Integrand size = 39, antiderivative size = 136 \[ \int \frac {1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {x}{2 d (2 c d-b e) \left (d+e x^2\right )}-\frac {(4 c d-b e) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e} (2 c d-b e)^2}-\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^2} \]
-1/2*x/d/(-b*e+2*c*d)/(e*x^2+d)-1/2*(-b*e+4*c*d)*arctan(x*e^(1/2)/d^(1/2)) /d^(3/2)/(-b*e+2*c*d)^2/e^(1/2)-c^(3/2)*arctanh(x*c^(1/2)*e^(1/2)/(-b*e+c* d)^(1/2))/(-b*e+2*c*d)^2/e^(1/2)/(-b*e+c*d)^(1/2)
Time = 0.14 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {x}{2 d (2 c d-b e) \left (d+e x^2\right )}+\frac {(-4 c d+b e) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e} (2 c d-b e)^2}+\frac {c^{3/2} \arctan \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {-c d+b e}}\right )}{\sqrt {e} (-2 c d+b e)^2 \sqrt {-c d+b e}} \]
-1/2*x/(d*(2*c*d - b*e)*(d + e*x^2)) + ((-4*c*d + b*e)*ArcTan[(Sqrt[e]*x)/ Sqrt[d]])/(2*d^(3/2)*Sqrt[e]*(2*c*d - b*e)^2) + (c^(3/2)*ArcTan[(Sqrt[c]*S qrt[e]*x)/Sqrt[-(c*d) + b*e]])/(Sqrt[e]*(-2*c*d + b*e)^2*Sqrt[-(c*d) + b*e ])
Time = 0.31 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1387, 316, 25, 27, 397, 218, 221}
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.
| \(\displaystyle \int \frac {1}{\left (d+e x^2\right ) \left (b d e+b e^2 x^2-c d^2+c e^2 x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1387 |
| \(\displaystyle \int \frac {1}{\left (d+e x^2\right )^2 \left (\frac {b d e-c d^2}{d}+c e x^2\right )}dx\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\int -\frac {e \left (-c e x^2+3 c d-b e\right )}{\left (e x^2+d\right ) \left (-c e x^2+c d-b e\right )}dx}{2 d e (2 c d-b e)}-\frac {x}{2 d \left (d+e x^2\right ) (2 c d-b e)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {e \left (-c e x^2+3 c d-b e\right )}{\left (e x^2+d\right ) \left (-c e x^2+c d-b e\right )}dx}{2 d e (2 c d-b e)}-\frac {x}{2 d \left (d+e x^2\right ) (2 c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {-c e x^2+3 c d-b e}{\left (e x^2+d\right ) \left (-c e x^2+c d-b e\right )}dx}{2 d (2 c d-b e)}-\frac {x}{2 d \left (d+e x^2\right ) (2 c d-b e)}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {\frac {2 c^2 d \int \frac {1}{-c e x^2+c d-b e}dx}{2 c d-b e}+\frac {(4 c d-b e) \int \frac {1}{e x^2+d}dx}{2 c d-b e}}{2 d (2 c d-b e)}-\frac {x}{2 d \left (d+e x^2\right ) (2 c d-b e)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {2 c^2 d \int \frac {1}{-c e x^2+c d-b e}dx}{2 c d-b e}+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (4 c d-b e)}{\sqrt {d} \sqrt {e} (2 c d-b e)}}{2 d (2 c d-b e)}-\frac {x}{2 d \left (d+e x^2\right ) (2 c d-b e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (4 c d-b e)}{\sqrt {d} \sqrt {e} (2 c d-b e)}+\frac {2 c^{3/2} d \text {arctanh}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)}}{2 d (2 c d-b e)}-\frac {x}{2 d \left (d+e x^2\right ) (2 c d-b e)}\) |
-1/2*x/(d*(2*c*d - b*e)*(d + e*x^2)) - (((4*c*d - b*e)*ArcTan[(Sqrt[e]*x)/ Sqrt[d]])/(Sqrt[d]*Sqrt[e]*(2*c*d - b*e)) + (2*c^(3/2)*d*ArcTanh[(Sqrt[c]* Sqrt[e]*x)/Sqrt[c*d - b*e]])/(Sqrt[e]*Sqrt[c*d - b*e]*(2*c*d - b*e)))/(2*d *(2*c*d - b*e))
