3.658 \(\int \frac {1}{(d f+e f x) (a+b (d+e x)^2+c (d+e x)^4)^3} \, dx\)

Optimal. Leaf size=270 \[ -\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^3 e f}+\frac {\log (d+e x)}{a^3 e f}+\frac {16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2-15 a b^2 c+2 b^4}{4 a^2 e f \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 e f \left (b^2-4 a c\right )^{5/2}}+\frac {-2 a c+b^2+b c (d+e x)^2}{4 a e f \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \]

[Out]

1/4*(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)^2+1/4*(2*b^4-15*a*b^2*c+16*a^2*c^
2+2*b*c*(-7*a*c+b^2)*(e*x+d)^2)/a^2/(-4*a*c+b^2)^2/e/f/(a+b*(e*x+d)^2+c*(e*x+d)^4)+1/2*b*(30*a^2*c^2-10*a*b^2*
c+b^4)*arctanh((b+2*c*(e*x+d)^2)/(-4*a*c+b^2)^(1/2))/a^3/(-4*a*c+b^2)^(5/2)/e/f+ln(e*x+d)/a^3/e/f-1/4*ln(a+b*(
e*x+d)^2+c*(e*x+d)^4)/a^3/e/f

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Rubi [A]  time = 0.50, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1142, 1114, 740, 822, 800, 634, 618, 206, 628} \[ \frac {16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2-15 a b^2 c+2 b^4}{4 a^2 e f \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 e f \left (b^2-4 a c\right )^{5/2}}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^3 e f}+\frac {\log (d+e x)}{a^3 e f}+\frac {-2 a c+b^2+b c (d+e x)^2}{4 a e f \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((d*f + e*f*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]

[Out]

(b^2 - 2*a*c + b*c*(d + e*x)^2)/(4*a*(b^2 - 4*a*c)*e*f*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (2*b^4 - 15*a*
b^2*c + 16*a^2*c^2 + 2*b*c*(b^2 - 7*a*c)*(d + e*x)^2)/(4*a^2*(b^2 - 4*a*c)^2*e*f*(a + b*(d + e*x)^2 + c*(d + e
*x)^4)) + (b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/(2*a^3*(b^2 - 4
*a*c)^(5/2)*e*f) + Log[d + e*x]/(a^3*e*f) - Log[a + b*(d + e*x)^2 + c*(d + e*x)^4]/(4*a^3*e*f)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

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

Rule 740

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 800

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 822

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

Rule 1114

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 1142

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*} \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \left (a+b x^2+c x^4\right )^3} \, dx,x,d+e x\right )}{e f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \left (a+b x+c x^2\right )^3} \, dx,x,(d+e x)^2\right )}{2 e f}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {-2 \left (b^2-4 a c\right )-3 b c x}{x \left (a+b x+c x^2\right )^2} \, dx,x,(d+e x)^2\right )}{4 a \left (b^2-4 a c\right ) e f}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\operatorname {Subst}\left (\int \frac {2 \left (b^2-4 a c\right )^2+2 b c \left (b^2-7 a c\right ) x}{x \left (a+b x+c x^2\right )} \, dx,x,(d+e x)^2\right )}{4 a^2 \left (b^2-4 a c\right )^2 e f}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\operatorname {Subst}\left (\int \left (\frac {2 \left (-b^2+4 a c\right )^2}{a x}+\frac {2 \left (-b \left (b^4-9 a b^2 c+23 a^2 c^2\right )-c \left (b^2-4 a c\right )^2 x\right )}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,(d+e x)^2\right )}{4 a^2 \left (b^2-4 a c\right )^2 e f}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\log (d+e x)}{a^3 e f}+\frac {\operatorname {Subst}\left (\int \frac {-b \left (b^4-9 a b^2 c+23 a^2 c^2\right )-c \left (b^2-4 a c\right )^2 x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 e f}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\log (d+e x)}{a^3 e f}-\frac {\operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a^3 e f}-\frac {\left (b \left (b^4-10 a b^2 c+30 a^2 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a^3 \left (b^2-4 a c\right )^2 e f}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\log (d+e x)}{a^3 e f}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^3 e f}+\frac {\left (b \left (b^4-10 a b^2 c+30 a^2 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 e f}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {b \left (b^4-10 a b^2 c+30 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{5/2} e f}+\frac {\log (d+e x)}{a^3 e f}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^3 e f}\\ \end {align*}

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Mathematica [A]  time = 3.93, size = 394, normalized size = 1.46 \[ \frac {\frac {a^2 \left (2 a c-b^2-b c (d+e x)^2\right )}{\left (4 a c-b^2\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {a \left (16 a^2 c^2-15 a b^2 c-14 a b c^2 (d+e x)^2+2 b^4+2 b^3 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^2 \left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )}-\frac {\left (16 a^2 c^2 \sqrt {b^2-4 a c}+30 a^2 b c^2-10 a b^3 c-8 a b^2 c \sqrt {b^2-4 a c}+b^4 \sqrt {b^2-4 a c}+b^5\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {\left (-16 a^2 c^2 \sqrt {b^2-4 a c}+30 a^2 b c^2-10 a b^3 c+8 a b^2 c \sqrt {b^2-4 a c}-b^4 \sqrt {b^2-4 a c}+b^5\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{5/2}}+4 \log (d+e x)}{4 a^3 e f} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d*f + e*f*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]

[Out]

((a^2*(-b^2 + 2*a*c - b*c*(d + e*x)^2))/((-b^2 + 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (a*(2*b^4 - 1
5*a*b^2*c + 16*a^2*c^2 + 2*b^3*c*(d + e*x)^2 - 14*a*b*c^2*(d + e*x)^2))/((b^2 - 4*a*c)^2*(a + (d + e*x)^2*(b +
c*(d + e*x)^2))) + 4*Log[d + e*x] - ((b^5 - 10*a*b^3*c + 30*a^2*b*c^2 + b^4*Sqrt[b^2 - 4*a*c] - 8*a*b^2*c*Sqr
t[b^2 - 4*a*c] + 16*a^2*c^2*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)^(5/
2) + ((b^5 - 10*a*b^3*c + 30*a^2*b*c^2 - b^4*Sqrt[b^2 - 4*a*c] + 8*a*b^2*c*Sqrt[b^2 - 4*a*c] - 16*a^2*c^2*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)^(5/2))/(4*a^3*e*f)

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fricas [B]  time = 4.92, size = 9926, normalized size = 36.76 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="fricas")

[Out]

