\(\int \frac {1}{(d+e x^2) (-c d^2+b d e+b e^2 x^2+c e^2 x^4)} \, dx\) [218]

Optimal result

Integrand size = 39, antiderivative size = 136 \[ \int \frac {1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {x}{2 d (2 c d-b e) \left (d+e x^2\right )}-\frac {(4 c d-b e) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e} (2 c d-b e)^2}-\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^2} \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {1163, 425, 536, 211, 214} \[ \int \frac {1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (4 c d-b e)}{2 d^{3/2} \sqrt {e} (2 c d-b e)^2}-\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^2}-\frac {x}{2 d \left (d+e x^2\right ) (2 c d-b e)} \]

[In]

Int[1/((d + e*x^2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

-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)*ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]])/(Sqrt[e]*Sqrt[c*d - b*e]*(2*c*d - b*e)^2)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

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]

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 1163

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(d + e*x^2)^(p +
q)*(a/d + (c/e)*x^2)^p, x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2
, 0] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (d+e x^2\right )^2 \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx \\ & = -\frac {x}{2 d (2 c d-b e) \left (d+e x^2\right )}+\frac {\int \frac {e (3 c d-b e)-c e^2 x^2}{\left (d+e x^2\right ) \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx}{2 d e (2 c d-b e)} \\ & = -\frac {x}{2 d (2 c d-b e) \left (d+e x^2\right )}+\frac {c^2 \int \frac {1}{\frac {-c d^2+b d e}{d}+c e x^2} \, dx}{(2 c d-b e)^2}-\frac {(4 c d-b e) \int \frac {1}{d+e x^2} \, dx}{2 d (2 c d-b e)^2} \\ & = -\frac {x}{2 d (2 c d-b e) \left (d+e x^2\right )}-\frac {(4 c d-b e) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e} (2 c d-b e)^2}-\frac {c^{3/2} \tanh ^{-1}\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {x}{2 d (2 c d-b e) \left (d+e x^2\right )}+\frac {(-4 c d+b e) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e} (2 c d-b e)^2}+\frac {c^{3/2} \arctan \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {-c d+b e}}\right )}{\sqrt {e} (-2 c d+b e)^2 \sqrt {-c d+b e}} \]

[In]

Integrate[1/((d + e*x^2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

-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]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/(Sqrt[e]*(-2*c*d + b*e)^2*Sqrt[-(c*d) + b
*e])

Maple [A] (verified)

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	\(\frac {x}{2 d \left (b e -2 c d \right ) \left (e \,x^{2}+d \right )}+\frac {\sqrt {-\left (b e -c d \right ) e c}\, c \ln \left (\left (-\sqrt {-\left (b e -c d \right ) e c}\, b^{4} e^{4}+10 \sqrt {-\left (b e -c d \right ) e c}\, b^{3} c d \,e^{3}-37 \sqrt {-\left (b e -c d \right ) e c}\, b^{2} c^{2} d^{2} e^{2}+48 \sqrt {-\left (b e -c d \right ) e c}\, b \,c^{3} d^{3} e -20 \sqrt {-\left (b e -c d \right ) e c}\, c^{4} d^{4}-4 \left (-\left (b e -c d \right ) e c \right )^{\frac {3}{2}} b c \,d^{2}\right ) x +b^{5} e^{5}-11 b^{4} c d \,e^{4}+43 b^{3} c^{2} d^{2} e^{3}-77 b^{2} c^{3} d^{3} e^{2}+64 b \,c^{4} d^{4} e -20 c^{5} d^{5}\right )}{2 e \left (b e -c d \right ) \left (b e -2 c d \right )^{2}}-\frac {\sqrt {-\left (b e -c d \right ) e c}\, c \ln \left (\left (\sqrt {-\left (b e -c d \right ) e c}\, b^{4} e^{4}-10 \sqrt {-\left (b e -c d \right ) e c}\, b^{3} c d \,e^{3}+37 \sqrt {-\left (b e -c d \right ) e c}\, b^{2} c^{2} d^{2} e^{2}-48 \sqrt {-\left (b e -c d \right ) e c}\, b \,c^{3} d^{3} e +20 \sqrt {-\left (b e -c d \right ) e c}\, c^{4} d^{4}+4 \left (-\left (b e -c d \right ) e c \right )^{\frac {3}{2}} b c \,d^{2}\right ) x +b^{5} e^{5}-11 b^{4} c d \,e^{4}+43 b^{3} c^{2} d^{2} e^{3}-77 b^{2} c^{3} d^{3} e^{2}+64 b \,c^{4} d^{4} e -20 c^{5} d^{5}\right )}{2 e \left (b e -c d \right ) \left (b e -2 c d \right )^{2}}-\frac {\ln \left (d \,e^{2} x -\left (-e d \right )^{\frac {3}{2}}\right ) b e}{4 \sqrt {-e d}\, \left (b e -2 c d \right )^{2} d}+\frac {\ln \left (d \,e^{2} x -\left (-e d \right )^{\frac {3}{2}}\right ) c}{\sqrt {-e d}\, \left (b e -2 c d \right )^{2}}+\frac {\ln \left (-d \,e^{2} x -\left (-e d \right )^{\frac {3}{2}}\right ) b e}{4 \sqrt {-e d}\, \left (b e -2 c d \right )^{2} d}-\frac {\ln \left (-d \,e^{2} x -\left (-e d \right )^{\frac {3}{2}}\right ) c}{\sqrt {-e d}\, \left (b e -2 c d \right )^{2}}\)	\(673\)

