\(\int \frac {1}{(d f+e f x) (a+b (d+e x)^2+c (d+e x)^4)^2} \, dx\) [650]

Optimal result

Integrand size = 33, antiderivative size = 174 \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {b \left (b^2-6 a c\right ) \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2} e f}+\frac {\log (d+e x)}{a^2 e f}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f} \]

[Out]

1/2*(b^2-2*a*c+b*c*(e*x+d)^2)/a/(-4*a*c+b^2)/e/f/(a+b*(e*x+d)^2+c*(e*x+d)^4)+1/2*b*(-6*a*c+b^2)*arctanh((b+2*c
*(e*x+d)^2)/(-4*a*c+b^2)^(1/2))/a^2/(-4*a*c+b^2)^(3/2)/e/f+ln(e*x+d)/a^2/e/f-1/4*ln(a+b*(e*x+d)^2+c*(e*x+d)^4)
/a^2/e/f

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {1156, 1128, 754, 814, 648, 632, 212, 642} \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\frac {b \left (b^2-6 a c\right ) \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 e f \left (b^2-4 a c\right )^{3/2}}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f}+\frac {\log (d+e x)}{a^2 e f}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a e f \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \]

[In]

Int[1/((d*f + e*f*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]

[Out]

(b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*e*f*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (b*(b^2 - 6*a*c)
*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/(2*a^2*(b^2 - 4*a*c)^(3/2)*e*f) + Log[d + e*x]/(a^2*e*f) -
Log[a + b*(d + e*x)^2 + c*(d + e*x)^4]/(4*a^2*e*f)

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 754

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

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 1128

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

Rule 1156

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x^2+c x^4\right )^2} \, dx,x,d+e x\right )}{e f} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x+c x^2\right )^2} \, dx,x,(d+e x)^2\right )}{2 e f} \\ & = \frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\text {Subst}\left (\int \frac {-b^2+4 a c-b c x}{x \left (a+b x+c x^2\right )} \, dx,x,(d+e x)^2\right )}{2 a \left (b^2-4 a c\right ) e f} \\ & = \frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\text {Subst}\left (\int \left (\frac {-b^2+4 a c}{a x}+\frac {b \left (b^2-5 a c\right )+c \left (b^2-4 a c\right ) x}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,(d+e x)^2\right )}{2 a \left (b^2-4 a c\right ) e f} \\ & = \frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\log (d+e x)}{a^2 e f}-\frac {\text {Subst}\left (\int \frac {b \left (b^2-5 a c\right )+c \left (b^2-4 a c\right ) x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 a^2 \left (b^2-4 a c\right ) e f} \\ & = \frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\log (d+e x)}{a^2 e f}-\frac {\text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a^2 e f}-\frac {\left (b \left (b^2-6 a c\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a^2 \left (b^2-4 a c\right ) e f} \\ & = \frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\log (d+e x)}{a^2 e f}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f}+\frac {\left (b \left (b^2-6 a c\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{2 a^2 \left (b^2-4 a c\right ) e f} \\ & = \frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2} e f}+\frac {\log (d+e x)}{a^2 e f}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\frac {\frac {2 a \left (b^2-2 a c+b c (d+e x)^2\right )}{\left (b^2-4 a c\right ) \left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )}+4 \log (d+e x)-\frac {\left (b^3-6 a b c+b^2 \sqrt {b^2-4 a c}-4 a c \sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (b^3-6 a b c-b^2 \sqrt {b^2-4 a c}+4 a c \sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}}{4 a^2 e f} \]

[In]

Integrate[1/((d*f + e*f*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]

[Out]

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

Maple [C] (verified)

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

Time = 0.74 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.32

method	result	size
default	\(\frac {-\frac {\frac {\frac {a e b c \,x^{2}}{8 a c -2 b^{2}}+\frac {b c d a x}{4 a c -b^{2}}-\frac {a \left (-b c \,d^{2}+2 a c -b^{2}\right )}{2 e \left (4 a c -b^{2}\right )}}{c \,x^{4} e^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 b d e x +b \,d^{2}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,e^{4} \textit {\_Z}^{4}+4 c d \,e^{3} \textit {\_Z}^{3}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 b d e \right ) \textit {\_Z} +d^{4} c +b \,d^{2}+a \right )}{\sum }\frac {\left (e^{3} c \left (4 a c -b^{2}\right ) \textit {\_R}^{3}+3 c d \,e^{2} \left (4 a c -b^{2}\right ) \textit {\_R}^{2}+e \left (12 a \,c^{2} d^{2}-3 b^{2} c \,d^{2}+5 a b c -b^{3}\right ) \textit {\_R} +4 a \,c^{2} d^{3}-b^{2} c \,d^{3}+5 a b c d -b^{3} d \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 c d \,e^{2} \textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 d^{3} c +b e \textit {\_R} +b d}}{2 \left (4 a c -b^{2}\right ) e}}{a^{2}}+\frac {\ln \left (e x +d \right )}{a^{2} e}}{f}\)	\(403\)
risch	\(\frac {-\frac {c \,x^{2} b e}{2 a \left (4 a c -b^{2}\right )}-\frac {x b c d}{\left (4 a c -b^{2}\right ) a}+\frac {-b c \,d^{2}+2 a c -b^{2}}{2 e a \left (4 a c -b^{2}\right )}}{f \left (c \,x^{4} e^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 b d e x +b \,d^{2}+a \right )}+\frac {\ln \left (e x +d \right )}{a^{2} e f}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 a^{5} c^{3} e^{2} f^{2}-48 a^{4} b^{2} c^{2} e^{2} f^{2}+12 a^{3} b^{4} c \,e^{2} f^{2}-a^{2} b^{6} e^{2} f^{2}\right ) \textit {\_Z}^{2}+\left (64 c^{3} a^{3} e f -48 a^{2} b^{2} c^{2} e f +12 a \,b^{4} c e f -b^{6} e f \right ) \textit {\_Z} +16 a \,c^{3}-3 b^{2} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (160 a^{5} c^{3} e^{4} f^{2}-128 a^{4} b^{2} c^{2} e^{4} f^{2}+34 a^{3} b^{4} c \,e^{4} f^{2}-3 a^{2} b^{6} e^{4} f^{2}\right ) \textit {\_R}^{2}+\left (80 a^{3} c^{3} e^{3} f -36 a^{2} b^{2} c^{2} e^{3} f +4 a \,b^{4} c \,e^{3} f \right ) \textit {\_R} +2 b^{2} c^{2} e^{2}\right ) x^{2}+\left (\left (320 a^{5} c^{3} d \,e^{3} f^{2}-256 a^{4} b^{2} c^{2} d \,e^{3} f^{2}+68 a^{3} b^{4} c d \,e^{3} f^{2}-6 a^{2} b^{6} d \,e^{3} f^{2}\right ) \textit {\_R}^{2}+\left (160 a^{3} c^{3} d \,e^{2} f -72 a^{2} b^{2} c^{2} d \,e^{2} f +8 a \,b^{4} c d \,e^{2} f \right ) \textit {\_R} +4 b^{2} c^{2} d e \right ) x +\left (160 a^{5} c^{3} d^{2} e^{2} f^{2}-128 a^{4} b^{2} c^{2} d^{2} e^{2} f^{2}+34 a^{3} b^{4} c \,d^{2} e^{2} f^{2}-3 a^{2} b^{6} d^{2} e^{2} f^{2}-16 a^{5} b \,c^{2} e^{2} f^{2}+8 a^{4} b^{3} c \,e^{2} f^{2}-a^{3} b^{5} e^{2} f^{2}\right ) \textit {\_R}^{2}+\left (80 a^{3} c^{3} d^{2} e f -36 a^{2} b^{2} c^{2} d^{2} e f +4 a \,b^{4} c \,d^{2} e f +36 a^{3} b \,c^{2} e f -17 a^{2} b^{3} c e f +2 a \,b^{5} e f \right ) \textit {\_R} +2 d^{2} c^{2} b^{2}-8 a b \,c^{2}+2 b^{3} c \right )\right )}{2}\)	\(769\)

