3.7.60 \(\int \frac {1}{(d f+e f x)^3 (a+b (d+e x)^2+c (d+e x)^4)^3} \, dx\) [660]

Optimal. Leaf size=343 \[ -\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 \left (b^2-4 a c\right )^2 e f^3 (d+e x)^2}+\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e f^3 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e f^3 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{5/2} e f^3}-\frac {3 b \log (d+e x)}{a^4 e f^3}+\frac {3 b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^4 e f^3} \]

[Out]

-3/2*(-5*a*c+b^2)*(-2*a*c+b^2)/a^3/(-4*a*c+b^2)^2/e/f^3/(e*x+d)^2+1/4*(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)^2+1/4*(3*b^4-20*a*b^2*c+20*a^2*c^2+3*b*c*(-6*a*c+b^2)*(e*x+d)^2)/
a^2/(-4*a*c+b^2)^2/e/f^3/(e*x+d)^2/(a+b*(e*x+d)^2+c*(e*x+d)^4)-3/2*(-20*a^3*c^3+30*a^2*b^2*c^2-10*a*b^4*c+b^6)
*arctanh((b+2*c*(e*x+d)^2)/(-4*a*c+b^2)^(1/2))/a^4/(-4*a*c+b^2)^(5/2)/e/f^3-3*b*ln(e*x+d)/a^4/e/f^3+3/4*b*ln(a
+b*(e*x+d)^2+c*(e*x+d)^4)/a^4/e/f^3

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Rubi [A]
time = 0.38, antiderivative size = 343, 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 = {1156, 1128, 754, 836, 814, 648, 632, 212, 642} \begin {gather*} \frac {3 b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^4 e f^3}-\frac {3 b \log (d+e x)}{a^4 e f^3}-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 e f^3 \left (b^2-4 a c\right )^2 (d+e x)^2}+\frac {20 a^2 c^2+3 b c \left (b^2-6 a c\right ) (d+e x)^2-20 a b^2 c+3 b^4}{4 a^2 e f^3 \left (b^2-4 a c\right )^2 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {3 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 e f^3 \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^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 )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(b^2 - 5*a*c)*(b^2 - 2*a*c))/(2*a^3*(b^2 - 4*a*c)^2*e*f^3*(d + e*x)^2) + (b^2 - 2*a*c + b*c*(d + e*x)^2)/(
4*a*(b^2 - 4*a*c)*e*f^3*(d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (3*b^4 - 20*a*b^2*c + 20*a^2*c^2
+ 3*b*c*(b^2 - 6*a*c)*(d + e*x)^2)/(4*a^2*(b^2 - 4*a*c)^2*e*f^3*(d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4
)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/(2*
a^4*(b^2 - 4*a*c)^(5/2)*e*f^3) - (3*b*Log[d + e*x])/(a^4*e*f^3) + (3*b*Log[a + b*(d + e*x)^2 + c*(d + e*x)^4])
/(4*a^4*e*f^3)

Rule 212

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

Rule 632

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

Rule 642

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

Rule 648

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

Rule 754

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

Rule 814

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

Rule 836

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 1128

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

Rule 1156

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

Rubi steps

\begin {align*} \int \frac {1}{(d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx,x,d+e x\right )}{e f^3}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (a+b x+c x^2\right )^3} \, dx,x,(d+e x)^2\right )}{2 e f^3}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 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 )^2}-\frac {\text {Subst}\left (\int \frac {-3 b^2+10 a c-4 b c x}{x^2 \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^3}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 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 )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e f^3 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\text {Subst}\left (\int \frac {6 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )+6 b c \left (b^2-6 a c\right ) x}{x^2 \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^3}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 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 )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e f^3 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\text {Subst}\left (\int \left (\frac {6 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a x^2}-\frac {6 b \left (-b^2+4 a c\right )^2}{a^2 x}+\frac {6 \left (b^6-9 a b^4 c+23 a^2 b^2 c^2-10 a^3 c^3+b c \left (b^2-4 a c\right )^2 x\right )}{a^2 \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^3}\\ &=-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 \left (b^2-4 a c\right )^2 e f^3 (d+e x)^2}+\frac {b^2-2 a c+b c (d+e x)^2}{4 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 )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e f^3 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {3 b \log (d+e x)}{a^4 e f^3}+\frac {3 \text {Subst}\left (\int \frac {b^6-9 a b^4 c+23 a^2 b^2 c^2-10 a^3 c^3+b 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^4 \left (b^2-4 a c\right )^2 e f^3}\\ &=-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 \left (b^2-4 a c\right )^2 e f^3 (d+e x)^2}+\frac {b^2-2 a c+b c (d+e x)^2}{4 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 )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e f^3 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {3 b \log (d+e x)}{a^4 e f^3}+\frac {(3 b) \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a^4 e f^3}+\frac {\left (3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a^4 \left (b^2-4 a c\right )^2 e f^3}\\ &=-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 \left (b^2-4 a c\right )^2 e f^3 (d+e x)^2}+\frac {b^2-2 a c+b c (d+e x)^2}{4 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 )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e f^3 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {3 b \log (d+e x)}{a^4 e f^3}+\frac {3 b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^4 e f^3}-\frac {\left (3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{2 a^4 \left (b^2-4 a c\right )^2 e f^3}\\ &=-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 \left (b^2-4 a c\right )^2 e f^3 (d+e x)^2}+\frac {b^2-2 a c+b c (d+e x)^2}{4 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 )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e f^3 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{5/2} e f^3}-\frac {3 b \log (d+e x)}{a^4 e f^3}+\frac {3 b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^4 e f^3}\\ \end {align*}

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Mathematica [A]
time = 6.10, size = 509, normalized size = 1.48 \begin {gather*} -\frac {1}{2 a^3 e f^3 (d+e x)^2}+\frac {b^3-3 a b c+b^2 c (d+e x)^2-2 a c^2 (d+e x)^2}{4 a^2 \left (-b^2+4 a c\right ) e f^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {-4 b^5+29 a b^3 c-46 a^2 b c^2-4 b^4 c (d+e x)^2+26 a b^2 c^2 (d+e x)^2-28 a^2 c^3 (d+e x)^2}{4 a^3 \left (-b^2+4 a c\right )^2 e f^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {3 b \log (d+e x)}{a^4 e f^3}+\frac {3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3+b^5 \sqrt {b^2-4 a c}-8 a b^3 c \sqrt {b^2-4 a c}+16 a^2 b c^2 \sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{4 a^4 \left (b^2-4 a c\right )^{5/2} e f^3}+\frac {3 \left (-b^6+10 a b^4 c-30 a^2 b^2 c^2+20 a^3 c^3+b^5 \sqrt {b^2-4 a c}-8 a b^3 c \sqrt {b^2-4 a c}+16 a^2 b c^2 \sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{4 a^4 \left (b^2-4 a c\right )^{5/2} e f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*1/(a^3*e*f^3*(d + e*x)^2) + (b^3 - 3*a*b*c + b^2*c*(d + e*x)^2 - 2*a*c^2*(d + e*x)^2)/(4*a^2*(-b^2 + 4*a*
c)*e*f^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (-4*b^5 + 29*a*b^3*c - 46*a^2*b*c^2 - 4*b^4*c*(d + e*x)^2 +
26*a*b^2*c^2*(d + e*x)^2 - 28*a^2*c^3*(d + e*x)^2)/(4*a^3*(-b^2 + 4*a*c)^2*e*f^3*(a + b*(d + e*x)^2 + c*(d + e
*x)^4)) - (3*b*Log[d + e*x])/(a^4*e*f^3) + (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3 + b^5*Sqrt[b^2 -
4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*c^2*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e
*x)^2])/(4*a^4*(b^2 - 4*a*c)^(5/2)*e*f^3) + (3*(-b^6 + 10*a*b^4*c - 30*a^2*b^2*c^2 + 20*a^3*c^3 + b^5*Sqrt[b^2
- 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*c^2*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d +
e*x)^2])/(4*a^4*(b^2 - 4*a*c)^(5/2)*e*f^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.34, size = 1145, normalized size = 3.34

