3.880 \(\int \frac {1}{x^3 (a+b x^2+c x^4)^3} \, dx\)

Optimal. Leaf size=255 \[ \frac {3 b \log \left (a+b x^2+c x^4\right )}{4 a^4}-\frac {3 b \log (x)}{a^4}-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 x^2 \left (b^2-4 a c\right )^2}+\frac {20 a^2 c^2+3 b c x^2 \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{4 a^2 x^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {3 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{5/2}}+\frac {-2 a c+b^2+b c x^2}{4 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.39, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1114, 740, 822, 800, 634, 618, 206, 628} \[ \frac {20 a^2 c^2+3 b c x^2 \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{4 a^2 x^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {3 \left (30 a^2 b^2 c^2-20 a^3 c^3-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{5/2}}-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 x^2 \left (b^2-4 a c\right )^2}+\frac {3 b \log \left (a+b x^2+c x^4\right )}{4 a^4}-\frac {3 b \log (x)}{a^4}+\frac {-2 a c+b^2+b c x^2}{4 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x^2 + c*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*x^2) + (b^2 - 2*a*c + b*c*x^2)/(4*a*(b^2 - 4*a*c)*x^2*
(a + b*x^2 + c*x^4)^2) + (3*b^4 - 20*a*b^2*c + 20*a^2*c^2 + 3*b*c*(b^2 - 6*a*c)*x^2)/(4*a^2*(b^2 - 4*a*c)^2*x^
2*(a + b*x^2 + c*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*x^2)/Sqrt[b^2 -
4*a*c]])/(2*a^4*(b^2 - 4*a*c)^(5/2)) - (3*b*Log[x])/a^4 + (3*b*Log[a + b*x^2 + c*x^4])/(4*a^4)

Rule 206

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

Rule 618

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

Rule 628

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

Rule 634

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

Rule 740

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

Rule 800

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

Rule 822

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

Rule 1114

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

Rubi steps

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

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Mathematica [A]  time = 0.62, size = 402, normalized size = 1.58 \[ \frac {\frac {a^2 \left (-3 a b c-2 a c^2 x^2+b^3+b^2 c x^2\right )}{\left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )^2}-\frac {a \left (46 a^2 b c^2+28 a^2 c^3 x^2-29 a b^3 c-26 a b^2 c^2 x^2+4 b^5+4 b^4 c x^2\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 \left (-20 a^3 c^3+30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt {b^2-4 a c}-10 a b^4 c+b^5 \sqrt {b^2-4 a c}-8 a b^3 c \sqrt {b^2-4 a c}+b^6\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {3 \left (20 a^3 c^3-30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt {b^2-4 a c}+10 a b^4 c+b^5 \sqrt {b^2-4 a c}-8 a b^3 c \sqrt {b^2-4 a c}-b^6\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac {2 a}{x^2}-12 b \log (x)}{4 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a)/x^2 + (a^2*(b^3 - 3*a*b*c + b^2*c*x^2 - 2*a*c^2*x^2))/((-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) - (a*(4*b
^5 - 29*a*b^3*c + 46*a^2*b*c^2 + 4*b^4*c*x^2 - 26*a*b^2*c^2*x^2 + 28*a^2*c^3*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2
+ c*x^4)) - 12*b*Log[x] + (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*x^2])/(b^2 - 4*a*c)^(5
/2) + (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*x^2])/(b^2 - 4*a*c)^(5/2))/(4*a^4)