[1/4*(2*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*e^6*x^6 + 12*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*d
*e^5*x^5 + (4*a*b^6*c - 45*a^2*b^4*c^2 + 132*a^3*b^2*c^3 - 64*a^4*c^4 + 30*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^
3*b*c^4)*d^2)*e^4*x^4 + 3*a^2*b^6 - 33*a^3*b^4*c + 108*a^4*b^2*c^2 - 96*a^5*c^3 + 2*(a*b^5*c^2 - 11*a^2*b^3*c^
3 + 28*a^3*b*c^4)*d^6 + 4*(10*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*d^3 + (4*a*b^6*c - 45*a^2*b^4*c^2 +
132*a^3*b^2*c^3 - 64*a^4*c^4)*d)*e^3*x^3 + (4*a*b^6*c - 45*a^2*b^4*c^2 + 132*a^3*b^2*c^3 - 64*a^4*c^4)*d^4 + 2
*(a*b^7 - 10*a^2*b^5*c + 23*a^3*b^3*c^2 + 4*a^4*b*c^3 + 15*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*d^4 + 3
*(4*a*b^6*c - 45*a^2*b^4*c^2 + 132*a^3*b^2*c^3 - 64*a^4*c^4)*d^2)*e^2*x^2 + 2*(a*b^7 - 10*a^2*b^5*c + 23*a^3*b
^3*c^2 + 4*a^4*b*c^3)*d^2 + 4*(3*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*d^5 + (4*a*b^6*c - 45*a^2*b^4*c^2
+ 132*a^3*b^2*c^3 - 64*a^4*c^4)*d^3 + (a*b^7 - 10*a^2*b^5*c + 23*a^3*b^3*c^2 + 4*a^4*b*c^3)*d)*e*x + ((b^5*c^
2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*e^8*x^8 + 8*(b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d*e^7*x^7 + 2*(b^6*c - 10
*a*b^4*c^2 + 30*a^2*b^2*c^3 + 14*(b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^2)*e^6*x^6 + 4*(14*(b^5*c^2 - 10*a*
b^3*c^3 + 30*a^2*b*c^4)*d^3 + 3*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d)*e^5*x^5 + (b^5*c^2 - 10*a*b^3*c^3 +
30*a^2*b*c^4)*d^8 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3 + 70*(b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*
c^4)*d^4 + 30*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^2)*e^4*x^4 + a^2*b^5 - 10*a^3*b^3*c + 30*a^4*b*c^2 + 2
*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^6 + 4*(14*(b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^5 + 10*(b^6*c -
10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^3 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*d)*e^3*x^3 + (b^7 - 8*
a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*d^4 + 2*(a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2 + 14*(b^5*c^2 - 10*a*b
^3*c^3 + 30*a^2*b*c^4)*d^6 + 15*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^4 + 3*(b^7 - 8*a*b^5*c + 10*a^2*b^3*
c^2 + 60*a^3*b*c^3)*d^2)*e^2*x^2 + 2*(a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*d^2 + 4*(2*(b^5*c^2 - 10*a*b^3*c^
3 + 30*a^2*b*c^4)*d^7 + 3*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^5 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60
*a^3*b*c^3)*d^3 + (a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*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^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*e^8*x^8
+ 8*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d*e^7*x^7 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c
^3 - 64*a^3*b*c^4 + 14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2)*e^6*x^6 + 4*(14*(b^6*c^2 -
12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^3 + 3*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d)*
e^5*x^5 + (b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^8 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32
*a^3*b^2*c^3 - 128*a^4*c^4 + 70*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 + 30*(b^7*c - 12*a*
b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^2)*e^4*x^4 + a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 +
2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^6 + 4*(14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4
- 64*a^3*c^5)*d^5 + 10*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3 + (b^8 - 10*a*b^6*c + 24*a^
2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d)*e^3*x^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 1
28*a^4*c^4)*d^4 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3 + 14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^
2*b^2*c^4 - 64*a^3*c^5)*d^6 + 15*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^4 + 3*(b^8 - 10*a*b^
6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2)*e^2*x^2 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 -
64*a^4*b*c^3)*d^2 + 4*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^7 + 3*(b^7*c - 12*a*b^5*c^2
+ 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^5 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^3
+ (a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*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^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*
c^5)*e^8*x^8 + 8*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d*e^7*x^7 + 2*(b^7*c - 12*a*b^5*c^2 +
48*a^2*b^3*c^3 - 64*a^3*b*c^4 + 14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2)*e^6*x^6 + 4*(14
*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^3 + 3*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^
3*b*c^4)*d)*e^5*x^5 + (b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^8 + (b^8 - 10*a*b^6*c + 24*a^2*
b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4 + 70*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 + 30*(b
^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^2)*e^4*x^4 + a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 -
64*a^5*c^3 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^6 + 4*(14*(b^6*c^2 - 12*a*b^4*c^3 + 48
*a^2*b^2*c^4 - 64*a^3*c^5)*d^5 + 10*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3 + (b^8 - 10*a*b
^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d)*e^3*x^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3
*b^2*c^3 - 128*a^4*c^4)*d^4 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3 + 14*(b^6*c^2 - 12*a*b^4
*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^6 + 15*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^4 + 3*(b
^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2)*e^2*x^2 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a
^3*b^3*c^2 - 64*a^4*b*c^3)*d^2 + 4*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^7 + 3*(b^7*c -
12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^5 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*
a^4*c^4)*d^3 + (a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d)*e*x)*log(e*x + d))/((a^3*b^6*c^2 - 12
*a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*e^9*f*x^8 + 8*(a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*
a^6*c^5)*d*e^8*f*x^7 + 2*(a^3*b^7*c - 12*a^4*b^5*c^2 + 48*a^5*b^3*c^3 - 64*a^6*b*c^4 + 14*(a^3*b^6*c^2 - 12*a^
4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*d^2)*e^7*f*x^6 + 4*(14*(a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^5*b^2*c^4
- 64*a^6*c^5)*d^3 + 3*(a^3*b^7*c - 12*a^4*b^5*c^2 + 48*a^5*b^3*c^3 - 64*a^6*b*c^4)*d)*e^6*f*x^5 + (a^3*b^8 -
10*a^4*b^6*c + 24*a^5*b^4*c^2 + 32*a^6*b^2*c^3 - 128*a^7*c^4 + 70*(a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^5*b^2*c
^4 - 64*a^6*c^5)*d^4 + 30*(a^3*b^7*c - 12*a^4*b^5*c^2 + 48*a^5*b^3*c^3 - 64*a^6*b*c^4)*d^2)*e^5*f*x^4 + 4*(14*
(a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*d^5 + 10*(a^3*b^7*c - 12*a^4*b^5*c^2 + 48*a^5*b^3
*c^3 - 64*a^6*b*c^4)*d^3 + (a^3*b^8 - 10*a^4*b^6*c + 24*a^5*b^4*c^2 + 32*a^6*b^2*c^3 - 128*a^7*c^4)*d)*e^4*f*x
^3 + 2*(a^4*b^7 - 12*a^5*b^5*c + 48*a^6*b^3*c^2 - 64*a^7*b*c^3 + 14*(a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^5*b^2
*c^4 - 64*a^6*c^5)*d^6 + 15*(a^3*b^7*c - 12*a^4*b^5*c^2 + 48*a^5*b^3*c^3 - 64*a^6*b*c^4)*d^4 + 3*(a^3*b^8 - 10
*a^4*b^6*c + 24*a^5*b^4*c^2 + 32*a^6*b^2*c^3 - 128*a^7*c^4)*d^2)*e^3*f*x^2 + 4*(2*(a^3*b^6*c^2 - 12*a^4*b^4*c^
3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*d^7 + 3*(a^3*b^7*c - 12*a^4*b^5*c^2 + 48*a^5*b^3*c^3 - 64*a^6*b*c^4)*d^5 + (a
^3*b^8 - 10*a^4*b^6*c + 24*a^5*b^4*c^2 + 32*a^6*b^2*c^3 - 128*a^7*c^4)*d^3 + (a^4*b^7 - 12*a^5*b^5*c + 48*a^6*
b^3*c^2 - 64*a^7*b*c^3)*d)*e^2*f*x + (a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3 + (a^3*b^6*c^2 - 12
*a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*d^8 + 2*(a^3*b^7*c - 12*a^4*b^5*c^2 + 48*a^5*b^3*c^3 - 64*a^6*b*c^
4)*d^6 + (a^3*b^8 - 10*a^4*b^6*c + 24*a^5*b^4*c^2 + 32*a^6*b^2*c^3 - 128*a^7*c^4)*d^4 + 2*(a^4*b^7 - 12*a^5*b^
5*c + 48*a^6*b^3*c^2 - 64*a^7*b*c^3)*d^2)*e*f), 1/4*(2*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*e^6*x^6 + 1
2*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*d*e^5*x^5 + (4*a*b^6*c - 45*a^2*b^4*c^2 + 132*a^3*b^2*c^3 - 64*a
^4*c^4 + 30*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*d^2)*e^4*x^4 + 3*a^2*b^6 - 33*a^3*b^4*c + 108*a^4*b^2*
c^2 - 96*a^5*c^3 + 2*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*d^6 + 4*(10*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*
a^3*b*c^4)*d^3 + (4*a*b^6*c - 45*a^2*b^4*c^2 + 132*a^3*b^2*c^3 - 64*a^4*c^4)*d)*e^3*x^3 + (4*a*b^6*c - 45*a^2*
b^4*c^2 + 132*a^3*b^2*c^3 - 64*a^4*c^4)*d^4 + 2*(a*b^7 - 10*a^2*b^5*c + 23*a^3*b^3*c^2 + 4*a^4*b*c^3 + 15*(a*b
^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*d^4 + 3*(4*a*b^6*c - 45*a^2*b^4*c^2 + 132*a^3*b^2*c^3 - 64*a^4*c^4)*d^
2)*e^2*x^2 + 2*(a*b^7 - 10*a^2*b^5*c + 23*a^3*b^3*c^2 + 4*a^4*b*c^3)*d^2 + 4*(3*(a*b^5*c^2 - 11*a^2*b^3*c^3 +
28*a^3*b*c^4)*d^5 + (4*a*b^6*c - 45*a^2*b^4*c^2 + 132*a^3*b^2*c^3 - 64*a^4*c^4)*d^3 + (a*b^7 - 10*a^2*b^5*c +
23*a^3*b^3*c^2 + 4*a^4*b*c^3)*d)*e*x + 2*((b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*e^8*x^8 + 8*(b^5*c^2 - 10*a*
b^3*c^3 + 30*a^2*b*c^4)*d*e^7*x^7 + 2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3 + 14*(b^5*c^2 - 10*a*b^3*c^3 + 30
*a^2*b*c^4)*d^2)*e^6*x^6 + 4*(14*(b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^3 + 3*(b^6*c - 10*a*b^4*c^2 + 30*a^
2*b^2*c^3)*d)*e^5*x^5 + (b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^8 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a
^3*b*c^3 + 70*(b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^4 + 30*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^2)*e^
4*x^4 + a^2*b^5 - 10*a^3*b^3*c + 30*a^4*b*c^2 + 2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^6 + 4*(14*(b^5*c^2
- 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^5 + 10*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*d^3 + (b^7 - 8*a*b^5*c + 10*a
^2*b^3*c^2 + 60*a^3*b*c^3)*d)*e^3*x^3 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*d^4 + 2*(a*b^6 - 10*
a^2*b^4*c + 30*a^3*b^2*c^2 + 14*(b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^6 + 15*(b^6*c - 10*a*b^4*c^2 + 30*a^
2*b^2*c^3)*d^4 + 3*(b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*d^2)*e^2*x^2 + 2*(a*b^6 - 10*a^2*b^4*c +
30*a^3*b^2*c^2)*d^2 + 4*(2*(b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*d^7 + 3*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*
c^3)*d^5 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*d^3 + (a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*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^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*e^8*x^8 + 8*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 6
4*a^3*c^5)*d*e^7*x^7 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4 + 14*(b^6*c^2 - 12*a*b^4*c^3 +
48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2)*e^6*x^6 + 4*(14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^3 +
3*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d)*e^5*x^5 + (b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^
4 - 64*a^3*c^5)*d^8 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4 + 70*(b^6*c^2 - 12*a*b
^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 + 30*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^2)*e^4
*x^4 + a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^
3*b*c^4)*d^6 + 4*(14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^5 + 10*(b^7*c - 12*a*b^5*c^2 + 4
8*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d)*e^3*
x^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^4 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^
3*b^3*c^2 - 64*a^4*b*c^3 + 14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^6 + 15*(b^7*c - 12*a*b^
5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^4 + 3*(b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c
^4)*d^2)*e^2*x^2 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d^2 + 4*(2*(b^6*c^2 - 12*a*b^4*c^3
+ 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^7 + 3*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^5 + (b^8 - 10
*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^3 + (a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^
4*b*c^3)*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^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*e^8*x^8 + 8*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2
*b^2*c^4 - 64*a^3*c^5)*d*e^7*x^7 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4 + 14*(b^6*c^2 - 12*
a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2)*e^6*x^6 + 4*(14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^
3*c^5)*d^3 + 3*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d)*e^5*x^5 + (b^6*c^2 - 12*a*b^4*c^3 + 4
8*a^2*b^2*c^4 - 64*a^3*c^5)*d^8 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4 + 70*(b^6*
c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 + 30*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c
^4)*d^2)*e^4*x^4 + a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3
*c^3 - 64*a^3*b*c^4)*d^6 + 4*(14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^5 + 10*(b^7*c - 12*a
*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*
c^4)*d)*e^3*x^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^4 + 2*(a*b^7 - 12*a^2*b
^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3 + 14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^6 + 15*(b^7
*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^4 + 3*(b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3
- 128*a^4*c^4)*d^2)*e^2*x^2 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d^2 + 4*(2*(b^6*c^2 -
12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^7 + 3*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^5
+ (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^3 + (a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3
*c^2 - 64*a^4*b*c^3)*d)*e*x)*log(e*x + d))/((a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*e^9*f
*x^8 + 8*(a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*d*e^8*f*x^7 + 2*(a^3*b^7*c - 12*a^4*b^5*
c^2 + 48*a^5*b^3*c^3 - 64*a^6*b*c^4 + 14*(a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*d^2)*e^7
*f*x^6 + 4*(14*(a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*d^3 + 3*(a^3*b^7*c - 12*a^4*b^5*c^
2 + 48*a^5*b^3*c^3 - 64*a^6*b*c^4)*d)*e^6*f*x^5 + (a^3*b^8 - 10*a^4*b^6*c + 24*a^5*b^4*c^2 + 32*a^6*b^2*c^3 -
128*a^7*c^4 + 70*(a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*d^4 + 30*(a^3*b^7*c - 12*a^4*b^5
*c^2 + 48*a^5*b^3*c^3 - 64*a^6*b*c^4)*d^2)*e^5*f*x^4 + 4*(14*(a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^5*b^2*c^4 -
64*a^6*c^5)*d^5 + 10*(a^3*b^7*c - 12*a^4*b^5*c^2 + 48*a^5*b^3*c^3 - 64*a^6*b*c^4)*d^3 + (a^3*b^8 - 10*a^4*b^6*
c + 24*a^5*b^4*c^2 + 32*a^6*b^2*c^3 - 128*a^7*c^4)*d)*e^4*f*x^3 + 2*(a^4*b^7 - 12*a^5*b^5*c + 48*a^6*b^3*c^2 -
64*a^7*b*c^3 + 14*(a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*d^6 + 15*(a^3*b^7*c - 12*a^4*b
^5*c^2 + 48*a^5*b^3*c^3 - 64*a^6*b*c^4)*d^4 + 3*(a^3*b^8 - 10*a^4*b^6*c + 24*a^5*b^4*c^2 + 32*a^6*b^2*c^3 - 12
8*a^7*c^4)*d^2)*e^3*f*x^2 + 4*(2*(a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*d^7 + 3*(a^3*b^7
*c - 12*a^4*b^5*c^2 + 48*a^5*b^3*c^3 - 64*a^6*b*c^4)*d^5 + (a^3*b^8 - 10*a^4*b^6*c + 24*a^5*b^4*c^2 + 32*a^6*b
^2*c^3 - 128*a^7*c^4)*d^3 + (a^4*b^7 - 12*a^5*b^5*c + 48*a^6*b^3*c^2 - 64*a^7*b*c^3)*d)*e^2*f*x + (a^5*b^6 - 1
2*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3 + (a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*d^8 +
2*(a^3*b^7*c - 12*a^4*b^5*c^2 + 48*a^5*b^3*c^3 - 64*a^6*b*c^4)*d^6 + (a^3*b^8 - 10*a^4*b^6*c + 24*a^5*b^4*c^2
+ 32*a^6*b^2*c^3 - 128*a^7*c^4)*d^4 + 2*(a^4*b^7 - 12*a^5*b^5*c + 48*a^6*b^3*c^2 - 64*a^7*b*c^3)*d^2)*e*f)]