method	result	size
default	\(\frac {-\frac {1}{2 a^{3} e \left (e x +d \right )^{2}}-\frac {3 b \ln \left (e x +d \right )}{a^{4} e}-\frac {\frac {\frac {c^{2} e^{5} \left (14 a^{2} c^{2}-13 a \,b^{2} c +2 b^{4}\right ) a \,x^{6}}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}+\frac {3 \left (14 a^{2} c^{2}-13 a \,b^{2} c +2 b^{4}\right ) a \,c^{2} d \,e^{4} x^{5}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {e^{3} a c \left (420 a^{2} c^{3} d^{2}-390 a \,b^{2} c^{2} d^{2}+60 b^{4} c \,d^{2}+74 a^{2} b \,c^{2}-55 a \,b^{3} c +8 b^{5}\right ) x^{4}}{64 a^{2} c^{2}-32 a \,b^{2} c +4 b^{4}}+\frac {c d \,e^{2} a \left (140 a^{2} c^{3} d^{2}-130 a \,b^{2} c^{2} d^{2}+20 b^{4} c \,d^{2}+74 a^{2} b \,c^{2}-55 a \,b^{3} c +8 b^{5}\right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {e a \left (210 a^{2} c^{4} d^{4}-195 a \,b^{2} c^{3} d^{4}+30 b^{4} c^{2} d^{4}+222 a^{2} b \,c^{3} d^{2}-165 a \,b^{3} c^{2} d^{2}+24 b^{5} c \,d^{2}+18 a^{3} c^{3}+7 a^{2} b^{2} c^{2}-12 a \,b^{4} c +2 b^{6}\right ) x^{2}}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}+\frac {d a \left (42 a^{2} c^{4} d^{4}-39 a \,b^{2} c^{3} d^{4}+6 b^{4} c^{2} d^{4}+74 a^{2} b \,c^{3} d^{2}-55 a \,b^{3} c^{2} d^{2}+8 b^{5} c \,d^{2}+18 a^{3} c^{3}+7 a^{2} b^{2} c^{2}-12 a \,b^{4} c +2 b^{6}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {a \left (28 a^{2} c^{4} d^{6}-26 a \,b^{2} c^{3} d^{6}+4 b^{4} c^{2} d^{6}+74 a^{2} b \,c^{3} d^{4}-55 a \,b^{3} c^{2} d^{4}+8 b^{5} c \,d^{4}+36 a^{3} c^{3} d^{2}+14 a^{2} b^{2} c^{2} d^{2}-24 a \,b^{4} c \,d^{2}+4 b^{6} d^{2}+58 a^{3} b \,c^{2}-36 a^{2} b^{3} c +5 a \,b^{5}\right )}{4 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 d e b x +d^{2} b +a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (e^{3} b c \left (-16 a^{2} c^{2}+8 a \,b^{2} c -b^{4}\right ) \textit {\_R}^{3}+3 d \,e^{2} b c \left (-16 a^{2} c^{2}+8 a \,b^{2} c -b^{4}\right ) \textit {\_R}^{2}+e \left (-48 a^{2} b \,c^{3} d^{2}+24 a \,b^{3} c^{2} d^{2}-3 b^{5} c \,d^{2}+10 a^{3} c^{3}-23 a^{2} b^{2} c^{2}+9 a \,b^{4} c -b^{6}\right ) \textit {\_R} -16 a^{2} b \,c^{3} d^{3}+8 a \,b^{3} c^{2} d^{3}-b^{5} c \,d^{3}+10 a^{3} c^{3} d -23 a^{2} b^{2} c^{2} d +9 a \,b^{4} c d -b^{6} d \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}\right )}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) e}}{a^{4}}}{f^{3}}\)	\(1145\)
risch	\(\text {Expression too large to display}\)	\(2364\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

1/f^3*(-1/2/a^3/e/(e*x+d)^2-3*b*ln(e*x+d)/a^4/e-1/a^4*((1/2*c^2*e^5*(14*a^2*c^2-13*a*b^2*c+2*b^4)*a/(16*a^2*c^
2-8*a*b^2*c+b^4)*x^6+3*(14*a^2*c^2-13*a*b^2*c+2*b^4)*a*c^2*d*e^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5+1/4*e^3*a*c*(4
20*a^2*c^3*d^2-390*a*b^2*c^2*d^2+60*b^4*c*d^2+74*a^2*b*c^2-55*a*b^3*c+8*b^5)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4+c*
d*e^2*a*(140*a^2*c^3*d^2-130*a*b^2*c^2*d^2+20*b^4*c*d^2+74*a^2*b*c^2-55*a*b^3*c+8*b^5)/(16*a^2*c^2-8*a*b^2*c+b
^4)*x^3+1/2*e*a*(210*a^2*c^4*d^4-195*a*b^2*c^3*d^4+30*b^4*c^2*d^4+222*a^2*b*c^3*d^2-165*a*b^3*c^2*d^2+24*b^5*c
*d^2+18*a^3*c^3+7*a^2*b^2*c^2-12*a*b^4*c+2*b^6)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+d*a*(42*a^2*c^4*d^4-39*a*b^2*c^
3*d^4+6*b^4*c^2*d^4+74*a^2*b*c^3*d^2-55*a*b^3*c^2*d^2+8*b^5*c*d^2+18*a^3*c^3+7*a^2*b^2*c^2-12*a*b^4*c+2*b^6)/(
16*a^2*c^2-8*a*b^2*c+b^4)*x+1/4/e*a*(28*a^2*c^4*d^6-26*a*b^2*c^3*d^6+4*b^4*c^2*d^6+74*a^2*b*c^3*d^4-55*a*b^3*c
^2*d^4+8*b^5*c*d^4+36*a^3*c^3*d^2+14*a^2*b^2*c^2*d^2-24*a*b^4*c*d^2+4*b^6*d^2+58*a^3*b*c^2-36*a^2*b^3*c+5*a*b^
5)/(16*a^2*c^2-8*a*b^2*c+b^4))/(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+3/2/(16*a^2*c^2-8*a*b^2*c+b^4)/e*sum((e^3*b*c*(-16*a^2*c^2+8*a*b^2*c-b^4)*_R^3+3*d*e^2*b*c*(-16*a^2
*c^2+8*a*b^2*c-b^4)*_R^2+e*(-48*a^2*b*c^3*d^2+24*a*b^3*c^2*d^2-3*b^5*c*d^2+10*a^3*c^3-23*a^2*b^2*c^2+9*a*b^4*c
-b^6)*_R-16*a^2*b*c^3*d^3+8*a*b^3*c^2*d^3-b^5*c*d^3+10*a^3*c^3*d-23*a^2*b^2*c^2*d+9*a*b^4*c*d-b^6*d)/(2*_R^3*c
*e^3+6*_R^2*c*d*e^2+6*_R*c*d^2*e+2*c*d^3+_R*b*e+b*d)*ln(x-_R),_R=RootOf(e^4*c*_Z^4+4*d*e^3*c*_Z^3+(6*c*d^2*e^2
+b*e^2)*_Z^2+(4*c*d^3*e+2*b*d*e)*_Z+d^4*c+d^2*b+a))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)^3,x, algorithm="maxima")

