+c*e))/(c*x^2+b*x+a)-(b^5*(-A*e+B*d)*f^2-2*b^4*
f*(B*c*d*e-A*(a*f^2-c*d*f+c*e^2))-4*c^2*(A*(c^3*d^3-3*a^3*f^3-a^2*c*f*(-7*d*f+e^2)+a*c^2*d*(-5*d*f+3*e^2))-a*B
*e*(c^2*d^2-3*a^2*f^2-a*c*(-2*d*f+e^2)))-4*b^2*c*(B*c^2*d^2*e+A*f*(2*c^2*d^2+3*a^2*f^2+3*a*c*(-d*f+e^2)))+2*b*
c*(B*(c^3*d^3+3*a^3*f^3+a*c^2*d*(-7*d*f+e^2)+3*a^2*c*f*(d*f+e^2))+A*c*e*(3*c^2*d^2+3*a^2*f^2+a*c*(2*d*f+3*e^2)
))-b^3*(A*c*e*(-4*a*f^2-2*c*d*f+c*e^2)+B*(4*a*c*d*f^2+a^2*f^3-c^2*d*(5*d*f+e^2))))*arctanh((2*c*x+b)/(-4*a*c+b
^2)^(1/2))/(-4*a*c+b^2)^(3/2)/(c^2*d^2+f*(a^2*f-a*b*e+b^2*d)-c*(b*d*e-a*(-2*d*f+e^2)))^2+1/2*(A*(-b*f+c*e)*(f*
(-2*a*f+b*e)-c*(-2*d*f+e^2))-B*(2*c*d*f*(-a*f+b*e)-f^2*(-a^2*f+b^2*d)-c^2*d*(-d*f+e^2)))*ln(c*x^2+b*x+a)/(c^2*
d^2+f*(a^2*f-a*b*e+b^2*d)-c*(b*d*e-a*(-2*d*f+e^2)))^2-1/2*(A*(-b*f+c*e)*(f*(-2*a*f+b*e)-c*(-2*d*f+e^2))-B*(2*c
*d*f*(-a*f+b*e)-f^2*(-a^2*f+b^2*d)-c^2*d*(-d*f+e^2)))*ln(f*x^2+e*x+d)/(c^2*d^2+f*(a^2*f-a*b*e+b^2*d)-c*(b*d*e-
a*(-2*d*f+e^2)))^2+(B*(c^2*d*e*(-3*d*f+e^2)-2*c*d*f*(-a*e*f-2*b*d*f+b*e^2)+f^2*(a^2*e*f-4*a*b*d*f+b^2*d*e))-A*
(c^2*(2*d^2*f^2-4*d*e^2*f+e^4)-f^2*(2*a*b*e*f-2*a^2*f^2-b^2*(-2*d*f+e^2))+2*c*f*(a*f*(-2*d*f+e^2)-b*(-3*d*e*f+
e^3))))*arctanh((2*f*x+e)/(-4*d*f+e^2)^(1/2))/(c^2*d^2+f*(a^2*f-a*b*e+b^2*d)-c*(b*d*e-a*(-2*d*f+e^2)))^2/(-4*d
*f+e^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 2.53, antiderivative size = 1067, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1030, 1086, 648, 632, 212, 642} \begin {gather*} -\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )+c \left (A f b^2-(B c d+A c e+a B f) b+2 c (A c d+a B e-a A f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (c x^2+b x+a\right )}-\frac {\left ((B d-A e) f^2 b^5-2 f \left (-a A f^2+B c d e-A c \left (e^2-d f\right )\right ) b^4-\left (A c e \left (c e^2-4 a f^2-2 c d f\right )+B \left (a^2 f^3+4 a c d f^2-c^2 d \left (e^2+5 d f\right )\right )\right ) b^3-4 \left (B d^2 e c^3+A f \left (2 c^2 d^2+3 a^2 f^2+3 a c \left (e^2-d f\right )\right ) c\right ) b^2+2 c \left (B \left (c^3 d^3+a c^2 \left (e^2-7 d f\right ) d+3 a^3 f^3+3 a^2 c f \left (e^2+d f\right )\right )+A c e \left (3 c^2 d^2+3 a^2 f^2+a c \left (3 e^2+2 d f\right )\right )\right ) b-4 c^2 \left (A \left (c^3 d^3+a c^2 \left (3 e^2-5 d f\right ) d-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\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-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )\right )^2}+\frac {\left (B \left (d e \left (e^2-3 d f\right ) c^2-2 d f \left (b e^2-a f e-2 b d f\right ) c+f^2 \left (e f a^2-4 b d f a+b^2 d e\right )\right )-A \left (\left (e^4-4 d f e^2+2 d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-2 d f\right )-b \left (e^3-3 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-2 d f\right ) b^2\right )+2 a e f b-2 a^2 f^2\right )\right )\right ) \tanh ^{-1}\left (\frac {e+2 f x}{\sqrt {e^2-4 d f}}\right )}{\sqrt {e^2-4 d f} \left (c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )\right )^2}+\frac {\left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) \log \left (c x^2+b x+a\right )}{2 \left (c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )\right )^2}-\frac {\left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (-d \left (e^2-d f\right ) c^2+2 d f (b e-a f) c-f^2 \left (b^2 d-a^2 f\right )\right )\right ) \log \left (f x^2+e x+d\right )}{2 \left (c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((A*c*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*b^2*f + 2*c*(A*c*d +
a*B*e - a*A*f) - b*(B*c*d + A*c*e + a*B*f))*x)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(a +
b*x + c*x^2))) - ((b^5*(B*d - A*e)*f^2 - 2*b^4*f*(B*c*d*e - a*A*f^2 - A*c*(e^2 - d*f)) - 4*c^2*(A*(c^3*d^3 - 3
*a^3*f^3 - a^2*c*f*(e^2 - 7*d*f) + a*c^2*d*(3*e^2 - 5*d*f)) - a*B*e*(c^2*d^2 - 3*a^2*f^2 - a*c*(e^2 - 2*d*f)))
- 4*b^2*(B*c^3*d^2*e + A*c*f*(2*c^2*d^2 + 3*a^2*f^2 + 3*a*c*(e^2 - d*f))) + 2*b*c*(B*(c^3*d^3 + 3*a^3*f^3 + a
