Integrand size = 25, antiderivative size = 150 \[ \int \frac {A+B x^2}{x \left (a+b x^2+c x^4\right )^2} \, dx=-\frac {a b B-A \left (b^2-2 a c\right )-(A b-2 a B) c x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (4 a^2 B c+A \left (b^3-6 a b c\right )\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}+\frac {A \log (x)}{a^2}-\frac {A \log \left (a+b x^2+c x^4\right )}{4 a^2} \]
1/2*(-a*b*B+A*(-2*a*c+b^2)+(A*b-2*B*a)*c*x^2)/a/(-4*a*c+b^2)/(c*x^4+b*x^2+ a)+1/2*(4*a^2*B*c+A*(-6*a*b*c+b^3))*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2) )/a^2/(-4*a*c+b^2)^(3/2)+A*ln(x)/a^2-1/4*A*ln(c*x^4+b*x^2+a)/a^2
Time = 0.23 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.62 \[ \int \frac {A+B x^2}{x \left (a+b x^2+c x^4\right )^2} \, dx=\frac {-\frac {2 a \left (a B \left (b+2 c x^2\right )-A \left (b^2-2 a c+b c x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+4 A \log (x)-\frac {\left (4 a^2 B c+A \left (b^3-6 a b c+b^2 \sqrt {b^2-4 a c}-4 a c \sqrt {b^2-4 a c}\right )\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (4 a^2 B c+A \left (b^3-6 a b c-b^2 \sqrt {b^2-4 a c}+4 a c \sqrt {b^2-4 a c}\right )\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}}{4 a^2} \]
((-2*a*(a*B*(b + 2*c*x^2) - A*(b^2 - 2*a*c + b*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + 4*A*Log[x] - ((4*a^2*B*c + A*(b^3 - 6*a*b*c + b^2*Sqrt [b^2 - 4*a*c] - 4*a*c*Sqrt[b^2 - 4*a*c]))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c* x^2])/(b^2 - 4*a*c)^(3/2) + ((4*a^2*B*c + A*(b^3 - 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sqrt[b^2 - 4*a*c]))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2]) /(b^2 - 4*a*c)^(3/2))/(4*a^2)
Time = 0.44 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.20, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1578, 1235, 25, 1200, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {A+B x^2}{x \left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {B x^2+A}{x^2 \left (c x^4+b x^2+a\right )^2}dx^2\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {(A b-2 a B) c x^2+A \left (b^2-4 a c\right )}{x^2 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {(A b-2 a B) c x^2+A \left (b^2-4 a c\right )}{x^2 \left (c x^4+b x^2+a\right )}dx^2}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \left (\frac {-A b^3+5 a A c b-A c \left (b^2-4 a c\right ) x^2-2 a^2 B c}{a \left (c x^4+b x^2+a\right )}-\frac {A \left (4 a c-b^2\right )}{a x^2}\right )dx^2}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\left (4 a^2 B c+A \left (b^3-6 a b c\right )\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c}}+\frac {A \log \left (x^2\right ) \left (b^2-4 a c\right )}{a}-\frac {A \left (b^2-4 a c\right ) \log \left (a+b x^2+c x^4\right )}{2 a}}{a \left (b^2-4 a c\right )}+\frac {c x^2 (A b-2 a B)-2 a A c-a b B+A b^2}{a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
((A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x^2)/(a*(b^2 - 4*a*c)*(a + b*x ^2 + c*x^4)) + (((4*a^2*B*c + A*(b^3 - 6*a*b*c))*ArcTanh[(b + 2*c*x^2)/Sqr t[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]) + (A*(b^2 - 4*a*c)*Log[x^2])/a - (A *(b^2 - 4*a*c)*Log[a + b*x^2 + c*x^4])/(2*a))/(a*(b^2 - 4*a*c)))/2
3.2.16.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
