Integrand size = 30, antiderivative size = 436 \[ \int \frac {x^4 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {f x}{c^2}+\frac {x \left (a \left (2 c^2 d-b c e+b^2 f-2 a c f\right )-\left (b^2 c e-2 a c^2 e-b^3 f-b c (c d-3 a f)\right ) x^2\right )}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (b^2 c e-6 a c^2 e-3 b^3 f+b c (c d+13 a f)-\frac {b^3 c e-8 a b c^2 e-3 b^4 f+4 a c^2 (c d-5 a f)+b^2 c (c d+19 a f)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} c^{5/2} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (b^2 c e-6 a c^2 e-3 b^3 f+b c (c d+13 a f)+\frac {b^3 c e-8 a b c^2 e-3 b^4 f+4 a c^2 (c d-5 a f)+b^2 c (c d+19 a f)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} c^{5/2} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}} \]
f*x/c^2+1/2*x*(a*(-2*a*c*f+b^2*f-b*c*e+2*c^2*d)-(b^2*c*e-2*a*c^2*e-b^3*f-b *c*(-3*a*f+c*d))*x^2)/c^2/(-4*a*c+b^2)/(c*x^4+b*x^2+a)+1/4*arctan(x*2^(1/2 )*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*(b^2*c*e-6*a*c^2*e-3*b^3*f+b*c*(13 *a*f+c*d)+(-b^3*c*e+8*a*b*c^2*e+3*b^4*f-4*a*c^2*(-5*a*f+c*d)-b^2*c*(19*a*f +c*d))/(-4*a*c+b^2)^(1/2))/c^(5/2)/(-4*a*c+b^2)*2^(1/2)/(b-(-4*a*c+b^2)^(1 /2))^(1/2)+1/4*arctan(x*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*(b^2 *c*e-6*a*c^2*e-3*b^3*f+b*c*(13*a*f+c*d)+(b^3*c*e-8*a*b*c^2*e-3*b^4*f+4*a*c ^2*(-5*a*f+c*d)+b^2*c*(19*a*f+c*d))/(-4*a*c+b^2)^(1/2))/c^(5/2)/(-4*a*c+b^ 2)*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 0.92 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.17 \[ \int \frac {x^4 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {4 \sqrt {c} f x+\frac {2 \sqrt {c} x \left (-2 a^2 c f+b \left (c^2 d-b c e+b^2 f\right ) x^2+a \left (b^2 f+2 c^2 \left (d+e x^2\right )-b c \left (e+3 f x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\sqrt {2} \left (-3 b^4 f+2 a c^2 \left (2 c d+3 \sqrt {b^2-4 a c} e-10 a f\right )+b^2 c \left (c d-\sqrt {b^2-4 a c} e+19 a f\right )+b^3 \left (c e+3 \sqrt {b^2-4 a c} f\right )-b c \left (c \sqrt {b^2-4 a c} d+8 a c e+13 a \sqrt {b^2-4 a c} f\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (3 b^4 f+2 a c^2 \left (-2 c d+3 \sqrt {b^2-4 a c} e+10 a f\right )-b^2 c \left (c d+\sqrt {b^2-4 a c} e+19 a f\right )+b^3 \left (-c e+3 \sqrt {b^2-4 a c} f\right )-b c \left (c \sqrt {b^2-4 a c} d-8 a c e+13 a \sqrt {b^2-4 a c} f\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{4 c^{5/2}} \]
(4*Sqrt[c]*f*x + (2*Sqrt[c]*x*(-2*a^2*c*f + b*(c^2*d - b*c*e + b^2*f)*x^2 + a*(b^2*f + 2*c^2*(d + e*x^2) - b*c*(e + 3*f*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (Sqrt[2]*(-3*b^4*f + 2*a*c^2*(2*c*d + 3*Sqrt[b^2 - 4*a*c ]*e - 10*a*f) + b^2*c*(c*d - Sqrt[b^2 - 4*a*c]*e + 19*a*f) + b^3*(c*e + 3* Sqrt[b^2 - 4*a*c]*f) - b*c*(c*Sqrt[b^2 - 4*a*c]*d + 8*a*c*e + 13*a*Sqrt[b^ 2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(( b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(3*b^4*f + 2*a* c^2*(-2*c*d + 3*Sqrt[b^2 - 4*a*c]*e + 10*a*f) - b^2*c*(c*d + Sqrt[b^2 - 4* a*c]*e + 19*a*f) + b^3*(-(c*e) + 3*Sqrt[b^2 - 4*a*c]*f) - b*c*(c*Sqrt[b^2 - 4*a*c]*d - 8*a*c*e + 13*a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]* x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*c^(5/2))
Time = 3.15 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2197, 2205, 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.
