3.9.18 \(\int \frac {1}{(d+e x) (f+g x) (a+b x+c x^2)^2} \, dx\) [818]

Optimal. Leaf size=644 \[ -\frac {b^3 e g-b^2 c (e f+d g)+2 a c^2 (e f+d g)+b c (c d f-3 a e g)+c \left (2 c^2 d f+b^2 e g-c (b e f+b d g+2 a e g)\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (c f^2-g (b f-a g)\right ) \left (a+b x+c x^2\right )}+\frac {2 c \left (2 c^2 d f+b^2 e g-c (b e f+b d g+2 a e g)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right ) \left (c f^2-g (b f-a g)\right )}+\frac {\left (b^2 e^2 g^2 (b e f+b d g-2 a e g)-2 c^3 d f \left (e^2 f^2+d e f g+d^2 g^2\right )+2 c e g \left (a^2 e^2 g^2+a b e g (e f+d g)-b^2 (e f+d g)^2\right )-c^2 \left (4 a d e^2 f g^2-b \left (e^3 f^3+5 d e^2 f^2 g+5 d^2 e f g^2+d^3 g^3\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \left (c f^2-g (b f-a g)\right )^2}+\frac {e^4 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^2 (e f-d g)}-\frac {g^4 \log (f+g x)}{(e f-d g) \left (c f^2-b f g+a g^2\right )^2}-\frac {(c e f+c d g-b e g) \left (c \left (e^2 f^2+d^2 g^2\right )+e g (2 a e g-b (e f+d g))\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^2 \left (c f^2-g (b f-a g)\right )^2} \]

[Out]

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

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Rubi [A]
time = 1.26, antiderivative size = 644, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {907, 652, 632, 212, 648, 642} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (2 c e g \left (a^2 e^2 g^2+a b e g (d g+e f)-b^2 (d g+e f)^2\right )+b^2 e^2 g^2 (-2 a e g+b d g+b e f)-c^2 \left (4 a d e^2 f g^2-b \left (d^3 g^3+5 d^2 e f g^2+5 d e^2 f^2 g+e^3 f^3\right )\right )-2 c^3 d f \left (d^2 g^2+d e f g+e^2 f^2\right )\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^2 \left (c f^2-g (b f-a g)\right )^2}+\frac {2 c \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right ) \left (c f^2-g (b f-a g)\right )}-\frac {c x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )+b c (c d f-3 a e g)+2 a c^2 (d g+e f)+b^3 e g-b^2 c (d g+e f)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right ) \left (c f^2-g (b f-a g)\right )}-\frac {\log \left (a+b x+c x^2\right ) (-b e g+c d g+c e f) \left (e g (2 a e g-b (d g+e f))+c \left (d^2 g^2+e^2 f^2\right )\right )}{2 \left (a e^2-b d e+c d^2\right )^2 \left (c f^2-g (b f-a g)\right )^2}+\frac {e^4 \log (d+e x)}{(e f-d g) \left (a e^2-b d e+c d^2\right )^2}-\frac {g^4 \log (f+g x)}{(e f-d g) \left (a g^2-b f g+c f^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(f + g*x)*(a + b*x + c*x^2)^2),x]

[Out]

-((b^3*e*g - b^2*c*(e*f + d*g) + 2*a*c^2*(e*f + d*g) + b*c*(c*d*f - 3*a*e*g) + c*(2*c^2*d*f + b^2*e*g - c*(b*e
*f + b*d*g + 2*a*e*g))*x)/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(c*f^2 - g*(b*f - a*g))*(a + b*x + c*x^2))) +
(2*c*(2*c^2*d*f + b^2*e*g - c*(b*e*f + b*d*g + 2*a*e*g))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*
c)^(3/2)*(c*d^2 - b*d*e + a*e^2)*(c*f^2 - g*(b*f - a*g))) + ((b^2*e^2*g^2*(b*e*f + b*d*g - 2*a*e*g) - 2*c^3*d*
f*(e^2*f^2 + d*e*f*g + d^2*g^2) + 2*c*e*g*(a^2*e^2*g^2 + a*b*e*g*(e*f + d*g) - b^2*(e*f + d*g)^2) - c^2*(4*a*d
*e^2*f*g^2 - b*(e^3*f^3 + 5*d*e^2*f^2*g + 5*d^2*e*f*g^2 + d^3*g^3)))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(
Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)^2*(c*f^2 - g*(b*f - a*g))^2) + (e^4*Log[d + e*x])/((c*d^2 - b*d*e +
a*e^2)^2*(e*f - d*g)) - (g^4*Log[f + g*x])/((e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^2) - ((c*e*f + c*d*g - b*e*g)*
(c*(e^2*f^2 + d^2*g^2) + e*g*(2*a*e*g - b*(e*f + d*g)))*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^2*(c*
f^2 - g*(b*f - a*g))^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 652

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), x] - Dist[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 -
4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 907

