Integrand size = 24, antiderivative size = 459 \[ \int \frac {\left (d+e x^2\right )^4}{a+b x^2+c x^4} \, dx=\frac {e^2 \left (6 c^2 d^2+b^2 e^2-c e (4 b d+a e)\right ) x}{c^3}+\frac {e^3 (4 c d-b e) x^3}{3 c^2}+\frac {e^4 x^5}{5 c}+\frac {\left (e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+\frac {2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{7/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )-\frac {2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{7/2} \sqrt {b+\sqrt {b^2-4 a c}}} \]
e^2*(6*c^2*d^2+b^2*e^2-c*e*(a*e+4*b*d))*x/c^3+1/3*e^3*(-b*e+4*c*d)*x^3/c^2 +1/5*e^4*x^5/c+1/2*arctan(x*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))* (e*(-b*e+2*c*d)*(2*c^2*d^2+b^2*e^2-2*c*e*(a*e+b*d))+(2*c^4*d^4+b^4*e^4-4*b ^2*c*e^3*(a*e+b*d)-4*c^3*d^2*e*(3*a*e+b*d)+2*c^2*e^2*(a^2*e^2+6*a*b*d*e+3* b^2*d^2))/(-4*a*c+b^2)^(1/2))/c^(7/2)*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2) +1/2*arctan(x*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*(e*(-b*e+2*c*d )*(2*c^2*d^2+b^2*e^2-2*c*e*(a*e+b*d))+(-2*c^4*d^4-b^4*e^4+4*b^2*c*e^3*(a*e +b*d)+4*c^3*d^2*e*(3*a*e+b*d)-2*c^2*e^2*(a^2*e^2+6*a*b*d*e+3*b^2*d^2))/(-4 *a*c+b^2)^(1/2))/c^(7/2)*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 0.43 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.24 \[ \int \frac {\left (d+e x^2\right )^4}{a+b x^2+c x^4} \, dx=\frac {e^2 \left (6 c^2 d^2+b^2 e^2-c e (4 b d+a e)\right ) x}{c^3}+\frac {e^3 (4 c d-b e) x^3}{3 c^2}+\frac {e^4 x^5}{5 c}+\frac {\left (2 c^4 d^4+b^3 \left (b-\sqrt {b^2-4 a c}\right ) e^4+4 c^3 d^2 e \left (-b d+\sqrt {b^2-4 a c} d-3 a e\right )+2 b c e^3 \left (-2 b^2 d+2 b \sqrt {b^2-4 a c} d-2 a b e+a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (3 b^2 d^2-3 b d \left (\sqrt {b^2-4 a c} d-2 a e\right )+a e \left (-2 \sqrt {b^2-4 a c} d+a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{7/2} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (2 c^4 d^4+b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4-4 c^3 d^2 e \left (b d+\sqrt {b^2-4 a c} d+3 a e\right )-2 b c e^3 \left (2 b^2 d+a \sqrt {b^2-4 a c} e+2 b \left (\sqrt {b^2-4 a c} d+a e\right )\right )+2 c^2 e^2 \left (3 b^2 d^2+a e \left (2 \sqrt {b^2-4 a c} d+a e\right )+3 b d \left (\sqrt {b^2-4 a c} d+2 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{7/2} \sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}} \]
(e^2*(6*c^2*d^2 + b^2*e^2 - c*e*(4*b*d + a*e))*x)/c^3 + (e^3*(4*c*d - b*e) *x^3)/(3*c^2) + (e^4*x^5)/(5*c) + ((2*c^4*d^4 + b^3*(b - Sqrt[b^2 - 4*a*c] )*e^4 + 4*c^3*d^2*e*(-(b*d) + Sqrt[b^2 - 4*a*c]*d - 3*a*e) + 2*b*c*e^3*(-2 *b^2*d + 2*b*Sqrt[b^2 - 4*a*c]*d - 2*a*b*e + a*Sqrt[b^2 - 4*a*c]*e) + 2*c^ 2*e^2*(3*b^2*d^2 - 3*b*d*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + a*e*(-2*Sqrt[b^2 - 4*a*c]*d + a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]] ])/(Sqrt[2]*c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((2*c ^4*d^4 + b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 - 4*c^3*d^2*e*(b*d + Sqrt[b^2 - 4 *a*c]*d + 3*a*e) - 2*b*c*e^3*(2*b^2*d + a*Sqrt[b^2 - 4*a*c]*e + 2*b*(Sqrt[ b^2 - 4*a*c]*d + a*e)) + 2*c^2*e^2*(3*b^2*d^2 + a*e*(2*Sqrt[b^2 - 4*a*c]*d + a*e) + 3*b*d*(Sqrt[b^2 - 4*a*c]*d + 2*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x) /Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])
