### 3.2220 $$\int \frac{1}{x^2 (a+b x+c x^2)^4} \, dx$$

Optimal. Leaf size=352 $\frac{2 \left (3 b c x \left (29 a^2 c^2-10 a b^2 c+b^4\right )+105 a^2 b^2 c^2-70 a^3 c^3-32 a b^4 c+3 b^6\right )}{3 a^3 x \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{2 \left (7 a^2 c^2-7 a b^2 c+b^4\right )+b c x \left (2 b^2-13 a c\right )}{3 a^2 x \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{4 \left (38 a^2 b^2 c^2-35 a^3 c^3-11 a b^4 c+b^6\right )}{a^4 x \left (b^2-4 a c\right )^3}-\frac{4 \left (70 a^2 b^4 c^2-140 a^3 b^2 c^3+70 a^4 c^4-14 a b^6 c+b^8\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{7/2}}+\frac{2 b \log \left (a+b x+c x^2\right )}{a^5}-\frac{4 b \log (x)}{a^5}+\frac{-2 a c+b^2+b c x}{3 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}$

[Out]

(-4*(b^6 - 11*a*b^4*c + 38*a^2*b^2*c^2 - 35*a^3*c^3))/(a^4*(b^2 - 4*a*c)^3*x) + (b^2 - 2*a*c + b*c*x)/(3*a*(b^
2 - 4*a*c)*x*(a + b*x + c*x^2)^3) + (2*(b^4 - 7*a*b^2*c + 7*a^2*c^2) + b*c*(2*b^2 - 13*a*c)*x)/(3*a^2*(b^2 - 4
*a*c)^2*x*(a + b*x + c*x^2)^2) + (2*(3*b^6 - 32*a*b^4*c + 105*a^2*b^2*c^2 - 70*a^3*c^3 + 3*b*c*(b^4 - 10*a*b^2
*c + 29*a^2*c^2)*x))/(3*a^3*(b^2 - 4*a*c)^3*x*(a + b*x + c*x^2)) - (4*(b^8 - 14*a*b^6*c + 70*a^2*b^4*c^2 - 140
*a^3*b^2*c^3 + 70*a^4*c^4)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^5*(b^2 - 4*a*c)^(7/2)) - (4*b*Log[x])/a^
5 + (2*b*Log[a + b*x + c*x^2])/a^5

________________________________________________________________________________________

Rubi [A]  time = 0.51991, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.438, Rules used = {740, 822, 800, 634, 618, 206, 628} $\frac{2 \left (3 b c x \left (29 a^2 c^2-10 a b^2 c+b^4\right )+105 a^2 b^2 c^2-70 a^3 c^3-32 a b^4 c+3 b^6\right )}{3 a^3 x \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{2 \left (7 a^2 c^2-7 a b^2 c+b^4\right )+b c x \left (2 b^2-13 a c\right )}{3 a^2 x \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{4 \left (38 a^2 b^2 c^2-35 a^3 c^3-11 a b^4 c+b^6\right )}{a^4 x \left (b^2-4 a c\right )^3}-\frac{4 \left (70 a^2 b^4 c^2-140 a^3 b^2 c^3+70 a^4 c^4-14 a b^6 c+b^8\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{7/2}}+\frac{2 b \log \left (a+b x+c x^2\right )}{a^5}-\frac{4 b \log (x)}{a^5}+\frac{-2 a c+b^2+b c x}{3 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x^2*(a + b*x + c*x^2)^4),x]

[Out]

(-4*(b^6 - 11*a*b^4*c + 38*a^2*b^2*c^2 - 35*a^3*c^3))/(a^4*(b^2 - 4*a*c)^3*x) + (b^2 - 2*a*c + b*c*x)/(3*a*(b^
2 - 4*a*c)*x*(a + b*x + c*x^2)^3) + (2*(b^4 - 7*a*b^2*c + 7*a^2*c^2) + b*c*(2*b^2 - 13*a*c)*x)/(3*a^2*(b^2 - 4
*a*c)^2*x*(a + b*x + c*x^2)^2) + (2*(3*b^6 - 32*a*b^4*c + 105*a^2*b^2*c^2 - 70*a^3*c^3 + 3*b*c*(b^4 - 10*a*b^2
*c + 29*a^2*c^2)*x))/(3*a^3*(b^2 - 4*a*c)^3*x*(a + b*x + c*x^2)) - (4*(b^8 - 14*a*b^6*c + 70*a^2*b^4*c^2 - 140
*a^3*b^2*c^3 + 70*a^4*c^4)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^5*(b^2 - 4*a*c)^(7/2)) - (4*b*Log[x])/a^
5 + (2*b*Log[a + b*x + c*x^2])/a^5

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 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 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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (a+b x+c x^2\right )^4} \, dx &=\frac{b^2-2 a c+b c x}{3 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^3}-\frac{\int \frac{-2 \left (2 b^2-7 a c\right )-6 b c x}{x^2 \left (a+b x+c x^2\right )^3} \, dx}{3 a \left (b^2-4 a c\right )}\\ &=\frac{b^2-2 a c+b c x}{3 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^3}+\frac{2 \left (b^4-7 a b^2 c+7 a^2 c^2\right )+b c \left (2 b^2-13 a c\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )^2}+\frac{\int \frac{4 \left (3 b^2-7 a c\right ) \left (b^2-5 a c\right )+8 b c \left (2 b^2-13 a c\right ) x}{x^2 \left (a+b x+c x^2\right )^2} \, dx}{6 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac{b^2-2 a c+b c x}{3 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^3}+\frac{2 \left (b^4-7 a b^2 c+7 a^2 c^2\right )+b c \left (2 b^2-13 a c\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )^2}+\frac{2 \left (3 b^6-32 a b^4 c+105 a^2 b^2 c^2-70 a^3 c^3+3 b c \left (b^4-10 a b^2 c+29 a^2 c^2\right ) x\right )}{3 a^3 \left (b^2-4 a c\right )^3 x \left (a+b x+c x^2\right )}-\frac{\int \frac{-24 \left (b^6-11 a b^4 c+38 a^2 b^2 c^2-35 a^3 c^3\right )-24 b c \left (b^4-10 a b^2 