[Out]

-1/4*(6*(b^4*c^2 - 7*a*b^2*c^3 + 10*a^2*c^4)*d^8 + 48*(b^4*c^2*e^7 - 7*a*b^2*c^3*e^7 + 10*a^2*c^4*e^7)*d*x^7 +
6*(b^4*c^2*e^8 - 7*a*b^2*c^3*e^8 + 10*a^2*c^4*e^8)*x^8 + 3*(4*b^5*c - 29*a*b^3*c^2 + 46*a^2*b*c^3)*d^6 + 3*(4
*b^5*c*e^6 - 29*a*b^3*c^2*e^6 + 46*a^2*b*c^3*e^6 + 56*(b^4*c^2*e^6 - 7*a*b^2*c^3*e^6 + 10*a^2*c^4*e^6)*d^2)*x^
6 + 2*a^2*b^4 - 16*a^3*b^2*c + 32*a^4*c^2 + 6*(56*(b^4*c^2*e^5 - 7*a*b^2*c^3*e^5 + 10*a^2*c^4*e^5)*d^3 + 3*(4*
b^5*c*e^5 - 29*a*b^3*c^2*e^5 + 46*a^2*b*c^3*e^5)*d)*x^5 + 2*(3*b^6 - 18*a*b^4*c + 7*a^2*b^2*c^2 + 50*a^3*c^3)*
d^4 + (6*b^6*e^4 - 36*a*b^4*c*e^4 + 14*a^2*b^2*c^2*e^4 + 100*a^3*c^3*e^4 + 420*(b^4*c^2*e^4 - 7*a*b^2*c^3*e^4
+ 10*a^2*c^4*e^4)*d^4 + 45*(4*b^5*c*e^4 - 29*a*b^3*c^2*e^4 + 46*a^2*b*c^3*e^4)*d^2)*x^4 + 4*(84*(b^4*c^2*e^3 -
7*a*b^2*c^3*e^3 + 10*a^2*c^4*e^3)*d^5 + 15*(4*b^5*c*e^3 - 29*a*b^3*c^2*e^3 + 46*a^2*b*c^3*e^3)*d^3 + 2*(3*b^6
*e^3 - 18*a*b^4*c*e^3 + 7*a^2*b^2*c^2*e^3 + 50*a^3*c^3*e^3)*d)*x^3 + (9*a*b^5 - 68*a^2*b^3*c + 122*a^3*b*c^2)*
d^2 + (168*(b^4*c^2*e^2 - 7*a*b^2*c^3*e^2 + 10*a^2*c^4*e^2)*d^6 + 9*a*b^5*e^2 - 68*a^2*b^3*c*e^2 + 122*a^3*b*c
^2*e^2 + 45*(4*b^5*c*e^2 - 29*a*b^3*c^2*e^2 + 46*a^2*b*c^3*e^2)*d^4 + 12*(3*b^6*e^2 - 18*a*b^4*c*e^2 + 7*a^2*b
^2*c^2*e^2 + 50*a^3*c^3*e^2)*d^2)*x^2 + 2*(24*(b^4*c^2*e - 7*a*b^2*c^3*e + 10*a^2*c^4*e)*d^7 + 9*(4*b^5*c*e -
29*a*b^3*c^2*e + 46*a^2*b*c^3*e)*d^5 + 4*(3*b^6*e - 18*a*b^4*c*e + 7*a^2*b^2*c^2*e + 50*a^3*c^3*e)*d^3 + (9*a*
b^5*e - 68*a^2*b^3*c*e + 122*a^3*b*c^2*e)*d)*x)/(10*(a^3*b^4*c^2*e^10 - 8*a^4*b^2*c^3*e^10 + 16*a^5*c^4*e^10)*
d*f^3*x^9 + (a^3*b^4*c^2*e^11 - 8*a^4*b^2*c^3*e^11 + 16*a^5*c^4*e^11)*f^3*x^10 + (2*a^3*b^5*c*e^9 - 16*a^4*b^3
*c^2*e^9 + 32*a^5*b*c^3*e^9 + 45*(a^3*b^4*c^2*e^9 - 8*a^4*b^2*c^3*e^9 + 16*a^5*c^4*e^9)*d^2)*f^3*x^8 + 8*(15*(
a^3*b^4*c^2*e^8 - 8*a^4*b^2*c^3*e^8 + 16*a^5*c^4*e^8)*d^3 + 2*(a^3*b^5*c*e^8 - 8*a^4*b^3*c^2*e^8 + 16*a^5*b*c^
3*e^8)*d)*f^3*x^7 + (a^3*b^6*e^7 - 6*a^4*b^4*c*e^7 + 32*a^6*c^3*e^7 + 210*(a^3*b^4*c^2*e^7 - 8*a^4*b^2*c^3*e^7
+ 16*a^5*c^4*e^7)*d^4 + 56*(a^3*b^5*c*e^7 - 8*a^4*b^3*c^2*e^7 + 16*a^5*b*c^3*e^7)*d^2)*f^3*x^6 + 2*(126*(a^3*
b^4*c^2*e^6 - 8*a^4*b^2*c^3*e^6 + 16*a^5*c^4*e^6)*d^5 + 56*(a^3*b^5*c*e^6 - 8*a^4*b^3*c^2*e^6 + 16*a^5*b*c^3*e
^6)*d^3 + 3*(a^3*b^6*e^6 - 6*a^4*b^4*c*e^6 + 32*a^6*c^3*e^6)*d)*f^3*x^5 + (2*a^4*b^5*e^5 - 16*a^5*b^3*c*e^5 +
32*a^6*b*c^2*e^5 + 210*(a^3*b^4*c^2*e^5 - 8*a^4*b^2*c^3*e^5 + 16*a^5*c^4*e^5)*d^6 + 140*(a^3*b^5*c*e^5 - 8*a^4
*b^3*c^2*e^5 + 16*a^5*b*c^3*e^5)*d^4 + 15*(a^3*b^6*e^5 - 6*a^4*b^4*c*e^5 + 32*a^6*c^3*e^5)*d^2)*f^3*x^4 + 4*(3
0*(a^3*b^4*c^2*e^4 - 8*a^4*b^2*c^3*e^4 + 16*a^5*c^4*e^4)*d^7 + 28*(a^3*b^5*c*e^4 - 8*a^4*b^3*c^2*e^4 + 16*a^5*
b*c^3*e^4)*d^5 + 5*(a^3*b^6*e^4 - 6*a^4*b^4*c*e^4 + 32*a^6*c^3*e^4)*d^3 + 2*(a^4*b^5*e^4 - 8*a^5*b^3*c*e^4 + 1
6*a^6*b*c^2*e^4)*d)*f^3*x^3 + (a^5*b^4*e^3 - 8*a^6*b^2*c*e^3 + 16*a^7*c^2*e^3 + 45*(a^3*b^4*c^2*e^3 - 8*a^4*b^
2*c^3*e^3 + 16*a^5*c^4*e^3)*d^8 + 56*(a^3*b^5*c*e^3 - 8*a^4*b^3*c^2*e^3 + 16*a^5*b*c^3*e^3)*d^6 + 15*(a^3*b^6*
e^3 - 6*a^4*b^4*c*e^3 + 32*a^6*c^3*e^3)*d^4 + 12*(a^4*b^5*e^3 - 8*a^5*b^3*c*e^3 + 16*a^6*b*c^2*e^3)*d^2)*f^3*x
^2 + 2*(5*(a^3*b^4*c^2*e^2 - 8*a^4*b^2*c^3*e^2 + 16*a^5*c^4*e^2)*d^9 + 8*(a^3*b^5*c*e^2 - 8*a^4*b^3*c^2*e^2 +
16*a^5*b*c^3*e^2)*d^7 + 3*(a^3*b^6*e^2 - 6*a^4*b^4*c*e^2 + 32*a^6*c^3*e^2)*d^5 + 4*(a^4*b^5*e^2 - 8*a^5*b^3*c*
e^2 + 16*a^6*b*c^2*e^2)*d^3 + (a^5*b^4*e^2 - 8*a^6*b^2*c*e^2 + 16*a^7*c^2*e^2)*d)*f^3*x + ((a^3*b^4*c^2*e - 8*
a^4*b^2*c^3*e + 16*a^5*c^4*e)*d^10 + 2*(a^3*b^5*c*e - 8*a^4*b^3*c^2*e + 16*a^5*b*c^3*e)*d^8 + (a^3*b^6*e - 6*a
^4*b^4*c*e + 32*a^6*c^3*e)*d^6 + 2*(a^4*b^5*e - 8*a^5*b^3*c*e + 16*a^6*b*c^2*e)*d^4 + (a^5*b^4*e - 8*a^6*b^2*c
*e + 16*a^7*c^2*e)*d^2)*f^3) + 3*integrate(((b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3 + 3*(b^5*c*e^2 - 8*a*b^3*
c^2*e^2 + 16*a^2*b*c^3*e^2)*d*x^2 + (b^5*c*e^3 - 8*a*b^3*c^2*e^3 + 16*a^2*b*c^3*e^3)*x^3 + (b^6 - 9*a*b^4*c +
23*a^2*b^2*c^2 - 10*a^3*c^3)*d + (b^6*e - 9*a*b^4*c*e + 23*a^2*b^2*c^2*e - 10*a^3*c^3*e + 3*(b^5*c*e - 8*a*b^3
*c^2*e + 16*a^2*b*c^3*e)*d^2)*x)/(c*x^4*e^4 + 4*c*d*x^3*e^3 + c*d^4 + b*d^2 + (6*c*d^2*e^2 + b*e^2)*x^2 + 2*(2
*c*d^3*e + b*d*e)*x + a), x)/((a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*f^3) - 3*b*e^(-1)*log(x*e + d)/(a^4*f^3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 7510 vs. \(2 (336) = 672\).
time = 4.19, size = 15147, normalized size = 44.16 \begin {gather*} \text {Too large to display} \end {gather*}

