∫ √ ___ d+ex___ x3(a+bx+cx2) dx

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

Optimal. Leaf size=531 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (-a b e-a c d+b^2 d\right )}{a^3 \sqrt {d}}+\frac {\sqrt {2} \sqrt {c} \left (b^2 \left (d \sqrt {b^2-4 a c}-a e\right )-a b \left (e \sqrt {b^2-4 a c}+3 c d\right )-a c \left (d \sqrt {b^2-4 a c}-2 a e\right )+b^3 d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{a^3 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\sqrt {2} \sqrt {c} \left (-b^2 \left (d \sqrt {b^2-4 a c}+a e\right )-a b \left (3 c d-e \sqrt {b^2-4 a c}\right )+a c \left (d \sqrt {b^2-4 a c}+2 a e\right )+b^3 d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{a^3 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {e (b d-a e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a^2 d^{3/2}}+\frac {\sqrt {d+e x} (b d-a e)}{a^2 d x}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 a d^{3/2}}-\frac {\sqrt {d+e x}}{2 a x^2}+\frac {3 e \sqrt {d+e x}}{4 a d x} \]

________________________________________________________________________________________

Rubi [A] time = 3.57, antiderivative size = 531, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {897, 1287, 199, 206, 1166, 208} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (-a b e-a c d+b^2 d\right )}{a^3 \sqrt {d}}+\frac {\sqrt {2} \sqrt {c} \left (b^2 \left (d \sqrt {b^2-4 a c}-a e\right )-a b \left (e \sqrt {b^2-4 a c}+3 c d\right )-a c \left (d \sqrt {b^2-4 a c}-2 a e\right )+b^3 d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{a^3 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\sqrt {2} \sqrt {c} \left (-b^2 \left (d \sqrt {b^2-4 a c}+a e\right )-a b \left (3 c d-e \sqrt {b^2-4 a c}\right )+a c \left (d \sqrt {b^2-4 a c}+2 a e\right )+b^3 d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{a^3 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {e (b d-a e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a^2 d^{3/2}}+\frac {\sqrt {d+e x} (b d-a e)}{a^2 d x}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 a d^{3/2}}-\frac {\sqrt {d+e x}}{2 a x^2}+\frac {3 e \sqrt {d+e x}}{4 a d x} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Sqrt[d + e*x]/(2*a*x^2) + (3*e*Sqrt[d + e*x])/(4*a*d*x) + ((b*d - a*e)*Sqrt[d + e*x])/(a^2*d*x) - (3*e^2*ArcT

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

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

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

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

[b^2 - 4*a*c])*e]) - (Sqrt[2]*Sqrt[c]*(b^3*d - b^2*(Sqrt[b^2 - 4*a*c]*d + a*e) + a*c*(Sqrt[b^2 - 4*a*c]*d + 2*

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

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

________________________________________________________________________________________

Mathematica [A] time = 2.34, size = 516, normalized size = 0.97 \begin {gather*} -\frac {\frac {3 a^2 e \left (e x \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-\sqrt {d} \sqrt {d+e x}\right )}{d^{3/2} x}+\frac {2 a^2 \sqrt {d+e x}}{x^2}+\frac {8 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (-a b e-a c d+b^2 d\right )}{\sqrt {d}}+\frac {4 \sqrt {2} \sqrt {c} \left (b^2 \left (a e-d \sqrt {b^2-4 a c}\right )+a b \left (e \sqrt {b^2-4 a c}+3 c d\right )+a c \left (d \sqrt {b^2-4 a c}-2 a e\right )+b^3 (-d)\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )}{\sqrt {b^2-4 a c} \sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}+\frac {4 \sqrt {2} \sqrt {c} \left (-b^2 \left (d \sqrt {b^2-4 a c}+a e\right )+a b \left (e \sqrt {b^2-4 a c}-3 c d\right )+a c \left (d \sqrt {b^2-4 a c}+2 a e\right )+b^3 d\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {4 a e (a e-b d) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {4 a \sqrt {d+e x} (b d-a e)}{d x}}{4 a^3} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/4*((2*a^2*Sqrt[d + e*x])/x^2 - (4*a*(b*d - a*e)*Sqrt[d + e*x])/(d*x) - (4*a*e*(-(b*d) + a*e)*ArcTanh[Sqrt[d

+ e*x]/Sqrt[d]])/d^(3/2) + (8*(b^2*d - a*c*d - a*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d] + (3*a^2*e*(-(S

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

(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + b^2*(-(Sqrt[b^2 - 4*a*c]*d) + a*e) + a*b*(3*c*d + Sqrt[b^2 - 4*a*c]*e))*ArcTan

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

b + Sqrt[b^2 - 4*a*c])*e]) + (4*Sqrt[2]*Sqrt[c]*(b^3*d - b^2*(Sqrt[b^2 - 4*a*c]*d + a*e) + a*c*(Sqrt[b^2 - 4*a

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

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

________________________________________________________________________________________

IntegrateAlgebraic [A] time = 2.46, size = 577, normalized size = 1.09 \begin {gather*} -\frac {\sqrt {d+e x} \left (a e (d+e x)+a d e+4 b d^2-4 b d (d+e x)\right )}{4 a^2 d e x^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a^2 e^2+4 a b d e+8 a c d^2-8 b^2 d^2\right )}{4 a^3 d^{3/2}}+\frac {\left (-2 \sqrt {2} a^2 c^{3/2} e+\sqrt {2} a c^{3/2} d \sqrt {b^2-4 a c}-\sqrt {2} b^2 \sqrt {c} d \sqrt {b^2-4 a c}+\sqrt {2} a b^2 \sqrt {c} e+\sqrt {2} a b \sqrt {c} e \sqrt {b^2-4 a c}+3 \sqrt {2} a b c^{3/2} d-\sqrt {2} b^3 \sqrt {c} d\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-e \sqrt {b^2-4 a c}+b e-2 c d}}\right )}{a^3 \sqrt {b^2-4 a c} \sqrt {-e \sqrt {b^2-4 a c}+b e-2 c d}}+\frac {\left (2 \sqrt {2} a^2 c^{3/2} e+\sqrt {2} a c^{3/2} d \sqrt {b^2-4 a c}-\sqrt {2} b^2 \sqrt {c} d \sqrt {b^2-4 a c}-\sqrt {2} a b^2 \sqrt {c} e+\sqrt {2} a b \sqrt {c} e \sqrt {b^2-4 a c}-3 \sqrt {2} a b c^{3/2} d+\sqrt {2} b^3 \sqrt {c} d\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {b^2-4 a c}+b e-2 c d}}\right )}{a^3 \sqrt {b^2-4 a c} \sqrt {e \sqrt {b^2-4 a c}+b e-2 c d}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/4*(Sqrt[d + e*x]*(4*b*d^2 + a*d*e - 4*b*d*(d + e*x) + a*e*(d + e*x)))/(a^2*d*e*x^2) + ((-(Sqrt[2]*b^3*Sqrt[

c]*d) + 3*Sqrt[2]*a*b*c^(3/2)*d - Sqrt[2]*b^2*Sqrt[c]*Sqrt[b^2 - 4*a*c]*d + Sqrt[2]*a*c^(3/2)*Sqrt[b^2 - 4*a*c

]*d + Sqrt[2]*a*b^2*Sqrt[c]*e - 2*Sqrt[2]*a^2*c^(3/2)*e + Sqrt[2]*a*b*Sqrt[c]*Sqrt[b^2 - 4*a*c]*e)*ArcTan[(Sqr

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

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

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

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

]*e]])/(a^3*Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e]) + ((-8*b^2*d^2 + 8*a*c*d^2 + 4*a*b*d*e

+ a^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*a^3*d^(3/2))

________________________________________________________________________________________

fricas [B] time = 149.30, size = 7425, normalized size = 13.98

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(4*sqrt(2)*a^3*d^2*x^2*sqrt(((b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*d - (a*b^5 - 5*a^2*b^3*c + 5*a

^3*b*c^2)*e + (a^6*b^2 - 4*a^7*c)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d

^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c + 11

*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c)))/(a^6*b^2 - 4*a^7*c))*log(sqrt(2)*((b^9 -

