3.359 \(\int \frac {\sqrt {d+e x^2}}{x^3 (a+b x^2+c x^4)} \, dx\)

Optimal. Leaf size=382 \[ -\frac {\sqrt {c} \left (\sqrt {b^2-4 a c} (b d-a e)-a b e-2 a c d+b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {c} \left (-b \left (d \sqrt {b^2-4 a c}+a e\right )-a \left (2 c d-e \sqrt {b^2-4 a c}\right )+b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {(b d-a e) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a^2 \sqrt {d}}-\frac {\sqrt {d+e x^2}}{2 a x^2}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 a \sqrt {d}} \]

[Out]

1/2*e*arctanh((e*x^2+d)^(1/2)/d^(1/2))/a/d^(1/2)+(-a*e+b*d)*arctanh((e*x^2+d)^(1/2)/d^(1/2))/a^2/d^(1/2)-1/2*(
e*x^2+d)^(1/2)/a/x^2-1/2*arctanh(2^(1/2)*c^(1/2)*(e*x^2+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2))*c^(1/
2)*(b^2*d-2*a*c*d-a*b*e+(-a*e+b*d)*(-4*a*c+b^2)^(1/2))/a^2*2^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)
^(1/2)))^(1/2)+1/2*arctanh(2^(1/2)*c^(1/2)*(e*x^2+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*c^(1/2)*(b^
2*d-2*a*c*d-a*b*e-(-a*e+b*d)*(-4*a*c+b^2)^(1/2))/a^2*2^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)
))^(1/2)

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Rubi [A]  time = 4.13, antiderivative size = 370, normalized size of antiderivative = 0.97, number of steps used = 10, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1251, 897, 1287, 199, 206, 1166, 208} \[ -\frac {\sqrt {c} \left (\sqrt {b^2-4 a c} (b d-a e)-a b e-2 a c d+b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {c} \left (-\sqrt {b^2-4 a c} (b d-a e)-a b e-2 a c d+b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {(b d-a e) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a^2 \sqrt {d}}-\frac {\sqrt {d+e x^2}}{2 a x^2}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 a \sqrt {d}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x^2]/(x^3*(a + b*x^2 + c*x^4)),x]

[Out]

-Sqrt[d + e*x^2]/(2*a*x^2) + (e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*a*Sqrt[d]) + ((b*d - a*e)*ArcTanh[Sqrt[d
+ e*x^2]/Sqrt[d]])/(a^2*Sqrt[d]) - (Sqrt[c]*(b^2*d - 2*a*c*d - a*b*e + Sqrt[b^2 - 4*a*c]*(b*d - a*e))*ArcTanh[
(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*a^2*Sqrt[b^2 - 4*a*c]*Sqr
t[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[c]*(b^2*d - 2*a*c*d - a*b*e - Sqrt[b^2 - 4*a*c]*(b*d - a*e))*Arc
Tanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*a^2*Sqrt[b^2 - 4*a*c
]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 208

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

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[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]

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 1287

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

Rubi steps

\begin {align*} \int \frac {\sqrt {d+e x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {d+e x}}{x^2 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{\left (-\frac {d}{e}+\frac {x^2}{e}\right )^2 \left (\frac {c d^2-b d e+a e^2}{e^2}-\frac {(2 c d-b e) x^2}{e^2}+\frac {c x^4}{e^2}\right )} \, dx,x,\sqrt {d+e x^2}\right )}{e}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {d e^2}{a \left (d-x^2\right )^2}-\frac {e (-b d+a e)}{a^2 \left (d-x^2\right )}+\frac {e \left (-b \left (c d^2-b d e+a e^2\right )+c (b d-a e) x^2\right )}{a^2 \left (c d^2-b d e+a e^2-(2 c d-b e) x^2+c x^4\right )}\right ) \, dx,x,\sqrt {d+e x^2}\right )}{e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {-b \left (c d^2-b d e+a e^2\right )+c (b d-a e) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x^2}\right )}{a^2}+\frac {(d e) \operatorname {Subst}\left (\int \frac {1}{\left (d-x^2\right )^2} \, dx,x,\sqrt {d+e x^2}\right )}{a}+\frac {(b d-a e) \operatorname {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d+e x^2}\right )}{a^2}\\ &=-\frac {\sqrt {d+e x^2}}{2 a x^2}+\frac {(b d-a e) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a^2 \sqrt {d}}+\frac {e \operatorname {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d+e x^2}\right )}{2 a}-\frac {\left (c \left (b^2 d-2 a c d-a b e-\sqrt {b^2-4 a c} (b d-a e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x^2}\right )}{2 a^2 \sqrt {b^2-4 a c}}+\frac {\left (c \left (b^2 d-2 a c d-a b e+\sqrt {b^2-4 a c} (b d-a e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x^2}\right )}{2 a^2 \sqrt {b^2-4 a c}}\\ &=-\frac {\sqrt {d+e x^2}}{2 a x^2}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 a \sqrt {d}}+\frac {(b d-a e) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a^2 \sqrt {d}}-\frac {\sqrt {c} \left (b^2 d-2 a c d-a b e+\sqrt {b^2-4 a c} (b d-a e)\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {c} \left (b^2 d-2 a c d-a b e-\sqrt {b^2-4 a c} (b d-a e)\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}

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Mathematica [A]  time = 1.39, size = 349, normalized size = 0.91 \[ \frac {\frac {\sqrt {2} \sqrt {c} \left (\frac {\left (-b d \sqrt {b^2-4 a c}+a e \sqrt {b^2-4 a c}+a b e+2 a c d+b^2 (-d)\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )}{\sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}-\frac {\left (b d \sqrt {b^2-4 a c}-a e \sqrt {b^2-4 a c}+a b e+2 a c d+b^2 (-d)\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {b^2-4 a c}}+\frac {(2 b d-a e) \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right )}{\sqrt {d}}+\frac {\log (x) (a e-2 b d)}{\sqrt {d}}-\frac {a \sqrt {d+e x^2}}{x^2}}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x^2]/(x^3*(a + b*x^2 + c*x^4)),x]

[Out]

(-((a*Sqrt[d + e*x^2])/x^2) + (Sqrt[2]*Sqrt[c]*(((-(b^2*d) + 2*a*c*d - b*Sqrt[b^2 - 4*a*c]*d + a*b*e + a*Sqrt[
b^2 - 4*a*c]*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d
+ (-b + Sqrt[b^2 - 4*a*c])*e] - ((-(b^2*d) + 2*a*c*d + b*Sqrt[b^2 - 4*a*c]*d + a*b*e - a*Sqrt[b^2 - 4*a*c]*e)
*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[b^
2 - 4*a*c])*e]))/Sqrt[b^2 - 4*a*c] + ((-2*b*d + a*e)*Log[x])/Sqrt[d] + ((2*b*d - a*e)*Log[d + Sqrt[d]*Sqrt[d +
e*x^2]])/Sqrt[d])/(2*a^2)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(1/2)/x^3/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(1/2)/x^3/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const ge
n & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueDone

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maple [C]  time = 0.03, size = 401, normalized size = 1.05 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^(1/2)/x^3/(c*x^4+b*x^2+a),x)

[Out]

