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

Optimal. Leaf size=346 \[ -\frac {\left (-c d \left (d \sqrt {b^2-4 a c}-4 a e\right )+a e^2 \sqrt {b^2-4 a c}-b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} a \sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (-c d \left (d \sqrt {b^2-4 a c}+4 a e\right )+a e^2 \sqrt {b^2-4 a c}+b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} a \sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a} \]

[Out]

-d^(3/2)*arctanh((e*x^2+d)^(1/2)/d^(1/2))/a-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))*(-b*(a*e^2+c*d^2)+a*e^2*(-4*a*c+b^2)^(1/2)-c*d*(-4*a*e+d*(-4*a*c+b^2)^(1/2)))/a*2^(1/2)/c^(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))*(b*(a*e^2+c*d^2)+a*e^2*(-4*a*c+b^2)^(1/2)-c*d*(4*a*e+d*(-4*a*c+b^2)^(1/2))
)/a*2^(1/2)/c^(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 = 1.74, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1251, 897, 1287, 206, 1166, 208} \[ -\frac {\left (-c d \left (d \sqrt {b^2-4 a c}-4 a e\right )+a e^2 \sqrt {b^2-4 a c}-b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} a \sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (-c d \left (d \sqrt {b^2-4 a c}+4 a e\right )+a e^2 \sqrt {b^2-4 a c}+b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} a \sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d^(3/2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/a) - ((a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d - 4*a*e
) - b*(c*d^2 + a*e^2))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sq
rt[2]*a*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - ((a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(
Sqrt[b^2 - 4*a*c]*d + 4*a*e) + b*(c*d^2 + a*e^2))*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*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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 {\left (d+e x^2\right )^{3/2}}{x \left (a+b x^2+c x^4\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(d+e x)^{3/2}}{x \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{\left (-\frac {d}{e}+\frac {x^2}{e}\right ) \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^2 e}{a \left (d-x^2\right )}+\frac {e \left (d \left (c d^2-b d e+a e^2\right )-\left (c d^2-a e^2\right ) x^2\right )}{a \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 {d \left (c d^2-b d e+a e^2\right )+\left (-c d^2+a e^2\right ) 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}-\frac {d^2 \operatorname {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d+e x^2}\right )}{a}\\ &=-\frac {d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a}+\frac {\left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d-4 a e\right )-b \left (c d^2+a e^2\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 \sqrt {b^2-4 a c}}+\frac {\left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d+4 a e\right )+b \left (c d^2+a e^2\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 \sqrt {b^2-4 a c}}\\ &=-\frac {d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a}-\frac {\left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d-4 a e\right )-b \left (c d^2+a e^2\right )\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 \sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d+4 a e\right )+b \left (c d^2+a e^2\right )\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 \sqrt {c} \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 = 1.38, size = 333, normalized size = 0.96 \[ -\frac {\frac {\left (-c d \left (d \sqrt {b^2-4 a c}+4 a e\right )+a e^2 \sqrt {b^2-4 a c}+b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\left (c d \left (d \sqrt {b^2-4 a c}-4 a e\right )-a e^2 \sqrt {b^2-4 a c}+b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )}{\sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}}{\sqrt {2} a \sqrt {c} \sqrt {b^2-4 a c}}-\frac {d^{3/2} \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right )}{a}+\frac {d^{3/2} \log (x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-(((-(a*Sqrt[b^2 - 4*a*c]*e^2) + c*d*(Sqrt[b^2 - 4*a*c]*d - 4*a*e) + b*(c*d^2 + a*e^2))*ArcTanh[(Sqrt[2]*Sq
rt[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]) + (
(a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) + b*(c*d^2 + a*e^2))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqr
t[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[2]*a*Sq
rt[c]*Sqrt[b^2 - 4*a*c])) + (d^(3/2)*Log[x])/a - (d^(3/2)*Log[d + Sqrt[d]*Sqrt[d + e*x^2]])/a

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

[Out]

Timed out

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giac [B]  time = 1.02, size = 827, normalized size = 2.39 \[ \frac {d^{2} \arctan \left (\frac {\sqrt {x^{2} e + d}}{\sqrt {-d}}\right )}{a \sqrt {-d}} - \frac {{\left ({\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} e - {\left (a b^{2} - 4 \, a^{2} c\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} a^{2} - 2 \, {\left (\sqrt {b^{2} - 4 \, a c} a c^{2} d^{3} - \sqrt {b^{2} - 4 \, a c} a b c d^{2} e + \sqrt {b^{2} - 4 \, a c} a^{2} c d e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | a \right |} - {\left (2 \, a^{2} b c^{2} d^{3} + 6 \, a^{3} b c d e^{2} - a^{3} b^{2} e^{3} - {\left (a^{2} b^{2} c + 8 \, a^{3} c^{2}\right )} d^{2} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d}}{\sqrt {-\frac {2 \, a c d - a b e + \sqrt {-4 \, {\left (a c d^{2} - a b d e + a^{2} e^{2}\right )} a c + {\left (2 \, a c d - a b e\right )}^{2}}}{a c}}}\right )}{8 \, {\left (\sqrt {b^{2} - 4 \, a c} a^{2} c^{2} d^{2} - \sqrt {b^{2} - 4 \, a c} a^{2} b c d e + \sqrt {b^{2} - 4 \, a c} a^{3} c e^{2}\right )} {\left | a \right |} {\left | c \right |}} + \frac {{\left ({\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} e - {\left (a b^{2} - 4 \, a^{2} c\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} a^{2} + 2 \, {\left (\sqrt {b^{2} - 4 \, a c} a c^{2} d^{3} - \sqrt {b^{2} - 4 \, a c} a b c d^{2} e + \sqrt {b^{2} - 4 \, a c} a^{2} c d e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | a \right |} - {\left (2 \, a^{2} b c^{2} d^{3} + 6 \, a^{3} b c d e^{2} - a^{3} b^{2} e^{3} - {\left (a^{2} b^{2} c + 8 \, a^{3} c^{2}\right )} d^{2} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d}}{\sqrt {-\frac {2 \, a c d - a b e - \sqrt {-4 \, {\left (a c d^{2} - a b d e + a^{2} e^{2}\right )} a c + {\left (2 \, a c d - a b e\right )}^{2}}}{a c}}}\right )}{8 \, {\left (\sqrt {b^{2} - 4 \, a c} a^{2} c^{2} d^{2} - \sqrt {b^{2} - 4 \, a c} a^{2} b c d e + \sqrt {b^{2} - 4 \, a c} a^{3} c e^{2}\right )} {\left | a \right |} {\left | c \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^2*arctan(sqrt(x^2*e + d)/sqrt(-d))/(a*sqrt(-d)) - 1/8*(((b^2*c - 4*a*c^2)*d^2*e - (a*b^2 - 4*a^2*c)*e^3)*sqr