c+29 a^2 c^2\right ) x}{x^2 \left (a+b x+c x^2\right )} \, dx}{6 a^3 \left (b^2-4 a c\right )^3}\\ &=\frac{b^2-2 a c+b c x}{3 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^3}+\frac{2 \left (b^4-7 a b^2 c+7 a^2 c^2\right )+b c \left (2 b^2-13 a c\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )^2}+\frac{2 \left (3 b^6-32 a b^4 c+105 a^2 b^2 c^2-70 a^3 c^3+3 b c \left (b^4-10 a b^2 c+29 a^2 c^2\right ) x\right )}{3 a^3 \left (b^2-4 a c\right )^3 x \left (a+b x+c x^2\right )}-\frac{\int \left (\frac{24 \left (-b^6+11 a b^4 c-38 a^2 b^2 c^2+35 a^3 c^3\right )}{a x^2}-\frac{24 b \left (-b^2+4 a c\right )^3}{a^2 x}+\frac{24 \left (-\left (b^2-5 a c\right ) \left (b^6-8 a b^4 c+19 a^2 b^2 c^2-7 a^3 c^3\right )-b c \left (b^2-4 a c\right )^3 x\right )}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx}{6 a^3 \left (b^2-4 a c\right )^3}\\ &=-\frac{4 \left (b^6-11 a b^4 c+38 a^2 b^2 c^2-35 a^3 c^3\right )}{a^4 \left (b^2-4 a c\right )^3 x}+\frac{b^2-2 a c+b c x}{3 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^3}+\frac{2 \left (b^4-7 a b^2 c+7 a^2 c^2\right )+b c \left (2 b^2-13 a c\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )^2}+\frac{2 \left (3 b^6-32 a b^4 c+105 a^2 b^2 c^2-70 a^3 c^3+3 b c \left (b^4-10 a b^2 c+29 a^2 c^2\right ) x\right )}{3 a^3 \left (b^2-4 a c\right )^3 x \left (a+b x+c x^2\right )}-\frac{4 b \log (x)}{a^5}-\frac{4 \int \frac{-\left (b^2-5 a c\right ) \left (b^6-8 a b^4 c+19 a^2 b^2 c^2-7 a^3 c^3\right )-b c \left (b^2-4 a c\right )^3 x}{a+b x+c x^2} \, dx}{a^5 \left (b^2-4 a c\right )^3}\\ &=-\frac{4 \left (b^6-11 a b^4 c+38 a^2 b^2 c^2-35 a^3 c^3\right )}{a^4 \left (b^2-4 a c\right )^3 x}+\frac{b^2-2 a c+b c x}{3 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^3}+\frac{2 \left (b^4-7 a b^2 c+7 a^2 c^2\right )+b c \left (2 b^2-13 a c\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )^2}+\frac{2 \left (3 b^6-32 a b^4 c+105 a^2 b^2 c^2-70 a^3 c^3+3 b c \left (b^4-10 a b^2 c+29 a^2 c^2\right ) x\right )}{3 a^3 \left (b^2-4 a c\right )^3 x \left (a+b x+c x^2\right )}-\frac{4 b \log (x)}{a^5}+\frac{(2 b) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{a^5}+\frac{\left (2 \left (b^8-14 a b^6 c+70 a^2 b^4 c^2-140 a^3 b^2 c^3+70 a^4 c^4\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{a^5 \left (b^2-4 a c\right )^3}\\ &=-\frac{4 \left (b^6-11 a b^4 c+38 a^2 b^2 c^2-35 a^3 c^3\right )}{a^4 \left (b^2-4 a c\right )^3 x}+\frac{b^2-2 a c+b c x}{3 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^3}+\frac{2 \left (b^4-7 a b^2 c+7 a^2 c^2\right )+b c \left (2 b^2-13 a c\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )^2}+\frac{2 \left (3 b^6-32 a b^4 c+105 a^2 b^2 c^2-70 a^3 c^3+3 b c \left (b^4-10 a b^2 c+29 a^2 c^2\right ) x\right )}{3 a^3 \left (b^2-4 a c\right )^3 x \left (a+b x+c x^2\right )}-\frac{4 b \log (x)}{a^5}+\frac{2 b \log \left (a+b x+c x^2\right )}{a^5}-\frac{\left (4 \left (b^8-14 a b^6 c+70 a^2 b^4 c^2-140 a^3 b^2 c^3+70 a^4 c^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^5 \left (b^2-4 a c\right )^3}\\ &=-\frac{4 \left (b^6-11 a b^4 c+38 a^2 b^2 c^2-35 a^3 c^3\right )}{a^4 \left (b^2-4 a c\right )^3 x}+\frac{b^2-2 a c+b c x}{3 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^3}+\frac{2 \left (b^4-7 a b^2 c+7 a^2 c^2\right )+b c \left (2 b^2-13 a c\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )^2}+\frac{2 \left (3 b^6-32 a b^4 c+105 a^2 b^2 c^2-70 a^3 c^3+3 b c \left (b^4-10 a b^2 c+29 a^2 c^2\right ) x\right )}{3 a^3 \left (b^2-4 a c\right )^3 x \left (a+b x+c x^2\right )}-\frac{4 \left (b^8-14 a b^6 c+70 a^2 b^4 c^2-140 a^3 b^2 c^3+70 a^4 c^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{7/2}}-\frac{4 b \log (x)}{a^5}+\frac{2 b \log \left (a+b x+c x^2\right )}{a^5}\\ \end{align*}

Mathematica [A]  time = 0.906965, size = 329, normalized size = 0.93 $\frac{\frac{a^3 \left (-3 a b c-2 a c^2 x+b^2 c x+b^3\right )}{\left (4 a c-b^2\right ) (a+x (b+c x))^3}-\frac{a^2 \left (35 a^2 b c^2+22 a^2 c^3 x-20 a b^2 c^2 x-22 a b^3 c+3 b^4 c x+3 b^5\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+\frac{3 a \left (-104 a^2 b^2 c^3 x-124 a^2 b^3 c^2+134 a^3 b c^3+76 a^3 c^4 x+32 a b^4 c^2 x+34 a b^5 c-3 b^6 c x-3 b^7\right )}{\left (b^2-4 a c\right )^3 (a+x (b+c x))}-\frac{12 \left (70 a^2 b^4 c^2-140 a^3 b^2 c^3+70 a^4 c^4-14 a b^6 c+b^8\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{7/2}}+6 b \log (a+x (b+c x))-\frac{3 a}{x}-12 b \log (x)}{3 a^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-3*a)/x + (a^3*(b^3 - 3*a*b*c + b^2*c*x - 2*a*c^2*x))/((-b^2 + 4*a*c)*(a + x*(b + c*x))^3) - (a^2*(3*b^5 - 2