9*a*b^7*c + 27*a^2*b^5*c^2 - 31*a^3*b^3*c^3 + 12*a^4*b*c^4)*d - (a*b^8 - 8*a^2*b^6*c + 20*a^3*b^4*c^2 - 17*a^4

*b^2*c^3 + 4*a^5*c^4)*e - (a^6*b^5 - 7*a^7*b^3*c + 12*a^8*b*c^2)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24

*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*

e + (a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c)))*sqrt(((b^6

- 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*d - (a*b^5 - 5*a^2*b^3*c + 5*a^3*b*c^2)*e + (a^6*b^2 - 4*a^7*c)*sqrt

(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b

^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4

)*e^2)/(a^12*b^2 - 4*a^13*c)))/(a^6*b^2 - 4*a^7*c)) - 4*((b^5*c^3 - 4*a*b^3*c^4 + 3*a^2*b*c^5)*d - (a*b^4*c^3

- 3*a^2*b^2*c^4 + a^3*c^5)*e)*sqrt(e*x + d)) - 4*sqrt(2)*a^3*d^2*x^2*sqrt(((b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 -

2*a^3*c^3)*d - (a*b^5 - 5*a^2*b^3*c + 5*a^3*b*c^2)*e + (a^6*b^2 - 4*a^7*c)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*b^

6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5

*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c)))/

(a^6*b^2 - 4*a^7*c))*log(-sqrt(2)*((b^9 - 9*a*b^7*c + 27*a^2*b^5*c^2 - 31*a^3*b^3*c^3 + 12*a^4*b*c^4)*d - (a*b

^8 - 8*a^2*b^6*c + 20*a^3*b^4*c^2 - 17*a^4*b^2*c^3 + 4*a^5*c^4)*e - (a^6*b^5 - 7*a^7*b^3*c + 12*a^8*b*c^2)*sqr

t(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*

b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^

4)*e^2)/(a^12*b^2 - 4*a^13*c)))*sqrt(((b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*d - (a*b^5 - 5*a^2*b^3*c +

5*a^3*b*c^2)*e + (a^6*b^2 - 4*a^7*c)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^

4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c

+ 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c)))/(a^6*b^2 - 4*a^7*c)) - 4*((b^5*c^3 -

4*a*b^3*c^4 + 3*a^2*b*c^5)*d - (a*b^4*c^3 - 3*a^2*b^2*c^4 + a^3*c^5)*e)*sqrt(e*x + d)) + 4*sqrt(2)*a^3*d^2*x^2

*sqrt(((b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*d - (a*b^5 - 5*a^2*b^3*c + 5*a^3*b*c^2)*e - (a^6*b^2 - 4*

a^7*c)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c

+ 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c^

3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c)))/(a^6*b^2 - 4*a^7*c))*log(sqrt(2)*((b^9 - 9*a*b^7*c + 27*a^2*b^5*c^2

- 31*a^3*b^3*c^3 + 12*a^4*b*c^4)*d - (a*b^8 - 8*a^2*b^6*c + 20*a^3*b^4*c^2 - 17*a^4*b^2*c^3 + 4*a^5*c^4)*e + (

a^6*b^5 - 7*a^7*b^3*c + 12*a^8*b*c^2)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^

4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c

+ 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c)))*sqrt(((b^6 - 6*a*b^4*c + 9*a^2*b^2*c^

2 - 2*a^3*c^3)*d - (a*b^5 - 5*a^2*b^3*c + 5*a^3*b*c^2)*e - (a^6*b^2 - 4*a^7*c)*sqrt(((b^10 - 8*a*b^8*c + 22*a^

2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3

*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c

)))/(a^6*b^2 - 4*a^7*c)) - 4*((b^5*c^3 - 4*a*b^3*c^4 + 3*a^2*b*c^5)*d - (a*b^4*c^3 - 3*a^2*b^2*c^4 + a^3*c^5)*

e)*sqrt(e*x + d)) - 4*sqrt(2)*a^3*d^2*x^2*sqrt(((b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*d - (a*b^5 - 5*a

^2*b^3*c + 5*a^3*b*c^2)*e - (a^6*b^2 - 4*a^7*c)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*

a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*

a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c)))/(a^6*b^2 - 4*a^7*c))*log(-s

qrt(2)*((b^9 - 9*a*b^7*c + 27*a^2*b^5*c^2 - 31*a^3*b^3*c^3 + 12*a^4*b*c^4)*d - (a*b^8 - 8*a^2*b^6*c + 20*a^3*b

^4*c^2 - 17*a^4*b^2*c^3 + 4*a^5*c^4)*e + (a^6*b^5 - 7*a^7*b^3*c + 12*a^8*b*c^2)*sqrt(((b^10 - 8*a*b^8*c + 22*a

^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 +

3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*

c)))*sqrt(((b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*d - (a*b^5 - 5*a^2*b^3*c + 5*a^3*b*c^2)*e - (a^6*b^2

- 4*a^7*c)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b

^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^

2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c)))/(a^6*b^2 - 4*a^7*c)) - 4*((b^5*c^3 - 4*a*b^3*c^4 + 3*a^2*b*c^5)*

d - (a*b^4*c^3 - 3*a^2*b^2*c^4 + a^3*c^5)*e)*sqrt(e*x + d)) + (4*a*b*d*e + a^2*e^2 - 8*(b^2 - a*c)*d^2)*sqrt(d

)*x^2*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*a^2*d^2 - (4*a*b*d^2 - a^2*d*e)*x)*sqrt(e*x + d))/(a

^3*d^2*x^2), 1/4*(2*sqrt(2)*a^3*d^2*x^2*sqrt(((b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*d - (a*b^5 - 5*a^2

*b^3*c + 5*a^3*b*c^2)*e + (a^6*b^2 - 4*a^7*c)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^

4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^

3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c)))/(a^6*b^2 - 4*a^7*c))*log(sqrt

(2)*((b^9 - 9*a*b^7*c + 27*a^2*b^5*c^2 - 31*a^3*b^3*c^3 + 12*a^4*b*c^4)*d - (a*b^8 - 8*a^2*b^6*c + 20*a^3*b^4*

c^2 - 17*a^4*b^2*c^3 + 4*a^5*c^4)*e - (a^6*b^5 - 7*a^7*b^3*c + 12*a^8*b*c^2)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*

b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a

^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c))

)*sqrt(((b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*d - (a*b^5 - 5*a^2*b^3*c + 5*a^3*b*c^2)*e + (a^6*b^2 - 4

*a^7*c)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*

c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c

^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c)))/(a^6*b^2 - 4*a^7*c)) - 4*((b^5*c^3 - 4*a*b^3*c^4 + 3*a^2*b*c^5)*d -

(a*b^4*c^3 - 3*a^2*b^2*c^4 + a^3*c^5)*e)*sqrt(e*x + d)) - 2*sqrt(2)*a^3*d^2*x^2*sqrt(((b^6 - 6*a*b^4*c + 9*a^

2*b^2*c^2 - 2*a^3*c^3)*d - (a*b^5 - 5*a^2*b^3*c + 5*a^3*b*c^2)*e + (a^6*b^2 - 4*a^7*c)*sqrt(((b^10 - 8*a*b^8*c

+ 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3

*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 -

4*a^13*c)))/(a^6*b^2 - 4*a^7*c))*log(-sqrt(2)*((b^9 - 9*a*b^7*c + 27*a^2*b^5*c^2 - 31*a^3*b^3*c^3 + 12*a^4*b*c

^4)*d - (a*b^8 - 8*a^2*b^6*c + 20*a^3*b^4*c^2 - 17*a^4*b^2*c^3 + 4*a^5*c^4)*e - (a^6*b^5 - 7*a^7*b^3*c + 12*a^

8*b*c^2)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7

*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*

c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c)))*sqrt(((b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*d - (a*b^5 - 5

*a^2*b^3*c + 5*a^3*b*c^2)*e + (a^6*b^2 - 4*a^7*c)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 +

9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 -

6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c)))/(a^6*b^2 - 4*a^7*c)) - 4*