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)^3,x, algorithm="fricas")

[Out]

[-1/4*(6*(a*b^6*c^2 - 11*a^2*b^4*c^3 + 38*a^3*b^2*c^4 - 40*a^4*c^5)*x^8*e^8 + 48*(a*b^6*c^2 - 11*a^2*b^4*c^3 +
38*a^3*b^2*c^4 - 40*a^4*c^5)*d*x^7*e^7 + 2*a^3*b^6 - 24*a^4*b^4*c + 96*a^5*b^2*c^2 - 128*a^6*c^3 + 6*(a*b^6*c
^2 - 11*a^2*b^4*c^3 + 38*a^3*b^2*c^4 - 40*a^4*c^5)*d^8 + 3*(4*a*b^7*c - 45*a^2*b^5*c^2 + 162*a^3*b^3*c^3 - 184
*a^4*b*c^4 + 56*(a*b^6*c^2 - 11*a^2*b^4*c^3 + 38*a^3*b^2*c^4 - 40*a^4*c^5)*d^2)*x^6*e^6 + 3*(4*a*b^7*c - 45*a^
2*b^5*c^2 + 162*a^3*b^3*c^3 - 184*a^4*b*c^4)*d^6 + 6*(56*(a*b^6*c^2 - 11*a^2*b^4*c^3 + 38*a^3*b^2*c^4 - 40*a^4
*c^5)*d^3 + 3*(4*a*b^7*c - 45*a^2*b^5*c^2 + 162*a^3*b^3*c^3 - 184*a^4*b*c^4)*d)*x^5*e^5 + (6*a*b^8 - 60*a^2*b^
6*c + 158*a^3*b^4*c^2 + 44*a^4*b^2*c^3 - 400*a^5*c^4 + 420*(a*b^6*c^2 - 11*a^2*b^4*c^3 + 38*a^3*b^2*c^4 - 40*a
^4*c^5)*d^4 + 45*(4*a*b^7*c - 45*a^2*b^5*c^2 + 162*a^3*b^3*c^3 - 184*a^4*b*c^4)*d^2)*x^4*e^4 + 2*(3*a*b^8 - 30
*a^2*b^6*c + 79*a^3*b^4*c^2 + 22*a^4*b^2*c^3 - 200*a^5*c^4)*d^4 + 4*(84*(a*b^6*c^2 - 11*a^2*b^4*c^3 + 38*a^3*b
^2*c^4 - 40*a^4*c^5)*d^5 + 15*(4*a*b^7*c - 45*a^2*b^5*c^2 + 162*a^3*b^3*c^3 - 184*a^4*b*c^4)*d^3 + 2*(3*a*b^8
- 30*a^2*b^6*c + 79*a^3*b^4*c^2 + 22*a^4*b^2*c^3 - 200*a^5*c^4)*d)*x^3*e^3 + (9*a^2*b^7 - 104*a^3*b^5*c + 394*
a^4*b^3*c^2 - 488*a^5*b*c^3 + 168*(a*b^6*c^2 - 11*a^2*b^4*c^3 + 38*a^3*b^2*c^4 - 40*a^4*c^5)*d^6 + 45*(4*a*b^7
*c - 45*a^2*b^5*c^2 + 162*a^3*b^3*c^3 - 184*a^4*b*c^4)*d^4 + 12*(3*a*b^8 - 30*a^2*b^6*c + 79*a^3*b^4*c^2 + 22*
a^4*b^2*c^3 - 200*a^5*c^4)*d^2)*x^2*e^2 + (9*a^2*b^7 - 104*a^3*b^5*c + 394*a^4*b^3*c^2 - 488*a^5*b*c^3)*d^2 +
2*(24*(a*b^6*c^2 - 11*a^2*b^4*c^3 + 38*a^3*b^2*c^4 - 40*a^4*c^5)*d^7 + 9*(4*a*b^7*c - 45*a^2*b^5*c^2 + 162*a^3
*b^3*c^3 - 184*a^4*b*c^4)*d^5 + 4*(3*a*b^8 - 30*a^2*b^6*c + 79*a^3*b^4*c^2 + 22*a^4*b^2*c^3 - 200*a^5*c^4)*d^3
+ (9*a^2*b^7 - 104*a^3*b^5*c + 394*a^4*b^3*c^2 - 488*a^5*b*c^3)*d)*x*e + 3*((b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*
b^2*c^4 - 20*a^3*c^5)*x^10*e^10 + 10*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d*x^9*e^9 + (b^6*c
^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^10 + (2*b^7*c - 20*a*b^5*c^2 + 60*a^2*b^3*c^3 - 40*a^3*b*c^
4 + 45*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^2)*x^8*e^8 + 2*(b^7*c - 10*a*b^5*c^2 + 30*a^2*
b^3*c^3 - 20*a^3*b*c^4)*d^8 + 8*(15*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^3 + 2*(b^7*c - 10
*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*d)*x^7*e^7 + (b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 -
40*a^4*c^4 + 210*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^4 + 56*(b^7*c - 10*a*b^5*c^2 + 30*a^
2*b^3*c^3 - 20*a^3*b*c^4)*d^2)*x^6*e^6 + (b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*d^6
+ 2*(126*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^5 + 56*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^
3 - 20*a^3*b*c^4)*d^3 + 3*(b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*d)*x^5*e^5 + (2*a*b
^7 - 20*a^2*b^5*c + 60*a^3*b^3*c^2 - 40*a^4*b*c^3 + 210*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)
*d^6 + 140*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*d^4 + 15*(b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 +
40*a^3*b^2*c^3 - 40*a^4*c^4)*d^2)*x^4*e^4 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3)*d^4 + 4*
(30*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^7 + 28*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 2
0*a^3*b*c^4)*d^5 + 5*(b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*d^3 + 2*(a*b^7 - 10*a^2*
b^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3)*d)*x^3*e^3 + (45*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)
*d^8 + a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3 + 56*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a
^3*b*c^4)*d^6 + 15*(b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*d^4 + 12*(a*b^7 - 10*a^2*b
^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3)*d^2)*x^2*e^2 + (a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3)*d^
2 + 2*(5*(b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*d^9 + 8*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3
- 20*a^3*b*c^4)*d^7 + 3*(b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*d^5 + 4*(a*b^7 - 10*
a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3)*d^3 + (a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3)*d)*x*e
)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^4*e^4 + 8*c^2*d*x^3*e^3 + 2*c^2*d^4 + 2*b*c*d^2 + 2*(6*c^2*d^2 + b*c)*x^2*e^2
+ 4*(2*c^2*d^3 + b*c*d)*x*e + b^2 - 2*a*c + (2*c*x^2*e^2 + 4*c*d*x*e + 2*c*d^2 + b)*sqrt(b^2 - 4*a*c))/(c*x^4
