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

Optimal. Leaf size=174 \[ \frac {b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 e f \left (b^2-4 a c\right )^{3/2}}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f}+\frac {\log (d+e x)}{a^2 e f}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a e f \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \]

[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

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Rubi [A]  time = 0.30, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {1142, 1114, 740, 800, 634, 618, 206, 628} \[ \frac {b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 e f \left (b^2-4 a c\right )^{3/2}}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f}+\frac {\log (d+e x)}{a^2 e f}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a e f \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \]

Antiderivative was successfully verified.

[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 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 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 )^2} \, dx &=\frac {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 ) \operatorname {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 ) \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^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*}

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Mathematica [A]  time = 0.44, size = 238, normalized size = 1.37 \[ \frac {\frac {2 a \left (-2 a c+b^2+b c (d+e x)^2\right )}{\left (b^2-4 a c\right ) \left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )}-\frac {\left (b^2 \sqrt {b^2-4 a c}-4 a c \sqrt {b^2-4 a c}-6 a b c+b^3\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (-b^2 \sqrt {b^2-4 a c}+4 a c \sqrt {b^2-4 a c}-6 a b c+b^3\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}+4 \log (d+e x)}{4 a^2 e f} \]

Antiderivative was successfully verified.

[In]

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

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fricas [B]  time = 1.72, size = 2486, normalized size = 14.29 \[ \text {result too large to display} \]

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)^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)]

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giac [B]  time = 1.43, size = 476, normalized size = 2.74 \[ -\frac {{\left (a^{2} b^{3} c f e^{3} - 6 \, a^{3} b c^{2} f e^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left ({\left | b x^{2} e^{2} + 2 \, b d x e + \sqrt {b^{2} - 4 \, a c} x^{2} e^{2} + 2 \, \sqrt {b^{2} - 4 \, a c} d x e + b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} + 2 \, a \right |}\right ) - {\left (a^{2} b^{3} c f e^{3} - 6 \, a^{3} b c^{2} f e^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left ({\left | -b x^{2} e^{2} - 2 \, b d x e + \sqrt {b^{2} - 4 \, a c} x^{2} e^{2} + 2 \, \sqrt {b^{2} - 4 \, a c} d x e - b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} - 2 \, a \right |}\right )}{4 \, {\left (a^{4} b^{4} c f^{2} e^{4} - 8 \, a^{5} b^{2} c^{2} f^{2} e^{4} + 16 \, a^{6} c^{3} f^{2} e^{4}\right )}} - \frac {e^{\left (-1\right )} \log \left ({\left | c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + 6 \, c d^{2} x^{2} e^{2} + 4 \, c d^{3} x e + c d^{4} + b x^{2} e^{2} + 2 \, b d x e + b d^{2} + a \right |}\right )}{4 \, a^{2} f} + \frac {e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\right )}{a^{2} f} + \frac {{\left (a b c x^{2} e^{2} + 2 \, a b c d x e + a b c d^{2} + a b^{2} - 2 \, a^{2} c\right )} e^{\left (-1\right )}}{2 \, {\left (c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + 6 \, c d^{2} x^{2} e^{2} + 4 \, c d^{3} x e + c d^{4} + b x^{2} e^{2} + 2 \, b d x e + b d^{2} + a\right )} {\left (b^{2} - 4 \, a c\right )} a^{2} f} \]

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)^2,x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.03, size = 714, normalized size = 4.10 \[ \text {result too large to display} \]

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)^2,x)

[Out]

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

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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)^2,x, algorithm="maxima")

[Out]

Timed out

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mupad [B]  time = 11.69, size = 13434, normalized size = 77.21 \[ \text {result too large to display} \]

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)^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))

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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)**2,x)

[Out]

Timed out

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