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

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

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Mathematica [A]
time = 1.66, size = 710, normalized size = 1.10 \begin {gather*} \frac {-b^3 e g+b^2 c (d g+e (f-g x))-2 c^2 (a d g+c d f x+a e (f-g x))+b c (3 a e g+c (-d f+e f x+d g x))}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) \left (-c f^2+g (b f-a g)\right ) (a+x (b+c x))}+\frac {\left (4 c^5 d^3 f^3+b^4 e^2 g^2 (b e f+b d g-2 a e g)-2 b^2 c e g \left (-6 a^2 e^2 g^2+2 a b e g (e f+d g)+b^2 \left (e^2 f^2+d e f g+d^2 g^2\right )\right )+2 c^4 d f \left (-3 b d f (e f+d g)+2 a \left (3 e^2 f^2+d e f g+3 d^2 g^2\right )\right )+c^2 \left (-12 a^3 e^3 g^3-6 a^2 b e^2 g^2 (e f+d g)+12 a b^2 e g \left (e^2 f^2+d e f g+d^2 g^2\right )+b^3 \left (e^3 f^3+d e^2 f^2 g+d^2 e f g^2+d^3 g^3\right )\right )-2 c^3 \left (-4 b^2 d^2 e f^2 g+2 a^2 e g \left (e^2 f^2-5 d e f g+d^2 g^2\right )+a b \left (3 e^3 f^3+11 d e^2 f^2 g+11 d^2 e f g^2+3 d^3 g^3\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} \left (c d^2+e (-b d+a e)\right )^2 \left (c f^2+g (-b f+a g)\right )^2}+\frac {e^4 \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^2 (e f-d g)}-\frac {g^4 \log (f+g x)}{(e f-d g) \left (c f^2+g (-b f+a g)\right )^2}-\frac {(c e f+c d g-b e g) \left (c \left (e^2 f^2+d^2 g^2\right )+e g (2 a e g-b (e f+d g))\right ) \log (a+x (b+c x))}{2 \left (c d^2+e (-b d+a e)\right )^2 \left (c f^2+g (-b f+a g)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*(f + g*x)*(a + b*x + c*x^2)^2),x]

[Out]

(-(b^3*e*g) + b^2*c*(d*g + e*(f - g*x)) - 2*c^2*(a*d*g + c*d*f*x + a*e*(f - g*x)) + b*c*(3*a*e*g + c*(-(d*f) +
e*f*x + d*g*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(-(c*f^2) + g*(b*f - a*g))*(a + x*(b + c*x))) + ((
4*c^5*d^3*f^3 + b^4*e^2*g^2*(b*e*f + b*d*g - 2*a*e*g) - 2*b^2*c*e*g*(-6*a^2*e^2*g^2 + 2*a*b*e*g*(e*f + d*g) +
b^2*(e^2*f^2 + d*e*f*g + d^2*g^2)) + 2*c^4*d*f*(-3*b*d*f*(e*f + d*g) + 2*a*(3*e^2*f^2 + d*e*f*g + 3*d^2*g^2))
+ c^2*(-12*a^3*e^3*g^3 - 6*a^2*b*e^2*g^2*(e*f + d*g) + 12*a*b^2*e*g*(e^2*f^2 + d*e*f*g + d^2*g^2) + b^3*(e^3*f
^3 + d*e^2*f^2*g + d^2*e*f*g^2 + d^3*g^3)) - 2*c^3*(-4*b^2*d^2*e*f^2*g + 2*a^2*e*g*(e^2*f^2 - 5*d*e*f*g + d^2*
g^2) + a*b*(3*e^3*f^3 + 11*d*e^2*f^2*g + 11*d^2*e*f*g^2 + 3*d^3*g^3)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])
/((-b^2 + 4*a*c)^(3/2)*(c*d^2 + e*(-(b*d) + a*e))^2*(c*f^2 + g*(-(b*f) + a*g))^2) + (e^4*Log[d + e*x])/((c*d^2
+ e*(-(b*d) + a*e))^2*(e*f - d*g)) - (g^4*Log[f + g*x])/((e*f - d*g)*(c*f^2 + g*(-(b*f) + a*g))^2) - ((c*e*f
+ c*d*g - b*e*g)*(c*(e^2*f^2 + d^2*g^2) + e*g*(2*a*e*g - b*(e*f + d*g)))*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(
-(b*d) + a*e))^2*(c*f^2 + g*(-(b*f) + a*g))^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2227\) vs. \(2(635)=1270\).
time = 0.94, size = 2228, normalized size = 3.46

method	result	size
default	\(\text {Expression too large to display}\)	\(2228\)
risch	\(\text {Expression too large to display}\)	\(29824\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(g*x+f)/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

