3.9.17 \(\int \frac {1}{(d+e x) (f+g x) (a+b x+c x^2)} \, dx\) [817]

Optimal. Leaf size=246 \[ -\frac {\left (2 c^2 d f+b^2 e g-c (b e f+b d g+2 a e g)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (c f^2-g (b f-a g)\right )}+\frac {e^2 \log (d+e x)}{\left (c d^2-b d e+a e^2\right ) (e f-d g)}-\frac {g^2 \log (f+g x)}{(e f-d g) \left (c f^2-b f g+a g^2\right )}-\frac {(c e f+c d g-b e g) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right ) \left (c f^2-g (b f-a g)\right )} \]

[Out]

e^2*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)/(-d*g+e*f)-g^2*ln(g*x+f)/(-d*g+e*f)/(a*g^2-b*f*g+c*f^2)-1/2*(-b*e*g+c*d*g+c*
e*f)*ln(c*x^2+b*x+a)/(a*e^2-b*d*e+c*d^2)/(c*f^2-g*(-a*g+b*f))-(2*c^2*d*f+b^2*e*g-c*(2*a*e*g+b*d*g+b*e*f))*arct
anh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^2)/(c*f^2-g*(-a*g+b*f))/(-4*a*c+b^2)^(1/2)

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Rubi [A]
time = 0.30, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {907, 648, 632, 212, 642} \begin {gather*} -\frac {\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 )}{\sqrt {b^2-4 a c} \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)}{2 \left (a e^2-b d e+c d^2\right ) \left (c f^2-g (b f-a g)\right )}+\frac {e^2 \log (d+e x)}{(e f-d g) \left (a e^2-b d e+c d^2\right )}-\frac {g^2 \log (f+g x)}{(e f-d g) \left (a g^2-b f g+c f^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.22, size = 246, normalized size = 1.00 \begin {gather*} \frac {\left (2 c^2 d f+b^2 e g-c (b e f+b d g+2 a e g)\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c} \left (c d^2+e (-b d+a e)\right ) \left (c f^2+g (-b f+a g)\right )}+\frac {e^2 \log (d+e x)}{\left (c d^2+e (-b d+a e)\right ) (e f-d g)}-\frac {g^2 \log (f+g x)}{(e f-d g) \left (c f^2+g (-b f+a g)\right )}-\frac {(c e f+c d g-b e g) \log (a+x (b+c x))}{2 \left (c d^2+e (-b d+a e)\right ) \left (c f^2+g (-b f+a g)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.28, size = 244, normalized size = 0.99

method	result	size
default	\(-\frac {e^{2} \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (d g -e f \right )}+\frac {g^{2} \ln \left (g x +f \right )}{\left (d g -e f \right ) \left (a \,g^{2}-b f g +c \,f^{2}\right )}+\frac {\frac {\left (b c e g -c^{2} d g -c^{2} e f \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-a c e g +b^{2} e g -b c d g -b c e f +c^{2} d f -\frac {\left (b c e g -c^{2} d g -c^{2} e f \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (a \,g^{2}-b f g +c \,f^{2}\right )}\)	\(244\)
risch	\(\text {Expression too large to display}\)	\(170453\)

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),x,method=_RETURNVERBOSE)

[Out]

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

[Out]

Timed out

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Giac [A]
time = 4.43, size = 392, normalized size = 1.59 \begin {gather*} \frac {g^{3} \log \left ({\left | g x + f \right |}\right )}{c d f^{2} g^{2} - b d f g^{3} + a d g^{4} - c f^{3} g e + b f^{2} g^{2} e - a f g^{3} e} - \frac {{\left (c d g + c f e - b g e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (c^{2} d^{2} f^{2} - b c d^{2} f g + a c d^{2} g^{2} - b c d f^{2} e + b^{2} d f g e - a b d g^{2} e + a c f^{2} e^{2} - a b f g e^{2} + a^{2} g^{2} e^{2}\right )}} - \frac {e^{3} \log \left ({\left | x e + d \right |}\right )}{c d^{3} g e - c d^{2} f e^{2} - b d^{2} g e^{2} + b d f e^{3} + a d g e^{3} - a f e^{4}} + \frac {{\left (2 \, c^{2} d f - b c d g - b c f e + b^{2} g e - 2 \, a c g e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (c^{2} d^{2} f^{2} - b c d^{2} f g + a c d^{2} g^{2} - b c d f^{2} e + b^{2} d f g e - a b d g^{2} e + a c f^{2} e^{2} - a b f g e^{2} + a^{2} g^{2} e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} \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),x, algorithm="giac")