1/a^2*b*d^(1/2)*ln((2*d+2*(e*x^2+d)^(1/2)*d^(1/2))/x)-1/2/a^2*b*(e*x^2+d)^(1/2)-1/2/a^2*e^(1/2)*x*b+1/4/a^2*su
m((c*(-a*e+b*d)*_R^6+(-4*a*b*e^2-a*c*d*e+4*b^2*d*e-3*b*c*d^2)*_R^4+d*(4*a*b*e^2+a*c*d*e-4*b^2*d*e+3*b*c*d^2)*_
R^2+a*c*d^3*e-b*c*d^4)/(_R^7*c+3*_R^5*b*e-3*_R^5*c*d+8*_R^3*a*e^2-4*_R^3*b*d*e+3*_R^3*c*d^2+_R*b*d^2*e-_R*c*d^
3)*ln(-e^(1/2)*x-_R+(e*x^2+d)^(1/2)),_R=RootOf(_Z^8*c+(4*b*e-4*c*d)*_Z^6+c*d^4+(16*a*e^2-8*b*d*e+6*c*d^2)*_Z^4
+(4*b*d^2*e-4*c*d^3)*_Z^2))+1/2/a^2*b*d/(-e^(1/2)*x+(e*x^2+d)^(1/2))-1/2/a/d/x^2*(e*x^2+d)^(3/2)-1/2/a*e/d^(1/
2)*ln((2*d+2*(e*x^2+d)^(1/2)*d^(1/2))/x)+1/2/a*e/d*(e*x^2+d)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x^{2} + d}}{{\left (c x^{4} + b x^{2} + a\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(1/2)/x^3/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

integrate(sqrt(e*x^2 + d)/((c*x^4 + b*x^2 + a)*x^3), x)

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mupad [B]  time = 5.46, size = 19959, normalized size = 52.25 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2)^(1/2)/(x^3*(a + b*x^2 + c*x^4)),x)

[Out]