-e^4/(d*g-e*f)/(a*e^2-b*d*e+c*d^2)^2*ln(e*x+d)+g^4/(d*g-e*f)/(a*g^2-b*f*g+c*f^2)^2*ln(g*x+f)-1/(a*e^2-b*d*e+c*
d^2)^2/(a*g^2-b*f*g+c*f^2)^2*((c*(2*a^3*c*e^3*g^3-a^2*b^2*e^3*g^3-a^2*b*c*d*e^2*g^3-a^2*b*c*e^3*f*g^2+2*a^2*c^
2*d^2*e*g^3-2*a^2*c^2*d*e^2*f*g^2+2*a^2*c^2*e^3*f^2*g+a*b^3*d*e^2*g^3+a*b^3*e^3*f*g^2-2*a*b^2*c*d^2*e*g^3-2*a*
b^2*c*e^3*f^2*g+a*b*c^2*d^3*g^3+a*b*c^2*d^2*e*f*g^2+a*b*c^2*d*e^2*f^2*g+a*b*c^2*e^3*f^3-2*a*c^3*d^3*f*g^2+2*a*
c^3*d^2*e*f^2*g-2*a*c^3*d*e^2*f^3-b^4*d*e^2*f*g^2+2*b^3*c*d^2*e*f*g^2+2*b^3*c*d*e^2*f^2*g-b^2*c^2*d^3*f*g^2-5*
b^2*c^2*d^2*e*f^2*g-b^2*c^2*d*e^2*f^3+3*b*c^3*d^3*f^2*g+3*b*c^3*d^2*e*f^3-2*c^4*d^3*f^3)/(4*a*c-b^2)*x+(3*a^3*
b*c*e^3*g^3-2*a^3*c^2*d*e^2*g^3-2*a^3*c^2*e^3*f*g^2-a^2*b^3*e^3*g^3-2*a^2*b^2*c*d*e^2*g^3-2*a^2*b^2*c*e^3*f*g^
2+5*a^2*b*c^2*d^2*e*g^3+3*a^2*b*c^2*d*e^2*f*g^2+5*a^2*b*c^2*e^3*f^2*g-2*a^2*c^3*d^3*g^3-2*a^2*c^3*d^2*e*f*g^2-
2*a^2*c^3*d*e^2*f^2*g-2*a^2*c^3*e^3*f^3+a*b^4*d*e^2*g^3+a*b^4*e^3*f*g^2-2*a*b^3*c*d^2*e*g^3+a*b^3*c*d*e^2*f*g^
2-2*a*b^3*c*e^3*f^2*g+a*b^2*c^2*d^3*g^3-3*a*b^2*c^2*d^2*e*f*g^2-3*a*b^2*c^2*d*e^2*f^2*g+a*b^2*c^2*e^3*f^3+a*b*
c^3*d^3*f*g^2+7*a*b*c^3*d^2*e*f^2*g+a*b*c^3*d*e^2*f^3-2*a*c^4*d^3*f^2*g-2*a*c^4*d^2*e*f^3-b^5*d*e^2*f*g^2+2*b^
4*c*d^2*e*f*g^2+2*b^4*c*d*e^2*f^2*g-b^3*c^2*d^3*f*g^2-4*b^3*c^2*d^2*e*f^2*g-b^3*c^2*d*e^2*f^3+2*b^2*c^3*d^3*f^
2*g+2*b^2*c^3*d^2*e*f^3-b*c^4*d^3*f^3)/(4*a*c-b^2))/(c*x^2+b*x+a)+1/(4*a*c-b^2)*(1/2*(-8*a^2*b*c^2*e^3*g^3+8*a
^2*c^3*d*e^2*g^3+8*a^2*c^3*e^3*f*g^2+2*a*b^3*c*e^3*g^3+2*a*b^2*c^2*d*e^2*g^3+2*a*b^2*c^2*e^3*f*g^2-8*a*b*c^3*d