*e^4 + 4*c*d*x^3*e^3 + c*d^4 + (6*c*d^2 + b)*x^2*e^2 + b*d^2 + 2*(2*c*d^3 + b*d)*x*e + a)) - 3*((b^7*c^2 - 12*
a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^10*e^10 + 10*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*
c^5)*d*x^9*e^9 + (b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^10 + (2*b^8*c - 24*a*b^6*c^2 + 96*
a^2*b^4*c^3 - 128*a^3*b^2*c^4 + 45*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^2)*x^8*e^8 + 2*(
b^8*c - 12*a*b^6*c^2 + 48*a^2*b^4*c^3 - 64*a^3*b^2*c^4)*d^8 + 8*(15*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 -
64*a^3*b*c^5)*d^3 + 2*(b^8*c - 12*a*b^6*c^2 + ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1735 vs. \(2 (336) = 672\).
time = 4.83, size = 1735, normalized size = 5.06 \begin {gather*} \frac {3 \, {\left ({\left (a^{4} b^{8} c f^{3} e^{3} - 14 \, a^{5} b^{6} c^{2} f^{3} e^{3} + 70 \, a^{6} b^{4} c^{3} f^{3} e^{3} - 140 \, a^{7} b^{2} c^{4} f^{3} e^{3} + 80 \, a^{8} c^{5} 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^{4} b^{8} c f^{3} e^{3} - 14 \, a^{5} b^{6} c^{2} f^{3} e^{3} + 70 \, a^{6} b^{4} c^{3} f^{3} e^{3} - 140 \, a^{7} b^{2} c^{4} f^{3} e^{3} + 80 \, a^{8} c^{5} 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 )\right )}}{4 \, {\left (a^{8} b^{8} c f^{6} e^{4} - 16 \, a^{9} b^{6} c^{2} f^{6} e^{4} + 96 \, a^{10} b^{4} c^{3} f^{6} e^{4} - 256 \, a^{11} b^{2} c^{4} f^{6} e^{4} + 256 \, a^{12} c^{5} f^{6} e^{4}\right )}} - \frac {6 \, b^{4} c^{2} x^{8} e^{8} - 42 \, a b^{2} c^{3} x^{8} e^{8} + 60 \, a^{2} c^{4} x^{8} e^{8} + 48 \, b^{4} c^{2} d x^{7} e^{7} - 336 \, a b^{2} c^{3} d x^{7} e^{7} + 480 \, a^{2} c^{4} d x^{7} e^{7} + 168 \, b^{4} c^{2} d^{2} x^{6} e^{6} - 1176 \, a b^{2} c^{3} d^{2} x^{6} e^{6} + 1680 \, a^{2} c^{4} d^{2} x^{6} e^{6} + 336 \, b^{4} c^{2} d^{3} x^{5} e^{5} - 2352 \, a b^{2} c^{3} d^{3} x^{5} e^{5} + 3360 \, a^{2} c^{4} d^{3} x^{5} e^{5} + 420 \, b^{4} c^{2} d^{4} x^{4} e^{4} - 2940 \, a b^{2} c^{3} d^{4} x^{4} e^{4} + 4200 \, a^{2} c^{4} d^{4} x^{4} e^{4} + 336 \, b^{4} c^{2} d^{5} x^{3} e^{3} - 2352 \, a b^{2} c^{3} d^{5} x^{3} e^{3} + 3360 \, a^{2} c^{4} d^{5} x^{3} e^{3} + 168 \, b^{4} c^{2} d^{6} x^{2} e^{2} - 1176 \, a b^{2} c^{3} d^{6} x^{2} e^{2} + 1680 \, a^{2} c^{4} d^{6} x^{2} e^{2} + 48 \, b^{4} c^{2} d^{7} x e - 336 \, a b^{2} c^{3} d^{7} x e + 480 \, a^{2} c^{4} d^{7} x e + 6 \, b^{4} c^{2} d^{8} - 42 \, a b^{2} c^{3} d^{8} + 60 \, a^{2} c^{4} d^{8} + 12 \, b^{5} c x^{6} e^{6} - 87 \, a b^{3} c^{2} x^{6} e^{6} + 138 \, a^{2} b c^{3} x^{6} e^{6} + 72 \, b^{5} c d x^{5} e^{5} - 522 \, a b^{3} c^{2} d x^{5} e^{5} + 828 \, a^{2} b c^{3} d x^{5} e^{5} + 180 \, b^{5} c d^{2} x^{4} e^{4} - 1305 \, a b^{3} c^{2} d^{2} x^{4} e^{4} + 2070 \, a^{2} b c^{3} d^{2} x^{4} e^{4} + 240 \, b^{5} c d^{3} x^{3} e^{3} - 1740 \, a b^{3} c^{2} d^{3} x^{3} e^{3} + 2760 \, a^{2} b c^{3} d^{3} x^{3} e^{3} + 180 \, b^{5} c d^{4} x^{2} e^{2} - 1305 \, a b^{3} c^{2} d^{4} x^{2} e^{2} + 2070 \, a^{2} b c^{3} d^{4} x^{2} e^{2} + 72 \, b^{5} c d^{5} x e - 522 \, a b^{3} c^{2} d^{5} x e + 828 \, a^{2} b c^{3} d^{5} x e + 12 \, b^{5} c d^{6} - 87 \, a b^{3} c^{2} d^{6} + 138 \, a^{2} b c^{3} d^{6} + 6 \, b^{6} x^{4} e^{4} - 36 \, a b^{4} c x^{4} e^{4} + 14 \, a^{2} b^{2} c^{2} x^{4} e^{4} + 100 \, a^{3} c^{3} x^{4} e^{4} + 24 \, b^{6} d x^{3} e^{3} - 144 \, a b^{4} c d x^{3} e^{3} + 56 \, a^{2} b^{2} c^{2} d x^{3} e^{3} + 400 \, a^{3} c^{3} d x^{3} e^{3} + 36 \, b^{6} d^{2} x^{2} e^{2} - 216 \, a b^{4} c d^{2} x^{2} e^{2} + 84 \, a^{2} b^{2} c^{2} d^{2} x^{2} e^{2} + 600 \, a^{3} c^{3} d^{2} x^{2} e^{2} + 24 \, b^{6} d^{3} x e - 144 \, a b^{4} c d^{3} x e + 56 \, a^{2} b^{2} c^{2} d^{3} x e + 400 \, a^{3} c^{3} d^{3} x e + 6 \, b^{6} d^{4} - 36 \, a b^{4} c d^{4} + 14 \, a^{2} b^{2} c^{2} d^{4} + 100 \, a^{3} c^{3} d^{4} + 9 \, a b^{5} x^{2} e^{2} - 68 \, a^{2} b^{3} c x^{2} e^{2} + 122 \, a^{3} b c^{2} x^{2} e^{2} + 18 \, a b^{5} d x e - 136 \, a^{2} b^{3} c d x e + 244 \, a^{3} b c^{2} d x e + 9 \, a b^{5} d^{2} - 68 \, a^{2} b^{3} c d^{2} + 122 \, a^{3} b c^{2} d^{2} + 2 \, a^{2} b^{4} - 16 \, a^{3} b^{2} c + 32 \, a^{4} c^{2}}{4 \, {\left (a^{3} b^{4} f^{3} e - 8 \, a^{4} b^{2} c f^{3} e + 16 \, a^{5} c^{2} f^{3} e\right )} {\left (c x^{5} e^{5} + 5 \, c d x^{4} e^{4} + 10 \, c d^{2} x^{3} e^{3} + 10 \, c d^{3} x^{2} e^{2} + 5 \, c d^{4} x e + c d^{5} + b x^{3} e^{3} + 3 \, b d x^{2} e^{2} + 3 \, b d^{2} x e + b d^{3} + a x e + a d\right )}^{2}} + \frac {3 \, 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 )}{4 \, a^{4} f^{3}} - \frac {3 \, b e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\right )}{a^{4} f^{3}} \end {gather*}