[In]

int(1/(e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x,method=_RETURNVERBOSE)

[Out]

1/f*(-1/a^2*((1/2*a/(4*a*c-b^2)*e*b*c*x^2+b*c*d*a/(4*a*c-b^2)*x-1/2/e*a*(-b*c*d^2+2*a*c-b^2)/(4*a*c-b^2))/(c*e
^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*e*x+b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+a)+1/2/(4*a*c-b^2)/e*sum((e^3
*c*(4*a*c-b^2)*_R^3+3*c*d*e^2*(4*a*c-b^2)*_R^2+e*(12*a*c^2*d^2-3*b^2*c*d^2+5*a*b*c-b^3)*_R+4*a*c^2*d^3-b^2*c*d
^3+5*a*b*c*d-b^3*d)/(2*_R^3*c*e^3+6*_R^2*c*d*e^2+6*_R*c*d^2*e+2*c*d^3+_R*b*e+b*d)*ln(x-_R),_R=RootOf(c*e^4*_Z^
4+4*c*d*e^3*_Z^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(4*c*d^3*e+2*b*d*e)*_Z+d^4*c+b*d^2+a)))+ln(e*x+d)/a^2/e)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1178 vs. \(2 (164) = 328\).

Time = 0.54 (sec) , antiderivative size = 2486, normalized size of antiderivative = 14.29 \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x, algorithm="fricas")

[Out]

[1/4*(2*a*b^4 - 12*a^2*b^2*c + 16*a^3*c^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*e^2*x^2 + 4*(a*b^3*c - 4*a^2*b*c^2)*d*e*
x + 2*(a*b^3*c - 4*a^2*b*c^2)*d^2 + ((b^3*c - 6*a*b*c^2)*e^4*x^4 + 4*(b^3*c - 6*a*b*c^2)*d*e^3*x^3 + (b^3*c -
6*a*b*c^2)*d^4 + (b^4 - 6*a*b^2*c + 6*(b^3*c - 6*a*b*c^2)*d^2)*e^2*x^2 + a*b^3 - 6*a^2*b*c + (b^4 - 6*a*b^2*c)
*d^2 + 2*(2*(b^3*c - 6*a*b*c^2)*d^3 + (b^4 - 6*a*b^2*c)*d)*e*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*e^4*x^4 + 8*c^2*d
*e^3*x^3 + 2*c^2*d^4 + 2*(6*c^2*d^2 + b*c)*e^2*x^2 + 2*b*c*d^2 + 4*(2*c^2*d^3 + b*c*d)*e*x + b^2 - 2*a*c + (2*
c*e^2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqrt(b^2 - 4*a*c))/(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^2
*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a)) - ((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*e^4*x^4 + 4*(b^4*c - 8*a*b^2*
c^2 + 16*a^2*c^3)*d*e^3*x^3 + a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4 + (b^5
- 8*a*b^3*c + 16*a^2*b*c^2 + 6*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^2)*e^2*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*
c^2)*d^2 + 2*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d)*e*x)*log(c*e^4*x^
4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a) + 4*((b^4*c - 8*a*b^2*c
^2 + 16*a^2*c^3)*e^4*x^4 + 4*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^3*x^3 + a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 +
(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2 + 6*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^
3)*d^2)*e^2*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^2 + 2*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3 + (b^5 -
8*a*b^3*c + 16*a^2*b*c^2)*d)*e*x)*log(e*x + d))/((a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*e^5*f*x^4 + 4*(a^2*b
^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d*e^4*f*x^3 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2 + 6*(a^2*b^4*c - 8*a^3*
b^2*c^2 + 16*a^4*c^3)*d^2)*e^3*f*x^2 + 2*(2*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d^3 + (a^2*b^5 - 8*a^3*b^
3*c + 16*a^4*b*c^2)*d)*e^2*f*x + (a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3
)*d^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d^2)*e*f), 1/4*(2*a*b^4 - 12*a^2*b^2*c + 16*a^3*c^2 + 2*(a*b^3*
c - 4*a^2*b*c^2)*e^2*x^2 + 4*(a*b^3*c - 4*a^2*b*c^2)*d*e*x + 2*(a*b^3*c - 4*a^2*b*c^2)*d^2 + 2*((b^3*c - 6*a*b
*c^2)*e^4*x^4 + 4*(b^3*c - 6*a*b*c^2)*d*e^3*x^3 + (b^3*c - 6*a*b*c^2)*d^4 + (b^4 - 6*a*b^2*c + 6*(b^3*c - 6*a*
b*c^2)*d^2)*e^2*x^2 + a*b^3 - 6*a^2*b*c + (b^4 - 6*a*b^2*c)*d^2 + 2*(2*(b^3*c - 6*a*b*c^2)*d^3 + (b^4 - 6*a*b^
2*c)*d)*e*x)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*e^2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*
c)) - ((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*e^4*x^4 + 4*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^3*x^3 + a*b^4 - 8
*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2 + 6*(b^4*c
- 8*a*b^2*c^2 + 16*a^2*c^3)*d^2)*e^2*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^2 + 2*(2*(b^4*c - 8*a*b^2*c^2 +
16*a^2*c^3)*d^3 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d)*e*x)*log(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 +
b)*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a) + 4*((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*e^4*x^4 + 4*(b^4*c - 8
*a*b^2*c^2 + 16*a^2*c^3)*d*e^3*x^3 + a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4
+ (b^5 - 8*a*b^3*c + 16*a^2*b*c^2 + 6*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^2)*e^2*x^2 + (b^5 - 8*a*b^3*c + 16
*a^2*b*c^2)*d^2 + 2*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d)*e*x)*log(e
*x + d))/((a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*e^5*f*x^4 + 4*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d*e^
4*f*x^3 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2 + 6*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d^2)*e^3*f*x^2 +
2*(2*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d^3 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d)*e^2*f*x + (a^3*b
^4 - 8*a^4*b^2*c + 16*a^5*c^2 + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4
*b*c^2)*d^2)*e*f)]