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fricas [B]  time = 3.06, size = 2312, normalized size = 9.07 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(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)*x^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)*x^6 + 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)*x^4 + (9*a^2*b^7 - 104*a^3*b^5*c + 394*a^4*b^3*c^2 -
488*a^5*b*c^3)*x^2 + 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 + 2*(b^7*c - 10*a*b^5*c^2
+ 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*x^8 + (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)*x^6 +
2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3)*x^4 + (a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*
a^5*c^3)*x^2)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^2 + b)*sqrt(b^2 - 4*a*c))/(c
*x^4 + b*x^2 + 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 + 2*(b^8*c - 12*a*b^6*c^
2 + 48*a^2*b^4*c^3 - 64*a^3*b^2*c^4)*x^8 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 + 32*a^3*b^3*c^3 - 128*a^4*b*c^4
)*x^6 + 2*(a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*c^2 - 64*a^4*b^2*c^3)*x^4 + (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*
c^2 - 64*a^5*b*c^3)*x^2)*log(c*x^4 + b*x^2 + a) + 12*((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 + 2*(b^8*c - 12*a*b^6*c^2 + 48*a^2*b^4*c^3 - 64*a^3*b^2*c^4)*x^8 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 +
32*a^3*b^3*c^3 - 128*a^4*b*c^4)*x^6 + 2*(a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*c^2 - 64*a^4*b^2*c^3)*x^4 + (a^2*b^
7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x^2)*log(x))/((a^4*b^6*c^2 - 12*a^5*b^4*c^3 + 48*a^6*b^2*c^4
- 64*a^7*c^5)*x^10 + 2*(a^4*b^7*c - 12*a^5*b^5*c^2 + 48*a^6*b^3*c^3 - 64*a^7*b*c^4)*x^8 + (a^4*b^8 - 10*a^5*b
^6*c + 24*a^6*b^4*c^2 + 32*a^7*b^2*c^3 - 128*a^8*c^4)*x^6 + 2*(a^5*b^7 - 12*a^6*b^5*c + 48*a^7*b^3*c^2 - 64*a^
8*b*c^3)*x^4 + (a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*x^2), -1/4*(2*a^3*b^6 - 24*a^4*b^4*c + 9
6*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)*x^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)*x^6 + 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)*x^4 + (9*a^2*b^7 - 104*a^3*b^5*c + 394*a^4*b^3*c^2 - 488*a^5*b*c^3)*x^2 + 6*((b^6*c^2 -
10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*x^10 + 2*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*x
^8 + (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)*x^6 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b
^3*c^2 - 20*a^4*b*c^3)*x^4 + (a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3)*x^2)*sqrt(-b^2 + 4*a*c)*ar
ctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - 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 + 2*(b^8*c - 12*a*b^6*c^2 + 48*a^2*b^4*c^3 - 64*a^3*b^2*c^4)*x^8 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c
^2 + 32*a^3*b^3*c^3 - 128*a^4*b*c^4)*x^6 + 2*(a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*c^2 - 64*a^4*b^2*c^3)*x^4 + (a
^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x^2)*log(c*x^4 + b*x^2 + a) + 12*((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 + 2*(b^8*c - 12*a*b^6*c^2 + 48*a^2*b^4*c^3 - 64*a^3*b^2*c^4)*x^8 + (b
^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 + 32*a^3*b^3*c^3 - 128*a^4*b*c^4)*x^6 + 2*(a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*
c^2 - 64*a^4*b^2*c^3)*x^4 + (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x^2)*log(x))/((a^4*b^6*c^
2 - 12*a^5*b^4*c^3 + 48*a^6*b^2*c^4 - 64*a^7*c^5)*x^10 + 2*(a^4*b^7*c - 12*a^5*b^5*c^2 + 48*a^6*b^3*c^3 - 64*a
^7*b*c^4)*x^8 + (a^4*b^8 - 10*a^5*b^6*c + 24*a^6*b^4*c^2 + 32*a^7*b^2*c^3 - 128*a^8*c^4)*x^6 + 2*(a^5*b^7 - 12
*a^6*b^5*c + 48*a^7*b^3*c^2 - 64*a^8*b*c^3)*x^4 + (a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*x^2)]

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giac [A]  time = 1.80, size = 382, normalized size = 1.50 \[ \frac {3 \, {\left (b^{6} - 10 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} - 20 \, a^{3} c^{3}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {9 \, b^{5} c^{2} x^{8} - 72 \, a b^{3} c^{3} x^{8} + 144 \, a^{2} b c^{4} x^{8} + 18 \, b^{6} c x^{6} - 136 \, a b^{4} c^{2} x^{6} + 236 \, a^{2} b^{2} c^{3} x^{6} + 56 \, a^{3} c^{4} x^{6} + 9 \, b^{7} x^{4} - 38 \, a b^{5} c x^{4} - 110 \, a^{2} b^{3} c^{2} x^{4} + 436 \, a^{3} b c^{3} x^{4} + 26 \, a b^{6} x^{2} - 192 \, a^{2} b^{4} c x^{2} + 316 \, a^{3} b^{2} c^{2} x^{2} + 72 \, a^{4} c^{3} x^{2} + 19 \, a^{2} b^{5} - 144 \, a^{3} b^{3} c + 260 \, a^{4} b c^{2}}{8 \, {\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{2}} + \frac {3 \, b \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} - \frac {3 \, b \log \left (x^{2}\right )}{2 \, a^{4}} + \frac {3 \, b x^{2} - a}{2 \, a^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((a^4*b^4 - 8*a^
5*b^2*c + 16*a^6*c^2)*sqrt(-b^2 + 4*a*c)) - 1/8*(9*b^5*c^2*x^8 - 72*a*b^3*c^3*x^8 + 144*a^2*b*c^4*x^8 + 18*b^6
*c*x^6 - 136*a*b^4*c^2*x^6 + 236*a^2*b^2*c^3*x^6 + 56*a^3*c^4*x^6 + 9*b^7*x^4 - 38*a*b^5*c*x^4 - 110*a^2*b^3*c
^2*x^4 + 436*a^3*b*c^3*x^4 + 26*a*b^6*x^2 - 192*a^2*b^4*c*x^2 + 316*a^3*b^2*c^2*x^2 + 72*a^4*c^3*x^2 + 19*a^2*
b^5 - 144*a^3*b^3*c + 260*a^4*b*c^2)/((a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*(c*x^4 + b*x^2 + a)^2) + 3/4*b*log(
c*x^4 + b*x^2 + a)/a^4 - 3/2*b*log(x^2)/a^4 + 1/2*(3*b*x^2 - a)/(a^4*x^2)

