Integrand size = 24, antiderivative size = 386 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {x \left (b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )+\left (b c d^2-4 a c d e+a b e^2\right ) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (b c d^2-4 a c d e+a b e^2+\frac {8 a b c d e+b^2 \left (c d^2-a e^2\right )-4 a c \left (3 c d^2+a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (b c d^2-4 a c d e+a b e^2-\frac {8 a b c d e+b^2 \left (c d^2-a e^2\right )-4 a c \left (3 c d^2+a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}} \]
1/2*x*(b^2*d^2-2*a*b*d*e-2*a*(-a*e^2+c*d^2)+(a*b*e^2-4*a*c*d*e+b*c*d^2)*x^ 2)/a/(-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*c*d^2-4*a*c*d*e+a*b*e^2+(8*a*b*c*d*e+b^2*(-a*e^2+c*d ^2)-4*a*c*(a*e^2+3*c*d^2))/(-4*a*c+b^2)^(1/2))/a/(-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))*(b*c*d^2-4*a*c*d*e+a*b*e^2+(-8*a*b*c*d*e-b^2*(-a*e^2+c*d ^2)+4*a*c*(a*e^2+3*c*d^2))/(-4*a*c+b^2)^(1/2))/a/(-4*a*c+b^2)*2^(1/2)/c^(1 /2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 0.68 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.08 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {\frac {2 x \left (b^2 d^2+2 a^2 e^2+b c d^2 x^2+a b e \left (-2 d+e x^2\right )-2 a c d \left (d+2 e x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \left (b^2 \left (c d^2-a e^2\right )-4 a c \left (3 c d^2+e \left (\sqrt {b^2-4 a c} d+a e\right )\right )+b \left (a \sqrt {b^2-4 a c} e^2+c d \left (\sqrt {b^2-4 a c} d+8 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (b^2 \left (-c d^2+a e^2\right )+b \left (a \sqrt {b^2-4 a c} e^2+c d \left (\sqrt {b^2-4 a c} d-8 a e\right )\right )+4 a c \left (3 c d^2+e \left (-\sqrt {b^2-4 a c} d+a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{4 a} \]
((2*x*(b^2*d^2 + 2*a^2*e^2 + b*c*d^2*x^2 + a*b*e*(-2*d + e*x^2) - 2*a*c*d* (d + 2*e*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(b^2*(c*d^2 - a*e^2) - 4*a*c*(3*c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d + a*e)) + b*(a*Sqrt[b^ 2 - 4*a*c]*e^2 + c*d*(Sqrt[b^2 - 4*a*c]*d + 8*a*e)))*ArcTan[(Sqrt[2]*Sqrt[ c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(b^2*(-(c*d^2) + a*e^2) + b*(a*Sqrt[b^2 - 4 *a*c]*e^2 + c*d*(Sqrt[b^2 - 4*a*c]*d - 8*a*e)) + 4*a*c*(3*c*d^2 + e*(-(Sqr t[b^2 - 4*a*c]*d) + a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*a)
Time = 0.68 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1517, 25, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle \frac {x \left (x^2 \left (a b e^2-4 a c d e+b c d^2\right )-2 a b d e-2 a \left (c d^2-a e^2\right )+b^2 d^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {b^2 d^2+2 a b e d+\left (b c d^2-4 a c e d+a b e^2\right ) x^2-2 a \left (3 c d^2+a e^2\right )}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {b^2 d^2+2 a b e d+\left (b c d^2-4 a c e d+a b e^2\right ) x^2-2 a \left (3 c d^2+a e^2\right )}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (x^2 \left (a b e^2-4 a c d e+b c d^2\right )-2 a b d e-2 a \left (c d^2-a e^2\right )+b^2 d^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {b^2 \left (c d^2-a e^2\right )+8 a b c d e-4 a c \left (a e^2+3 c d^2\right )}{\sqrt {b^2-4 a c}}+a b e^2-4 a c d e+b c d^2\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} \left (-\frac {b^2 \left (c d^2-a e^2\right )+8 a b c d e-4 a c \left (a e^2+3 c d^2\right )}{\sqrt {b^2-4 a c}}+a b e^2-4 a c d e+b c d^2\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (x^2 \left (a b e^2-4 a c d e+b c d^2\right )-2 a b d e-2 a \left (c d^2-a e^2\right )+b^2 d^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {b^2 \left (c d^2-a e^2\right )+8 a b c d e-4 a c \left (a e^2+3 c d^2\right )}{\sqrt {b^2-4 a c}}+a b e^2-4 a c d e+b c d^2\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {b^2 \left (c d^2-a e^2\right )+8 a b c d e-4 a c \left (a e^2+3 c d^2\right )}{\sqrt {b^2-4 a c}}+a b e^2-4 a c d e+b c d^2\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 a \left (b^2-4 a c\right )}+\frac {x \left (x^2 \left (a b e^2-4 a c d e+b c d^2\right )-2 a b d e-2 a \left (c d^2-a e^2\right )+b^2 d^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