((b^5*c^3 - 4*a*b^3*c^4 + 3*a^2*b*c^5)*d - (a*b^4*c^3 - 3*a^2*b^2*c^4 + a^3*c^5)*e)*sqrt(e*x + d)) + 2*sqrt(2)

*a^3*d^2*x^2*sqrt(((b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*d - (a*b^5 - 5*a^2*b^3*c + 5*a^3*b*c^2)*e - (

a^6*b^2 - 4*a^7*c)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 -

7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^2 -

6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c)))/(a^6*b^2 - 4*a^7*c))*log(sqrt(2)*((b^9 - 9*a*b^7*c + 27*

a^2*b^5*c^2 - 31*a^3*b^3*c^3 + 12*a^4*b*c^4)*d - (a*b^8 - 8*a^2*b^6*c + 20*a^3*b^4*c^2 - 17*a^4*b^2*c^3 + 4*a^

5*c^4)*e + (a^6*b^5 - 7*a^7*b^3*c + 12*a^8*b*c^2)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 +

9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 -

6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c)))*sqrt(((b^6 - 6*a*b^4*c +

9*a^2*b^2*c^2 - 2*a^3*c^3)*d - (a*b^5 - 5*a^2*b^3*c + 5*a^3*b*c^2)*e - (a^6*b^2 - 4*a^7*c)*sqrt(((b^10 - 8*a*b

^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4

*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^

2 - 4*a^13*c)))/(a^6*b^2 - 4*a^7*c)) - 4*((b^5*c^3 - 4*a*b^3*c^4 + 3*a^2*b*c^5)*d - (a*b^4*c^3 - 3*a^2*b^2*c^4

+ a^3*c^5)*e)*sqrt(e*x + d)) - 2*sqrt(2)*a^3*d^2*x^2*sqrt(((b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*d -

(a*b^5 - 5*a^2*b^3*c + 5*a^3*b*c^2)*e - (a^6*b^2 - 4*a^7*c)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*

b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (

a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c)))/(a^6*b^2 - 4*a^

7*c))*log(-sqrt(2)*((b^9 - 9*a*b^7*c + 27*a^2*b^5*c^2 - 31*a^3*b^3*c^3 + 12*a^4*b*c^4)*d - (a*b^8 - 8*a^2*b^6*

c + 20*a^3*b^4*c^2 - 17*a^4*b^2*c^3 + 4*a^5*c^4)*e + (a^6*b^5 - 7*a^7*b^3*c + 12*a^8*b*c^2)*sqrt(((b^10 - 8*a*

b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^

4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b

^2 - 4*a^13*c)))*sqrt(((b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 2*a^3*c^3)*d - (a*b^5 - 5*a^2*b^3*c + 5*a^3*b*c^2)*e

- (a^6*b^2 - 4*a^7*c)*sqrt(((b^10 - 8*a*b^8*c + 22*a^2*b^6*c^2 - 24*a^3*b^4*c^3 + 9*a^4*b^2*c^4)*d^2 - 2*(a*b

^9 - 7*a^2*b^7*c + 16*a^3*b^5*c^2 - 13*a^4*b^3*c^3 + 3*a^5*b*c^4)*d*e + (a^2*b^8 - 6*a^3*b^6*c + 11*a^4*b^4*c^

2 - 6*a^5*b^2*c^3 + a^6*c^4)*e^2)/(a^12*b^2 - 4*a^13*c)))/(a^6*b^2 - 4*a^7*c)) - 4*((b^5*c^3 - 4*a*b^3*c^4 + 3

*a^2*b*c^5)*d - (a*b^4*c^3 - 3*a^2*b^2*c^4 + a^3*c^5)*e)*sqrt(e*x + d)) - (4*a*b*d*e + a^2*e^2 - 8*(b^2 - a*c)

*d^2)*sqrt(-d)*x^2*arctan(sqrt(e*x + d)*sqrt(-d)/d) - (2*a^2*d^2 - (4*a*b*d^2 - a^2*d*e)*x)*sqrt(e*x + d))/(a^

3*d^2*x^2)]

________________________________________________________________________________________

giac [B] time = 0.59, size = 1041, normalized size = 1.96 \begin {gather*} -\frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d e - {\left (a b^{3} - 4 \, a^{2} b c\right )} e^{2}\right )} a^{2} - 2 \, {\left ({\left (a b^{2} c - a^{2} c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (a b^{3} - a^{2} b c\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a^{2} b^{2} - a^{3} c\right )} \sqrt {b^{2} - 4 \, a c} e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | a \right |} - \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, {\left (a^{2} b^{3} c - 3 \, a^{3} b c^{2}\right )} d^{2} - {\left (a^{2} b^{4} - a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} d e + {\left (a^{3} b^{3} - 2 \, a^{4} b c\right )} e^{2}\right )}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, a^{3} c d - a^{3} b e + \sqrt {-4 \, {\left (a^{3} c d^{2} - a^{3} b d e + a^{4} e^{2}\right )} a^{3} c + {\left (2 \, a^{3} c d - a^{3} b e\right )}^{2}}}{a^{3} c}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} a^{4} c d^{2} - \sqrt {b^{2} - 4 \, a c} a^{4} b d e + \sqrt {b^{2} - 4 \, a c} a^{5} e^{2}\right )} {\left | a \right |} {\left | c \right |}} + \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d e - {\left (a b^{3} - 4 \, a^{2} b c\right )} e^{2}\right )} a^{2} + 2 \, {\left ({\left (a b^{2} c - a^{2} c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (a b^{3} - a^{2} b c\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a^{2} b^{2} - a^{3} c\right )} \sqrt {b^{2} - 4 \, a c} e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | a \right |} - \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, {\left (a^{2} b^{3} c - 3 \, a^{3} b c^{2}\right )} d^{2} - {\left (a^{2} b^{4} - a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} d e + {\left (a^{3} b^{3} - 2 \, a^{4} b c\right )} e^{2}\right )}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, a^{3} c d - a^{3} b e - \sqrt {-4 \, {\left (a^{3} c d^{2} - a^{3} b d e + a^{4} e^{2}\right )} a^{3} c + {\left (2 \, a^{3} c d - a^{3} b e\right )}^{2}}}{a^{3} c}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} a^{4} c d^{2} - \sqrt {b^{2} - 4 \, a c} a^{4} b d e + \sqrt {b^{2} - 4 \, a c} a^{5} e^{2}\right )} {\left | a \right |} {\left | c \right |}} + \frac {{\left (8 \, b^{2} d^{2} - 8 \, a c d^{2} - 4 \, a b d e - a^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, a^{3} \sqrt {-d} d} + \frac {{\left (4 \, {\left (x e + d\right )}^{\frac {3}{2}} b d e - 4 \, \sqrt {x e + d} b d^{2} e - {\left (x e + d\right )}^{\frac {3}{2}} a e^{2} - \sqrt {x e + d} a d e^{2}\right )} e^{\left (-2\right )}}{4 \, a^{2} d x^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*((b^4 - 5*a*b^2*c + 4*a^2*c^2)*d*e - (a*b^3 - 4*a^2*b*c

)*e^2)*a^2 - 2*((a*b^2*c - a^2*c^2)*sqrt(b^2 - 4*a*c)*d^2 - (a*b^3 - a^2*b*c)*sqrt(b^2 - 4*a*c)*d*e + (a^2*b^2

- a^3*c)*sqrt(b^2 - 4*a*c)*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(a) - sqrt(-4*c^2*d + 2*(

b*c - sqrt(b^2 - 4*a*c)*c)*e)*(2*(a^2*b^3*c - 3*a^3*b*c^2)*d^2 - (a^2*b^4 - a^3*b^2*c - 4*a^4*c^2)*d*e + (a^3*

b^3 - 2*a^4*b*c)*e^2))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*a^3*c*d - a^3*b*e + sqrt(-4*(a^3*c*d^2 - a^3*

b*d*e + a^4*e^2)*a^3*c + (2*a^3*c*d - a^3*b*e)^2))/(a^3*c)))/((sqrt(b^2 - 4*a*c)*a^4*c*d^2 - sqrt(b^2 - 4*a*c)

*a^4*b*d*e + sqrt(b^2 - 4*a*c)*a^5*e^2)*abs(a)*abs(c)) + 1/4*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)