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)^3,x, algorithm="giac")

[Out]

3/4*((a^4*b^8*c*f^3*e^3 - 14*a^5*b^6*c^2*f^3*e^3 + 70*a^6*b^4*c^3*f^3*e^3 - 140*a^7*b^2*c^4*f^3*e^3 + 80*a^8*c
^5*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^4*b^8*c*f^3*e^3 - 14*a^5*b^6*c^2*f^3*e^3 + 70*a^6*b^4*c^3*f
^3*e^3 - 140*a^7*b^2*c^4*f^3*e^3 + 80*a^8*c^5*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^8*b^8*c*f^6*e^4
- 16*a^9*b^6*c^2*f^6*e^4 + 96*a^10*b^4*c^3*f^6*e^4 - 256*a^11*b^2*c^4*f^6*e^4 + 256*a^12*c^5*f^6*e^4) - 1/4*(6
*b^4*c^2*x^8*e^8 - 42*a*b^2*c^3*x^8*e^8 + 60*a^2*c^4*x^8*e^8 + 48*b^4*c^2*d*x^7*e^7 - 336*a*b^2*c^3*d*x^7*e^7
+ 480*a^2*c^4*d*x^7*e^7 + 168*b^4*c^2*d^2*x^6*e^6 - 1176*a*b^2*c^3*d^2*x^6*e^6 + 1680*a^2*c^4*d^2*x^6*e^6 + 33
6*b^4*c^2*d^3*x^5*e^5 - 2352*a*b^2*c^3*d^3*x^5*e^5 + 3360*a^2*c^4*d^3*x^5*e^5 + 420*b^4*c^2*d^4*x^4*e^4 - 2940
*a*b^2*c^3*d^4*x^4*e^4 + 4200*a^2*c^4*d^4*x^4*e^4 + 336*b^4*c^2*d^5*x^3*e^3 - 2352*a*b^2*c^3*d^5*x^3*e^3 + 336
0*a^2*c^4*d^5*x^3*e^3 + 168*b^4*c^2*d^6*x^2*e^2 - 1176*a*b^2*c^3*d^6*x^2*e^2 + 1680*a^2*c^4*d^6*x^2*e^2 + 48*b
^4*c^2*d^7*x*e - 336*a*b^2*c^3*d^7*x*e + 480*a^2*c^4*d^7*x*e + 6*b^4*c^2*d^8 - 42*a*b^2*c^3*d^8 + 60*a^2*c^4*d
^8 + 12*b^5*c*x^6*e^6 - 87*a*b^3*c^2*x^6*e^6 + 138*a^2*b*c^3*x^6*e^6 + 72*b^5*c*d*x^5*e^5 - 522*a*b^3*c^2*d*x^
5*e^5 + 828*a^2*b*c^3*d*x^5*e^5 + 180*b^5*c*d^2*x^4*e^4 - 1305*a*b^3*c^2*d^2*x^4*e^4 + 2070*a^2*b*c^3*d^2*x^4*
e^4 + 240*b^5*c*d^3*x^3*e^3 - 1740*a*b^3*c^2*d^3*x^3*e^3 + 2760*a^2*b*c^3*d^3*x^3*e^3 + 180*b^5*c*d^4*x^2*e^2
- 1305*a*b^3*c^2*d^4*x^2*e^2 + 2070*a^2*b*c^3*d^4*x^2*e^2 + 72*b^5*c*d^5*x*e - 522*a*b^3*c^2*d^5*x*e + 828*a^2
*b*c^3*d^5*x*e + 12*b^5*c*d^6 - 87*a*b^3*c^2*d^6 + 138*a^2*b*c^3*d^6 + 6*b^6*x^4*e^4 - 36*a*b^4*c*x^4*e^4 + 14
*a^2*b^2*c^2*x^4*e^4 + 100*a^3*c^3*x^4*e^4 + 24*b^6*d*x^3*e^3 - 144*a*b^4*c*d*x^3*e^3 + 56*a^2*b^2*c^2*d*x^3*e
^3 + 400*a^3*c^3*d*x^3*e^3 + 36*b^6*d^2*x^2*e^2 - 216*a*b^4*c*d^2*x^2*e^2 + 84*a^2*b^2*c^2*d^2*x^2*e^2 + 600*a
^3*c^3*d^2*x^2*e^2 + 24*b^6*d^3*x*e - 144*a*b^4*c*d^3*x*e + 56*a^2*b^2*c^2*d^3*x*e + 400*a^3*c^3*d^3*x*e + 6*b
^6*d^4 - 36*a*b^4*c*d^4 + 14*a^2*b^2*c^2*d^4 + 100*a^3*c^3*d^4 + 9*a*b^5*x^2*e^2 - 68*a^2*b^3*c*x^2*e^2 + 122*
a^3*b*c^2*x^2*e^2 + 18*a*b^5*d*x*e - 136*a^2*b^3*c*d*x*e + 244*a^3*b*c^2*d*x*e + 9*a*b^5*d^2 - 68*a^2*b^3*c*d^
2 + 122*a^3*b*c^2*d^2 + 2*a^2*b^4 - 16*a^3*b^2*c + 32*a^4*c^2)/((a^3*b^4*f^3*e - 8*a^4*b^2*c*f^3*e + 16*a^5*c^
2*f^3*e)*(c*x^5*e^5 + 5*c*d*x^4*e^4 + 10*c*d^2*x^3*e^3 + 10*c*d^3*x^2*e^2 + 5*c*d^4*x*e + c*d^5 + b*x^3*e^3 +
3*b*d*x^2*e^2 + 3*b*d^2*x*e + b*d^3 + a*x*e + a*d)^2) + 3/4*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^4*f^3) - 3*b*e^(-1)*log(abs(x*e + d)
)/(a^4*f^3)

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Mupad [B]
time = 24.91, size = 2500, normalized size = 7.29 \begin {gather*} \text {Too large to display} \end {gather*}

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)^3),x)

[Out]