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giac [B]  time = 1.70, size = 1044, normalized size = 3.87 \[ -\frac {{\left (a^{3} b^{7} c f e^{3} - 14 \, a^{4} b^{5} c^{2} f e^{3} + 70 \, a^{5} b^{3} c^{3} f e^{3} - 120 \, a^{6} b c^{4} f e^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left ({\left | b x^{2} e^{2} + 2 \, b d x e + \sqrt {b^{2} - 4 \, a c} x^{2} e^{2} + 2 \, \sqrt {b^{2} - 4 \, a c} d x e + b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} + 2 \, a \right |}\right ) - {\left (a^{3} b^{7} c f e^{3} - 14 \, a^{4} b^{5} c^{2} f e^{3} + 70 \, a^{5} b^{3} c^{3} f e^{3} - 120 \, a^{6} b c^{4} f e^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left ({\left | -b x^{2} e^{2} - 2 \, b d x e + \sqrt {b^{2} - 4 \, a c} x^{2} e^{2} + 2 \, \sqrt {b^{2} - 4 \, a c} d x e - b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} - 2 \, a \right |}\right )}{4 \, {\left (a^{6} b^{8} c f^{2} e^{4} - 16 \, a^{7} b^{6} c^{2} f^{2} e^{4} + 96 \, a^{8} b^{4} c^{3} f^{2} e^{4} - 256 \, a^{9} b^{2} c^{4} f^{2} e^{4} + 256 \, a^{10} c^{5} f^{2} e^{4}\right )}} - \frac {e^{\left (-1\right )} \log \left ({\left | c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + 6 \, c d^{2} x^{2} e^{2} + 4 \, c d^{3} x e + c d^{4} + b x^{2} e^{2} + 2 \, b d x e + b d^{2} + a \right |}\right )}{4 \, a^{3} f} + \frac {e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\right )}{a^{3} f} + \frac {{\left (2 \, a b^{3} c^{2} d^{6} - 14 \, a^{2} b c^{3} d^{6} + 4 \, a b^{4} c d^{4} - 29 \, a^{2} b^{2} c^{2} d^{4} + 16 \, a^{3} c^{3} d^{4} + 2 \, a b^{5} d^{2} - 12 \, a^{2} b^{3} c d^{2} - 2 \, a^{3} b c^{2} d^{2} + 2 \, {\left (a b^{3} c^{2} e^{6} - 7 \, a^{2} b c^{3} e^{6}\right )} x^{6} + 3 \, a^{2} b^{4} - 21 \, a^{3} b^{2} c + 24 \, a^{4} c^{2} + 12 \, {\left (a b^{3} c^{2} d e^{5} - 7 \, a^{2} b c^{3} d e^{5}\right )} x^{5} + {\left (30 \, a b^{3} c^{2} d^{2} e^{4} - 210 \, a^{2} b c^{3} d^{2} e^{4} + 4 \, a b^{4} c e^{4} - 29 \, a^{2} b^{2} c^{2} e^{4} + 16 \, a^{3} c^{3} e^{4}\right )} x^{4} + 4 \, {\left (10 \, a b^{3} c^{2} d^{3} e^{3} - 70 \, a^{2} b c^{3} d^{3} e^{3} + 4 \, a b^{4} c d e^{3} - 29 \, a^{2} b^{2} c^{2} d e^{3} + 16 \, a^{3} c^{3} d e^{3}\right )} x^{3} + 2 \, {\left (15 \, a b^{3} c^{2} d^{4} e^{2} - 105 \, a^{2} b c^{3} d^{4} e^{2} + 12 \, a b^{4} c d^{2} e^{2} - 87 \, a^{2} b^{2} c^{2} d^{2} e^{2} + 48 \, a^{3} c^{3} d^{2} e^{2} + a b^{5} e^{2} - 6 \, a^{2} b^{3} c e^{2} - a^{3} b c^{2} e^{2}\right )} x^{2} + 4 \, {\left (3 \, a b^{3} c^{2} d^{5} e - 21 \, a^{2} b c^{3} d^{5} e + 4 \, a b^{4} c d^{3} e - 29 \, a^{2} b^{2} c^{2} d^{3} e + 16 \, a^{3} c^{3} d^{3} e + a b^{5} d e - 6 \, a^{2} b^{3} c d e - a^{3} b c^{2} d e\right )} x\right )} e^{\left (-1\right )}}{4 \, {\left (c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + 6 \, c d^{2} x^{2} e^{2} + 4 \, c d^{3} x e + c d^{4} + b x^{2} e^{2} + 2 \, b d x e + b d^{2} + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} a^{3} f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="giac")

[Out]

-1/4*((a^3*b^7*c*f*e^3 - 14*a^4*b^5*c^2*f*e^3 + 70*a^5*b^3*c^3*f*e^3 - 120*a^6*b*c^4*f*e^3)*sqrt(b^2 - 4*a*c)*
log(abs(b*x^2*e^2 + 2*b*d*x*e + sqrt(b^2 - 4*a*c)*x^2*e^2 + 2*sqrt(b^2 - 4*a*c)*d*x*e + b*d^2 + sqrt(b^2 - 4*a
*c)*d^2 + 2*a)) - (a^3*b^7*c*f*e^3 - 14*a^4*b^5*c^2*f*e^3 + 70*a^5*b^3*c^3*f*e^3 - 120*a^6*b*c^4*f*e^3)*sqrt(b
^2 - 4*a*c)*log(abs(-b*x^2*e^2 - 2*b*d*x*e + sqrt(b^2 - 4*a*c)*x^2*e^2 + 2*sqrt(b^2 - 4*a*c)*d*x*e - b*d^2 + s
qrt(b^2 - 4*a*c)*d^2 - 2*a)))/(a^6*b^8*c*f^2*e^4 - 16*a^7*b^6*c^2*f^2*e^4 + 96*a^8*b^4*c^3*f^2*e^4 - 256*a^9*b
^2*c^4*f^2*e^4 + 256*a^10*c^5*f^2*e^4) - 1/4*e^(-1)*log(abs(c*x^4*e^4 + 4*c*d*x^3*e^3 + 6*c*d^2*x^2*e^2 + 4*c*
d^3*x*e + c*d^4 + b*x^2*e^2 + 2*b*d*x*e + b*d^2 + a))/(a^3*f) + e^(-1)*log(abs(x*e + d))/(a^3*f) + 1/4*(2*a*b^
3*c^2*d^6 - 14*a^2*b*c^3*d^6 + 4*a*b^4*c*d^4 - 29*a^2*b^2*c^2*d^4 + 16*a^3*c^3*d^4 + 2*a*b^5*d^2 - 12*a^2*b^3*
c*d^2 - 2*a^3*b*c^2*d^2 + 2*(a*b^3*c^2*e^6 - 7*a^2*b*c^3*e^6)*x^6 + 3*a^2*b^4 - 21*a^3*b^2*c + 24*a^4*c^2 + 12
*(a*b^3*c^2*d*e^5 - 7*a^2*b*c^3*d*e^5)*x^5 + (30*a*b^3*c^2*d^2*e^4 - 210*a^2*b*c^3*d^2*e^4 + 4*a*b^4*c*e^4 - 2
9*a^2*b^2*c^2*e^4 + 16*a^3*c^3*e^4)*x^4 + 4*(10*a*b^3*c^2*d^3*e^3 - 70*a^2*b*c^3*d^3*e^3 + 4*a*b^4*c*d*e^3 - 2
9*a^2*b^2*c^2*d*e^3 + 16*a^3*c^3*d*e^3)*x^3 + 2*(15*a*b^3*c^2*d^4*e^2 - 105*a^2*b*c^3*d^4*e^2 + 12*a*b^4*c*d^2
*e^2 - 87*a^2*b^2*c^2*d^2*e^2 + 48*a^3*c^3*d^2*e^2 + a*b^5*e^2 - 6*a^2*b^3*c*e^2 - a^3*b*c^2*e^2)*x^2 + 4*(3*a
*b^3*c^2*d^5*e - 21*a^2*b*c^3*d^5*e + 4*a*b^4*c*d^3*e - 29*a^2*b^2*c^2*d^3*e + 16*a^3*c^3*d^3*e + a*b^5*d*e -
6*a^2*b^3*c*d*e - a^3*b*c^2*d*e)*x)*e^(-1)/((c*x^4*e^4 + 4*c*d*x^3*e^3 + 6*c*d^2*x^2*e^2 + 4*c*d^3*x*e + c*d^4
+ b*x^2*e^2 + 2*b*d*x*e + b*d^2 + a)^2*(b^2 - 4*a*c)^2*a^3*f)

________________________________________________________________________________________

maple [C]  time = 0.08, size = 4606, normalized size = 17.06 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x)

[Out]