*((b^4 - 5*a*b^2*c + 4*a^2*c^2)*d*e - (a*b^3 - 4*a^2*b*c)*e^2)*a^2 + 2*((a*b^2*c - a^2*c^2)*sqrt(b^2 - 4*a*c)*

d^2 - (a*b^3 - a^2*b*c)*sqrt(b^2 - 4*a*c)*d*e + (a^2*b^2 - a^3*c)*sqrt(b^2 - 4*a*c)*e^2)*sqrt(-4*c^2*d + 2*(b*

c + sqrt(b^2 - 4*a*c)*c)*e)*abs(a) - sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(2*(a^2*b^3*c - 3*a^3*b*

c^2)*d^2 - (a^2*b^4 - a^3*b^2*c - 4*a^4*c^2)*d*e + (a^3*b^3 - 2*a^4*b*c)*e^2))*arctan(2*sqrt(1/2)*sqrt(x*e + d

)/sqrt(-(2*a^3*c*d - a^3*b*e - sqrt(-4*(a^3*c*d^2 - a^3*b*d*e + a^4*e^2)*a^3*c + (2*a^3*c*d - a^3*b*e)^2))/(a^

3*c)))/((sqrt(b^2 - 4*a*c)*a^4*c*d^2 - sqrt(b^2 - 4*a*c)*a^4*b*d*e + sqrt(b^2 - 4*a*c)*a^5*e^2)*abs(a)*abs(c))

+ 1/4*(8*b^2*d^2 - 8*a*c*d^2 - 4*a*b*d*e - a^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(a^3*sqrt(-d)*d) + 1/4*(4*

(x*e + d)^(3/2)*b*d*e - 4*sqrt(x*e + d)*b*d^2*e - (x*e + d)^(3/2)*a*e^2 - sqrt(x*e + d)*a*d*e^2)*e^(-2)/(a^2*d

*x^2)

________________________________________________________________________________________

maple [B] time = 0.05, size = 1486, normalized size = 2.80

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/a/x^2/d*(e*x+d)^(3/2)+1/e/a^2/x^2*(e*x+d)^(3/2)*b-1/e/a^2/x^2*(e*x+d)^(1/2)*b*d-1/4*(e*x+d)^(1/2)/a/x^2+1

/4*e^2*arctanh((e*x+d)^(1/2)/d^(1/2))/a/d^(3/2)+e/a^2/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*b+2/a^2*d^(1/2)*a

rctanh((e*x+d)^(1/2)/d^(1/2))*c-2/a^3*d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*b^2+2*e^2/a*c^2/(-(4*a*c-b^2)*e^2

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

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

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

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

/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b*d+e/a^3*c/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c

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

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

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

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

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

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

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

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

*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^2-3*e/a^2*c^2/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2

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

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

b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^3*d+e/a^2*c*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)

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

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

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

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B] time = 8.09, size = 33838, normalized size = 63.73

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((((((128*a^12*c^4*d*e^12 + 768*a^10*c^6*d^5*e^8 + 896*a^11*c^5*d^3*e^10 + 128*a^8*b^4*c^4*d^5*e^8 - 96*a

^8*b^5*c^3*d^4*e^9 - 32*a^8*b^6*c^2*d^3*e^10 - 704*a^9*b^2*c^5*d^5*e^8 + 448*a^9*b^3*c^4*d^4*e^9 + 392*a^9*b^4

*c^3*d^3*e^10 + 24*a^9*b^5*c^2*d^2*e^11 - 1280*a^10*b^2*c^4*d^3*e^10 - 192*a^10*b^3*c^3*d^2*e^11 - 256*a^10*b*

c^5*d^4*e^9 + 8*a^10*b^4*c^2*d*e^12 + 384*a^11*b*c^4*d^2*e^11 - 64*a^11*b^2*c^3*d*e^12)/(2*a^8*d^2) - ((d + e*

x)^(1/2)*((b^8*d + 8*a^4*c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*

d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) +

9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2)

- 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2)*(1536*a^12*c^5*d^4*

e^8 + 1024*a^13*c^4*d^2*e^10 + 128*a^10*b^4*c^3*d^4*e^8 - 128*a^10*b^5*c^2*d^3*e^9 - 896*a^11*b^2*c^4*d^4*e^8

+ 960*a^11*b^3*c^3*d^3*e^9 + 64*a^11*b^4*c^2*d^2*e^10 - 512*a^12*b^2*c^3*d^2*e^10 - 1792*a^12*b*c^4*d^3*e^9))/

(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*

c^3*d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2

) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(

1/2) - 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) + ((d + e*x)^(1

/2)*(8*a^10*c^5*d*e^12 - 12*a^10*b*c^4*e^13 - a^8*b^5*c^2*e^13 + 7*a^9*b^3*c^3*e^13 + 1152*a^8*c^7*d^5*e^8 + 5

12*a^9*c^6*d^3*e^10 + 128*a^4*b^8*c^3*d^5*e^8 - 128*a^4*b^9*c^2*d^4*e^9 - 1152*a^5*b^6*c^4*d^5*e^8 + 1088*a^5*

b^7*c^3*d^4*e^9 + 192*a^5*b^8*c^2*d^3*e^10 + 3520*a^6*b^4*c^5*d^5*e^8 - 2816*a^6*b^5*c^4*d^4*e^9 - 1728*a^6*b^

6*c^3*d^3*e^10 - 64*a^6*b^7*c^2*d^2*e^11 - 4096*a^7*b^2*c^6*d^5*e^8 + 1792*a^7*b^3*c^5*d^4*e^9 + 4944*a^7*b^4*

c^4*d^3*e^10 + 568*a^7*b^5*c^3*d^2*e^11 - 4512*a^8*b^2*c^5*d^3*e^10 - 1536*a^8*b^3*c^4*d^2*e^11 - 8*a^7*b^6*c^

2*d*e^12 + 896*a^8*b*c^6*d^4*e^9 + 57*a^8*b^4*c^3*d*e^12 + 1152*a^9*b*c^5*d^2*e^11 - 102*a^9*b^2*c^4*d*e^12))/

(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*

c^3*d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2

) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(

1/2) - 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) + (4*a^9*c^5*e^

14 - a^6*b^6*c^2*e^14 + 7*a^7*b^4*c^3*e^14 - 13*a^8*b^2*c^4*e^14 - 192*a^6*c^8*d^6*e^8 - 192*a^7*c^7*d^4*e^10

+ 4*a^8*c^6*d^2*e^12 - 128*a^2*b^8*c^4*d^6*e^8 + 96*a^2*b^9*c^3*d^5*e^9 + 32*a^2*b^10*c^2*d^4*e^10 + 960*a^3*b

^6*c^5*d^6*e^8 - 512*a^3*b^7*c^4*d^5*e^9 - 552*a^3*b^8*c^3*d^4*e^10 - 56*a^3*b^9*c^2*d^3*e^11 - 2176*a^4*b^4*c

^6*d^6*e^8 + 224*a^4*b^5*c^5*d^5*e^9 + 2688*a^4*b^6*c^4*d^4*e^10 + 672*a^4*b^7*c^3*d^3*e^11 + 24*a^4*b^8*c^2*d

^2*e^12 + 1600*a^5*b^2*c^7*d^6*e^8 + 1408*a^5*b^3*c^6*d^5*e^9 - 4536*a^5*b^4*c^5*d^4*e^10 - 2616*a^5*b^5*c^4*d

^3*e^11 - 209*a^5*b^6*c^3*d^2*e^12 + 2336*a^6*b^2*c^6*d^4*e^10 + 3648*a^6*b^3*c^5*d^3*e^11 + 559*a^6*b^4*c^4*d

^2*e^12 - 429*a^7*b^2*c^5*d^2*e^12 - 132*a^8*b*c^5*d*e^13 + a^5*b^7*c^2*d*e^13 - 1088*a^6*b*c^7*d^5*e^9 - 23*a

^6*b^5*c^3*d*e^13 - 1408*a^7*b*c^6*d^3*e^11 + 109*a^7*b^3*c^4*d*e^13)/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d - b^5