2*a*b^3*c + 35*a^2*b*c^2 + 3*b^4*c*x - 20*a*b^2*c^2*x + 22*a^2*c^3*x))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) +
(3*a*(-3*b^7 + 34*a*b^5*c - 124*a^2*b^3*c^2 + 134*a^3*b*c^3 - 3*b^6*c*x + 32*a*b^4*c^2*x - 104*a^2*b^2*c^3*x
+ 76*a^3*c^4*x))/((b^2 - 4*a*c)^3*(a + x*(b + c*x))) - (12*(b^8 - 14*a*b^6*c + 70*a^2*b^4*c^2 - 140*a^3*b^2*c^
3 + 70*a^4*c^4)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2) - 12*b*Log[x] + 6*b*Log[a + x*(b
+ c*x)])/(3*a^5)

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Maple [B]  time = 0.175, size = 2162, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a^4/x+3/a^4/(c*x^2+b*x+a)^3*b^9/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2-116*a/(c*x^2+b*x+a)^3/(64*a^
3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x*c^4+7/a^3/(c*x^2+b*x+a)^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x*
b^8-4/a^5/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^8
+128/a^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^3*ln(c*x^2+b*x+a)*b-590/3*a/(c*x^2+b*x+a)^3*b/(64*a^3*c^
3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^3-49/a/(c*x^2+b*x+a)^3*b^5/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c-76/
a/(c*x^2+b*x+a)^3*c^6/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^5-496/(c*x^2+b*x+a)^3*b/(64*a^3*c^3-48*a^2*
b^2*c^2+12*a*b^4*c-b^6)*x^2*c^4-166/(c*x^2+b*x+a)^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x*b^2*c^3-280/a
/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^4-96/a^3/(
64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^2*ln(c*x^2+b*x+a)*b^3+24/a^4/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c
-b^6)*c*ln(c*x^2+b*x+a)*b^5+535/3/(c*x^2+b*x+a)^3*b^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^2-544/3/(c*
x^2+b*x+a)^3*c^5/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3-2/a^5/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^
6)*ln(c*x^2+b*x+a)*b^7+13/3/a^2/(c*x^2+b*x+a)^3*b^7/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)-93/a^3/(c*x^2+b
*x+a)^3*c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3*b^6-98/a^3/(c*x^2+b*x+a)^3*b^5*c^3/(64*a^3*c^3-48*a
^2*b^2*c^2+12*a*b^4*c-b^6)*x^4+9/a^4/(c*x^2+b*x+a)^3*b^7*c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^4-10
2/a/(c*x^2+b*x+a)^3*c^4/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3*b^2-30/a^2/(c*x^2+b*x+a)^3*b^5/(64*a^3*
c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2*c^2-20/a^3/(c*x^2+b*x+a)^3*b^7/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^
6)*x^2*c+243/a/(c*x^2+b*x+a)^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x*b^4*c^2-75/a^2/(c*x^2+b*x+a)^3/(64
*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x*b^6*c-280/a^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b^2)
^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4*c^2+9/a^4/(c*x^2+b*x+a)^3*c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4
*c-b^6)*x^3*b^8+832/3/a^2/(c*x^2+b*x+a)^3*c^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3*b^4-286/a/(c*x^2+
b*x+a)^3*b*c^5/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^4+332/a^2/(c*x^2+b*x+a)^3*b^3*c^4/(64*a^3*c^3-48*a
^2*b^2*c^2+12*a*b^4*c-b^6)*x^4+3/a^4/(c*x^2+b*x+a)^3*c^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^5*b^6+56
0/a^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*c^3
+56/a^4/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^6*c
+397/a/(c*x^2+b*x+a)^3*b^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2*c^3+104/a^2/(c*x^2+b*x+a)^3*c^5/(64*
a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^5*b^2-32/a^3/(c*x^2+b*x+a)^3*c^4/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*
c-b^6)*x^5*b^4-4*b*ln(x)/a^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 17.9034, size = 8609, normalized size = 24.46 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/3*(3*a^4*b^8 - 48*a^5*b^6*c + 288*a^6*b^4*c^2 - 768*a^7*b^2*c^3 + 768*a^8*c^4 + 12*(a*b^8*c^3 - 15*a^2*b^6
*c^4 + 82*a^3*b^4*c^5 - 187*a^4*b^2*c^6 + 140*a^5*c^7)*x^6 + 6*(6*a*b^9*c^2 - 91*a^2*b^7*c^3 + 506*a^3*b^5*c^4
