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

Optimal. Leaf size=228 \[ \frac {b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{2 a^3 e f^3}-\frac {2 b \log (d+e x)}{a^3 e f^3}-\frac {b^2-3 a c}{a^2 e f^3 \left (b^2-4 a c\right ) (d+e x)^2}-\frac {\left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{a^3 e f^3 \left (b^2-4 a c\right )^{3/2}}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a e f^3 \left (b^2-4 a c\right ) (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \]

[Out]

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

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Rubi [A]  time = 0.37, antiderivative size = 228, 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} \[ -\frac {\left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{a^3 e f^3 \left (b^2-4 a c\right )^{3/2}}-\frac {b^2-3 a c}{a^2 e f^3 \left (b^2-4 a c\right ) (d+e x)^2}+\frac {b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{2 a^3 e f^3}-\frac {2 b \log (d+e x)}{a^3 e f^3}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a e f^3 \left (b^2-4 a c\right ) (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.52, size = 287, normalized size = 1.26 \[ \frac {\frac {\left (6 a^2 c^2-6 a b^2 c-4 a b c \sqrt {b^2-4 a c}+b^3 \sqrt {b^2-4 a c}+b^4\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (-6 a^2 c^2+6 a b^2 c-4 a b c \sqrt {b^2-4 a c}+b^3 \sqrt {b^2-4 a c}-b^4\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {a \left (-3 a b c-2 a c^2 (d+e x)^2+b^3+b^2 c (d+e x)^2\right )}{\left (4 a c-b^2\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {a}{(d+e x)^2}-4 b \log (d+e x)}{2 a^3 e f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 4.14, size = 4604, normalized size = 20.19 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*e^4*x^4 + 8*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d*e^3*x^3 +
a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + 2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d^4 + (2*a*b^5 - 15*a^2*b^3*c +
28*a^3*b*c^2 + 12*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d^2)*e^2*x^2 + (2*a*b^5 - 15*a^2*b^3*c + 28*a^3*b*c^
2)*d^2 + 2*(4*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d^3 + (2*a*b^5 - 15*a^2*b^3*c + 28*a^3*b*c^2)*d)*e*x + ((
b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*e^6*x^6 + 6*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d*e^5*x^5 + (b^5 - 6*a*b^3*c +
6*a^2*b*c^2 + 15*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d^2)*e^4*x^4 + (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d^6 + 4*(5
*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d^3 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d)*e^3*x^3 + (b^5 - 6*a*b^3*c + 6*a^2
*b*c^2)*d^4 + (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2 + 15*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d^4 + 6*(b^5 - 6*a*b^3*c
+ 6*a^2*b*c^2)*d^2)*e^2*x^2 + (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*d^2 + 2*(3*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*
d^5 + 2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d^3 + (a*b^4 - 6*a^2*b^2*c + 6*a^3*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^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e^6
*x^6 + 6*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e^5*x^5 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2 + 15*(b^5*c - 8*a*
b^3*c^2 + 16*a^2*b*c^3)*d^2)*e^4*x^4 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6 + 4*(5*(b^5*c - 8*a*b^3*c^2 +
16*a^2*b*c^3)*d^3 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d)*e^3*x^3 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d^4 + (
a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + 15*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^4 + 6*(b^6 - 8*a*b^4*c + 16*a^2
*b^2*c^2)*d^2)*e^2*x^2 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d^2 + 2*(3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*
d^5 + 2*(b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d^3 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*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^5*c - 8*a*b^3*c^2 +
16*a^2*b*c^3)*e^6*x^6 + 6*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e^5*x^5 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2