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maple [B]  time = 0.03, size = 1002, normalized size = 3.93 \[ -\frac {7 c^{4} x^{6}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {13 b^{2} c^{3} x^{6}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {b^{4} c^{2} x^{6}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{3}}-\frac {37 b \,c^{3} x^{4}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {55 b^{3} c^{2} x^{4}}{4 \left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {2 b^{5} c \,x^{4}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{3}}-\frac {7 b^{2} c^{2} x^{2}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {6 b^{4} c \,x^{2}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {b^{6} x^{2}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{3}}-\frac {9 c^{3} x^{2}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {9 b^{3} c}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}-\frac {30 c^{3} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, a}-\frac {5 b^{5}}{4 \left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}+\frac {45 b^{2} c^{2} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, a^{2}}-\frac {15 b^{4} c \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, a^{3}}+\frac {3 b^{6} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, a^{4}}-\frac {29 b \,c^{2}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {12 b \,c^{2} \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {6 b^{3} c \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{3}}+\frac {3 b^{5} \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{4}}-\frac {3 b \ln \relax (x )}{a^{4}}-\frac {1}{2 a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/a/(c*x^4+b*x^2+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+13/2/a^2/(c*x^4+b*x^2+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c
+b^4)*x^6*b^2-1/a^3/(c*x^4+b*x^2+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*b^4-37/2/a/(c*x^4+b*x^2+a)^2*b*c^3/(1
6*a^2*c^2-8*a*b^2*c+b^4)*x^4+55/4/a^2/(c*x^4+b*x^2+a)^2*b^3*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4-2/a^3/(c*x^4+b*
x^2+a)^2*b^5*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4-9/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*c^3-7/2/a/(c*
x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^2*c^2+6/a^2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b
^4*c-1/a^3/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^6-29/2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b
^4)*b*c^2+9/a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*b^3*c-5/4/a^2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2
*c+b^4)*b^5+12/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*ln(c*x^4+b*x^2+a)*b-6/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c*ln(c*
x^4+b*x^2+a)*b^3+3/4/a^4/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^4+b*x^2+a)*b^5-30/a/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a
*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*c^3+45/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*ar
ctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^2*c^2-15/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^
2+b)/(4*a*c-b^2)^(1/2))*b^4*c+3/2/a^4/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b
^2)^(1/2))*b^6-1/2/a^3/x^2-3*b*ln(x)/a^4

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 11.76, size = 10074, normalized size = 39.51 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x^2 + c*x^4)^3),x)

[Out]