*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(

4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a

*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^

(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) - ((d + e*x)^(1/2)*(a^6*b^2*c^5*e^14 - 2*a^7*c^6*e^14 +

192*a^4*c^9*d^6*e^8 + 32*a^5*c^8*d^4*e^10 + 34*a^6*c^7*d^2*e^12 + 64*b^8*c^5*d^6*e^8 + 704*a^2*b^4*c^7*d^6*e^

8 + 960*a^2*b^5*c^6*d^5*e^9 + 192*a^2*b^6*c^5*d^4*e^10 - 512*a^3*b^2*c^8*d^6*e^8 - 1280*a^3*b^3*c^7*d^5*e^9 -

752*a^3*b^4*c^6*d^4*e^10 - 56*a^3*b^5*c^5*d^3*e^11 + 704*a^4*b^2*c^7*d^4*e^10 + 128*a^4*b^3*c^6*d^3*e^11 - 15*

a^4*b^4*c^5*d^2*e^12 + 60*a^5*b^2*c^6*d^2*e^12 - 10*a^6*b*c^6*d*e^13 - 384*a*b^6*c^6*d^6*e^8 - 192*a*b^7*c^5*d

^5*e^9 + 384*a^4*b*c^8*d^5*e^9 - 144*a^5*b*c^7*d^3*e^11 + 6*a^5*b^3*c^5*d*e^13))/(2*a^8*d^2))*((b^8*d + 8*a^4*

c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a^

3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*

c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*c

- b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2)*1i - (((((128*a^12*c^4*d*e^12 + 768*a^10*c^6

*d^5*e^8 + 896*a^11*c^5*d^3*e^10 + 128*a^8*b^4*c^4*d^5*e^8 - 96*a^8*b^5*c^3*d^4*e^9 - 32*a^8*b^6*c^2*d^3*e^10

- 704*a^9*b^2*c^5*d^5*e^8 + 448*a^9*b^3*c^4*d^4*e^9 + 392*a^9*b^4*c^3*d^3*e^10 + 24*a^9*b^5*c^2*d^2*e^11 - 128

0*a^10*b^2*c^4*d^3*e^10 - 192*a^10*b^3*c^3*d^2*e^11 - 256*a^10*b*c^5*d^4*e^9 + 8*a^10*b^4*c^2*d*e^12 + 384*a^1

1*b*c^4*d^2*e^11 - 64*a^11*b^2*c^3*d*e^12)/(2*a^8*d^2) + ((d + e*x)^(1/2)*((b^8*d + 8*a^4*c^4*d - b^5*d*(-(4*a

*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c - b

^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d*

(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2

*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2)*(1536*a^12*c^5*d^4*e^8 + 1024*a^13*c^4*d^2*e^10 + 128*a^10*b^4*c

^3*d^4*e^8 - 128*a^10*b^5*c^2*d^3*e^9 - 896*a^11*b^2*c^4*d^4*e^8 + 960*a^11*b^3*c^3*d^3*e^9 + 64*a^11*b^4*c^2*

d^2*e^10 - 512*a^12*b^2*c^3*d^2*e^10 - 1792*a^12*b*c^4*d^3*e^9))/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d - b^5*d*(-

(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c

- b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*

c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2)

)/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) - ((d + e*x)^(1/2)*(8*a^10*c^5*d*e^12 - 12*a^10*b*c^4*e^13 -

a^8*b^5*c^2*e^13 + 7*a^9*b^3*c^3*e^13 + 1152*a^8*c^7*d^5*e^8 + 512*a^9*c^6*d^3*e^10 + 128*a^4*b^8*c^3*d^5*e^8

- 128*a^4*b^9*c^2*d^4*e^9 - 1152*a^5*b^6*c^4*d^5*e^8 + 1088*a^5*b^7*c^3*d^4*e^9 + 192*a^5*b^8*c^2*d^3*e^10 +

3520*a^6*b^4*c^5*d^5*e^8 - 2816*a^6*b^5*c^4*d^4*e^9 - 1728*a^6*b^6*c^3*d^3*e^10 - 64*a^6*b^7*c^2*d^2*e^11 - 40

96*a^7*b^2*c^6*d^5*e^8 + 1792*a^7*b^3*c^5*d^4*e^9 + 4944*a^7*b^4*c^4*d^3*e^10 + 568*a^7*b^5*c^3*d^2*e^11 - 451

2*a^8*b^2*c^5*d^3*e^10 - 1536*a^8*b^3*c^4*d^2*e^11 - 8*a^7*b^6*c^2*d*e^12 + 896*a^8*b*c^6*d^4*e^9 + 57*a^8*b^4

*c^3*d*e^12 + 1152*a^9*b*c^5*d^2*e^11 - 102*a^9*b^2*c^4*d*e^12))/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d - b^5*d*(-

(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c

- b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*

c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2)

)/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) + (4*a^9*c^5*e^14 - a^6*b^6*c^2*e^14 + 7*a^7*b^4*c^3*e^14 -

13*a^8*b^2*c^4*e^14 - 192*a^6*c^8*d^6*e^8 - 192*a^7*c^7*d^4*e^10 + 4*a^8*c^6*d^2*e^12 - 128*a^2*b^8*c^4*d^6*e^

8 + 96*a^2*b^9*c^3*d^5*e^9 + 32*a^2*b^10*c^2*d^4*e^10 + 960*a^3*b^6*c^5*d^6*e^8 - 512*a^3*b^7*c^4*d^5*e^9 - 55

2*a^3*b^8*c^3*d^4*e^10 - 56*a^3*b^9*c^2*d^3*e^11 - 2176*a^4*b^4*c^6*d^6*e^8 + 224*a^4*b^5*c^5*d^5*e^9 + 2688*a

^4*b^6*c^4*d^4*e^10 + 672*a^4*b^7*c^3*d^3*e^11 + 24*a^4*b^8*c^2*d^2*e^12 + 1600*a^5*b^2*c^7*d^6*e^8 + 1408*a^5

*b^3*c^6*d^5*e^9 - 4536*a^5*b^4*c^5*d^4*e^10 - 2616*a^5*b^5*c^4*d^3*e^11 - 209*a^5*b^6*c^3*d^2*e^12 + 2336*a^6

*b^2*c^6*d^4*e^10 + 3648*a^6*b^3*c^5*d^3*e^11 + 559*a^6*b^4*c^4*d^2*e^12 - 429*a^7*b^2*c^5*d^2*e^12 - 132*a^8*

b*c^5*d*e^13 + a^5*b^7*c^2*d*e^13 - 1088*a^6*b*c^7*d^5*e^9 - 23*a^6*b^5*c^3*d*e^13 - 1408*a^7*b*c^6*d^3*e^11 +

109*a^7*b^3*c^4*d*e^13)/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^

2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*

e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c

^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c

)))^(1/2) + ((d + e*x)^(1/2)*(a^6*b^2*c^5*e^14 - 2*a^7*c^6*e^14 + 192*a^4*c^9*d^6*e^8 + 32*a^5*c^8*d^4*e^10 +

34*a^6*c^7*d^2*e^12 + 64*b^8*c^5*d^6*e^8 + 704*a^2*b^4*c^7*d^6*e^8 + 960*a^2*b^5*c^6*d^5*e^9 + 192*a^2*b^6*c^5

*d^4*e^10 - 512*a^3*b^2*c^8*d^6*e^8 - 1280*a^3*b^3*c^7*d^5*e^9 - 752*a^3*b^4*c^6*d^4*e^10 - 56*a^3*b^5*c^5*d^3

*e^11 + 704*a^4*b^2*c^7*d^4*e^10 + 128*a^4*b^3*c^6*d^3*e^11 - 15*a^4*b^4*c^5*d^2*e^12 + 60*a^5*b^2*c^6*d^2*e^1

2 - 10*a^6*b*c^6*d*e^13 - 384*a*b^6*c^6*d^6*e^8 - 192*a*b^7*c^5*d^5*e^9 + 384*a^4*b*c^8*d^5*e^9 - 144*a^5*b*c^

7*d^3*e^11 + 6*a^5*b^3*c^5*d*e^13))/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^

7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c

*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2)