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] + Simp[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] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Time = 0.12 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.41
| method | result | size |
| default | \(\frac {A \ln \left (x \right )}{a^{2}}-\frac {\frac {\frac {a c \left (A b -2 B a \right ) x^{2}}{4 a c -b^{2}}-\frac {a \left (2 A a c -A \,b^{2}+a b B \right )}{4 a c -b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {\frac {\left (4 A a \,c^{2}-A \,b^{2} c \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (5 A a b c -A \,b^{3}-2 a^{2} B c -\frac {\left (4 A a \,c^{2}-A \,b^{2} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{2 a^{2}}\) | \(212\) |
| risch | \(\frac {-\frac {c \left (A b -2 B a \right ) x^{2}}{2 a \left (4 a c -b^{2}\right )}+\frac {2 A a c -A \,b^{2}+a b B}{2 \left (4 a c -b^{2}\right ) a}}{c \,x^{4}+b \,x^{2}+a}+\frac {A \ln \left (x \right )}{a^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 a^{5} c^{3}-48 b^{2} a^{4} c^{2}+12 b^{4} a^{3} c -a^{2} b^{6}\right ) \textit {\_Z}^{2}+\left (64 c^{3} a^{3} A -48 a^{2} b^{2} c^{2} A +12 a \,b^{4} c A -b^{6} A \right ) \textit {\_Z} +16 a \,c^{3} A^{2}-3 b^{2} c^{2} A^{2}-12 A a b \,c^{2} B +2 A \,b^{3} c B +4 a^{2} c^{2} B^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-160 a^{5} c^{3}+128 b^{2} a^{4} c^{2}-34 b^{4} a^{3} c +3 a^{2} b^{6}\right ) \textit {\_R}^{2}+\left (-80 c^{3} a^{3} A +36 a^{2} b^{2} c^{2} A -4 a \,b^{4} c A +8 B \,a^{3} b \,c^{2}-2 B \,a^{2} b^{3} c \right ) \textit {\_R} -2 b^{2} c^{2} A^{2}+8 A a b \,c^{2} B -8 a^{2} c^{2} B^{2}\right ) x^{2}+\left (16 a^{5} b \,c^{2}-8 a^{4} b^{3} c +a^{3} b^{5}\right ) \textit {\_R}^{2}+\left (-36 A \,a^{3} b \,c^{2}+17 A \,a^{2} b^{3} c -2 A a \,b^{5}+8 B \,a^{4} c^{2}-2 B \,a^{3} b^{2} c \right ) \textit {\_R} +8 A^{2} a b \,c^{2}-2 A^{2} b^{3} c -16 A B \,a^{2} c^{2}+4 A B a \,b^{2} c \right )\right )}{2}\) | \(471\) |
A*ln(x)/a^2-1/2/a^2*((a*c*(A*b-2*B*a)/(4*a*c-b^2)*x^2-a*(2*A*a*c-A*b^2+B*a *b)/(4*a*c-b^2))/(c*x^4+b*x^2+a)+1/(4*a*c-b^2)*(1/2*(4*A*a*c^2-A*b^2*c)/c* ln(c*x^4+b*x^2+a)+2*(5*A*a*b*c-A*b^3-2*a^2*B*c-1/2*(4*A*a*c^2-A*b^2*c)*b/c )/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (140) = 280\).
Time = 0.60 (sec) , antiderivative size = 1014, normalized size of antiderivative = 6.76 \[ \int \frac {A+B x^2}{x \left (a+b x^2+c x^4\right )^2} \, dx=\left [-\frac {2 \, B a^{2} b^{3} - 2 \, A a b^{4} - 16 \, A a^{3} c^{2} - 2 \, {\left (4 \, {\left (2 \, B a^{3} - A a^{2} b\right )} c^{2} - {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} c\right )} x^{2} - {\left (A a b^{3} + {\left (A b^{3} c + 2 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} c^{2}\right )} x^{4} + {\left (A b^{4} + 2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{2} + 2 \, {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - 4 \, {\left (2 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} c + {\left (A a b^{4} - 8 \, A a^{2} b^{2} c + 16 \, A a^{3} c^{2} + {\left (A b^{4} c - 8 \, A a b^{2} c^{2} + 16 \, A a^{2} c^{3}\right )} x^{4} + {\left (A b^{5} - 8 \, A a b^{3} c + 16 \, A a^{2} b c^{2}\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right ) - 