(log(((27*c^5*x^2*(b^4 + 10*a^2*c^2 - 7*a*b^2*c)^3)/(a^9*(4*a*c - b^2)^6) - ((3*b - 3*a^4*(-(b^6 - 20*a^3*c^3
+ 30*a^2*b^2*c^2 - 10*a*b^4*c)^2/(a^8*(4*a*c - b^2)^5))^(1/2))*((9*c^3*(4*b^10 - 100*a^5*c^5 + 342*a^2*b^6*c^2
- 837*a^3*b^4*c^3 + 780*a^4*b^2*c^4 - 61*a*b^8*c))/(a^6*(4*a*c - b^2)^4) - ((3*b - 3*a^4*(-(b^6 - 20*a^3*c^3
+ 30*a^2*b^2*c^2 - 10*a*b^4*c)^2/(a^8*(4*a*c - b^2)^5))^(1/2))*((6*c^3*x^2*(b^6 + 100*a^3*c^3 - 30*a^2*b^2*c^2
- 2*a*b^4*c))/(a^3*(4*a*c - b^2)^2) + (b*c^2*(3*b - 3*a^4*(-(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^
2/(a^8*(4*a*c - b^2)^5))^(1/2))*(a*b + 3*b^2*x^2 - 10*a*c*x^2))/a^4 + (12*b*c^2*(b^6 - 10*a^3*c^3 + 23*a^2*b^2
*c^2 - 9*a*b^4*c))/(a^3*(4*a*c - b^2)^2)))/(4*a^4) + (9*b*c^4*x^2*(6*b^8 + 900*a^4*c^4 + 479*a^2*b^4*c^2 - 110
0*a^3*b^2*c^3 - 89*a*b^6*c))/(a^6*(4*a*c - b^2)^4)))/(4*a^4) + (27*b*c^4*(b^4 + 10*a^2*c^2 - 7*a*b^2*c)^2)/(a^
9*(4*a*c - b^2)^4))*((27*c^5*x^2*(b^4 + 10*a^2*c^2 - 7*a*b^2*c)^3)/(a^9*(4*a*c - b^2)^6) - ((3*b + 3*a^4*(-(b^
6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^2/(a^8*(4*a*c - b^2)^5))^(1/2))*((9*c^3*(4*b^10 - 100*a^5*c^5 +
342*a^2*b^6*c^2 - 837*a^3*b^4*c^3 + 780*a^4*b^2*c^4 - 61*a*b^8*c))/(a^6*(4*a*c - b^2)^4) - ((3*b + 3*a^4*(-(b^
6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^2/(a^8*(4*a*c - b^2)^5))^(1/2))*((6*c^3*x^2*(b^6 + 100*a^3*c^3 -
30*a^2*b^2*c^2 - 2*a*b^4*c))/(a^3*(4*a*c - b^2)^2) + (b*c^2*(3*b + 3*a^4*(-(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2
- 10*a*b^4*c)^2/(a^8*(4*a*c - b^2)^5))^(1/2))*(a*b + 3*b^2*x^2 - 10*a*c*x^2))/a^4 + (12*b*c^2*(b^6 - 10*a^3*c
^3 + 23*a^2*b^2*c^2 - 9*a*b^4*c))/(a^3*(4*a*c - b^2)^2)))/(4*a^4) + (9*b*c^4*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*(4*a*c - b^2)^4)))/(4*a^4) + (27*b*c^4*(b^4 + 10*a^2*c^2 - 7*
a*b^2*c)^2)/(a^9*(4*a*c - b^2)^4)))*(6*b^11 - 6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b
^3*c^4 - 120*a*b^9*c))/(2*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 512
0*a^8*b^2*c^4)) - (3*b*log(x))/a^4 - (1/(2*a) + (x^4*(3*b^6 + 50*a^3*c^3 + 7*a^2*b^2*c^2 - 18*a*b^4*c))/(2*a^3
*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*x^6*(4*b^5*c - 29*a*b^3*c^2 + 46*a^2*b*c^3))/(4*a^3*(b^4 + 16*a^2*c^2 -
8*a*b^2*c)) + (x^2*(9*b^5 + 122*a^2*b*c^2 - 68*a*b^3*c))/(4*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*c^2*x^8*(
b^4 + 10*a^2*c^2 - 7*a*b^2*c))/(2*a^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^6*(2*a*c + b^2) + a^2*x^2 + c^2*x^10
+ 2*a*b*x^4 + 2*b*c*x^8) - (3*atan((x^2*((((27000*a^6*c^11 + 27*b^12*c^5 - 567*a*b^10*c^6 + 4779*a^2*b^8*c^7
- 20601*a^3*b^6*c^8 + 47790*a^4*b^4*c^9 - 56700*a^5*b^2*c^10)/(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^10*c + 240
*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5) - (((129600*a^9*b*c^10 + 54*a^3*b^1
3*c^4 - 1233*a^4*b^11*c^5 + 11583*a^5*b^9*c^6 - 57204*a^6*b^7*c^7 + 156276*a^7*b^5*c^8 - 223200*a^8*b^3*c^9)/(
a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^10*c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^
14*b^2*c^5) - (((153600*a^13*c^10 + 6*a^6*b^14*c^3 - 108*a^7*b^12*c^4 + 588*a^8*b^10*c^5 + 792*a^9*b^8*c^6 - 2
2272*a^10*b^6*c^7 + 100608*a^11*b^4*c^8 - 199680*a^12*b^2*c^9)/(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^10*c + 24
0*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5) - ((6*b^11 - 6144*a^5*b*c^5 + 960*
a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b^3*c^4 - 120*a*b^9*c)*(163840*a^16*b*c^9 - 12*a^9*b^15*c^2 + 328*a^