- 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 -

8*a^7*b^2*c)))^(1/2)*1i)/((((((128*a^12*c^4*d*e^12 + 768*a^10*c^6*d^5*e^8 + 896*a^11*c^5*d^3*e^10 + 128*a^8*b^

4*c^4*d^5*e^8 - 96*a^8*b^5*c^3*d^4*e^9 - 32*a^8*b^6*c^2*d^3*e^10 - 704*a^9*b^2*c^5*d^5*e^8 + 448*a^9*b^3*c^4*d

^4*e^9 + 392*a^9*b^4*c^3*d^3*e^10 + 24*a^9*b^5*c^2*d^2*e^11 - 1280*a^10*b^2*c^4*d^3*e^10 - 192*a^10*b^3*c^3*d^

2*e^11 - 256*a^10*b*c^5*d^4*e^9 + 8*a^10*b^4*c^2*d*e^12 + 384*a^11*b*c^4*d^2*e^11 - 64*a^11*b^2*c^3*d*e^12)/(2

*a^8*d^2) - ((d + e*x)^(1/2)*((b^8*d + 8*a^4*c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2

*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*

c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4

*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2)

*(1536*a^12*c^5*d^4*e^8 + 1024*a^13*c^4*d^2*e^10 + 128*a^10*b^4*c^3*d^4*e^8 - 128*a^10*b^5*c^2*d^3*e^9 - 896*a

^11*b^2*c^4*d^4*e^8 + 960*a^11*b^3*c^3*d^3*e^9 + 64*a^11*b^4*c^2*d^2*e^10 - 512*a^12*b^2*c^3*d^2*e^10 - 1792*a

^12*b*c^4*d^3*e^9))/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4

*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(

4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*

(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(

1/2) + ((d + e*x)^(1/2)*(8*a^10*c^5*d*e^12 - 12*a^10*b*c^4*e^13 - a^8*b^5*c^2*e^13 + 7*a^9*b^3*c^3*e^13 + 1152

*a^8*c^7*d^5*e^8 + 512*a^9*c^6*d^3*e^10 + 128*a^4*b^8*c^3*d^5*e^8 - 128*a^4*b^9*c^2*d^4*e^9 - 1152*a^5*b^6*c^4

*d^5*e^8 + 1088*a^5*b^7*c^3*d^4*e^9 + 192*a^5*b^8*c^2*d^3*e^10 + 3520*a^6*b^4*c^5*d^5*e^8 - 2816*a^6*b^5*c^4*d

^4*e^9 - 1728*a^6*b^6*c^3*d^3*e^10 - 64*a^6*b^7*c^2*d^2*e^11 - 4096*a^7*b^2*c^6*d^5*e^8 + 1792*a^7*b^3*c^5*d^4

*e^9 + 4944*a^7*b^4*c^4*d^3*e^10 + 568*a^7*b^5*c^3*d^2*e^11 - 4512*a^8*b^2*c^5*d^3*e^10 - 1536*a^8*b^3*c^4*d^2

*e^11 - 8*a^7*b^6*c^2*d*e^12 + 896*a^8*b*c^6*d^4*e^9 + 57*a^8*b^4*c^3*d*e^12 + 1152*a^9*b*c^5*d^2*e^11 - 102*a

^9*b^2*c^4*d*e^12))/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4

*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(

4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*

(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(

1/2) + (4*a^9*c^5*e^14 - a^6*b^6*c^2*e^14 + 7*a^7*b^4*c^3*e^14 - 13*a^8*b^2*c^4*e^14 - 192*a^6*c^8*d^6*e^8 - 1

92*a^7*c^7*d^4*e^10 + 4*a^8*c^6*d^2*e^12 - 128*a^2*b^8*c^4*d^6*e^8 + 96*a^2*b^9*c^3*d^5*e^9 + 32*a^2*b^10*c^2*

d^4*e^10 + 960*a^3*b^6*c^5*d^6*e^8 - 512*a^3*b^7*c^4*d^5*e^9 - 552*a^3*b^8*c^3*d^4*e^10 - 56*a^3*b^9*c^2*d^3*e

^11 - 2176*a^4*b^4*c^6*d^6*e^8 + 224*a^4*b^5*c^5*d^5*e^9 + 2688*a^4*b^6*c^4*d^4*e^10 + 672*a^4*b^7*c^3*d^3*e^1

1 + 24*a^4*b^8*c^2*d^2*e^12 + 1600*a^5*b^2*c^7*d^6*e^8 + 1408*a^5*b^3*c^6*d^5*e^9 - 4536*a^5*b^4*c^5*d^4*e^10

- 2616*a^5*b^5*c^4*d^3*e^11 - 209*a^5*b^6*c^3*d^2*e^12 + 2336*a^6*b^2*c^6*d^4*e^10 + 3648*a^6*b^3*c^5*d^3*e^11

+ 559*a^6*b^4*c^4*d^2*e^12 - 429*a^7*b^2*c^5*d^2*e^12 - 132*a^8*b*c^5*d*e^13 + a^5*b^7*c^2*d*e^13 - 1088*a^6*

b*c^7*d^5*e^9 - 23*a^6*b^5*c^3*d*e^13 - 1408*a^7*b*c^6*d^3*e^11 + 109*a^7*b^3*c^4*d*e^13)/(2*a^8*d^2))*((b^8*d

+ 8*a^4*c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c

^2*e + a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e +

20*a^4*b*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e

*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) - ((d + e*x)^(1/2)*(a^6*b^2*c^5*e^1

4 - 2*a^7*c^6*e^14 + 192*a^4*c^9*d^6*e^8 + 32*a^5*c^8*d^4*e^10 + 34*a^6*c^7*d^2*e^12 + 64*b^8*c^5*d^6*e^8 + 70

4*a^2*b^4*c^7*d^6*e^8 + 960*a^2*b^5*c^6*d^5*e^9 + 192*a^2*b^6*c^5*d^4*e^10 - 512*a^3*b^2*c^8*d^6*e^8 - 1280*a^

3*b^3*c^7*d^5*e^9 - 752*a^3*b^4*c^6*d^4*e^10 - 56*a^3*b^5*c^5*d^3*e^11 + 704*a^4*b^2*c^7*d^4*e^10 + 128*a^4*b^

3*c^6*d^3*e^11 - 15*a^4*b^4*c^5*d^2*e^12 + 60*a^5*b^2*c^6*d^2*e^12 - 10*a^6*b*c^6*d*e^13 - 384*a*b^6*c^6*d^6*e

^8 - 192*a*b^7*c^5*d^5*e^9 + 384*a^4*b*c^8*d^5*e^9 - 144*a^5*b*c^7*d^3*e^11 + 6*a^5*b^3*c^5*d*e^13))/(2*a^8*d^

2))*((b^8*d + 8*a^4*c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 2

5*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2

*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*

a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) + (((((128*a^12*c^4*d*e^

12 + 768*a^10*c^6*d^5*e^8 + 896*a^11*c^5*d^3*e^10 + 128*a^8*b^4*c^4*d^5*e^8 - 96*a^8*b^5*c^3*d^4*e^9 - 32*a^8*

b^6*c^2*d^3*e^10 - 704*a^9*b^2*c^5*d^5*e^8 + 448*a^9*b^3*c^4*d^4*e^9 + 392*a^9*b^4*c^3*d^3*e^10 + 24*a^9*b^5*c

^2*d^2*e^11 - 1280*a^10*b^2*c^4*d^3*e^10 - 192*a^10*b^3*c^3*d^2*e^11 - 256*a^10*b*c^5*d^4*e^9 + 8*a^10*b^4*c^2

*d*e^12 + 384*a^11*b*c^4*d^2*e^11 - 64*a^11*b^2*c^3*d*e^12)/(2*a^8*d^2) + ((d + e*x)^(1/2)*((b^8*d + 8*a^4*c^4

*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a^3*c

^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3

*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*c -

b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2)*(1536*a^12*c^5*d^4*e^8 + 1024*a^13*c^4*d^2*e^10

+ 128*a^10*b^4*c^3*d^4*e^8 - 128*a^10*b^5*c^2*d^3*e^9 - 896*a^11*b^2*c^4*d^4*e^8 + 960*a^11*b^3*c^3*d^3*e^9 +