+ 15*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2)*e^4*x^4 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6 + 4*(5*(b^5*
c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d)*e^3*x^3 + (b^6 - 8*a*b^4*c + 16*a^
2*b^2*c^2)*d^4 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + 15*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^4 + 6*(b^6 -
8*a*b^4*c + 16*a^2*b^2*c^2)*d^2)*e^2*x^2 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d^2 + 2*(3*(b^5*c - 8*a*b^3*c^
2 + 16*a^2*b*c^3)*d^5 + 2*(b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d^3 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d)*e*x
)*log(e*x + d))/((a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*e^7*f^3*x^6 + 6*(a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*
c^3)*d*e^6*f^3*x^5 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2 + 15*(a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*d^2)*
e^5*f^3*x^4 + 4*(5*(a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*d^3 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d)*e^
4*f^3*x^3 + (a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 + 15*(a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*d^4 + 6*(a^3*b^5
- 8*a^4*b^3*c + 16*a^5*b*c^2)*d^2)*e^3*f^3*x^2 + 2*(3*(a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*d^5 + 2*(a^3*b
^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d^3 + (a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*d)*e^2*f^3*x + ((a^3*b^4*c - 8*a^4
*b^2*c^2 + 16*a^5*c^3)*d^6 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d^4 + (a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)
*d^2)*e*f^3), -1/2*(2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*e^4*x^4 + 8*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3
)*d*e^3*x^3 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + 2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d^4 + (2*a*b^5 - 1
5*a^2*b^3*c + 28*a^3*b*c^2 + 12*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d^2)*e^2*x^2 + (2*a*b^5 - 15*a^2*b^3*c
+ 28*a^3*b*c^2)*d^2 + 2*(4*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d^3 + (2*a*b^5 - 15*a^2*b^3*c + 28*a^3*b*c^2
)*d)*e*x + 2*((b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*e^6*x^6 + 6*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d*e^5*x^5 + (b^5
- 6*a*b^3*c + 6*a^2*b*c^2 + 15*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d^2)*e^4*x^4 + (b^4*c - 6*a*b^2*c^2 + 6*a^2*
c^3)*d^6 + 4*(5*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d^3 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d)*e^3*x^3 + (b^5 - 6*
a*b^3*c + 6*a^2*b*c^2)*d^4 + (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2 + 15*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d^4 + 6*(
b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d^2)*e^2*x^2 + (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*d^2 + 2*(3*(b^4*c - 6*a*b^2*c^
2 + 6*a^2*c^3)*d^5 + 2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d^3 + (a*b^4 - 6*a^2*b^2*c + 6*a^3*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^5*c - 8*a*
b^3*c^2 + 16*a^2*b*c^3)*e^6*x^6 + 6*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e^5*x^5 + (b^6 - 8*a*b^4*c + 16*a^2
*b^2*c^2 + 15*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2)*e^4*x^4 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6 + 4
*(5*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d)*e^3*x^3 + (b^6 - 8*a*b^4*
c + 16*a^2*b^2*c^2)*d^4 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + 15*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^4 +
6*(b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d^2)*e^2*x^2 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d^2 + 2*(3*(b^5*c - 8
*a*b^3*c^2 + 16*a^2*b*c^3)*d^5 + 2*(b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d^3 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*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^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e^6*x^6 + 6*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e^5*x^5 + (b^6 -
8*a*b^4*c + 16*a^2*b^2*c^2 + 15*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2)*e^4*x^4 + (b^5*c - 8*a*b^3*c^2 + 16*
a^2*b*c^3)*d^6 + 4*(5*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d)*e^3*x^3
