\(\int \frac {1}{x^2 (a+b x+c x^2)^4} \, dx\) [2220]

Optimal result

Integrand size = 16, antiderivative size = 352 \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx=-\frac {4 \left (b^6-11 a b^4 c+38 a^2 b^2 c^2-35 a^3 c^3\right )}{a^4 \left (b^2-4 a c\right )^3 x}+\frac {b^2-2 a c+b c x}{3 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^3}+\frac {2 \left (b^4-7 a b^2 c+7 a^2 c^2\right )+b c \left (2 b^2-13 a c\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )^2}+\frac {2 \left (3 b^6-32 a b^4 c+105 a^2 b^2 c^2-70 a^3 c^3+3 b c \left (b^4-10 a b^2 c+29 a^2 c^2\right ) x\right )}{3 a^3 \left (b^2-4 a c\right )^3 x \left (a+b x+c x^2\right )}-\frac {4 \left (b^8-14 a b^6 c+70 a^2 b^4 c^2-140 a^3 b^2 c^3+70 a^4 c^4\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{7/2}}-\frac {4 b \log (x)}{a^5}+\frac {2 b \log \left (a+b x+c x^2\right )}{a^5} \]

[Out]

-4*(-35*a^3*c^3+38*a^2*b^2*c^2-11*a*b^4*c+b^6)/a^4/(-4*a*c+b^2)^3/x+1/3*(b*c*x-2*a*c+b^2)/a/(-4*a*c+b^2)/x/(c*
x^2+b*x+a)^3+1/3*(14*a^2*c^2-14*a*b^2*c+2*b^4+b*c*(-13*a*c+2*b^2)*x)/a^2/(-4*a*c+b^2)^2/x/(c*x^2+b*x+a)^2+2/3*
(3*b^6-32*a*b^4*c+105*a^2*b^2*c^2-70*a^3*c^3+3*b*c*(29*a^2*c^2-10*a*b^2*c+b^4)*x)/a^3/(-4*a*c+b^2)^3/x/(c*x^2+
b*x+a)-4*(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)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/a^5/
(-4*a*c+b^2)^(7/2)-4*b*ln(x)/a^5+2*b*ln(c*x^2+b*x+a)/a^5

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {754, 836, 814, 648, 632, 212, 642} \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx=\frac {2 b \log \left (a+b x+c x^2\right )}{a^5}-\frac {4 b \log (x)}{a^5}+\frac {2 \left (7 a^2 c^2-7 a b^2 c+b^4\right )+b c x \left (2 b^2-13 a c\right )}{3 a^2 x \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {2 \left (-70 a^3 c^3+105 a^2 b^2 c^2+3 b c x \left (29 a^2 c^2-10 a b^2 c+b^4\right )-32 a b^4 c+3 b^6\right )}{3 a^3 x \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {4 \left (-35 a^3 c^3+38 a^2 b^2 c^2-11 a b^4 c+b^6\right )}{a^4 x \left (b^2-4 a c\right )^3}-\frac {4 \left (70 a^4 c^4-140 a^3 b^2 c^3+70 a^2 b^4 c^2-14 a b^6 c+b^8\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{7/2}}+\frac {-2 a c+b^2+b c x}{3 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3} \]

[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 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])

Rubi steps \begin{align*} \text {integral}& = \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 (\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 )\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 (\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 )\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 ) \text {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] (verified)

Time = 0.46 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx=\frac {-\frac {3 a}{x}+\frac {a^3 \left (b^3-3 a b c+b^2 c x-2 a c^2 x\right )}{\left (-b^2+4 a c\right ) (a+x (b+c x))^3}-\frac {a^2 \left (3 b^5-22 a b^3 c+35 a^2 b c^2+3 b^4 c x-20 a b^2 c^2 x+22 a^2 c^3 x\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+\frac {3 a \left (-3 b^7+34 a b^5 c-124 a^2 b^3 c^2+134 a^3 b c^3-3 b^6 c x+32 a b^4 c^2 x-104 a^2 b^2 c^3 x+76 a^3 c^4 x\right )}{\left (b^2-4 a c\right )^3 (a+x (b+c x))}-\frac {12 \left (b^8-14 a b^6 c+70 a^2 b^4 c^2-140 a^3 b^2 c^3+70 a^4 c^4\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{7/2}}-12 b \log (x)+6 b \log (a+x (b+c x))}{3 a^5} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(711\) vs. \(2(341)=682\).