| \(\displaystyle \int \frac {x^4 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2197 |
| \(\displaystyle \frac {x \left (a \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-x^2 \left (-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )\right )}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {2 a \left (4 a-\frac {b^2}{c}\right ) f x^4-\frac {a \left (-f b^3+c e b^2+c (c d+5 a f) b-6 a c^2 e\right ) x^2}{c^2}+\frac {a^2 \left (f b^2+2 c^2 d-c (b e+2 a f)\right )}{c^2}}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 2205 |
| \(\displaystyle \frac {x \left (a \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-x^2 \left (-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )\right )}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \left (\frac {a^2 \left (3 f b^2-c e b+2 c^2 d-10 a c f\right )-a \left (-3 f b^3+c e b^2+c (c d+13 a f) b-6 a c^2 e\right ) x^2}{c^2 \left (c x^4+b x^2+a\right )}-\frac {2 a \left (b^2-4 a c\right ) f}{c^2}\right )dx}{2 a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x \left (a \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-x^2 \left (-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )\right )}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {-\frac {a \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (-\frac {b^2 c (19 a f+c d)-8 a b c^2 e+4 a c^2 (c d-5 a f)-3 b^4 f+b^3 c e}{\sqrt {b^2-4 a c}}+b c (13 a f+c d)-6 a c^2 e-3 b^3 f+b^2 c e\right )}{\sqrt {2} c^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {a \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (\frac {b^2 c (19 a f+c d)-8 a b c^2 e+4 a c^2 (c d-5 a f)-3 b^4 f+b^3 c e}{\sqrt {b^2-4 a c}}+b c (13 a f+c d)-6 a c^2 e-3 b^3 f+b^2 c e\right )}{\sqrt {2} c^{5/2} \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {2 a f x \left (b^2-4 a c\right )}{c^2}}{2 a \left (b^2-4 a c\right )}\) |
(x*(a*(2*c^2*d - b*c*e + b^2*f - 2*a*c*f) - (b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*x^2))/(2*c^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((-2 *a*(b^2 - 4*a*c)*f*x)/c^2 - (a*(b^2*c*e - 6*a*c^2*e - 3*b^3*f + b*c*(c*d + 13*a*f) - (b^3*c*e - 8*a*b*c^2*e - 3*b^4*f + 4*a*c^2*(c*d - 5*a*f) + b^2* c*(c*d + 19*a*f))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - S qrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (a*(b^ 2*c*e - 6*a*c^2*e - 3*b^3*f + b*c*(c*d + 13*a*f) + (b^3*c*e - 8*a*b*c^2*e - 3*b^4*f + 4*a*c^2*(c*d - 5*a*f) + b^2*c*(c*d + 19*a*f))/Sqrt[b^2 - 4*a*c ])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(5/ 2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2*a*(b^2 - 4*a*c))
3.1.69.3.1 Defintions of rubi rules used
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x], d = Coeff[Pol ynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Polynomial Remainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4) ^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c*x ^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*Qx + b^2*d*(2*p + 3) - 2* a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; Fre eQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x^2], 1] && NeQ[b^2 - 4 *a*c, 0] && LtQ[p, -1] && IGtQ[m/2, 0]