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*f*x+d*f)/(a+b*(e*x+d)**2+c*(e*x+d)**4)**2,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\int { \frac {1}{{\left ({\left (e x + d\right )}^{4} c + {\left (e x + d\right )}^{2} b + a\right )}^{2} {\left (e f x + d f\right )}} \,d x } \]

[In]

integrate(1/(e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x, algorithm="maxima")

[Out]

1/2*(b*c*e^2*x^2 + 2*b*c*d*e*x + b*c*d^2 + b^2 - 2*a*c)/((a*b^2*c - 4*a^2*c^2)*e^5*f*x^4 + 4*(a*b^2*c - 4*a^2*
c^2)*d*e^4*f*x^3 + (a*b^3 - 4*a^2*b*c + 6*(a*b^2*c - 4*a^2*c^2)*d^2)*e^3*f*x^2 + 2*(2*(a*b^2*c - 4*a^2*c^2)*d^
3 + (a*b^3 - 4*a^2*b*c)*d)*e^2*f*x + ((a*b^2*c - 4*a^2*c^2)*d^4 + a^2*b^2 - 4*a^3*c + (a*b^3 - 4*a^2*b*c)*d^2)
*e*f) - integrate(((b^2*c - 4*a*c^2)*e^3*x^3 + 3*(b^2*c - 4*a*c^2)*d*e^2*x^2 + (b^2*c - 4*a*c^2)*d^3 + (b^3 -
5*a*b*c + 3*(b^2*c - 4*a*c^2)*d^2)*e*x + (b^3 - 5*a*b*c)*d)/((b^2*c - 4*a*c^2)*e^4*x^4 + 4*(b^2*c - 4*a*c^2)*d
*e^3*x^3 + (b^2*c - 4*a*c^2)*d^4 + (b^3 - 4*a*b*c + 6*(b^2*c - 4*a*c^2)*d^2)*e^2*x^2 + a*b^2 - 4*a^2*c + (b^3
- 4*a*b*c)*d^2 + 2*(2*(b^2*c - 4*a*c^2)*d^3 + (b^3 - 4*a*b*c)*d)*e*x), x)/(a^2*f) + log(e*x + d)/(a^2*e*f)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (164) = 328\).

Time = 0.40 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.81 \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=-\frac {{\left (a^{2} b^{3} c e^{3} f - 6 \, a^{3} b c^{2} e^{3} f\right )} \sqrt {b^{2} - 4 \, a c} \log \left ({\left | b e^{2} x^{2} + \sqrt {b^{2} - 4 \, a c} e^{2} x^{2} + 2 \, b d e x + 2 \, \sqrt {b^{2} - 4 \, a c} d e x + b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} + 2 \, a \right |}\right ) - {\left (a^{2} b^{3} c e^{3} f - 6 \, a^{3} b c^{2} e^{3} f\right )} \sqrt {b^{2} - 4 \, a c} \log \left ({\left | -b e^{2} x^{2} + \sqrt {b^{2} - 4 \, a c} e^{2} x^{2} - 2 \, b d e x + 2 \, \sqrt {b^{2} - 4 \, a c} d e x - b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} - 2 \, a \right |}\right )}{4 \, {\left (a^{4} b^{4} c e^{4} f^{2} - 8 \, a^{5} b^{2} c^{2} e^{4} f^{2} + 16 \, a^{6} c^{3} e^{4} f^{2}\right )}} - \frac {\log \left ({\left | c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + 6 \, c d^{2} e^{2} x^{2} + 4 \, c d^{3} e x + c d^{4} + b e^{2} x^{2} + 2 \, b d e x + b d^{2} + a \right |}\right )}{4 \, a^{2} e f} + \frac {\log \left ({\left | e x + d \right |}\right )}{a^{2} e f} + \frac {a b c e^{2} x^{2} + 2 \, a b c d e x + a b c d^{2} + a b^{2} - 2 \, a^{2} c}{2 \, {\left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + 6 \, c d^{2} e^{2} x^{2} + 4 \, c d^{3} e x + c d^{4} + b e^{2} x^{2} + 2 \, b d e x + b d^{2} + a\right )} {\left (b^{2} - 4 \, a c\right )} a^{2} e f} \]

[In]

integrate(1/(e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 14.69 (sec) , antiderivative size = 13434, normalized size of antiderivative = 77.21 \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\text {Too large to display} \]

[In]

int(1/((d*f + e*f*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x)

[Out]