(log(((27*c^5*e^16*x^2*(b^4 + 10*a^2*c^2 - 7*a*b^2*c)^3)/(a^9*f^9*(4*a*c - b^2)^6) - ((3*b - 3*a^4*e*f^3*(-(b^
6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^2/(a^8*e^2*f^6*(4*a*c - b^2)^5))^(1/2))*((9*c^3*e^15*(b^4 + 10*a
^2*c^2 - 7*a*b^2*c)*(4*b^6 - 10*a^3*c^3 + 6*b^5*c*d^2 + 71*a^2*b^2*c^2 - 33*a*b^4*c - 47*a*b^3*c^2*d^2 + 90*a^
2*b*c^3*d^2))/(a^6*f^6*(4*a*c - b^2)^4) - ((3*b - 3*a^4*e*f^3*(-(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*
c)^2/(a^8*e^2*f^6*(4*a*c - b^2)^5))^(1/2))*((6*c^2*e^16*(2*b^7 - 20*a^3*b*c^3 + b^6*c*d^2 + 46*a^2*b^3*c^2 + 1
00*a^3*c^4*d^2 - 18*a*b^5*c - 2*a*b^4*c^2*d^2 - 30*a^2*b^2*c^3*d^2))/(a^3*f^3*(4*a*c - b^2)^2) + (b*c^2*e^16*(
3*b - 3*a^4*e*f^3*(-(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^2/(a^8*e^2*f^6*(4*a*c - b^2)^5))^(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^4*f^3) + (6*c^
3*e^18*x^2*(b^6 + 100*a^3*c^3 - 30*a^2*b^2*c^2 - 2*a*b^4*c))/(a^3*f^3*(4*a*c - b^2)^2) + (12*c^3*d*e^17*x*(b^6
+ 100*a^3*c^3 - 30*a^2*b^2*c^2 - 2*a*b^4*c))/(a^3*f^3*(4*a*c - b^2)^2)))/(4*a^4*e*f^3) + (9*b*c^4*e^17*x^2*(6
*b^8 + 900*a^4*c^4 + 479*a^2*b^4*c^2 - 1100*a^3*b^2*c^3 - 89*a*b^6*c))/(a^6*f^6*(4*a*c - b^2)^4) + (18*b*c^4*d
*e^16*x*(6*b^8 + 900*a^4*c^4 + 479*a^2*b^4*c^2 - 1100*a^3*b^2*c^3 - 89*a*b^6*c))/(a^6*f^6*(4*a*c - b^2)^4)))/(
4*a^4*e*f^3) + (27*c^4*e^14*(b^4 + 10*a^2*c^2 - 7*a*b^2*c)^2*(b^5 + 16*a^2*b*c^2 + b^4*c*d^2 + 10*a^2*c^3*d^2
- 8*a*b^3*c - 7*a*b^2*c^2*d^2))/(a^9*f^9*(4*a*c - b^2)^6) + (54*c^5*d*e^15*x*(b^4 + 10*a^2*c^2 - 7*a*b^2*c)^3)
/(a^9*f^9*(4*a*c - b^2)^6))*((27*c^5*e^16*x^2*(b^4 + 10*a^2*c^2 - 7*a*b^2*c)^3)/(a^9*f^9*(4*a*c - b^2)^6) - ((
3*b + 3*a^4*e*f^3*(-(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^2/(a^8*e^2*f^6*(4*a*c - b^2)^5))^(1/2))*(
(9*c^3*e^15*(b^4 + 10*a^2*c^2 - 7*a*b^2*c)*(4*b^6 - 10*a^3*c^3 + 6*b^5*c*d^2 + 71*a^2*b^2*c^2 - 33*a*b^4*c - 4
7*a*b^3*c^2*d^2 + 90*a^2*b*c^3*d^2))/(a^6*f^6*(4*a*c - b^2)^4) - ((3*b + 3*a^4*e*f^3*(-(b^6 - 20*a^3*c^3 + 30*
a^2*b^2*c^2 - 10*a*b^4*c)^2/(a^8*e^2*f^6*(4*a*c - b^2)^5))^(1/2))*((6*c^2*e^16*(2*b^7 - 20*a^3*b*c^3 + b^6*c*d
^2 + 46*a^2*b^3*c^2 + 100*a^3*c^4*d^2 - 18*a*b^5*c - 2*a*b^4*c^2*d^2 - 30*a^2*b^2*c^3*d^2))/(a^3*f^3*(4*a*c -
b^2)^2) + (b*c^2*e^16*(3*b + 3*a^4*e*f^3*(-(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^2/(a^8*e^2*f^6*(4*
a*c - b^2)^5))^(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^4*f^3) + (6*c^3*e^18*x^2*(b^6 + 100*a^3*c^3 - 30*a^2*b^2*c^2 - 2*a*b^4*c))/(a^3*f^3*(4*a*c - b^2)^2)
+ (12*c^3*d*e^17*x*(b^6 + 100*a^3*c^3 - 30*a^2*b^2*c^2 - 2*a*b^4*c))/(a^3*f^3*(4*a*c - b^2)^2)))/(4*a^4*e*f^3)