Int[(Px_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandInte grand[Px/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x^ 2] && Expon[Px, x^2] > 1
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.56
| method | result | size |
| risch | \(\frac {f x}{c^{2}}+\frac {\frac {\left (3 a b c f -2 a \,c^{2} e -b^{3} f +b^{2} c e -b \,c^{2} d \right ) x^{3}}{8 a c -2 b^{2}}+\frac {a \left (2 a c f -b^{2} f +e b c -2 c^{2} d \right ) x}{8 a c -2 b^{2}}}{c^{2} \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (-\frac {\left (13 a b c f -6 a \,c^{2} e -3 b^{3} f +b^{2} c e +b \,c^{2} d \right ) \textit {\_R}^{2}}{4 a c -b^{2}}-\frac {a \left (10 a c f -3 b^{2} f +e b c -2 c^{2} d \right )}{4 a c -b^{2}}\right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}}{4 c^{2}}\) | \(242\) |
| default | \(\frac {f x}{c^{2}}-\frac {\frac {-\frac {\left (3 a b c f -2 a \,c^{2} e -b^{3} f +b^{2} c e -b \,c^{2} d \right ) x^{3}}{2 \left (4 a c -b^{2}\right )}-\frac {a \left (2 a c f -b^{2} f +e b c -2 c^{2} d \right ) x}{2 \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {2 c \left (\frac {\left (13 \sqrt {-4 a c +b^{2}}\, a b c f -6 \sqrt {-4 a c +b^{2}}\, a \,c^{2} e -3 b^{3} f \sqrt {-4 a c +b^{2}}+b^{2} c e \sqrt {-4 a c +b^{2}}+b \,c^{2} d \sqrt {-4 a c +b^{2}}-20 a^{2} c^{2} f +19 a \,b^{2} c f -8 a b \,c^{2} e +4 a \,c^{3} d -3 b^{4} f +b^{3} c e +b^{2} c^{2} d \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (13 \sqrt {-4 a c +b^{2}}\, a b c f -6 \sqrt {-4 a c +b^{2}}\, a \,c^{2} e -3 b^{3} f \sqrt {-4 a c +b^{2}}+b^{2} c e \sqrt {-4 a c +b^{2}}+b \,c^{2} d \sqrt {-4 a c +b^{2}}+20 a^{2} c^{2} f -19 a \,b^{2} c f +8 a b \,c^{2} e -4 a \,c^{3} d +3 b^{4} f -b^{3} c e -b^{2} c^{2} d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}}{c^{2}}\) | \(522\) |
f*x/c^2+(1/2*(3*a*b*c*f-2*a*c^2*e-b^3*f+b^2*c*e-b*c^2*d)/(4*a*c-b^2)*x^3+1 /2*a*(2*a*c*f-b^2*f+b*c*e-2*c^2*d)/(4*a*c-b^2)*x)/c^2/(c*x^4+b*x^2+a)+1/4/ c^2*sum((-(13*a*b*c*f-6*a*c^2*e-3*b^3*f+b^2*c*e+b*c^2*d)/(4*a*c-b^2)*_R^2- a*(10*a*c*f-3*b^2*f+b*c*e-2*c^2*d)/(4*a*c-b^2))/(2*_R^3*c+_R*b)*ln(x-_R),_ R=RootOf(_Z^4*c+_Z^2*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 12597 vs. \(2 (394) = 788\).
Time = 23.67 (sec) , antiderivative size = 12597, normalized size of antiderivative = 28.89 \[ \int \frac {x^4 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^4 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {x^4 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {{\left (f x^{4} + e x^{2} + d\right )} x^{4}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \]
1/2*((b*c^2*d - (b^2*c - 2*a*c^2)*e + (b^3 - 3*a*b*c)*f)*x^3 + (2*a*c^2*d - a*b*c*e + (a*b^2 - 2*a^2*c)*f)*x)/(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4* a*c^4)*x^4 + (b^3*c^2 - 4*a*b*c^3)*x^2) + f*x/c^2 + 1/2*integrate(-(2*a*c^ 2*d - a*b*c*e - (b*c^2*d + (b^2*c - 6*a*c^2)*e - (3*b^3 - 13*a*b*c)*f)*x^2 + (3*a*b^2 - 10*a^2*c)*f)/(c*x^4 + b*x^2 + a), x)/(b^2*c^2 - 4*a*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 7479 vs. \(2 (394) = 788\).