10*b^13*c^3 - 3840*a^11*b^11*c^4 + 24960*a^12*b^9*c^5 - 97280*a^13*b^7*c^6 + 227328*a^14*b^5*c^7 - 294912*a^15
*b^3*c^8))/(2*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^
4)*(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^10*c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 614
4*a^14*b^2*c^5)))*(6*b^11 - 6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b^3*c^4 - 120*a*b^9
*c))/(2*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)))*(
6*b^11 - 6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b^3*c^4 - 120*a*b^9*c))/(2*(4*a^4*b^10
- 4096*a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)) - (3*((3*((153600*a^1
3*c^10 + 6*a^6*b^14*c^3 - 108*a^7*b^12*c^4 + 588*a^8*b^10*c^5 + 792*a^9*b^8*c^6 - 22272*a^10*b^6*c^7 + 100608*
a^11*b^4*c^8 - 199680*a^12*b^2*c^9)/(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^10*c + 240*a^11*b^8*c^2 - 1280*a^12*
b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5) - ((6*b^11 - 6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*
c^3 + 7680*a^4*b^3*c^4 - 120*a*b^9*c)*(163840*a^16*b*c^9 - 12*a^9*b^15*c^2 + 328*a^10*b^13*c^3 - 3840*a^11*b^1
1*c^4 + 24960*a^12*b^9*c^5 - 97280*a^13*b^7*c^6 + 227328*a^14*b^5*c^7 - 294912*a^15*b^3*c^8))/(2*(4*a^4*b^10 -
4096*a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)*(a^9*b^12 + 4096*a^15*c^
6 - 24*a^10*b^10*c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5)))*(b^6 - 20
*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c))/(4*a^4*(4*a*c - b^2)^(5/2)) - (3*(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 -
10*a*b^4*c)*(6*b^11 - 6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b^3*c^4 - 120*a*b^9*c)*(
163840*a^16*b*c^9 - 12*a^9*b^15*c^2 + 328*a^10*b^13*c^3 - 3840*a^11*b^11*c^4 + 24960*a^12*b^9*c^5 - 97280*a^13
*b^7*c^6 + 227328*a^14*b^5*c^7 - 294912*a^15*b^3*c^8))/(8*a^4*(4*a*c - b^2)^(5/2)*(4*a^4*b^10 - 4096*a^9*c^5 -
80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)*(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^1
0*c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5)))*(b^6 - 20*a^3*c^3 + 30*a
^2*b^2*c^2 - 10*a*b^4*c))/(4*a^4*(4*a*c - b^2)^(5/2)) + (9*(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^2*
(6*b^11 - 6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b^3*c^4 - 120*a*b^9*c)*(163840*a^16*b
*c^9 - 12*a^9*b^15*c^2 + 328*a^10*b^13*c^3 - 3840*a^11*b^11*c^4 + 24960*a^12*b^9*c^5 - 97280*a^13*b^7*c^6 + 22
7328*a^14*b^5*c^7 - 294912*a^15*b^3*c^8))/(32*a^8*(4*a*c - b^2)^5*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^8*c +
640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)*(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^10*c + 240*a^11*b
^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5)))*(3*b^8 + 10*a^4*c^4 + 120*a^2*b^4*c^2 -
145*a^3*b^2*c^3 - 33*a*b^6*c))/(8*a^3*c^2*(4*a*c - b^2)^6*(100*a^6*c^6 - 6*b^12 - 960*a^2*b^8*c^2 + 3840*a^3*b
^6*c^3 - 7675*a^4*b^4*c^4 + 6100*a^5*b^2*c^5 + 120*a*b^10*c)) + (b*((((3*((153600*a^13*c^10 + 6*a^6*b^14*c^3 -
108*a^7*b^12*c^4 + 588*a^8*b^10*c^5 + 792*a^9*b^8*c^6 - 22272*a^10*b^6*c^7 + 100608*a^11*b^4*c^8 - 199680*a^1
2*b^2*c^9)/(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^10*c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c
^4 - 6144*a^14*b^2*c^5) - ((6*b^11 - 6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b^3*c^4 -
120*a*b^9*c)*(163840*a^16*b*c^9 - 12*a^9*b^15*c^2 + 328*a^10*b^13*c^3 - 3840*a^11*b^11*c^4 + 24960*a^12*b^9*c^