^2*e*g^3-8*a*b*c^3*d*e^2*f*g^2-8*a*b*c^3*e^3*f^2*g+4*a*c^4*d^3*g^3+4*a*c^4*d^2*e*f*g^2+4*a*c^4*d*e^2*f^2*g+4*a
*c^4*e^3*f^3-b^4*c*d*e^2*g^3-b^4*c*e^3*f*g^2+2*b^3*c^2*d^2*e*g^3+2*b^3*c^2*d*e^2*f*g^2+2*b^3*c^2*e^3*f^2*g-b^2
*c^3*d^3*g^3-b^2*c^3*d^2*e*f*g^2-b^2*c^3*d*e^2*f^2*g-b^2*c^3*e^3*f^3)/c*ln(c*x^2+b*x+a)+2*(13*a*b*c^3*d*e^2*f^
2*g+2*a*b^4*e^3*g^3-b^5*d*e^2*g^3-b^5*e^3*f*g^2-b^3*c^2*d^3*g^3-b^3*c^2*e^3*f^3+6*a^3*c^2*e^3*g^3-2*c^5*d^3*f^
3-10*a^2*c^3*d*e^2*f*g^2+3*a*b^3*c*d*e^2*g^3+3*a*b^3*c*e^3*f*g^2-10*a*b^2*c^2*d^2*e*g^3-10*a*b^2*c^2*e^3*f^2*g
-2*a*c^4*d^2*e*f^2*g+2*b^4*c*d*e^2*f*g^2-b^3*c^2*d^2*e*f*g^2-b^3*c^2*d*e^2*f^2*g-4*b^2*c^3*d^2*e*f^2*g-1/2*(-8
*a^2*b*c^2*e^3*g^3+8*a^2*c^3*d*e^2*g^3+8*a^2*c^3*e^3*f*g^2+2*a*b^3*c*e^3*g^3+2*a*b^2*c^2*d*e^2*g^3+2*a*b^2*c^2
*e^3*f*g^2-8*a*b*c^3*d^2*e*g^3-8*a*b*c^3*d*e^2*f*g^2-8*a*b*c^3*e^3*f^2*g+4*a*c^4*d^3*g^3+4*a*c^4*d^2*e*f*g^2+4
*a*c^4*d*e^2*f^2*g+4*a*c^4*e^3*f^3-b^4*c*d*e^2*g^3-b^4*c*e^3*f*g^2+2*b^3*c^2*d^2*e*g^3+2*b^3*c^2*d*e^2*f*g^2+2
*b^3*c^2*e^3*f^2*g-b^2*c^3*d^3*g^3-b^2*c^3*d^2*e*f*g^2-b^2*c^3*d*e^2*f^2*g-b^2*c^3*e^3*f^3)*b/c+7*a^2*b*c^2*d*
e^2*g^3+7*a^2*b*c^2*e^3*f*g^2+2*b^4*c*d^2*e*g^3+2*b^4*c*e^3*f^2*g-10*a*b^2*c^2*d*e^2*f*g^2+13*a*b*c^3*d^2*e*f*
g^2-10*a^2*b^2*c*e^3*g^3+2*a^2*c^3*d^2*e*g^3+2*a^2*c^3*e^3*f^2*g+5*a*b*c^3*d^3*g^3+5*a*b*c^3*e^3*f^3-6*a*c^4*d
^3*f*g^2-6*a*c^4*d*e^2*f^3+3*b*c^4*d^3*f^2*g+3*b*c^4*d^2*e*f^3)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)
^(1/2))))