- 1191*a^4*b^3*c^5 + 956*a^5*b*c^6)*x^5 + 2*(18*a*b^10*c - 261*a^2*b^8*c^2 + 1334*a^3*b^6*c^3 - 2537*a^4*b^4*
c^4 + 340*a^5*b^2*c^5 + 2240*a^6*c^6)*x^4 + 3*(4*a*b^11 - 42*a^2*b^9*c + 50*a^3*b^7*c^2 + 837*a^4*b^5*c^3 - 33
64*a^5*b^3*c^4 + 3520*a^6*b*c^5)*x^3 + 3*(10*a^2*b^10 - 148*a^3*b^8*c + 783*a^4*b^6*c^2 - 1618*a^5*b^4*c^3 + 5
48*a^6*b^2*c^4 + 1232*a^7*c^5)*x^2 + 6*((b^8*c^3 - 14*a*b^6*c^4 + 70*a^2*b^4*c^5 - 140*a^3*b^2*c^6 + 70*a^4*c^
7)*x^7 + 3*(b^9*c^2 - 14*a*b^7*c^3 + 70*a^2*b^5*c^4 - 140*a^3*b^3*c^5 + 70*a^4*b*c^6)*x^6 + 3*(b^10*c - 13*a*b
^8*c^2 + 56*a^2*b^6*c^3 - 70*a^3*b^4*c^4 - 70*a^4*b^2*c^5 + 70*a^5*c^6)*x^5 + (b^11 - 8*a*b^9*c - 14*a^2*b^7*c
^2 + 280*a^3*b^5*c^3 - 770*a^4*b^3*c^4 + 420*a^5*b*c^5)*x^4 + 3*(a*b^10 - 13*a^2*b^8*c + 56*a^3*b^6*c^2 - 70*a
^4*b^4*c^3 - 70*a^5*b^2*c^4 + 70*a^6*c^5)*x^3 + 3*(a^2*b^9 - 14*a^3*b^7*c + 70*a^4*b^5*c^2 - 140*a^5*b^3*c^3 +
70*a^6*b*c^4)*x^2 + (a^3*b^8 - 14*a^4*b^6*c + 70*a^5*b^4*c^2 - 140*a^6*b^2*c^3 + 70*a^7*c^4)*x)*sqrt(b^2 - 4*
a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + (22*a^3*b^9
- 343*a^4*b^7*c + 1987*a^5*b^5*c^2 - 5034*a^6*b^3*c^3 + 4664*a^7*b*c^4)*x - 6*((b^9*c^3 - 16*a*b^7*c^4 + 96*a^
2*b^5*c^5 - 256*a^3*b^3*c^6 + 256*a^4*b*c^7)*x^7 + 3*(b^10*c^2 - 16*a*b^8*c^3 + 96*a^2*b^6*c^4 - 256*a^3*b^4*c
^5 + 256*a^4*b^2*c^6)*x^6 + 3*(b^11*c - 15*a*b^9*c^2 + 80*a^2*b^7*c^3 - 160*a^3*b^5*c^4 + 256*a^5*b*c^6)*x^5 +
(b^12 - 10*a*b^10*c + 320*a^3*b^6*c^3 - 1280*a^4*b^4*c^4 + 1536*a^5*b^2*c^5)*x^4 + 3*(a*b^11 - 15*a^2*b^9*c +
80*a^3*b^7*c^2 - 160*a^4*b^5*c^3 + 256*a^6*b*c^5)*x^3 + 3*(a^2*b^10 - 16*a^3*b^8*c + 96*a^4*b^6*c^2 - 256*a^5
*b^4*c^3 + 256*a^6*b^2*c^4)*x^2 + (a^3*b^9 - 16*a^4*b^7*c + 96*a^5*b^5*c^2 - 256*a^6*b^3*c^3 + 256*a^7*b*c^4)*
x)*log(c*x^2 + b*x + a) + 12*((b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256*a^3*b^3*c^6 + 256*a^4*b*c^7)*x^7
+ 3*(b^10*c^2 - 16*a*b^8*c^3 + 96*a^2*b^6*c^4 - 256*a^3*b^4*c^5 + 256*a^4*b^2*c^6)*x^6 + 3*(b^11*c - 15*a*b^9*
c^2 + 80*a^2*b^7*c^3 - 160*a^3*b^5*c^4 + 256*a^5*b*c^6)*x^5 + (b^12 - 10*a*b^10*c + 320*a^3*b^6*c^3 - 1280*a^4
*b^4*c^4 + 1536*a^5*b^2*c^5)*x^4 + 3*(a*b^11 - 15*a^2*b^9*c + 80*a^3*b^7*c^2 - 160*a^4*b^5*c^3 + 256*a^6*b*c^5
)*x^3 + 3*(a^2*b^10 - 16*a^3*b^8*c + 96*a^4*b^6*c^2 - 256*a^5*b^4*c^3 + 256*a^6*b^2*c^4)*x^2 + (a^3*b^9 - 16*a
^4*b^7*c + 96*a^5*b^5*c^2 - 256*a^6*b^3*c^3 + 256*a^7*b*c^4)*x)*log(x))/((a^5*b^8*c^3 - 16*a^6*b^6*c^4 + 96*a^
7*b^4*c^5 - 256*a^8*b^2*c^6 + 256*a^9*c^7)*x^7 + 3*(a^5*b^9*c^2 - 16*a^6*b^7*c^3 + 96*a^7*b^5*c^4 - 256*a^8*b^
3*c^5 + 256*a^9*b*c^6)*x^6 + 3*(a^5*b^10*c - 15*a^6*b^8*c^2 + 80*a^7*b^6*c^3 - 160*a^8*b^4*c^4 + 256*a^10*c^6)
*x^5 + (a^5*b^11 - 10*a^6*b^9*c + 320*a^8*b^5*c^3 - 1280*a^9*b^3*c^4 + 1536*a^10*b*c^5)*x^4 + 3*(a^6*b^10 - 15
*a^7*b^8*c + 80*a^8*b^6*c^2 - 160*a^9*b^4*c^3 + 256*a^11*c^5)*x^3 + 3*(a^7*b^9 - 16*a^8*b^7*c + 96*a^9*b^5*c^2
- 256*a^10*b^3*c^3 + 256*a^11*b*c^4)*x^2 + (a^8*b^8 - 16*a^9*b^6*c + 96*a^10*b^4*c^2 - 256*a^11*b^2*c^3 + 256
*a^12*c^4)*x), -1/3*(3*a^4*b^8 - 48*a^5*b^6*c + 288*a^6*b^4*c^2 - 768*a^7*b^2*c^3 + 768*a^8*c^4 + 12*(a*b^8*c^
3 - 15*a^2*b^6*c^4 + 82*a^3*b^4*c^5 - 187*a^4*b^2*c^6 + 140*a^5*c^7)*x^6 + 6*(6*a*b^9*c^2 - 91*a^2*b^7*c^3 + 5
06*a^3*b^5*c^4 - 1191*a^4*b^3*c^5 + 956*a^5*b*c^6)*x^5 + 2*(18*a*b^10*c - 261*a^2*b^8*c^2 + 1334*a^3*b^6*c^3 -
2537*a^4*b^4*c^4 + 340*a^5*b^2*c^5 + 2240*a^6*c^6)*x^4 + 3*(4*a*b^11 - 42*a^2*b^9*c + 50*a^3*b^7*c^2 + 837*a^
4*b^5*c^3 - 3364*a^5*b^3*c^4 + 3520*a^6*b*c^5)*x^3 + 3*(10*a^2*b^10 - 148*a^3*b^8*c + 783*a^4*b^6*c^2 - 1618*a
^5*b^4*c^3 + 548*a^6*b^2*c^4 + 1232*a^7*c^5)*x^2 + 12*((b^8*c^3 - 14*a*b^6*c^4 + 70*a^2*b^4*c^5 - 140*a^3*b^2*
c^6 + 70*a^4*c^7)*x^7 + 3*(b^9*c^2 - 14*a*b^7*c^3 + 70*a^2*b^5*c^4 - 140*a^3*b^3*c^5 + 70*a^4*b*c^6)*x^6 + 3*(
b^10*c - 13*a*b^8*c^2 + 56*a^2*b^6*c^3 - 70*a^3*b^4*c^4 - 70*a^4*b^2*c^5 + 70*a^5*c^6)*x^5 + (b^11 - 8*a*b^9*c
- 14*a^2*b^7*c^2 + 280*a^3*b^5*c^3 - 770*a^4*b^3*c^4 + 420*a^5*b*c^5)*x^4 + 3*(a*b^10 - 13*a^2*b^8*c + 56*a^3
*b^6*c^2 - 70*a^4*b^4*c^3 - 70*a^5*b^2*c^4 + 70*a^6*c^5)*x^3 + 3*(a^2*b^9 - 14*a^3*b^7*c + 70*a^4*b^5*c^2 - 14
0*a^5*b^3*c^3 + 70*a^6*b*c^4)*x^2 + (a^3*b^8 - 14*a^4*b^6*c + 70*a^5*b^4*c^2 - 140*a^6*b^2*c^3 + 70*a^7*c^4)*x
)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (22*a^3*b^9 - 343*a^4*b^7*c + 198