(atan((((a*e - 2*b*d)*(((d + e*x^2)^(1/2)*(6*a^4*c^5*e^12 + 4*a^2*c^7*d^4*e^8 + 6*a^3*c^6*d^2*e^10 + 4*b^4*c^5
*d^4*e^8 + 21*a^2*b^2*c^5*d^2*e^10 - 18*a^3*b*c^5*d*e^11 - 8*a*b^2*c^6*d^4*e^8 - 12*a*b^3*c^5*d^3*e^9))/(2*a^4
) - (((16*a^5*b*c^4*e^12 + 20*a^5*c^5*d*e^11 + a^3*b^5*c^2*e^12 - 8*a^4*b^3*c^3*e^12 + 20*a^4*c^6*d^3*e^9 + 40
*a^2*b^3*c^5*d^4*e^8 - 20*a^2*b^4*c^4*d^3*e^9 - 27*a^2*b^5*c^3*d^2*e^10 - 20*a^3*b^2*c^5*d^3*e^9 + 84*a^3*b^3*
c^4*d^2*e^10 - 8*a*b^5*c^4*d^4*e^8 + 6*a*b^6*c^3*d^3*e^9 + 2*a*b^7*c^2*d^2*e^10 - 3*a^2*b^6*c^2*d*e^11 - 32*a^
3*b*c^6*d^4*e^8 + 28*a^3*b^4*c^3*d*e^11 - 36*a^4*b*c^5*d^2*e^10 - 68*a^4*b^2*c^4*d*e^11)/a^4 - ((a*e - 2*b*d)*
(((d + e*x^2)^(1/2)*(240*a^6*b*c^4*e^11 + 64*a^6*c^5*d*e^10 + 20*a^4*b^5*c^2*e^11 - 140*a^5*b^3*c^3*e^11 + 160
*a^5*c^6*d^3*e^8 - 32*a^2*b^6*c^3*d^3*e^8 + 32*a^2*b^7*c^2*d^2*e^9 + 224*a^3*b^4*c^4*d^3*e^8 - 208*a^3*b^5*c^3
*d^2*e^9 - 432*a^4*b^2*c^5*d^3*e^8 + 272*a^4*b^3*c^4*d^2*e^9 - 48*a^3*b^6*c^2*d*e^10 + 348*a^4*b^4*c^3*d*e^10
+ 224*a^5*b*c^5*d^2*e^9 - 648*a^5*b^2*c^4*d*e^10))/(2*a^4) - ((a*e - 2*b*d)*((128*a^8*c^4*e^11 + 8*a^6*b^4*c^2
*e^11 - 64*a^7*b^2*c^3*e^11 + 128*a^7*c^5*d^2*e^9 + 32*a^5*b^3*c^4*d^3*e^8 - 24*a^5*b^4*c^3*d^2*e^9 + 64*a^6*b
^2*c^4*d^2*e^9 - 256*a^7*b*c^4*d*e^10 - 8*a^5*b^5*c^2*d*e^10 - 128*a^6*b*c^5*d^3*e^8 + 96*a^6*b^3*c^3*d*e^10)/
a^4 - ((d + e*x^2)^(1/2)*(a*e - 2*b*d)*(1024*a^9*c^4*e^10 + 64*a^7*b^4*c^2*e^10 - 512*a^8*b^2*c^3*e^10 + 1536*
a^8*c^5*d^2*e^8 + 128*a^6*b^4*c^3*d^2*e^8 - 896*a^7*b^2*c^4*d^2*e^8 - 1792*a^8*b*c^4*d*e^9 - 128*a^6*b^5*c^2*d
*e^9 + 960*a^7*b^3*c^3*d*e^9))/(8*a^6*d^(1/2))))/(4*a^2*d^(1/2))))/(4*a^2*d^(1/2)))*(a*e - 2*b*d))/(4*a^2*d^(1
/2)))*1i)/(4*a^2*d^(1/2)) + ((a*e - 2*b*d)*(((d + e*x^2)^(1/2)*(6*a^4*c^5*e^12 + 4*a^2*c^7*d^4*e^8 + 6*a^3*c^6
*d^2*e^10 + 4*b^4*c^5*d^4*e^8 + 21*a^2*b^2*c^5*d^2*e^10 - 18*a^3*b*c^5*d*e^11 - 8*a*b^2*c^6*d^4*e^8 - 12*a*b^3
*c^5*d^3*e^9))/(2*a^4) + (((16*a^5*b*c^4*e^12 + 20*a^5*c^5*d*e^11 + a^3*b^5*c^2*e^12 - 8*a^4*b^3*c^3*e^12 + 20
*a^4*c^6*d^3*e^9 + 40*a^2*b^3*c^5*d^4*e^8 - 20*a^2*b^4*c^4*d^3*e^9 - 27*a^2*b^5*c^3*d^2*e^10 - 20*a^3*b^2*c^5*
d^3*e^9 + 84*a^3*b^3*c^4*d^2*e^10 - 8*a*b^5*c^4*d^4*e^8 + 6*a*b^6*c^3*d^3*e^9 + 2*a*b^7*c^2*d^2*e^10 - 3*a^2*b
^6*c^2*d*e^11 - 32*a^3*b*c^6*d^4*e^8 + 28*a^3*b^4*c^3*d*e^11 - 36*a^4*b*c^5*d^2*e^10 - 68*a^4*b^2*c^4*d*e^11)/
a^4 + ((a*e - 2*b*d)*(((d + e*x^2)^(1/2)*(240*a^6*b*c^4*e^11 + 64*a^6*c^5*d*e^10 + 20*a^4*b^5*c^2*e^11 - 140*a
^5*b^3*c^3*e^11 + 160*a^5*c^6*d^3*e^8 - 32*a^2*b^6*c^3*d^3*e^8 + 32*a^2*b^7*c^2*d^2*e^9 + 224*a^3*b^4*c^4*d^3*
e^8 - 208*a^3*b^5*c^3*d^2*e^9 - 432*a^4*b^2*c^5*d^3*e^8 + 272*a^4*b^3*c^4*d^2*e^9 - 48*a^3*b^6*c^2*d*e^10 + 34
8*a^4*b^4*c^3*d*e^10 + 224*a^5*b*c^5*d^2*e^9 - 648*a^5*b^2*c^4*d*e^10))/(2*a^4) + ((a*e - 2*b*d)*((128*a^8*c^4
*e^11 + 8*a^6*b^4*c^2*e^11 - 64*a^7*b^2*c^3*e^11 + 128*a^7*c^5*d^2*e^9 + 32*a^5*b^3*c^4*d^3*e^8 - 24*a^5*b^4*c
^3*d^2*e^9 + 64*a^6*b^2*c^4*d^2*e^9 - 256*a^7*b*c^4*d*e^10 - 8*a^5*b^5*c^2*d*e^10 - 128*a^6*b*c^5*d^3*e^8 + 96
*a^6*b^3*c^3*d*e^10)/a^4 + ((d + e*x^2)^(1/2)*(a*e - 2*b*d)*(1024*a^9*c^4*e^10 + 64*a^7*b^4*c^2*e^10 - 512*a^8
*b^2*c^3*e^10 + 1536*a^8*c^5*d^2*e^8 + 128*a^6*b^4*c^3*d^2*e^8 - 896*a^7*b^2*c^4*d^2*e^8 - 1792*a^8*b*c^4*d*e^
9 - 128*a^6*b^5*c^2*d*e^9 + 960*a^7*b^3*c^3*d*e^9))/(8*a^6*d^(1/2))))/(4*a^2*d^(1/2))))/(4*a^2*d^(1/2)))*(a*e
- 2*b*d))/(4*a^2*d^(1/2)))*1i)/(4*a^2*d^(1/2)))/(((a^3*c^5*e^13)/2 + a*c^7*d^4*e^9 - 2*b*c^7*d^5*e^8 + (3*a^2*
c^6*d^2*e^11)/2 + 2*b^2*c^6*d^4*e^9 - 4*a*b*c^6*d^3*e^10 - (3*a^2*b*c^5*d*e^12)/2 + a*b^2*c^5*d^2*e^11)/a^4 -
((a*e - 2*b*d)*(((d + e*x^2)^(1/2)*(6*a^4*c^5*e^12 + 4*a^2*c^7*d^4*e^8 + 6*a^3*c^6*d^2*e^10 + 4*b^4*c^5*d^4*e^
8 + 21*a^2*b^2*c^5*d^2*e^10 - 18*a^3*b*c^5*d*e^11 - 8*a*b^2*c^6*d^4*e^8 - 12*a*b^3*c^5*d^3*e^9))/(2*a^4) - (((
16*a^5*b*c^4*e^12 + 20*a^5*c^5*d*e^11 + a^3*b^5*c^2*e^12 - 8*a^4*b^3*c^3*e^12 + 20*a^4*c^6*d^3*e^9 + 40*a^2*b^
3*c^5*d^4*e^8 - 20*a^2*b^4*c^4*d^3*e^9 - 27*a^2*b^5*c^3*d^2*e^10 - 20*a^3*b^2*c^5*d^3*e^9 + 84*a^3*b^3*c^4*d^2
*e^10 - 8*a*b^5*c^4*d^4*e^8 + 6*a*b^6*c^3*d^3*e^9 + 2*a*b^7*c^2*d^2*e^10 - 3*a^2*b^6*c^2*d*e^11 - 32*a^3*b*c^6
*d^4*e^8 + 28*a^3*b^4*c^3*d*e^11 - 36*a^4*b*c^5*d^2*e^10 - 68*a^4*b^2*c^4*d*e^11)/a^4 - ((a*e - 2*b*d)*(((d +
e*x^2)^(1/2)*(240*a^6*b*c^4*e^11 + 64*a^6*c^5*d*e^10 + 20*a^4*b^5*c^2*e^11 - 140*a^5*b^3*c^3*e^11 + 160*a^5*c^
6*d^3*e^8 - 32*a^2*b^6*c^3*d^3*e^8 + 32*a^2*b^7*c^2*d^2*e^9 + 224*a^3*b^4*c^4*d^3*e^8 - 208*a^3*b^5*c^3*d^2*e^
9 - 432*a^4*b^2*c^5*d^3*e^8 + 272*a^4*b^3*c^4*d^2*e^9 - 48*a^3*b^6*c^2*d*e^10 + 348*a^4*b^4*c^3*d*e^10 + 224*a
^5*b*c^5*d^2*e^9 - 648*a^5*b^2*c^4*d*e^10))/(2*a^4) - ((a*e - 2*b*d)*((128*a^8*c^4*e^11 + 8*a^6*b^4*c^2*e^11 -
64*a^7*b^2*c^3*e^11 + 128*a^7*c^5*d^2*e^9 + 32*a^5*b^3*c^4*d^3*e^8 - 24*a^5*b^4*c^3*d^2*e^9 + 64*a^6*b^2*c^4*
d^2*e^9 - 256*a^7*b*c^4*d*e^10 - 8*a^5*b^5*c^2*d*e^10 - 128*a^6*b*c^5*d^3*e^8 + 96*a^6*b^3*c^3*d*e^10)/a^4 - (