[In]

int(1/(e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x,method=_RETURNVERBOSE)

[Out]

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)))

Fricas [A] (verification not implemented)

none

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 ] \]

[In]

integrate(1/(e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="fricas")

[Out]

[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)*ar
ctan(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*sqr
t(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(s
qrt(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)]

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x**2+d)/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e*(b*e-c*d)>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\frac {c^{2} \arctan \left (\frac {c e x}{\sqrt {-c^{2} d e + b c e^{2}}}\right )}{{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \sqrt {-c^{2} d e + b c e^{2}}} - \frac {{\left (4 \, c d - b e\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, {\left (4 \, c^{2} d^{3} - 4 \, b c d^{2} e + b^{2} d e^{2}\right )} \sqrt {d e}} - \frac {x}{2 \, {\left (2 \, c d^{2} - b d e\right )} {\left (e x^{2} + d\right )}} \]

[In]

integrate(1/(e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="giac")

[Out]

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))

Mupad [B] (verification not implemented)

Time = 9.26 (sec) , antiderivative size = 3901, normalized size of antiderivative = 28.68 \[ \int \frac {1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\text {Too large to display} \]

[In]

int(1/((d + e*x^2)*(b*e^2*x^2 - c*d^2 + c*e^2*x^4 + b*d*e)),x)

[Out]

- 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 + 20
8*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))/((((((96*c^7*d^6*e^6 - 224*b*c^6*d^5*e^7 - 2*b^5*c^2*d*e^11 + 20
8*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))/(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) - ((b*c^4*e^6
)/2 - 2*c^5*d*e^5)/(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) + (((((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))/(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)*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) - (atan(((((x*(b^2*c^3*e^8 + 20*c^5*d^2*e^6 - 8*b*c^4*d*e^7))/(2*(4*c^2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3*e)) -
((-d^3*e)^(1/2)*((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)/(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*(-d^3*e)^(1/2)*
(b*e - 4*c*d)*(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)*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2)))*(b*e -
4*c*d))/(4*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2)))*(-d^3*e)^(1/2)*(b*e - 4*c*d)*1i)/(4*(4*c^2*d^5*e + b
^2*d^3*e^3 - 4*b*c*d^4*e^2)) + (((x*(b^2*c^3*e^8 + 20*c^5*d^2*e^6 - 8*b*c^4*d*e^7))/(2*(4*c^2*d^4 + b^2*d^2*e^
2 - 4*b*c*d^3*e)) + ((-d^3*e)^(1/2)*((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)/(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*(-d^3*e)^(1/2)*(b*e - 4*c*d)*(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)*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*
c*d^4*e^2)))*(b*e - 4*c*d))/(4*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2)))*(-d^3*e)^(1/2)*(b*e - 4*c*d)*1i)/
(4*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2)))/(((b*c^4*e^6)/2 - 2*c^5*d*e^5)/(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*(b^2*c^3*e^8 + 20*c^5*d^2*e^6 - 8*b*c^4*d*e^7))/(2*(4*c^2*d^4 + b^2*d^2*
e^2 - 4*b*c*d^3*e)) - ((-d^3*e)^(1/2)*((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)/(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*(-d^3*e)^(1/2)*(b*e - 4*c*d)*(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)*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*
b*c*d^4*e^2)))*(b*e - 4*c*d))/(4*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2)))*(-d^3*e)^(1/2)*(b*e - 4*c*d))/(
4*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2)) - (((x*(b^2*c^3*e^8 + 20*c^5*d^2*e^6 - 8*b*c^4*d*e^7))/(2*(4*c^
2*d^4 + b^2*d^2*e^2 - 4*b*c*d^3*e)) + ((-d^3*e)^(1/2)*((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)/(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*(-d^3*e)^(1/2)*(b*e - 4*c*d)*(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)*(4*c^2*d^5*e +
b^2*d^3*e^3 - 4*b*c*d^4*e^2)))*(b*e - 4*c*d))/(4*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2)))*(-d^3*e)^(1/2)*
(b*e - 4*c*d))/(4*(4*c^2*d^5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2))))*(-d^3*e)^(1/2)*(b*e - 4*c*d)*1i)/(2*(4*c^2*d^
5*e + b^2*d^3*e^3 - 4*b*c*d^4*e^2))