Time = 1.34 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1484, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^4}{a+b x^2+c x^4} \, dx\) |
| \(\Big \downarrow \) 1484 |
| \(\displaystyle \int \left (\frac {e^2 \left (-c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right )}{c^3}+\frac {e x^2 (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )-a b^2 e^4+a c e^3 (a e+4 b d)-6 a c^2 d^2 e^2+c^3 d^4}{c^3 \left (a+b x^2+c x^4\right )}+\frac {e^3 x^2 (4 c d-b e)}{c^2}+\frac {e^4 x^4}{c}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4}{\sqrt {b^2-4 a c}}+e (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )\right )}{\sqrt {2} c^{7/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (e (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )-\frac {2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4}{\sqrt {b^2-4 a c}}\right )}{\sqrt {2} c^{7/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {e^2 x \left (-c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right )}{c^3}+\frac {e^3 x^3 (4 c d-b e)}{3 c^2}+\frac {e^4 x^5}{5 c}\) |
(e^2*(6*c^2*d^2 + b^2*e^2 - c*e*(4*b*d + a*e))*x)/c^3 + (e^3*(4*c*d - b*e) *x^3)/(3*c^2) + (e^4*x^5)/(5*c) + ((e*(2*c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)) + (2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c ^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))/Sqrt [b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(S qrt[2]*c^(7/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((e*(2*c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)) - (2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^ 2*e^2))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])
3.3.63.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symb ol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; Fre eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.49
| method | result | size |
| risch | \(\frac {e^{4} x^{5}}{5 c}-\frac {e^{4} b \,x^{3}}{3 c^{2}}+\frac {4 d \,e^{3} x^{3}}{3 c}-\frac {e^{4} a x}{c^{2}}+\frac {e^{4} b^{2} x}{c^{3}}-\frac {4 e^{3} b d x}{c^{2}}+\frac {6 e^{2} d^{2} x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (e \left (2 a b c \,e^{3}-4 a \,c^{2} d \,e^{2}-b^{3} e^{3}+4 b^{2} c d \,e^{2}-6 b \,c^{2} d^{2} e +4 c^{3} d^{3}\right ) \textit {\_R}^{2}+a^{2} c \,e^{4}-a \,b^{2} e^{4}+4 a b c d \,e^{3}-6 a \,c^{2} d^{2} e^{2}+c^{3} d^{4}\right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}}{2 c^{3}}\) | \(227\) |
| default | \(-\frac {e^{2} \left (-\frac {1}{5} e^{2} x^{5} c^{2}+\frac {1}{3} b c \,e^{2} x^{3}-\frac {4}{3} c^{2} d e \,x^{3}+e^{2} a c x -b^{2} e^{2} x +4 b c d e x -6 c^{2} d^{2} x \right )}{c^{3}}+\frac {\frac {\left (2 a b c \,e^{4} \sqrt {-4 a c +b^{2}}-4 a \,c^{2} d \,e^{3} \sqrt {-4 a c +b^{2}}-b^{3} e^{4} \sqrt {-4 a c +b^{2}}+4 b^{2} c d \,e^{3} \sqrt {-4 a c +b^{2}}-6 b \,c^{2} d^{2} e^{2} \sqrt {-4 a c +b^{2}}+4 c^{3} d^{3} e \sqrt {-4 a c +b^{2}}-2 a^{2} c^{2} e^{4}+4 a \,b^{2} e^{4} c -12 a b \,c^{2} d \,e^{3}+12 a \,c^{3} d^{2} e^{2}-b^{4} e^{4}+4 b^{3} c d \,e^{3}-6 b^{2} c^{2} d^{2} e^{2}+4 b \,c^{3} e \,d^{3}-2 c^{4} d^{4}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (2 a b c \,e^{4} \sqrt {-4 a c +b^{2}}-4 a \,c^{2} d \,e^{3} \sqrt {-4 a c +b^{2}}-b^{3} e^{4} \sqrt {-4 a c +b^{2}}+4 b^{2} c d \,e^{3} \sqrt {-4 a c +b^{2}}-6 b \,c^{2} d^{2} e^{2} \sqrt {-4 a c +b^{2}}+4 c^{3} d^{3} e \sqrt {-4 a c +b^{2}}+2 a^{2} c^{2} e^{4}-4 a \,b^{2} e^{4} c +12 a b \,c^{2} d \,e^{3}-12 a \,c^{3} d^{2} e^{2}+b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} e \,d^{3}+2 c^{4} d^{4}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{c^{2}}\) | \(621\) |