+ (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d^4 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + 15*(b^5*c - 8*a*b^3*c^2 + 16
*a^2*b*c^3)*d^4 + 6*(b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d^2)*e^2*x^2 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d^2
+ 2*(3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^5 + 2*(b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d^3 + (a*b^5 - 8*a^2*b
^3*c + 16*a^3*b*c^2)*d)*e*x)*log(e*x + d))/((a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*e^7*f^3*x^6 + 6*(a^3*b^4*
c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*d*e^6*f^3*x^5 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2 + 15*(a^3*b^4*c - 8*a^4*
b^2*c^2 + 16*a^5*c^3)*d^2)*e^5*f^3*x^4 + 4*(5*(a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*d^3 + (a^3*b^5 - 8*a^4*
b^3*c + 16*a^5*b*c^2)*d)*e^4*f^3*x^3 + (a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 + 15*(a^3*b^4*c - 8*a^4*b^2*c^2 + 1
6*a^5*c^3)*d^4 + 6*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d^2)*e^3*f^3*x^2 + 2*(3*(a^3*b^4*c - 8*a^4*b^2*c^2 +
16*a^5*c^3)*d^5 + 2*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d^3 + (a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*d)*e^2*
f^3*x + ((a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*d^6 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d^4 + (a^4*b^4
- 8*a^5*b^2*c + 16*a^6*c^2)*d^2)*e*f^3)]

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giac [B]  time = 1.30, size = 687, normalized size = 3.01 \[ \frac {{\left (a^{3} b^{4} c f^{3} e^{3} - 6 \, a^{4} b^{2} c^{2} f^{3} e^{3} + 6 \, a^{5} c^{3} f^{3} e^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left ({\left | b x^{2} e^{2} + 2 \, b d x e + \sqrt {b^{2} - 4 \, a c} x^{2} e^{2} + 2 \, \sqrt {b^{2} - 4 \, a c} d x e + b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} + 2 \, a \right |}\right ) - {\left (a^{3} b^{4} c f^{3} e^{3} - 6 \, a^{4} b^{2} c^{2} f^{3} e^{3} + 6 \, a^{5} c^{3} f^{3} e^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left ({\left | -b x^{2} e^{2} - 2 \, b d x e + \sqrt {b^{2} - 4 \, a c} x^{2} e^{2} + 2 \, \sqrt {b^{2} - 4 \, a c} d x e - b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} - 2 \, a \right |}\right )}{2 \, {\left (a^{6} b^{4} c f^{6} e^{4} - 8 \, a^{7} b^{2} c^{2} f^{6} e^{4} + 16 \, a^{8} c^{3} f^{6} e^{4}\right )}} + \frac {b e^{\left (-1\right )} \log \left ({\left | c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + 6 \, c d^{2} x^{2} e^{2} + 4 \, c d^{3} x e + c d^{4} + b x^{2} e^{2} + 2 \, b d x e + b d^{2} + a \right |}\right )}{2 \, a^{3} f^{3}} - \frac {2 \, b e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\right )}{a^{3} f^{3}} - \frac {{\left (2 \, a b^{2} c d^{4} - 6 \, a^{2} c^{2} d^{4} + 2 \, a b^{3} d^{2} - 7 \, a^{2} b c d^{2} + 2 \, {\left (a b^{2} c e^{4} - 3 \, a^{2} c^{2} e^{4}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + 8 \, {\left (a b^{2} c d e^{3} - 3 \, a^{2} c^{2} d e^{3}\right )} x^{3} + {\left (12 \, a b^{2} c d^{2} e^{2} - 36 \, a^{2} c^{2} d^{2} e^{2} + 2 \, a b^{3} e^{2} - 7 \, a^{2} b c e^{2}\right )} x^{2} + 2 \, {\left (4 \, a b^{2} c d^{3} e - 12 \, a^{2} c^{2} d^{3} e + 2 \, a b^{3} d e - 7 \, a^{2} b c d e\right )} x\right )} e^{\left (-1\right )}}{2 \, {\left (c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + 6 \, c d^{2} x^{2} e^{2} + 4 \, c d^{3} x e + c d^{4} + b x^{2} e^{2} + 2 \, b d x e + b d^{2} + a\right )} {\left (b^{2} - 4 \, a c\right )} {\left (x e + d\right )}^{2} a^{3} f^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((a^3*b^4*c*f^3*e^3 - 6*a^4*b^2*c^2*f^3*e^3 + 6*a^5*c^3*f^3*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^4*c*f^3*e^3 - 6*a^4*b^2*c^2*f^3*e^3 + 6*a^5*c^3*f^3*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^6*b^4*c*f^6*
e^4 - 8*a^7*b^2*c^2*f^6*e^4 + 16*a^8*c^3*f^6*e^4) + 1/2*b*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^3) - 2*b*e^(-1)*log(abs(x*e + d))/(a
^3*f^3) - 1/2*(2*a*b^2*c*d^4 - 6*a^2*c^2*d^4 + 2*a*b^3*d^2 - 7*a^2*b*c*d^2 + 2*(a*b^2*c*e^4 - 3*a^2*c^2*e^4)*x
^4 + a^2*b^2 - 4*a^3*c + 8*(a*b^2*c*d*e^3 - 3*a^2*c^2*d*e^3)*x^3 + (12*a*b^2*c*d^2*e^2 - 36*a^2*c^2*d^2*e^2 +
2*a*b^3*e^2 - 7*a^2*b*c*e^2)*x^2 + 2*(4*a*b^2*c*d^3*e - 12*a^2*c^2*d^3*e + 2*a*b^3*d*e - 7*a^2*b*c*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)*(
b^2 - 4*a*c)*(x*e + d)^2*a^3*f^3)