t(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*a^2 - 2*(sqrt(b^2 - 4*a*c)*a*c^2*d^3 - sqrt(b^2 - 4*a*c)*a*b*c*d
^2*e + sqrt(b^2 - 4*a*c)*a^2*c*d*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(a) - (2*a^2*b*c^2*d
^3 + 6*a^3*b*c*d*e^2 - a^3*b^2*e^3 - (a^2*b^2*c + 8*a^3*c^2)*d^2*e)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)
*c)*e))*arctan(2*sqrt(1/2)*sqrt(x^2*e + d)/sqrt(-(2*a*c*d - a*b*e + sqrt(-4*(a*c*d^2 - a*b*d*e + a^2*e^2)*a*c
+ (2*a*c*d - a*b*e)^2))/(a*c)))/((sqrt(b^2 - 4*a*c)*a^2*c^2*d^2 - sqrt(b^2 - 4*a*c)*a^2*b*c*d*e + sqrt(b^2 - 4
*a*c)*a^3*c*e^2)*abs(a)*abs(c)) + 1/8*(((b^2*c - 4*a*c^2)*d^2*e - (a*b^2 - 4*a^2*c)*e^3)*sqrt(-4*c^2*d + 2*(b*
c + sqrt(b^2 - 4*a*c)*c)*e)*a^2 + 2*(sqrt(b^2 - 4*a*c)*a*c^2*d^3 - sqrt(b^2 - 4*a*c)*a*b*c*d^2*e + sqrt(b^2 -
4*a*c)*a^2*c*d*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(a) - (2*a^2*b*c^2*d^3 + 6*a^3*b*c*d*e
^2 - a^3*b^2*e^3 - (a^2*b^2*c + 8*a^3*c^2)*d^2*e)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*s
qrt(1/2)*sqrt(x^2*e + d)/sqrt(-(2*a*c*d - a*b*e - sqrt(-4*(a*c*d^2 - a*b*d*e + a^2*e^2)*a*c + (2*a*c*d - a*b*e
)^2))/(a*c)))/((sqrt(b^2 - 4*a*c)*a^2*c^2*d^2 - sqrt(b^2 - 4*a*c)*a^2*b*c*d*e + sqrt(b^2 - 4*a*c)*a^3*c*e^2)*a
bs(a)*abs(c))

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maple [C]  time = 0.03, size = 388, normalized size = 1.12 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/24/a*(e*x^2+d)^(3/2)-1/a*d^(3/2)*ln((2*d+2*(e*x^2+d)^(1/2)*d^(1/2))/x)+3/8/a*(e*x^2+d)^(1/2)*d+1/6/a*e^(3/2)
*x^3-1/8/a*e*(e*x^2+d)^(1/2)*x^2+3/4/a*e^(1/2)*x*d-1/4/a*sum(((-a*e^2+c*d^2)*_R^6+d*(-5*a*e^2+4*b*d*e-3*c*d^2)
*_R^4+d^2*(5*a*e^2-4*b*d*e+3*c*d^2)*_R^2+a*d^3*e^2-c*d^5)/(_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))-5/8/a*d^2/(-e^(1/2)*x+(e*x^2+d)^(1/2))-1/24/a*d
^3/(-e^(1/2)*x+(e*x^2+d)^(1/2))^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 7.67, size = 28434, normalized size = 82.18 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan((((d + e*x^2)^(1/2)*(2*a^4*c*e^16 + 6*c^5*d^8*e^8 - 16*a*c^4*d^6*e^10 - 16*b*c^4*d^7*e^9 + 4*b^4*c*d^4*e^
12 + 16*a^2*c^3*d^4*e^12 + 8*a^3*c^2*d^2*e^14 + 24*b^2*c^3*d^6*e^10 - 16*b^3*c^2*d^5*e^11 - 8*a^3*b*c*d*e^15 -
8*a*b^3*c*d^3*e^13 + 16*a*b^2*c^2*d^4*e^12 - 24*a^2*b*c^2*d^3*e^13 + 12*a^2*b^2*c*d^2*e^14) + (-(((4*b^4*c*d^
3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e +
48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*
d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*
b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) - 2*b^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c^2*d^3 + 48*a
^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b^3*c*d^2*e - 24*a^2*b*c^2*d^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^
2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*(((-(((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a
^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3
+ 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^
2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) - 2*b^4*c*d^3 + 2*a^2*
b^3*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c^2*d^3 + 48*a^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b^3*c*d^2*e - 24*a^2*b*c^
2*d^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((d + e*x^2)^(1/2)*(-(((4*b
^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*
d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6
+ c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e
^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) - 2*b^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c^2*d^3
+ 48*a^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b^3*c*d^2*e - 24*a^2*b*c^2*d^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c
^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*(512*a^5*c^4*e^10 + 32*a^3*b^4*c^2*e^10 - 256*a^4*b^2*c^3*e^10 + 768*a
^4*c^5*d^2*e^8 + 64*a^2*b^4*c^3*d^2*e^8 - 448*a^3*b^2*c^4*d^2*e^8 - 896*a^4*b*c^4*d*e^9 - 64*a^2*b^5*c^2*d*e^9
+ 480*a^3*b^3*c^3*d*e^9) - 192*a^3*c^5*d^4*e^8 - 192*a^4*c^4*d^2*e^10 + 48*a^2*b^2*c^4*d^4*e^8 - 64*a^2*b^3*c
^3*d^3*e^9 + 16*a^2*b^4*c^2*d^2*e^10 - 16*a^3*b^2*c^3*d^2*e^10 + 64*a^4*b*c^3*d*e^11 + 256*a^3*b*c^4*d^3*e^9 -
16*a^3*b^3*c^2*d*e^11) + (d + e*x^2)^(1/2)*(8*a^3*b^3*c*e^13 - 32*a^4*b*c^2*e^13 + 176*a^4*c^3*d*e^12 - 144*a
^2*c^5*d^5*e^8 + 224*a^3*c^4*d^3*e^10 - 16*b^4*c^3*d^5*e^8 + 16*b^5*c^2*d^4*e^9 + 48*a^2*b^2*c^3*d^3*e^10 + 11
2*a^2*b^3*c^2*d^2*e^11 - 16*a^2*b^4*c*d*e^12 + 96*a*b^2*c^4*d^5*e^8 - 80*a*b^3*c^3*d^4*e^9 - 32*a*b^4*c^2*d^3*
e^10 + 96*a^2*b*c^4*d^4*e^9 - 416*a^3*b*c^3*d^2*e^11 + 16*a^3*b^2*c^2*d*e^12))*(-(((4*b^4*c*d^3 - 4*a^2*b^3*e^
3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^
2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^
3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*
a*b*c*d^3*e^3))^(1/2) - 2*b^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c^2*d^3 + 48*a^3*c^2*d*e^2 - 8
*a^3*b*c*e^3 + 6*a*b^3*c*d^2*e - 24*a^2*b*c^2*d^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*
b^2*c^2)))^(1/2) + 12*a*c^5*d^7*e^8 + 4*a^4*c^2*d*e^14 - 84*a^2*c^4*d^5*e^10 - 92*a^3*c^3*d^3*e^12 - 4*b^2*c^4
*d^7*e^8 - 4*b^3*c^3*d^6*e^9 + 8*b^4*c^2*d^5*e^10 - 12*a^2*b^2*c^2*d^3*e^12 + 32*a*b*c^4*d^6*e^9 - 4*a^3*b^2*c
*d*e^14 - 36*a*b^2*c^3*d^5*e^10 - 20*a*b^3*c^2*d^4*e^11 + 160*a^2*b*c^3*d^4*e^11 + 4*a^2*b^3*c*d^2*e^13 + 16*a