1/5*e^4*x^5/c-1/3*e^4/c^2*b*x^3+4/3*d*e^3*x^3/c-e^4/c^2*a*x+e^4/c^3*b^2*x- 4*e^3/c^2*b*d*x+6*e^2/c*d^2*x+1/2/c^3*sum((e*(2*a*b*c*e^3-4*a*c^2*d*e^2-b^ 3*e^3+4*b^2*c*d*e^2-6*b*c^2*d^2*e+4*c^3*d^3)*_R^2+a^2*c*e^4-a*b^2*e^4+4*a* b*c*d*e^3-6*a*c^2*d^2*e^2+c^3*d^4)/(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. 16218 vs. \(2 (421) = 842\).
Time = 270.80 (sec) , antiderivative size = 16218, normalized size of antiderivative = 35.33 \[ \int \frac {\left (d+e x^2\right )^4}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^4}{a+b x^2+c x^4} \, dx=\text {Timed out} \]
\[ \int \frac {\left (d+e x^2\right )^4}{a+b x^2+c x^4} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{4}}{c x^{4} + b x^{2} + a} \,d x } \]
1/15*(3*c^2*e^4*x^5 + 5*(4*c^2*d*e^3 - b*c*e^4)*x^3 + 15*(6*c^2*d^2*e^2 - 4*b*c*d*e^3 + (b^2 - a*c)*e^4)*x)/c^3 + integrate((c^3*d^4 - 6*a*c^2*d^2*e ^2 + 4*a*b*c*d*e^3 - (a*b^2 - a^2*c)*e^4 + (4*c^3*d^3*e - 6*b*c^2*d^2*e^2 + 4*(b^2*c - a*c^2)*d*e^3 - (b^3 - 2*a*b*c)*e^4)*x^2)/(c*x^4 + b*x^2 + a), x)/c^3
Leaf count of result is larger than twice the leaf count of optimal. 9306 vs. \(2 (421) = 842\).
Time = 1.30 (sec) , antiderivative size = 9306, normalized size of antiderivative = 20.27 \[ \int \frac {\left (d+e x^2\right )^4}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
1/8*(4*(2*b^4*c^5 - 16*a*b^2*c^6 + 32*a^2*c^7 - sqrt(2)*sqrt(b^2 - 4*a*c)* sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^3 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt (b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b *c + sqrt(b^2 - 4*a*c)*c)*b^3*c^4 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^5 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq rt(b^2 - 4*a*c)*c)*a*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^5 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4 *a*c)*c)*a*c^6 - 2*(b^2 - 4*a*c)*b^2*c^5 + 8*(b^2 - 4*a*c)*a*c^6)*c^2*d^3* e - 6*(2*b^5*c^4 - 16*a*b^3*c^5 + 32*a^2*b*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c) *sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c^2 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr t(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt( b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^3 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 - 8*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 + sqr t(b^2 - 4*a*c)*c)*b^3*c^4 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^ 2 - 4*a*c)*c)*a*b*c^5 - 2*(b^2 - 4*a*c)*b^3*c^4 + 8*(b^2 - 4*a*c)*a*b*c^5) *c^2*d^2*e^2 + 4*(2*b^6*c^3 - 18*a*b^4*c^4 + 48*a^2*b^2*c^5 - 32*a^3*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6*c + 9*sqrt( 2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 + 2*sqrt(2) *sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c^2 - 24*sqrt(2)...