4 \, {\left (A a b^{4} - 8 \, A a^{2} b^{2} c + 16 \, A a^{3} c^{2} + {\left (A b^{4} c - 8 \, A a b^{2} c^{2} + 16 \, A a^{2} c^{3}\right )} x^{4} + {\left (A b^{5} - 8 \, A a b^{3} c + 16 \, A a^{2} b c^{2}\right )} x^{2}\right )} \log \left (x\right )}{4 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2} + {\left (a^{2} b^{4} c - 8 \, a^{3} b^{2} c^{2} + 16 \, a^{4} c^{3}\right )} x^{4} + {\left (a^{2} b^{5} - 8 \, a^{3} b^{3} c + 16 \, a^{4} b c^{2}\right )} x^{2}\right )}}, -\frac {2 \, B a^{2} b^{3} - 2 \, A a b^{4} - 16 \, A a^{3} c^{2} - 2 \, {\left (4 \, {\left (2 \, B a^{3} - A a^{2} b\right )} c^{2} - {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} c\right )} x^{2} - 2 \, {\left (A a b^{3} + {\left (A b^{3} c + 2 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} c^{2}\right )} x^{4} + {\left (A b^{4} + 2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{2} + 2 \, {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - 4 \, {\left (2 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} c + {\left (A a b^{4} - 8 \, A a^{2} b^{2} c + 16 \, A a^{3} c^{2} + {\left (A b^{4} c - 8 \, A a b^{2} c^{2} + 16 \, A a^{2} c^{3}\right )} x^{4} + {\left (A b^{5} - 8 \, A a b^{3} c + 16 \, A a^{2} b c^{2}\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right ) - 4 \, {\left (A a b^{4} - 8 \, A a^{2} b^{2} c + 16 \, A a^{3} c^{2} + {\left (A b^{4} c - 8 \, A a b^{2} c^{2} + 16 \, A a^{2} c^{3}\right )} x^{4} + {\left (A b^{5} - 8 \, A a b^{3} c + 16 \, A a^{2} b c^{2}\right )} x^{2}\right )} \log \left (x\right )}{4 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2} + {\left (a^{2} b^{4} c - 8 \, a^{3} b^{2} c^{2} + 16 \, a^{4} c^{3}\right )} x^{4} + {\left (a^{2} b^{5} - 8 \, a^{3} b^{3} c + 16 \, a^{4} b c^{2}\right )} x^{2}\right )}}\right ] \]
[-1/4*(2*B*a^2*b^3 - 2*A*a*b^4 - 16*A*a^3*c^2 - 2*(4*(2*B*a^3 - A*a^2*b)*c ^2 - (2*B*a^2*b^2 - A*a*b^3)*c)*x^2 - (A*a*b^3 + (A*b^3*c + 2*(2*B*a^2 - 3 *A*a*b)*c^2)*x^4 + (A*b^4 + 2*(2*B*a^2*b - 3*A*a*b^2)*c)*x^2 + 2*(2*B*a^3 - 3*A*a^2*b)*c)*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)) - 4*(2*B*a^3*b - 3*A*a^2*b^2)*c + (A*a*b^4 - 8*A*a^2*b^2*c + 16*A*a^3*c^2 + (A*b^4*c - 8*A* a*b^2*c^2 + 16*A*a^2*c^3)*x^4 + (A*b^5 - 8*A*a*b^3*c + 16*A*a^2*b*c^2)*x^2 )*log(c*x^4 + b*x^2 + a) - 4*(A*a*b^4 - 8*A*a^2*b^2*c + 16*A*a^3*c^2 + (A* b^4*c - 8*A*a*b^2*c^2 + 16*A*a^2*c^3)*x^4 + (A*b^5 - 8*A*a*b^3*c + 16*A*a^ 2*b*c^2)*x^2)*log(x))/(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 + (a^2*b^4*c - 8 *a^3*b^2*c^2 + 16*a^4*c^3)*x^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^ 2), -1/4*(2*B*a^2*b^3 - 2*A*a*b^4 - 16*A*a^3*c^2 - 2*(4*(2*B*a^3 - A*a^2*b )*c^2 - (2*B*a^2*b^2 - A*a*b^3)*c)*x^2 - 2*(A*a*b^3 + (A*b^3*c + 2*(2*B*a^ 2 - 3*A*a*b)*c^2)*x^4 + (A*b^4 + 2*(2*B*a^2*b - 3*A*a*b^2)*c)*x^2 + 2*(2*B *a^3 - 3*A*a^2*b)*c)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - 4*(2*B*a^3*b - 3*A*a^2*b^2)*c + (A*a*b^4 - 8*A*a^2 *b^2*c + 16*A*a^3*c^2 + (A*b^4*c - 8*A*a*b^2*c^2 + 16*A*a^2*c^3)*x^4 + (A* b^5 - 8*A*a*b^3*c + 16*A*a^2*b*c^2)*x^2)*log(c*x^4 + b*x^2 + a) - 4*(A*a*b ^4 - 8*A*a^2*b^2*c + 16*A*a^3*c^2 + (A*b^4*c - 8*A*a*b^2*c^2 + 16*A*a^2*c^ 3)*x^4 + (A*b^5 - 8*A*a*b^3*c + 16*A*a^2*b*c^2)*x^2)*log(x))/(a^3*b^4 -...