(d + e*x^2)^(1/2)*(a*e - 2*b*d)*(1024*a^9*c^4*e^10 + 64*a^7*b^4*c^2*e^10 - 512*a^8*b^2*c^3*e^10 + 1536*a^8*c^5
*d^2*e^8 + 128*a^6*b^4*c^3*d^2*e^8 - 896*a^7*b^2*c^4*d^2*e^8 - 1792*a^8*b*c^4*d*e^9 - 128*a^6*b^5*c^2*d*e^9 +
960*a^7*b^3*c^3*d*e^9))/(8*a^6*d^(1/2))))/(4*a^2*d^(1/2))))/(4*a^2*d^(1/2)))*(a*e - 2*b*d))/(4*a^2*d^(1/2))))/
(4*a^2*d^(1/2)) + ((a*e - 2*b*d)*(((d + e*x^2)^(1/2)*(6*a^4*c^5*e^12 + 4*a^2*c^7*d^4*e^8 + 6*a^3*c^6*d^2*e^10
+ 4*b^4*c^5*d^4*e^8 + 21*a^2*b^2*c^5*d^2*e^10 - 18*a^3*b*c^5*d*e^11 - 8*a*b^2*c^6*d^4*e^8 - 12*a*b^3*c^5*d^3*e
^9))/(2*a^4) + (((16*a^5*b*c^4*e^12 + 20*a^5*c^5*d*e^11 + a^3*b^5*c^2*e^12 - 8*a^4*b^3*c^3*e^12 + 20*a^4*c^6*d
^3*e^9 + 40*a^2*b^3*c^5*d^4*e^8 - 20*a^2*b^4*c^4*d^3*e^9 - 27*a^2*b^5*c^3*d^2*e^10 - 20*a^3*b^2*c^5*d^3*e^9 +
84*a^3*b^3*c^4*d^2*e^10 - 8*a*b^5*c^4*d^4*e^8 + 6*a*b^6*c^3*d^3*e^9 + 2*a*b^7*c^2*d^2*e^10 - 3*a^2*b^6*c^2*d*e
^11 - 32*a^3*b*c^6*d^4*e^8 + 28*a^3*b^4*c^3*d*e^11 - 36*a^4*b*c^5*d^2*e^10 - 68*a^4*b^2*c^4*d*e^11)/a^4 + ((a*
e - 2*b*d)*(((d + e*x^2)^(1/2)*(240*a^6*b*c^4*e^11 + 64*a^6*c^5*d*e^10 + 20*a^4*b^5*c^2*e^11 - 140*a^5*b^3*c^3
*e^11 + 160*a^5*c^6*d^3*e^8 - 32*a^2*b^6*c^3*d^3*e^8 + 32*a^2*b^7*c^2*d^2*e^9 + 224*a^3*b^4*c^4*d^3*e^8 - 208*
a^3*b^5*c^3*d^2*e^9 - 432*a^4*b^2*c^5*d^3*e^8 + 272*a^4*b^3*c^4*d^2*e^9 - 48*a^3*b^6*c^2*d*e^10 + 348*a^4*b^4*
c^3*d*e^10 + 224*a^5*b*c^5*d^2*e^9 - 648*a^5*b^2*c^4*d*e^10))/(2*a^4) + ((a*e - 2*b*d)*((128*a^8*c^4*e^11 + 8*
a^6*b^4*c^2*e^11 - 64*a^7*b^2*c^3*e^11 + 128*a^7*c^5*d^2*e^9 + 32*a^5*b^3*c^4*d^3*e^8 - 24*a^5*b^4*c^3*d^2*e^9
+ 64*a^6*b^2*c^4*d^2*e^9 - 256*a^7*b*c^4*d*e^10 - 8*a^5*b^5*c^2*d*e^10 - 128*a^6*b*c^5*d^3*e^8 + 96*a^6*b^3*c
^3*d*e^10)/a^4 + ((d + e*x^2)^(1/2)*(a*e - 2*b*d)*(1024*a^9*c^4*e^10 + 64*a^7*b^4*c^2*e^10 - 512*a^8*b^2*c^3*e
^10 + 1536*a^8*c^5*d^2*e^8 + 128*a^6*b^4*c^3*d^2*e^8 - 896*a^7*b^2*c^4*d^2*e^8 - 1792*a^8*b*c^4*d*e^9 - 128*a^
6*b^5*c^2*d*e^9 + 960*a^7*b^3*c^3*d*e^9))/(8*a^6*d^(1/2))))/(4*a^2*d^(1/2))))/(4*a^2*d^(1/2)))*(a*e - 2*b*d))/
(4*a^2*d^(1/2))))/(4*a^2*d^(1/2))))*(a*e - 2*b*d)*1i)/(2*a^2*d^(1/2)) - atan(((((64*a^5*b*c^4*e^12 + 80*a^5*c^
5*d*e^11 + 4*a^3*b^5*c^2*e^12 - 32*a^4*b^3*c^3*e^12 + 80*a^4*c^6*d^3*e^9 + 160*a^2*b^3*c^5*d^4*e^8 - 80*a^2*b^
4*c^4*d^3*e^9 - 108*a^2*b^5*c^3*d^2*e^10 - 80*a^3*b^2*c^5*d^3*e^9 + 336*a^3*b^3*c^4*d^2*e^10 - 32*a*b^5*c^4*d^
4*e^8 + 24*a*b^6*c^3*d^3*e^9 + 8*a*b^7*c^2*d^2*e^10 - 12*a^2*b^6*c^2*d*e^11 - 128*a^3*b*c^6*d^4*e^8 + 112*a^3*
b^4*c^3*d*e^11 - 144*a^4*b*c^5*d^2*e^10 - 272*a^4*b^2*c^4*d*e^11)/(4*a^4) + (((512*a^8*c^4*e^11 + 32*a^6*b^4*c
^2*e^11 - 256*a^7*b^2*c^3*e^11 + 512*a^7*c^5*d^2*e^9 + 128*a^5*b^3*c^4*d^3*e^8 - 96*a^5*b^4*c^3*d^2*e^9 + 256*
a^6*b^2*c^4*d^2*e^9 - 1024*a^7*b*c^4*d*e^10 - 32*a^5*b^5*c^2*d*e^10 - 512*a^6*b*c^5*d^3*e^8 + 384*a^6*b^3*c^3*
d*e^10)/(4*a^4) - ((d + e*x^2)^(1/2)*(-(8*a^3*c^3*d - b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^
2*b^2*c^2*d + 8*a*b^4*c*d + a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a
*c - b^2)^3)^(1/2) + 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*(1024
*a^9*c^4*e^10 + 64*a^7*b^4*c^2*e^10 - 512*a^8*b^2*c^3*e^10 + 1536*a^8*c^5*d^2*e^8 + 128*a^6*b^4*c^3*d^2*e^8 -
896*a^7*b^2*c^4*d^2*e^8 - 1792*a^8*b*c^4*d*e^9 - 128*a^6*b^5*c^2*d*e^9 + 960*a^7*b^3*c^3*d*e^9))/(2*a^4))*(-(8
*a^3*c^3*d - b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d + a*b^2*e*(-(4*
a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c*d*(-(4*a*c -
b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) - ((d + e*x^2)^(1/2)*(240*a^6*b*c^4*e^11 + 64*
a^6*c^5*d*e^10 + 20*a^4*b^5*c^2*e^11 - 140*a^5*b^3*c^3*e^11 + 160*a^5*c^6*d^3*e^8 - 32*a^2*b^6*c^3*d^3*e^8 + 3
2*a^2*b^7*c^2*d^2*e^9 + 224*a^3*b^4*c^4*d^3*e^8 - 208*a^3*b^5*c^3*d^2*e^9 - 432*a^4*b^2*c^5*d^3*e^8 + 272*a^4*
b^3*c^4*d^2*e^9 - 48*a^3*b^6*c^2*d*e^10 + 348*a^4*b^4*c^3*d*e^10 + 224*a^5*b*c^5*d^2*e^9 - 648*a^5*b^2*c^4*d*e
^10))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*
c*d + a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) + 2
*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2))*(-(8*a^3*c^3*d - b^6*d - b
^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d + a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) -
7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^
4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) - ((d + e*x^2)^(1/2)*(6*a^4*c^5*e^12 + 4*a^2*c^7*d^4*e^8 + 6*a^3*c^6
*d^2*e^10 + 4*b^4*c^5*d^4*e^8 + 21*a^2*b^2*c^5*d^2*e^10 - 18*a^3*b*c^5*d*e^11 - 8*a*b^2*c^6*d^4*e^8 - 12*a*b^3
*c^5*d^3*e^9))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d +
8*a*b^4*c*d + a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c - b^2)^3)^
(1/2) + 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*1i - (((64*a^5*b*c
^4*e^12 + 80*a^5*c^5*d*e^11 + 4*a^3*b^5*c^2*e^12 - 32*a^4*b^3*c^3*e^12 + 80*a^4*c^6*d^3*e^9 + 160*a^2*b^3*c^5*