-1/2/f/(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)^2*e/(16*a^2*c^2
-8*a*b^2*c+b^4)*x^2*b*c^2-1/f/(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)^2*d/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b*c^2-1/2/f/(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)^2/e/(16*a^2*c^2-8*a*b^2*c+b^4)*b*c^2*d^2+1/2/f/a^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)^2*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^5+1/f/a^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)^2*d/(16*a^2*c^2-8*a*b^2*c
+b^4)*x*b^5+1/2/f/a^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)^
2/e/(16*a^2*c^2-8*a*b^2*c+b^4)*b^5*d^2+ln(e*x+d)/a^3/e/f-7/2/f/a/(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)^2*c^3*e^5*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/2/f/a^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)^2*c^2*e^5*b^3/(16*a^2*c^2-8*a*b^2*c+
b^4)*x^6+6/f*a/(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)^2/e/(16
*a^2*c^2-8*a*b^2*c+b^4)*c^2+3/4/f/a/(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)^2/e/(16*a^2*c^2-8*a*b^2*c+b^4)*b^4+4/f/(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)^2*e^3*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4+16/f/(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)^2*d^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x*c^3+4/f/(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)^2/e/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3
*d^4-21/4/f/(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)^2/e/(16*a^
2*c^2-8*a*b^2*c+b^4)*b^2*c-1/2/f/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)/e*sum((c*e^3*(16*a^2*c^2-8*a*b^2*c+b^4)*_R^3+3
*c*d*e^2*(16*a^2*c^2-8*a*b^2*c+b^4)*_R^2+e*(48*a^2*c^3*d^2-24*a*b^2*c^2*d^2+3*b^4*c*d^2+23*a^2*b*c^2-9*a*b^3*c
+b^5)*_R+16*a^2*c^3*d^3-8*a*b^2*c^2*d^3+b^4*c*d^3+23*a^2*b*c^2*d-9*a*b^3*c*d+b^5*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(-_R+x),_R=RootOf(_Z^4*c*e^4+4*_Z^3*c*d*e^3+c*d^4+b*d^2+(6*c*d^2*e^2+b*e
^2)*_Z^2+(4*c*d^3*e+2*b*d*e)*_Z+a))+10/f/a^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)^2*c^2*d^3*e^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*b^3-29/f/a/(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)^2*c^2*d*e^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*b^2+4/f/a^
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)^2*c*d*e^2/(16*a^2*c^
2-8*a*b^2*c+b^4)*x^3*b^4-105/2/f/a/(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)^2*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b*c^3*d^4+15/2/f/a^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)^2*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^3*c^2*d^4-87/2/f/a/(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)^2*e/(16*a^2*c^2-8*a*b^2*c+b^4
)*x^2*b^2*c^2*d^2+6/f/a^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)^2*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^4*c*d^2-21/f/a/(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)^2*b*c^3*d*e^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5+3/f/a^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)^2*b^3*c^2*d*e^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^
5-105/2/f/a/(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)^2*e^3*c^3/
(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*b*d^2+15/2/f/a^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)^2*e^3*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*b^3*d^2-70/f/a/(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)^2*c^3*d^3*e^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*b+16/
f/(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)^2*c^3*d*e^2/(16*a^2*
c^2-8*a*b^2*c+b^4)*x^3+24/f/(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)^2*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*c^3*d^2+1/f/a^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)^2*e^3*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*b^4-3/f/a/(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)^2*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^3*c-21/f/a/
(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)^2*d^5/(16*a^2*c^2-8*a*
b^2*c+b^4)*x*b*c^3+3/f/a^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)^2*d^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^3*c^2-29/f/a/(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)^2*d^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^2*c^2+4/f/a^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)^2*d^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^4*c-6/f/a/(
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)^2*d/(16*a^2*c^2-8*a*b^2
*c+b^4)*x*b^3*c-7/2/f/a/(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
)^2/e/(16*a^2*c^2-8*a*b^2*c+b^4)*b*c^3*d^6+1/2/f/a^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)^2/e/(16*a^2*c^2-8*a*b^2*c+b^4)*b^3*c^2*d^6-29/4/f/a/(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)^2/e/(16*a^2*c^2-8*a*b^2*c+b^4)*b^2*c^2*d^4+1/f/a^
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)^2/e/(16*a^2*c^2-8*a*
b^2*c+b^4)*b^4*c*d^4-3/f/a/(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)^2/e/(16*a^2*c^2-8*a*b^2*c+b^4)*b^3*c*d^2-29/4/f/a/(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)^2*e^3*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*b^2

________________________________________________________________________________________

maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

mupad [B]  time = 18.49, size = 22621, normalized size = 83.78 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d*f + e*f*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x)

[Out]