3.3.18.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)* (x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^ p, x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 0.48 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.80
| method | result | size |
| default | \(\frac {c^{2} \arctan \left (\frac {x c e}{\sqrt {\left (b e -c d \right ) e c}}\right )}{\left (b e -2 c d \right )^{2} \sqrt {\left (b e -c d \right ) e c}}+\frac {\frac {\left (b e -2 c d \right ) x}{2 d \left (e \,x^{2}+d \right )}+\frac {\left (b e -4 c d \right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 d \sqrt {e d}}}{\left (b e -2 c d \right )^{2}}\) | \(109\) |
| risch | \(\frac {x}{2 d \left (b e -2 c d \right ) \left (e \,x^{2}+d \right )}+\frac {\sqrt {-\left (b e -c d \right ) e c}\, c \ln \left (\left (-\sqrt {-\left (b e -c d \right ) e c}\, b^{4} e^{4}+10 \sqrt {-\left (b e -c d \right ) e c}\, b^{3} c d \,e^{3}-37 \sqrt {-\left (b e -c d \right ) e c}\, b^{2} c^{2} d^{2} e^{2}+48 \sqrt {-\left (b e -c d \right ) e c}\, b \,c^{3} d^{3} e -20 \sqrt {-\left (b e -c d \right ) e c}\, c^{4} d^{4}-4 \left (-\left (b e -c d \right ) e c \right )^{\frac {3}{2}} b c \,d^{2}\right ) x +b^{5} e^{5}-11 b^{4} c d \,e^{4}+43 b^{3} c^{2} d^{2} e^{3}-77 b^{2} c^{3} d^{3} e^{2}+64 b \,c^{4} d^{4} e -20 c^{5} d^{5}\right )}{2 e \left (b e -c d \right ) \left (b e -2 c d \right )^{2}}-\frac {\sqrt {-\left (b e -c d \right ) e c}\, c \ln \left (\left (\sqrt {-\left (b e -c d \right ) e c}\, b^{4} e^{4}-10 \sqrt {-\left (b e -c d \right ) e c}\, b^{3} c d \,e^{3}+37 \sqrt {-\left (b e -c d \right ) e c}\, b^{2} c^{2} d^{2} e^{2}-48 \sqrt {-\left (b e -c d \right ) e c}\, b \,c^{3} d^{3} e +20 \sqrt {-\left (b e -c d \right ) e c}\, c^{4} d^{4}+4 \left (-\left (b e -c d \right ) e c \right )^{\frac {3}{2}} b c \,d^{2}\right ) x +b^{5} e^{5}-11 b^{4} c d \,e^{4}+43 b^{3} c^{2} d^{2} e^{3}-77 b^{2} c^{3} d^{3} e^{2}+64 b \,c^{4} d^{4} e -20 c^{5} d^{5}\right )}{2 e \left (b e -c d \right ) \left (b e -2 c d \right )^{2}}-\frac {\ln \left (d \,e^{2} x -\left (-e d \right )^{\frac {3}{2}}\right ) b e}{4 \sqrt {-e d}\, \left (b e -2 c d \right )^{2} d}+\frac {\ln \left (d \,e^{2} x -\left (-e d \right )^{\frac {3}{2}}\right ) c}{\sqrt {-e d}\, \left (b e -2 c d \right )^{2}}+\frac {\ln \left (-d \,e^{2} x -\left (-e d \right )^{\frac {3}{2}}\right ) b e}{4 \sqrt {-e d}\, \left (b e -2 c d \right )^{2} d}-\frac {\ln \left (-d \,e^{2} x -\left (-e d \right )^{\frac {3}{2}}\right ) c}{\sqrt {-e d}\, \left (b e -2 c d \right )^{2}}\) | \(673\) |
c^2/(b*e-2*c*d)^2/((b*e-c*d)*e*c)^(1/2)*arctan(x*c*e/((b*e-c*d)*e*c)^(1/2) )+1/(b*e-2*c*d)^2*(1/2*(b*e-2*c*d)/d*x/(e*x^2+d)+1/2*(b*e-4*c*d)/d/(e*d)^( 1/2)*arctan(e*x/(e*d)^(1/2)))