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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(1/(e*x+d)/(g*x+f)/(c*x^2+b*x+a)^2,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 deta

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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(1/(e*x+d)/(g*x+f)/(c*x^2+b*x+a)^2,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(1/(e*x+d)/(g*x+f)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3315 vs. \(2 (653) = 1306\).
time = 4.82, size = 3315, normalized size = 5.15 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(g*x+f)/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

g^5*log(abs(g*x + f))/(c^2*d*f^4*g^2 - 2*b*c*d*f^3*g^3 + b^2*d*f^2*g^4 + 2*a*c*d*f^2*g^4 - 2*a*b*d*f*g^5 + a^2
*d*g^6 - c^2*f^5*g*e + 2*b*c*f^4*g^2*e - b^2*f^3*g^3*e - 2*a*c*f^3*g^3*e + 2*a*b*f^2*g^4*e - a^2*f*g^5*e) - 1/
2*(c^2*d^3*g^3 + c^2*d^2*f*g^2*e - 2*b*c*d^2*g^3*e + c^2*d*f^2*g*e^2 - 2*b*c*d*f*g^2*e^2 + b^2*d*g^3*e^2 + 2*a
*c*d*g^3*e^2 + c^2*f^3*e^3 - 2*b*c*f^2*g*e^3 + b^2*f*g^2*e^3 + 2*a*c*f*g^2*e^3 - 2*a*b*g^3*e^3)*log(c*x^2 + b*
x + a)/(c^4*d^4*f^4 - 2*b*c^3*d^4*f^3*g + b^2*c^2*d^4*f^2*g^2 + 2*a*c^3*d^4*f^2*g^2 - 2*a*b*c^2*d^4*f*g^3 + a^
2*c^2*d^4*g^4 - 2*b*c^3*d^3*f^4*e + 4*b^2*c^2*d^3*f^3*g*e - 2*b^3*c*d^3*f^2*g^2*e - 4*a*b*c^2*d^3*f^2*g^2*e +
4*a*b^2*c*d^3*f*g^3*e - 2*a^2*b*c*d^3*g^4*e + b^2*c^2*d^2*f^4*e^2 + 2*a*c^3*d^2*f^4*e^2 - 2*b^3*c*d^2*f^3*g*e^
2 - 4*a*b*c^2*d^2*f^3*g*e^2 + b^4*d^2*f^2*g^2*e^2 + 4*a*b^2*c*d^2*f^2*g^2*e^2 + 4*a^2*c^2*d^2*f^2*g^2*e^2 - 2*
a*b^3*d^2*f*g^3*e^2 - 4*a^2*b*c*d^2*f*g^3*e^2 + a^2*b^2*d^2*g^4*e^2 + 2*a^3*c*d^2*g^4*e^2 - 2*a*b*c^2*d*f^4*e^
3 + 4*a*b^2*c*d*f^3*g*e^3 - 2*a*b^3*d*f^2*g^2*e^3 - 4*a^2*b*c*d*f^2*g^2*e^3 + 4*a^2*b^2*d*f*g^3*e^3 - 2*a^3*b*
d*g^4*e^3 + a^2*c^2*f^4*e^4 - 2*a^2*b*c*f^3*g*e^4 + a^2*b^2*f^2*g^2*e^4 + 2*a^3*c*f^2*g^2*e^4 - 2*a^3*b*f*g^3*
e^4 + a^4*g^4*e^4) - e^5*log(abs(x*e + d))/(c^2*d^5*g*e - c^2*d^4*f*e^2 - 2*b*c*d^4*g*e^2 + 2*b*c*d^3*f*e^3 +
b^2*d^3*g*e^3 + 2*a*c*d^3*g*e^3 - b^2*d^2*f*e^4 - 2*a*c*d^2*f*e^4 - 2*a*b*d^2*g*e^4 + 2*a*b*d*f*e^5 + a^2*d*g*
e^5 - a^2*f*e^6) - (4*c^5*d^3*f^3 - 6*b*c^4*d^3*f^2*g + 12*a*c^4*d^3*f*g^2 + b^3*c^2*d^3*g^3 - 6*a*b*c^3*d^3*g
^3 - 6*b*c^4*d^2*f^3*e + 8*b^2*c^3*d^2*f^2*g*e + 4*a*c^4*d^2*f^2*g*e + b^3*c^2*d^2*f*g^2*e - 22*a*b*c^3*d^2*f*
g^2*e - 2*b^4*c*d^2*g^3*e + 12*a*b^2*c^2*d^2*g^3*e - 4*a^2*c^3*d^2*g^3*e + 12*a*c^4*d*f^3*e^2 + b^3*c^2*d*f^2*
g*e^2 - 22*a*b*c^3*d*f^2*g*e^2 - 2*b^4*c*d*f*g^2*e^2 + 12*a*b^2*c^2*d*f*g^2*e^2 + 20*a^2*c^3*d*f*g^2*e^2 + b^5
*d*g^3*e^2 - 4*a*b^3*c*d*g^3*e^2 - 6*a^2*b*c^2*d*g^3*e^2 + b^3*c^2*f^3*e^3 - 6*a*b*c^3*f^3*e^3 - 2*b^4*c*f^2*g
*e^3 + 12*a*b^2*c^2*f^2*g*e^3 - 4*a^2*c^3*f^2*g*e^3 + b^5*f*g^2*e^3 - 4*a*b^3*c*f*g^2*e^3 - 6*a^2*b*c^2*f*g^2*
e^3 - 2*a*b^4*g^3*e^3 + 12*a^2*b^2*c*g^3*e^3 - 12*a^3*c^2*g^3*e^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^