((x^2*(b^5*e + 48*a^2*c^3*d^2*e + 15*b^3*c^2*d^4*e - 6*a*b^3*c*e - a^2*b*c^2*e + 12*b^4*c*d^2*e - 105*a*b*c^3*
d^4*e - 87*a*b^2*c^2*d^2*e))/(2*(a^2*b^4 + 16*a^4*c^2 - 8*a^3*b^2*c)) + (x^4*(4*b^4*c*e^3 + 16*a^2*c^3*e^3 - 2
9*a*b^2*c^2*e^3 + 30*b^3*c^2*d^2*e^3 - 210*a*b*c^3*d^2*e^3))/(4*(a^2*b^4 + 16*a^4*c^2 - 8*a^3*b^2*c)) + (x^3*(
16*a^2*c^3*d*e^2 + 10*b^3*c^2*d^3*e^2 + 4*b^4*c*d*e^2 - 29*a*b^2*c^2*d*e^2 - 70*a*b*c^3*d^3*e^2))/(a^2*b^4 + 1
6*a^4*c^2 - 8*a^3*b^2*c) + (3*x^5*(b^3*c^2*d*e^4 - 7*a*b*c^3*d*e^4))/(a^2*b^4 + 16*a^4*c^2 - 8*a^3*b^2*c) + (x
^6*(b^3*c^2*e^5 - 7*a*b*c^3*e^5))/(2*(a^2*b^4 + 16*a^4*c^2 - 8*a^3*b^2*c)) + (x*(b^5*d + 4*b^4*c*d^3 + 16*a^2*
c^3*d^3 + 3*b^3*c^2*d^5 - 29*a*b^2*c^2*d^3 - 6*a*b^3*c*d - a^2*b*c^2*d - 21*a*b*c^3*d^5))/(a^2*b^4 + 16*a^4*c^
2 - 8*a^3*b^2*c) + (3*a*b^4 + 24*a^3*c^2 + 2*b^5*d^2 - 21*a^2*b^2*c + 4*b^4*c*d^4 + 16*a^2*c^3*d^4 + 2*b^3*c^2
*d^6 - 2*a^2*b*c^2*d^2 - 29*a*b^2*c^2*d^4 - 12*a*b^3*c*d^2 - 14*a*b*c^3*d^6)/(4*e*(a^2*b^4 + 16*a^4*c^2 - 8*a^
3*b^2*c)))/(x^3*(56*c^2*d^5*e^3*f + 4*b^2*d*e^3*f + 40*b*c*d^3*e^3*f + 8*a*c*d*e^3*f) + x^2*(6*b^2*d^2*e^2*f +
28*c^2*d^6*e^2*f + 2*a*b*e^2*f + 12*a*c*d^2*e^2*f + 30*b*c*d^4*e^2*f) + x*(4*b^2*d^3*e*f + 8*c^2*d^7*e*f + 4*
a*b*d*e*f + 8*a*c*d^3*e*f + 12*b*c*d^5*e*f) + x^4*(b^2*e^4*f + 70*c^2*d^4*e^4*f + 2*a*c*e^4*f + 30*b*c*d^2*e^4
*f) + x^5*(56*c^2*d^3*e^5*f + 12*b*c*d*e^5*f) + a^2*f + x^6*(28*c^2*d^2*e^6*f + 2*b*c*e^6*f) + b^2*d^4*f + c^2
*d^8*f + c^2*e^8*f*x^8 + 2*a*b*d^2*f + 2*a*c*d^4*f + 2*b*c*d^6*f + 8*c^2*d*e^7*f*x^7) - (log((((a^3*e*f*(-(b^2
*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(a^6*e^2*f^2*(4*a*c - b^2)^5))^(1/2) + 1)*(((a^3*e*f*(-(b^2*(b^4 + 30*a^2*
c^2 - 10*a*b^2*c)^2)/(a^6*e^2*f^2*(4*a*c - b^2)^5))^(1/2) + 1)*((2*b*c^2*e^16*(2*b^5 + 46*a^2*b*c^2 + b^4*c*d^
2 + 10*a^2*c^3*d^2 - 18*a*b^3*c - 2*a*b^2*c^2*d^2))/(a^2*f*(4*a*c - b^2)^2) + (b*c^2*e^16*(a^3*e*f*(-(b^2*(b^4
+ 30*a^2*c^2 - 10*a*b^2*c)^2)/(a^6*e^2*f^2*(4*a*c - b^2)^5))^(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^3*f) + (2*b*c^3*e^18*x^2*(b^4 + 10*a^2*c^2 - 2*a*b
^2*c))/(a^2*f*(4*a*c - b^2)^2) + (4*b*c^3*d*e^17*x*(b^4 + 10*a^2*c^2 - 2*a*b^2*c))/(a^2*f*(4*a*c - b^2)^2)))/(
4*a^3*e*f) + (b*c^3*e^15*(7*a*c - b^2)*(4*b^5 + 71*a^2*b*c^2 + 6*b^4*c*d^2 + 80*a^2*c^3*d^2 - 33*a*b^3*c - 47*
a*b^2*c^2*d^2))/(a^4*f^2*(4*a*c - b^2)^4) - (b*c^4*e^17*x^2*(6*b^6 - 560*a^3*c^3 + 409*a^2*b^2*c^2 - 89*a*b^4*
c))/(a^4*f^2*(4*a*c - b^2)^4) - (2*b*c^4*d*e^16*x*(6*b^6 - 560*a^3*c^3 + 409*a^2*b^2*c^2 - 89*a*b^4*c))/(a^4*f
^2*(4*a*c - b^2)^4)))/(4*a^3*e*f) - (b^3*c^5*e^16*x^2*(7*a*c - b^2)^3)/(a^6*f^3*(4*a*c - b^2)^6) + (b^2*c^4*e^
14*(7*a*c - b^2)^2*(b^4 + 16*a^2*c^2 + b^3*c*d^2 - 8*a*b^2*c - 7*a*b*c^2*d^2))/(a^6*f^3*(4*a*c - b^2)^6) - (2*
b^3*c^5*d*e^15*x*(7*a*c - b^2)^3)/(a^6*f^3*(4*a*c - b^2)^6))*(((a^3*e*f*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)
^2)/(a^6*e^2*f^2*(4*a*c - b^2)^5))^(1/2) - 1)*(((a^3*e*f*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(a^6*e^2*f^
2*(4*a*c - b^2)^5))^(1/2) - 1)*((2*b*c^2*e^16*(2*b^5 + 46*a^2*b*c^2 + b^4*c*d^2 + 10*a^2*c^3*d^2 - 18*a*b^3*c
- 2*a*b^2*c^2*d^2))/(a^2*f*(4*a*c - b^2)^2) - (b*c^2*e^16*(a^3*e*f*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(
a^6*e^2*f^2*(4*a*c - b^2)^5))^(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^3*f) + (2*b*c^3*e^18*x^2*(b^4 + 10*a^2*c^2 - 2*a*b^2*c))/(a^2*f*(4*a*c - b^2)^2) +
(4*b*c^3*d*e^17*x*(b^4 + 10*a^2*c^2 - 2*a*b^2*c))/(a^2*f*(4*a*c - b^2)^2)))/(4*a^3*e*f) - (b*c^3*e^15*(7*a*c
- b^2)*(4*b^5 + 71*a^2*b*c^2 + 6*b^4*c*d^2 + 80*a^2*c^3*d^2 - 33*a*b^3*c - 47*a*b^2*c^2*d^2))/(a^4*f^2*(4*a*c
- b^2)^4) + (b*c^4*e^17*x^2*(6*b^6 - 560*a^3*c^3 + 409*a^2*b^2*c^2 - 89*a*b^4*c))/(a^4*f^2*(4*a*c - b^2)^4) +
(2*b*c^4*d*e^16*x*(6*b^6 - 560*a^3*c^3 + 409*a^2*b^2*c^2 - 89*a*b^4*c))/(a^4*f^2*(4*a*c - b^2)^4)))/(4*a^3*e*f
) - (b^3*c^5*e^16*x^2*(7*a*c - b^2)^3)/(a^6*f^3*(4*a*c - b^2)^6) + (b^2*c^4*e^14*(7*a*c - b^2)^2*(b^4 + 16*a^2
*c^2 + b^3*c*d^2 - 8*a*b^2*c - 7*a*b*c^2*d^2))/(a^6*f^3*(4*a*c - b^2)^6) - (2*b^3*c^5*d*e^15*x*(7*a*c - b^2)^3
)/(a^6*f^3*(4*a*c - b^2)^6)))*(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 25
60*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f))/(2*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2
- 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)) + log(d + e*x)/(a^3*e*f) - (b*a
tan((x*((((((b*((2*(5120*a^10*b*c^9*d*e^17*f^2 + 2*a^4*b^13*c^3*d*e^17*f^2 - 36*a^5*b^11*c^4*d*e^17*f^2 + 276*
a^6*b^9*c^5*d*e^17*f^2 - 1216*a^7*b^7*c^6*d*e^17*f^2 + 3456*a^8*b^5*c^7*d*e^17*f^2 - 6144*a^9*b^3*c^8*d*e^17*f
^2))/(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840
*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3) - ((2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*
b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f)*(163840*a^13*b*c^9*d*e^18*f^3 - 12*a^6*b^15*c^2*d*e^18*f^
3 + 328*a^7*b^13*c^3*d*e^18*f^3 - 3840*a^8*b^11*c^4*d*e^18*f^3 + 24960*a^9*b^9*c^5*d*e^18*f^3 - 97280*a^10*b^7
*c^6*d*e^18*f^3 + 227328*a^11*b^5*c^7*d*e^18*f^3 - 294912*a^12*b^3*c^8*d*e^18*f^3))/((4*a^3*b^10*e^2*f^2 - 409
6*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8
*c*e^2*f^2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3
+ 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(4*a^3*e*f*(4*a*c - b^2)^
(5/2)) - (b*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)*(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^
4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f)*(163840*a^13*b*c^9*d*e^18*f^3 - 12*a^6*b^15*c^2*d*e^18*f^3
+ 328*a^7*b^13*c^3*d*e^18*f^3 - 3840*a^8*b^11*c^4*d*e^18*f^3 + 24960*a^9*b^9*c^5*d*e^18*f^3 - 97280*a^10*b^7*c
^6*d*e^18*f^3 + 227328*a^11*b^5*c^7*d*e^18*f^3 - 294912*a^12*b^3*c^8*d*e^18*f^3))/(4*a^3*e*f*(4*a*c - b^2)^(5/
2)*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*
b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^
2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*(2*b^10*e*f - 2048*a^5*c^5*e*f
+ 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f))/(2*(4*a^3*b^10*e^2*f^2
- 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a
^4*b^8*c*e^2*f^2)) - (b*((2*(6*a^2*b^11*c^4*d*e^16*f - 137*a^3*b^9*c^5*d*e^16*f + 1217*a^4*b^7*c^6*d*e^16*f -
5256*a^5*b^5*c^7*d*e^16*f + 11024*a^6*b^3*c^8*d*e^16*f - 8960*a^7*b*c^9*d*e^16*f))/(a^6*b^12*f^3 + 4096*a^12*c
^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^
2*c^5*f^3) - (((2*(5120*a^10*b*c^9*d*e^17*f^2 + 2*a^4*b^13*c^3*d*e^17*f^2 - 36*a^5*b^11*c^4*d*e^17*f^2 + 276*a
^6*b^9*c^5*d*e^17*f^2 - 1216*a^7*b^7*c^6*d*e^17*f^2 + 3456*a^8*b^5*c^7*d*e^17*f^2 - 6144*a^9*b^3*c^8*d*e^17*f^
2))/(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*
a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3) - ((2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b
^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f)*(163840*a^13*b*c^9*d*e^18*f^3 - 12*a^6*b^15*c^2*d*e^18*f^3
+ 328*a^7*b^13*c^3*d*e^18*f^3 - 3840*a^8*b^11*c^4*d*e^18*f^3 + 24960*a^9*b^9*c^5*d*e^18*f^3 - 97280*a^10*b^7*
c^6*d*e^18*f^3 + 227328*a^11*b^5*c^7*d*e^18*f^3 - 294912*a^12*b^3*c^8*d*e^18*f^3))/((4*a^3*b^10*e^2*f^2 - 4096
*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*
c*e^2*f^2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3
+ 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280
*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f))/(2*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640
*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)))*(b^4 + 30
*a^2*c^2 - 10*a*b^2*c))/(4*a^3*e*f*(4*a*c - b^2)^(5/2)) + (b^3*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^3*(163840*a^13*
b*c^9*d*e^18*f^3 - 12*a^6*b^15*c^2*d*e^18*f^3 + 328*a^7*b^13*c^3*d*e^18*f^3 - 3840*a^8*b^11*c^4*d*e^18*f^3 + 2
4960*a^9*b^9*c^5*d*e^18*f^3 - 97280*a^10*b^7*c^6*d*e^18*f^3 + 227328*a^11*b^5*c^7*d*e^18*f^3 - 294912*a^12*b^3
*c^8*d*e^18*f^3))/(32*a^9*e^3*f^3*(4*a*c - b^2)^(15/2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 +
240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*(3*b^8 + 160*a^
4*c^4 + 180*a^2*b^4*c^2 - 325*a^3*b^2*c^3 - 39*a*b^6*c))/(8*a^3*c^2*(4*a*c - b^2)^(13/2)*(6*b^10 - 6400*a^5*c^
5 + 960*a^2*b^6*c^2 - 3850*a^3*b^4*c^3 + 7775*a^4*b^2*c^4 - 120*a*b^8*c)) + (3*b*((2*(b^9*c^5*d*e^15 - 21*a*b^
7*c^6*d*e^15 + 147*a^2*b^5*c^7*d*e^15 - 343*a^3*b^3*c^8*d*e^15))/(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^