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maple [C]  time = 0.03, size = 1047, normalized size = 4.59 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^3/(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 = 13.52, size = 14830, normalized size = 65.04 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(2*b^3*d - 12*a*c^2*d^3 + 4*b^2*c*d^3 - 7*a*b*c*d))/(4*a^3*c - a^2*b^2) - (x^4*(3*a*c^2*e^3 - b^2*c*e^3))/
(4*a^3*c - a^2*b^2) - (4*x^3*(3*a*c^2*d*e^2 - b^2*c*d*e^2))/(4*a^3*c - a^2*b^2) + (a*b^2 - 4*a^2*c + 2*b^3*d^2
- 6*a*c^2*d^4 + 2*b^2*c*d^4 - 7*a*b*c*d^2)/(2*e*(4*a^3*c - a^2*b^2)) + (x^2*(2*b^3*e - 36*a*c^2*d^2*e + 12*b^
2*c*d^2*e - 7*a*b*c*e))/(2*(4*a^3*c - a^2*b^2)))/(x^3*(20*c*d^3*e^3*f^3 + 4*b*d*e^3*f^3) + x*(2*a*d*e*f^3 + 4*
b*d^3*e*f^3 + 6*c*d^5*e*f^3) + x^4*(b*e^4*f^3 + 15*c*d^2*e^4*f^3) + x^2*(a*e^2*f^3 + 6*b*d^2*e^2*f^3 + 15*c*d^
4*e^2*f^3) + a*d^2*f^3 + b*d^4*f^3 + c*d^6*f^3 + c*e^6*f^3*x^6 + 6*c*d*e^5*f^3*x^5) + (log((((b + a^3*e*f^3*(-
(b^4 + 6*a^2*c^2 - 6*a*b^2*c)^2/(a^6*e^2*f^6*(4*a*c - b^2)^3))^(1/2))*(((b + a^3*e*f^3*(-(b^4 + 6*a^2*c^2 - 6*
a*b^2*c)^2/(a^6*e^2*f^6*(4*a*c - b^2)^3))^(1/2))*((4*c^2*e^16*(2*b^5 + 6*a^2*b*c^2 + b^4*c*d^2 - 30*a^2*c^3*d^
2 - 10*a*b^3*c + 2*a*b^2*c^2*d^2))/(a^2*f^3*(4*a*c - b^2)) + (4*c^3*e^18*x^2*(b^4 - 30*a^2*c^2 + 2*a*b^2*c))/(
a^2*f^3*(4*a*c - b^2)) - (2*b*c^2*e^16*(b + a^3*e*f^3*(-(b^4 + 6*a^2*c^2 - 6*a*b^2*c)^2/(a^6*e^2*f^6*(4*a*c -
b^2)^3))^(1/2))*(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^3) + (8*c^3*d*e^17*x*(b^4 - 30*a^2*c^2 + 2*a*b^2*c))/(a^2*f^3*(4*a*c - b^2))))/(2*a^3*e*f^3) - (4*c^3*e
^15*(3*a*c - b^2)*(4*b^4 + 3*a^2*c^2 + 6*b^3*c*d^2 - 17*a*b^2*c - 23*a*b*c^2*d^2))/(a^4*f^6*(4*a*c - b^2)^2) +
(4*b*c^4*e^17*x^2*(6*b^4 + 69*a^2*c^2 - 41*a*b^2*c))/(a^4*f^6*(4*a*c - b^2)^2) + (8*b*c^4*d*e^16*x*(6*b^4 + 6
9*a^2*c^2 - 41*a*b^2*c))/(a^4*f^6*(4*a*c - b^2)^2)))/(2*a^3*e*f^3) - (8*c^5*e^16*x^2*(3*a*c - b^2)^3)/(a^6*f^9
*(4*a*c - b^2)^3) + (8*c^4*e^14*(3*a*c - b^2)^2*(b^3 - 3*a*c^2*d^2 + b^2*c*d^2 - 4*a*b*c))/(a^6*f^9*(4*a*c - b
^2)^3) - (16*c^5*d*e^15*x*(3*a*c - b^2)^3)/(a^6*f^9*(4*a*c - b^2)^3))*(((b - a^3*e*f^3*(-(b^4 + 6*a^2*c^2 - 6*
a*b^2*c)^2/(a^6*e^2*f^6*(4*a*c - b^2)^3))^(1/2))*(((b - a^3*e*f^3*(-(b^4 + 6*a^2*c^2 - 6*a*b^2*c)^2/(a^6*e^2*f
^6*(4*a*c - b^2)^3))^(1/2))*((4*c^2*e^16*(2*b^5 + 6*a^2*b*c^2 + b^4*c*d^2 - 30*a^2*c^3*d^2 - 10*a*b^3*c + 2*a*
b^2*c^2*d^2))/(a^2*f^3*(4*a*c - b^2)) + (4*c^3*e^18*x^2*(b^4 - 30*a^2*c^2 + 2*a*b^2*c))/(a^2*f^3*(4*a*c - b^2)
) - (2*b*c^2*e^16*(b - a^3*e*f^3*(-(b^4 + 6*a^2*c^2 - 6*a*b^2*c)^2/(a^6*e^2*f^6*(4*a*c - b^2)^3))^(1/2))*(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^3) + (8*c^3*d*
e^17*x*(b^4 - 30*a^2*c^2 + 2*a*b^2*c))/(a^2*f^3*(4*a*c - b^2))))/(2*a^3*e*f^3) - (4*c^3*e^15*(3*a*c - b^2)*(4*
b^4 + 3*a^2*c^2 + 6*b^3*c*d^2 - 17*a*b^2*c - 23*a*b*c^2*d^2))/(a^4*f^6*(4*a*c - b^2)^2) + (4*b*c^4*e^17*x^2*(6
*b^4 + 69*a^2*c^2 - 41*a*b^2*c))/(a^4*f^6*(4*a*c - b^2)^2) + (8*b*c^4*d*e^16*x*(6*b^4 + 69*a^2*c^2 - 41*a*b^2*
c))/(a^4*f^6*(4*a*c - b^2)^2)))/(2*a^3*e*f^3) - (8*c^5*e^16*x^2*(3*a*c - b^2)^3)/(a^6*f^9*(4*a*c - b^2)^3) + (
8*c^4*e^14*(3*a*c - b^2)^2*(b^3 - 3*a*c^2*d^2 + b^2*c*d^2 - 4*a*b*c))/(a^6*f^9*(4*a*c - b^2)^3) - (16*c^5*d*e^
15*x*(3*a*c - b^2)^3)/(a^6*f^9*(4*a*c - b^2)^3)))*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*
b^3*c^2*e*f^3))/(2*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)) - (
2*b*log(d + e*x))/(a^3*e*f^3) - (atan(((2*a^9*b^6*f^9*(4*a*c - b^2)^(9/2) - 128*a^12*c^3*f^9*(4*a*c - b^2)^(9/
2) - 24*a^10*b^4*c*f^9*(4*a*c - b^2)^(9/2) + 96*a^11*b^2*c^2*f^9*(4*a*c - b^2)^(9/2))*(x*((((8*(54*a^3*c^8*d*e
^15 - 2*b^6*c^5*d*e^15 + 18*a*b^4*c^6*d*e^15 - 54*a^2*b^2*c^7*d*e^15))/(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*
b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) - (((8*(276*a^5*b*c^7*d*e^16*f^3 - 6*a^2*b^7*c^4*d*e^16*f^3 + 65*a^3*b^5*c^5*d
*e^16*f^3 - 233*a^4*b^3*c^6*d*e^16*f^3))/(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9
) - (((8*(480*a^8*c^7*d*e^17*f^6 - a^4*b^8*c^3*d*e^17*f^6 + 6*a^5*b^6*c^4*d*e^17*f^6 + 30*a^6*b^4*c^5*d*e^17*f
^6 - 272*a^7*b^2*c^6*d*e^17*f^6))/(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) - (4*
(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3)*(640*a^10*b*c^6*d*e^18*f^9 + 3*a^6*
