Integrand size = 30, antiderivative size = 362 \[ \int \frac {x^2 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=-\frac {x \left (b c d-2 a c e+a b f+\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (2 c d-b e+6 a f-\frac {b^2 f}{c}+\frac {b^2 c e+4 a c^2 e+b^3 f-4 b c (c d+2 a f)}{c \sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (2 c d-b e+6 a f-\frac {b^2 f}{c}-\frac {b^2 c e+4 a c^2 e+b^3 f-4 b c (c d+2 a f)}{c \sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}} \]
-1/2*x*(b*c*d-2*a*c*e+a*b*f+(-2*a*c*f+b^2*f-b*c*e+2*c^2*d)*x^2)/c/(-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))*(2*c*d-b*e+6*a*f-b^2*f/c+(b^2*c*e+4*a*c^2*e+b^3*f-4*b*c*(2*a*f+c*d)) /c/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)*2^(1/2)/c^(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))*(2*c*d-b *e+6*a*f-b^2*f/c+(-b^2*c*e-4*a*c^2*e-b^3*f+4*b*c*(2*a*f+c*d))/c/(-4*a*c+b^ 2)^(1/2))/(-4*a*c+b^2)*2^(1/2)/c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 0.66 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.14 \[ \int \frac {x^2 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {-\frac {2 \sqrt {c} x \left (a b f+2 c^2 d x^2+b^2 f x^2+b c \left (d-e x^2\right )-2 a c \left (e+f x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \left (-b^3 f+b c \left (4 c d+\sqrt {b^2-4 a c} e+8 a f\right )+b^2 \left (-c e+\sqrt {b^2-4 a c} f\right )-2 c \left (c \sqrt {b^2-4 a c} d+2 a c e+3 a \sqrt {b^2-4 a c} f\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (b^3 f+b c \left (-4 c d+\sqrt {b^2-4 a c} e-8 a f\right )+b^2 \left (c e+\sqrt {b^2-4 a c} f\right )-2 c \left (c \sqrt {b^2-4 a c} d-2 a c e+3 a \sqrt {b^2-4 a c} f\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{4 c^{3/2}} \]
((-2*Sqrt[c]*x*(a*b*f + 2*c^2*d*x^2 + b^2*f*x^2 + b*c*(d - e*x^2) - 2*a*c* (e + f*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-(b^3*f) + b *c*(4*c*d + Sqrt[b^2 - 4*a*c]*e + 8*a*f) + b^2*(-(c*e) + Sqrt[b^2 - 4*a*c] *f) - 2*c*(c*Sqrt[b^2 - 4*a*c]*d + 2*a*c*e + 3*a*Sqrt[b^2 - 4*a*c]*f))*Arc Tan[(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]*(b^3*f + b*c*(-4*c*d + Sqrt[b^2 - 4*a*c]*e - 8*a*f) + b^2*(c*e + Sqrt[b^2 - 4*a*c]*f) - 2*c*(c*Sqrt[b^2 - 4 *a*c]*d - 2*a*c*e + 3*a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/S qrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a* c]]))/(4*c^(3/2))
Time = 0.74 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2197, 25, 27, 1480, 218}
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.
| \(\displaystyle \int \frac {x^2 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2197 |
| \(\displaystyle -\frac {\int -\frac {a \left (-c \left (-\frac {f b^2}{c}-e b+2 c d+6 a f\right ) x^2+b c d-2 a c e+a b f\right )}{c \left (c x^4+b x^2+a\right )}dx}{2 a \left (b^2-4 a c\right )}-\frac {x \left (x^2 \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )+a b f-2 a c e+b c d\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {a \left (-\left (\left (-f b^2-c e b+2 c^2 d+6 a c f\right ) x^2\right )+b c d-2 a c e+a b f\right )}{c \left (c x^4+b x^2+a\right )}dx}{2 a \left (b^2-4 a c\right )}-\frac {x \left (x^2 \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )+a b f-2 a c e+b c d\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-\left (\left (-f b^2-c e b+2 c^2 d+6 a c f\right ) x^2\right )+b c d-2 a c e+a b f}{c x^4+b x^2+a}dx}{2 c \left (b^2-4 a c\right )}-\frac {x \left (x^2 \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )+a b f-2 a c e+b c d\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {-\frac {1}{2} \left (\frac {-4 b c (2 a f+c d)+4 a c^2 e+b^3 f+b^2 c e}{\sqrt {b^2-4 a c}}+6 a c f+b^2 (-f)-b c e+2 c^2 d\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx-\frac {1}{2} \left (-\frac {-4 b c (2 a f+c d)+4 a c^2 e+b^3 f+b^2 c e}{\sqrt {b^2-4 a c}}+6 a c f+b^2 (-f)-b c e+2 c^2 d\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{2 c \left (b^2-4 a c\right )}-\frac {x \left (x^2 \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )+a b f-2 a c e+b c d\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {-4 b c (2 a f+c d)+4 a c^2 e+b^3 f+b^2 c e}{\sqrt {b^2-4 a c}}+6 a c f+b^2 (-f)-b c e+2 c^2 d\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {-4 b c (2 a f+c d)+4 a c^2 e+b^3 f+b^2 c e}{\sqrt {b^2-4 a c}}+6 a c f+b^2 (-f)-b c e+2 c^2 d\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 c \left (b^2-4 a c\right )}-\frac {x \left (x^2 \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )+a b f-2 a c e+b c d\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
-1/2*(x*(b*c*d - 2*a*c*e + a*b*f + (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x^2 ))/(c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (-(((2*c^2*d - b*c*e - b^2*f + 6*a*c*f + (b^2*c*e + 4*a*c^2*e + b^3*f - 4*b*c*(c*d + 2*a*f))/Sqrt[b^2 - 4 *a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*S qrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])) - ((2*c^2*d - b*c*e - b^2*f + 6*a*c*f - (b^2*c*e + 4*a*c^2*e + b^3*f - 4*b*c*(c*d + 2*a*f))/Sqrt[b^2 - 4*a*c])* ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]* Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2*c*(b^2 - 4*a*c))
3.1.70.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.14 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.55
| method | result | size |
| risch | \(\frac {-\frac {\left (2 a c f -b^{2} f +e b c -2 c^{2} d \right ) x^{3}}{2 c \left (4 a c -b^{2}\right )}+\frac {\left (a b f -2 a c e +b c d \right ) x}{2 c \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\frac {\left (6 a c f -b^{2} f -e b c +2 c^{2} d \right ) \textit {\_R}^{2}}{4 a c -b^{2}}-\frac {a b f -2 a c e +b c d}{4 a c -b^{2}}\right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}}{4 c}\) | \(200\) |
| default | \(\frac {-\frac {\left (2 a c f -b^{2} f +e b c -2 c^{2} d \right ) x^{3}}{2 c \left (4 a c -b^{2}\right )}+\frac {\left (a b f -2 a c e +b c d \right ) x}{2 c \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {\frac {\left (6 a c f \sqrt {-4 a c +b^{2}}-b^{2} f \sqrt {-4 a c +b^{2}}-e b c \sqrt {-4 a c +b^{2}}+2 c^{2} d \sqrt {-4 a c +b^{2}}+8 a b c f -4 a \,c^{2} e -b^{3} f -b^{2} c e +4 b \,c^{2} d \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (6 a c f \sqrt {-4 a c +b^{2}}-b^{2} f \sqrt {-4 a c +b^{2}}-e b c \sqrt {-4 a c +b^{2}}+2 c^{2} d \sqrt {-4 a c +b^{2}}-8 a b c f +4 a \,c^{2} e +b^{3} f +b^{2} c e -4 b \,c^{2} d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{4 a c -b^{2}}\) | \(415\) |
(-1/2*(2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c/(4*a*c-b^2)*x^3+1/2/c*(a*b*f-2*a*c*e +b*c*d)/(4*a*c-b^2)*x)/(c*x^4+b*x^2+a)+1/4/c*sum(((6*a*c*f-b^2*f-b*c*e+2*c ^2*d)/(4*a*c-b^2)*_R^2-(a*b*f-2*a*c*e+b*c*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. 8951 vs. \(2 (320) = 640\).
Time = 13.28 (sec) , antiderivative size = 8951, normalized size of antiderivative = 24.73 \[ \int \frac {x^2 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^2 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {{\left (f x^{4} + e x^{2} + d\right )} x^{2}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \]
-1/2*((2*c^2*d - b*c*e + (b^2 - 2*a*c)*f)*x^3 + (b*c*d - 2*a*c*e + a*b*f)* x)/((b^2*c^2 - 4*a*c^3)*x^4 + a*b^2*c - 4*a^2*c^2 + (b^3*c - 4*a*b*c^2)*x^ 2) - 1/2*integrate(-(b*c*d - 2*a*c*e + a*b*f - (2*c^2*d - b*c*e - (b^2 - 6 *a*c)*f)*x^2)/(c*x^4 + b*x^2 + a), x)/(b^2*c - 4*a*c^2)
Leaf count of result is larger than twice the leaf count of optimal. 6200 vs. \(2 (320) = 640\).