^3*b*c^2*d^2*e^13))*(-(((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 +
16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c
- 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3
*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) - 2*b^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^
2*c^3*d^3 + 12*a*b^2*c^2*d^3 + 48*a^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b^3*c*d^2*e - 24*a^2*b*c^2*d^2*e - 12*a^
2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*1i + ((d + e*x^2)^(1/2)*(2*a^4*c*e^16 + 6*
c^5*d^8*e^8 - 16*a*c^4*d^6*e^10 - 16*b*c^4*d^7*e^9 + 4*b^4*c*d^4*e^12 + 16*a^2*c^3*d^4*e^12 + 8*a^3*c^2*d^2*e^
14 + 24*b^2*c^3*d^6*e^10 - 16*b^3*c^2*d^5*e^11 - 8*a^3*b*c*d*e^15 - 8*a*b^3*c*d^3*e^13 + 16*a*b^2*c^2*d^4*e^12
- 24*a^2*b*c^2*d^3*e^13 + 12*a^2*b^2*c*d^2*e^14) + (-(((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b
^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2
/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c
^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) - 2*b
^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c^2*d^3 + 48*a^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b^3*c*d^
2*e - 24*a^2*b*c^2*d^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*(((-(((4*b
^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*
d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6
+ c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e
^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) - 2*b^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c^2*d^3
+ 48*a^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b^3*c*d^2*e - 24*a^2*b*c^2*d^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c
^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((d + e*x^2)^(1/2)*(-(((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 -
24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d
*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4
+ 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2
) - 2*b^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c^2*d^3 + 48*a^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b
^3*c*d^2*e - 24*a^2*b*c^2*d^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*(51
2*a^5*c^4*e^10 + 32*a^3*b^4*c^2*e^10 - 256*a^4*b^2*c^3*e^10 + 768*a^4*c^5*d^2*e^8 + 64*a^2*b^4*c^3*d^2*e^8 - 4
48*a^3*b^2*c^4*d^2*e^8 - 896*a^4*b*c^4*d*e^9 - 64*a^2*b^5*c^2*d*e^9 + 480*a^3*b^3*c^3*d*e^9) + 192*a^3*c^5*d^4
*e^8 + 192*a^4*c^4*d^2*e^10 - 48*a^2*b^2*c^4*d^4*e^8 + 64*a^2*b^3*c^3*d^3*e^9 - 16*a^2*b^4*c^2*d^2*e^10 + 16*a
^3*b^2*c^3*d^2*e^10 - 64*a^4*b*c^3*d*e^11 - 256*a^3*b*c^4*d^3*e^9 + 16*a^3*b^3*c^2*d*e^11) + (d + e*x^2)^(1/2)
*(8*a^3*b^3*c*e^13 - 32*a^4*b*c^2*e^13 + 176*a^4*c^3*d*e^12 - 144*a^2*c^5*d^5*e^8 + 224*a^3*c^4*d^3*e^10 - 16*
b^4*c^3*d^5*e^8 + 16*b^5*c^2*d^4*e^9 + 48*a^2*b^2*c^3*d^3*e^10 + 112*a^2*b^3*c^2*d^2*e^11 - 16*a^2*b^4*c*d*e^1
2 + 96*a*b^2*c^4*d^5*e^8 - 80*a*b^3*c^3*d^4*e^9 - 32*a*b^4*c^2*d^3*e^10 + 96*a^2*b*c^4*d^4*e^9 - 416*a^3*b*c^3
*d^2*e^11 + 16*a^3*b^2*c^2*d*e^12))*(-(((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*
a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^
3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a
^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) - 2*b^4*c*d^3 + 2*a^2
*b^3*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c^2*d^3 + 48*a^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b^3*c*d^2*e - 24*a^2*b*c
^2*d^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) - 12*a*c^5*d^7*e^8 - 4*a^4
*c^2*d*e^14 + 84*a^2*c^4*d^5*e^10 + 92*a^3*c^3*d^3*e^12 + 4*b^2*c^4*d^7*e^8 + 4*b^3*c^3*d^6*e^9 - 8*b^4*c^2*d^
5*e^10 + 12*a^2*b^2*c^2*d^3*e^12 - 32*a*b*c^4*d^6*e^9 + 4*a^3*b^2*c*d*e^14 + 36*a*b^2*c^3*d^5*e^10 + 20*a*b^3*
c^2*d^4*e^11 - 160*a^2*b*c^3*d^4*e^11 - 4*a^2*b^3*c*d^2*e^13 - 16*a^3*b*c^2*d^2*e^13))*(-(((4*b^4*c*d^3 - 4*a^
2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*
b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^
3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^
5*e - 6*a*b*c*d^3*e^3))^(1/2) - 2*b^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c^2*d^3 + 48*a^3*c^2*d
*e^2 - 8*a^3*b*c*e^3 + 6*a*b^3*c*d^2*e - 24*a^2*b*c^2*d^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c
- 8*a^3*b^2*c^2)))^(1/2)*1i)/(((d + e*x^2)^(1/2)*(2*a^4*c*e^16 + 6*c^5*d^8*e^8 - 16*a*c^4*d^6*e^10 - 16*b*c^4*
d^7*e^9 + 4*b^4*c*d^4*e^12 + 16*a^2*c^3*d^4*e^12 + 8*a^3*c^2*d^2*e^14 + 24*b^2*c^3*d^6*e^10 - 16*b^3*c^2*d^5*e
^11 - 8*a^3*b*c*d*e^15 - 8*a*b^3*c*d^3*e^13 + 16*a*b^2*c^2*d^4*e^12 - 24*a^2*b*c^2*d^3*e^13 + 12*a^2*b^2*c*d^2
*e^14) + (-(((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*
e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*
b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*
e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) - 2*b^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^2*c^3*d^3 +
12*a*b^2*c^2*d^3 + 48*a^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b^3*c*d^2*e - 24*a^2*b*c^2*d^2*e - 12*a^2*b^2*c*d*e
^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*(((-(((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 -
24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d
*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4
+ 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2
) - 2*b^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c^2*d^3 + 48*a^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b
^3*c*d^2*e - 24*a^2*b*c^2*d^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((d
+ e*x^2)^(1/2)*(-(((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a
^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 1
28*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2
*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) - 2*b^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^2*c^
3*d^3 + 12*a*b^2*c^2*d^3 + 48*a^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b^3*c*d^2*e - 24*a^2*b*c^2*d^2*e - 12*a^2*b^
2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*(512*a^5*c^4*e^10 + 32*a^3*b^4*c^2*e^10 - 256*