7*a^5*b^5*c^2 - 5034*a^6*b^3*c^3 + 4664*a^7*b*c^4)*x - 6*((b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256*a^3*b
^3*c^6 + 256*a^4*b*c^7)*x^7 + 3*(b^10*c^2 - 16*a*b^8*c^3 + 96*a^2*b^6*c^4 - 256*a^3*b^4*c^5 + 256*a^4*b^2*c^6)
*x^6 + 3*(b^11*c - 15*a*b^9*c^2 + 80*a^2*b^7*c^3 - 160*a^3*b^5*c^4 + 256*a^5*b*c^6)*x^5 + (b^12 - 10*a*b^10*c
+ 320*a^3*b^6*c^3 - 1280*a^4*b^4*c^4 + 1536*a^5*b^2*c^5)*x^4 + 3*(a*b^11 - 15*a^2*b^9*c + 80*a^3*b^7*c^2 - 160
*a^4*b^5*c^3 + 256*a^6*b*c^5)*x^3 + 3*(a^2*b^10 - 16*a^3*b^8*c + 96*a^4*b^6*c^2 - 256*a^5*b^4*c^3 + 256*a^6*b^
2*c^4)*x^2 + (a^3*b^9 - 16*a^4*b^7*c + 96*a^5*b^5*c^2 - 256*a^6*b^3*c^3 + 256*a^7*b*c^4)*x)*log(c*x^2 + b*x +
a) + 12*((b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256*a^3*b^3*c^6 + 256*a^4*b*c^7)*x^7 + 3*(b^10*c^2 - 16*a*
b^8*c^3 + 96*a^2*b^6*c^4 - 256*a^3*b^4*c^5 + 256*a^4*b^2*c^6)*x^6 + 3*(b^11*c - 15*a*b^9*c^2 + 80*a^2*b^7*c^3
- 160*a^3*b^5*c^4 + 256*a^5*b*c^6)*x^5 + (b^12 - 10*a*b^10*c + 320*a^3*b^6*c^3 - 1280*a^4*b^4*c^4 + 1536*a^5*b
^2*c^5)*x^4 + 3*(a*b^11 - 15*a^2*b^9*c + 80*a^3*b^7*c^2 - 160*a^4*b^5*c^3 + 256*a^6*b*c^5)*x^3 + 3*(a^2*b^10 -
16*a^3*b^8*c + 96*a^4*b^6*c^2 - 256*a^5*b^4*c^3 + 256*a^6*b^2*c^4)*x^2 + (a^3*b^9 - 16*a^4*b^7*c + 96*a^5*b^5
*c^2 - 256*a^6*b^3*c^3 + 256*a^7*b*c^4)*x)*log(x))/((a^5*b^8*c^3 - 16*a^6*b^6*c^4 + 96*a^7*b^4*c^5 - 256*a^8*b
^2*c^6 + 256*a^9*c^7)*x^7 + 3*(a^5*b^9*c^2 - 16*a^6*b^7*c^3 + 96*a^7*b^5*c^4 - 256*a^8*b^3*c^5 + 256*a^9*b*c^6
)*x^6 + 3*(a^5*b^10*c - 15*a^6*b^8*c^2 + 80*a^7*b^6*c^3 - 160*a^8*b^4*c^4 + 256*a^10*c^6)*x^5 + (a^5*b^11 - 10
*a^6*b^9*c + 320*a^8*b^5*c^3 - 1280*a^9*b^3*c^4 + 1536*a^10*b*c^5)*x^4 + 3*(a^6*b^10 - 15*a^7*b^8*c + 80*a^8*b
^6*c^2 - 160*a^9*b^4*c^3 + 256*a^11*c^5)*x^3 + 3*(a^7*b^9 - 16*a^8*b^7*c + 96*a^9*b^5*c^2 - 256*a^10*b^3*c^3 +
256*a^11*b*c^4)*x^2 + (a^8*b^8 - 16*a^9*b^6*c + 96*a^10*b^4*c^2 - 256*a^11*b^2*c^3 + 256*a^12*c^4)*x)]

________________________________________________________________________________________

Sympy [B]  time = 114.677, size = 9418, normalized size = 26.76 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b/a**5 - 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c +
b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b
**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)))*log(x + (-6864896*a**21*b*c**11*(2*b/a**5 - 2*sqrt(-(
4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a*
*7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b
**10*c**2 + 28*a*b**12*c - b**14)))**2 + 19451904*a**20*b**3*c**10*(2*b/a**5 - 2*sqrt(-(4*a*c - b**2)**7)*(70*
a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b
**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12
*c - b**14)))**2 - 24960000*a**19*b**5*c**9*(2*b/a**5 - 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b*
*2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b
**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)))**2 + 1915
7248*a**18*b**7*c**8*(2*b/a**5 - 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*
c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b*
*6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)))**2 - 9777216*a**17*b**9*c**7*(2*
b/a**5 - 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**
8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8
*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)))**2 - 1254400*a**17*c**12*(2*b/a**5 - 2*sqrt(-(4*a*c - b*
*2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 -
28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2
+ 28*a*b**12*c - b**14))) + 3485552*a**16*b**11*c**6*(2*b/a**5 - 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 14
0*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 215
04*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)))*
*2 - 4017152*a**16*b**2*c**11*(2*b/a**5 - 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a
**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 896
0*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14))) - 886004*a**15*b**13*c*
*5*(2*b/a**5 - 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c
+ b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**
3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)))**2 + 12987008*a**15*b**4*c**10*(2*b/a**5 - 2*sqrt(
-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*
a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2