*c^2*d*(e^2 - 7*d*f) + 3*a^2*c*f*(e^2 + d*f)) + A*c*e*(3*c^2*d^2 + 3*a^2*f^2 + a*c*(3*e^2 + 2*d*f))) - b^3*(A*
c*e*(c*e^2 - 2*c*d*f - 4*a*f^2) + B*(4*a*c*d*f^2 + a^2*f^3 - c^2*d*(e^2 + 5*d*f))))*ArcTanh[(b + 2*c*x)/Sqrt[b
^2 - 4*a*c]])/((b^2 - 4*a*c)^(3/2)*(c^2*d^2 - b*c*d*e + f*(b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f))^2) + ((
B*(c^2*d*e*(e^2 - 3*d*f) - 2*c*d*f*(b*e^2 - 2*b*d*f - a*e*f) + f^2*(b^2*d*e - 4*a*b*d*f + a^2*e*f)) - A*(c^2*(
e^4 - 4*d*e^2*f + 2*d^2*f^2) - f^2*(2*a*b*e*f - 2*a^2*f^2 - b^2*(e^2 - 2*d*f)) + 2*c*f*(a*f*(e^2 - 2*d*f) - b*
(e^3 - 3*d*e*f))))*ArcTanh[(e + 2*f*x)/Sqrt[e^2 - 4*d*f]])/(Sqrt[e^2 - 4*d*f]*(c^2*d^2 - b*c*d*e + f*(b^2*d -
a*b*e + a^2*f) + a*c*(e^2 - 2*d*f))^2) + ((A*(c*e - b*f)*(f*(b*e - 2*a*f) - c*(e^2 - 2*d*f)) - B*(2*c*d*f*(b*e
- a*f) - f^2*(b^2*d - a^2*f) - c^2*d*(e^2 - d*f)))*Log[a + b*x + c*x^2])/(2*(c^2*d^2 - b*c*d*e + f*(b^2*d - a
*b*e + a^2*f) + a*c*(e^2 - 2*d*f))^2) - ((A*(c*e - b*f)*(f*(b*e - 2*a*f) - c*(e^2 - 2*d*f)) - B*(2*c*d*f*(b*e
- a*f) - f^2*(b^2*d - a^2*f) - c^2*d*(e^2 - d*f)))*Log[d + e*x + f*x^2])/(2*(c^2*d^2 - b*c*d*e + f*(b^2*d - a*
b*e + a^2*f) + a*c*(e^2 - 2*d*f))^2)

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 642

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 648

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 1030

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Sy
mbol] :> Simp[(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)
*(c*e - b*f))*(p + 1)))*(g*c*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(
g*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - h*(b*c*d - 2*a*c*e + a*b*f))*x), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*
f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*h - 2*g*c)
*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*
f) - a*((-h)*c*e)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d +
b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*g*f - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a*((-h)*c*e)))*(b
*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) + a*h*c*e
))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2
- 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1
])

Rule 1086

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)
), x_Symbol] :> With[{q = c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*d*f - a*b*e*f + a^2*f^2}, Dist[1/q, In
t[(A*c^2*d - a*c*C*d - A*b*c*e + a*B*c*e + A*b^2*f - a*b*B*f - a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d - A*c*e +
a*C*e + A*b*f - a*B*f)*x)/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[(c*C*d^2 - B*c*d*e + A*c*e^2 + b*B*d*f - A
*c*d*f - a*C*d*f - A*b*e*f + a*A*f^2 - f*(B*c*d - b*C*d - A*c*e + a*C*e + A*b*f - a*B*f)*x)/(d + e*x + f*x^2),
x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]

Rubi steps

\begin {align*} \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+e x+f x^2\right )} \, dx &=-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )+c \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )+\left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right ) x+c f \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right ) x^2}{\left (a+b x+c x^2\right ) \left (d+e x+f x^2\right )} \, dx}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\\ &=-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )+c \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {-a c^2 d f \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right )+a^2 c f^2 \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right )+a c e \left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right )-a b f \left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right )+c^2 d \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )-b c e \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )+b^2 f \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )-a c f \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )+c \left (-b c d f \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right )+a c e f \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right )+c d \left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right )-a f \left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right )-c e \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )+b f \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )\right ) x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )^2}-\frac {\int \frac {c^2 d^2 f \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right )-a c d f^2 \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right )-c d e \left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right )+b d f \left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right )+c e^2 \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )-c d f \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )-b e f \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )+a f^2 \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )-f \left (-b c d f \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right )+a c e f \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right )+c d \left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right )-a f \left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right )-c e \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )+b f \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )\right ) x}{d+e x+f x^2} \, dx}{\left (b^2-4 a c\right ) \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )^2}\\ &=-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )+c \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (a+b x+c x^2\right )}+\frac {\left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )^2}-\frac {\left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right ) \int \frac {e+2 f x}{d+e x+f x^2} \, dx}{2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )^2}-\frac {\left (B \left (c^2 d e \left (e^2-3 d f\right )-2 c d f \left (b e^2-2 b d f-a e f\right )+f^2 \left (b^2 d e-4 a b d f+a^2 e f\right )\right )-A \left (c^2 \left (e^4-4 d e^2 f+2 d^2 f^2\right )-f^2 \left (2 a b e f-2 a^2 f^2-b^2 \left (e^2-2 d f\right )\right )+2 c f \left (a f \left (e^2-2 d f\right )-b \left (e^3-3 d e f\right )\right )\right )\right ) \int \frac {1}{d+e x+f x^2} \, dx}{2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )^2}-\frac {\left (-b c \left (-b c d f \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right )+a c e f \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right )+c d \left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right )-a f \left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right )-c e \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )+b f \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )\right )+2 c \left (-a c^2 d f \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right )+a^2 c f^2 \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right )+a c e \left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right )-a b f \left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right )+c^2 d \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )-b c e \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )+b^2 f \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )-a c f \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c \left (b^2-4 a c\right ) \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )^2}\\ &=-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )+c \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (a+b x+c x^2\right )}+\frac {\left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )^2}-\frac {\left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right ) \log \left (d+e x+f x^2\right )}{2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )^2}+\frac {\left (B \left (c^2 d e \left (e^2-3 d f\right )-2 c d f \left (b e^2-2 b d f-a e f\right )+f^2 \left (b^2 d e-4 a b d f+a^2 e f\right )\right )-A \left (c^2 \left (e^4-4 d