Time = 0.35 (sec) , antiderivative size = 895, normalized size of antiderivative = 6.58 \[ \int \frac {1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\left [\frac {2 \, {\left (c d^{2} e^{2} x^{2} + c d^{3} e\right )} \sqrt {\frac {c}{c d e - b e^{2}}} \log \left (\frac {c e x^{2} - 2 \, {\left (c d e - b e^{2}\right )} x \sqrt {\frac {c}{c d e - b e^{2}}} + c d - b e}{c e x^{2} - c d + b e}\right ) + {\left (4 \, c d^{2} - b d e + {\left (4 \, c d e - b e^{2}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) - 2 \, {\left (2 \, c d^{2} e - b d e^{2}\right )} x}{4 \, {\left (4 \, c^{2} d^{5} e - 4 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3} + {\left (4 \, c^{2} d^{4} e^{2} - 4 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2}\right )}}, -\frac {{\left (4 \, c d^{2} - b d e + {\left (4 \, c d e - b e^{2}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - {\left (c d^{2} e^{2} x^{2} + c d^{3} e\right )} \sqrt {\frac {c}{c d e - b e^{2}}} \log \left (\frac {c e x^{2} - 2 \, {\left (c d e - b e^{2}\right )} x \sqrt {\frac {c}{c d e - b e^{2}}} + c d - b e}{c e x^{2} - c d + b e}\right ) + {\left (2 \, c d^{2} e - b d e^{2}\right )} x}{2 \, {\left (4 \, c^{2} d^{5} e - 4 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3} + {\left (4 \, c^{2} d^{4} e^{2} - 4 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2}\right )}}, \frac {4 \, {\left (c d^{2} e^{2} x^{2} + c d^{3} e\right )} \sqrt {-\frac {c}{c d e - b e^{2}}} \arctan \left (e x \sqrt {-\frac {c}{c d e - b e^{2}}}\right ) + {\left (4 \, c d^{2} - b d e + {\left (4 \, c d e - b e^{2}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) - 2 \, {\left (2 \, c d^{2} e - b d e^{2}\right )} x}{4 \, {\left (4 \, c^{2} d^{5} e - 4 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3} + {\left (4 \, c^{2} d^{4} e^{2} - 4 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2}\right )}}, \frac {2 \, {\left (c d^{2} e^{2} x^{2} + c d^{3} e\right )} \sqrt {-\frac {c}{c d e - b e^{2}}} \arctan \left (e x \sqrt {-\frac {c}{c d e - b e^{2}}}\right ) - {\left (4 \, c d^{2} - b d e + {\left (4 \, c d e - b e^{2}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - {\left (2 \, c d^{2} e - b d e^{2}\right )} x}{2 \, {\left (4 \, c^{2} d^{5} e - 4 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3} + {\left (4 \, c^{2} d^{4} e^{2} - 4 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2}\right )}}\right ] \]
[1/4*(2*(c*d^2*e^2*x^2 + c*d^3*e)*sqrt(c/(c*d*e - b*e^2))*log((c*e*x^2 - 2 *(c*d*e - b*e^2)*x*sqrt(c/(c*d*e - b*e^2)) + c*d - b*e)/(c*e*x^2 - c*d + b *e)) + (4*c*d^2 - b*d*e + (4*c*d*e - b*e^2)*x^2)*sqrt(-d*e)*log((e*x^2 - 2 *sqrt(-d*e)*x - d)/(e*x^2 + d)) - 2*(2*c*d^2*e - b*d*e^2)*x)/(4*c^2*d^5*e - 4*b*c*d^4*e^2 + b^2*d^3*e^3 + (4*c^2*d^4*e^2 - 4*b*c*d^3*e^3 + b^2*d^2*e ^4)*x^2), -1/2*((4*c*d^2 - b*d*e + (4*c*d*e - b*e^2)*x^2)*sqrt(d*e)*arctan (sqrt(d*e)*x/d) - (c*d^2*e^2*x^2 + c*d^3*e)*sqrt(c/(c*d*e - b*e^2))*log((c *e*x^2 - 2*(c*d*e - b*e^2)*x*sqrt(c/(c*d*e - b*e^2)) + c*d - b*e)/(c*e*x^2 - c*d + b*e)) + (2*c*d^2*e - b*d*e^2)*x)/(4*c^2*d^5*e - 4*b*c*d^4*e^2 + b ^2*d^3*e^3 + (4*c^2*d^4*e^2 - 4*b*c*d^3*e^3 + b^2*d^2*e^4)*x^2), 1/4*(4*(c *d^2*e^2*x^2 + c*d^3*e)*sqrt(-c/(c*d*e - b*e^2))*arctan(e*x*sqrt(-c/(c*d*e - b*e^2))) + (4*c*d^2 - b*d*e + (4*c*d*e - b*e^2)*x^2)*sqrt(-d*e)*log((e* x^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)) - 2*(2*c*d^2*e - b*d*e^2)*x)/(4*c^2 *d^5*e - 4*b*c*d^4*e^2 + b^2*d^3*e^3 + (4*c^2*d^4*e^2 - 4*b*c*d^3*e^3 + b^ 2*d^2*e^4)*x^2), 1/2*(2*(c*d^2*e^2*x^2 + c*d^3*e)*sqrt(-c/(c*d*e - b*e^2)) *arctan(e*x*sqrt(-c/(c*d*e - b*e^2))) - (4*c*d^2 - b*d*e + (4*c*d*e - b*e^ 2)*x^2)*sqrt(d*e)*arctan(sqrt(d*e)*x/d) - (2*c*d^2*e - b*d*e^2)*x)/(4*c^2* d^5*e - 4*b*c*d^4*e^2 + b^2*d^3*e^3 + (4*c^2*d^4*e^2 - 4*b*c*d^3*e^3 + b^2 *d^2*e^4)*x^2)]