((b^2 - 2*a*c + b*c*d^2)/(2*e*(a*b^2 - 4*a^2*c)) + (b*c*e*x^2)/(2*(a*b^2 - 4*a^2*c)) + (b*c*d*x)/(a*b^2 - 4*a^
2*c))/(a*f + x^2*(b*e^2*f + 6*c*d^2*e^2*f) + x*(4*c*d^3*e*f + 2*b*d*e*f) + b*d^2*f + c*d^4*f + c*e^4*f*x^4 + 4
*c*d*e^3*f*x^3) - (log((((a^2*e*f*(-(b^2*(6*a*c - b^2)^2)/(a^4*e^2*f^2*(4*a*c - b^2)^3))^(1/2) - 1)*(((a^2*e*f
*(-(b^2*(6*a*c - b^2)^2)/(a^4*e^2*f^2*(4*a*c - b^2)^3))^(1/2) - 1)*((2*b*c^2*e^16*(2*b^3 - 10*a*c^2*d^2 + b^2*
c*d^2 - 10*a*b*c))/(a*f*(4*a*c - b^2)) + (b*c^2*e^16*(a^2*e*f*(-(b^2*(6*a*c - b^2)^2)/(a^4*e^2*f^2*(4*a*c - b^
2)^3))^(1/2) - 1)*(a*b + 3*b^2*d^2 + 3*b^2*e^2*x^2 - 10*a*c*d^2 + 6*b^2*d*e*x - 10*a*c*e^2*x^2 - 20*a*c*d*e*x)
)/(a^2*f) - (2*b*c^3*e^18*x^2*(10*a*c - b^2))/(a*f*(4*a*c - b^2)) - (4*b*c^3*d*e^17*x*(10*a*c - b^2))/(a*f*(4*
a*c - b^2))))/(4*a^2*e*f) - (b*c^3*e^15*(4*b^3 - 20*a*c^2*d^2 + 6*b^2*c*d^2 - 17*a*b*c))/(a^2*f^2*(4*a*c - b^2
)^2) + (2*b*c^4*e^17*x^2*(10*a*c - 3*b^2))/(a^2*f^2*(4*a*c - b^2)^2) + (4*b*c^4*d*e^16*x*(10*a*c - 3*b^2))/(a^
2*f^2*(4*a*c - b^2)^2)))/(4*a^2*e*f) + (b^3*c^5*e^16*x^2)/(a^3*f^3*(4*a*c - b^2)^3) + (b^2*c^4*e^14*(b^2 - 4*a
*c + b*c*d^2))/(a^3*f^3*(4*a*c - b^2)^3) + (2*b^3*c^5*d*e^15*x)/(a^3*f^3*(4*a*c - b^2)^3))*((b^3*c^5*e^16*x^2)
/(a^3*f^3*(4*a*c - b^2)^3) - ((a^2*e*f*(-(b^2*(6*a*c - b^2)^2)/(a^4*e^2*f^2*(4*a*c - b^2)^3))^(1/2) + 1)*(((a^
2*e*f*(-(b^2*(6*a*c - b^2)^2)/(a^4*e^2*f^2*(4*a*c - b^2)^3))^(1/2) + 1)*((b*c^2*e^16*(a^2*e*f*(-(b^2*(6*a*c -
b^2)^2)/(a^4*e^2*f^2*(4*a*c - b^2)^3))^(1/2) + 1)*(a*b + 3*b^2*d^2 + 3*b^2*e^2*x^2 - 10*a*c*d^2 + 6*b^2*d*e*x
- 10*a*c*e^2*x^2 - 20*a*c*d*e*x))/(a^2*f) - (2*b*c^2*e^16*(2*b^3 - 10*a*c^2*d^2 + b^2*c*d^2 - 10*a*b*c))/(a*f*
(4*a*c - b^2)) + (2*b*c^3*e^18*x^2*(10*a*c - b^2))/(a*f*(4*a*c - b^2)) + (4*b*c^3*d*e^17*x*(10*a*c - b^2))/(a*
f*(4*a*c - b^2))))/(4*a^2*e*f) - (b*c^3*e^15*(4*b^3 - 20*a*c^2*d^2 + 6*b^2*c*d^2 - 17*a*b*c))/(a^2*f^2*(4*a*c
- b^2)^2) + (2*b*c^4*e^17*x^2*(10*a*c - 3*b^2))/(a^2*f^2*(4*a*c - b^2)^2) + (4*b*c^4*d*e^16*x*(10*a*c - 3*b^2)
)/(a^2*f^2*(4*a*c - b^2)^2)))/(4*a^2*e*f) + (b^2*c^4*e^14*(b^2 - 4*a*c + b*c*d^2))/(a^3*f^3*(4*a*c - b^2)^3) +
(2*b^3*c^5*d*e^15*x)/(a^3*f^3*(4*a*c - b^2)^3)))*(2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4
*c*e*f))/(2*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)) + log(
d + e*x)/(a^2*e*f) + (b*atan((x^2*((((b*(6*a*c - b^2)*((6*a*b^5*c^4*e^17*f + 80*a^3*b*c^6*e^17*f - 44*a^2*b^3*
c^5*e^17*f)/(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3) - (((2*a^2*b^7*c^3*e^18*f^2
- 36*a^3*b^5*c^4*e^18*f^2 + 192*a^4*b^3*c^5*e^18*f^2 - 320*a^5*b*c^6*e^18*f^2)/(a^3*b^6*f^3 - 64*a^6*c^3*f^3
- 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3) + ((2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f
)*(12*a^3*b^9*c^2*e^19*f^3 - 184*a^4*b^7*c^3*e^19*f^3 + 1056*a^5*b^5*c^4*e^19*f^3 - 2688*a^6*b^3*c^5*e^19*f^3
+ 2560*a^7*b*c^6*e^19*f^3))/(2*(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)*(4*a^2*b
^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)))*(2*b^6*e*f - 128*a^3*c^3*
e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f))/(2*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*
f^2 - 48*a^3*b^4*c*e^2*f^2))))/(4*a^2*e*f*(4*a*c - b^2)^(3/2)) - (((b*((2*a^2*b^7*c^3*e^18*f^2 - 36*a^3*b^5*c^
4*e^18*f^2 + 192*a^4*b^3*c^5*e^18*f^2 - 320*a^5*b*c^6*e^18*f^2)/(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f
^3 + 48*a^5*b^2*c^2*f^3) + ((2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(12*a^3*b^9*c^
2*e^19*f^3 - 184*a^4*b^7*c^3*e^19*f^3 + 1056*a^5*b^5*c^4*e^19*f^3 - 2688*a^6*b^3*c^5*e^19*f^3 + 2560*a^7*b*c^6
*e^19*f^3))/(2*(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)*(4*a^2*b^6*e^2*f^2 - 256
*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)))*(6*a*c - b^2))/(4*a^2*e*f*(4*a*c - b^2)^(
3/2)) + (b*(6*a*c - b^2)*(2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(12*a^3*b^9*c^2*e
^19*f^3 - 184*a^4*b^7*c^3*e^19*f^3 + 1056*a^5*b^5*c^4*e^19*f^3 - 2688*a^6*b^3*c^5*e^19*f^3 + 2560*a^7*b*c^6*e^