(x*(b^2*d^2 - 2*a*b*d*e - 2*a*(c*d^2 - a*e^2) + (b*c*d^2 - 4*a*c*d*e + a*b *e^2)*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (((b*c*d^2 - 4*a*c*d *e + a*b*e^2 + (8*a*b*c*d*e + b^2*(c*d^2 - a*e^2) - 4*a*c*(3*c*d^2 + a*e^2 ))/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]]) + ((b*c*d^2 - 4*a*c*d*e + a*b*e^2 - (8*a*b*c*d*e + b^2*(c*d^2 - a*e^2) - 4*a*c*(3*c*d^2 + a*e^2)) /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*a*(b^2 - 4*a*c))
3.3.71.3.1 Defintions of rubi rules used
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[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> With[{f = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2* a*c) - c*(b*f - 2*a*g)*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)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)* x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ [c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.58
| method | result | size |
| risch | \(\frac {-\frac {\left (a b \,e^{2}-4 a c d e +b c \,d^{2}\right ) x^{3}}{2 a \left (4 a c -b^{2}\right )}-\frac {\left (2 e^{2} a^{2}-2 a b d e -2 d^{2} a c +b^{2} d^{2}\right ) x}{2 a \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (-\frac {\left (a b \,e^{2}-4 a c d e +b c \,d^{2}\right ) \textit {\_R}^{2}}{4 a c -b^{2}}+\frac {2 e^{2} a^{2}-2 a b d e +6 d^{2} a c -b^{2} d^{2}}{4 a c -b^{2}}\right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}}{4 a}\) | \(224\) |
| default | \(\frac {-\frac {\left (a b \,e^{2}-4 a c d e +b c \,d^{2}\right ) x^{3}}{2 a \left (4 a c -b^{2}\right )}-\frac {\left (2 e^{2} a^{2}-2 a b d e -2 d^{2} a c +b^{2} d^{2}\right ) x}{2 a \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {2 c \left (\frac {\left (-a b \,e^{2} \sqrt {-4 a c +b^{2}}+4 a c d e \sqrt {-4 a c +b^{2}}-b c \,d^{2} \sqrt {-4 a c +b^{2}}-4 a^{2} c \,e^{2}-a \,b^{2} e^{2}+8 a b c d e -12 a \,c^{2} d^{2}+b^{2} c \,d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (-a b \,e^{2} \sqrt {-4 a c +b^{2}}+4 a c d e \sqrt {-4 a c +b^{2}}-b c \,d^{2} \sqrt {-4 a c +b^{2}}+4 a^{2} c \,e^{2}+a \,b^{2} e^{2}-8 a b c d e +12 a \,c^{2} d^{2}-b^{2} c \,d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a \left (4 a c -b^{2}\right )}\) | \(427\) |
(-1/2/a*(a*b*e^2-4*a*c*d*e+b*c*d^2)/(4*a*c-b^2)*x^3-1/2*(2*a^2*e^2-2*a*b*d *e-2*a*c*d^2+b^2*d^2)/a/(4*a*c-b^2)*x)/(c*x^4+b*x^2+a)+1/4/a*sum((-(a*b*e^ 2-4*a*c*d*e+b*c*d^2)/(4*a*c-b^2)*_R^2+(2*a^2*e^2-2*a*b*d*e+6*a*c*d^2-b^2*d ^2)/(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. 7338 vs. \(2 (344) = 688\).
Time = 12.70 (sec) , antiderivative size = 7338, normalized size of antiderivative = 19.01 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \]
1/2*((b*c*d^2 - 4*a*c*d*e + a*b*e^2)*x^3 - (2*a*b*d*e - 2*a^2*e^2 - (b^2 - 2*a*c)*d^2)*x)/((a*b^2*c - 4*a^2*c^2)*x^4 + a^2*b^2 - 4*a^3*c + (a*b^3 - 4*a^2*b*c)*x^2) + 1/2*integrate((2*a*b*d*e - 2*a^2*e^2 + (b^2 - 6*a*c)*d^2 + (b*c*d^2 - 4*a*c*d*e + a*b*e^2)*x^2)/(c*x^4 + b*x^2 + a), x)/(a*b^2 - 4 *a^2*c)
Leaf count of result is larger than twice the leaf count of optimal. 6390 vs. \(2 (344) = 688\).