b^9*c^2*d*e^18*f^9 - 46*a^7*b^7*c^3*d*e^18*f^9 + 264*a^8*b^5*c^4*d*e^18*f^9 - 672*a^9*b^3*c^5*d*e^18*f^9))/((a
^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 4
8*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^
3*c^2*e*f^3))/(2*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^7
*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3))/(2*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f
^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)) - (((((8*(480*a^8*c^7*d*e^17*f^6 - a^4*b^8*c^3*d*e^17*f^6
+ 6*a^5*b^6*c^4*d*e^17*f^6 + 30*a^6*b^4*c^5*d*e^17*f^6 - 272*a^7*b^2*c^6*d*e^17*f^6))/(a^6*b^6*f^9 - 64*a^9*c
^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) - (4*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a
^2*b^3*c^2*e*f^3)*(640*a^10*b*c^6*d*e^18*f^9 + 3*a^6*b^9*c^2*d*e^18*f^9 - 46*a^7*b^7*c^3*d*e^18*f^9 + 264*a^8*
b^5*c^4*d*e^18*f^9 - 672*a^9*b^3*c^5*d*e^18*f^9))/((a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b
^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^4 + 6*
a^2*c^2 - 6*a*b^2*c))/(2*a^3*e*f^3*(4*a*c - b^2)^(3/2)) - (2*(b^4 + 6*a^2*c^2 - 6*a*b^2*c)*(b^7*e*f^3 - 12*a*b
^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3)*(640*a^10*b*c^6*d*e^18*f^9 + 3*a^6*b^9*c^2*d*e^18*f^9
- 46*a^7*b^7*c^3*d*e^18*f^9 + 264*a^8*b^5*c^4*d*e^18*f^9 - 672*a^9*b^3*c^5*d*e^18*f^9))/(a^3*e*f^3*(4*a*c - b^
2)^(3/2)*(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*
e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^4 + 6*a^2*c^2 - 6*a*b^2*c))/(2*a^3*e*f^3*(4*a*c
- b^2)^(3/2)) + ((b^4 + 6*a^2*c^2 - 6*a*b^2*c)^2*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b
^3*c^2*e*f^3)*(640*a^10*b*c^6*d*e^18*f^9 + 3*a^6*b^9*c^2*d*e^18*f^9 - 46*a^7*b^7*c^3*d*e^18*f^9 + 264*a^8*b^5*
c^4*d*e^18*f^9 - 672*a^9*b^3*c^5*d*e^18*f^9))/(a^6*e^2*f^6*(4*a*c - b^2)^3*(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*
a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^
4*c*e^2*f^6)))*(3*b^6 - 3*a^3*c^3 + 36*a^2*b^2*c^2 - 21*a*b^4*c))/(8*a^3*c^2*(4*a*c - b^2)^3*(9*a^4*c^4 - 6*b^
8 - 288*a^2*b^4*c^2 + 382*a^3*b^2*c^3 + 72*a*b^6*c)) - (b*((((((8*(480*a^8*c^7*d*e^17*f^6 - a^4*b^8*c^3*d*e^17
*f^6 + 6*a^5*b^6*c^4*d*e^17*f^6 + 30*a^6*b^4*c^5*d*e^17*f^6 - 272*a^7*b^2*c^6*d*e^17*f^6))/(a^6*b^6*f^9 - 64*a
^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) - (4*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 +
48*a^2*b^3*c^2*e*f^3)*(640*a^10*b*c^6*d*e^18*f^9 + 3*a^6*b^9*c^2*d*e^18*f^9 - 46*a^7*b^7*c^3*d*e^18*f^9 + 264*
a^8*b^5*c^4*d*e^18*f^9 - 672*a^9*b^3*c^5*d*e^18*f^9))/((a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a
^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^4
+ 6*a^2*c^2 - 6*a*b^2*c))/(2*a^3*e*f^3*(4*a*c - b^2)^(3/2)) - (2*(b^4 + 6*a^2*c^2 - 6*a*b^2*c)*(b^7*e*f^3 - 12
*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3)*(640*a^10*b*c^6*d*e^18*f^9 + 3*a^6*b^9*c^2*d*e^18*
f^9 - 46*a^7*b^7*c^3*d*e^18*f^9 + 264*a^8*b^5*c^4*d*e^18*f^9 - 672*a^9*b^3*c^5*d*e^18*f^9))/(a^3*e*f^3*(4*a*c
- b^2)^(3/2)*(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*
c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*
f^3 + 48*a^2*b^3*c^2*e*f^3))/(2*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*
e^2*f^6)) - (((8*(276*a^5*b*c^7*d*e^16*f^3 - 6*a^2*b^7*c^4*d*e^16*f^3 + 65*a^3*b^5*c^5*d*e^16*f^3 - 233*a^4*b^
3*c^6*d*e^16*f^3))/(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) - (((8*(480*a^8*c^7*
d*e^17*f^6 - a^4*b^8*c^3*d*e^17*f^6 + 6*a^5*b^6*c^4*d*e^17*f^6 + 30*a^6*b^4*c^5*d*e^17*f^6 - 272*a^7*b^2*c^6*d
*e^17*f^6))/(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) - (4*(b^7*e*f^3 - 12*a*b^5*
c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3)*(640*a^10*b*c^6*d*e^18*f^9 + 3*a^6*b^9*c^2*d*e^18*f^9 - 4
6*a^7*b^7*c^3*d*e^18*f^9 + 264*a^8*b^5*c^4*d*e^18*f^9 - 672*a^9*b^3*c^5*d*e^18*f^9))/((a^6*b^6*f^9 - 64*a^9*c^
3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6
- 12*a^4*b^4*c*e^2*f^6)))*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3))/(2*(a^3*
b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^4 + 6*a^2*c^2 - 6*a*b^2
*c))/(2*a^3*e*f^3*(4*a*c - b^2)^(3/2)) + ((b^4 + 6*a^2*c^2 - 6*a*b^2*c)^3*(640*a^10*b*c^6*d*e^18*f^9 + 3*a^6*b
^9*c^2*d*e^18*f^9 - 46*a^7*b^7*c^3*d*e^18*f^9 + 264*a^8*b^5*c^4*d*e^18*f^9 - 672*a^9*b^3*c^5*d*e^18*f^9))/(a^9