a^4*b^2*c^3*e^10 + 768*a^4*c^5*d^2*e^8 + 64*a^2*b^4*c^3*d^2*e^8 - 448*a^3*b^2*c^4*d^2*e^8 - 896*a^4*b*c^4*d*e^
9 - 64*a^2*b^5*c^2*d*e^9 + 480*a^3*b^3*c^3*d*e^9) + 192*a^3*c^5*d^4*e^8 + 192*a^4*c^4*d^2*e^10 - 48*a^2*b^2*c^
4*d^4*e^8 + 64*a^2*b^3*c^3*d^3*e^9 - 16*a^2*b^4*c^2*d^2*e^10 + 16*a^3*b^2*c^3*d^2*e^10 - 64*a^4*b*c^3*d*e^11 -
256*a^3*b*c^4*d^3*e^9 + 16*a^3*b^3*c^2*d*e^11) + (d + e*x^2)^(1/2)*(8*a^3*b^3*c*e^13 - 32*a^4*b*c^2*e^13 + 17
6*a^4*c^3*d*e^12 - 144*a^2*c^5*d^5*e^8 + 224*a^3*c^4*d^3*e^10 - 16*b^4*c^3*d^5*e^8 + 16*b^5*c^2*d^4*e^9 + 48*a
^2*b^2*c^3*d^3*e^10 + 112*a^2*b^3*c^2*d^2*e^11 - 16*a^2*b^4*c*d*e^12 + 96*a*b^2*c^4*d^5*e^8 - 80*a*b^3*c^3*d^4
*e^9 - 32*a*b^4*c^2*d^3*e^10 + 96*a^2*b*c^4*d^4*e^9 - 416*a^3*b*c^3*d^2*e^11 + 16*a^3*b^2*c^2*d*e^12))*(-(((4*
b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c
*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^
6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*
e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) - 2*b^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c^2*d^
3 + 48*a^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b^3*c*d^2*e - 24*a^2*b*c^2*d^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*
c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) - 12*a*c^5*d^7*e^8 - 4*a^4*c^2*d*e^14 + 84*a^2*c^4*d^5*e^10 + 92*a^3*
c^3*d^3*e^12 + 4*b^2*c^4*d^7*e^8 + 4*b^3*c^3*d^6*e^9 - 8*b^4*c^2*d^5*e^10 + 12*a^2*b^2*c^2*d^3*e^12 - 32*a*b*c
^4*d^6*e^9 + 4*a^3*b^2*c*d*e^14 + 36*a*b^2*c^3*d^5*e^10 + 20*a*b^3*c^2*d^4*e^11 - 160*a^2*b*c^3*d^4*e^11 - 4*a
^2*b^3*c*d^2*e^13 - 16*a^3*b*c^2*d^2*e^13))*(-(((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d
^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (25
6*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e
^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) - 2*b^4*c*d^3
+ 2*a^2*b^3*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c^2*d^3 + 48*a^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b^3*c*d^2*e - 24
*a^2*b*c^2*d^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) - ((d + e*x^2)^(1/
2)*(2*a^4*c*e^16 + 6*c^5*d^8*e^8 - 16*a*c^4*d^6*e^10 - 16*b*c^4*d^7*e^9 + 4*b^4*c*d^4*e^12 + 16*a^2*c^3*d^4*e^
12 + 8*a^3*c^2*d^2*e^14 + 24*b^2*c^3*d^6*e^10 - 16*b^3*c^2*d^5*e^11 - 8*a^3*b*c*d*e^15 - 8*a*b^3*c*d^3*e^13 +
16*a*b^2*c^2*d^4*e^12 - 24*a^2*b*c^2*d^3*e^13 + 12*a^2*b^2*c*d^2*e^14) + (-(((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32
*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e +
24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*
a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*
d^3*e^3))^(1/2) - 2*b^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c^2*d^3 + 48*a^3*c^2*d*e^2 - 8*a^3*b
*c*e^3 + 6*a*b^3*c*d^2*e - 24*a^2*b*c^2*d^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^
2)))^(1/2)*(((-(((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*
b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*
a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*
d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) - 2*b^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^2*c^3*d
^3 + 12*a*b^2*c^2*d^3 + 48*a^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b^3*c*d^2*e - 24*a^2*b*c^2*d^2*e - 12*a^2*b^2*c
*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((d + e*x^2)^(1/2)*(-(((4*b^4*c*d^3 - 4*a^2*b^3*e
^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d
^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e
^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6
*a*b*c*d^3*e^3))^(1/2) - 2*b^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c^2*d^3 + 48*a^3*c^2*d*e^2 -
8*a^3*b*c*e^3 + 6*a*b^3*c*d^2*e - 24*a^2*b*c^2*d^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3
*b^2*c^2)))^(1/2)*(512*a^5*c^4*e^10 + 32*a^3*b^4*c^2*e^10 - 256*a^4*b^2*c^3*e^10 + 768*a^4*c^5*d^2*e^8 + 64*a^
2*b^4*c^3*d^2*e^8 - 448*a^3*b^2*c^4*d^2*e^8 - 896*a^4*b*c^4*d*e^9 - 64*a^2*b^5*c^2*d*e^9 + 480*a^3*b^3*c^3*d*e
^9) - 192*a^3*c^5*d^4*e^8 - 192*a^4*c^4*d^2*e^10 + 48*a^2*b^2*c^4*d^4*e^8 - 64*a^2*b^3*c^3*d^3*e^9 + 16*a^2*b^
4*c^2*d^2*e^10 - 16*a^3*b^2*c^3*d^2*e^10 + 64*a^4*b*c^3*d*e^11 + 256*a^3*b*c^4*d^3*e^9 - 16*a^3*b^3*c^2*d*e^11
) + (d + e*x^2)^(1/2)*(8*a^3*b^3*c*e^13 - 32*a^4*b*c^2*e^13 + 176*a^4*c^3*d*e^12 - 144*a^2*c^5*d^5*e^8 + 224*a
^3*c^4*d^3*e^10 - 16*b^4*c^3*d^5*e^8 + 16*b^5*c^2*d^4*e^9 + 48*a^2*b^2*c^3*d^3*e^10 + 112*a^2*b^3*c^2*d^2*e^11
- 16*a^2*b^4*c*d*e^12 + 96*a*b^2*c^4*d^5*e^8 - 80*a*b^3*c^3*d^4*e^9 - 32*a*b^4*c^2*d^3*e^10 + 96*a^2*b*c^4*d^
4*e^9 - 416*a^3*b*c^3*d^2*e^11 + 16*a^3*b^2*c^2*d*e^12))*(-(((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 2
4*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e
^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 +
3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)
- 2*b^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c^2*d^3 + 48*a^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b^3
*c*d^2*e - 24*a^2*b*c^2*d^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) + 12*
a*c^5*d^7*e^8 + 4*a^4*c^2*d*e^14 - 84*a^2*c^4*d^5*e^10 - 92*a^3*c^3*d^3*e^12 - 4*b^2*c^4*d^7*e^8 - 4*b^3*c^3*d
^6*e^9 + 8*b^4*c^2*d^5*e^10 - 12*a^2*b^2*c^2*d^3*e^12 + 32*a*b*c^4*d^6*e^9 - 4*a^3*b^2*c*d*e^14 - 36*a*b^2*c^3
*d^5*e^10 - 20*a*b^3*c^2*d^4*e^11 + 160*a^2*b*c^3*d^4*e^11 + 4*a^2*b^3*c*d^2*e^13 + 16*a^3*b*c^2*d^2*e^13))*(-
(((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*
b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a
^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2
*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) - 2*b^4*c*d^3 + 2*a^2*b^3*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c
^2*d^3 + 48*a^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b^3*c*d^2*e - 24*a^2*b*c^2*d^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16
*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) + 6*c^4*d^8*e^10 + 14*a*c^3*d^6*e^12 + 2*a^3*c*d^2*e^16 - 16*b*c
^3*d^7*e^11 - 4*b^3*c*d^5*e^13 + 10*a^2*c^2*d^4*e^14 + 14*b^2*c^2*d^6*e^12 - 24*a*b*c^2*d^5*e^13 + 10*a*b^2*c*