*b**10*c**2 + 28*a*b**12*c - b**14))) + 160635*a**14*b**15*c**4*(2*b/a**5 - 2*sqrt(-(4*a*c - b**2)**7)*(70*a**
4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2
*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c
- b**14)))**2 - 14915520*a**14*b**6*c**9*(2*b/a**5 - 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*
c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4
*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14))) - 20362*a**1
3*b**17*c**3*(2*b/a**5 - 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 1
4*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4
+ 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)))**2 + 9737948*a**13*b**8*c**8*(2*b/a**5 -
2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5
*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 -
336*a**2*b**10*c**2 + 28*a*b**12*c - b**14))) + 1719*a**12*b**19*c**2*(2*b/a**5 - 2*sqrt(-(4*a*c - b**2)**7)*(
70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**
6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b*
*12*c - b**14)))**2 - 4124656*a**12*b**10*c**7*(2*b/a**5 - 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3
*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**
5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14))) - 5017
600*a**12*b*c**12 - 87*a**11*b**21*c*(2*b/a**5 - 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3
+ 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**
5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)))**2 + 1194984*a**
11*b**12*c**6*(2*b/a**5 - 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 -
14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4
+ 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14))) + 93769728*a**11*b**3*c**11 + 2*a**10*b
**23*(2*b/a**5 - 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6
*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a
**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)))**2 - 240632*a**10*b**14*c**5*(2*b/a**5 - 2*sqrt(
-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*
a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2
*b**10*c**2 + 28*a*b**12*c - b**14))) - 259282432*a**10*b**5*c**10 + 33300*a**9*b**16*c**4*(2*b/a**5 - 2*sqrt(
-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*
a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2
*b**10*c**2 + 28*a*b**12*c - b**14))) + 339697920*a**9*b**7*c**9 - 3032*a**8*b**18*c**3*(2*b/a**5 - 2*sqrt(-(4
*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**
7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b*
*10*c**2 + 28*a*b**12*c - b**14))) - 267564176*a**8*b**9*c**8 + 164*a**7*b**20*c**2*(2*b/a**5 - 2*sqrt(-(4*a*c
- b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c*
*7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*
c**2 + 28*a*b**12*c - b**14))) + 139936832*a**7*b**11*c**7 - 4*a**6*b**22*c*(2*b/a**5 - 2*sqrt(-(4*a*c - b**2)
**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 286
72*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 2
8*a*b**12*c - b**14))) - 50988896*a**6*b**13*c**6 + 13213536*a**5*b**15*c**5 - 2436960*a**4*b**17*c**4 + 31366
4*a**3*b**19*c**3 - 26848*a**2*b**21*c**2 + 1376*a*b**23*c - 32*b**25)/(1372000*a**12*c**13 + 33055680*a**11*b
**2*c**12 - 134248800*a**10*b**4*c**11 + 211721440*a**9*b**6*c**10 - 187538736*a**8*b**8*c**9 + 106627392*a**7
*b**10*c**8 - 41403488*a**6*b**12*c**7 + 11287584*a**5*b**14*c**6 - 2170560*a**4*b**16*c**5 + 289408*a**3*b**1
8*c**4 - 25536*a**2*b**20*c**3 + 1344*a*b**22*c**2 - 32*b**24*c)) + (2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70
*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*
b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**1
2*c - b**14)))*log(x + (-6864896*a**21*b*c**11*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3
*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**
5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)))**2 + 1
9451904*a**20*b**3*c**10*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b
**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**
4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)))**2 - 24960000*a**19*b**5*c**
9*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c
+ b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3
*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)))**2 + 19157248*a**18*b**7*c**8*(2*b/a**5 + 2*sqrt(-(