5 - 97280*a^13*b^7*c^6 + 227328*a^14*b^5*c^7 - 294912*a^15*b^3*c^8))/(2*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^
8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)*(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^10*c + 240*
a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5)))*(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2
- 10*a*b^4*c))/(4*a^4*(4*a*c - b^2)^(5/2)) - (3*(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)*(6*b^11 - 61
44*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b^3*c^4 - 120*a*b^9*c)*(163840*a^16*b*c^9 - 12*a^
9*b^15*c^2 + 328*a^10*b^13*c^3 - 3840*a^11*b^11*c^4 + 24960*a^12*b^9*c^5 - 97280*a^13*b^7*c^6 + 227328*a^14*b^
5*c^7 - 294912*a^15*b^3*c^8))/(8*a^4*(4*a*c - b^2)^(5/2)*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b
^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)*(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^10*c + 240*a^11*b^8*c^2 -
1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5)))*(6*b^11 - 6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840
*a^3*b^5*c^3 + 7680*a^4*b^3*c^4 - 120*a*b^9*c))/(2*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2
- 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)) - (3*((129600*a^9*b*c^10 + 54*a^3*b^13*c^4 - 1233*a^4*b^11*c^5 + 1158
3*a^5*b^9*c^6 - 57204*a^6*b^7*c^7 + 156276*a^7*b^5*c^8 - 223200*a^8*b^3*c^9)/(a^9*b^12 + 4096*a^15*c^6 - 24*a^
10*b^10*c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5) - (((153600*a^13*c^1
0 + 6*a^6*b^14*c^3 - 108*a^7*b^12*c^4 + 588*a^8*b^10*c^5 + 792*a^9*b^8*c^6 - 22272*a^10*b^6*c^7 + 100608*a^11*
b^4*c^8 - 199680*a^12*b^2*c^9)/(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^10*c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c
^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5) - ((6*b^11 - 6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 +
7680*a^4*b^3*c^4 - 120*a*b^9*c)*(163840*a^16*b*c^9 - 12*a^9*b^15*c^2 + 328*a^10*b^13*c^3 - 3840*a^11*b^11*c^4
+ 24960*a^12*b^9*c^5 - 97280*a^13*b^7*c^6 + 227328*a^14*b^5*c^7 - 294912*a^15*b^3*c^8))/(2*(4*a^4*b^10 - 4096
*a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)*(a^9*b^12 + 4096*a^15*c^6 - 2
4*a^10*b^10*c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5)))*(6*b^11 - 6144
*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b^3*c^4 - 120*a*b^9*c))/(2*(4*a^4*b^10 - 4096*a^9*c
^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)))*(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^
2 - 10*a*b^4*c))/(4*a^4*(4*a*c - b^2)^(5/2)) + (27*(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^3*(163840*
a^16*b*c^9 - 12*a^9*b^15*c^2 + 328*a^10*b^13*c^3 - 3840*a^11*b^11*c^4 + 24960*a^12*b^9*c^5 - 97280*a^13*b^7*c^
6 + 227328*a^14*b^5*c^7 - 294912*a^15*b^3*c^8))/(64*a^12*(4*a*c - b^2)^(15/2)*(a^9*b^12 + 4096*a^15*c^6 - 24*a
^10*b^10*c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5)))*(3*b^8 + 190*a^4*
c^4 + 180*a^2*b^4*c^2 - 335*a^3*b^2*c^3 - 39*a*b^6*c))/(8*a^3*c^2*(4*a*c - b^2)^(13/2)*(100*a^6*c^6 - 6*b^12 -
960*a^2*b^8*c^2 + 3840*a^3*b^6*c^3 - 7675*a^4*b^4*c^4 + 6100*a^5*b^2*c^5 + 120*a*b^10*c)))*(16*a^12*b^12*(4*a
*c - b^2)^(15/2) + 65536*a^18*c^6*(4*a*c - b^2)^(15/2) - 384*a^13*b^10*c*(4*a*c - b^2)^(15/2) + 3840*a^14*b^8*
c^2*(4*a*c - b^2)^(15/2) - 20480*a^15*b^6*c^3*(4*a*c - b^2)^(15/2) + 61440*a^16*b^4*c^4*(4*a*c - b^2)^(15/2) -
98304*a^17*b^2*c^5*(4*a*c - b^2)^(15/2)))/(10800*a^6*c^8 + 27*b^12*c^2 - 540*a*b^10*c^3 + 4320*a^2*b^8*c^4 -