64*a^11*b^4*c^2*d^2*e^10 - 512*a^12*b^2*c^3*d^2*e^10 - 1792*a^12*b*c^4*d^3*e^9))/(2*a^8*d^2))*((b^8*d + 8*a^4

*c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a

^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b

*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*

c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) - ((d + e*x)^(1/2)*(8*a^10*c^5*d*e^12 - 12*

a^10*b*c^4*e^13 - a^8*b^5*c^2*e^13 + 7*a^9*b^3*c^3*e^13 + 1152*a^8*c^7*d^5*e^8 + 512*a^9*c^6*d^3*e^10 + 128*a^

4*b^8*c^3*d^5*e^8 - 128*a^4*b^9*c^2*d^4*e^9 - 1152*a^5*b^6*c^4*d^5*e^8 + 1088*a^5*b^7*c^3*d^4*e^9 + 192*a^5*b^

8*c^2*d^3*e^10 + 3520*a^6*b^4*c^5*d^5*e^8 - 2816*a^6*b^5*c^4*d^4*e^9 - 1728*a^6*b^6*c^3*d^3*e^10 - 64*a^6*b^7*

c^2*d^2*e^11 - 4096*a^7*b^2*c^6*d^5*e^8 + 1792*a^7*b^3*c^5*d^4*e^9 + 4944*a^7*b^4*c^4*d^3*e^10 + 568*a^7*b^5*c

^3*d^2*e^11 - 4512*a^8*b^2*c^5*d^3*e^10 - 1536*a^8*b^3*c^4*d^2*e^11 - 8*a^7*b^6*c^2*d*e^12 + 896*a^8*b*c^6*d^4

*e^9 + 57*a^8*b^4*c^3*d*e^12 + 1152*a^9*b*c^5*d^2*e^11 - 102*a^9*b^2*c^4*d*e^12))/(2*a^8*d^2))*((b^8*d + 8*a^4

*c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a

^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b

*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*

c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) + (4*a^9*c^5*e^14 - a^6*b^6*c^2*e^14 + 7*a^

7*b^4*c^3*e^14 - 13*a^8*b^2*c^4*e^14 - 192*a^6*c^8*d^6*e^8 - 192*a^7*c^7*d^4*e^10 + 4*a^8*c^6*d^2*e^12 - 128*a

^2*b^8*c^4*d^6*e^8 + 96*a^2*b^9*c^3*d^5*e^9 + 32*a^2*b^10*c^2*d^4*e^10 + 960*a^3*b^6*c^5*d^6*e^8 - 512*a^3*b^7

*c^4*d^5*e^9 - 552*a^3*b^8*c^3*d^4*e^10 - 56*a^3*b^9*c^2*d^3*e^11 - 2176*a^4*b^4*c^6*d^6*e^8 + 224*a^4*b^5*c^5

*d^5*e^9 + 2688*a^4*b^6*c^4*d^4*e^10 + 672*a^4*b^7*c^3*d^3*e^11 + 24*a^4*b^8*c^2*d^2*e^12 + 1600*a^5*b^2*c^7*d

^6*e^8 + 1408*a^5*b^3*c^6*d^5*e^9 - 4536*a^5*b^4*c^5*d^4*e^10 - 2616*a^5*b^5*c^4*d^3*e^11 - 209*a^5*b^6*c^3*d^

2*e^12 + 2336*a^6*b^2*c^6*d^4*e^10 + 3648*a^6*b^3*c^5*d^3*e^11 + 559*a^6*b^4*c^4*d^2*e^12 - 429*a^7*b^2*c^5*d^

2*e^12 - 132*a^8*b*c^5*d*e^13 + a^5*b^7*c^2*d*e^13 - 1088*a^6*b*c^7*d^5*e^9 - 23*a^6*b^5*c^3*d*e^13 - 1408*a^7

*b*c^6*d^3*e^11 + 109*a^7*b^3*c^4*d*e^13)/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2)

- a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a

*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^

(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*

c^2 - 8*a^7*b^2*c)))^(1/2) + ((d + e*x)^(1/2)*(a^6*b^2*c^5*e^14 - 2*a^7*c^6*e^14 + 192*a^4*c^9*d^6*e^8 + 32*a^

5*c^8*d^4*e^10 + 34*a^6*c^7*d^2*e^12 + 64*b^8*c^5*d^6*e^8 + 704*a^2*b^4*c^7*d^6*e^8 + 960*a^2*b^5*c^6*d^5*e^9

+ 192*a^2*b^6*c^5*d^4*e^10 - 512*a^3*b^2*c^8*d^6*e^8 - 1280*a^3*b^3*c^7*d^5*e^9 - 752*a^3*b^4*c^6*d^4*e^10 - 5

6*a^3*b^5*c^5*d^3*e^11 + 704*a^4*b^2*c^7*d^4*e^10 + 128*a^4*b^3*c^6*d^3*e^11 - 15*a^4*b^4*c^5*d^2*e^12 + 60*a^

5*b^2*c^6*d^2*e^12 - 10*a^6*b*c^6*d*e^13 - 384*a*b^6*c^6*d^6*e^8 - 192*a*b^7*c^5*d^5*e^9 + 384*a^4*b*c^8*d^5*e

^9 - 144*a^5*b*c^7*d^3*e^11 + 6*a^5*b^3*c^5*d*e^13))/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d - b^5*d*(-(4*a*c - b^2

)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c - b^2)^3)^(

1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d*(-(4*a*c

- b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^

4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) + (7*a^5*c^7*d*e^14 + 56*a^3*c^9*d^5*e^10 + 63*a^4*c^8*d^3*e^12 - 64*b^4

*c^8*d^7*e^8 + 64*b^5*c^7*d^6*e^9 + 64*a^2*b^2*c^8*d^5*e^10 + 224*a^2*b^3*c^7*d^4*e^11 - 112*a^3*b^2*c^7*d^3*e

^12 + 64*a*b^2*c^9*d^7*e^8 + 64*a*b^3*c^8*d^6*e^9 - 192*a*b^4*c^7*d^5*e^10 - 96*a^2*b*c^9*d^6*e^9 - 136*a^3*b*

c^8*d^4*e^11 + 9*a^4*b*c^7*d^2*e^13)/(a^8*d^2)))*((b^8*d + 8*a^4*c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^

7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c

*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2)

- 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 -

8*a^7*b^2*c)))^(1/2)*2i - (((a*e^2 + 4*b*d*e)*(d + e*x)^(1/2))/(4*a^2) + ((a*e^2 - 4*b*d*e)*(d + e*x)^(3/2))/(

4*a^2*d))/((d + e*x)^2 - 2*d*(d + e*x) + d^2) + atan(((((((128*a^12*c^4*d*e^12 + 768*a^10*c^6*d^5*e^8 + 896*a^

11*c^5*d^3*e^10 + 128*a^8*b^4*c^4*d^5*e^8 - 96*a^8*b^5*c^3*d^4*e^9 - 32*a^8*b^6*c^2*d^3*e^10 - 704*a^9*b^2*c^5

*d^5*e^8 + 448*a^9*b^3*c^4*d^4*e^9 + 392*a^9*b^4*c^3*d^3*e^10 + 24*a^9*b^5*c^2*d^2*e^11 - 1280*a^10*b^2*c^4*d^

3*e^10 - 192*a^10*b^3*c^3*d^2*e^11 - 256*a^10*b*c^5*d^4*e^9 + 8*a^10*b^4*c^2*d*e^12 + 384*a^11*b*c^4*d^2*e^11

- 64*a^11*b^2*c^3*d*e^12)/(2*a^8*d^2) - ((d + e*x)^(1/2)*((b^8*d + 8*a^4*c^4*d + b^5*d*(-(4*a*c - b^2)^3)^(1/2

) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e - a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10

*a*b^6*c*d - a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e - 4*a*b^3*c*d*(-(4*a*c - b^2)^3