2*c^4*d^4*f^4 - 4*a*c^5*d^4*f^4 - 2*b^3*c^3*d^4*f^3*g + 8*a*b*c^4*d^4*f^3*g + b^4*c^2*d^4*f^2*g^2 - 2*a*b^2*c^
3*d^4*f^2*g^2 - 8*a^2*c^4*d^4*f^2*g^2 - 2*a*b^3*c^2*d^4*f*g^3 + 8*a^2*b*c^3*d^4*f*g^3 + a^2*b^2*c^2*d^4*g^4 -
4*a^3*c^3*d^4*g^4 - 2*b^3*c^3*d^3*f^4*e + 8*a*b*c^4*d^3*f^4*e + 4*b^4*c^2*d^3*f^3*g*e - 16*a*b^2*c^3*d^3*f^3*g
*e - 2*b^5*c*d^3*f^2*g^2*e + 4*a*b^3*c^2*d^3*f^2*g^2*e + 16*a^2*b*c^3*d^3*f^2*g^2*e + 4*a*b^4*c*d^3*f*g^3*e -
16*a^2*b^2*c^2*d^3*f*g^3*e - 2*a^2*b^3*c*d^3*g^4*e + 8*a^3*b*c^2*d^3*g^4*e + b^4*c^2*d^2*f^4*e^2 - 2*a*b^2*c^3
*d^2*f^4*e^2 - 8*a^2*c^4*d^2*f^4*e^2 - 2*b^5*c*d^2*f^3*g*e^2 + 4*a*b^3*c^2*d^2*f^3*g*e^2 + 16*a^2*b*c^3*d^2*f^
3*g*e^2 + b^6*d^2*f^2*g^2*e^2 - 12*a^2*b^2*c^2*d^2*f^2*g^2*e^2 - 16*a^3*c^3*d^2*f^2*g^2*e^2 - 2*a*b^5*d^2*f*g^
3*e^2 + 4*a^2*b^3*c*d^2*f*g^3*e^2 + 16*a^3*b*c^2*d^2*f*g^3*e^2 + a^2*b^4*d^2*g^4*e^2 - 2*a^3*b^2*c*d^2*g^4*e^2
- 8*a^4*c^2*d^2*g^4*e^2 - 2*a*b^3*c^2*d*f^4*e^3 + 8*a^2*b*c^3*d*f^4*e^3 + 4*a*b^4*c*d*f^3*g*e^3 - 16*a^2*b^2*
c^2*d*f^3*g*e^3 - 2*a*b^5*d*f^2*g^2*e^3 + 4*a^2*b^3*c*d*f^2*g^2*e^3 + 16*a^3*b*c^2*d*f^2*g^2*e^3 + 4*a^2*b^4*d
*f*g^3*e^3 - 16*a^3*b^2*c*d*f*g^3*e^3 - 2*a^3*b^3*d*g^4*e^3 + 8*a^4*b*c*d*g^4*e^3 + a^2*b^2*c^2*f^4*e^4 - 4*a^
3*c^3*f^4*e^4 - 2*a^2*b^3*c*f^3*g*e^4 + 8*a^3*b*c^2*f^3*g*e^4 + a^2*b^4*f^2*g^2*e^4 - 2*a^3*b^2*c*f^2*g^2*e^4
- 8*a^4*c^2*f^2*g^2*e^4 - 2*a^3*b^3*f*g^3*e^4 + 8*a^4*b*c*f*g^3*e^4 + a^4*b^2*g^4*e^4 - 4*a^5*c*g^4*e^4)*sqrt(
-b^2 + 4*a*c)) - (b*c^4*d^3*f^3 - 2*b^2*c^3*d^3*f^2*g + 2*a*c^4*d^3*f^2*g + b^3*c^2*d^3*f*g^2 - a*b*c^3*d^3*f*
g^2 - a*b^2*c^2*d^3*g^3 + 2*a^2*c^3*d^3*g^3 - 2*b^2*c^3*d^2*f^3*e + 2*a*c^4*d^2*f^3*e + 4*b^3*c^2*d^2*f^2*g*e
- 7*a*b*c^3*d^2*f^2*g*e - 2*b^4*c*d^2*f*g^2*e + 3*a*b^2*c^2*d^2*f*g^2*e + 2*a^2*c^3*d^2*f*g^2*e + 2*a*b^3*c*d^
2*g^3*e - 5*a^2*b*c^2*d^2*g^3*e + b^3*c^2*d*f^3*e^2 - a*b*c^3*d*f^3*e^2 - 2*b^4*c*d*f^2*g*e^2 + 3*a*b^2*c^2*d*
f^2*g*e^2 + 2*a^2*c^3*d*f^2*g*e^2 + b^5*d*f*g^2*e^2 - a*b^3*c*d*f*g^2*e^2 - 3*a^2*b*c^2*d*f*g^2*e^2 - a*b^4*d*
g^3*e^2 + 2*a^2*b^2*c*d*g^3*e^2 + 2*a^3*c^2*d*g^3*e^2 - a*b^2*c^2*f^3*e^3 + 2*a^2*c^3*f^3*e^3 + 2*a*b^3*c*f^2*
g*e^3 - 5*a^2*b*c^2*f^2*g*e^3 - a*b^4*f*g^2*e^3 + 2*a^2*b^2*c*f*g^2*e^3 + 2*a^3*c^2*f*g^2*e^3 + a^2*b^3*g^3*e^
3 - 3*a^3*b*c*g^3*e^3 + (2*c^5*d^3*f^3 - 3*b*c^4*d^3*f^2*g + b^2*c^3*d^3*f*g^2 + 2*a*c^4*d^3*f*g^2 - a*b*c^3*d
^3*g^3 - 3*b*c^4*d^2*f^3*e + 5*b^2*c^3*d^2*f^2*g*e - 2*a*c^4*d^2*f^2*g*e - 2*b^3*c^2*d^2*f*g^2*e - a*b*c^3*d^2
*f*g^2*e + 2*a*b^2*c^2*d^2*g^3*e - 2*a^2*c^3*d^2*g^3*e + b^2*c^3*d*f^3*e^2 + 2*a*c^4*d*f^3*e^2 - 2*b^3*c^2*d*f
^2*g*e^2 - a*b*c^3*d*f^2*g*e^2 + b^4*c*d*f*g^2*e^2 + 2*a^2*c^3*d*f*g^2*e^2 - a*b^3*c*d*g^3*e^2 + a^2*b*c^2*d*g
^3*e^2 - a*b*c^3*f^3*e^3 + 2*a*b^2*c^2*f^2*g*e^3 - 2*a^2*c^3*f^2*g*e^3 - a*b^3*c*f*g^2*e^3 + a^2*b*c^2*f*g^2*e
^3 + a^2*b^2*c*g^3*e^3 - 2*a^3*c^2*g^3*e^3)*x)/...