d^4*e^8 - 80*a^2*b^4*c^4*d^3*e^9 - 108*a^2*b^5*c^3*d^2*e^10 - 80*a^3*b^2*c^5*d^3*e^9 + 336*a^3*b^3*c^4*d^2*e^1
0 - 32*a*b^5*c^4*d^4*e^8 + 24*a*b^6*c^3*d^3*e^9 + 8*a*b^7*c^2*d^2*e^10 - 12*a^2*b^6*c^2*d*e^11 - 128*a^3*b*c^6
*d^4*e^8 + 112*a^3*b^4*c^3*d*e^11 - 144*a^4*b*c^5*d^2*e^10 - 272*a^4*b^2*c^4*d*e^11)/(4*a^4) + (((512*a^8*c^4*
e^11 + 32*a^6*b^4*c^2*e^11 - 256*a^7*b^2*c^3*e^11 + 512*a^7*c^5*d^2*e^9 + 128*a^5*b^3*c^4*d^3*e^8 - 96*a^5*b^4
*c^3*d^2*e^9 + 256*a^6*b^2*c^4*d^2*e^9 - 1024*a^7*b*c^4*d*e^10 - 32*a^5*b^5*c^2*d*e^10 - 512*a^6*b*c^5*d^3*e^8
+ 384*a^6*b^3*c^3*d*e^10)/(4*a^4) + ((d + e*x^2)^(1/2)*(-(8*a^3*c^3*d - b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2
) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d + a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2
*e - a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b
^2*c)))^(1/2)*(1024*a^9*c^4*e^10 + 64*a^7*b^4*c^2*e^10 - 512*a^8*b^2*c^3*e^10 + 1536*a^8*c^5*d^2*e^8 + 128*a^6
*b^4*c^3*d^2*e^8 - 896*a^7*b^2*c^4*d^2*e^8 - 1792*a^8*b*c^4*d*e^9 - 128*a^6*b^5*c^2*d*e^9 + 960*a^7*b^3*c^3*d*
e^9))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*
c*d + a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) + 2
*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + ((d + e*x^2)^(1/2)*(240*a
^6*b*c^4*e^11 + 64*a^6*c^5*d*e^10 + 20*a^4*b^5*c^2*e^11 - 140*a^5*b^3*c^3*e^11 + 160*a^5*c^6*d^3*e^8 - 32*a^2*
b^6*c^3*d^3*e^8 + 32*a^2*b^7*c^2*d^2*e^9 + 224*a^3*b^4*c^4*d^3*e^8 - 208*a^3*b^5*c^3*d^2*e^9 - 432*a^4*b^2*c^5
*d^3*e^8 + 272*a^4*b^3*c^4*d^2*e^9 - 48*a^3*b^6*c^2*d*e^10 + 348*a^4*b^4*c^3*d*e^10 + 224*a^5*b*c^5*d^2*e^9 -
648*a^5*b^2*c^4*d*e^10))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b
^2*c^2*d + 8*a*b^4*c*d + a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c
- b^2)^3)^(1/2) + 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2))*(-(8*a^
3*c^3*d - b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d + a*b^2*e*(-(4*a*c
- b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c*d*(-(4*a*c - b^
2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + ((d + e*x^2)^(1/2)*(6*a^4*c^5*e^12 + 4*a^2*c^7*
d^4*e^8 + 6*a^3*c^6*d^2*e^10 + 4*b^4*c^5*d^4*e^8 + 21*a^2*b^2*c^5*d^2*e^10 - 18*a^3*b*c^5*d*e^11 - 8*a*b^2*c^6
*d^4*e^8 - 12*a*b^3*c^5*d^3*e^9))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e -
18*a^2*b^2*c^2*d + 8*a*b^4*c*d + a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*
(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)
*1i)/((a^3*c^5*e^13 + 2*a*c^7*d^4*e^9 - 4*b*c^7*d^5*e^8 + 3*a^2*c^6*d^2*e^11 + 4*b^2*c^6*d^4*e^9 - 8*a*b*c^6*d
^3*e^10 - 3*a^2*b*c^5*d*e^12 + 2*a*b^2*c^5*d^2*e^11)/(2*a^4) + (((64*a^5*b*c^4*e^12 + 80*a^5*c^5*d*e^11 + 4*a^
3*b^5*c^2*e^12 - 32*a^4*b^3*c^3*e^12 + 80*a^4*c^6*d^3*e^9 + 160*a^2*b^3*c^5*d^4*e^8 - 80*a^2*b^4*c^4*d^3*e^9 -
108*a^2*b^5*c^3*d^2*e^10 - 80*a^3*b^2*c^5*d^3*e^9 + 336*a^3*b^3*c^4*d^2*e^10 - 32*a*b^5*c^4*d^4*e^8 + 24*a*b^
6*c^3*d^3*e^9 + 8*a*b^7*c^2*d^2*e^10 - 12*a^2*b^6*c^2*d*e^11 - 128*a^3*b*c^6*d^4*e^8 + 112*a^3*b^4*c^3*d*e^11
- 144*a^4*b*c^5*d^2*e^10 - 272*a^4*b^2*c^4*d*e^11)/(4*a^4) + (((512*a^8*c^4*e^11 + 32*a^6*b^4*c^2*e^11 - 256*a
^7*b^2*c^3*e^11 + 512*a^7*c^5*d^2*e^9 + 128*a^5*b^3*c^4*d^3*e^8 - 96*a^5*b^4*c^3*d^2*e^9 + 256*a^6*b^2*c^4*d^2
*e^9 - 1024*a^7*b*c^4*d*e^10 - 32*a^5*b^5*c^2*d*e^10 - 512*a^6*b*c^5*d^3*e^8 + 384*a^6*b^3*c^3*d*e^10)/(4*a^4)
- ((d + e*x^2)^(1/2)*(-(8*a^3*c^3*d - b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8
*a*b^4*c*d + a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c - b^2)^3)^(1
/2) + 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*(1024*a^9*c^4*e^10 +
64*a^7*b^4*c^2*e^10 - 512*a^8*b^2*c^3*e^10 + 1536*a^8*c^5*d^2*e^8 + 128*a^6*b^4*c^3*d^2*e^8 - 896*a^7*b^2*c^4
*d^2*e^8 - 1792*a^8*b*c^4*d*e^9 - 128*a^6*b^5*c^2*d*e^9 + 960*a^7*b^3*c^3*d*e^9))/(2*a^4))*(-(8*a^3*c^3*d - b^
6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d + a*b^2*e*(-(4*a*c - b^2)^3)^(
1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))
/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) - ((d + e*x^2)^(1/2)*(240*a^6*b*c^4*e^11 + 64*a^6*c^5*d*e^10
+ 20*a^4*b^5*c^2*e^11 - 140*a^5*b^3*c^3*e^11 + 160*a^5*c^6*d^3*e^8 - 32*a^2*b^6*c^3*d^3*e^8 + 32*a^2*b^7*c^2*d
^2*e^9 + 224*a^3*b^4*c^4*d^3*e^8 - 208*a^3*b^5*c^3*d^2*e^9 - 432*a^4*b^2*c^5*d^3*e^8 + 272*a^4*b^3*c^4*d^2*e^9
- 48*a^3*b^6*c^2*d*e^10 + 348*a^4*b^4*c^3*d*e^10 + 224*a^5*b*c^5*d^2*e^9 - 648*a^5*b^2*c^4*d*e^10))/(2*a^4))*
(-(8*a^3*c^3*d - b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d + a*b^2*e*(
-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c*d*(-(4*a
*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2))*(-(8*a^3*c^3*d - b^6*d - b^3*d*(-(4*a*c -
b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d + a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e +
12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*
c^2 - 8*a^5*b^2*c)))^(1/2) - ((d + e*x^2)^(1/2)*(6*a^4*c^5*e^12 + 4*a^2*c^7*d^4*e^8 + 6*a^3*c^6*d^2*e^10 + 4*b