17280*a^3*b^6*c^5 + 35100*a^4*b^4*c^6 - 32400*a^5*b^2*c^7) + (((27*b^9*c^4 - 378*a*b^7*c^5 + 2700*a^4*b*c^8 +
1863*a^2*b^5*c^6 - 3780*a^3*b^3*c^7)/(a^9*b^8 + 256*a^13*c^4 - 16*a^10*b^6*c + 96*a^11*b^4*c^2 - 256*a^12*b^2*
c^3) + (((900*a^8*c^8 - 36*a^3*b^10*c^3 + 549*a^4*b^8*c^4 - 3078*a^5*b^6*c^5 + 7533*a^6*b^4*c^6 - 7020*a^7*b^2
*c^7)/(a^9*b^8 + 256*a^13*c^4 - 16*a^10*b^6*c + 96*a^11*b^4*c^2 - 256*a^12*b^2*c^3) - (((1920*a^11*b*c^7 - 12*
a^6*b^11*c^2 + 204*a^7*b^9*c^3 - 1332*a^8*b^7*c^4 + 4056*a^9*b^5*c^5 - 5376*a^10*b^3*c^6)/(a^9*b^8 + 256*a^13*
c^4 - 16*a^10*b^6*c + 96*a^11*b^4*c^2 - 256*a^12*b^2*c^3) - ((4*a^10*b^10*c^2 - 64*a^11*b^8*c^3 + 384*a^12*b^6
*c^4 - 1024*a^13*b^4*c^5 + 1024*a^14*b^2*c^6)*(6*b^11 - 6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 +
7680*a^4*b^3*c^4 - 120*a*b^9*c))/(2*(a^9*b^8 + 256*a^13*c^4 - 16*a^10*b^6*c + 96*a^11*b^4*c^2 - 256*a^12*b^2*c
^3)*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)))*(6*b^
11 - 6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b^3*c^4 - 120*a*b^9*c))/(2*(4*a^4*b^10 - 4
096*a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)))*(6*b^11 - 6144*a^5*b*c^5
+ 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b^3*c^4 - 120*a*b^9*c))/(2*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^
5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)) + (3*((3*((1920*a^11*b*c^7 - 12*a^6*b^11*c^2
+ 204*a^7*b^9*c^3 - 1332*a^8*b^7*c^4 + 4056*a^9*b^5*c^5 - 5376*a^10*b^3*c^6)/(a^9*b^8 + 256*a^13*c^4 - 16*a^1
0*b^6*c + 96*a^11*b^4*c^2 - 256*a^12*b^2*c^3) - ((4*a^10*b^10*c^2 - 64*a^11*b^8*c^3 + 384*a^12*b^6*c^4 - 1024*
a^13*b^4*c^5 + 1024*a^14*b^2*c^6)*(6*b^11 - 6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b^3
*c^4 - 120*a*b^9*c))/(2*(a^9*b^8 + 256*a^13*c^4 - 16*a^10*b^6*c + 96*a^11*b^4*c^2 - 256*a^12*b^2*c^3)*(4*a^4*b
^10 - 4096*a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)))*(b^6 - 20*a^3*c^3
+ 30*a^2*b^2*c^2 - 10*a*b^4*c))/(4*a^4*(4*a*c - b^2)^(5/2)) - (3*(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^
4*c)*(4*a^10*b^10*c^2 - 64*a^11*b^8*c^3 + 384*a^12*b^6*c^4 - 1024*a^13*b^4*c^5 + 1024*a^14*b^2*c^6)*(6*b^11 -
6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b^3*c^4 - 120*a*b^9*c))/(8*a^4*(4*a*c - b^2)^(5
/2)*(a^9*b^8 + 256*a^13*c^4 - 16*a^10*b^6*c + 96*a^11*b^4*c^2 - 256*a^12*b^2*c^3)*(4*a^4*b^10 - 4096*a^9*c^5 -
80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)))*(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 -
10*a*b^4*c))/(4*a^4*(4*a*c - b^2)^(5/2)) - (9*(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^2*(4*a^10*b^10*
c^2 - 64*a^11*b^8*c^3 + 384*a^12*b^6*c^4 - 1024*a^13*b^4*c^5 + 1024*a^14*b^2*c^6)*(6*b^11 - 6144*a^5*b*c^5 + 9
60*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b^3*c^4 - 120*a*b^9*c))/(32*a^8*(4*a*c - b^2)^5*(a^9*b^8 + 256*a^
13*c^4 - 16*a^10*b^6*c + 96*a^11*b^4*c^2 - 256*a^12*b^2*c^3)*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^8*c + 640*a
^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)))*(3*b^8 + 10*a^4*c^4 + 120*a^2*b^4*c^2 - 145*a^3*b^2*c^3 -
33*a*b^6*c)*(16*a^12*b^12*(4*a*c - b^2)^(15/2) + 65536*a^18*c^6*(4*a*c - b^2)^(15/2) - 384*a^13*b^10*c*(4*a*c
- b^2)^(15/2) + 3840*a^14*b^8*c^2*(4*a*c - b^2)^(15/2) - 20480*a^15*b^6*c^3*(4*a*c - b^2)^(15/2) + 61440*a^16*
b^4*c^4*(4*a*c - b^2)^(15/2) - 98304*a^17*b^2*c^5*(4*a*c - b^2)^(15/2)))/(8*a^3*c^2*(4*a*c - b^2)^6*(100*a^6*c
^6 - 6*b^12 - 960*a^2*b^8*c^2 + 3840*a^3*b^6*c^3 - 7675*a^4*b^4*c^4 + 6100*a^5*b^2*c^5 + 120*a*b^10*c)*(10800*