+ (9*b*c^4*e^17*x^2*(6*b^8 + 900*a^4*c^4 + 479*a^2*b^4*c^2 - 1100*a^3*b^2*c^3 - 89*a*b^6*c))/(a^6*f^6*(4*a*c
- b^2)^4) + (18*b*c^4*d*e^16*x*(6*b^8 + 900*a^4*c^4 + 479*a^2*b^4*c^2 - 1100*a^3*b^2*c^3 - 89*a*b^6*c))/(a^6*f
^6*(4*a*c - b^2)^4)))/(4*a^4*e*f^3) + (27*c^4*e^14*(b^4 + 10*a^2*c^2 - 7*a*b^2*c)^2*(b^5 + 16*a^2*b*c^2 + b^4*
c*d^2 + 10*a^2*c^3*d^2 - 8*a*b^3*c - 7*a*b^2*c^2*d^2))/(a^9*f^9*(4*a*c - b^2)^6) + (54*c^5*d*e^15*x*(b^4 + 10*
a^2*c^2 - 7*a*b^2*c)^3)/(a^9*f^9*(4*a*c - b^2)^6)))*(6*b^11*e*f^3 - 120*a*b^9*c*e*f^3 - 6144*a^5*b*c^5*e*f^3 +
960*a^2*b^7*c^2*e*f^3 - 3840*a^3*b^5*c^3*e*f^3 + 7680*a^4*b^3*c^4*e*f^3))/(2*(4*a^4*b^10*e^2*f^6 - 4096*a^9*c
^5*e^2*f^6 + 640*a^6*b^6*c^2*e^2*f^6 - 2560*a^7*b^4*c^3*e^2*f^6 + 5120*a^8*b^2*c^4*e^2*f^6 - 80*a^5*b^8*c*e^2*
f^6)) - ((x^4*(6*b^6*e^3 + 100*a^3*c^3*e^3 + 180*b^5*c*d^2*e^3 + 14*a^2*b^2*c^2*e^3 + 4200*a^2*c^4*d^4*e^3 + 4
20*b^4*c^2*d^4*e^3 - 36*a*b^4*c*e^3 - 1305*a*b^3*c^2*d^2*e^3 + 2070*a^2*b*c^3*d^2*e^3 - 2940*a*b^2*c^3*d^4*e^3
))/(4*(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2*c)) + (3*x^6*(4*b^5*c*e^5 - 29*a*b^3*c^2*e^5 + 46*a^2*b*c^3*e^5 + 560*
a^2*c^4*d^2*e^5 + 56*b^4*c^2*d^2*e^5 - 392*a*b^2*c^3*d^2*e^5))/(4*(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2*c)) + (x*(
12*b^6*d^3 + 36*b^5*c*d^5 + 200*a^3*c^3*d^3 + 240*a^2*c^4*d^7 + 24*b^4*c^2*d^7 + 9*a*b^5*d - 261*a*b^3*c^2*d^5
+ 414*a^2*b*c^3*d^5 - 168*a*b^2*c^3*d^7 + 28*a^2*b^2*c^2*d^3 - 68*a^2*b^3*c*d + 122*a^3*b*c^2*d - 72*a*b^4*c*
d^3))/(2*(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2*c)) + (3*x^5*(560*a^2*c^4*d^3*e^4 + 56*b^4*c^2*d^3*e^4 + 12*b^5*c*d
*e^4 - 87*a*b^3*c^2*d*e^4 + 138*a^2*b*c^3*d*e^4 - 392*a*b^2*c^3*d^3*e^4))/(2*(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2
*c)) + (3*x^8*(10*a^2*c^4*e^7 + b^4*c^2*e^7 - 7*a*b^2*c^3*e^7))/(2*(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2*c)) + (x^
2*(36*b^6*d^2*e + 9*a*b^5*e + 600*a^3*c^3*d^2*e + 1680*a^2*c^4*d^6*e + 168*b^4*c^2*d^6*e - 68*a^2*b^3*c*e + 12
2*a^3*b*c^2*e + 180*b^5*c*d^4*e - 216*a*b^4*c*d^2*e - 1305*a*b^3*c^2*d^4*e + 2070*a^2*b*c^3*d^4*e - 1176*a*b^2
*c^3*d^6*e + 84*a^2*b^2*c^2*d^2*e))/(4*(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2*c)) + (x^3*(6*b^6*d*e^2 + 100*a^3*c^3
*d*e^2 + 60*b^5*c*d^3*e^2 + 840*a^2*c^4*d^5*e^2 + 84*b^4*c^2*d^5*e^2 - 36*a*b^4*c*d*e^2 + 14*a^2*b^2*c^2*d*e^2
- 435*a*b^3*c^2*d^3*e^2 + 690*a^2*b*c^3*d^3*e^2 - 588*a*b^2*c^3*d^5*e^2))/(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2*c
) + (12*x^7*(10*a^2*c^4*d*e^6 + b^4*c^2*d*e^6 -...

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