^4*c^5*d^4*e^8 + 21*a^2*b^2*c^5*d^2*e^10 - 18*a^3*b*c^5*d*e^11 - 8*a*b^2*c^6*d^4*e^8 - 12*a*b^3*c^5*d^3*e^9))/
(2*a^4))*(-(8*a^3*c^3*d - b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d +
a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c
*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (((64*a^5*b*c^4*e^12 + 80*a^5*c
^5*d*e^11 + 4*a^3*b^5*c^2*e^12 - 32*a^4*b^3*c^3*e^12 + 80*a^4*c^6*d^3*e^9 + 160*a^2*b^3*c^5*d^4*e^8 - 80*a^2*b
^4*c^4*d^3*e^9 - 108*a^2*b^5*c^3*d^2*e^10 - 80*a^3*b^2*c^5*d^3*e^9 + 336*a^3*b^3*c^4*d^2*e^10 - 32*a*b^5*c^4*d
^4*e^8 + 24*a*b^6*c^3*d^3*e^9 + 8*a*b^7*c^2*d^2*e^10 - 12*a^2*b^6*c^2*d*e^11 - 128*a^3*b*c^6*d^4*e^8 + 112*a^3
*b^4*c^3*d*e^11 - 144*a^4*b*c^5*d^2*e^10 - 272*a^4*b^2*c^4*d*e^11)/(4*a^4) + (((512*a^8*c^4*e^11 + 32*a^6*b^4*
c^2*e^11 - 256*a^7*b^2*c^3*e^11 + 512*a^7*c^5*d^2*e^9 + 128*a^5*b^3*c^4*d^3*e^8 - 96*a^5*b^4*c^3*d^2*e^9 + 256
*a^6*b^2*c^4*d^2*e^9 - 1024*a^7*b*c^4*d*e^10 - 32*a^5*b^5*c^2*d*e^10 - 512*a^6*b*c^5*d^3*e^8 + 384*a^6*b^3*c^3
*d*e^10)/(4*a^4) + ((d + e*x^2)^(1/2)*(-(8*a^3*c^3*d - b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a
^2*b^2*c^2*d + 8*a*b^4*c*d + a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*
a*c - b^2)^3)^(1/2) + 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*(102
4*a^9*c^4*e^10 + 64*a^7*b^4*c^2*e^10 - 512*a^8*b^2*c^3*e^10 + 1536*a^8*c^5*d^2*e^8 + 128*a^6*b^4*c^3*d^2*e^8 -
896*a^7*b^2*c^4*d^2*e^8 - 1792*a^8*b*c^4*d*e^9 - 128*a^6*b^5*c^2*d*e^9 + 960*a^7*b^3*c^3*d*e^9))/(2*a^4))*(-(
8*a^3*c^3*d - b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d + a*b^2*e*(-(4
*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c*d*(-(4*a*c
- b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + ((d + e*x^2)^(1/2)*(240*a^6*b*c^4*e^11 + 64
*a^6*c^5*d*e^10 + 20*a^4*b^5*c^2*e^11 - 140*a^5*b^3*c^3*e^11 + 160*a^5*c^6*d^3*e^8 - 32*a^2*b^6*c^3*d^3*e^8 +
32*a^2*b^7*c^2*d^2*e^9 + 224*a^3*b^4*c^4*d^3*e^8 - 208*a^3*b^5*c^3*d^2*e^9 - 432*a^4*b^2*c^5*d^3*e^8 + 272*a^4
*b^3*c^4*d^2*e^9 - 48*a^3*b^6*c^2*d*e^10 + 348*a^4*b^4*c^3*d*e^10 + 224*a^5*b*c^5*d^2*e^9 - 648*a^5*b^2*c^4*d*
e^10))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4
*c*d + a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) +
2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2))*(-(8*a^3*c^3*d - b^6*d -
b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d + a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) -
7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a
^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + ((d + e*x^2)^(1/2)*(6*a^4*c^5*e^12 + 4*a^2*c^7*d^4*e^8 + 6*a^3*c^
6*d^2*e^10 + 4*b^4*c^5*d^4*e^8 + 21*a^2*b^2*c^5*d^2*e^10 - 18*a^3*b*c^5*d*e^11 - 8*a*b^2*c^6*d^4*e^8 - 12*a*b^
3*c^5*d^3*e^9))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d
+ 8*a*b^4*c*d + a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c - b^2)^3)
^(1/2) + 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)))*(-(8*a^3*c^3*d
- b^6*d - b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d + a*b^2*e*(-(4*a*c - b^2)^
3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e - a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1
/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*2i - (d + e*x^2)^(1/2)/(2*a*x^2) - atan(((((64*a^5*b*c^4*
e^12 + 80*a^5*c^5*d*e^11 + 4*a^3*b^5*c^2*e^12 - 32*a^4*b^3*c^3*e^12 + 80*a^4*c^6*d^3*e^9 + 160*a^2*b^3*c^5*d^4
*e^8 - 80*a^2*b^4*c^4*d^3*e^9 - 108*a^2*b^5*c^3*d^2*e^10 - 80*a^3*b^2*c^5*d^3*e^9 + 336*a^3*b^3*c^4*d^2*e^10 -
32*a*b^5*c^4*d^4*e^8 + 24*a*b^6*c^3*d^3*e^9 + 8*a*b^7*c^2*d^2*e^10 - 12*a^2*b^6*c^2*d*e^11 - 128*a^3*b*c^6*d^
4*e^8 + 112*a^3*b^4*c^3*d*e^11 - 144*a^4*b*c^5*d^2*e^10 - 272*a^4*b^2*c^4*d*e^11)/(4*a^4) + (((512*a^8*c^4*e^1
1 + 32*a^6*b^4*c^2*e^11 - 256*a^7*b^2*c^3*e^11 + 512*a^7*c^5*d^2*e^9 + 128*a^5*b^3*c^4*d^3*e^8 - 96*a^5*b^4*c^
3*d^2*e^9 + 256*a^6*b^2*c^4*d^2*e^9 - 1024*a^7*b*c^4*d*e^10 - 32*a^5*b^5*c^2*d*e^10 - 512*a^6*b*c^5*d^3*e^8 +
384*a^6*b^3*c^3*d*e^10)/(4*a^4) - ((d + e*x^2)^(1/2)*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) +
a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d - a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e
+ a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*
c)))^(1/2)*(1024*a^9*c^4*e^10 + 64*a^7*b^4*c^2*e^10 - 512*a^8*b^2*c^3*e^10 + 1536*a^8*c^5*d^2*e^8 + 128*a^6*b^
4*c^3*d^2*e^8 - 896*a^7*b^2*c^4*d^2*e^8 - 1792*a^8*b*c^4*d*e^9 - 128*a^6*b^5*c^2*d*e^9 + 960*a^7*b^3*c^3*d*e^9
))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d
- a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*
b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) - ((d + e*x^2)^(1/2)*(240*a^6*
b*c^4*e^11 + 64*a^6*c^5*d*e^10 + 20*a^4*b^5*c^2*e^11 - 140*a^5*b^3*c^3*e^11 + 160*a^5*c^6*d^3*e^8 - 32*a^2*b^6
*c^3*d^3*e^8 + 32*a^2*b^7*c^2*d^2*e^9 + 224*a^3*b^4*c^4*d^3*e^8 - 208*a^3*b^5*c^3*d^2*e^9 - 432*a^4*b^2*c^5*d^