a^6*c^8 + 27*b^12*c^2 - 540*a*b^10*c^3 + 4320*a^2*b^8*c^4 - 17280*a^3*b^6*c^5 + 35100*a^4*b^4*c^6 - 32400*a^5*
b^2*c^7)) - (b*((((3*((1920*a^11*b*c^7 - 12*a^6*b^11*c^2 + 204*a^7*b^9*c^3 - 1332*a^8*b^7*c^4 + 4056*a^9*b^5*c
^5 - 5376*a^10*b^3*c^6)/(a^9*b^8 + 256*a^13*c^4 - 16*a^10*b^6*c + 96*a^11*b^4*c^2 - 256*a^12*b^2*c^3) - ((4*a^
10*b^10*c^2 - 64*a^11*b^8*c^3 + 384*a^12*b^6*c^4 - 1024*a^13*b^4*c^5 + 1024*a^14*b^2*c^6)*(6*b^11 - 6144*a^5*b
*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b^3*c^4 - 120*a*b^9*c))/(2*(a^9*b^8 + 256*a^13*c^4 - 16*a
^10*b^6*c + 96*a^11*b^4*c^2 - 256*a^12*b^2*c^3)*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2 -
2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)))*(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c))/(4*a^4*(4*a*c - b^2)^
(5/2)) - (3*(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)*(4*a^10*b^10*c^2 - 64*a^11*b^8*c^3 + 384*a^12*b^6
*c^4 - 1024*a^13*b^4*c^5 + 1024*a^14*b^2*c^6)*(6*b^11 - 6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 +
7680*a^4*b^3*c^4 - 120*a*b^9*c))/(8*a^4*(4*a*c - b^2)^(5/2)*(a^9*b^8 + 256*a^13*c^4 - 16*a^10*b^6*c + 96*a^11*
b^4*c^2 - 256*a^12*b^2*c^3)*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5
120*a^8*b^2*c^4)))*(6*b^11 - 6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b^3*c^4 - 120*a*b^
9*c))/(2*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)) -
(3*((900*a^8*c^8 - 36*a^3*b^10*c^3 + 549*a^4*b^8*c^4 - 3078*a^5*b^6*c^5 + 7533*a^6*b^4*c^6 - 7020*a^7*b^2*c^7
)/(a^9*b^8 + 256*a^13*c^4 - 16*a^10*b^6*c + 96*a^11*b^4*c^2 - 256*a^12*b^2*c^3) - (((1920*a^11*b*c^7 - 12*a^6*
b^11*c^2 + 204*a^7*b^9*c^3 - 1332*a^8*b^7*c^4 + 4056*a^9*b^5*c^5 - 5376*a^10*b^3*c^6)/(a^9*b^8 + 256*a^13*c^4
- 16*a^10*b^6*c + 96*a^11*b^4*c^2 - 256*a^12*b^2*c^3) - ((4*a^10*b^10*c^2 - 64*a^11*b^8*c^3 + 384*a^12*b^6*c^4
- 1024*a^13*b^4*c^5 + 1024*a^14*b^2*c^6)*(6*b^11 - 6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680
*a^4*b^3*c^4 - 120*a*b^9*c))/(2*(a^9*b^8 + 256*a^13*c^4 - 16*a^10*b^6*c + 96*a^11*b^4*c^2 - 256*a^12*b^2*c^3)*
(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)))*(6*b^11 -
6144*a^5*b*c^5 + 960*a^2*b^7*c^2 - 3840*a^3*b^5*c^3 + 7680*a^4*b^3*c^4 - 120*a*b^9*c))/(2*(4*a^4*b^10 - 4096*
a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)))*(b^6 - 20*a^3*c^3 + 30*a^2*b
^2*c^2 - 10*a*b^4*c))/(4*a^4*(4*a*c - b^2)^(5/2)) + (27*(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c)^3*(4*
a^10*b^10*c^2 - 64*a^11*b^8*c^3 + 384*a^12*b^6*c^4 - 1024*a^13*b^4*c^5 + 1024*a^14*b^2*c^6))/(64*a^12*(4*a*c -
b^2)^(15/2)*(a^9*b^8 + 256*a^13*c^4 - 16*a^10*b^6*c + 96*a^11*b^4*c^2 - 256*a^12*b^2*c^3)))*(3*b^8 + 190*a^4*
c^4 + 180*a^2*b^4*c^2 - 335*a^3*b^2*c^3 - 39*a*b^6*c)*(16*a^12*b^12*(4*a*c - b^2)^(15/2) + 65536*a^18*c^6*(4*a
*c - b^2)^(15/2) - 384*a^13*b^10*c*(4*a*c - b^2)^(15/2) + 3840*a^14*b^8*c^2*(4*a*c - b^2)^(15/2) - 20480*a^15*
b^6*c^3*(4*a*c - b^2)^(15/2) + 61440*a^16*b^4*c^4*(4*a*c - b^2)^(15/2) - 98304*a^17*b^2*c^5*(4*a*c - b^2)^(15/
2)))/(8*a^3*c^2*(4*a*c - b^2)^(13/2)*(100*a^6*c^6 - 6*b^12 - 960*a^2*b^8*c^2 + 3840*a^3*b^6*c^3 - 7675*a^4*b^4
*c^4 + 6100*a^5*b^2*c^5 + 120*a*b^10*c)*(10800*a^6*c^8 + 27*b^12*c^2 - 540*a*b^10*c^3 + 4320*a^2*b^8*c^4 - 172
80*a^3*b^6*c^5 + 35100*a^4*b^4*c^6 - 32400*a^5*b^2*c^7)))*(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c))/(2
*a^4*(4*a*c - b^2)^(5/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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