4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a*
*7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b
**10*c**2 + 28*a*b**12*c - b**14)))**2 - 9777216*a**17*b**9*c**7*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a*
*4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**
2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c
- b**14)))**2 - 1254400*a**17*c**12*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3
+ 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**
5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14))) + 3485552*a**16*
b**11*c**6*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*
a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 +
2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)))**2 - 4017152*a**16*b**2*c**11*(2*b/a**5 +
2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*
(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 3
36*a**2*b**10*c**2 + 28*a*b**12*c - b**14))) - 886004*a**15*b**13*c**5*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*
(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a*
*6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b
**12*c - b**14)))**2 + 12987008*a**15*b**4*c**10*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a*
*3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a
**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14))) + 16
0635*a**14*b**15*c**4*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4
*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b
**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)))**2 - 14915520*a**14*b**6*c**9*(
2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b
**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b*
*8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14))) - 20362*a**13*b**17*c**3*(2*b/a**5 + 2*sqrt(-(4*a*c -
b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7
- 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**
2 + 28*a*b**12*c - b**14)))**2 + 9737948*a**13*b**8*c**8*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4
- 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 +
21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14
))) + 1719*a**12*b**19*c**2*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**
2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*
a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)))**2 - 4124656*a**12*b**10*
c**7*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6
*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a
**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14))) - 5017600*a**12*b*c**12 - 87*a**11*b**21*c*(2*b/
a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)
/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c
**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)))**2 + 1194984*a**11*b**12*c**6*(2*b/a**5 + 2*sqrt(-(4*a*c -
b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7
- 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c*
*2 + 28*a*b**12*c - b**14))) + 93769728*a**11*b**3*c**11 + 2*a**10*b**23*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7
)*(70*a**4*c**4 - 140*a**3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*
a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a
*b**12*c - b**14)))**2 - 240632*a**10*b**14*c**5*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a*
*3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a
**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14))) - 25
9282432*a**10*b**5*c**10 + 33300*a**9*b**16*c**4*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a*
*3*b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a
**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14))) + 33
9697920*a**9*b**7*c**9 - 3032*a**8*b**18*c**3*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*
b**2*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5
*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14))) - 26756
4176*a**8*b**9*c**8 + 164*a**7*b**20*c**2*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2
*c**3 + 70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**
4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14))) + 139936832
*a**7*b**11*c**7 - 4*a**6*b**22*c*(2*b/a**5 + 2*sqrt(-(4*a*c - b**2)**7)*(70*a**4*c**4 - 140*a**3*b**2*c**3 +