d^4*e^14 - 8*a^2*b*c*d^3*e^15))*(-(((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*
c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 +
16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c
*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) - 2*b^4*c*d^3 + 2*a^2*b^3
*e^3 - 16*a^2*c^3*d^3 + 12*a*b^2*c^2*d^3 + 48*a^3*c^2*d*e^2 - 8*a^3*b*c*e^3 + 6*a*b^3*c*d^2*e - 24*a^2*b*c^2*d
^2*e - 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*2i + atan((((d + e*x^2)^(1/2)*
(2*a^4*c*e^16 + 6*c^5*d^8*e^8 - 16*a*c^4*d^6*e^10 - 16*b*c^4*d^7*e^9 + 4*b^4*c*d^4*e^12 + 16*a^2*c^3*d^4*e^12
+ 8*a^3*c^2*d^2*e^14 + 24*b^2*c^3*d^6*e^10 - 16*b^3*c^2*d^5*e^11 - 8*a^3*b*c*d*e^15 - 8*a*b^3*c*d^3*e^13 + 16*
a*b^2*c^2*d^4*e^12 - 24*a^2*b*c^2*d^3*e^13 + 12*a^2*b^2*c*d^2*e^14) + ((((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2
*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a
^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^
2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*
e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 + 16*a^2*c^3*d^3 - 12*a*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2 + 8*a^3*b*c*e
^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e + 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))
^(1/2)*((((((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e
^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b
^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e
^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 + 16*a^2*c^3*d^3 -
12*a*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2 + 8*a^3*b*c*e^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e + 12*a^2*b^2*c*d*e^
2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((d + e*x^2)^(1/2)*((((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 3
2*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e +
24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3
*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c
*d^3*e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 + 16*a^2*c^3*d^3 - 12*a*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2 + 8*a^3*
b*c*e^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e + 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c
^2)))^(1/2)*(512*a^5*c^4*e^10 + 32*a^3*b^4*c^2*e^10 - 256*a^4*b^2*c^3*e^10 + 768*a^4*c^5*d^2*e^8 + 64*a^2*b^4*
c^3*d^2*e^8 - 448*a^3*b^2*c^4*d^2*e^8 - 896*a^4*b*c^4*d*e^9 - 64*a^2*b^5*c^2*d*e^9 + 480*a^3*b^3*c^3*d*e^9) -
192*a^3*c^5*d^4*e^8 - 192*a^4*c^4*d^2*e^10 + 48*a^2*b^2*c^4*d^4*e^8 - 64*a^2*b^3*c^3*d^3*e^9 + 16*a^2*b^4*c^2*
d^2*e^10 - 16*a^3*b^2*c^3*d^2*e^10 + 64*a^4*b*c^3*d*e^11 + 256*a^3*b*c^4*d^3*e^9 - 16*a^3*b^3*c^2*d*e^11) + (d
+ e*x^2)^(1/2)*(8*a^3*b^3*c*e^13 - 32*a^4*b*c^2*e^13 + 176*a^4*c^3*d*e^12 - 144*a^2*c^5*d^5*e^8 + 224*a^3*c^4
*d^3*e^10 - 16*b^4*c^3*d^5*e^8 + 16*b^5*c^2*d^4*e^9 + 48*a^2*b^2*c^3*d^3*e^10 + 112*a^2*b^3*c^2*d^2*e^11 - 16*
a^2*b^4*c*d*e^12 + 96*a*b^2*c^4*d^5*e^8 - 80*a*b^3*c^3*d^4*e^9 - 32*a*b^4*c^2*d^3*e^10 + 96*a^2*b*c^4*d^4*e^9
- 416*a^3*b*c^3*d^2*e^11 + 16*a^3*b^2*c^2*d*e^12))*((((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2
*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4
- (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2
*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) + 2*b^4
*c*d^3 - 2*a^2*b^3*e^3 + 16*a^2*c^3*d^3 - 12*a*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2 + 8*a^3*b*c*e^3 - 6*a*b^3*c*d^2*
e + 24*a^2*b*c^2*d^2*e + 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) + 12*a*c^5*d
^7*e^8 + 4*a^4*c^2*d*e^14 - 84*a^2*c^4*d^5*e^10 - 92*a^3*c^3*d^3*e^12 - 4*b^2*c^4*d^7*e^8 - 4*b^3*c^3*d^6*e^9
+ 8*b^4*c^2*d^5*e^10 - 12*a^2*b^2*c^2*d^3*e^12 + 32*a*b*c^4*d^6*e^9 - 4*a^3*b^2*c*d*e^14 - 36*a*b^2*c^3*d^5*e^
10 - 20*a*b^3*c^2*d^4*e^11 + 160*a^2*b*c^3*d^4*e^11 + 4*a^2*b^3*c*d^2*e^13 + 16*a^3*b*c^2*d^2*e^13))*((((4*b^4
*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^
2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 +
c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5
- 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 + 16*a^2*c^3*d^3 - 12*a*b^2*c^2*d^3 -
48*a^3*c^2*d*e^2 + 8*a^3*b*c*e^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e + 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3
+ a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*1i + ((d + e*x^2)^(1/2)*(2*a^4*c*e^16 + 6*c^5*d^8*e^8 - 16*a*c^4*d^6*e^1
0 - 16*b*c^4*d^7*e^9 + 4*b^4*c*d^4*e^12 + 16*a^2*c^3*d^4*e^12 + 8*a^3*c^2*d^2*e^14 + 24*b^2*c^3*d^6*e^10 - 16*
b^3*c^2*d^5*e^11 - 8*a^3*b*c*d*e^15 - 8*a*b^3*c*d^3*e^13 + 16*a*b^2*c^2*d^4*e^12 - 24*a^2*b*c^2*d^3*e^13 + 12*
a^2*b^2*c*d^2*e^14) + ((((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 +
16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*
c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 +
3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 + 16*a
^2*c^3*d^3 - 12*a*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2 + 8*a^3*b*c*e^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e + 12*a
^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((((((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^
2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*
a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b
^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3
*e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 + 16*a^2*c^3*d^3 - 12*a*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2 + 8*a^3*b*c*
e^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e + 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2))
)^(1/2)*((d + e*x^2)^(1/2)*((((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*
e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2
*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e
^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 +
16*a^2*c^3*d^3 - 12*a*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2 + 8*a^3*b*c*e^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e +
12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*(512*a^5*c^4*e^10 + 32*a^3*b^4*c^2*e