Timed out. \[ \int \frac {A+B x^2}{x \left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x^2}{x \left (a+b x^2+c x^4\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.61 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.34 \[ \int \frac {A+B x^2}{x \left (a+b x^2+c x^4\right )^2} \, dx=-\frac {{\left (A b^{3} + 4 \, B a^{2} c - 6 \, A a b c\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {A \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{2}} + \frac {A \log \left (x^{2}\right )}{2 \, a^{2}} + \frac {A b^{2} c x^{4} - 4 \, A a c^{2} x^{4} + A b^{3} x^{2} - 4 \, B a^{2} c x^{2} - 2 \, A a b c x^{2} - 2 \, B a^{2} b + 3 \, A a b^{2} - 8 \, A a^{2} c}{4 \, {\left (c x^{4} + b x^{2} + a\right )} {\left (a^{2} b^{2} - 4 \, a^{3} c\right )}} \]
-1/2*(A*b^3 + 4*B*a^2*c - 6*A*a*b*c)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a* c))/((a^2*b^2 - 4*a^3*c)*sqrt(-b^2 + 4*a*c)) - 1/4*A*log(c*x^4 + b*x^2 + a )/a^2 + 1/2*A*log(x^2)/a^2 + 1/4*(A*b^2*c*x^4 - 4*A*a*c^2*x^4 + A*b^3*x^2 - 4*B*a^2*c*x^2 - 2*A*a*b*c*x^2 - 2*B*a^2*b + 3*A*a*b^2 - 8*A*a^2*c)/((c*x ^4 + b*x^2 + a)*(a^2*b^2 - 4*a^3*c))
Time = 12.30 (sec) , antiderivative size = 7119, normalized size of antiderivative = 47.46 \[ \int \frac {A+B x^2}{x \left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
((2*A*a*c - A*b^2 + B*a*b)/(2*a*(4*a*c - b^2)) - (c*x^2*(A*b - 2*B*a))/(2* a*(4*a*c - b^2)))/(a + b*x^2 + c*x^4) + (A*log(x))/a^2 - (log((((A + a^2*( -(A*b^3 + 4*B*a^2*c - 6*A*a*b*c)^2/(a^4*(4*a*c - b^2)^3))^(1/2))*(((A + a^ 2*(-(A*b^3 + 4*B*a^2*c - 6*A*a*b*c)^2/(a^4*(4*a*c - b^2)^3))^(1/2))*((4*b* c^2*(A*b^3 + 2*B*a^2*c - 5*A*a*b*c))/(a*(4*a*c - b^2)) - (b*c^2*(A + a^2*( -(A*b^3 + 4*B*a^2*c - 6*A*a*b*c)^2/(a^4*(4*a*c - b^2)^3))^(1/2))*(a*b + 3* b^2*x^2 - 10*a*c*x^2))/a^2 + (2*c^3*x^2*(A*b^3 + 8*B*a*b^2 - 20*B*a^2*c - 10*A*a*b*c))/(a*(4*a*c - b^2))))/(4*a^2) + (c^3*(A*b - 2*B*a)*(4*A*b^3 + 2 *B*a^2*c - 17*A*a*b*c))/(a^2*(4*a*c - b^2)^2) - (2*c^4*x^2*(A*b - 2*B*a)*( 10*A*a*c - 3*A*b^2 + B*a*b))/(a^2*(4*a*c - b^2)^2)))/(4*a^2) + (c^5*x^2*(A *b - 2*B*a)^3)/(a^3*(4*a*c - b^2)^3) - (A*c^4*(A*b - 2*B*a)^2)/(a^3*(4*a*c - b^2)^2))*(((A - a^2*(-(A*b^3 + 4*B*a^2*c - 6*A*a*b*c)^2/(a^4*(4*a*c - b ^2)^3))^(1/2))*(((A - a^2*(-(A*b^3 + 4*B*a^2*c - 6*A*a*b*c)^2/(a^4*(4*a*c - b^2)^3))^(1/2))*((4*b*c^2*(A*b^3 + 2*B*a^2*c - 5*A*a*b*c))/(a*(4*a*c - b ^2)) - (b*c^2*(A - a^2*(-(A*b^3 + 4*B*a^2*c - 6*A*a*b*c)^2/(a^4*(4*a*c - b ^2)^3))^(1/2))*(a*b + 3*b^2*x^2 - 10*a*c*x^2))/a^2 + (2*c^3*x^2*(A*b^3 + 8 *B*a*b^2 - 20*B*a^2*c - 10*A*a*b*c))/(a*(4*a*c - b^2))))/(4*a^2) + (c^3*(A *b - 2*B*a)*(4*A*b^3 + 2*B*a^2*c - 17*A*a*b*c))/(a^2*(4*a*c - b^2)^2) - (2 *c^4*x^2*(A*b - 2*B*a)*(10*A*a*c - 3*A*b^2 + B*a*b))/(a^2*(4*a*c - b^2)^2) ))/(4*a^2) + (c^5*x^2*(A*b - 2*B*a)^3)/(a^3*(4*a*c - b^2)^3) - (A*c^4*(...