Time = 17.21 (sec) , antiderivative size = 712, normalized size of antiderivative = 2.02

method	result	size
default	\(-\frac {1}{a^{4} x}-\frac {4 b \ln \left (x \right )}{a^{5}}-\frac {\frac {\frac {c^{3} a \left (76 c^{3} a^{3}-104 a^{2} b^{2} c^{2}+32 a \,b^{4} c -3 b^{6}\right ) x^{5}}{64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {b \,c^{2} a \left (286 c^{3} a^{3}-332 a^{2} b^{2} c^{2}+98 a \,b^{4} c -9 b^{6}\right ) x^{4}}{64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {a c \left (544 a^{4} c^{4}+306 a^{3} b^{2} c^{3}-832 a^{2} b^{4} c^{2}+279 a \,b^{6} c -27 b^{8}\right ) x^{3}}{192 c^{3} a^{3}-144 a^{2} b^{2} c^{2}+36 a \,b^{4} c -3 b^{6}}+\frac {b a \left (496 a^{4} c^{4}-397 a^{3} b^{2} c^{3}+30 a^{2} b^{4} c^{2}+20 a \,b^{6} c -3 b^{8}\right ) x^{2}}{64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {a^{2} \left (116 a^{4} c^{4}+166 a^{3} b^{2} c^{3}-243 a^{2} b^{4} c^{2}+75 a \,b^{6} c -7 b^{8}\right ) x}{64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {\left (590 c^{3} a^{3}-535 a^{2} b^{2} c^{2}+147 a \,b^{4} c -13 b^{6}\right ) a^{3} b}{192 c^{3} a^{3}-144 a^{2} b^{2} c^{2}+36 a \,b^{4} c -3 b^{6}}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {\frac {2 \left (-64 a^{3} b \,c^{4}+48 a^{2} b^{3} c^{3}-12 a \,b^{5} c^{2}+b^{7} c \right ) \ln \left (c \,x^{2}+b x +a \right )}{c}+\frac {8 \left (35 a^{4} c^{4}-102 a^{3} b^{2} c^{3}+59 a^{2} b^{4} c^{2}-13 a \,b^{6} c +b^{8}-\frac {\left (-64 a^{3} b \,c^{4}+48 a^{2} b^{3} c^{3}-12 a \,b^{5} c^{2}+b^{7} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}}{a^{5}}\)	\(712\)
risch	\(\text {Expression too large to display}\)	\(1183\)

[In]

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

[Out]

-1/a^4/x-4*b*ln(x)/a^5-1/a^5*((c^3*a*(76*a^3*c^3-104*a^2*b^2*c^2+32*a*b^4*c-3*b^6)/(64*a^3*c^3-48*a^2*b^2*c^2+
12*a*b^4*c-b^6)*x^5+b*c^2*a*(286*a^3*c^3-332*a^2*b^2*c^2+98*a*b^4*c-9*b^6)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4
*c-b^6)*x^4+1/3*a*c*(544*a^4*c^4+306*a^3*b^2*c^3-832*a^2*b^4*c^2+279*a*b^6*c-27*b^8)/(64*a^3*c^3-48*a^2*b^2*c^
2+12*a*b^4*c-b^6)*x^3+b*a*(496*a^4*c^4-397*a^3*b^2*c^3+30*a^2*b^4*c^2+20*a*b^6*c-3*b^8)/(64*a^3*c^3-48*a^2*b^2
*c^2+12*a*b^4*c-b^6)*x^2+a^2*(116*a^4*c^4+166*a^3*b^2*c^3-243*a^2*b^4*c^2+75*a*b^6*c-7*b^8)/(64*a^3*c^3-48*a^2
*b^2*c^2+12*a*b^4*c-b^6)*x+1/3*(590*a^3*c^3-535*a^2*b^2*c^2+147*a*b^4*c-13*b^6)*a^3*b/(64*a^3*c^3-48*a^2*b^2*c
^2+12*a*b^4*c-b^6))/(c*x^2+b*x+a)^3+4/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*(1/2*(-64*a^3*b*c^4+48*a^2*b^
3*c^3-12*a*b^5*c^2+b^7*c)/c*ln(c*x^2+b*x+a)+2*(35*a^4*c^4-102*a^3*b^2*c^3+59*a^2*b^4*c^2-13*a*b^6*c+b^8-1/2*(-
64*a^3*b*c^4+48*a^2*b^3*c^3-12*a*b^5*c^2+b^7*c)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1957 vs. \(2 (341) = 682\).