3*e^8 + 272*a^4*b^3*c^4*d^2*e^9 - 48*a^3*b^6*c^2*d*e^10 + 348*a^4*b^4*c^3*d*e^10 + 224*a^5*b*c^5*d^2*e^9 - 648
*a^5*b^2*c^4*d*e^10))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*
c^2*d + 8*a*b^4*c*d - a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a^2*c*e*(-(4*a*c - b
^2)^3)^(1/2) - 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2))*(-(8*a^3*c
^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d - a*b^2*e*(-(4*a*c -
b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c*d*(-(4*a*c - b^2)^
3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) - ((d + e*x^2)^(1/2)*(6*a^4*c^5*e^12 + 4*a^2*c^7*d^4
*e^8 + 6*a^3*c^6*d^2*e^10 + 4*b^4*c^5*d^4*e^8 + 21*a^2*b^2*c^5*d^2*e^10 - 18*a^3*b*c^5*d*e^11 - 8*a*b^2*c^6*d^
4*e^8 - 12*a*b^3*c^5*d^3*e^9))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18
*a^2*b^2*c^2*d + 8*a*b^4*c*d - a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a^2*c*e*(-(
4*a*c - b^2)^3)^(1/2) - 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*1i
- (((64*a^5*b*c^4*e^12 + 80*a^5*c^5*d*e^11 + 4*a^3*b^5*c^2*e^12 - 32*a^4*b^3*c^3*e^12 + 80*a^4*c^6*d^3*e^9 +
160*a^2*b^3*c^5*d^4*e^8 - 80*a^2*b^4*c^4*d^3*e^9 - 108*a^2*b^5*c^3*d^2*e^10 - 80*a^3*b^2*c^5*d^3*e^9 + 336*a^3
*b^3*c^4*d^2*e^10 - 32*a*b^5*c^4*d^4*e^8 + 24*a*b^6*c^3*d^3*e^9 + 8*a*b^7*c^2*d^2*e^10 - 12*a^2*b^6*c^2*d*e^11
- 128*a^3*b*c^6*d^4*e^8 + 112*a^3*b^4*c^3*d*e^11 - 144*a^4*b*c^5*d^2*e^10 - 272*a^4*b^2*c^4*d*e^11)/(4*a^4) +
(((512*a^8*c^4*e^11 + 32*a^6*b^4*c^2*e^11 - 256*a^7*b^2*c^3*e^11 + 512*a^7*c^5*d^2*e^9 + 128*a^5*b^3*c^4*d^3*
e^8 - 96*a^5*b^4*c^3*d^2*e^9 + 256*a^6*b^2*c^4*d^2*e^9 - 1024*a^7*b*c^4*d*e^10 - 32*a^5*b^5*c^2*d*e^10 - 512*a
^6*b*c^5*d^3*e^8 + 384*a^6*b^3*c^3*d*e^10)/(4*a^4) + ((d + e*x^2)^(1/2)*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*
c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d - a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*
e + 12*a^3*b*c^2*e + a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a
^6*c^2 - 8*a^5*b^2*c)))^(1/2)*(1024*a^9*c^4*e^10 + 64*a^7*b^4*c^2*e^10 - 512*a^8*b^2*c^3*e^10 + 1536*a^8*c^5*d
^2*e^8 + 128*a^6*b^4*c^3*d^2*e^8 - 896*a^7*b^2*c^4*d^2*e^8 - 1792*a^8*b*c^4*d*e^9 - 128*a^6*b^5*c^2*d*e^9 + 96
0*a^7*b^3*c^3*d*e^9))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*
c^2*d + 8*a*b^4*c*d - a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a^2*c*e*(-(4*a*c - b
^2)^3)^(1/2) - 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + ((d + e*x
^2)^(1/2)*(240*a^6*b*c^4*e^11 + 64*a^6*c^5*d*e^10 + 20*a^4*b^5*c^2*e^11 - 140*a^5*b^3*c^3*e^11 + 160*a^5*c^6*d
^3*e^8 - 32*a^2*b^6*c^3*d^3*e^8 + 32*a^2*b^7*c^2*d^2*e^9 + 224*a^3*b^4*c^4*d^3*e^8 - 208*a^3*b^5*c^3*d^2*e^9 -
432*a^4*b^2*c^5*d^3*e^8 + 272*a^4*b^3*c^4*d^2*e^9 - 48*a^3*b^6*c^2*d*e^10 + 348*a^4*b^4*c^3*d*e^10 + 224*a^5*
b*c^5*d^2*e^9 - 648*a^5*b^2*c^4*d*e^10))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*
b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d - a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a
^2*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c))
)^(1/2))*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d -
a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c
*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + ((d + e*x^2)^(1/2)*(6*a^4*c^5*e
^12 + 4*a^2*c^7*d^4*e^8 + 6*a^3*c^6*d^2*e^10 + 4*b^4*c^5*d^4*e^8 + 21*a^2*b^2*c^5*d^2*e^10 - 18*a^3*b*c^5*d*e^
11 - 8*a*b^2*c^6*d^4*e^8 - 12*a*b^3*c^5*d^3*e^9))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(
1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d - a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*
c^2*e + a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^
5*b^2*c)))^(1/2)*1i)/((a^3*c^5*e^13 + 2*a*c^7*d^4*e^9 - 4*b*c^7*d^5*e^8 + 3*a^2*c^6*d^2*e^11 + 4*b^2*c^6*d^4*e
^9 - 8*a*b*c^6*d^3*e^10 - 3*a^2*b*c^5*d*e^12 + 2*a*b^2*c^5*d^2*e^11)/(2*a^4) + (((64*a^5*b*c^4*e^12 + 80*a^5*c
^5*d*e^11 + 4*a^3*b^5*c^2*e^12 - 32*a^4*b^3*c^3*e^12 + 80*a^4*c^6*d^3*e^9 + 160*a^2*b^3*c^5*d^4*e^8 - 80*a^2*b
^4*c^4*d^3*e^9 - 108*a^2*b^5*c^3*d^2*e^10 - 80*a^3*b^2*c^5*d^3*e^9 + 336*a^3*b^3*c^4*d^2*e^10 - 32*a*b^5*c^4*d
^4*e^8 + 24*a*b^6*c^3*d^3*e^9 + 8*a*b^7*c^2*d^2*e^10 - 12*a^2*b^6*c^2*d*e^11 - 128*a^3*b*c^6*d^4*e^8 + 112*a^3
*b^4*c^3*d*e^11 - 144*a^4*b*c^5*d^2*e^10 - 272*a^4*b^2*c^4*d*e^11)/(4*a^4) + (((512*a^8*c^4*e^11 + 32*a^6*b^4*
c^2*e^11 - 256*a^7*b^2*c^3*e^11 + 512*a^7*c^5*d^2*e^9 + 128*a^5*b^3*c^4*d^3*e^8 - 96*a^5*b^4*c^3*d^2*e^9 + 256
*a^6*b^2*c^4*d^2*e^9 - 1024*a^7*b*c^4*d*e^10 - 32*a^5*b^5*c^2*d*e^10 - 512*a^6*b*c^5*d^3*e^8 + 384*a^6*b^3*c^3
*d*e^10)/(4*a^4) - ((d + e*x^2)^(1/2)*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a
^2*b^2*c^2*d + 8*a*b^4*c*d - a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a^2*c*e*(-(4*
a*c - b^2)^3)^(1/2) - 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*(102