Timed out. \[ \int \frac {1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*(b*e-c*d)>0)', see `assume?` f or more de
Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\frac {c^{2} \arctan \left (\frac {c e x}{\sqrt {-c^{2} d e + b c e^{2}}}\right )}{{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \sqrt {-c^{2} d e + b c e^{2}}} - \frac {{\left (4 \, c d - b e\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, {\left (4 \, c^{2} d^{3} - 4 \, b c d^{2} e + b^{2} d e^{2}\right )} \sqrt {d e}} - \frac {x}{2 \, {\left (2 \, c d^{2} - b d e\right )} {\left (e x^{2} + d\right )}} \]
c^2*arctan(c*e*x/sqrt(-c^2*d*e + b*c*e^2))/((4*c^2*d^2 - 4*b*c*d*e + b^2*e ^2)*sqrt(-c^2*d*e + b*c*e^2)) - 1/2*(4*c*d - b*e)*arctan(e*x/sqrt(d*e))/(( 4*c^2*d^3 - 4*b*c*d^2*e + b^2*d*e^2)*sqrt(d*e)) - 1/2*x/((2*c*d^2 - b*d*e) *(e*x^2 + d))
Time = 9.26 (sec) , antiderivative size = 3901, normalized size of antiderivative = 28.68 \[ \int \frac {1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\text {Too large to display} \]
- x/(2*(d + e*x^2)*(2*c*d^2 - b*d*e)) - (atan(((((((96*c^7*d^6*e^6 - 224*b *c^6*d^5*e^7 - 2*b^5*c^2*d*e^11 + 208*b^2*c^5*d^4*e^8 - 96*b^3*c^4*d^3*e^9 + 22*b^4*c^3*d^2*e^10)/(2*(8*c^3*d^5 - b^3*d^2*e^3 + 6*b^2*c*d^3*e^2 - 12 *b*c^2*d^4*e)) - (x*(-c^3*e*(b*e - c*d))^(1/2)*(256*b*c^6*d^6*e^8 - 512*b^ 2*c^5*d^5*e^9 + 384*b^3*c^4*d^4*e^10 - 128*b^4*c^3*d^3*e^11 + 16*b^5*c^2*d ^2*e^12))/(8*(4*c^2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3*e)*(b^3*e^4 - 4*c^3*d^3* e + 8*b*c^2*d^2*e^2 - 5*b^2*c*d*e^3)))*(-c^3*e*(b*e - c*d))^(1/2))/(2*(b^3 *e^4 - 4*c^3*d^3*e + 8*b*c^2*d^2*e^2 - 5*b^2*c*d*e^3)) - (x*(b^2*c^3*e^8 + 20*c^5*d^2*e^6 - 8*b*c^4*d*e^7))/(4*(4*c^2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3* e)))*(-c^3*e*(b*e - c*d))^(1/2)*1i)/(b^3*e^4 - 4*c^3*d^3*e + 8*b*c^2*d^2*e ^2 - 5*b^2*c*d*e^3) - (((((96*c^7*d^6*e^6 - 224*b*c^6*d^5*e^7 - 2*b^5*c^2* d*e^11 + 208*b^2*c^5*d^4*e^8 - 96*b^3*c^4*d^3*e^9 + 22*b^4*c^3*d^2*e^10)/( 2*(8*c^3*d^5 - b^3*d^2*e^3 + 6*b^2*c*d^3*e^2 - 12*b*c^2*d^4*e)) + (x*(-c^3 *e*(b*e - c*d))^(1/2)*(256*b*c^6*d^6*e^8 - 512*b^2*c^5*d^5*e^9 + 384*b^3*c ^4*d^4*e^10 - 128*b^4*c^3*d^3*e^11 + 16*b^5*c^2*d^2*e^12))/(8*(4*c^2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3*e)*(b^3*e^4 - 4*c^3*d^3*e + 8*b*c^2*d^2*e^2 - 5*b ^2*c*d*e^3)))*(-c^3*e*(b*e - c*d))^(1/2))/(2*(b^3*e^4 - 4*c^3*d^3*e + 8*b* c^2*d^2*e^2 - 5*b^2*c*d*e^3)) + (x*(b^2*c^3*e^8 + 20*c^5*d^2*e^6 - 8*b*c^4 *d*e^7))/(4*(4*c^2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3*e)))*(-c^3*e*(b*e - c*d)) ^(1/2)*1i)/(b^3*e^4 - 4*c^3*d^3*e + 8*b*c^2*d^2*e^2 - 5*b^2*c*d*e^3))/(...