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Mupad [B]
time = 32.63, size = 2500, normalized size = 3.88 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((f + g*x)*(d + e*x)*(a + b*x + c*x^2)^2),x)

[Out]

((b^3*e*g + 2*a*c^2*d*g + 2*a*c^2*e*f + b*c^2*d*f - b^2*c*d*g - b^2*c*e*f - 3*a*b*c*e*g)/(4*a*c^3*d^2*f^2 + 4*
a^3*c*e^2*g^2 - a^2*b^2*e^2*g^2 + 4*a^2*c^2*d^2*g^2 + 4*a^2*c^2*e^2*f^2 - b^2*c^2*d^2*f^2 + a*b^3*d*e*g^2 + b^
3*c*d*e*f^2 + a*b^3*e^2*f*g + b^3*c*d^2*f*g - a*b^2*c*d^2*g^2 - a*b^2*c*e^2*f^2 - b^4*d*e*f*g - 4*a*b*c^2*d*e*
f^2 - 4*a^2*b*c*d*e*g^2 - 4*a*b*c^2*d^2*f*g - 4*a^2*b*c*e^2*f*g + 4*a*b^2*c*d*e*f*g) - (x*(2*a*c^2*e*g - 2*c^3
*d*f + b*c^2*d*g + b*c^2*e*f - b^2*c*e*g))/(4*a*c^3*d^2*f^2 + 4*a^3*c*e^2*g^2 - a^2*b^2*e^2*g^2 + 4*a^2*c^2*d^
2*g^2 + 4*a^2*c^2*e^2*f^2 - b^2*c^2*d^2*f^2 + a*b^3*d*e*g^2 + b^3*c*d*e*f^2 + a*b^3*e^2*f*g + b^3*c*d^2*f*g -
a*b^2*c*d^2*g^2 - a*b^2*c*e^2*f^2 - b^4*d*e*f*g - 4*a*b*c^2*d*e*f^2 - 4*a^2*b*c*d*e*g^2 - 4*a*b*c^2*d^2*f*g -
4*a^2*b*c*e^2*f*g + 4*a*b^2*c*d*e*f*g))/(a + b*x + c*x^2) + symsum(log((12*a^2*c^5*e^6*g^6 - 3*b^2*c^5*d^2*e^4
*g^6 - 3*b^2*c^5*e^6*f^2*g^4 + 4*c^7*d^2*e^4*f^2*g^4 - 2*a*b^2*c^4*e^6*g^6 + 16*a*c^6*d^2*e^4*g^6 + 3*b^3*c^4*
d*e^5*g^6 + 16*a*c^6*e^6*f^2*g^4 + 3*b^3*c^4*e^6*f*g^5 - 4*b*c^6*d*e^5*f^2*g^4 - 4*b*c^6*d^2*e^4*f*g^5 - 16*a*
b*c^5*d*e^5*g^6 - 16*a*b*c^5*e^6*f*g^5 + 16*a*c^6*d*e^5*f*g^5)/(16*a^2*c^6*d^4*f^4 + a^4*b^4*e^4*g^4 + 16*a^4*
c^4*d^4*g^4 + 16*a^4*c^4*e^4*f^4 + b^4*c^4*d^4*f^4 + 16*a^6*c^2*e^4*g^4 + a^2*b^4*c^2*d^4*g^4 + a^2*b^4*c^2*e^
4*f^4 - 8*a^3*b^2*c^3*d^4*g^4 - 8*a^3*b^2*c^3*e^4*f^4 + a^2*b^6*d^2*e^2*g^4 + 32*a^3*c^5*d^2*e^2*f^4 + 32*a^5*
c^3*d^2*e^2*g^4 + b^6*c^2*d^2*e^2*f^4 + a^2*b^6*e^4*f^2*g^2 + 32*a^3*c^5*d^4*f^2*g^2 + 32*a^5*c^3*e^4*f^2*g^2
+ b^6*c^2*d^4*f^2*g^2 + b^8*d^2*e^2*f^2*g^2 - 8*a*b^2*c^5*d^4*f^4 - 8*a^5*b^2*c*e^4*g^4 - 2*a^3*b^5*d*e^3*g^4
- 2*b^5*c^3*d^3*e*f^4 - 2*a^3*b^5*e^4*f*g^3 - 2*b^5*c^3*d^4*f^3*g + 16*a*b^3*c^4*d^3*e*f^4 - 2*a*b^5*c^2*d*e^3
*f^4 - 32*a^2*b*c^5*d^3*e*f^4 - 32*a^3*b*c^4*d*e^3*f^4 - 2*a^2*b^5*c*d^3*e*g^4 - 32*a^4*b*c^3*d^3*e*g^4 + 16*a
^4*b^3*c*d*e^3*g^4 - 32*a^5*b*c^2*d*e^3*g^4 + 16*a*b^3*c^4*d^4*f^3*g - 2*a*b^5*c^2*d^4*f*g^3 - 32*a^2*b*c^5*d^
4*f^3*g - 32*a^3*b*c^4*d^4*f*g^3 - 2*a^2*b^5*c*e^4*f^3*g - 32*a^4*b*c^3*e^4*f^3*g + 16*a^4*b^3*c*e^4*f*g^3 - 3
2*a^5*b*c^2*e^4*f*g^3 - 2*a*b^7*d*e^3*f^2*g^2 - 2*a*b^7*d^2*e^2*f*g^3 + 4*a^2*b^6*d*e^3*f*g^3 + 4*b^6*c^2*d^3*
e*f^3*g - 2*b^7*c*d^2*e^2*f^3*g - 2*b^7*c*d^3*e*f^2*g^2 - 6*a*b^4*c^3*d^2*e^2*f^4 + 16*a^2*b^3*c^3*d*e^3*f^4 +
16*a^3*b^3*c^2*d^3*e*g^4 - 6*a^3*b^4*c*d^2*e^2*g^4 - 6*a*b^4*c^3*d^4*f^2*g^2 + 16*a^2*b^3*c^3*d^4*f*g^3 + 16*