4*a^9*c^4*e^10 + 64*a^7*b^4*c^2*e^10 - 512*a^8*b^2*c^3*e^10 + 1536*a^8*c^5*d^2*e^8 + 128*a^6*b^4*c^3*d^2*e^8 -
896*a^7*b^2*c^4*d^2*e^8 - 1792*a^8*b*c^4*d*e^9 - 128*a^6*b^5*c^2*d*e^9 + 960*a^7*b^3*c^3*d*e^9))/(2*a^4))*(-(
8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d - a*b^2*e*(-(4
*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c*d*(-(4*a*c
- b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) - ((d + e*x^2)^(1/2)*(240*a^6*b*c^4*e^11 + 64
*a^6*c^5*d*e^10 + 20*a^4*b^5*c^2*e^11 - 140*a^5*b^3*c^3*e^11 + 160*a^5*c^6*d^3*e^8 - 32*a^2*b^6*c^3*d^3*e^8 +
32*a^2*b^7*c^2*d^2*e^9 + 224*a^3*b^4*c^4*d^3*e^8 - 208*a^3*b^5*c^3*d^2*e^9 - 432*a^4*b^2*c^5*d^3*e^8 + 272*a^4
*b^3*c^4*d^2*e^9 - 48*a^3*b^6*c^2*d*e^10 + 348*a^4*b^4*c^3*d*e^10 + 224*a^5*b*c^5*d^2*e^9 - 648*a^5*b^2*c^4*d*
e^10))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4
*c*d - a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) -
2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2))*(-(8*a^3*c^3*d - b^6*d +
b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d - a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) -
7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a
^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) - ((d + e*x^2)^(1/2)*(6*a^4*c^5*e^12 + 4*a^2*c^7*d^4*e^8 + 6*a^3*c^
6*d^2*e^10 + 4*b^4*c^5*d^4*e^8 + 21*a^2*b^2*c^5*d^2*e^10 - 18*a^3*b*c^5*d*e^11 - 8*a*b^2*c^6*d^4*e^8 - 12*a*b^
3*c^5*d^3*e^9))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d
+ 8*a*b^4*c*d - a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a^2*c*e*(-(4*a*c - b^2)^3)
^(1/2) - 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + (((64*a^5*b*c^4
*e^12 + 80*a^5*c^5*d*e^11 + 4*a^3*b^5*c^2*e^12 - 32*a^4*b^3*c^3*e^12 + 80*a^4*c^6*d^3*e^9 + 160*a^2*b^3*c^5*d^
4*e^8 - 80*a^2*b^4*c^4*d^3*e^9 - 108*a^2*b^5*c^3*d^2*e^10 - 80*a^3*b^2*c^5*d^3*e^9 + 336*a^3*b^3*c^4*d^2*e^10
- 32*a*b^5*c^4*d^4*e^8 + 24*a*b^6*c^3*d^3*e^9 + 8*a*b^7*c^2*d^2*e^10 - 12*a^2*b^6*c^2*d*e^11 - 128*a^3*b*c^6*d
^4*e^8 + 112*a^3*b^4*c^3*d*e^11 - 144*a^4*b*c^5*d^2*e^10 - 272*a^4*b^2*c^4*d*e^11)/(4*a^4) + (((512*a^8*c^4*e^
11 + 32*a^6*b^4*c^2*e^11 - 256*a^7*b^2*c^3*e^11 + 512*a^7*c^5*d^2*e^9 + 128*a^5*b^3*c^4*d^3*e^8 - 96*a^5*b^4*c
^3*d^2*e^9 + 256*a^6*b^2*c^4*d^2*e^9 - 1024*a^7*b*c^4*d*e^10 - 32*a^5*b^5*c^2*d*e^10 - 512*a^6*b*c^5*d^3*e^8 +
384*a^6*b^3*c^3*d*e^10)/(4*a^4) + ((d + e*x^2)^(1/2)*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2)
+ a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d - a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e
+ a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2
*c)))^(1/2)*(1024*a^9*c^4*e^10 + 64*a^7*b^4*c^2*e^10 - 512*a^8*b^2*c^3*e^10 + 1536*a^8*c^5*d^2*e^8 + 128*a^6*b
^4*c^3*d^2*e^8 - 896*a^7*b^2*c^4*d^2*e^8 - 1792*a^8*b*c^4*d*e^9 - 128*a^6*b^5*c^2*d*e^9 + 960*a^7*b^3*c^3*d*e^
9))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*
d - a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a
*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + ((d + e*x^2)^(1/2)*(240*a^6
*b*c^4*e^11 + 64*a^6*c^5*d*e^10 + 20*a^4*b^5*c^2*e^11 - 140*a^5*b^3*c^3*e^11 + 160*a^5*c^6*d^3*e^8 - 32*a^2*b^
6*c^3*d^3*e^8 + 32*a^2*b^7*c^2*d^2*e^9 + 224*a^3*b^4*c^4*d^3*e^8 - 208*a^3*b^5*c^3*d^2*e^9 - 432*a^4*b^2*c^5*d
^3*e^8 + 272*a^4*b^3*c^4*d^2*e^9 - 48*a^3*b^6*c^2*d*e^10 + 348*a^4*b^4*c^3*d*e^10 + 224*a^5*b*c^5*d^2*e^9 - 64
8*a^5*b^2*c^4*d*e^10))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2
*c^2*d + 8*a*b^4*c*d - a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a^2*c*e*(-(4*a*c -
b^2)^3)^(1/2) - 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2))*(-(8*a^3*
c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d - a*b^2*e*(-(4*a*c -
b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c*d*(-(4*a*c - b^2)
^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2) + ((d + e*x^2)^(1/2)*(6*a^4*c^5*e^12 + 4*a^2*c^7*d^
4*e^8 + 6*a^3*c^6*d^2*e^10 + 4*b^4*c^5*d^4*e^8 + 21*a^2*b^2*c^5*d^2*e^10 - 18*a^3*b*c^5*d*e^11 - 8*a*b^2*c^6*d
^4*e^8 - 12*a*b^3*c^5*d^3*e^9))/(2*a^4))*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 1
8*a^2*b^2*c^2*d + 8*a*b^4*c*d - a*b^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a^2*c*e*(-
(4*a*c - b^2)^3)^(1/2) - 2*a*b*c*d*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)))
*(-(8*a^3*c^3*d - b^6*d + b^3*d*(-(4*a*c - b^2)^3)^(1/2) + a*b^5*e - 18*a^2*b^2*c^2*d + 8*a*b^4*c*d - a*b^2*e*
(-(4*a*c - b^2)^3)^(1/2) - 7*a^2*b^3*c*e + 12*a^3*b*c^2*e + a^2*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b*c*d*(-(4*
a*c - b^2)^3)^(1/2))/(8*(a^4*b^4 + 16*a^6*c^2 - 8*a^5*b^2*c)))^(1/2)*2i

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d + e x^{2}}}{x^{3} \left (a + b x^{2} + c x^{4}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**(1/2)/x**3/(c*x**4+b*x**2+a),x)

[Out]

Integral(sqrt(d + e*x**2)/(x**3*(a + b*x**2 + c*x**4)), x)

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