e^2 f+2 d^2 f^2\right )-f^2 \left (2 a b e f-2 a^2 f^2-b^2 \left (e^2-2 d f\right )\right )+2 c f \left (a f \left (e^2-2 d f\right )-b \left (e^3-3 d e f\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{e^2-4 d f-x^2} \, dx,x,e+2 f x\right )}{\left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )^2}+\frac {\left (-b c \left (-b c d f \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right )+a c e f \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right )+c d \left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right )-a f \left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right )-c e \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )+b f \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )\right )+2 c \left (-a c^2 d f \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right )+a^2 c f^2 \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right )+a c e \left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right )-a b f \left (A b^3 f^2+b^2 B f (c d-a f)-b c \left (B c d e+A c e^2+a B e f+4 a A f^2\right )+2 c \left (A c e (c d+a f)+a B \left (c e^2-2 c d f+2 a f^2\right )\right )\right )+c^2 d \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )-b c e \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )+b^2 f \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )-a c f \left (-b^3 (B d f-A e f)-b c (B d (c d-3 a f)+A e (c d+4 a f))+b^2 \left (B c d e-A \left (c e^2-2 c d f+a f^2\right )\right )-2 c \left (a B c d e-A \left (c^2 d^2+2 a c e^2-3 a c d f+2 a^2 f^2\right )\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c \left (b^2-4 a c\right ) \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )^2}\\ &=-\frac {A c (2 a c e-b (c d+a f))+(A b-a B) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )+c \left (A b^2 f+2 c (A c d+a B e-a A f)-b (B c d+A c e+a B f)\right ) x}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \left (a+b x+c x^2\right )}-\frac {\left (b^5 (B d-A e) f^2-2 b^4 f \left (B c d e-a A f^2-A c \left (e^2-d f\right )\right )-4 c^2 \left (A \left (c^3 d^3-3 a^3 f^3-a^2 c f \left (e^2-7 d f\right )+a c^2 d \left (3 e^2-5 d f\right )\right )-a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )\right )-4 b^2 \left (B c^3 d^2 e+A c f \left (2 c^2 d^2+3 a^2 f^2+3 a c \left (e^2-d f\right )\right )\right )+2 b c \left (B \left (c^3 d^3+3 a^3 f^3+a c^2 d \left (e^2-7 d f\right )+3 a^2 c f \left (e^2+d f\right )\right )+A c e \left (3 c^2 d^2+3 a^2 f^2+a c \left (3 e^2+2 d f\right )\right )\right )-b^3 \left (A c e \left (c e^2-2 c d f-4 a f^2\right )+B \left (4 a c d f^2+a^2 f^3-c^2 d \left (e^2+5 d f\right )\right )\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-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )^2}+\frac {\left (B \left (c^2 d e \left (e^2-3 d f\right )-2 c d f \left (b e^2-2 b d f-a e f\right )+f^2 \left (b^2 d e-4 a b d f+a^2 e f\right )\right )-A \left (c^2 \left (e^4-4 d e^2 f+2 d^2 f^2\right )-f^2 \left (2 a b e f-2 a^2 f^2-b^2 \left (e^2-2 d f\right )\right )+2 c f \left (a f \left (e^2-2 d f\right )-b \left (e^3-3 d e f\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {e+2 f x}{\sqrt {e^2-4 d f}}\right )}{\sqrt {e^2-4 d f} \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )^2}+\frac {\left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )^2}-\frac {\left (A (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-B \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right ) \log \left (d+e x+f x^2\right )}{2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )^2}\\ \end {align*}

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Mathematica [A]