^10 - 256*a^4*b^2*c^3*e^10 + 768*a^4*c^5*d^2*e^8 + 64*a^2*b^4*c^3*d^2*e^8 - 448*a^3*b^2*c^4*d^2*e^8 - 896*a^4*
b*c^4*d*e^9 - 64*a^2*b^5*c^2*d*e^9 + 480*a^3*b^3*c^3*d*e^9) + 192*a^3*c^5*d^4*e^8 + 192*a^4*c^4*d^2*e^10 - 48*
a^2*b^2*c^4*d^4*e^8 + 64*a^2*b^3*c^3*d^3*e^9 - 16*a^2*b^4*c^2*d^2*e^10 + 16*a^3*b^2*c^3*d^2*e^10 - 64*a^4*b*c^
3*d*e^11 - 256*a^3*b*c^4*d^3*e^9 + 16*a^3*b^3*c^2*d*e^11) + (d + e*x^2)^(1/2)*(8*a^3*b^3*c*e^13 - 32*a^4*b*c^2
*e^13 + 176*a^4*c^3*d*e^12 - 144*a^2*c^5*d^5*e^8 + 224*a^3*c^4*d^3*e^10 - 16*b^4*c^3*d^5*e^8 + 16*b^5*c^2*d^4*
e^9 + 48*a^2*b^2*c^3*d^3*e^10 + 112*a^2*b^3*c^2*d^2*e^11 - 16*a^2*b^4*c*d*e^12 + 96*a*b^2*c^4*d^5*e^8 - 80*a*b
^3*c^3*d^4*e^9 - 32*a*b^4*c^2*d^3*e^10 + 96*a^2*b*c^4*d^4*e^9 - 416*a^3*b*c^3*d^2*e^11 + 16*a^3*b^2*c^2*d*e^12
))*((((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 1
2*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2
)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3
*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 + 16*a^2*c^3*d^3 - 12*a*b
^2*c^2*d^3 - 48*a^3*c^2*d*e^2 + 8*a^3*b*c*e^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e + 12*a^2*b^2*c*d*e^2)/(16
*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) - 12*a*c^5*d^7*e^8 - 4*a^4*c^2*d*e^14 + 84*a^2*c^4*d^5*e^10
+ 92*a^3*c^3*d^3*e^12 + 4*b^2*c^4*d^7*e^8 + 4*b^3*c^3*d^6*e^9 - 8*b^4*c^2*d^5*e^10 + 12*a^2*b^2*c^2*d^3*e^12 -
32*a*b*c^4*d^6*e^9 + 4*a^3*b^2*c*d*e^14 + 36*a*b^2*c^3*d^5*e^10 + 20*a*b^3*c^2*d^4*e^11 - 160*a^2*b*c^3*d^4*e
^11 - 4*a^2*b^3*c*d^2*e^13 - 16*a^3*b*c^2*d^2*e^13))*((((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b
^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2
/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c
^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) + 2*b
^4*c*d^3 - 2*a^2*b^3*e^3 + 16*a^2*c^3*d^3 - 12*a*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2 + 8*a^3*b*c*e^3 - 6*a*b^3*c*d^
2*e + 24*a^2*b*c^2*d^2*e + 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*1i)/(((d +
e*x^2)^(1/2)*(2*a^4*c*e^16 + 6*c^5*d^8*e^8 - 16*a*c^4*d^6*e^10 - 16*b*c^4*d^7*e^9 + 4*b^4*c*d^4*e^12 + 16*a^2
*c^3*d^4*e^12 + 8*a^3*c^2*d^2*e^14 + 24*b^2*c^3*d^6*e^10 - 16*b^3*c^2*d^5*e^11 - 8*a^3*b*c*d*e^15 - 8*a*b^3*c*
d^3*e^13 + 16*a*b^2*c^2*d^4*e^12 - 24*a^2*b*c^2*d^3*e^13 + 12*a^2*b^2*c*d^2*e^14) + ((((4*b^4*c*d^3 - 4*a^2*b^
3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^
2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^
3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e
- 6*a*b*c*d^3*e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 + 16*a^2*c^3*d^3 - 12*a*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2
+ 8*a^3*b*c*e^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e + 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*
a^3*b^2*c^2)))^(1/2)*((((((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2
+ 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4
*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 +
3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 + 16*
a^2*c^3*d^3 - 12*a*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2 + 8*a^3*b*c*e^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e + 12*
a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((d + e*x^2)^(1/2)*((((4*b^4*c*d^3 - 4*a
^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2
*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b
^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d
^5*e - 6*a*b*c*d^3*e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 + 16*a^2*c^3*d^3 - 12*a*b^2*c^2*d^3 - 48*a^3*c^2*
d*e^2 + 8*a^3*b*c*e^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e + 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c
- 8*a^3*b^2*c^2)))^(1/2)*(512*a^5*c^4*e^10 + 32*a^3*b^4*c^2*e^10 - 256*a^4*b^2*c^3*e^10 + 768*a^4*c^5*d^2*e^8
+ 64*a^2*b^4*c^3*d^2*e^8 - 448*a^3*b^2*c^4*d^2*e^8 - 896*a^4*b*c^4*d*e^9 - 64*a^2*b^5*c^2*d*e^9 + 480*a^3*b^3
*c^3*d*e^9) + 192*a^3*c^5*d^4*e^8 + 192*a^4*c^4*d^2*e^10 - 48*a^2*b^2*c^4*d^4*e^8 + 64*a^2*b^3*c^3*d^3*e^9 - 1
6*a^2*b^4*c^2*d^2*e^10 + 16*a^3*b^2*c^3*d^2*e^10 - 64*a^4*b*c^3*d*e^11 - 256*a^3*b*c^4*d^3*e^9 + 16*a^3*b^3*c^
2*d*e^11) + (d + e*x^2)^(1/2)*(8*a^3*b^3*c*e^13 - 32*a^4*b*c^2*e^13 + 176*a^4*c^3*d*e^12 - 144*a^2*c^5*d^5*e^8
+ 224*a^3*c^4*d^3*e^10 - 16*b^4*c^3*d^5*e^8 + 16*b^5*c^2*d^4*e^9 + 48*a^2*b^2*c^3*d^3*e^10 + 112*a^2*b^3*c^2*
d^2*e^11 - 16*a^2*b^4*c*d*e^12 + 96*a*b^2*c^4*d^5*e^8 - 80*a*b^3*c^3*d^4*e^9 - 32*a*b^4*c^2*d^3*e^10 + 96*a^2*
b*c^4*d^4*e^9 - 416*a^3*b*c^3*d^2*e^11 + 16*a^3*b^2*c^2*d*e^12))*((((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*
d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^
2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2
*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))
^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 + 16*a^2*c^3*d^3 - 12*a*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2 + 8*a^3*b*c*e^3 -
6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e + 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2
) - 12*a*c^5*d^7*e^8 - 4*a^4*c^2*d*e^14 + 84*a^2*c^4*d^5*e^10 + 92*a^3*c^3*d^3*e^12 + 4*b^2*c^4*d^7*e^8 + 4*b^
3*c^3*d^6*e^9 - 8*b^4*c^2*d^5*e^10 + 12*a^2*b^2*c^2*d^3*e^12 - 32*a*b*c^4*d^6*e^9 + 4*a^3*b^2*c*d*e^14 + 36*a*
b^2*c^3*d^5*e^10 + 20*a*b^3*c^2*d^4*e^11 - 160*a^2*b*c^3*d^4*e^11 - 4*a^2*b^3*c*d^2*e^13 - 16*a^3*b*c^2*d^2*e^
13))*((((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 -
12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c
^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 -
3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 + 16*a^2*c^3*d^3 - 12*a
*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2 + 8*a^3*b*c*e^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e + 12*a^2*b^2*c*d*e^2)/(
16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) - ((d + e*x^2)^(1/2)*(2*a^4*c*e^16 + 6*c^5*d^8*e^8 - 16*a*
c^4*d^6*e^10 - 16*b*c^4*d^7*e^9 + 4*b^4*c*d^4*e^12 + 16*a^2*c^3*d^4*e^12 + 8*a^3*c^2*d^2*e^14 + 24*b^2*c^3*d^6