*e^3*f^9*(4*a*c - b^2)^(9/2)*(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9)))*(3*b^6 -
49*a^3*c^3 + 72*a^2*b^2*c^2 - 27*a*b^4*c))/(8*a^3*c^2*(4*a*c - b^2)^(7/2)*(9*a^4*c^4 - 6*b^8 - 288*a^2*b^4*c^
2 + 382*a^3*b^2*c^3 + 72*a*b^6*c))) + x^2*((((4*(54*a^3*c^8*e^16 - 2*b^6*c^5*e^16 + 18*a*b^4*c^6*e^16 - 54*a^2
*b^2*c^7*e^16))/(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) + (((4*(6*a^2*b^7*c^4*e
^17*f^3 - 65*a^3*b^5*c^5*e^17*f^3 + 233*a^4*b^3*c^6*e^17*f^3 - 276*a^5*b*c^7*e^17*f^3))/(a^6*b^6*f^9 - 64*a^9*
c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) + (((4*(480*a^8*c^7*e^18*f^6 - a^4*b^8*c^3*e^18*f^6 + 6*a^5*b
^6*c^4*e^18*f^6 + 30*a^6*b^4*c^5*e^18*f^6 - 272*a^7*b^2*c^6*e^18*f^6))/(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*
b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) - (2*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3
)*(3*a^6*b^9*c^2*e^19*f^9 - 46*a^7*b^7*c^3*e^19*f^9 + 264*a^8*b^5*c^4*e^19*f^9 - 672*a^9*b^3*c^5*e^19*f^9 + 64
0*a^10*b*c^6*e^19*f^9))/((a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f
^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*
a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3))/(2*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 1
2*a^4*b^4*c*e^2*f^6)))*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3))/(2*(a^3*b^6
*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)) - (((((4*(480*a^8*c^7*e^18*f^6
- a^4*b^8*c^3*e^18*f^6 + 6*a^5*b^6*c^4*e^18*f^6 + 30*a^6*b^4*c^5*e^18*f^6 - 272*a^7*b^2*c^6*e^18*f^6))/(a^6*b
^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) - (2*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b
*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3)*(3*a^6*b^9*c^2*e^19*f^9 - 46*a^7*b^7*c^3*e^19*f^9 + 264*a^8*b^5*c^4*e^19*f^
9 - 672*a^9*b^3*c^5*e^19*f^9 + 640*a^10*b*c^6*e^19*f^9))/((a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 4
8*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b
^4 + 6*a^2*c^2 - 6*a*b^2*c))/(2*a^3*e*f^3*(4*a*c - b^2)^(3/2)) - ((b^4 + 6*a^2*c^2 - 6*a*b^2*c)*(b^7*e*f^3 - 1
2*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3)*(3*a^6*b^9*c^2*e^19*f^9 - 46*a^7*b^7*c^3*e^19*f^9
+ 264*a^8*b^5*c^4*e^19*f^9 - 672*a^9*b^3*c^5*e^19*f^9 + 640*a^10*b*c^6*e^19*f^9))/(a^3*e*f^3*(4*a*c - b^2)^(3
/2)*(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f
^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^4 + 6*a^2*c^2 - 6*a*b^2*c))/(2*a^3*e*f^3*(4*a*c - b^2
)^(3/2)) + ((b^4 + 6*a^2*c^2 - 6*a*b^2*c)^2*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^
2*e*f^3)*(3*a^6*b^9*c^2*e^19*f^9 - 46*a^7*b^7*c^3*e^19*f^9 + 264*a^8*b^5*c^4*e^19*f^9 - 672*a^9*b^3*c^5*e^19*f
^9 + 640*a^10*b*c^6*e^19*f^9))/(2*a^6*e^2*f^6*(4*a*c - b^2)^3*(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9
+ 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6))
)*(3*b^6 - 3*a^3*c^3 + 36*a^2*b^2*c^2 - 21*a*b^4*c))/(8*a^3*c^2*(4*a*c - b^2)^3*(9*a^4*c^4 - 6*b^8 - 288*a^2*b
^4*c^2 + 382*a^3*b^2*c^3 + 72*a*b^6*c)) - (b*((((((4*(480*a^8*c^7*e^18*f^6 - a^4*b^8*c^3*e^18*f^6 + 6*a^5*b^6*
c^4*e^18*f^6 + 30*a^6*b^4*c^5*e^18*f^6 - 272*a^7*b^2*c^6*e^18*f^6))/(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4
*c*f^9 + 48*a^8*b^2*c^2*f^9) - (2*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3)*(
3*a^6*b^9*c^2*e^19*f^9 - 46*a^7*b^7*c^3*e^19*f^9 + 264*a^8*b^5*c^4*e^19*f^9 - 672*a^9*b^3*c^5*e^19*f^9 + 640*a
^10*b*c^6*e^19*f^9))/((a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6
- 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^4 + 6*a^2*c^2 - 6*a*b^2*c))/(2*a^3*
e*f^3*(4*a*c - b^2)^(3/2)) - ((b^4 + 6*a^2*c^2 - 6*a*b^2*c)*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3
+ 48*a^2*b^3*c^2*e*f^3)*(3*a^6*b^9*c^2*e^19*f^9 - 46*a^7*b^7*c^3*e^19*f^9 + 264*a^8*b^5*c^4*e^19*f^9 - 672*a^
9*b^3*c^5*e^19*f^9 + 640*a^10*b*c^6*e^19*f^9))/(a^3*e*f^3*(4*a*c - b^2)^(3/2)*(a^6*b^6*f^9 - 64*a^9*c^3*f^9 -
12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4
*b^4*c*e^2*f^6)))*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3))/(2*(a^3*b^6*e^2*
f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)) + (((4*(6*a^2*b^7*c^4*e^17*f^3 - 65
*a^3*b^5*c^5*e^17*f^3 + 233*a^4*b^3*c^6*e^17*f^3 - 276*a^5*b*c^7*e^17*f^3))/(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12