time = 4.41, size = 952, normalized size = 0.89 \begin {gather*} \frac {-\frac {2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right ) \left (A \left (b^3 f+b^2 c (-e+f x)+b c (-3 a f+c (d-e x))+2 c^2 (c d x+a (e-f x))\right )+B \left (2 a^2 c f-b c^2 d x-a \left (b^2 f+2 c^2 (d-e x)+b c (-e+f x)\right )\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac {2 \left (b^5 (B d-A e) f^2+2 b^4 f \left (-B c d e+a A f^2+A c \left (e^2-d f\right )\right )-4 b^2 \left (B c^3 d^2 e+A c f \left (2 c^2 d^2+3 a^2 f^2+3 a c \left (e^2-d f\right )\right )\right )+2 b c \left (B \left (c^3 d^3+3 a^3 f^3+a c^2 d \left (e^2-7 d f\right )+3 a^2 c f \left (e^2+d f\right )\right )+A c e \left (3 c^2 d^2+3 a^2 f^2+a c \left (3 e^2+2 d f\right )\right )\right )+4 c^2 \left (a B e \left (c^2 d^2-3 a^2 f^2-a c \left (e^2-2 d f\right )\right )+A \left (-c^3 d^3+3 a^3 f^3+a^2 c f \left (e^2-7 d f\right )+a c^2 d \left (-3 e^2+5 d f\right )\right )\right )+b^3 \left (A c e \left (-c e^2+2 c d f+4 a f^2\right )+B \left (-4 a c d f^2-a^2 f^3+c^2 d \left (e^2+5 d f\right )\right )\right )\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+\frac {2 \left (B \left (c^2 d e \left (-e^2+3 d f\right )-2 c d f \left (-b e^2+2 b d f+a e f\right )+f^2 \left (-b^2 d e+4 a b d f-a^2 e f\right )\right )+A \left (c^2 \left (e^4-4 d e^2 f+2 d^2 f^2\right )+f^2 \left (-2 a b e f+2 a^2 f^2+b^2 \left (e^2-2 d f\right )\right )+2 c f \left (a f \left (e^2-2 d f\right )-b \left (e^3-3 d e f\right )\right )\right )\right ) \tan ^{-1}\left (\frac {e+2 f x}{\sqrt {-e^2+4 d f}}\right )}{\sqrt {-e^2+4 d f}}-\left (A (c e-b f) \left (f (-b e+2 a f)+c \left (e^2-2 d f\right )\right )+B \left (2 c d f (b e-a f)+f^2 \left (-b^2 d+a^2 f\right )+c^2 d \left (-e^2+d f\right )\right )\right ) \log (a+x (b+c x))+\left (A (c e-b f) \left (f (-b e+2 a f)+c \left (e^2-2 d f\right )\right )+B \left (2 c d f (b e-a f)+f^2 \left (-b^2 d+a^2 f\right )+c^2 d \left (-e^2+d f\right )\right )\right ) \log (d+x (e+f x))}{2 \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*(c^2*d^2 - b*c*d*e + f*(b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f))*(A*(b^3*f + b^2*c*(-e + f*x) + b*c*(-
3*a*f + c*(d - e*x)) + 2*c^2*(c*d*x + a*(e - f*x))) + B*(2*a^2*c*f - b*c^2*d*x - a*(b^2*f + 2*c^2*(d - e*x) +
b*c*(-e + f*x)))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) - (2*(b^5*(B*d - A*e)*f^2 + 2*b^4*f*(-(B*c*d*e) + a*A*f^2
+ A*c*(e^2 - d*f)) - 4*b^2*(B*c^3*d^2*e + A*c*f*(2*c^2*d^2 + 3*a^2*f^2 + 3*a*c*(e^2 - d*f))) + 2*b*c*(B*(c^3*
d^3 + 3*a^3*f^3 + a*c^2*d*(e^2 - 7*d*f) + 3*a^2*c*f*(e^2 + d*f)) + A*c*e*(3*c^2*d^2 + 3*a^2*f^2 + a*c*(3*e^2 +
2*d*f))) + 4*c^2*(a*B*e*(c^2*d^2 - 3*a^2*f^2 - a*c*(e^2 - 2*d*f)) + A*(-(c^3*d^3) + 3*a^3*f^3 + a^2*c*f*(e^2
- 7*d*f) + a*c^2*d*(-3*e^2 + 5*d*f))) + b^3*(A*c*e*(-(c*e^2) + 2*c*d*f + 4*a*f^2) + B*(-4*a*c*d*f^2 - a^2*f^3
+ c^2*d*(e^2 + 5*d*f))))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (2*(B*(c^2*d*e*(-e^2 +
3*d*f) - 2*c*d*f*(-(b*e^2) + 2*b*d*f + a*e*f) + f^2*(-(b^2*d*e) + 4*a*b*d*f - a^2*e*f)) + A*(c^2*(e^4 - 4*d*e
^2*f + 2*d^2*f^2) + f^2*(-2*a*b*e*f + 2*a^2*f^2 + b^2*(e^2 - 2*d*f)) + 2*c*f*(a*f*(e^2 - 2*d*f) - b*(e^3 - 3*d
*e*f))))*ArcTan[(e + 2*f*x)/Sqrt[-e^2 + 4*d*f]])/Sqrt[-e^2 + 4*d*f] - (A*(c*e - b*f)*(f*(-(b*e) + 2*a*f) + c*(
e^2 - 2*d*f)) + B*(2*c*d*f*(b*e - a*f) + f^2*(-(b^2*d) + a^2*f) + c^2*d*(-e^2 + d*f)))*Log[a + x*(b + c*x)] +
(A*(c*e - b*f)*(f*(-(b*e) + 2*a*f) + c*(e^2 - 2*d*f)) + B*(2*c*d*f*(b*e - a*f) + f^2*(-(b^2*d) + a^2*f) + c^2*
d*(-e^2 + d*f)))*Log[d + x*(e + f*x)])/(2*(c^2*d^2 - b*c*d*e + f*(b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f))^
2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(51469\) vs. \(2(1065)=2130\).
time = 0.04, size = 51470, normalized size = 47.88 \[\text {output too large to display}\]

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+e*x+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+e*x+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*d*f-%e^2>0)', see `assume?`
for more det

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Fricas [F(-1)] Timed out
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+e*x+d),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