a^3*b^3*c^2*e^4*f^3*g - 6*a^3*b^4*c*e^4*f^2*g^2 + 64*a^4*c^4*d^2*e^2*f^2*g^2 + 4*a*b^6*c*d*e^3*f^3*g + 4*a*b^6
*c*d^3*e*f*g^3 - 32*a*b^4*c^3*d^3*e*f^3*g - 32*a^3*b^4*c*d*e^3*f*g^3 - 12*a^2*b^4*c^2*d^2*e^2*f^2*g^2 + 32*a^3
*b^2*c^3*d^2*e^2*f^2*g^2 + 12*a*b^5*c^2*d^2*e^2*f^3*g + 12*a*b^5*c^2*d^3*e*f^2*g^2 - 4*a*b^6*c*d^2*e^2*f^2*g^2
+ 64*a^2*b^2*c^4*d^3*e*f^3*g - 32*a^2*b^4*c^2*d*e^3*f^3*g - 32*a^2*b^4*c^2*d^3*e*f*g^3 + 12*a^2*b^5*c*d*e^3*f
^2*g^2 + 12*a^2*b^5*c*d^2*e^2*f*g^3 - 64*a^3*b*c^4*d^2*e^2*f^3*g - 64*a^3*b*c^4*d^3*e*f^2*g^2 + 64*a^3*b^2*c^3
*d*e^3*f^3*g + 64*a^3*b^2*c^3*d^3*e*f*g^3 - 64*a^4*b*c^3*d*e^3*f^2*g^2 - 64*a^4*b*c^3*d^2*e^2*f*g^3 + 64*a^4*b
^2*c^2*d*e^3*f*g^3) - root(1120*a^6*b^2*c^6*d^9*e*f*g^9*z^4 + 1120*a^6*b^2*c^6*d*e^9*f^9*g*z^4 - 792*a^5*b^4*c
^5*d^9*e*f*g^9*z^4 - 792*a^5*b^4*c^5*d*e^9*f^9*g*z^4 + 512*a^9*b*c^4*d^4*e^6*f*g^9*z^4 + 512*a^9*b*c^4*d*e^9*f
^4*g^6*z^4 - 512*a^7*b*c^6*d^8*e^2*f*g^9*z^4 - 512*a^7*b*c^6*d*e^9*f^8*g^2*z^4 - 512*a^6*b*c^7*d^9*e*f^2*g^8*z
^4 - 512*a^6*b*c^7*d^2*e^8*f^9*g*z^4 + 512*a^4*b*c^9*d^9*e*f^6*g^4*z^4 + 512*a^4*b*c^9*d^6*e^4*f^9*g*z^4 + 256
*a^10*b*c^3*d^2*e^8*f*g^9*z^4 + 256*a^10*b*c^3*d*e^9*f^2*g^8*z^4 + 256*a^3*b*c^10*d^9*e*f^8*g^2*z^4 + 256*a^3*
b*c^10*d^8*e^2*f^9*g*z^4 - 200*a^6*b^7*c*d^4*e^6*f*g^9*z^4 - 200*a^6*b^7*c*d*e^9*f^4*g^6*z^4 - 200*a*b^7*c^6*d
^9*e*f^6*g^4*z^4 - 200*a*b^7*c^6*d^6*e^4*f^9*g*z^4 + 194*a^4*b^6*c^4*d^9*e*f*g^9*z^4 + 194*a^4*b^6*c^4*d*e^9*f
^9*g*z^4 + 144*a^5*b^8*c*d^5*e^5*f*g^9*z^4 + 144*a^5*b^8*c*d*e^9*f^5*g^5*z^4 + 144*a*b^8*c^5*d^9*e*f^5*g^5*z^4
+ 144*a*b^8*c^5*d^5*e^5*f^9*g*z^4 + 96*a^10*b^2*c^2*d*e^9*f*g^9*z^4 + 96*a^2*b^2*c^10*d^9*e*f^9*g*z^4 + 56*a^
7*b^6*c*d^3*e^7*f*g^9*z^4 + 56*a^7*b^6*c*d*e^9*f^3*g^7*z^4 + 56*a*b^6*c^7*d^9*e*f^7*g^3*z^4 + 56*a*b^6*c^7*d^7
*e^3*f^9*g*z^4 + 48*a^8*b^5*c*d^2*e^8*f*g^9*z^4 + 48*a^8*b^5*c*d*e^9*f^2*g^8*z^4 + 48*a*b^5*c^8*d^9*e*f^8*g^2*
z^4 + 48*a*b^5*c^8*d^8*e^2*f^9*g*z^4 + 20*a*b^12*c*d^6*e^4*f^4*g^6*z^4 + 20*a*b^12*c*d^4*e^6*f^6*g^4*z^4 - 16*
a^3*b^10*c*d^7*e^3*f*g^9*z^4 - 16*a^3*b^10*c*d*e^9*f^7*g^3*z^4 - 16*a^3*b^8*c^3*d^9*e*f*g^9*z^4 - 16*a^3*b^8*c
^3*d*e^9*f^9*g*z^4 - 16*a*b^12*c*d^7*e^3*f^3*g^7*z^4 - 16*a*b^12*c*d^3*e^7*f^7*g^3*z^4 - 16*a*b^10*c^3*d^9*e*f
^3*g^7*z^4 - 16*a*b^10*c^3*d^3*e^7*f^9*g*z^4 - 8*a^4*b^9*c*d^6*e^4*f*g^9*z^4 - 8*a^4*b^9*c*d*e^9*f^6*g^4*z^4 -
8*a*b^12*c*d^5*e^5*f^5*g^5*z^4 - 8*a*b^9*c^4*d^9*e*f^4*g^6*z^4 - 8*a*b^9*c^4*d^4*e^6*f^9*g*z^4 - 9984*a^7*b^2
*c^5*d^4*e^6*f^4*g^6*z^4 - 9984*a^5*b^2*c^7*d^6*e^4*f^6*g^4*z^4 - 8640*a^6*b^2*c^6*d^6*e^4*f^4*g^6*z^4 - 8640*
a^6*b^2*c^6*d^4*e^6*f^6*g^4*z^4 - 8544*a^5*b^4*...

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