10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3) - (((2*
(6*a^2*b^11*c^4*d*e^16*f - 137*a^3*b^9*c^5*d*e^16*f + 1217*a^4*b^7*c^6*d*e^16*f - 5256*a^5*b^5*c^7*d*e^16*f +
11024*a^6*b^3*c^8*d*e^16*f - 8960*a^7*b*c^9*d*e^16*f))/(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 +
240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3) - (((2*(5120*a^10
*b*c^9*d*e^17*f^2 + 2*a^4*b^13*c^3*d*e^17*f^2 - 36*a^5*b^11*c^4*d*e^17*f^2 + 276*a^6*b^9*c^5*d*e^17*f^2 - 1216
*a^7*b^7*c^6*d*e^17*f^2 + 3456*a^8*b^5*c^7*d*e^17*f^2 - 6144*a^9*b^3*c^8*d*e^17*f^2))/(a^6*b^12*f^3 + 4096*a^1
2*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11
*b^2*c^5*f^3) - ((2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^
4*e*f - 40*a*b^8*c*e*f)*(163840*a^13*b*c^9*d*e^18*f^3 - 12*a^6*b^15*c^2*d*e^18*f^3 + 328*a^7*b^13*c^3*d*e^18*f
^3 - 3840*a^8*b^11*c^4*d*e^18*f^3 + 24960*a^9*b^9*c^5*d*e^18*f^3 - 97280*a^10*b^7*c^6*d*e^18*f^3 + 227328*a^11
*b^5*c^7*d*e^18*f^3 - 294912*a^12*b^3*c^8*d*e^18*f^3))/((4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b
^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)*(a^6*b^12*f^3 + 4
096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 61
44*a^11*b^2*c^5*f^3)))*(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*
b^2*c^4*e*f - 40*a*b^8*c*e*f))/(2*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*
a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)))*(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*
a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f))/(2*(4*a^3*b^10*e^2*f^2 - 4096
*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*
c*e^2*f^2)) - (b*((b*((2*(5120*a^10*b*c^9*d*e^17*f^2 + 2*a^4*b^13*c^3*d*e^17*f^2 - 36*a^5*b^11*c^4*d*e^17*f^2
+ 276*a^6*b^9*c^5*d*e^17*f^2 - 1216*a^7*b^7*c^6*d*e^17*f^2 + 3456*a^8*b^5*c^7*d*e^17*f^2 - 6144*a^9*b^3*c^8*d*
e^17*f^2))/(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3
+ 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3) - ((2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 128
0*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f)*(163840*a^13*b*c^9*d*e^18*f^3 - 12*a^6*b^15*c^2*d*e
^18*f^3 + 328*a^7*b^13*c^3*d*e^18*f^3 - 3840*a^8*b^11*c^4*d*e^18*f^3 + 24960*a^9*b^9*c^5*d*e^18*f^3 - 97280*a^
10*b^7*c^6*d*e^18*f^3 + 227328*a^11*b^5*c^7*d*e^18*f^3 - 294912*a^12*b^3*c^8*d*e^18*f^3))/((4*a^3*b^10*e^2*f^2
- 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a
^4*b^8*c*e^2*f^2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c
^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(4*a^3*e*f*(4*a*c -
b^2)^(5/2)) - (b*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)*(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*
a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f)*(163840*a^13*b*c^9*d*e^18*f^3 - 12*a^6*b^15*c^2*d*e^1
8*f^3 + 328*a^7*b^13*c^3*d*e^18*f^3 - 3840*a^8*b^11*c^4*d*e^18*f^3 + 24960*a^9*b^9*c^5*d*e^18*f^3 - 97280*a^10
*b^7*c^6*d*e^18*f^3 + 227328*a^11*b^5*c^7*d*e^18*f^3 - 294912*a^12*b^3*c^8*d*e^18*f^3))/(4*a^3*e*f*(4*a*c - b^
2)^(5/2)*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 512
0*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*
b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*(b^4 + 30*a^2*c^2 - 10*a
*b^2*c))/(4*a^3*e*f*(4*a*c - b^2)^(5/2)) + (b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2*(2*b^10*e*f - 2048*a^5*c^5*e
*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f)*(163840*a^13*b*c^9*d*
e^18*f^3 - 12*a^6*b^15*c^2*d*e^18*f^3 + 328*a^7*b^13*c^3*d*e^18*f^3 - 3840*a^8*b^11*c^4*d*e^18*f^3 + 24960*a^9
*b^9*c^5*d*e^18*f^3 - 97280*a^10*b^7*c^6*d*e^18*f^3 + 227328*a^11*b^5*c^7*d*e^18*f^3 - 294912*a^12*b^3*c^8*d*e
^18*f^3))/(16*a^6*e^2*f^2*(4*a*c - b^2)^5*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2
- 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f
^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^
5*f^3)))*(b^6 - 45*a^3*c^3 + 40*a^2*b^2*c^2 - 11*a*b^4*c))/(8*a^3*c^2*(4*a*c - b^2)^6*(6*b^10 - 6400*a^5*c^5 +
960*a^2*b^6*c^2 - 3850*a^3*b^4*c^3 + 7775*a^4*b^2*c^4 - 120*a*b^8*c)))*(16*a^9*b^12*f^3*(4*a*c - b^2)^(15/2)
+ 65536*a^15*c^6*f^3*(4*a*c - b^2)^(15/2) - 384*a^10*b^10*c*f^3*(4*a*c - b^2)^(15/2) + 3840*a^11*b^8*c^2*f^3*(
4*a*c - b^2)^(15/2) - 20480*a^12*b^6*c^3*f^3*(4*a*c - b^2)^(15/2) + 61440*a^13*b^4*c^4*f^3*(4*a*c - b^2)^(15/2
) - 98304*a^14*b^2*c^5*f^3*(4*a*c - b^2)^(15/2)))/(b^10*c^2*e^14 - 20*a*b^8*c^3*e^14 + 160*a^2*b^6*c^4*e^14 -
600*a^3*b^4*c^5*e^14 + 900*a^4*b^2*c^6*e^14) + (x^2*((((((b*((2*a^4*b^13*c^3*e^18*f^2 - 36*a^5*b^11*c^4*e^18*f
^2 + 276*a^6*b^9*c^5*e^18*f^2 - 1216*a^7*b^7*c^6*e^18*f^2 + 3456*a^8*b^5*c^7*e^18*f^2 - 6144*a^9*b^3*c^8*e^18*
f^2 + 5120*a^10*b*c^9*e^18*f^2)/(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 -
1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3) + ((2*b^10*e*f - 2048*a^5*c^5*e*f + 320*
a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f)*(12*a^6*b^15*c^2*e^19*f^3 - 32
8*a^7*b^13*c^3*e^19*f^3 + 3840*a^8*b^11*c^4*e^19*f^3 - 24960*a^9*b^9*c^5*e^19*f^3 + 97280*a^10*b^7*c^6*e^19*f^
3 - 227328*a^11*b^5*c^7*e^19*f^3 + 294912*a^12*b^3*c^8*e^19*f^3 - 163840*a^13*b*c^9*e^19*f^3))/(2*(4*a^3*b^10*
e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2
- 80*a^4*b^8*c*e^2*f^2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^
9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(4*a^3*e*f*(
4*a*c - b^2)^(5/2)) + (b*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)*(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f
- 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f)*(12*a^6*b^15*c^2*e^19*f^3 - 328*a^7*b^13*c^3*e
^19*f^3 + 3840*a^8*b^11*c^4*e^19*f^3 - 24960*a^9*b^9*c^5*e^19*f^3 + 97280*a^10*b^7*c^6*e^19*f^3 - 227328*a^11*
b^5*c^7*e^19*f^3 + 294912*a^12*b^3*c^8*e^19*f^3 - 163840*a^13*b*c^9*e^19*f^3))/(8*a^3*e*f*(4*a*c - b^2)^(5/2)*
(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2
*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f
^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*(2*b^10*e*f - 2048*a^5*c^5*e*f +
320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f))/(2*(4*a^3*b^10*e^2*f^2 -
4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*
b^8*c*e^2*f^2)) + (b*((8960*a^7*b*c^9*e^17*f - 6*a^2*b^11*c^4*e^17*f + 137*a^3*b^9*c^5*e^17*f - 1217*a^4*b^7*c
^6*e^17*f + 5256*a^5*b^5*c^7*e^17*f - 11024*a^6*b^3*c^8*e^17*f)/(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^1
0*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3) + (((2*a
^4*b^13*c^3*e^18*f^2 - 36*a^5*b^11*c^4*e^18*f^2 + 276*a^6*b^9*c^5*e^18*f^2 - 1216*a^7*b^7*c^6*e^18*f^2 + 3456*
a^8*b^5*c^7*e^18*f^2 - 6144*a^9*b^3*c^8*e^18*f^2 + 5120*a^10*b*c^9*e^18*f^2)/(a^6*b^12*f^3 + 4096*a^12*c^6*f^3
- 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*
f^3) + ((2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 4
0*a*b^8*c*e*f)*(12*a^6*b^15*c^2*e^19*f^3 - 328*a^7*b^13*c^3*e^19*f^3 + 3840*a^8*b^11*c^4*e^19*f^3 - 24960*a^9*
b^9*c^5*e^19*f^3 + 97280*a^10*b^7*c^6*e^19*f^3 - 227328*a^11*b^5*c^7*e^19*f^3 + 294912*a^12*b^3*c^8*e^19*f^3 -
163840*a^13*b*c^9*e^19*f^3))/(2*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a
^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a
^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*
(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*
c*e*f))/(2*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5
120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(4*a^3*e*f*(4*a*c - b^2)^(5
/2)) - (b^3*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^3*(12*a^6*b^15*c^2*e^19*f^3 - 328*a^7*b^13*c^3*e^19*f^3 + 3840*a^8
*b^11*c^4*e^19*f^3 - 24960*a^9*b^9*c^5*e^19*f^3 + 97280*a^10*b^7*c^6*e^19*f^3 - 227328*a^11*b^5*c^7*e^19*f^3 +
294912*a^12*b^3*c^8*e^19*f^3 - 163840*a^13*b*c^9*e^19*f^3))/(64*a^9*e^3*f^3*(4*a*c - b^2)^(15/2)*(a^6*b^12*f^
3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3
- 6144*a^11*b^2*c^5*f^3)))*(3*b^8 + 160*a^4*c^4 + 180*a^2*b^4*c^2 - 325*a^3*b^2*c^3 - 39*a*b^6*c))/(8*a^3*c^2
*(4*a*c - b^2)^(13/2)*(6*b^10 - 6400*a^5*c^5 + 960*a^2*b^6*c^2 - 3850*a^3*b^4*c^3 + 7775*a^4*b^2*c^4 - 120*a*b
^8*c)) + (3*b*((b^9*c^5*e^16 - 21*a*b^7*c^6*e^16 + 147*a^2*b^5*c^7*e^16 - 343*a^3*b^3*c^8*e^16)/(a^6*b^12*f^3
+ 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 -