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+e*x+d),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3226 vs. \(2 (1095) = 2190\).
time = 2.30, size = 3226, normalized size = 3.00 \begin {gather*} \text {Too large to display} \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+e*x+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 + 2*B*b*c*d*f*e - 2*
A*c^2*d*f*e + A*b^2*f^2*e + 2*A*a*c*f^2*e - B*c^2*d*e^2 - 2*A*b*c*f*e^2 + A*c^2*e^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 - 2*b*c^3*d^3*e - 2*b^3*c*d^2*f*e + 2*a*b*c^2*d^2*f*e - 2*a*b^3*d*f^2*e + 2*a^2*b*c
*d*f^2*e - 2*a^3*b*f^3*e + b^2*c^2*d^2*e^2 + 2*a*c^3*d^2*e^2 + 4*a*b^2*c*d*f*e^2 - 4*a^2*c^2*d*f*e^2 + a^2*b^2
*f^2*e^2 + 2*a^3*c*f^2*e^2 - 2*a*b*c^2*d*e^3 - 2*a^2*b*c*f*e^3 + a^2*c^2*e^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 + 2*B*b*c*d*f*e - 2*A*c^2*d*f*e + A*b^2*f^2*e + 2*A
*a*c*f^2*e - B*c^2*d*e^2 - 2*A*b*c*f*e^2 + A*c^2*e^3)*log(f*x^2 + x*e + 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*c^3*d^3*e - 2*b^3*c*d^2*f*e + 2*a*b*c^2*d^2*f*e - 2*a*b^3*d*f^2*e + 2*a^2*b*c*d*f^2*e - 2*a^3*b*f^3*e + b^2*
c^2*d^2*e^2 + 2*a*c^3*d^2*e^2 + 4*a*b^2*c*d*f*e^2 - 4*a^2*c^2*d*f*e^2 + a^2*b^2*f^2*e^2 + 2*a^3*c*f^2*e^2 - 2*
a*b*c^2*d*e^3 - 2*a^2*b*c*f*e^3 + a^2*c^2*e^4) + (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 - 4*B*b^2*c^3*d^2*e + 4*B*a*c^4*d^2*e + 6*A*b*c^4*d^2*e - 2*B*b^4*c*d*f*e
+ 2*A*b^3*c^2*d*f*e + 8*B*a^2*c^3*d*f*e + 4*A*a*b*c^3*d*f*e - A*b^5*f^2*e + 4*A*a*b^3*c*f^2*e - 12*B*a^3*c^2*f
^2*e + 6*A*a^2*b*c^2*f^2*e + B*b^3*c^2*d*e^2 + 2*B*a*b*c^3*d*e^2 - 12*A*a*c^4*d*e^2 + 2*A*b^4*c*f*e^2 + 6*B*a^
2*b*c^2*f*e^2 - 12*A*a*b^2*c^2*f*e^2 + 4*A*a^2*c^3*f*e^2 - A*b^3*c^2*e^3 - 4*B*a^2*c^3*e^3 + 6*A*a*b*c^3*e^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 - 2*b^3*c^3*d^3*e + 8*a*b*c^4*d^3*e - 2
*b^5*c*d^2*f*e + 10*a*b^3*c^2*d^2*f*e - 8*a^2*b*c^3*d^2*f*e - 2*a*b^5*d*f^2*e + 10*a^2*b^3*c*d*f^2*e - 8*a^3*b
*c^2*d*f^2*e - 2*a^3*b^3*f^3*e + 8*a^4*b*c*f^3*e + b^4*c^2*d^2*e^2 - 2*a*b^2*c^3*d^2*e^2 - 8*a^2*c^4*d^2*e^2 +
4*a*b^4*c*d*f*e^2 - 20*a^2*b^2*c^2*d*f*e^2 + 16*a^3*c^3*d*f*e^2 + a^2*b^4*f^2*e^2 - 2*a^3*b^2*c*f^2*e^2 - 8*a
^4*c^2*f^2*e^2 - 2*a*b^3*c^2*d*e^3 + 8*a^2*b*c^3*d*e^3 - 2*a^2*b^3*c*f*e^3 + 8*a^3*b*c^2*f*e^3 + a^2*b^2*c^2*e
^4 - 4*a^3*c^3*e^4)*sqrt(-b^2 + 4*a*c)) - (4*B*b*c*d^2*f^2 - 2*A*c^2*d^2*f^2 - 4*B*a*b*d*f^3 + 2*A*b^2*d*f^3 +
4*A*a*c*d*f^3 - 2*A*a^2*f^4 - 3*B*c^2*d^2*f*e + B*b^2*d*f^2*e + 2*B*a*c*d*f^2*e - 6*A*b*c*d*f^2*e + B*a^2*f^3
*e + 2*A*a*b*f^3*e - 2*B*b*c*d*f*e^2 + 4*A*c^2*d*f*e^2 - A*b^2*f^2*e^2 - 2*A*a*c*f^2*e^2 + B*c^2*d*e^3 + 2*A*b
*c*f*e^3 - A*c^2*e^4)*arctan((2*f*x + e)/sqrt(4*d*f - e^2))/((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*c^3*d^3*e -
2*b^3*c*d^2*f*e + 2*a*b*c^2*d^2*f*e - 2*a*b^3*d*f^2*e + 2*a^2*b*c*d*f^2*e - 2*a^3*b*f^3*e + b^2*c^2*d^2*e^2 +
2*a*c^3*d^2*e^2 + 4*a*b^2*c*d*f*e^2 - 4*a^2*c^2*d*f*e^2 + a^2*b^2*f^2*e^2 + 2*a^3*c*f^2*e^2 - 2*a*b*c^2*d*e^3
- 2*a^2*b*c*f*e^3 + a^2*c^2*e^4)*sqrt(4*d*f - e^2)) + (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 - 3*B*a*b*c^3*d^2*e + 2*A*b^2*c^3*d^2*e - 2*A*a*c^4*d^2*e - 2*B*a*b^3*c*d*f*e + 2*A*b^4*c*d*f*e
+ 2*B*a^2*b*c^2*d*f*e - 6*A*a*b^2*c^2*d*f*e + 4*A*a^2*c^3*d*f*e - B*a^2*b^3*f^2*e + A*a*b^4*f^2*e + B*a^3*b*c
*f^2*e - 2*A*a^2*b^2*c*f^2*e - 2*A*a^3*c^2*f^2*e + B*a*b^2*c^2*d*e^2 - A*b^3*c^2*d*e^2 + 2*B*a^2*c^3*d*e^2 + A
*a*b*c^3*d*e^2 + 2*B*a^2*b^2*c*f*e^2 - 2*A*a*b^3*c*f*e^2 - 2*B*a^3*c^2*f*e^2 + 5*A*a^2*b*c^2*f*e^2 - B*a^2*b*c
^2*e^3 + A*a*b^2*c^2*e^3 - 2*A*a^2*c^3*e^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 - B*b^2*c^3*d^2*e - 2*B*a*c^4*d^2*e
+ 3*A*b*c^4*d^2*e - 4*B*a*b^2*c^2*d*f*e + 2*A*b^3*c^2*d*f*e + 4*B*a^2*c^3*d*f*e - 2*A*a*b*c^3*d*f*e - B*a^2*b^
2*c*f^2*e + A*a*b^3*c*f^2*e - 2*B*a^3*c^2*f^2*e - A*a^2*b*c^2*f^2*e + 3*B*a*b*c^3*d*e^2 - A*b^2*c^3*d*e^2 - 2*
A*a*c^4*d*e^2 + 3*B*a^2*b*c^2*f*e^2 - 2*A*a*b^2*c^2*f*e^2 + 2*A*a^2*c^3*f*e^2 - 2*B*a^2*c^3*e^3 + A*a*b*c^3*e^
3)*x)/((c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2...