*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) + (((4*(480*a^8*c^7*e^18*f^6 - a^4*b^8*c^3*e^18*f^6 + 6*a^5*b^6*c^4*e^18*
f^6 + 30*a^6*b^4*c^5*e^18*f^6 - 272*a^7*b^2*c^6*e^18*f^6))/(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 +
48*a^8*b^2*c^2*f^9) - (2*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3)*(3*a^6*b^9
*c^2*e^19*f^9 - 46*a^7*b^7*c^3*e^19*f^9 + 264*a^8*b^5*c^4*e^19*f^9 - 672*a^9*b^3*c^5*e^19*f^9 + 640*a^10*b*c^6
*e^19*f^9))/((a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*
c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*
f^3 + 48*a^2*b^3*c^2*e*f^3))/(2*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*
e^2*f^6)))*(b^4 + 6*a^2*c^2 - 6*a*b^2*c))/(2*a^3*e*f^3*(4*a*c - b^2)^(3/2)) + ((b^4 + 6*a^2*c^2 - 6*a*b^2*c)^3
*(3*a^6*b^9*c^2*e^19*f^9 - 46*a^7*b^7*c^3*e^19*f^9 + 264*a^8*b^5*c^4*e^19*f^9 - 672*a^9*b^3*c^5*e^19*f^9 + 640
*a^10*b*c^6*e^19*f^9))/(2*a^9*e^3*f^9*(4*a*c - b^2)^(9/2)*(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 4
8*a^8*b^2*c^2*f^9)))*(3*b^6 - 49*a^3*c^3 + 72*a^2*b^2*c^2 - 27*a*b^4*c))/(8*a^3*c^2*(4*a*c - b^2)^(7/2)*(9*a^4
*c^4 - 6*b^8 - 288*a^2*b^4*c^2 + 382*a^3*b^2*c^3 + 72*a*b^6*c))) + (((((4*(36*a^6*c^7*e^15*f^3 + 4*a^2*b^8*c^3
*e^15*f^3 - 45*a^3*b^6*c^4*e^15*f^3 + 170*a^4*b^4*c^5*e^15*f^3 - 225*a^5*b^2*c^6*e^15*f^3 - 276*a^5*b*c^7*d^2*
e^15*f^3 + 6*a^2*b^7*c^4*d^2*e^15*f^3 - 65*a^3*b^5*c^5*d^2*e^15*f^3 + 233*a^4*b^3*c^6*d^2*e^15*f^3))/(a^6*b^6*
f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) - (((4*(2*a^4*b^9*c^2*e^16*f^6 - 26*a^5*b^7*c^3*
e^16*f^6 + 118*a^6*b^5*c^4*e^16*f^6 - 208*a^7*b^3*c^5*e^16*f^6 - 480*a^8*c^7*d^2*e^16*f^6 + 96*a^8*b*c^6*e^16*
f^6 + a^4*b^8*c^3*d^2*e^16*f^6 - 6*a^5*b^6*c^4*d^2*e^16*f^6 - 30*a^6*b^4*c^5*d^2*e^16*f^6 + 272*a^7*b^2*c^6*d^
2*e^16*f^6))/(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) + (2*(b^7*e*f^3 - 12*a*b^5
*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3)*(a^7*b^8*c^2*e^17*f^9 - 12*a^8*b^6*c^3*e^17*f^9 + 48*a^9
*b^4*c^4*e^17*f^9 - 64*a^10*b^2*c^5*e^17*f^9 + 640*a^10*b*c^6*d^2*e^17*f^9 + 3*a^6*b^9*c^2*d^2*e^17*f^9 - 46*a
^7*b^7*c^3*d^2*e^17*f^9 + 264*a^8*b^5*c^4*d^2*e^17*f^9 - 672*a^9*b^3*c^5*d^2*e^17*f^9))/((a^6*b^6*f^9 - 64*a^9
*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f
^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3))/(2*(a
^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^7*e*f^3 - 12*a*b^5*c
*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3))/(2*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2
*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)) - (4*(2*b^7*c^4*e^14 - 20*a*b^5*c^5*e^14 - 72*a^3*b*c^7*e^14 + 66*a^2*b^3*c^
6*e^14 - 54*a^3*c^8*d^2*e^14 + 2*b^6*c^5*d^2*e^14 + 54*a^2*b^2*c^7*d^2*e^14 - 18*a*b^4*c^6*d^2*e^14))/(a^6*b^6
*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) + (((((4*(2*a^4*b^9*c^2*e^16*f^6 - 26*a^5*b^7*c
^3*e^16*f^6 + 118*a^6*b^5*c^4*e^16*f^6 - 208*a^7*b^3*c^5*e^16*f^6 - 480*a^8*c^7*d^2*e^16*f^6 + 96*a^8*b*c^6*e^
16*f^6 + a^4*b^8*c^3*d^2*e^16*f^6 - 6*a^5*b^6*c^4*d^2*e^16*f^6 - 30*a^6*b^4*c^5*d^2*e^16*f^6 + 272*a^7*b^2*c^6
*d^2*e^16*f^6))/(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) + (2*(b^7*e*f^3 - 12*a*
b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3)*(a^7*b^8*c^2*e^17*f^9 - 12*a^8*b^6*c^3*e^17*f^9 + 48*
a^9*b^4*c^4*e^17*f^9 - 64*a^10*b^2*c^5*e^17*f^9 + 640*a^10*b*c^6*d^2*e^17*f^9 + 3*a^6*b^9*c^2*d^2*e^17*f^9 - 4
6*a^7*b^7*c^3*d^2*e^17*f^9 + 264*a^8*b^5*c^4*d^2*e^17*f^9 - 672*a^9*b^3*c^5*d^2*e^17*f^9))/((a^6*b^6*f^9 - 64*
a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^
2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^4 + 6*a^2*c^2 - 6*a*b^2*c))/(2*a^3*e*f^3*(4*a*c - b^2)^(3/2)) + ((b^4 + 6*a
^2*c^2 - 6*a*b^2*c)*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3)*(a^7*b^8*c^2*e^
17*f^9 - 12*a^8*b^6*c^3*e^17*f^9 + 48*a^9*b^4*c^4*e^17*f^9 - 64*a^10*b^2*c^5*e^17*f^9 + 640*a^10*b*c^6*d^2*e^1
7*f^9 + 3*a^6*b^9*c^2*d^2*e^17*f^9 - 46*a^7*b^7*c^3*d^2*e^17*f^9 + 264*a^8*b^5*c^4*d^2*e^17*f^9 - 672*a^9*b^3*
c^5*d^2*e^17*f^9))/(a^3*e*f^3*(4*a*c - b^2)^(3/2)*(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^
2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^4 + 6*a
^2*c^2 - 6*a*b^2*c))/(2*a^3*e*f^3*(4*a*c - b^2)^(3/2)) + ((b^4 + 6*a^2*c^2 - 6*a*b^2*c)^2*(b^7*e*f^3 - 12*a*b^