Time = 1.78 (sec) , antiderivative size = 3934, normalized size of antiderivative = 11.18 \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \]

[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 [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/x^2/(c*x^2+b*x+a)^4,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 deta

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.41 \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx=\frac {4 \, {\left (b^{8} - 14 \, a b^{6} c + 70 \, a^{2} b^{4} c^{2} - 140 \, a^{3} b^{2} c^{3} + 70 \, a^{4} c^{4}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{5} b^{6} - 12 \, a^{6} b^{4} c + 48 \, a^{7} b^{2} c^{2} - 64 \, a^{8} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {2 \, b \log \left (c x^{2} + b x + a\right )}{a^{5}} - \frac {4 \, b \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {3 \, a^{4} b^{6} - 36 \, a^{5} b^{4} c + 144 \, a^{6} b^{2} c^{2} - 192 \, a^{7} c^{3} + 12 \, {\left (a b^{6} c^{3} - 11 \, a^{2} b^{4} c^{4} + 38 \, a^{3} b^{2} c^{5} - 35 \, a^{4} c^{6}\right )} x^{6} + 6 \, {\left (6 \, a b^{7} c^{2} - 67 \, a^{2} b^{5} c^{3} + 238 \, a^{3} b^{3} c^{4} - 239 \, a^{4} b c^{5}\right )} x^{5} + 2 \, {\left (18 \, a b^{8} c - 189 \, a^{2} b^{6} c^{2} + 578 \, a^{3} b^{4} c^{3} - 225 \, a^{4} b^{2} c^{4} - 560 \, a^{5} c^{5}\right )} x^{4} + 3 \, {\left (4 \, a b^{9} - 26 \, a^{2} b^{7} c - 54 \, a^{3} b^{5} c^{2} + 621 \, a^{4} b^{3} c^{3} - 880 \, a^{5} b c^{4}\right )} x^{3} + 3 \, {\left (10 \, a^{2} b^{8} - 108 \, a^{3} b^{6} c + 351 \, a^{4} b^{4} c^{2} - 214 \, a^{5} b^{2} c^{3} - 308 \, a^{6} c^{4}\right )} x^{2} + {\left (22 \, a^{3} b^{7} - 255 \, a^{4} b^{5} c + 967 \, a^{5} b^{3} c^{2} - 1166 \, a^{6} b c^{3}\right )} x}{3 \, {\left (c x^{2} + b x + a\right )}^{3} {\left (b^{2} - 4 \, a c\right )}^{3} a^{5} x} \]

[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)

Mupad [B] (verification not implemented)

Time = 11.80 (sec) , antiderivative size = 1856, normalized size of antiderivative = 5.27 \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \]

[In]

int(1/(x^2*(a + b*x + c*x^2)^4),x)

[Out]