Time = 10.92 (sec) , antiderivative size = 29551, normalized size of antiderivative = 64.38 \[ \int \frac {\left (d+e x^2\right )^4}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
x*((b*((b*e^4)/c^2 - (4*d*e^3)/c))/c - (a*e^4)/c^2 + (6*d^2*e^2)/c) - x^3* ((b*e^4)/(3*c^2) - (4*d*e^3)/(3*c)) + atan(((((16*a*c^8*d^4 + 16*a^3*c^6*e ^4 - 4*b^2*c^7*d^4 + 4*a*b^4*c^4*e^4 - 20*a^2*b^2*c^5*e^4 - 96*a^2*c^7*d^2 *e^2 - 16*a*b^3*c^5*d*e^3 + 64*a^2*b*c^6*d*e^3 + 24*a*b^2*c^6*d^2*e^2)/c^5 - (2*x*(4*b^3*c^7 - 16*a*b*c^8)*(-(a*b^9*e^8 + b^3*c^7*d^8 + c^7*d^8*(-(4 *a*c - b^2)^3)^(1/2) - a*b^6*e^8*(-(4*a*c - b^2)^3)^(1/2) - 11*a^2*b^7*c*e ^8 + 28*a^5*b*c^4*e^8 + 64*a^2*c^8*d^7*e - 64*a^5*c^5*d*e^7 + 42*a^3*b^5*c ^2*e^8 - 63*a^4*b^3*c^3*e^8 + a^4*c^3*e^8*(-(4*a*c - b^2)^3)^(1/2) - 448*a ^3*c^7*d^5*e^3 + 448*a^4*c^6*d^3*e^5 - 4*a*b*c^8*d^8 - 8*a*b^8*c*d*e^7 - 6 *a^3*b^2*c^2*e^8*(-(4*a*c - b^2)^3)^(1/2) + 336*a^2*b^2*c^6*d^5*e^3 - 490* a^2*b^3*c^5*d^4*e^4 + 448*a^2*b^4*c^4*d^3*e^5 - 252*a^2*b^5*c^3*d^2*e^6 - 1008*a^3*b^2*c^5*d^3*e^5 + 700*a^3*b^3*c^4*d^2*e^6 + 70*a^2*c^5*d^4*e^4*(- (4*a*c - b^2)^3)^(1/2) - 28*a^3*c^4*d^2*e^6*(-(4*a*c - b^2)^3)^(1/2) - 16* a*b^2*c^7*d^7*e + 5*a^2*b^4*c*e^8*(-(4*a*c - b^2)^3)^(1/2) + 28*a*b^3*c^6* d^6*e^2 - 56*a*b^4*c^5*d^5*e^3 + 70*a*b^5*c^4*d^4*e^4 - 56*a*b^6*c^3*d^3*e ^5 + 28*a*b^7*c^2*d^2*e^6 - 112*a^2*b*c^7*d^6*e^2 + 80*a^2*b^6*c^2*d*e^7 + 840*a^3*b*c^6*d^4*e^4 - 264*a^3*b^4*c^3*d*e^7 - 560*a^4*b*c^5*d^2*e^6 + 3 04*a^4*b^2*c^4*d*e^7 - 28*a*c^6*d^6*e^2*(-(4*a*c - b^2)^3)^(1/2) + 56*a*b* c^5*d^5*e^3*(-(4*a*c - b^2)^3)^(1/2) + 24*a^3*b*c^3*d*e^7*(-(4*a*c - b^2)^ 3)^(1/2) - 70*a*b^2*c^4*d^4*e^4*(-(4*a*c - b^2)^3)^(1/2) + 56*a*b^3*c^3...