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Mupad [B]
time = 30.31, size = 2500, normalized size = 2.33 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 + 4*B^3*a^2*c^5*e^2*f^4 + B^3*b^
2*c^5*d^2*f^4 - 16*A^3*a*b*c^5*f^6 + 16*A^3*a*c^6*e*f^5 + 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 - 4*A^3*b^2*c^5*e*f^5 + 6*B^3*a*b^2*c^4*d*f^5 - 4*B^3*a^2*b*c^4*e*f^5 + A^
2*B*b^2*c^5*e^2*f^4 - 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 + 8*A*B^2*a^2*c^5*e*f^5 - 6*A*B^2*b^3*c^4*d*f^5 + 8*A^2*B*b^2*c^5*d*f^5 + 2*A^2*
B*b^3*c^4*e*f^5 - 4*B^3*a*b*c^5*d*e*f^4 - 4*A*B^2*a*b*c^5*e^2*f^4 + 2*A*B^2*a*b^2*c^4*e*f^5 + 2*A*B^2*b^2*c^5*
d*e*f^4 + 16*A*B^2*a*b*c^5*d*f^5 - 12*A^2*B*a*b*c^5*e*f^5 + 8*A*B^2*a*c^6*d*e*f^4 - 4*A^2*B*b*c^6*d*e*f^4))/(1
6*a^2*c^6*d^4 + a^4*b^4*f^4 + 16*a^4*c^4*e^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 - 2*a^3*b^5*e*f^3 - 64*a^3*c^5*d^3*f - 64*a^5*c^3*d*f^3 - 2*b^5*c^3*d^3*e +
2*b^6*c^2*d^3*f + a^2*b^4*c^2*e^4 - 8*a^3*b^2*c^3*e^4 + 32*a^3*c^5*d^2*e^2 + a^2*b^6*e^2*f^2 + 96*a^4*c^4*d^2*
f^2 + b^6*c^2*d^2*e^2 + 32*a^5*c^3*e^2*f^2 - 2*a*b^7*d*e*f^2 - 2*b^7*c*d^2*e*f + 54*a^2*b^4*c^2*d^2*f^2 - 112*
a^3*b^2*c^3*d^2*f^2 + 16*a*b^3*c^4*d^3*e - 2*a*b^5*c^2*d*e^3 - 32*a^2*b*c^5*d^3*e - 32*a^3*b*c^4*d*e^3 - 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 - 2*a^2*b^5*c*e^3*f - 32*a^4*b*c^3*e^3*f + 16*a^4*b^3*
c*e*f^3 - 32*a^5*b*c^2*e*f^3 - 64*a^4*c^4*d*e^2*f - 6*a*b^4*c^3*d^2*e^2 + 16*a^2*b^3*c^3*d*e^3 + 64*a^2*b^2*c^
4*d^3*f + 64*a^4*b^2*c^2*d*f^3 + 16*a^3*b^3*c^2*e^3*f - 6*a^3*b^4*c*e^2*f^2 - 48*a^2*b^3*c^3*d^2*e*f - 36*a^2*
b^4*c^2*d*e^2*f + 96*a^3*b^2*c^3*d*e^2*f - 48*a^3*b^3*c^2*d*e*f^2 + 4*a*b^6*c*d*e^2*f + 18*a*b^5*c^2*d^2*e*f +
18*a^2*b^5*c*d*e*f^2 + 32*a^3*b*c^4*d^2*e*f + 32*a^4*b*c^3*d*e*f^2) - root(48416*a^6*b^2*c^6*d^4*e^2*f^4*z^4
- 41544*a^5*b^4*c^5*d^4*e^2*f^4*z^4 - 31872*a^7*b^2*c^5*d^3*e^2*f^5*z^4 - 31872*a^5*b^2*c^7*d^5*e^2*f^3*z^4 -
29184*a^6*b^2*c^6*d^3*e^4*f^3*z^4 + 28800*a^5*b^4*c^5*d^3*e^4*f^3*z^4 + 21510*a^4*b^6*c^4*d^4*e^2*f^4*z^4 + 21
408*a^6*b^4*c^4*d^3*e^2*f^5*z^4 + 21408*a^4*b^4*c^6*d^5*e^2*f^3*z^4 - 18112*a^7*b^3*c^4*d^2*e^3*f^5*z^4 - 1811
2*a^4*b^3*c^7*d^5*e^3*f^2*z^4 - 15600*a^5*b^5*c^4*d^3*e^3*f^4*z^4 - 15600*a^4*b^5*c^5*d^4*e^3*f^3*z^4 + 15296*
a^6*b^3*c^5*d^3*e^3*f^4*z^4 + 15296*a^5*b^3*c^6*d^4*e^3*f^3*z^4 + 14016*a^7*b^2*c^5*d^2*e^4*f^4*z^4 + 14016*a^
5*b^2*c^7*d^4*e^4*f^2*z^4 - 13920*a^4*b^6*c^4*d^3*e^4*f^3*z^4 - 11648*a^6*b^3*c^5*d^2*e^5*f^3*z^4 - 11648*a^5*