(2*log(2*a*b^8*(-(4*a*c - b^2)^7)^(1/2) - 2*b^16*x - 2*a*b^15 + 55*a^2*b^13*c + 26816*a^8*b*c^7 - 4480*a^8*c^8
*x + 2*b^9*x*(-(4*a*c - b^2)^7)^(1/2) - 647*a^3*b^11*c^2 + 4218*a^4*b^9*c^3 - 16443*a^5*b^7*c^4 + 38276*a^6*b^
5*c^5 - 49168*a^7*b^3*c^6 + 35*a^5*c^4*(-(4*a*c - b^2)^7)^(1/2) - 25*a^2*b^6*c*(-(4*a*c - b^2)^7)^(1/2) - 673*
a^2*b^12*c^2*x + 4504*a^3*b^10*c^3*x - 18159*a^4*b^8*c^4*x + 44282*a^5*b^6*c^5*x - 61208*a^6*b^4*c^6*x + 39136
*a^7*b^2*c^7*x + 56*a*b^14*c*x + 107*a^3*b^4*c^2*(-(4*a*c - b^2)^7)^(1/2) - 166*a^4*b^2*c^3*(-(4*a*c - b^2)^7)
^(1/2) - 28*a*b^7*c*x*(-(4*a*c - b^2)^7)^(1/2) + 227*a^4*b*c^4*x*(-(4*a*c - b^2)^7)^(1/2) + 143*a^2*b^5*c^2*x*
(-(4*a*c - b^2)^7)^(1/2) - 310*a^3*b^3*c^3*x*(-(4*a*c - b^2)^7)^(1/2))*(b^8*(-(4*a*c - b^2)^7)^(1/2) - b^15 +
16384*a^7*b*c^7 - 336*a^2*b^11*c^2 + 2240*a^3*b^9*c^3 - 8960*a^4*b^7*c^4 + 21504*a^5*b^5*c^5 - 28672*a^6*b^3*c
^6 + 70*a^4*c^4*(-(4*a*c - b^2)^7)^(1/2) + 28*a*b^13*c + 70*a^2*b^4*c^2*(-(4*a*c - b^2)^7)^(1/2) - 140*a^3*b^2
*c^3*(-(4*a*c - b^2)^7)^(1/2) - 14*a*b^6*c*(-(4*a*c - b^2)^7)^(1/2)))/(a^5*(4*a*c - b^2)^7) - (4*b*log(x))/a^5
- (2*log(2*a*b^15 + 2*b^16*x + 2*a*b^8*(-(4*a*c - b^2)^7)^(1/2) - 55*a^2*b^13*c - 26816*a^8*b*c^7 + 4480*a^8*
c^8*x + 2*b^9*x*(-(4*a*c - b^2)^7)^(1/2) + 647*a^3*b^11*c^2 - 4218*a^4*b^9*c^3 + 16443*a^5*b^7*c^4 - 38276*a^6
*b^5*c^5 + 49168*a^7*b^3*c^6 + 35*a^5*c^4*(-(4*a*c - b^2)^7)^(1/2) - 25*a^2*b^6*c*(-(4*a*c - b^2)^7)^(1/2) + 6
73*a^2*b^12*c^2*x - 4504*a^3*b^10*c^3*x + 18159*a^4*b^8*c^4*x - 44282*a^5*b^6*c^5*x + 61208*a^6*b^4*c^6*x - 39
136*a^7*b^2*c^7*x - 56*a*b^14*c*x + 107*a^3*b^4*c^2*(-(4*a*c - b^2)^7)^(1/2) - 166*a^4*b^2*c^3*(-(4*a*c - b^2)
^7)^(1/2) - 28*a*b^7*c*x*(-(4*a*c - b^2)^7)^(1/2) + 227*a^4*b*c^4*x*(-(4*a*c - b^2)^7)^(1/2) + 143*a^2*b^5*c^2
*x*(-(4*a*c - b^2)^7)^(1/2) - 310*a^3*b^3*c^3*x*(-(4*a*c - b^2)^7)^(1/2))*(b^15 + b^8*(-(4*a*c - b^2)^7)^(1/2)
- 16384*a^7*b*c^7 + 336*a^2*b^11*c^2 - 2240*a^3*b^9*c^3 + 8960*a^4*b^7*c^4 - 21504*a^5*b^5*c^5 + 28672*a^6*b^
3*c^6 + 70*a^4*c^4*(-(4*a*c - b^2)^7)^(1/2) - 28*a*b^13*c + 70*a^2*b^4*c^2*(-(4*a*c - b^2)^7)^(1/2) - 140*a^3*
b^2*c^3*(-(4*a*c - b^2)^7)^(1/2) - 14*a*b^6*c*(-(4*a*c - b^2)^7)^(1/2)))/(a^5*(4*a*c - b^2)^7) - (1/a - (2*x^4
*(560*a^4*c^5 - 18*b^8*c + 189*a*b^6*c^2 - 578*a^2*b^4*c^3 + 225*a^3*b^2*c^4))/(3*a^4*(b^6 - 64*a^3*c^3 + 48*a
^2*b^2*c^2 - 12*a*b^4*c)) + (2*x^5*(6*b^7*c^2 - 67*a*b^5*c^3 - 239*a^3*b*c^5 + 238*a^2*b^3*c^4))/(a^4*(b^6 - 6
4*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - (x^2*(308*a^4*c^4 - 10*b^8 - 351*a^2*b^4*c^2 + 214*a^3*b^2*c^3 + 1
08*a*b^6*c))/(a^3*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (x*(22*b^7 - 1166*a^3*b*c^3 + 967*a^2*b^
3*c^2 - 255*a*b^5*c))/(3*a^2*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - (x^3*(880*a^4*b*c^4 - 4*b^9 +
54*a^2*b^5*c^2 - 621*a^3*b^3*c^3 + 26*a*b^7*c))/(a^4*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (4*c
^3*x^6*(b^6 - 35*a^3*c^3 + 38*a^2*b^2*c^2 - 11*a*b^4*c))/(a^4*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)
))/(a^3*x + x^3*(3*a*b^2 + 3*a^2*c) + x^5*(3*a*c^2 + 3*b^2*c) + x^4*(b^3 + 6*a*b*c) + c^3*x^7 + 3*a^2*b*x^2 +
3*b*c^2*x^6)