6144*a^11*b^2*c^5*f^3) + (((8960*a^7*b*c^9*e^17*f - 6*a^2*b^11*c^4*e^17*f + 137*a^3*b^9*c^5*e^17*f - 1217*a^4
*b^7*c^6*e^17*f + 5256*a^5*b^5*c^7*e^17*f - 11024*a^6*b^3*c^8*e^17*f)/(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a
^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3) +
(((2*a^4*b^13*c^3*e^18*f^2 - 36*a^5*b^11*c^4*e^18*f^2 + 276*a^6*b^9*c^5*e^18*f^2 - 1216*a^7*b^7*c^6*e^18*f^2 +
3456*a^8*b^5*c^7*e^18*f^2 - 6144*a^9*b^3*c^8*e^18*f^2 + 5120*a^10*b*c^9*e^18*f^2)/(a^6*b^12*f^3 + 4096*a^12*c
^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^
2*c^5*f^3) + ((2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e
*f - 40*a*b^8*c*e*f)*(12*a^6*b^15*c^2*e^19*f^3 - 328*a^7*b^13*c^3*e^19*f^3 + 3840*a^8*b^11*c^4*e^19*f^3 - 2496
0*a^9*b^9*c^5*e^19*f^3 + 97280*a^10*b^7*c^6*e^19*f^3 - 227328*a^11*b^5*c^7*e^19*f^3 + 294912*a^12*b^3*c^8*e^19
*f^3 - 163840*a^13*b*c^9*e^19*f^3))/(2*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 -
2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f^3
- 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f
^3)))*(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*
a*b^8*c*e*f))/(2*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f
^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)))*(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f -
1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f))/(2*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2
+ 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)) - (b*
((b*((2*a^4*b^13*c^3*e^18*f^2 - 36*a^5*b^11*c^4*e^18*f^2 + 276*a^6*b^9*c^5*e^18*f^2 - 1216*a^7*b^7*c^6*e^18*f^
2 + 3456*a^8*b^5*c^7*e^18*f^2 - 6144*a^9*b^3*c^8*e^18*f^2 + 5120*a^10*b*c^9*e^18*f^2)/(a^6*b^12*f^3 + 4096*a^1
2*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11
*b^2*c^5*f^3) + ((2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^
4*e*f - 40*a*b^8*c*e*f)*(12*a^6*b^15*c^2*e^19*f^3 - 328*a^7*b^13*c^3*e^19*f^3 + 3840*a^8*b^11*c^4*e^19*f^3 - 2
4960*a^9*b^9*c^5*e^19*f^3 + 97280*a^10*b^7*c^6*e^19*f^3 - 227328*a^11*b^5*c^7*e^19*f^3 + 294912*a^12*b^3*c^8*e
^19*f^3 - 163840*a^13*b*c^9*e^19*f^3))/(2*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2
- 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f
^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^
5*f^3)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(4*a^3*e*f*(4*a*c - b^2)^(5/2)) + (b*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)
*(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8
*c*e*f)*(12*a^6*b^15*c^2*e^19*f^3 - 328*a^7*b^13*c^3*e^19*f^3 + 3840*a^8*b^11*c^4*e^19*f^3 - 24960*a^9*b^9*c^5
*e^19*f^3 + 97280*a^10*b^7*c^6*e^19*f^3 - 227328*a^11*b^5*c^7*e^19*f^3 + 294912*a^12*b^3*c^8*e^19*f^3 - 163840
*a^13*b*c^9*e^19*f^3))/(8*a^3*e*f*(4*a*c - b^2)^(5/2)*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6
*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)*(a^6*b^12*f^3 + 409
6*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144
*a^11*b^2*c^5*f^3)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(4*a^3*e*f*(4*a*c - b^2)^(5/2)) - (b^2*(b^4 + 30*a^2*c^2
- 10*a*b^2*c)^2*(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^
4*e*f - 40*a*b^8*c*e*f)*(12*a^6*b^15*c^2*e^19*f^3 - 328*a^7*b^13*c^3*e^19*f^3 + 3840*a^8*b^11*c^4*e^19*f^3 - 2
4960*a^9*b^9*c^5*e^19*f^3 + 97280*a^10*b^7*c^6*e^19*f^3 - 227328*a^11*b^5*c^7*e^19*f^3 + 294912*a^12*b^3*c^8*e
^19*f^3 - 163840*a^13*b*c^9*e^19*f^3))/(32*a^6*e^2*f^2*(4*a*c - b^2)^5*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*
f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)*(a
^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b
^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*(b^6 - 45*a^3*c^3 + 40*a^2*b^2*c^2 - 11*a*b^4*c))/(8*a^3*c^2*(4*a*c - b^
2)^6*(6*b^10 - 6400*a^5*c^5 + 960*a^2*b^6*c^2 - 3850*a^3*b^4*c^3 + 7775*a^4*b^2*c^4 - 120*a*b^8*c)))*(16*a^9*b
^12*f^3*(4*a*c - b^2)^(15/2) + 65536*a^15*c^6*f^3*(4*a*c - b^2)^(15/2) - 384*a^10*b^10*c*f^3*(4*a*c - b^2)^(15
/2) + 3840*a^11*b^8*c^2*f^3*(4*a*c - b^2)^(15/2) - 20480*a^12*b^6*c^3*f^3*(4*a*c - b^2)^(15/2) + 61440*a^13*b^
4*c^4*f^3*(4*a*c - b^2)^(15/2) - 98304*a^14*b^2*c^5*f^3*(4*a*c - b^2)^(15/2)))/(b^10*c^2*e^14 - 20*a*b^8*c^3*e
^14 + 160*a^2*b^6*c^4*e^14 - 600*a^3*b^4*c^5*e^14 + 900*a^4*b^2*c^6*e^14) - (((b*((4*a^2*b^12*c^3*e^15*f - 93*
a^3*b^10*c^4*e^15*f + 854*a^4*b^8*c^5*e^15*f - 3889*a^5*b^6*c^6*e^15*f + 8808*a^6*b^4*c^7*e^15*f - 7952*a^7*b^
2*c^8*e^15*f - 8960*a^7*b*c^9*d^2*e^15*f + 6*a^2*b^11*c^4*d^2*e^15*f - 137*a^3*b^9*c^5*d^2*e^15*f + 1217*a^4*b
^7*c^6*d^2*e^15*f - 5256*a^5*b^5*c^7*d^2*e^15*f + 11024*a^6*b^3*c^8*d^2*e^15*f)/(a^6*b^12*f^3 + 4096*a^12*c^6*
f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c
^5*f^3) - (((4*a^4*b^14*c^2*e^16*f^2 - 100*a^5*b^12*c^3*e^16*f^2 + 1052*a^6*b^10*c^4*e^16*f^2 - 5952*a^7*b^8*c
^5*e^16*f^2 + 19072*a^8*b^6*c^6*e^16*f^2 - 32768*a^9*b^4*c^7*e^16*f^2 + 23552*a^10*b^2*c^8*e^16*f^2 + 5120*a^1
0*b*c^9*d^2*e^16*f^2 + 2*a^4*b^13*c^3*d^2*e^16*f^2 - 36*a^5*b^11*c^4*d^2*e^16*f^2 + 276*a^6*b^9*c^5*d^2*e^16*f
^2 - 1216*a^7*b^7*c^6*d^2*e^16*f^2 + 3456*a^8*b^5*c^7*d^2*e^16*f^2 - 6144*a^9*b^3*c^8*d^2*e^16*f^2)/(a^6*b^12*
f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f
^3 - 6144*a^11*b^2*c^5*f^3) + ((2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2
560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f)*(4*a^7*b^14*c^2*e^17*f^3 - 96*a^8*b^12*c^3*e^17*f^3 + 960*a^9*b^10*c^4*e
^17*f^3 - 5120*a^10*b^8*c^5*e^17*f^3 + 15360*a^11*b^6*c^6*e^17*f^3 - 24576*a^12*b^4*c^7*e^17*f^3 + 16384*a^13*
b^2*c^8*e^17*f^3 - 163840*a^13*b*c^9*d^2*e^17*f^3 + 12*a^6*b^15*c^2*d^2*e^17*f^3 - 328*a^7*b^13*c^3*d^2*e^17*f
^3 + 3840*a^8*b^11*c^4*d^2*e^17*f^3 - 24960*a^9*b^9*c^5*d^2*e^17*f^3 + 97280*a^10*b^7*c^6*d^2*e^17*f^3 - 22732
8*a^11*b^5*c^7*d^2*e^17*f^3 + 294912*a^12*b^3*c^8*d^2*e^17*f^3))/(2*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2
+ 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)*(a^6*
b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*
c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f
+ 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f))/(2*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2
*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)))*(b^4 + 30*a^2*c^2 - 10*a*
b^2*c))/(4*a^3*e*f*(4*a*c - b^2)^(5/2)) - (((b*((4*a^4*b^14*c^2*e^16*f^2 - 100*a^5*b^12*c^3*e^16*f^2 + 1052*a^
6*b^10*c^4*e^16*f^2 - 5952*a^7*b^8*c^5*e^16*f^2 + 19072*a^8*b^6*c^6*e^16*f^2 - 32768*a^9*b^4*c^7*e^16*f^2 + 23
552*a^10*b^2*c^8*e^16*f^2 + 5120*a^10*b*c^9*d^2*e^16*f^2 + 2*a^4*b^13*c^3*d^2*e^16*f^2 - 36*a^5*b^11*c^4*d^2*e
^16*f^2 + 276*a^6*b^9*c^5*d^2*e^16*f^2 - 1216*a^7*b^7*c^6*d^2*e^16*f^2 + 3456*a^8*b^5*c^7*d^2*e^16*f^2 - 6144*
a^9*b^3*c^8*d^2*e^16*f^2)/(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a
^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3) + ((2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^
6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f)*(4*a^7*b^14*c^2*e^17*f^3 - 96*a^8*b^
12*c^3*e^17*f^3 + 960*a^9*b^10*c^4*e^17*f^3 - 5120*a^10*b^8*c^5*e^17*f^3 + 15360*a^11*b^6*c^6*e^17*f^3 - 24576
*a^12*b^4*c^7*e^17*f^3 + 16384*a^13*b^2*c^8*e^17*f^3 - 163840*a^13*b*c^9*d^2*e^17*f^3 + 12*a^6*b^15*c^2*d^2*e^
17*f^3 - 328*a^7*b^13*c^3*d^2*e^17*f^3 + 3840*a^8*b^11*c^4*d^2*e^17*f^3 - 24960*a^9*b^9*c^5*d^2*e^17*f^3 + 972
80*a^10*b^7*c^6*d^2*e^17*f^3 - 227328*a^11*b^5*c^7*d^2*e^17*f^3 + 294912*a^12*b^3*c^8*d^2*e^17*f^3))/(2*(4*a^3
*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e
^2*f^2 - 80*a^4*b^8*c*e^2*f^2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1
280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(4*a^3
*e*f*(4*a*c - b^2)^(5/2)) + (b*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)*(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^
2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f)*(4*a^7*b^14*c^2*e^17*f^3 - 96*a^8*b^12*c
^3*e^17*f^3 + 960*a^9*b^10*c^4*e^17*f^3 - 5120*a^10*b^8*c^5*e^17*f^3 + 15360*a^11*b^6*c^6*e^17*f^3 - 24576*a^1
2*b^4*c^7*e^17*f^3 + 16384*a^13*b^2*c^8*e^17*f^3 - 163840*a^13*b*c^9*d^2*e^17*f^3 + 12*a^6*b^15*c^2*d^2*e^17*f
^3 - 328*a^7*b^13*c^3*d^2*e^17*f^3 + 3840*a^8*b^11*c^4*d^2*e^17*f^3 - 24960*a^9*b^9*c^5*d^2*e^17*f^3 + 97280*a
^10*b^7*c^6*d^2*e^17*f^3 - 227328*a^11*b^5*c^7*d^2*e^17*f^3 + 294912*a^12*b^3*c^8*d^2*e^17*f^3))/(8*a^3*e*f*(4