[Out]

g^3*log(abs(g*x + f))/(c*d*f^2*g^2 - b*d*f*g^3 + a*d*g^4 - c*f^3*g*e + b*f^2*g^2*e - a*f*g^3*e) - 1/2*(c*d*g +
c*f*e - b*g*e)*log(c*x^2 + b*x + a)/(c^2*d^2*f^2 - b*c*d^2*f*g + a*c*d^2*g^2 - b*c*d*f^2*e + b^2*d*f*g*e - a*
b*d*g^2*e + a*c*f^2*e^2 - a*b*f*g*e^2 + a^2*g^2*e^2) - e^3*log(abs(x*e + d))/(c*d^3*g*e - c*d^2*f*e^2 - b*d^2*
g*e^2 + b*d*f*e^3 + a*d*g*e^3 - a*f*e^4) + (2*c^2*d*f - b*c*d*g - b*c*f*e + b^2*g*e - 2*a*c*g*e)*arctan((2*c*x
+ b)/sqrt(-b^2 + 4*a*c))/((c^2*d^2*f^2 - b*c*d^2*f*g + a*c*d^2*g^2 - b*c*d*f^2*e + b^2*d*f*g*e - a*b*d*g^2*e
+ a*c*f^2*e^2 - a*b*f*g*e^2 + a^2*g^2*e^2)*sqrt(-b^2 + 4*a*c))

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Mupad [B]
time = 19.25, size = 2500, normalized size = 10.16 \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)),x)

[Out]