19*f^3))/(8*a^2*e*f*(4*a*c - b^2)^(3/2)*(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)
*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)))*(2*b^6*e*f - 128
*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f))/(2*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2
*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)) + (b^3*(6*a*c - b^2)^3*(12*a^3*b^9*c^2*e^19*f^3 - 184*a^4*b^7*c^3*e^19*f
^3 + 1056*a^5*b^5*c^4*e^19*f^3 - 2688*a^6*b^3*c^5*e^19*f^3 + 2560*a^7*b*c^6*e^19*f^3))/(64*a^6*e^3*f^3*(4*a*c
- b^2)^(9/2)*(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)))*(3*b^6 - 40*a^3*c^3 + 69
*a^2*b^2*c^2 - 27*a*b^4*c))/(8*a^3*c^2*(4*a*c - b^2)^(7/2)*(6*b^6 - 400*a^3*c^3 + 291*a^2*b^2*c^2 - 72*a*b^4*c
)) + (3*b*(b^4 + 11*a^2*c^2 - 7*a*b^2*c)*((((6*a*b^5*c^4*e^17*f + 80*a^3*b*c^6*e^17*f - 44*a^2*b^3*c^5*e^17*f)
/(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3) - (((2*a^2*b^7*c^3*e^18*f^2 - 36*a^3*b
^5*c^4*e^18*f^2 + 192*a^4*b^3*c^5*e^18*f^2 - 320*a^5*b*c^6*e^18*f^2)/(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^
4*c*f^3 + 48*a^5*b^2*c^2*f^3) + ((2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(12*a^3*b
^9*c^2*e^19*f^3 - 184*a^4*b^7*c^3*e^19*f^3 + 1056*a^5*b^5*c^4*e^19*f^3 - 2688*a^6*b^3*c^5*e^19*f^3 + 2560*a^7*
b*c^6*e^19*f^3))/(2*(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)*(4*a^2*b^6*e^2*f^2
- 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)))*(2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^
2*b^2*c^2*e*f - 24*a*b^4*c*e*f))/(2*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^
3*b^4*c*e^2*f^2)))*(2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f))/(2*(4*a^2*b^6*e^2*f^2
- 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)) - (b^3*c^5*e^16)/(a^3*b^6*f^3 - 64*a^
6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3) + (b*((b*((2*a^2*b^7*c^3*e^18*f^2 - 36*a^3*b^5*c^4*e^18*f^2
+ 192*a^4*b^3*c^5*e^18*f^2 - 320*a^5*b*c^6*e^18*f^2)/(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^
5*b^2*c^2*f^3) + ((2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(12*a^3*b^9*c^2*e^19*f^3
- 184*a^4*b^7*c^3*e^19*f^3 + 1056*a^5*b^5*c^4*e^19*f^3 - 2688*a^6*b^3*c^5*e^19*f^3 + 2560*a^7*b*c^6*e^19*f^3)
)/(2*(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e
^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)))*(6*a*c - b^2))/(4*a^2*e*f*(4*a*c - b^2)^(3/2)) + (b
*(6*a*c - b^2)*(2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(12*a^3*b^9*c^2*e^19*f^3 -
184*a^4*b^7*c^3*e^19*f^3 + 1056*a^5*b^5*c^4*e^19*f^3 - 2688*a^6*b^3*c^5*e^19*f^3 + 2560*a^7*b*c^6*e^19*f^3))/(
8*a^2*e*f*(4*a*c - b^2)^(3/2)*(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)*(4*a^2*b^
6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)))*(6*a*c - b^2))/(4*a^2*e*f*
(4*a*c - b^2)^(3/2)) + (b^2*(6*a*c - b^2)^2*(2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f
)*(12*a^3*b^9*c^2*e^19*f^3 - 184*a^4*b^7*c^3*e^19*f^3 + 1056*a^5*b^5*c^4*e^19*f^3 - 2688*a^6*b^3*c^5*e^19*f^3
+ 2560*a^7*b*c^6*e^19*f^3))/(32*a^4*e^2*f^2*(4*a*c - b^2)^3*(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 +
48*a^5*b^2*c^2*f^3)*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2
))))/(8*a^3*c^2*(4*a*c - b^2)^3*(6*b^6 - 400*a^3*c^3 + 291*a^2*b^2*c^2 - 72*a*b^4*c)))*(16*a^6*b^6*f^3*(4*a*c
- b^2)^(9/2) - 1024*a^9*c^3*f^3*(4*a*c - b^2)^(9/2) - 192*a^7*b^4*c*f^3*(4*a*c - b^2)^(9/2) + 768*a^8*b^2*c^2*
f^3*(4*a*c - b^2)^(9/2)))/(b^6*c^2*e^14 - 12*a*b^4*c^3*e^14 + 36*a^2*b^2*c^4*e^14) + (x*((((((b*(6*a*c - b^2)*
((2*(320*a^5*b*c^6*d*e^17*f^2 - 2*a^2*b^7*c^3*d*e^17*f^2 + 36*a^3*b^5*c^4*d*e^17*f^2 - 192*a^4*b^3*c^5*d*e^17*
f^2))/(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3) - ((2*b^6*e*f - 128*a^3*c^3*e*f +
96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(2560*a^7*b*c^6*d*e^18*f^3 + 12*a^3*b^9*c^2*d*e^18*f^3 - 184*a^4*b^7*c^3
*d*e^18*f^3 + 1056*a^5*b^5*c^4*d*e^18*f^3 - 2688*a^6*b^3*c^5*d*e^18*f^3))/((a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*