*a*c - b^2)^(5/2)*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*
f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 +
240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*(2*b^10*e*f - 2
048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f))/(2*(4*a
^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4
*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)) + (b^3*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^3*(4*a^7*b^14*c^2*e^17*f^3 - 96*a^8*b
^12*c^3*e^17*f^3 + 960*a^9*b^10*c^4*e^17*f^3 - 5120*a^10*b^8*c^5*e^17*f^3 + 15360*a^11*b^6*c^6*e^17*f^3 - 2457
6*a^12*b^4*c^7*e^17*f^3 + 16384*a^13*b^2*c^8*e^17*f^3 - 163840*a^13*b*c^9*d^2*e^17*f^3 + 12*a^6*b^15*c^2*d^2*e
^17*f^3 - 328*a^7*b^13*c^3*d^2*e^17*f^3 + 3840*a^8*b^11*c^4*d^2*e^17*f^3 - 24960*a^9*b^9*c^5*d^2*e^17*f^3 + 97
280*a^10*b^7*c^6*d^2*e^17*f^3 - 227328*a^11*b^5*c^7*d^2*e^17*f^3 + 294912*a^12*b^3*c^8*d^2*e^17*f^3))/(64*a^9*
e^3*f^3*(4*a*c - b^2)^(15/2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 128
0*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*(3*b^8 + 160*a^4*c^4 + 180*a^2*b^4*c^2 -
325*a^3*b^2*c^3 - 39*a*b^6*c)*(16*a^9*b^12*f^3*(4*a*c - b^2)^(15/2) + 65536*a^15*c^6*f^3*(4*a*c - b^2)^(15/2)
- 384*a^10*b^10*c*f^3*(4*a*c - b^2)^(15/2) + 3840*a^11*b^8*c^2*f^3*(4*a*c - b^2)^(15/2) - 20480*a^12*b^6*c^3*f
^3*(4*a*c - b^2)^(15/2) + 61440*a^13*b^4*c^4*f^3*(4*a*c - b^2)^(15/2) - 98304*a^14*b^2*c^5*f^3*(4*a*c - b^2)^(
15/2)))/(8*a^3*c^2*(4*a*c - b^2)^(13/2)*(b^10*c^2*e^14 - 20*a*b^8*c^3*e^14 + 160*a^2*b^6*c^4*e^14 - 600*a^3*b^
4*c^5*e^14 + 900*a^4*b^2*c^6*e^14)*(6*b^10 - 6400*a^5*c^5 + 960*a^2*b^6*c^2 - 3850*a^3*b^4*c^3 + 7775*a^4*b^2*
c^4 - 120*a*b^8*c)) - (3*b*((((4*a^2*b^12*c^3*e^15*f - 93*a^3*b^10*c^4*e^15*f + 854*a^4*b^8*c^5*e^15*f - 3889*
a^5*b^6*c^6*e^15*f + 8808*a^6*b^4*c^7*e^15*f - 7952*a^7*b^2*c^8*e^15*f - 8960*a^7*b*c^9*d^2*e^15*f + 6*a^2*b^1
1*c^4*d^2*e^15*f - 137*a^3*b^9*c^5*d^2*e^15*f + 1217*a^4*b^7*c^6*d^2*e^15*f - 5256*a^5*b^5*c^7*d^2*e^15*f + 11
024*a^6*b^3*c^8*d^2*e^15*f)/(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280
*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3) - (((4*a^4*b^14*c^2*e^16*f^2 - 100*a^5*b^12*
c^3*e^16*f^2 + 1052*a^6*b^10*c^4*e^16*f^2 - 5952*a^7*b^8*c^5*e^16*f^2 + 19072*a^8*b^6*c^6*e^16*f^2 - 32768*a^9
*b^4*c^7*e^16*f^2 + 23552*a^10*b^2*c^8*e^16*f^2 + 5120*a^10*b*c^9*d^2*e^16*f^2 + 2*a^4*b^13*c^3*d^2*e^16*f^2 -
36*a^5*b^11*c^4*d^2*e^16*f^2 + 276*a^6*b^9*c^5*d^2*e^16*f^2 - 1216*a^7*b^7*c^6*d^2*e^16*f^2 + 3456*a^8*b^5*c^
7*d^2*e^16*f^2 - 6144*a^9*b^3*c^8*d^2*e^16*f^2)/(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^
8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3) + ((2*b^10*e*f - 2048*a^
5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f)*(4*a^7*b^14*c^
2*e^17*f^3 - 96*a^8*b^12*c^3*e^17*f^3 + 960*a^9*b^10*c^4*e^17*f^3 - 5120*a^10*b^8*c^5*e^17*f^3 + 15360*a^11*b^
6*c^6*e^17*f^3 - 24576*a^12*b^4*c^7*e^17*f^3 + 16384*a^13*b^2*c^8*e^17*f^3 - 163840*a^13*b*c^9*d^2*e^17*f^3 +
12*a^6*b^15*c^2*d^2*e^17*f^3 - 328*a^7*b^13*c^3*d^2*e^17*f^3 + 3840*a^8*b^11*c^4*d^2*e^17*f^3 - 24960*a^9*b^9*
c^5*d^2*e^17*f^3 + 97280*a^10*b^7*c^6*d^2*e^17*f^3 - 227328*a^11*b^5*c^7*d^2*e^17*f^3 + 294912*a^12*b^3*c^8*d^
2*e^17*f^3))/(2*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^
2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 2
40*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*(2*b^10*e*f - 204
8*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f))/(2*(4*a^3
*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e
^2*f^2 - 80*a^4*b^8*c*e^2*f^2)))*(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f +
2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f))/(2*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f
^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)) - (b^10*c^4*e^14 - 22*a*b^8*
c^5*e^14 + 177*a^2*b^6*c^6*e^14 - 616*a^3*b^4*c^7*e^14 + 784*a^4*b^2*c^8*e^14 + b^9*c^5*d^2*e^14 + 147*a^2*b^5
*c^7*d^2*e^14 - 343*a^3*b^3*c^8*d^2*e^14 - 21*a*b^7*c^6*d^2*e^14)/(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b
^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3) + (b*(
(b*((4*a^4*b^14*c^2*e^16*f^2 - 100*a^5*b^12*c^3*e^16*f^2 + 1052*a^6*b^10*c^4*e^16*f^2 - 5952*a^7*b^8*c^5*e^16*
f^2 + 19072*a^8*b^6*c^6*e^16*f^2 - 32768*a^9*b^4*c^7*e^16*f^2 + 23552*a^10*b^2*c^8*e^16*f^2 + 5120*a^10*b*c^9*
d^2*e^16*f^2 + 2*a^4*b^13*c^3*d^2*e^16*f^2 - 36*a^5*b^11*c^4*d^2*e^16*f^2 + 276*a^6*b^9*c^5*d^2*e^16*f^2 - 121
6*a^7*b^7*c^6*d^2*e^16*f^2 + 3456*a^8*b^5*c^7*d^2*e^16*f^2 - 6144*a^9*b^3*c^8*d^2*e^16*f^2)/(a^6*b^12*f^3 + 40
96*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 614
4*a^11*b^2*c^5*f^3) + ((2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*
b^2*c^4*e*f - 40*a*b^8*c*e*f)*(4*a^7*b^14*c^2*e^17*f^3 - 96*a^8*b^12*c^3*e^17*f^3 + 960*a^9*b^10*c^4*e^17*f^3
- 5120*a^10*b^8*c^5*e^17*f^3 + 15360*a^11*b^6*c^6*e^17*f^3 - 24576*a^12*b^4*c^7*e^17*f^3 + 16384*a^13*b^2*c^8*
e^17*f^3 - 163840*a^13*b*c^9*d^2*e^17*f^3 + 12*a^6*b^15*c^2*d^2*e^17*f^3 - 328*a^7*b^13*c^3*d^2*e^17*f^3 + 384
0*a^8*b^11*c^4*d^2*e^17*f^3 - 24960*a^9*b^9*c^5*d^2*e^17*f^3 + 97280*a^10*b^7*c^6*d^2*e^17*f^3 - 227328*a^11*b
^5*c^7*d^2*e^17*f^3 + 294912*a^12*b^3*c^8*d^2*e^17*f^3))/(2*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a
^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)*(a^6*b^12*f^3
+ 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3
- 6144*a^11*b^2*c^5*f^3)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(4*a^3*e*f*(4*a*c - b^2)^(5/2)) + (b*(b^4 + 30*a^2
*c^2 - 10*a*b^2*c)*(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3*b^4*c^3*e*f + 2560*a^4*b^2*
c^4*e*f - 40*a*b^8*c*e*f)*(4*a^7*b^14*c^2*e^17*f^3 - 96*a^8*b^12*c^3*e^17*f^3 + 960*a^9*b^10*c^4*e^17*f^3 - 51
20*a^10*b^8*c^5*e^17*f^3 + 15360*a^11*b^6*c^6*e^17*f^3 - 24576*a^12*b^4*c^7*e^17*f^3 + 16384*a^13*b^2*c^8*e^17
*f^3 - 163840*a^13*b*c^9*d^2*e^17*f^3 + 12*a^6*b^15*c^2*d^2*e^17*f^3 - 328*a^7*b^13*c^3*d^2*e^17*f^3 + 3840*a^
8*b^11*c^4*d^2*e^17*f^3 - 24960*a^9*b^9*c^5*d^2*e^17*f^3 + 97280*a^10*b^7*c^6*d^2*e^17*f^3 - 227328*a^11*b^5*c
^7*d^2*e^17*f^3 + 294912*a^12*b^3*c^8*d^2*e^17*f^3))/(8*a^3*e*f*(4*a*c - b^2)^(5/2)*(4*a^3*b^10*e^2*f^2 - 4096
*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*b^2*c^4*e^2*f^2 - 80*a^4*b^8*
c*e^2*f^2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^2*f^3 - 1280*a^9*b^6*c^3*f^3
+ 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(4*a^3*e*f*(4*a*c - b^2)^(
5/2)) + (b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2*(2*b^10*e*f - 2048*a^5*c^5*e*f + 320*a^2*b^6*c^2*e*f - 1280*a^3
*b^4*c^3*e*f + 2560*a^4*b^2*c^4*e*f - 40*a*b^8*c*e*f)*(4*a^7*b^14*c^2*e^17*f^3 - 96*a^8*b^12*c^3*e^17*f^3 + 96
0*a^9*b^10*c^4*e^17*f^3 - 5120*a^10*b^8*c^5*e^17*f^3 + 15360*a^11*b^6*c^6*e^17*f^3 - 24576*a^12*b^4*c^7*e^17*f
^3 + 16384*a^13*b^2*c^8*e^17*f^3 - 163840*a^13*b*c^9*d^2*e^17*f^3 + 12*a^6*b^15*c^2*d^2*e^17*f^3 - 328*a^7*b^1
3*c^3*d^2*e^17*f^3 + 3840*a^8*b^11*c^4*d^2*e^17*f^3 - 24960*a^9*b^9*c^5*d^2*e^17*f^3 + 97280*a^10*b^7*c^6*d^2*
e^17*f^3 - 227328*a^11*b^5*c^7*d^2*e^17*f^3 + 294912*a^12*b^3*c^8*d^2*e^17*f^3))/(32*a^6*e^2*f^2*(4*a*c - b^2)
^5*(4*a^3*b^10*e^2*f^2 - 4096*a^8*c^5*e^2*f^2 + 640*a^5*b^6*c^2*e^2*f^2 - 2560*a^6*b^4*c^3*e^2*f^2 + 5120*a^7*
b^2*c^4*e^2*f^2 - 80*a^4*b^8*c*e^2*f^2)*(a^6*b^12*f^3 + 4096*a^12*c^6*f^3 - 24*a^7*b^10*c*f^3 + 240*a^8*b^8*c^
2*f^3 - 1280*a^9*b^6*c^3*f^3 + 3840*a^10*b^4*c^4*f^3 - 6144*a^11*b^2*c^5*f^3)))*(b^6 - 45*a^3*c^3 + 40*a^2*b^2
*c^2 - 11*a*b^4*c)*(16*a^9*b^12*f^3*(4*a*c - b^2)^(15/2) + 65536*a^15*c^6*f^3*(4*a*c - b^2)^(15/2) - 384*a^10*
b^10*c*f^3*(4*a*c - b^2)^(15/2) + 3840*a^11*b^8*c^2*f^3*(4*a*c - b^2)^(15/2) - 20480*a^12*b^6*c^3*f^3*(4*a*c -
b^2)^(15/2) + 61440*a^13*b^4*c^4*f^3*(4*a*c - b^2)^(15/2) - 98304*a^14*b^2*c^5*f^3*(4*a*c - b^2)^(15/2)))/(8*
a^3*c^2*(4*a*c - b^2)^6*(b^10*c^2*e^14 - 20*a*b^8*c^3*e^14 + 160*a^2*b^6*c^4*e^14 - 600*a^3*b^4*c^5*e^14 + 900
*a^4*b^2*c^6*e^14)*(6*b^10 - 6400*a^5*c^5 + 960*a^2*b^6*c^2 - 3850*a^3*b^4*c^3 + 7775*a^4*b^2*c^4 - 120*a*b^8*
c)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(2*a^3*e*f*(4*a*c - b^2)^(5/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*f*x+d*f)/(a+b*(e*x+d)**2+c*(e*x+d)**4)**3,x)

[Out]

Timed out

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