Time = 1.52 (sec) , antiderivative size = 6200, normalized size of antiderivative = 17.13 \[ \int \frac {x^2 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
-1/2*(2*c^2*d*x^3 - b*c*e*x^3 + b^2*f*x^3 - 2*a*c*f*x^3 + b*c*d*x - 2*a*c* e*x + a*b*f*x)/((c*x^4 + b*x^2 + a)*(b^2*c - 4*a*c^2)) - 1/16*(2*(2*b^2*c^ 4 - 8*a*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^ 2*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^3 - sqrt (2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*c^4 - 2*(b^2 - 4*a*c )*c^4)*(b^2*c - 4*a*c^2)^2*d - (2*b^3*c^3 - 8*a*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c )*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sq rt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^3 - 2*(b^2 - 4*a*c)*b*c^3)*(b^2*c - 4*a*c^2)^2 *e - (2*b^4*c^2 - 20*a*b^2*c^3 + 48*a^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sq rt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq rt(b^2 - 4*a*c)*c)*b^3*c - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^ 2 - 4*a*c)*c)*a^2*c^2 - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c )*c)*b^2*c^2 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c) *a*c^3 - 2*(b^2 - 4*a*c)*b^2*c^2 + 12*(b^2 - 4*a*c)*a*c^3)*(b^2*c - 4*a*c^ 2)^2*f - 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c^3 - 8*sqrt(2)...
Time = 12.24 (sec) , antiderivative size = 19494, normalized size of antiderivative = 53.85 \[ \int \frac {x^2 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
((x^3*(2*c^2*d + b^2*f - 2*a*c*f - b*c*e))/(2*c*(4*a*c - b^2)) + (x*(a*b*f - 2*a*c*e + b*c*d))/(2*c*(4*a*c - b^2)))/(a + b*x^2 + c*x^4) - atan(((((2 048*a^4*c^6*e + 16*b^7*c^3*d + 768*a^2*b^3*c^5*d + 384*a^2*b^4*c^4*e - 153 6*a^3*b^2*c^5*e - 192*a^2*b^5*c^3*f + 768*a^3*b^3*c^4*f - 192*a*b^5*c^4*d - 1024*a^3*b*c^6*d - 32*a*b^6*c^3*e + 16*a*b^7*c^2*f - 1024*a^4*b*c^5*f)/( 8*(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3)) - (x*((768*a^4*b*c ^7*d^2 - b^9*c^3*d^2 - c^3*d^2*(-(4*a*c - b^2)^9)^(1/2) - a*b^11*f^2 - a*b ^9*c^2*e^2 + 768*a^5*b*c^6*e^2 + a*b^2*f^2*(-(4*a*c - b^2)^9)^(1/2) + a*c^ 2*e^2*(-(4*a*c - b^2)^9)^(1/2) + 27*a^2*b^9*c*f^2 + 3840*a^6*b*c^5*f^2 - 9 *a^2*c*f^2*(-(4*a*c - b^2)^9)^(1/2) + 96*a^2*b^5*c^5*d^2 - 512*a^3*b^3*c^6 *d^2 + 96*a^3*b^5*c^4*e^2 - 512*a^4*b^3*c^5*e^2 - 288*a^3*b^7*c^2*f^2 + 15 04*a^4*b^5*c^3*f^2 - 3840*a^5*b^3*c^4*f^2 - 1024*a^5*c^7*d*e - 3072*a^6*c^ 6*e*f + 12*a*b^8*c^3*d*e + 6*a*b^9*c^2*d*f + 3584*a^5*b*c^6*d*f - 6*a*c^2* d*f*(-(4*a*c - b^2)^9)^(1/2) - 128*a^2*b^6*c^4*d*e + 384*a^3*b^4*c^5*d*e - 128*a^2*b^7*c^3*d*f + 960*a^3*b^5*c^4*d*f - 3072*a^4*b^3*c^5*d*f + 36*a^2 *b^8*c^2*e*f - 192*a^3*b^6*c^3*e*f + 128*a^4*b^4*c^4*e*f + 1536*a^5*b^2*c^ 5*e*f - 2*a*b^10*c*e*f + 2*a*b*c*e*f*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a ^7*c^9 + a*b^12*c^3 - 24*a^2*b^10*c^4 + 240*a^3*b^8*c^5 - 1280*a^4*b^6*c^6 + 3840*a^5*b^4*c^7 - 6144*a^6*b^2*c^8)))^(1/2)*(16*b^7*c^3 - 192*a*b^5*c^ 4 - 1024*a^3*b*c^6 + 768*a^2*b^3*c^5))/(2*(b^4*c + 16*a^2*c^3 - 8*a*b^2...