5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3)*(a^7*b^8*c^2*e^17*f^9 - 12*a^8*b^6*c^3*e^17*f^9 + 48*a^
9*b^4*c^4*e^17*f^9 - 64*a^10*b^2*c^5*e^17*f^9 + 640*a^10*b*c^6*d^2*e^17*f^9 + 3*a^6*b^9*c^2*d^2*e^17*f^9 - 46*
a^7*b^7*c^3*d^2*e^17*f^9 + 264*a^8*b^5*c^4*d^2*e^17*f^9 - 672*a^9*b^3*c^5*d^2*e^17*f^9))/(2*a^6*e^2*f^6*(4*a*c
- b^2)^3*(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*c^3
*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(3*b^6 - 3*a^3*c^3 + 36*a^2*b^2*c^2 - 21*a*b^4*c))
/(8*a^3*c^2*(4*a*c - b^2)^3*(9*a^4*c^4 - 6*b^8 - 288*a^2*b^4*c^2 + 382*a^3*b^2*c^3 + 72*a*b^6*c)) - (b*((((4*(
36*a^6*c^7*e^15*f^3 + 4*a^2*b^8*c^3*e^15*f^3 - 45*a^3*b^6*c^4*e^15*f^3 + 170*a^4*b^4*c^5*e^15*f^3 - 225*a^5*b^
2*c^6*e^15*f^3 - 276*a^5*b*c^7*d^2*e^15*f^3 + 6*a^2*b^7*c^4*d^2*e^15*f^3 - 65*a^3*b^5*c^5*d^2*e^15*f^3 + 233*a
^4*b^3*c^6*d^2*e^15*f^3))/(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) - (((4*(2*a^4
*b^9*c^2*e^16*f^6 - 26*a^5*b^7*c^3*e^16*f^6 + 118*a^6*b^5*c^4*e^16*f^6 - 208*a^7*b^3*c^5*e^16*f^6 - 480*a^8*c^
7*d^2*e^16*f^6 + 96*a^8*b*c^6*e^16*f^6 + a^4*b^8*c^3*d^2*e^16*f^6 - 6*a^5*b^6*c^4*d^2*e^16*f^6 - 30*a^6*b^4*c^
5*d^2*e^16*f^6 + 272*a^7*b^2*c^6*d^2*e^16*f^6))/(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*
c^2*f^9) + (2*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3)*(a^7*b^8*c^2*e^17*f^9
- 12*a^8*b^6*c^3*e^17*f^9 + 48*a^9*b^4*c^4*e^17*f^9 - 64*a^10*b^2*c^5*e^17*f^9 + 640*a^10*b*c^6*d^2*e^17*f^9
+ 3*a^6*b^9*c^2*d^2*e^17*f^9 - 46*a^7*b^7*c^3*d^2*e^17*f^9 + 264*a^8*b^5*c^4*d^2*e^17*f^9 - 672*a^9*b^3*c^5*d^
2*e^17*f^9))/((a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6
*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e
*f^3 + 48*a^2*b^3*c^2*e*f^3))/(2*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c
*e^2*f^6)))*(b^4 + 6*a^2*c^2 - 6*a*b^2*c))/(2*a^3*e*f^3*(4*a*c - b^2)^(3/2)) - (((((4*(2*a^4*b^9*c^2*e^16*f^6
- 26*a^5*b^7*c^3*e^16*f^6 + 118*a^6*b^5*c^4*e^16*f^6 - 208*a^7*b^3*c^5*e^16*f^6 - 480*a^8*c^7*d^2*e^16*f^6 + 9
6*a^8*b*c^6*e^16*f^6 + a^4*b^8*c^3*d^2*e^16*f^6 - 6*a^5*b^6*c^4*d^2*e^16*f^6 - 30*a^6*b^4*c^5*d^2*e^16*f^6 + 2
72*a^7*b^2*c^6*d^2*e^16*f^6))/(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9) + (2*(b^7
*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3)*(a^7*b^8*c^2*e^17*f^9 - 12*a^8*b^6*c^3*
e^17*f^9 + 48*a^9*b^4*c^4*e^17*f^9 - 64*a^10*b^2*c^5*e^17*f^9 + 640*a^10*b*c^6*d^2*e^17*f^9 + 3*a^6*b^9*c^2*d^
2*e^17*f^9 - 46*a^7*b^7*c^3*d^2*e^17*f^9 + 264*a^8*b^5*c^4*d^2*e^17*f^9 - 672*a^9*b^3*c^5*d^2*e^17*f^9))/((a^6
*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*
a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)))*(b^4 + 6*a^2*c^2 - 6*a*b^2*c))/(2*a^3*e*f^3*(4*a*c - b^2)^(3/2))
+ ((b^4 + 6*a^2*c^2 - 6*a*b^2*c)*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3)*(
a^7*b^8*c^2*e^17*f^9 - 12*a^8*b^6*c^3*e^17*f^9 + 48*a^9*b^4*c^4*e^17*f^9 - 64*a^10*b^2*c^5*e^17*f^9 + 640*a^10
*b*c^6*d^2*e^17*f^9 + 3*a^6*b^9*c^2*d^2*e^17*f^9 - 46*a^7*b^7*c^3*d^2*e^17*f^9 + 264*a^8*b^5*c^4*d^2*e^17*f^9
- 672*a^9*b^3*c^5*d^2*e^17*f^9))/(a^3*e*f^3*(4*a*c - b^2)^(3/2)*(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f
^9 + 48*a^8*b^2*c^2*f^9)*(a^3*b^6*e^2*f^6 - 64*a^6*c^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6
)))*(b^7*e*f^3 - 12*a*b^5*c*e*f^3 - 64*a^3*b*c^3*e*f^3 + 48*a^2*b^3*c^2*e*f^3))/(2*(a^3*b^6*e^2*f^6 - 64*a^6*c
^3*e^2*f^6 + 48*a^5*b^2*c^2*e^2*f^6 - 12*a^4*b^4*c*e^2*f^6)) + ((b^4 + 6*a^2*c^2 - 6*a*b^2*c)^3*(a^7*b^8*c^2*e
^17*f^9 - 12*a^8*b^6*c^3*e^17*f^9 + 48*a^9*b^4*c^4*e^17*f^9 - 64*a^10*b^2*c^5*e^17*f^9 + 640*a^10*b*c^6*d^2*e^
17*f^9 + 3*a^6*b^9*c^2*d^2*e^17*f^9 - 46*a^7*b^7*c^3*d^2*e^17*f^9 + 264*a^8*b^5*c^4*d^2*e^17*f^9 - 672*a^9*b^3
*c^5*d^2*e^17*f^9))/(2*a^9*e^3*f^9*(4*a*c - b^2)^(9/2)*(a^6*b^6*f^9 - 64*a^9*c^3*f^9 - 12*a^7*b^4*c*f^9 + 48*a
^8*b^2*c^2*f^9)))*(3*b^6 - 49*a^3*c^3 + 72*a^2*b^2*c^2 - 27*a*b^4*c))/(8*a^3*c^2*(4*a*c - b^2)^(7/2)*(9*a^4*c^
4 - 6*b^8 - 288*a^2*b^4*c^2 + 382*a^3*b^2*c^3 + 72*a*b^6*c))))/(36*a^4*c^6*e^14 + b^8*c^2*e^14 - 12*a*b^6*c^3*
e^14 + 48*a^2*b^4*c^4*e^14 - 72*a^3*b^2*c^5*e^14))*(b^4 + 6*a^2*c^2 - 6*a*b^2*c))/(a^3*e*f^3*(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)**3/(a+b*(e*x+d)**2+c*(e*x+d)**4)**2,x)

[Out]

Timed out

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