(log(6*a^2*c^4*d^5*g^5 + 6*a^2*c^4*e^5*f^5 - a^3*b^3*e^5*g^5 - a^3*b^2*e^5*g^5*(b^2 - 4*a*c)^(1/2) - c^5*d^3*e
^2*f^5*(b^2 - 4*a*c)^(1/2) - c^5*d^5*f^3*g^2*(b^2 - 4*a*c)^(1/2) - 18*a^3*c^3*d^3*e^2*g^5 + b^2*c^4*d^2*e^3*f^
5 - 18*a^3*c^3*e^5*f^3*g^2 + b^2*c^4*d^5*f^2*g^3 + 4*a^4*b*c*e^5*g^5 + 4*a^4*c*e^5*g^5*(b^2 - 4*a*c)^(1/2) - 2
*a*b^2*c^3*d^5*g^5 - 2*a*b^2*c^3*e^5*f^5 + 2*a*b^5*d^2*e^3*g^5 - 10*a*c^5*d^2*e^3*f^5 + a^2*b^4*d*e^4*g^5 + b*
c^5*d^3*e^2*f^5 - 8*a^4*c^2*d*e^4*g^5 + 2*a*b^5*e^5*f^2*g^3 - 10*a*c^5*d^5*f^2*g^3 + a^2*b^4*e^5*f*g^4 + b*c^5
*d^5*f^3*g^2 - 8*a^4*c^2*e^5*f*g^4 - a^2*b^4*e^5*g^5*x - 8*a^4*c^2*e^5*g^5*x - 2*b^3*c^3*d^5*g^5*x - 2*b^3*c^3
*e^5*f^5*x + 2*b^6*d^2*e^3*g^5*x + 2*c^6*d^3*e^2*f^5*x + 2*b^6*e^5*f^2*g^3*x + 2*c^6*d^5*f^3*g^2*x - 2*a*b*c^3
*d^5*g^5*(b^2 - 4*a*c)^(1/2) - 2*a*b*c^3*e^5*f^5*(b^2 - 4*a*c)^(1/2) + 7*a*c^4*d*e^4*f^5*(b^2 - 4*a*c)^(1/2) +
7*a*c^4*d^5*f*g^4*(b^2 - 4*a*c)^(1/2) + 2*c^5*d^4*e*f^4*g*(b^2 - 4*a*c)^(1/2) + 3*a*c^4*d^5*g^5*x*(b^2 - 4*a*
c)^(1/2) + 3*a*c^4*e^5*f^5*x*(b^2 - 4*a*c)^(1/2) + 6*a*b^3*c^2*d^4*e*g^5 - 6*a*b^4*c*d^3*e^2*g^5 - 21*a^2*b*c^
3*d^4*e*g^5 - 2*a^3*b^2*c*d*e^4*g^5 + 6*a*b^3*c^2*e^5*f^4*g - 6*a*b^4*c*e^5*f^3*g^2 - 21*a^2*b*c^3*e^5*f^4*g -
2*a^3*b^2*c*e^5*f*g^4 + 10*a*c^5*d^3*e^2*f^4*g + 10*a*c^5*d^4*e*f^3*g^2 + 26*a^2*c^4*d*e^4*f^4*g + 26*a^2*c^4
*d^4*e*f*g^4 + 6*a^3*b^2*c*e^5*g^5*x - 3*b*c^5*d^2*e^3*f^5*x + 14*a^2*c^4*d^4*e*g^5*x + 5*b^2*c^4*d*e^4*f^5*x
+ 6*b^4*c^2*d^4*e*g^5*x - 6*b^5*c*d^3*e^2*g^5*x - 3*b*c^5*d^5*f^2*g^3*x + 14*a^2*c^4*e^5*f^4*g*x + 5*b^2*c^4*d
^5*f*g^4*x + 6*b^4*c^2*e^5*f^4*g*x - 6*b^5*c*e^5*f^3*g^2*x + 2*a*b^4*d^2*e^3*g^5*(b^2 - 4*a*c)^(1/2) + a^2*b^3
*d*e^4*g^5*(b^2 - 4*a*c)^(1/2) - b*c^4*d^2*e^3*f^5*(b^2 - 4*a*c)^(1/2) - 7*a^2*c^3*d^4*e*g^5*(b^2 - 4*a*c)^(1/
2) + 2*a*b^4*e^5*f^2*g^3*(b^2 - 4*a*c)^(1/2) + a^2*b^3*e^5*f*g^4*(b^2 - 4*a*c)^(1/2) - b*c^4*d^5*f^2*g^3*(b^2
- 4*a*c)^(1/2) - 7*a^2*c^3*e^5*f^4*g*(b^2 - 4*a*c)^(1/2) - a^2*b^3*e^5*g^5*x*(b^2 - 4*a*c)^(1/2) - 2*b^2*c^3*d
^5*g^5*x*(b^2 - 4*a*c)^(1/2) - 2*b^2*c^3*e^5*f^5*x*(b^2 - 4*a*c)^(1/2) + 2*b^5*d^2*e^3*g^5*x*(b^2 - 4*a*c)^(1/
2) - 5*c^5*d^2*e^3*f^5*x*(b^2 - 4*a*c)^(1/2) + 2*b^5*e^5*f^2*g^3*x*(b^2 - 4*a*c)^(1/2) - 5*c^5*d^5*f^2*g^3*x*(
b^2 - 4*a*c)^(1/2) - 13*a^2*b^3*c*d^2*e^3*g^5 + 21*a^3*b*c^2*d^2*e^3*g^5 - 13*a^2*b^3*c*e^5*f^2*g^3 + 21*a^3*b
*c^2*e^5*f^2*g^3 + 2*a^3*c^3*d*e^4*f^2*g^3 + 2*a^3*c^3*d^2*e^3*f*g^4 - b^2*c^4*d^3*e^2*f^4*g - b^2*c^4*d^4*e*f
^3*g^2 - b^3*c^3*d^2*e^3*f^4*g - b^3*c^3*d^4*e*f^2*g^3 - b^5*c*d^2*e^3*f^2*g^3 - 10*a^3*c^3*d^2*e^3*g^5*x - 10