Time = 1.78 (sec) , antiderivative size = 7479, normalized size of antiderivative = 17.15 \[ \int \frac {x^4 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
f*x/c^2 + 1/2*(b*c^2*d*x^3 - b^2*c*e*x^3 + 2*a*c^2*e*x^3 + b^3*f*x^3 - 3*a *b*c*f*x^3 + 2*a*c^2*d*x - a*b*c*e*x + a*b^2*f*x - 2*a^2*c*f*x)/((c*x^4 + b*x^2 + a)*(b^2*c^2 - 4*a*c^3)) - 1/16*((2*b^3*c^4 - 8*a*b*c^5 - sqrt(2)*s qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 4*sqrt(2)*sqrt( b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^3 - sqrt(2)*sqrt(b^2 - 4*a* c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^4 - 2*(b^2 - 4*a*c)*b*c^4)*(b^2*c^2 - 4*a*c^3)^2*d + (2*b^4*c^3 - 20*a*b^2*c^4 + 48*a^2*c^5 - sqrt(2)*sqrt(b^ 2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c + 10*sqrt(2)*sqrt(b^2 - 4 *a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a *c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 - 24*sqrt(2)*sqrt(b^2 - 4*a*c) *sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^3 - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sq rt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^3 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + s qrt(b^2 - 4*a*c)*c)*a*c^4 - 2*(b^2 - 4*a*c)*b^2*c^3 + 12*(b^2 - 4*a*c)*a*c ^4)*(b^2*c^2 - 4*a*c^3)^2*e - (6*b^5*c^2 - 50*a*b^3*c^3 + 104*a^2*b*c^4 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5 + 25*sqrt( 2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 6*sqrt(2)*s qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c - 52*sqrt(2)*sqrt(b ^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - 26*sqrt(2)*sqrt...
Time = 1.56 (sec) , antiderivative size = 25862, normalized size of antiderivative = 59.32 \[ \int \frac {x^4 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
(f*x)/c^2 - atan(((((10240*a^5*c^7*f - 2048*a^4*c^8*d - 384*a^2*b^4*c^6*d + 1536*a^3*b^2*c^7*d + 192*a^2*b^5*c^5*e - 768*a^3*b^3*c^6*e - 736*a^2*b^6 *c^4*f + 4224*a^3*b^4*c^5*f - 10752*a^4*b^2*c^6*f + 32*a*b^6*c^5*d - 16*a* b^7*c^4*e + 1024*a^4*b*c^7*e + 48*a*b^8*c^3*f)/(8*(64*a^3*c^6 - b^6*c^3 + 12*a*b^4*c^4 - 48*a^2*b^2*c^5)) - (x*((768*a^4*b*c^8*d^2 - b^9*c^4*d^2 - c ^4*d^2*(-(4*a*c - b^2)^9)^(1/2) - b^11*c^2*e^2 - 9*b^4*f^2*(-(4*a*c - b^2) ^9)^(1/2) - 9*b^13*f^2 + 27*a*b^9*c^3*e^2 + 3840*a^5*b*c^7*e^2 + 9*a*c^3*e ^2*(-(4*a*c - b^2)^9)^(1/2) - 26880*a^6*b*c^6*f^2 + 6*b^12*c*e*f + 96*a^2* b^5*c^6*d^2 - 512*a^3*b^3*c^7*d^2 - 288*a^2*b^7*c^4*e^2 + 1504*a^3*b^5*c^5 *e^2 - 3840*a^4*b^3*c^6*e^2 - 2077*a^2*b^9*c^2*f^2 + 10656*a^3*b^7*c^3*f^2 - 30240*a^4*b^5*c^4*f^2 + 44800*a^5*b^3*c^5*f^2 - 25*a^2*c^2*f^2*(-(4*a*c - b^2)^9)^(1/2) - b^2*c^2*e^2*(-(4*a*c - b^2)^9)^(1/2) + 213*a*b^11*c*f^2 - 3072*a^5*c^8*d*e - 2*b^10*c^3*d*e + 15360*a^6*c^7*e*f + 6*b^11*c^2*d*f + 36*a*b^8*c^4*d*e - 98*a*b^9*c^3*d*f + 1536*a^5*b*c^7*d*f + 10*a*c^3*d*f* (-(4*a*c - b^2)^9)^(1/2) - 2*b*c^3*d*e*(-(4*a*c - b^2)^9)^(1/2) - 152*a*b^ 10*c^2*e*f + 6*b^3*c*e*f*(-(4*a*c - b^2)^9)^(1/2) + 51*a*b^2*c*f^2*(-(4*a* c - b^2)^9)^(1/2) - 192*a^2*b^6*c^5*d*e + 128*a^3*b^4*c^6*d*e + 1536*a^4*b ^2*c^7*d*e + 576*a^2*b^7*c^4*d*f - 1344*a^3*b^5*c^5*d*f + 512*a^4*b^3*c^6* d*f + 1548*a^2*b^8*c^3*e*f - 8064*a^3*b^6*c^4*e*f + 22400*a^4*b^4*c^5*e*f - 30720*a^5*b^2*c^6*e*f + 6*b^2*c^2*d*f*(-(4*a*c - b^2)^9)^(1/2) - 44*a...