*e^10 - 16*b^3*c^2*d^5*e^11 - 8*a^3*b*c*d*e^15 - 8*a*b^3*c*d^3*e^13 + 16*a*b^2*c^2*d^4*e^12 - 24*a^2*b*c^2*d^3
*e^13 + 12*a^2*b^2*c*d^2*e^14) + ((((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*
c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 +
16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c
*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3
*e^3 + 16*a^2*c^3*d^3 - 12*a*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2 + 8*a^3*b*c*e^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d
^2*e + 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((((((4*b^4*c*d^3 - 4*a^2*b^3*
e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*
d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*
e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e -
6*a*b*c*d^3*e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 + 16*a^2*c^3*d^3 - 12*a*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2 +
8*a^3*b*c*e^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e + 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^
3*b^2*c^2)))^(1/2)*((d + e*x^2)^(1/2)*((((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96
*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c
^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*
a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^
2*b^3*e^3 + 16*a^2*c^3*d^3 - 12*a*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2 + 8*a^3*b*c*e^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*
c^2*d^2*e + 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*(512*a^5*c^4*e^10 + 32*a^
3*b^4*c^2*e^10 - 256*a^4*b^2*c^3*e^10 + 768*a^4*c^5*d^2*e^8 + 64*a^2*b^4*c^3*d^2*e^8 - 448*a^3*b^2*c^4*d^2*e^8
- 896*a^4*b*c^4*d*e^9 - 64*a^2*b^5*c^2*d*e^9 + 480*a^3*b^3*c^3*d*e^9) - 192*a^3*c^5*d^4*e^8 - 192*a^4*c^4*d^2
*e^10 + 48*a^2*b^2*c^4*d^4*e^8 - 64*a^2*b^3*c^3*d^3*e^9 + 16*a^2*b^4*c^2*d^2*e^10 - 16*a^3*b^2*c^3*d^2*e^10 +
64*a^4*b*c^3*d*e^11 + 256*a^3*b*c^4*d^3*e^9 - 16*a^3*b^3*c^2*d*e^11) + (d + e*x^2)^(1/2)*(8*a^3*b^3*c*e^13 - 3
2*a^4*b*c^2*e^13 + 176*a^4*c^3*d*e^12 - 144*a^2*c^5*d^5*e^8 + 224*a^3*c^4*d^3*e^10 - 16*b^4*c^3*d^5*e^8 + 16*b
^5*c^2*d^4*e^9 + 48*a^2*b^2*c^3*d^3*e^10 + 112*a^2*b^3*c^2*d^2*e^11 - 16*a^2*b^4*c*d*e^12 + 96*a*b^2*c^4*d^5*e
^8 - 80*a*b^3*c^3*d^4*e^9 - 32*a*b^4*c^2*d^3*e^10 + 96*a^2*b*c^4*d^4*e^9 - 416*a^3*b*c^3*d^2*e^11 + 16*a^3*b^2
*c^2*d*e^12))*((((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*
b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*
a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*
d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 + 16*a^2*c^3*d
^3 - 12*a*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2 + 8*a^3*b*c*e^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e + 12*a^2*b^2*c
*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) + 12*a*c^5*d^7*e^8 + 4*a^4*c^2*d*e^14 - 84*a^2*c^
4*d^5*e^10 - 92*a^3*c^3*d^3*e^12 - 4*b^2*c^4*d^7*e^8 - 4*b^3*c^3*d^6*e^9 + 8*b^4*c^2*d^5*e^10 - 12*a^2*b^2*c^2
*d^3*e^12 + 32*a*b*c^4*d^6*e^9 - 4*a^3*b^2*c*d*e^14 - 36*a*b^2*c^3*d^5*e^10 - 20*a*b^3*c^2*d^4*e^11 + 160*a^2*
b*c^3*d^4*e^11 + 4*a^2*b^3*c*d^2*e^13 + 16*a^3*b*c^2*d^2*e^13))*((((4*b^4*c*d^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d
^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e + 48*a^2*b*c^2*d^2*e + 24*a^2*b^2
*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*
e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^
(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 + 16*a^2*c^3*d^3 - 12*a*b^2*c^2*d^3 - 48*a^3*c^2*d*e^2 + 8*a^3*b*c*e^3 - 6
*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e + 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)
+ 6*c^4*d^8*e^10 + 14*a*c^3*d^6*e^12 + 2*a^3*c*d^2*e^16 - 16*b*c^3*d^7*e^11 - 4*b^3*c*d^5*e^13 + 10*a^2*c^2*d
^4*e^14 + 14*b^2*c^2*d^6*e^12 - 24*a*b*c^2*d^5*e^13 + 10*a*b^2*c*d^4*e^14 - 8*a^2*b*c*d^3*e^15))*((((4*b^4*c*d
^3 - 4*a^2*b^3*e^3 + 32*a^2*c^3*d^3 - 24*a*b^2*c^2*d^3 - 96*a^3*c^2*d*e^2 + 16*a^3*b*c*e^3 - 12*a*b^3*c*d^2*e
+ 48*a^2*b*c^2*d^2*e + 24*a^2*b^2*c*d*e^2)^2/4 - (256*a^4*c^3 + 16*a^2*b^4*c - 128*a^3*b^2*c^2)*(a^3*e^6 + c^3
*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3
*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2) + 2*b^4*c*d^3 - 2*a^2*b^3*e^3 + 16*a^2*c^3*d^3 - 12*a*b^2*c^2*d^3 - 48*
a^3*c^2*d*e^2 + 8*a^3*b*c*e^3 - 6*a*b^3*c*d^2*e + 24*a^2*b*c^2*d^2*e + 12*a^2*b^2*c*d*e^2)/(16*(16*a^4*c^3 + a
^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*2i - (atanh((72*c^4*d^6*e^10*(d + e*x^2)^(1/2)*(d^3)^(1/2))/(72*c^4*d^8*e^10
+ 60*a*c^3*d^6*e^12 + 2*a^3*c*d^2*e^16 - 104*b*c^3*d^7*e^11 - 6*b^3*c*d^5*e^13 + 8*a^2*c^2*d^4*e^14 + (18*c^5
*d^10*e^8)/a + 20*b^2*c^2*d^6*e^12 + (20*b^2*c^3*d^8*e^10)/a + (12*b^3*c^2*d^7*e^11)/a - (4*b^2*c^4*d^10*e^8)/
a^2 + (10*b^3*c^3*d^9*e^9)/a^2 - (6*b^4*c^2*d^8*e^10)/a^2 - 32*a*b*c^2*d^5*e^13 + 12*a*b^2*c*d^4*e^14 - 8*a^2*
b*c*d^3*e^15 - (48*b*c^4*d^9*e^9)/a) + (2*a^3*c*e^16*(d + e*x^2)^(1/2)*(d^3)^(1/2))/(72*c^4*d^8*e^10 + 60*a*c^
3*d^6*e^12 + 2*a^3*c*d^2*e^16 - 104*b*c^3*d^7*e^11 - 6*b^3*c*d^5*e^13 + 8*a^2*c^2*d^4*e^14 + (18*c^5*d^10*e^8)
/a + 20*b^2*c^2*d^6*e^12 + (20*b^2*c^3*d^8*e^10)/a + (12*b^3*c^2*d^7*e^11)/a - (4*b^2*c^4*d^10*e^8)/a^2 + (10*
b^3*c^3*d^9*e^9)/a^2 - (6*b^4*c^2*d^8*e^10)/a^2 - 32*a*b*c^2*d^5*e^13 + 12*a*b^2*c*d^4*e^14 - 8*a^2*b*c*d^3*e^
15 - (48*b*c^4*d^9*e^9)/a) + (18*c^5*d^8*e^8*(d + e*x^2)^(1/2)*(d^3)^(1/2))/(18*c^5*d^10*e^8 + 72*a*c^4*d^8*e^
10 + 2*a^4*c*d^2*e^16 - 48*b*c^4*d^9*e^9 + 60*a^2*c^3*d^6*e^12 + 8*a^3*c^2*d^4*e^14 + 20*b^2*c^3*d^8*e^10 + 12
*b^3*c^2*d^7*e^11 - (4*b^2*c^4*d^10*e^8)/a + (10*b^3*c^3*d^9*e^9)/a - (6*b^4*c^2*d^8*e^10)/a - 104*a*b*c^3*d^7
*e^11 - 6*a*b^3*c*d^5*e^13 - 8*a^3*b*c*d^3*e^15 + 20*a*b^2*c^2*d^6*e^12 - 32*a^2*b*c^2*d^5*e^13 + 12*a^2*b^2*c
*d^4*e^14) + (8*a^2*c^2*d^2*e^14*(d + e*x^2)^(1/2)*(d^3)^(1/2))/(72*c^4*d^8*e^10 + 60*a*c^3*d^6*e^12 + 2*a^3*c
*d^2*e^16 - 104*b*c^3*d^7*e^11 - 6*b^3*c*d^5*e^13 + 8*a^2*c^2*d^4*e^14 + (18*c^5*d^10*e^8)/a + 20*b^2*c^2*d^6*
e^12 + (20*b^2*c^3*d^8*e^10)/a + (12*b^3*c^2*d^7*e^11)/a - (4*b^2*c^4*d^10*e^8)/a^2 + (10*b^3*c^3*d^9*e^9)/a^2