70*a**2*b**4*c**2 - 14*a*b**6*c + b**8)/(a**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 -
8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14))) - 50988896*a**6*b**
13*c**6 + 13213536*a**5*b**15*c**5 - 2436960*a**4*b**17*c**4 + 313664*a**3*b**19*c**3 - 26848*a**2*b**21*c**2
+ 1376*a*b**23*c - 32*b**25)/(1372000*a**12*c**13 + 33055680*a**11*b**2*c**12 - 134248800*a**10*b**4*c**11 + 2
11721440*a**9*b**6*c**10 - 187538736*a**8*b**8*c**9 + 106627392*a**7*b**10*c**8 - 41403488*a**6*b**12*c**7 + 1
1287584*a**5*b**14*c**6 - 2170560*a**4*b**16*c**5 + 289408*a**3*b**18*c**4 - 25536*a**2*b**20*c**3 + 1344*a*b*
*22*c**2 - 32*b**24*c)) - (192*a**6*c**3 - 144*a**5*b**2*c**2 + 36*a**4*b**4*c - 3*a**3*b**6 + x**6*(420*a**3*
c**6 - 456*a**2*b**2*c**5 + 132*a*b**4*c**4 - 12*b**6*c**3) + x**5*(1434*a**3*b*c**5 - 1428*a**2*b**3*c**4 + 4
02*a*b**5*c**3 - 36*b**7*c**2) + x**4*(1120*a**4*c**5 + 450*a**3*b**2*c**4 - 1156*a**2*b**4*c**3 + 378*a*b**6*
c**2 - 36*b**8*c) + x**3*(2640*a**4*b*c**4 - 1863*a**3*b**3*c**3 + 162*a**2*b**5*c**2 + 78*a*b**7*c - 12*b**9)
+ x**2*(924*a**5*c**4 + 642*a**4*b**2*c**3 - 1053*a**3*b**4*c**2 + 324*a**2*b**6*c - 30*a*b**8) + x*(1166*a**
5*b*c**3 - 967*a**4*b**3*c**2 + 255*a**3*b**5*c - 22*a**2*b**7))/(x**7*(192*a**7*c**6 - 144*a**6*b**2*c**5 + 3
6*a**5*b**4*c**4 - 3*a**4*b**6*c**3) + x**6*(576*a**7*b*c**5 - 432*a**6*b**3*c**4 + 108*a**5*b**5*c**3 - 9*a**
4*b**7*c**2) + x**5*(576*a**8*c**5 + 144*a**7*b**2*c**4 - 324*a**6*b**4*c**3 + 99*a**5*b**6*c**2 - 9*a**4*b**8
*c) + x**4*(1152*a**8*b*c**4 - 672*a**7*b**3*c**3 + 72*a**6*b**5*c**2 + 18*a**5*b**7*c - 3*a**4*b**9) + x**3*(
576*a**9*c**4 + 144*a**8*b**2*c**3 - 324*a**7*b**4*c**2 + 99*a**6*b**6*c - 9*a**5*b**8) + x**2*(576*a**9*b*c**
3 - 432*a**8*b**3*c**2 + 108*a**7*b**5*c - 9*a**6*b**7) + x*(192*a**10*c**3 - 144*a**9*b**2*c**2 + 36*a**8*b**
4*c - 3*a**7*b**6)) - 4*b*log(x)/a**5

________________________________________________________________________________________

Giac [A]  time = 1.13254, size = 670, normalized size = 1.9 \begin{align*} \frac{4 \,{\left (b^{8} - 14 \, a b^{6} c + 70 \, a^{2} b^{4} c^{2} - 140 \, a^{3} b^{2} c^{3} + 70 \, a^{4} c^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{5} b^{6} - 12 \, a^{6} b^{4} c + 48 \, a^{7} b^{2} c^{2} - 64 \, a^{8} c^{3}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{2 \, b \log \left (c x^{2} + b x + a\right )}{a^{5}} - \frac{4 \, b \log \left ({\left | x \right |}\right )}{a^{5}} - \frac{3 \, a^{4} b^{6} - 36 \, a^{5} b^{4} c + 144 \, a^{6} b^{2} c^{2} - 192 \, a^{7} c^{3} + 12 \,{\left (a b^{6} c^{3} - 11 \, a^{2} b^{4} c^{4} + 38 \, a^{3} b^{2} c^{5} - 35 \, a^{4} c^{6}\right )} x^{6} + 6 \,{\left (6 \, a b^{7} c^{2} - 67 \, a^{2} b^{5} c^{3} + 238 \, a^{3} b^{3} c^{4} - 239 \, a^{4} b c^{5}\right )} x^{5} + 2 \,{\left (18 \, a b^{8} c - 189 \, a^{2} b^{6} c^{2} + 578 \, a^{3} b^{4} c^{3} - 225 \, a^{4} b^{2} c^{4} - 560 \, a^{5} c^{5}\right )} x^{4} + 3 \,{\left (4 \, a b^{9} - 26 \, a^{2} b^{7} c - 54 \, a^{3} b^{5} c^{2} + 621 \, a^{4} b^{3} c^{3} - 880 \, a^{5} b c^{4}\right )} x^{3} + 3 \,{\left (10 \, a^{2} b^{8} - 108 \, a^{3} b^{6} c + 351 \, a^{4} b^{4} c^{2} - 214 \, a^{5} b^{2} c^{3} - 308 \, a^{6} c^{4}\right )} x^{2} +{\left (22 \, a^{3} b^{7} - 255 \, a^{4} b^{5} c + 967 \, a^{5} b^{3} c^{2} - 1166 \, a^{6} b c^{3}\right )} x}{3 \,{\left (c x^{2} + b x + a\right )}^{3}{\left (b^{2} - 4 \, a c\right )}^{3} a^{5} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*(b^8 - 14*a*b^6*c + 70*a^2*b^4*c^2 - 140*a^3*b^2*c^3 + 70*a^4*c^4)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((
a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*sqrt(-b^2 + 4*a*c)) + 2*b*log(c*x^2 + b*x + a)/a^5 - 4*b
*log(abs(x))/a^5 - 1/3*(3*a^4*b^6 - 36*a^5*b^4*c + 144*a^6*b^2*c^2 - 192*a^7*c^3 + 12*(a*b^6*c^3 - 11*a^2*b^4*
c^4 + 38*a^3*b^2*c^5 - 35*a^4*c^6)*x^6 + 6*(6*a*b^7*c^2 - 67*a^2*b^5*c^3 + 238*a^3*b^3*c^4 - 239*a^4*b*c^5)*x^
5 + 2*(18*a*b^8*c - 189*a^2*b^6*c^2 + 578*a^3*b^4*c^3 - 225*a^4*b^2*c^4 - 560*a^5*c^5)*x^4 + 3*(4*a*b^9 - 26*a
^2*b^7*c - 54*a^3*b^5*c^2 + 621*a^4*b^3*c^3 - 880*a^5*b*c^4)*x^3 + 3*(10*a^2*b^8 - 108*a^3*b^6*c + 351*a^4*b^4
*c^2 - 214*a^5*b^2*c^3 - 308*a^6*c^4)*x^2 + (22*a^3*b^7 - 255*a^4*b^5*c + 967*a^5*b^3*c^2 - 1166*a^6*b*c^3)*x)
/((c*x^2 + b*x + a)^3*(b^2 - 4*a*c)^3*a^5*x)