*a^3*c^3*e^5*f^2*g^3*x + 3*a*b*c^4*d*e^4*f^5 + 5*a^3*c^2*d^2*e^3*g^5*(b^2 - 4*a*c)^(1/2) + 3*a*b*c^4*d^5*f*g^4
+ 5*a^3*c^2*e^5*f^2*g^3*(b^2 - 4*a*c)^(1/2) - 5*a*b^5*d*e^4*f*g^4 - 2*b*c^5*d^4*e*f^4*g + 7*a*b*c^4*d^5*g^5*x
+ 7*a*b*c^4*e^5*f^5*x + a*b^5*d*e^4*g^5*x - 14*a*c^5*d*e^4*f^5*x + a*b^5*e^5*f*g^4*x - 14*a*c^5*d^5*f*g^4*x -
5*b^6*d*e^4*f*g^4*x - 4*c^6*d^4*e*f^4*g*x + 27*a^2*b^2*c^2*d^3*e^2*g^5 + 27*a^2*b^2*c^2*e^5*f^3*g^2 - 40*a^2*
c^4*d^2*e^3*f^3*g^2 - 40*a^2*c^4*d^3*e^2*f^2*g^3 + b^3*c^3*d^3*e^2*f^3*g^2 + b^4*c^2*d^2*e^3*f^3*g^2 + b^4*c^2
*d^3*e^2*f^2*g^3 + 32*a*b^3*c^2*d^3*e^2*g^5*x - 35*a^2*b*c^3*d^3*e^2*g^5*x + 32*a*b^3*c^2*e^5*f^3*g^2*x - 35*a
^2*b*c^3*e^5*f^3*g^2*x + 48*a*c^5*d^3*e^2*f^3*g^2*x + 14*a^2*c^4*d*e^4*f^3*g^2*x + 14*a^2*c^4*d^3*e^2*f*g^4*x
+ 3*b^2*c^4*d^2*e^3*f^4*g*x + 3*b^2*c^4*d^4*e*f^2*g^3*x + 4*b^4*c^2*d*e^4*f^3*g^2*x + 4*b^4*c^2*d^3*e^2*f*g^4*
x + 13*a^2*b*c^2*d^3*e^2*g^5*(b^2 - 4*a*c)^(1/2) - 7*a^2*b^2*c*d^2*e^3*g^5*(b^2 - 4*a*c)^(1/2) + 13*a^2*b*c^2*
e^5*f^3*g^2*(b^2 - 4*a*c)^(1/2) - 7*a^2*b^2*c*e^5*f^2*g^3*(b^2 - 4*a*c)^(1/2) - 24*a*c^4*d^3*e^2*f^3*g^2*(b^2
- 4*a*c)^(1/2) - 7*a^2*c^3*d*e^4*f^3*g^2*(b^2 - 4*a*c)^(1/2) - 7*a^2*c^3*d^3*e^2*f*g^4*(b^2 - 4*a*c)^(1/2) + b
^2*c^3*d^2*e^3*f^4*g*(b^2 - 4*a*c)^(1/2) + b^2*c^3*d^4*e*f^2*g^3*(b^2 - 4*a*c)^(1/2) + b^4*c*d^2*e^3*f^2*g^3*(
b^2 - 4*a*c)^(1/2) - 9*a^2*c^3*d^3*e^2*g^5*x*(b^2 - 4*a*c)^(1/2) - 9*a^2*c^3*e^5*f^3*g^2*x*(b^2 - 4*a*c)^(1/2)
+ 10*a*b^2*c^3*d^2*e^3*f^3*g^2 + 10*a*b^2*c^3*d^3*e^2*f^2*g^3 - 23*a*b^3*c^2*d^2*e^3*f^2*g^3 + 96*a^2*b*c^3*d
^2*e^3*f^2*g^3 - 39*a^2*b^2*c^2*d*e^4*f^2*g^3 - 39*a^2*b^2*c^2*d^2*e^3*f*g^4 + 27*a^2*b^2*c^2*d^2*e^3*g^5*x +
27*a^2*b^2*c^2*e^5*f^2*g^3*x - 48*a^2*c^4*d^2*e^3*f^2*g^3*x - 18*b^2*c^4*d^3*e^2*f^3*g^2*x + 17*b^3*c^3*d^2*e^
3*f^3*g^2*x + 17*b^3*c^3*d^3*e^2*f^2*g^3*x - 27*b^4*c^2*d^2*e^3*f^2*g^3*x - 4*a^3*b*c*d*e^4*g^5*(b^2 - 4*a*c)^
(1/2) - 4*a^3*b*c*e^5*f*g^4*(b^2 - 4*a*c)^(1/2) - 5*a*b^4*d*e^4*f*g^4*(b^2 - 4*a*c)^(1/2) + 4*a^3*b*c*e^5*g^5*
x*(b^2 - 4*a*c)^(1/2) + a*b^4*d*e^4*g^5*x*(b^2 - 4*a*c)^(1/2) + 5*b*c^4*d*e^4*f^5*x*(b^2 - 4*a*c)^(1/2) + a*b^
4*e^5*f*g^4*x*(b^2 - 4*a*c)^(1/2) + 5*b*c^4*d^5*f*g^4*x*(b^2 - 4*a*c)^(1/2) - 5*b^5*d*e^4*f*g^4*x*(b^2 - 4*a*c
)^(1/2) + 7*a*b*c^4*d^2*e^3*f^4*g + 7*a*b*c^4*d^4*e*f^2*g^3 - 10*a*b^2*c^3*d*e^4*f^4*g - 10*a*b^2*c^3*d^4*e*f*
g^4 + 10*a*b^4*c*d*e^4*f^2*g^3 + 10*a*b^4*c*d^2*e^3*f*g^4 + 19*a^2*b^3*c*d*e^4*f*g^4 + 2*a^3*b*c^2*d*e^4*f*g^4
+ 24*a^2*c^3*d^2*e^3*f^2*g^3*(b^2 - 4*a*c)^(1/...

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