a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^
3*b^4*c*e^2*f^2))))/(4*a^2*e*f*(4*a*c - b^2)^(3/2)) - (b*(6*a*c - b^2)*(2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b
^2*c^2*e*f - 24*a*b^4*c*e*f)*(2560*a^7*b*c^6*d*e^18*f^3 + 12*a^3*b^9*c^2*d*e^18*f^3 - 184*a^4*b^7*c^3*d*e^18*f
^3 + 1056*a^5*b^5*c^4*d*e^18*f^3 - 2688*a^6*b^3*c^5*d*e^18*f^3))/(4*a^2*e*f*(4*a*c - b^2)^(3/2)*(a^3*b^6*f^3 -
64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^
2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)))*(2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f))/(
2*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)) + (b*((2*(6*a*b^
5*c^4*d*e^16*f - 44*a^2*b^3*c^5*d*e^16*f + 80*a^3*b*c^6*d*e^16*f))/(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*
c*f^3 + 48*a^5*b^2*c^2*f^3) + (((2*(320*a^5*b*c^6*d*e^17*f^2 - 2*a^2*b^7*c^3*d*e^17*f^2 + 36*a^3*b^5*c^4*d*e^1
7*f^2 - 192*a^4*b^3*c^5*d*e^17*f^2))/(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3) -
((2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(2560*a^7*b*c^6*d*e^18*f^3 + 12*a^3*b^9*c
^2*d*e^18*f^3 - 184*a^4*b^7*c^3*d*e^18*f^3 + 1056*a^5*b^5*c^4*d*e^18*f^3 - 2688*a^6*b^3*c^5*d*e^18*f^3))/((a^3
*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 +
192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)))*(2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4
*c*e*f))/(2*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)))*(6*a*
c - b^2))/(4*a^2*e*f*(4*a*c - b^2)^(3/2)) + (b^3*(6*a*c - b^2)^3*(2560*a^7*b*c^6*d*e^18*f^3 + 12*a^3*b^9*c^2*d
*e^18*f^3 - 184*a^4*b^7*c^3*d*e^18*f^3 + 1056*a^5*b^5*c^4*d*e^18*f^3 - 2688*a^6*b^3*c^5*d*e^18*f^3))/(32*a^6*e
^3*f^3*(4*a*c - b^2)^(9/2)*(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)))*(3*b^6 - 4
0*a^3*c^3 + 69*a^2*b^2*c^2 - 27*a*b^4*c))/(8*a^3*c^2*(4*a*c - b^2)^(7/2)*(6*b^6 - 400*a^3*c^3 + 291*a^2*b^2*c^
2 - 72*a*b^4*c)) + (3*b*(b^4 + 11*a^2*c^2 - 7*a*b^2*c)*((((2*(6*a*b^5*c^4*d*e^16*f - 44*a^2*b^3*c^5*d*e^16*f +
80*a^3*b*c^6*d*e^16*f))/(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3) + (((2*(320*a^
5*b*c^6*d*e^17*f^2 - 2*a^2*b^7*c^3*d*e^17*f^2 + 36*a^3*b^5*c^4*d*e^17*f^2 - 192*a^4*b^3*c^5*d*e^17*f^2))/(a^3*
b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3) - ((2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2
*c^2*e*f - 24*a*b^4*c*e*f)*(2560*a^7*b*c^6*d*e^18*f^3 + 12*a^3*b^9*c^2*d*e^18*f^3 - 184*a^4*b^7*c^3*d*e^18*f^3
+ 1056*a^5*b^5*c^4*d*e^18*f^3 - 2688*a^6*b^3*c^5*d*e^18*f^3))/((a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f
^3 + 48*a^5*b^2*c^2*f^3)*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2
*f^2)))*(2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f))/(2*(4*a^2*b^6*e^2*f^2 - 256*a^5*c
^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)))*(2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e
*f - 24*a*b^4*c*e*f))/(2*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2
*f^2)) - (2*b^3*c^5*d*e^15)/(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3) - (b*(6*a*c
- b^2)*((b*(6*a*c - b^2)*((2*(320*a^5*b*c^6*d*e^17*f^2 - 2*a^2*b^7*c^3*d*e^17*f^2 + 36*a^3*b^5*c^4*d*e^17*f^2
- 192*a^4*b^3*c^5*d*e^17*f^2))/(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3) - ((2*b
^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(2560*a^7*b*c^6*d*e^18*f^3 + 12*a^3*b^9*c^2*d*
e^18*f^3 - 184*a^4*b^7*c^3*d*e^18*f^3 + 1056*a^5*b^5*c^4*d*e^18*f^3 - 2688*a^6*b^3*c^5*d*e^18*f^3))/((a^3*b^6*
f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a
^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2))))/(4*a^2*e*f*(4*a*c - b^2)^(3/2)) - (b*(6*a*c - b^2)*(2*b^6*e*f -
128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(2560*a^7*b*c^6*d*e^18*f^3 + 12*a^3*b^9*c^2*d*e^18*f^3
- 184*a^4*b^7*c^3*d*e^18*f^3 + 1056*a^5*b^5*c^4*d*e^18*f^3 - 2688*a^6*b^3*c^5*d*e^18*f^3))/(4*a^2*e*f*(4*a*c -
b^2)^(3/2)*(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)*(4*a^2*b^6*e^2*f^2 - 256*a^