Time = 1.38 (sec) , antiderivative size = 6390, normalized size of antiderivative = 16.55 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
1/2*(b*c*d^2*x^3 - 4*a*c*d*e*x^3 + a*b*e^2*x^3 + b^2*d^2*x - 2*a*c*d^2*x - 2*a*b*d*e*x + 2*a^2*e^2*x)/((c*x^4 + b*x^2 + a)*(a*b^2 - 4*a^2*c)) + 1/16 *((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)*sqrt(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)*(a*b^2 - 4*a^2*c)^2*d^2 - 4*(2*a*b^2*c^3 - 8*a^2 *c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^2 + 2*s qrt(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)*a*c^3 - 2*(b^2 - 4*a*c) *a*c^3)*(a*b^2 - 4*a^2*c)^2*d*e + (2*a*b^3*c^2 - 8*a^2*b*c^3 - sqrt(2)*sqr t(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c + 2*sqrt(2)*sqrt(b^2 - 4* a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c - sqrt(2)*sqrt(b^2 - 4*a*c)*s qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - 2*(b^2 - 4*a*c)*a*b*c^2)*(a*b^2 - 4*a^2*c)^2*e^2 + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6*c - 14* sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 - 2*a*b^6*c^2 + 64*sqrt(2)*sqrt(b*c + sqrt (b^2 - 4*a*c)*c)*a^3*b^2*c^3 + 20*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*...
Time = 11.08 (sec) , antiderivative size = 18785, normalized size of antiderivative = 48.67 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
atan(((((6144*a^5*c^6*d^2 + 2048*a^6*c^5*e^2 + 16*a*b^8*c^2*d^2 - 288*a^2* b^6*c^3*d^2 + 1920*a^3*b^4*c^4*d^2 - 5632*a^4*b^2*c^5*d^2 - 32*a^3*b^6*c^2 *e^2 + 384*a^4*b^4*c^3*e^2 - 1536*a^5*b^2*c^4*e^2 - 2048*a^5*b*c^5*d*e + 3 2*a^2*b^7*c^2*d*e - 384*a^3*b^5*c^3*d*e + 1536*a^4*b^3*c^4*d*e)/(8*(a^2*b^ 6 - 64*a^5*c^3 - 12*a^3*b^4*c + 48*a^4*b^2*c^2)) - (x*(-(b^11*c*d^4 + a^3* b^9*e^4 + a^3*e^4*(-(4*a*c - b^2)^9)^(1/2) - 27*a*b^9*c^2*d^4 - 3840*a^5*b *c^6*d^4 + 9*a*c^2*d^4*(-(4*a*c - b^2)^9)^(1/2) - 768*a^7*b*c^4*e^4 - b^2* c*d^4*(-(4*a*c - b^2)^9)^(1/2) + 6144*a^6*c^6*d^3*e + 2048*a^7*c^5*d*e^3 + 288*a^2*b^7*c^3*d^4 - 1504*a^3*b^5*c^4*d^4 + 3840*a^4*b^3*c^5*d^4 - 96*a^ 5*b^5*c^2*e^4 + 512*a^6*b^3*c^3*e^4 + 4*a*b^10*c*d^3*e + 128*a^3*b^7*c^2*d ^2*e^2 - 1344*a^4*b^5*c^3*d^2*e^2 + 5120*a^5*b^3*c^4*d^2*e^2 - 24*a^3*b^8* c*d*e^3 - 72*a^2*b^8*c^2*d^3*e - 2*a^2*b^9*c*d^2*e^2 + 384*a^3*b^6*c^3*d^3 *e - 256*a^4*b^4*c^4*d^3*e + 256*a^4*b^6*c^2*d*e^3 - 3072*a^5*b^2*c^5*d^3* e - 768*a^5*b^4*c^3*d*e^3 - 6656*a^6*b*c^5*d^2*e^2 + 2*a^2*c*d^2*e^2*(-(4* a*c - b^2)^9)^(1/2) - 4*a*b*c*d^3*e*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^ 9*c^7 + a^3*b^12*c - 24*a^4*b^10*c^2 + 240*a^5*b^8*c^3 - 1280*a^6*b^6*c^4 + 3840*a^7*b^4*c^5 - 6144*a^8*b^2*c^6)))^(1/2)*(1024*a^5*b*c^5 - 16*a^2*b^ 7*c^2 + 192*a^3*b^5*c^3 - 768*a^4*b^3*c^4))/(2*(a^2*b^4 + 16*a^4*c^2 - 8*a ^3*b^2*c)))*(-(b^11*c*d^4 + a^3*b^9*e^4 + a^3*e^4*(-(4*a*c - b^2)^9)^(1/2) - 27*a*b^9*c^2*d^4 - 3840*a^5*b*c^6*d^4 + 9*a*c^2*d^4*(-(4*a*c - b^2)^...