b^3*c^6*d^3*e^5*f^2*z^4 + 10432*a^6*b^2*c^6*d^2*e^6*f^2*z^4 + 9008*a^6*b^5*c^3*d^2*e^3*f^5*z^4 + 9008*a^3*b^5*
c^6*d^5*e^3*f^2*z^4 + 8544*a^5*b^5*c^4*d^2*e^5*f^3*z^4 + 8544*a^4*b^5*c^5*d^3*e^5*f^2*z^4 - 8496*a^5*b^4*c^5*d
^2*e^6*f^2*z^4 + 7488*a^8*b^2*c^4*d^2*e^2*f^6*z^4 + 7488*a^4*b^2*c^8*d^6*e^2*f^2*z^4 + 7380*a^4*b^7*c^3*d^3*e^
3*f^4*z^4 + 7380*a^3*b^7*c^4*d^4*e^3*f^3*z^4 - 6720*a^3*b^8*c^3*d^4*e^2*f^4*z^4 - 5784*a^5*b^6*c^3*d^3*e^2*f^5
*z^4 - 5784*a^3*b^6*c^5*d^5*e^2*f^3*z^4 - 3440*a^6*b^4*c^4*d^2*e^4*f^4*z^4 - 3440*a^4*b^4*c^6*d^4*e^4*f^2*z^4
+ 3360*a^3*b^8*c^3*d^3*e^4*f^3*z^4 + 3140*a^4*b^6*c^4*d^2*e^6*f^2*z^4 - 2760*a^4*b^7*c^3*d^2*e^5*f^3*z^4 - 276
0*a^3*b^7*c^4*d^3*e^5*f^2*z^4 - 1764*a^5*b^7*c^2*d^2*e^3*f^5*z^4 - 1764*a^2*b^7*c^5*d^5*e^3*f^2*z^4 - 1640*a^3
*b^9*c^2*d^3*e^3*f^4*z^4 - 1640*a^2*b^9*c^3*d^4*e^3*f^3*z^4 - 1604*a^6*b^6*c^2*d^2*e^2*f^6*z^4 - 1604*a^2*b^6*
c^6*d^6*e^2*f^2*z^4 - 1500*a^5*b^6*c^3*d^2*e^4*f^4*z^4 - 1500*a^3*b^6*c^5*d^4*e^4*f^2*z^4 + 1140*a^2*b^10*c^2*
d^4*e^2*f^4*z^4 + 810*a^4*b^8*c^2*d^2*e^4*f^4*z^4 + 810*a^2*b^8*c^4*d^4*e^4*f^2*z^4 - 544*a^3*b^8*c^3*d^2*e^6*
f^2*z^4 + 416*a^3*b^9*c^2*d^2*e^5*f^3*z^4 + 416*a^2*b^9*c^3*d^3*e^5*f^2*z^4 - 384*a^2*b^10*c^2*d^3*e^4*f^3*z^4
+ 180*a^4*b^8*c^2*d^3*e^2*f^5*z^4 + 180*a^2*b^8*c^4*d^5*e^2*f^3*z^4 + 48*a^7*b^4*c^3*d^2*e^2*f^6*z^4 + 48*a^3
*b^4*c^7*d^6*e^2*f^2*z^4 + 36*a^2*b^10*c^2*d^2*e^6*f^2*z^4 - 1024*a^10*b*c^3*d*e*f^8*z^4 - 1024*a^3*b*c^10*d^8
*e*f*z^4 - 192*a^8*b^5*c*d*e*f^8*z^4 - 192*a*b^5*c^8*d^8*e*f*z^4 + 16128*a^7*b^3*c^4*d^3*e*f^6*z^4 + 16128*a^4
*b^3*c^7*d^6*e*f^3*z^4 - 11712*a^6*b^5*c^3*d^3*e*f^6*z^4 - 11712*a^3*b^5*c^6*d^6*e*f^3*z^4 + 11520*a^8*b*c^5*d
^2*e^3*f^5*z^4 + 11520*a^5*b*c^8*d^5*e^3*f^2*z^4 - 9984*a^6*b^3*c^5*d^4*e*f^5*z^4 - 9984*a^5*b^3*c^6*d^5*e*f^4
*z^4 + 8640*a^5*b^5*c^4*d^4*e*f^5*z^4 + 8640*a^4*b^5*c^5*d^5*e*f^4*z^4 - 7424*a^7*b*c^6*d^3*e^3*f^4*z^4 - 7424
*a^6*b*c^7*d^4*e^3*f^3*z^4 - 6912*a^8*b^3*c^3*d^2*e*f^7*z^4 - 6912*a^3*b^3*c^8*d^7*e*f^2*z^4 + 4800*a^7*b^3*c^
4*d*e^5*f^4*z^4 + 4800*a^4*b^3*c^7*d^4*e^5*f*z^4 + 4608*a^7*b*c^6*d^2*e^5*f^3*z^4 + 4608*a^6*b*c^7*d^3*e^5*f^2
*z^4 - 4560*a^4*b^7*c^3*d^4*e*f^5*z^4 - 4560*a^3*b^7*c^4*d^5*e*f^4*z^4 + 4176*a^5*b^7*c^2*d^3*e*f^6*z^4 + 4176
*a^2*b^7*c^5*d^6*e*f^3*z^4 + 3264*a^7*b^5*c^2*d^2*e*f^7*z^4 + 3264*a^2*b^5*c^7*d^7*e*f^2*z^4 + 3008*a^8*b^3*c^
3*d*e^3*f^6*z^4 + 3008*a^3*b^3*c^8*d^6*e^3*f*z^4 + 2880*a^6*b^3*c^5*d*e^7*f^2*z^4 + 2880*a^5*b^3*c^6*d^2*e^7*f
*z^4 - 2240*a^7*b^4*c^3*d*e^4*f^5*z^4 - 2240*a^...

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