5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2))))/(4*a^2*e*f*(4*a*c - b^2)^(3/2)) + (b^2*(6*a
*c - b^2)^2*(2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(2560*a^7*b*c^6*d*e^18*f^3 + 1
2*a^3*b^9*c^2*d*e^18*f^3 - 184*a^4*b^7*c^3*d*e^18*f^3 + 1056*a^5*b^5*c^4*d*e^18*f^3 - 2688*a^6*b^3*c^5*d*e^18*
f^3))/(16*a^4*e^2*f^2*(4*a*c - b^2)^3*(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)*(
4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2))))/(8*a^3*c^2*(4*a*c
- b^2)^3*(6*b^6 - 400*a^3*c^3 + 291*a^2*b^2*c^2 - 72*a*b^4*c)))*(16*a^6*b^6*f^3*(4*a*c - b^2)^(9/2) - 1024*a^
9*c^3*f^3*(4*a*c - b^2)^(9/2) - 192*a^7*b^4*c*f^3*(4*a*c - b^2)^(9/2) + 768*a^8*b^2*c^2*f^3*(4*a*c - b^2)^(9/2
)))/(b^6*c^2*e^14 - 12*a*b^4*c^3*e^14 + 36*a^2*b^2*c^4*e^14) + (((b*((4*a*b^6*c^3*e^15*f - 33*a^2*b^4*c^4*e^15
*f + 68*a^3*b^2*c^5*e^15*f + 6*a*b^5*c^4*d^2*e^15*f + 80*a^3*b*c^6*d^2*e^15*f - 44*a^2*b^3*c^5*d^2*e^15*f)/(a^
3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3) - (((4*a^2*b^8*c^2*e^16*f^2 - 52*a^3*b^6*c
^3*e^16*f^2 + 224*a^4*b^4*c^4*e^16*f^2 - 320*a^5*b^2*c^5*e^16*f^2 - 320*a^5*b*c^6*d^2*e^16*f^2 + 2*a^2*b^7*c^3
*d^2*e^16*f^2 - 36*a^3*b^5*c^4*d^2*e^16*f^2 + 192*a^4*b^3*c^5*d^2*e^16*f^2)/(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12
*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3) + ((2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(4
*a^4*b^8*c^2*e^17*f^3 - 48*a^5*b^6*c^3*e^17*f^3 + 192*a^6*b^4*c^4*e^17*f^3 - 256*a^7*b^2*c^5*e^17*f^3 + 2560*a
^7*b*c^6*d^2*e^17*f^3 + 12*a^3*b^9*c^2*d^2*e^17*f^3 - 184*a^4*b^7*c^3*d^2*e^17*f^3 + 1056*a^5*b^5*c^4*d^2*e^17
*f^3 - 2688*a^6*b^3*c^5*d^2*e^17*f^3))/(2*(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^
3)*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)))*(2*b^6*e*f - 1
28*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f))/(2*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b
^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)))*(6*a*c - b^2))/(4*a^2*e*f*(4*a*c - b^2)^(3/2)) - (((b*(6*a*c - b^2)*(
(4*a^2*b^8*c^2*e^16*f^2 - 52*a^3*b^6*c^3*e^16*f^2 + 224*a^4*b^4*c^4*e^16*f^2 - 320*a^5*b^2*c^5*e^16*f^2 - 320*
a^5*b*c^6*d^2*e^16*f^2 + 2*a^2*b^7*c^3*d^2*e^16*f^2 - 36*a^3*b^5*c^4*d^2*e^16*f^2 + 192*a^4*b^3*c^5*d^2*e^16*f
^2)/(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3) + ((2*b^6*e*f - 128*a^3*c^3*e*f + 9
6*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(4*a^4*b^8*c^2*e^17*f^3 - 48*a^5*b^6*c^3*e^17*f^3 + 192*a^6*b^4*c^4*e^17*f
^3 - 256*a^7*b^2*c^5*e^17*f^3 + 2560*a^7*b*c^6*d^2*e^17*f^3 + 12*a^3*b^9*c^2*d^2*e^17*f^3 - 184*a^4*b^7*c^3*d^
2*e^17*f^3 + 1056*a^5*b^5*c^4*d^2*e^17*f^3 - 2688*a^6*b^3*c^5*d^2*e^17*f^3))/(2*(a^3*b^6*f^3 - 64*a^6*c^3*f^3
- 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 -
48*a^3*b^4*c*e^2*f^2))))/(4*a^2*e*f*(4*a*c - b^2)^(3/2)) + (b*(6*a*c - b^2)*(2*b^6*e*f - 128*a^3*c^3*e*f + 96*
a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(4*a^4*b^8*c^2*e^17*f^3 - 48*a^5*b^6*c^3*e^17*f^3 + 192*a^6*b^4*c^4*e^17*f^3
- 256*a^7*b^2*c^5*e^17*f^3 + 2560*a^7*b*c^6*d^2*e^17*f^3 + 12*a^3*b^9*c^2*d^2*e^17*f^3 - 184*a^4*b^7*c^3*d^2*
e^17*f^3 + 1056*a^5*b^5*c^4*d^2*e^17*f^3 - 2688*a^6*b^3*c^5*d^2*e^17*f^3))/(8*a^2*e*f*(4*a*c - b^2)^(3/2)*(a^3
*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 +
192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)))*(2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4
*c*e*f))/(2*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)) + (b^3
*(6*a*c - b^2)^3*(4*a^4*b^8*c^2*e^17*f^3 - 48*a^5*b^6*c^3*e^17*f^3 + 192*a^6*b^4*c^4*e^17*f^3 - 256*a^7*b^2*c^
5*e^17*f^3 + 2560*a^7*b*c^6*d^2*e^17*f^3 + 12*a^3*b^9*c^2*d^2*e^17*f^3 - 184*a^4*b^7*c^3*d^2*e^17*f^3 + 1056*a
^5*b^5*c^4*d^2*e^17*f^3 - 2688*a^6*b^3*c^5*d^2*e^17*f^3))/(64*a^6*e^3*f^3*(4*a*c - b^2)^(9/2)*(a^3*b^6*f^3 - 6
4*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)))*(16*a^6*b^6*f^3*(4*a*c - b^2)^(9/2) - 1024*a^9*c^3*f^
3*(4*a*c - b^2)^(9/2) - 192*a^7*b^4*c*f^3*(4*a*c - b^2)^(9/2) + 768*a^8*b^2*c^2*f^3*(4*a*c - b^2)^(9/2))*(3*b^