- (6*b^4*c^2*d^8*e^10)/a^2 - 32*a*b*c^2*d^5*e^13 + 12*a*b^2*c*d^4*e^14 - 8*a^2*b*c*d^3*e^15 - (48*b*c^4*d^9*e
^9)/a) + (20*b^2*c^2*d^4*e^12*(d + e*x^2)^(1/2)*(d^3)^(1/2))/(72*c^4*d^8*e^10 + 60*a*c^3*d^6*e^12 + 2*a^3*c*d^
2*e^16 - 104*b*c^3*d^7*e^11 - 6*b^3*c*d^5*e^13 + 8*a^2*c^2*d^4*e^14 + (18*c^5*d^10*e^8)/a + 20*b^2*c^2*d^6*e^1
2 + (20*b^2*c^3*d^8*e^10)/a + (12*b^3*c^2*d^7*e^11)/a - (4*b^2*c^4*d^10*e^8)/a^2 + (10*b^3*c^3*d^9*e^9)/a^2 -
(6*b^4*c^2*d^8*e^10)/a^2 - 32*a*b*c^2*d^5*e^13 + 12*a*b^2*c*d^4*e^14 - 8*a^2*b*c*d^3*e^15 - (48*b*c^4*d^9*e^9)
/a) - (48*b*c^4*d^7*e^9*(d + e*x^2)^(1/2)*(d^3)^(1/2))/(18*c^5*d^10*e^8 + 72*a*c^4*d^8*e^10 + 2*a^4*c*d^2*e^16
- 48*b*c^4*d^9*e^9 + 60*a^2*c^3*d^6*e^12 + 8*a^3*c^2*d^4*e^14 + 20*b^2*c^3*d^8*e^10 + 12*b^3*c^2*d^7*e^11 - (
4*b^2*c^4*d^10*e^8)/a + (10*b^3*c^3*d^9*e^9)/a - (6*b^4*c^2*d^8*e^10)/a - 104*a*b*c^3*d^7*e^11 - 6*a*b^3*c*d^5
*e^13 - 8*a^3*b*c*d^3*e^15 + 20*a*b^2*c^2*d^6*e^12 - 32*a^2*b*c^2*d^5*e^13 + 12*a^2*b^2*c*d^4*e^14) - (4*b^2*c
^4*d^8*e^8*(d + e*x^2)^(1/2)*(d^3)^(1/2))/(18*a*c^5*d^10*e^8 + 2*a^5*c*d^2*e^16 + 72*a^2*c^4*d^8*e^10 + 60*a^3
*c^3*d^6*e^12 + 8*a^4*c^2*d^4*e^14 - 4*b^2*c^4*d^10*e^8 + 10*b^3*c^3*d^9*e^9 - 6*b^4*c^2*d^8*e^10 + 20*a^2*b^2
*c^2*d^6*e^12 - 48*a*b*c^4*d^9*e^9 - 8*a^4*b*c*d^3*e^15 + 20*a*b^2*c^3*d^8*e^10 + 12*a*b^3*c^2*d^7*e^11 - 104*
a^2*b*c^3*d^7*e^11 - 6*a^2*b^3*c*d^5*e^13 - 32*a^3*b*c^2*d^5*e^13 + 12*a^3*b^2*c*d^4*e^14) + (10*b^3*c^3*d^7*e
^9*(d + e*x^2)^(1/2)*(d^3)^(1/2))/(18*a*c^5*d^10*e^8 + 2*a^5*c*d^2*e^16 + 72*a^2*c^4*d^8*e^10 + 60*a^3*c^3*d^6
*e^12 + 8*a^4*c^2*d^4*e^14 - 4*b^2*c^4*d^10*e^8 + 10*b^3*c^3*d^9*e^9 - 6*b^4*c^2*d^8*e^10 + 20*a^2*b^2*c^2*d^6
*e^12 - 48*a*b*c^4*d^9*e^9 - 8*a^4*b*c*d^3*e^15 + 20*a*b^2*c^3*d^8*e^10 + 12*a*b^3*c^2*d^7*e^11 - 104*a^2*b*c^
3*d^7*e^11 - 6*a^2*b^3*c*d^5*e^13 - 32*a^3*b*c^2*d^5*e^13 + 12*a^3*b^2*c*d^4*e^14) - (6*b^4*c^2*d^6*e^10*(d +
e*x^2)^(1/2)*(d^3)^(1/2))/(18*a*c^5*d^10*e^8 + 2*a^5*c*d^2*e^16 + 72*a^2*c^4*d^8*e^10 + 60*a^3*c^3*d^6*e^12 +
8*a^4*c^2*d^4*e^14 - 4*b^2*c^4*d^10*e^8 + 10*b^3*c^3*d^9*e^9 - 6*b^4*c^2*d^8*e^10 + 20*a^2*b^2*c^2*d^6*e^12 -
48*a*b*c^4*d^9*e^9 - 8*a^4*b*c*d^3*e^15 + 20*a*b^2*c^3*d^8*e^10 + 12*a*b^3*c^2*d^7*e^11 - 104*a^2*b*c^3*d^7*e^
11 - 6*a^2*b^3*c*d^5*e^13 - 32*a^3*b*c^2*d^5*e^13 + 12*a^3*b^2*c*d^4*e^14) + (60*a*c^3*d^4*e^12*(d + e*x^2)^(1
/2)*(d^3)^(1/2))/(72*c^4*d^8*e^10 + 60*a*c^3*d^6*e^12 + 2*a^3*c*d^2*e^16 - 104*b*c^3*d^7*e^11 - 6*b^3*c*d^5*e^
13 + 8*a^2*c^2*d^4*e^14 + (18*c^5*d^10*e^8)/a + 20*b^2*c^2*d^6*e^12 + (20*b^2*c^3*d^8*e^10)/a + (12*b^3*c^2*d^
7*e^11)/a - (4*b^2*c^4*d^10*e^8)/a^2 + (10*b^3*c^3*d^9*e^9)/a^2 - (6*b^4*c^2*d^8*e^10)/a^2 - 32*a*b*c^2*d^5*e^
13 + 12*a*b^2*c*d^4*e^14 - 8*a^2*b*c*d^3*e^15 - (48*b*c^4*d^9*e^9)/a) - (104*b*c^3*d^5*e^11*(d + e*x^2)^(1/2)*
(d^3)^(1/2))/(72*c^4*d^8*e^10 + 60*a*c^3*d^6*e^12 + 2*a^3*c*d^2*e^16 - 104*b*c^3*d^7*e^11 - 6*b^3*c*d^5*e^13 +
8*a^2*c^2*d^4*e^14 + (18*c^5*d^10*e^8)/a + 20*b^2*c^2*d^6*e^12 + (20*b^2*c^3*d^8*e^10)/a + (12*b^3*c^2*d^7*e^
11)/a - (4*b^2*c^4*d^10*e^8)/a^2 + (10*b^3*c^3*d^9*e^9)/a^2 - (6*b^4*c^2*d^8*e^10)/a^2 - 32*a*b*c^2*d^5*e^13 +
12*a*b^2*c*d^4*e^14 - 8*a^2*b*c*d^3*e^15 - (48*b*c^4*d^9*e^9)/a) - (6*b^3*c*d^3*e^13*(d + e*x^2)^(1/2)*(d^3)^
(1/2))/(72*c^4*d^8*e^10 + 60*a*c^3*d^6*e^12 + 2*a^3*c*d^2*e^16 - 104*b*c^3*d^7*e^11 - 6*b^3*c*d^5*e^13 + 8*a^2
*c^2*d^4*e^14 + (18*c^5*d^10*e^8)/a + 20*b^2*c^2*d^6*e^12 + (20*b^2*c^3*d^8*e^10)/a + (12*b^3*c^2*d^7*e^11)/a
- (4*b^2*c^4*d^10*e^8)/a^2 + (10*b^3*c^3*d^9*e^9)/a^2 - (6*b^4*c^2*d^8*e^10)/a^2 - 32*a*b*c^2*d^5*e^13 + 12*a*
b^2*c*d^4*e^14 - 8*a^2*b*c*d^3*e^15 - (48*b*c^4*d^9*e^9)/a) + (20*b^2*c^3*d^6*e^10*(d + e*x^2)^(1/2)*(d^3)^(1/
2))/(18*c^5*d^10*e^8 + 72*a*c^4*d^8*e^10 + 2*a^4*c*d^2*e^16 - 48*b*c^4*d^9*e^9 + 60*a^2*c^3*d^6*e^12 + 8*a^3*c
^2*d^4*e^14 + 20*b^2*c^3*d^8*e^10 + 12*b^3*c^2*d^7*e^11 - (4*b^2*c^4*d^10*e^8)/a + (10*b^3*c^3*d^9*e^9)/a - (6
*b^4*c^2*d^8*e^10)/a - 104*a*b*c^3*d^7*e^11 - 6*a*b^3*c*d^5*e^13 - 8*a^3*b*c*d^3*e^15 + 20*a*b^2*c^2*d^6*e^12
- 32*a^2*b*c^2*d^5*e^13 + 12*a^2*b^2*c*d^4*e^14) + (12*b^3*c^2*d^5*e^11*(d + e*x^2)^(1/2)*(d^3)^(1/2))/(18*c^5
*d^10*e^8 + 72*a*c^4*d^8*e^10 + 2*a^4*c*d^2*e^16 - 48*b*c^4*d^9*e^9 + 60*a^2*c^3*d^6*e^12 + 8*a^3*c^2*d^4*e^14
+ 20*b^2*c^3*d^8*e^10 + 12*b^3*c^2*d^7*e^11 - (4*b^2*c^4*d^10*e^8)/a + (10*b^3*c^3*d^9*e^9)/a - (6*b^4*c^2*d^
8*e^10)/a - 104*a*b*c^3*d^7*e^11 - 6*a*b^3*c*d^5*e^13 - 8*a^3*b*c*d^3*e^15 + 20*a*b^2*c^2*d^6*e^12 - 32*a^2*b*
c^2*d^5*e^13 + 12*a^2*b^2*c*d^4*e^14) - (32*a*b*c^2*d^3*e^13*(d + e*x^2)^(1/2)*(d^3)^(1/2))/(72*c^4*d^8*e^10 +
60*a*c^3*d^6*e^12 + 2*a^3*c*d^2*e^16 - 104*b*c^3*d^7*e^11 - 6*b^3*c*d^5*e^13 + 8*a^2*c^2*d^4*e^14 + (18*c^5*d
^10*e^8)/a + 20*b^2*c^2*d^6*e^12 + (20*b^2*c^3*d^8*e^10)/a + (12*b^3*c^2*d^7*e^11)/a - (4*b^2*c^4*d^10*e^8)/a^
2 + (10*b^3*c^3*d^9*e^9)/a^2 - (6*b^4*c^2*d^8*e^10)/a^2 - 32*a*b*c^2*d^5*e^13 + 12*a*b^2*c*d^4*e^14 - 8*a^2*b*
c*d^3*e^15 - (48*b*c^4*d^9*e^9)/a) + (12*a*b^2*c*d^2*e^14*(d + e*x^2)^(1/2)*(d^3)^(1/2))/(72*c^4*d^8*e^10 + 60
*a*c^3*d^6*e^12 + 2*a^3*c*d^2*e^16 - 104*b*c^3*d^7*e^11 - 6*b^3*c*d^5*e^13 + 8*a^2*c^2*d^4*e^14 + (18*c^5*d^10
*e^8)/a + 20*b^2*c^2*d^6*e^12 + (20*b^2*c^3*d^8*e^10)/a + (12*b^3*c^2*d^7*e^11)/a - (4*b^2*c^4*d^10*e^8)/a^2 +
(10*b^3*c^3*d^9*e^9)/a^2 - (6*b^4*c^2*d^8*e^10)/a^2 - 32*a*b*c^2*d^5*e^13 + 12*a*b^2*c*d^4*e^14 - 8*a^2*b*c*d
^3*e^15 - (48*b*c^4*d^9*e^9)/a) - (8*a^2*b*c*d*e^15*(d + e*x^2)^(1/2)*(d^3)^(1/2))/(72*c^4*d^8*e^10 + 60*a*c^3
*d^6*e^12 + 2*a^3*c*d^2*e^16 - 104*b*c^3*d^7*e^11 - 6*b^3*c*d^5*e^13 + 8*a^2*c^2*d^4*e^14 + (18*c^5*d^10*e^8)/
a + 20*b^2*c^2*d^6*e^12 + (20*b^2*c^3*d^8*e^10)/a + (12*b^3*c^2*d^7*e^11)/a - (4*b^2*c^4*d^10*e^8)/a^2 + (10*b
^3*c^3*d^9*e^9)/a^2 - (6*b^4*c^2*d^8*e^10)/a^2 - 32*a*b*c^2*d^5*e^13 + 12*a*b^2*c*d^4*e^14 - 8*a^2*b*c*d^3*e^1
5 - (48*b*c^4*d^9*e^9)/a))*(d^3)^(1/2))/a

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sympy [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)**(3/2)/x/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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