6 - 40*a^3*c^3 + 69*a^2*b^2*c^2 - 27*a*b^4*c))/(8*a^3*c^2*(4*a*c - b^2)^(7/2)*(b^6*c^2*e^14 - 12*a*b^4*c^3*e^1
4 + 36*a^2*b^2*c^4*e^14)*(6*b^6 - 400*a^3*c^3 + 291*a^2*b^2*c^2 - 72*a*b^4*c)) + (3*b*(b^4 + 11*a^2*c^2 - 7*a*
b^2*c)*(16*a^6*b^6*f^3*(4*a*c - b^2)^(9/2) - 1024*a^9*c^3*f^3*(4*a*c - b^2)^(9/2) - 192*a^7*b^4*c*f^3*(4*a*c -
b^2)^(9/2) + 768*a^8*b^2*c^2*f^3*(4*a*c - b^2)^(9/2))*((((4*a*b^6*c^3*e^15*f - 33*a^2*b^4*c^4*e^15*f + 68*a^3
*b^2*c^5*e^15*f + 6*a*b^5*c^4*d^2*e^15*f + 80*a^3*b*c^6*d^2*e^15*f - 44*a^2*b^3*c^5*d^2*e^15*f)/(a^3*b^6*f^3 -
64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3) - (((4*a^2*b^8*c^2*e^16*f^2 - 52*a^3*b^6*c^3*e^16*f^2
+ 224*a^4*b^4*c^4*e^16*f^2 - 320*a^5*b^2*c^5*e^16*f^2 - 320*a^5*b*c^6*d^2*e^16*f^2 + 2*a^2*b^7*c^3*d^2*e^16*f
^2 - 36*a^3*b^5*c^4*d^2*e^16*f^2 + 192*a^4*b^3*c^5*d^2*e^16*f^2)/(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*
f^3 + 48*a^5*b^2*c^2*f^3) + ((2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(4*a^4*b^8*c^
2*e^17*f^3 - 48*a^5*b^6*c^3*e^17*f^3 + 192*a^6*b^4*c^4*e^17*f^3 - 256*a^7*b^2*c^5*e^17*f^3 + 2560*a^7*b*c^6*d^
2*e^17*f^3 + 12*a^3*b^9*c^2*d^2*e^17*f^3 - 184*a^4*b^7*c^3*d^2*e^17*f^3 + 1056*a^5*b^5*c^4*d^2*e^17*f^3 - 2688
*a^6*b^3*c^5*d^2*e^17*f^3))/(2*(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)*(4*a^2*b
^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)))*(2*b^6*e*f - 128*a^3*c^3*
e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f))/(2*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*
f^2 - 48*a^3*b^4*c*e^2*f^2)))*(2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f))/(2*(4*a^2*b
^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)) - (b^4*c^4*e^14 - 4*a*b^2*
c^5*e^14 + b^3*c^5*d^2*e^14)/(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3) + (b*((b*(
6*a*c - b^2)*((4*a^2*b^8*c^2*e^16*f^2 - 52*a^3*b^6*c^3*e^16*f^2 + 224*a^4*b^4*c^4*e^16*f^2 - 320*a^5*b^2*c^5*e
^16*f^2 - 320*a^5*b*c^6*d^2*e^16*f^2 + 2*a^2*b^7*c^3*d^2*e^16*f^2 - 36*a^3*b^5*c^4*d^2*e^16*f^2 + 192*a^4*b^3*
c^5*d^2*e^16*f^2)/(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3) + ((2*b^6*e*f - 128*a
^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(4*a^4*b^8*c^2*e^17*f^3 - 48*a^5*b^6*c^3*e^17*f^3 + 192*a^6*
b^4*c^4*e^17*f^3 - 256*a^7*b^2*c^5*e^17*f^3 + 2560*a^7*b*c^6*d^2*e^17*f^3 + 12*a^3*b^9*c^2*d^2*e^17*f^3 - 184*
a^4*b^7*c^3*d^2*e^17*f^3 + 1056*a^5*b^5*c^4*d^2*e^17*f^3 - 2688*a^6*b^3*c^5*d^2*e^17*f^3))/(2*(a^3*b^6*f^3 - 6
4*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*
c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2))))/(4*a^2*e*f*(4*a*c - b^2)^(3/2)) + (b*(6*a*c - b^2)*(2*b^6*e*f - 128*a^3
*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(4*a^4*b^8*c^2*e^17*f^3 - 48*a^5*b^6*c^3*e^17*f^3 + 192*a^6*b^
4*c^4*e^17*f^3 - 256*a^7*b^2*c^5*e^17*f^3 + 2560*a^7*b*c^6*d^2*e^17*f^3 + 12*a^3*b^9*c^2*d^2*e^17*f^3 - 184*a^
4*b^7*c^3*d^2*e^17*f^3 + 1056*a^5*b^5*c^4*d^2*e^17*f^3 - 2688*a^6*b^3*c^5*d^2*e^17*f^3))/(8*a^2*e*f*(4*a*c - b
^2)^(3/2)*(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a^5*b^2*c^2*f^3)*(4*a^2*b^6*e^2*f^2 - 256*a^5*
c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2)))*(6*a*c - b^2))/(4*a^2*e*f*(4*a*c - b^2)^(3/2))
+ (b^2*(6*a*c - b^2)^2*(2*b^6*e*f - 128*a^3*c^3*e*f + 96*a^2*b^2*c^2*e*f - 24*a*b^4*c*e*f)*(4*a^4*b^8*c^2*e^1
7*f^3 - 48*a^5*b^6*c^3*e^17*f^3 + 192*a^6*b^4*c^4*e^17*f^3 - 256*a^7*b^2*c^5*e^17*f^3 + 2560*a^7*b*c^6*d^2*e^1
7*f^3 + 12*a^3*b^9*c^2*d^2*e^17*f^3 - 184*a^4*b^7*c^3*d^2*e^17*f^3 + 1056*a^5*b^5*c^4*d^2*e^17*f^3 - 2688*a^6*
b^3*c^5*d^2*e^17*f^3))/(32*a^4*e^2*f^2*(4*a*c - b^2)^3*(a^3*b^6*f^3 - 64*a^6*c^3*f^3 - 12*a^4*b^4*c*f^3 + 48*a
^5*b^2*c^2*f^3)*(4*a^2*b^6*e^2*f^2 - 256*a^5*c^3*e^2*f^2 + 192*a^4*b^2*c^2*e^2*f^2 - 48*a^3*b^4*c*e^2*f^2))))/
(8*a^3*c^2*(4*a*c - b^2)^3*(b^6*c^2*e^14 - 12*a*b^4*c^3*e^14 + 36*a^2*b^2*c^4*e^14)*(6*b^6 - 400*a^3*c^3 + 291
*a^2*b^2*c^2 - 72*a*b^4*c)))*(6*a*c - b^2))/(2*a^2*e*f*(4*a*c - b^2)^(3/2))