)^(1/2) + 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^

8*c^2 - 8*a^7*b^2*c)))^(1/2)*(1536*a^12*c^5*d^4*e^8 + 1024*a^13*c^4*d^2*e^10 + 128*a^10*b^4*c^3*d^4*e^8 - 128*

a^10*b^5*c^2*d^3*e^9 - 896*a^11*b^2*c^4*d^4*e^8 + 960*a^11*b^3*c^3*d^3*e^9 + 64*a^11*b^4*c^2*d^2*e^10 - 512*a^

12*b^2*c^3*d^2*e^10 - 1792*a^12*b*c^4*d^3*e^9))/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d + b^5*d*(-(4*a*c - b^2)^3)^

(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e - a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2)

- 10*a*b^6*c*d - a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e - 4*a*b^3*c*d*(-(4*a*c - b^

2)^3)^(1/2) + 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 1

6*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) + ((d + e*x)^(1/2)*(8*a^10*c^5*d*e^12 - 12*a^10*b*c^4*e^13 - a^8*b^5*c^2*e^13

+ 7*a^9*b^3*c^3*e^13 + 1152*a^8*c^7*d^5*e^8 + 512*a^9*c^6*d^3*e^10 + 128*a^4*b^8*c^3*d^5*e^8 - 128*a^4*b^9*c^

2*d^4*e^9 - 1152*a^5*b^6*c^4*d^5*e^8 + 1088*a^5*b^7*c^3*d^4*e^9 + 192*a^5*b^8*c^2*d^3*e^10 + 3520*a^6*b^4*c^5*

d^5*e^8 - 2816*a^6*b^5*c^4*d^4*e^9 - 1728*a^6*b^6*c^3*d^3*e^10 - 64*a^6*b^7*c^2*d^2*e^11 - 4096*a^7*b^2*c^6*d^

5*e^8 + 1792*a^7*b^3*c^5*d^4*e^9 + 4944*a^7*b^4*c^4*d^3*e^10 + 568*a^7*b^5*c^3*d^2*e^11 - 4512*a^8*b^2*c^5*d^3

*e^10 - 1536*a^8*b^3*c^4*d^2*e^11 - 8*a^7*b^6*c^2*d*e^12 + 896*a^8*b*c^6*d^4*e^9 + 57*a^8*b^4*c^3*d*e^12 + 115

2*a^9*b*c^5*d^2*e^11 - 102*a^9*b^2*c^4*d*e^12))/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d + b^5*d*(-(4*a*c - b^2)^3)^

(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e - a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2)

- 10*a*b^6*c*d - a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e - 4*a*b^3*c*d*(-(4*a*c - b^

2)^3)^(1/2) + 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 1

6*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) + (4*a^9*c^5*e^14 - a^6*b^6*c^2*e^14 + 7*a^7*b^4*c^3*e^14 - 13*a^8*b^2*c^4*e^

14 - 192*a^6*c^8*d^6*e^8 - 192*a^7*c^7*d^4*e^10 + 4*a^8*c^6*d^2*e^12 - 128*a^2*b^8*c^4*d^6*e^8 + 96*a^2*b^9*c^

3*d^5*e^9 + 32*a^2*b^10*c^2*d^4*e^10 + 960*a^3*b^6*c^5*d^6*e^8 - 512*a^3*b^7*c^4*d^5*e^9 - 552*a^3*b^8*c^3*d^4

*e^10 - 56*a^3*b^9*c^2*d^3*e^11 - 2176*a^4*b^4*c^6*d^6*e^8 + 224*a^4*b^5*c^5*d^5*e^9 + 2688*a^4*b^6*c^4*d^4*e^

10 + 672*a^4*b^7*c^3*d^3*e^11 + 24*a^4*b^8*c^2*d^2*e^12 + 1600*a^5*b^2*c^7*d^6*e^8 + 1408*a^5*b^3*c^6*d^5*e^9

- 4536*a^5*b^4*c^5*d^4*e^10 - 2616*a^5*b^5*c^4*d^3*e^11 - 209*a^5*b^6*c^3*d^2*e^12 + 2336*a^6*b^2*c^6*d^4*e^10

+ 3648*a^6*b^3*c^5*d^3*e^11 + 559*a^6*b^4*c^4*d^2*e^12 - 429*a^7*b^2*c^5*d^2*e^12 - 132*a^8*b*c^5*d*e^13 + a^

5*b^7*c^2*d*e^13 - 1088*a^6*b*c^7*d^5*e^9 - 23*a^6*b^5*c^3*d*e^13 - 1408*a^7*b*c^6*d^3*e^11 + 109*a^7*b^3*c^4*

d*e^13)/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d + b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*

a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e - a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d - a*b^4*e*(-(4*a*c - b^2)

^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e - 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^2*d*(-(4*a*c - b

^2)^3)^(1/2) + 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) - ((d +

e*x)^(1/2)*(a^6*b^2*c^5*e^14 - 2*a^7*c^6*e^14 + 192*a^4*c^9*d^6*e^8 + 32*a^5*c^8*d^4*e^10 + 34*a^6*c^7*d^2*e^

12 + 64*b^8*c^5*d^6*e^8 + 704*a^2*b^4*c^7*d^6*e^8 + 960*a^2*b^5*c^6*d^5*e^9 + 192*a^2*b^6*c^5*d^4*e^10 - 512*a

^3*b^2*c^8*d^6*e^8 - 1280*a^3*b^3*c^7*d^5*e^9 - 752*a^3*b^4*c^6*d^4*e^10 - 56*a^3*b^5*c^5*d^3*e^11 + 704*a^4*b

^2*c^7*d^4*e^10 + 128*a^4*b^3*c^6*d^3*e^11 - 15*a^4*b^4*c^5*d^2*e^12 + 60*a^5*b^2*c^6*d^2*e^12 - 10*a^6*b*c^6*

d*e^13 - 384*a*b^6*c^6*d^6*e^8 - 192*a*b^7*c^5*d^5*e^9 + 384*a^4*b*c^8*d^5*e^9 - 144*a^5*b*c^7*d^3*e^11 + 6*a^

5*b^3*c^5*d*e^13))/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d + b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*

c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e - a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d - a*b^4*e*(-(4

*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e - 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^2*d*(

-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1

/2)*1i - (((((128*a^12*c^4*d*e^12 + 768*a^10*c^6*d^5*e^8 + 896*a^11*c^5*d^3*e^10 + 128*a^8*b^4*c^4*d^5*e^8 - 9

6*a^8*b^5*c^3*d^4*e^9 - 32*a^8*b^6*c^2*d^3*e^10 - 704*a^9*b^2*c^5*d^5*e^8 + 448*a^9*b^3*c^4*d^4*e^9 + 392*a^9*

b^4*c^3*d^3*e^10 + 24*a^9*b^5*c^2*d^2*e^11 - 1280*a^10*b^2*c^4*d^3*e^10 - 192*a^10*b^3*c^3*d^2*e^11 - 256*a^10

*b*c^5*d^4*e^9 + 8*a^10*b^4*c^2*d*e^12 + 384*a^11*b*c^4*d^2*e^11 - 64*a^11*b^2*c^3*d*e^12)/(2*a^8*d^2) + ((d +

e*x)^(1/2)*((b^8*d + 8*a^4*c^4*d + b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c

^3*d - 25*a^3*b^3*c^2*e - a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d - a*b^4*e*(-(4*a*c - b^2)^3)^(1/2)

+ 9*a^2*b^5*c*e + 20*a^4*b*c^3*e - 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1

/2) + 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2)*(1536*a^12*c^5*d

^4*e^8 + 1024*a^13*c^4*d^2*e^10 + 128*a^10*b^4*c^3*d^4*e^8 - 128*a^10*b^5*c^2*d^3*e^9 - 896*a^11*b^2*c^4*d^4*e

^8 + 960*a^11*b^3*c^3*d^3*e^9 + 64*a^11*b^4*c^2*d^2*e^10 - 512*a^12*b^2*c^3*d^2*e^10 - 1792*a^12*b*c^4*d^3*e^9

))/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d + b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b

^2*c^3*d - 25*a^3*b^3*c^2*e - a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d - a*b^4*e*(-(4*a*c - b^2)^3)^(

1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e - 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3

)^(1/2) + 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) - ((d + e*x)

^(1/2)*(8*a^10*c^5*d*e^12 - 12*a^10*b*c^4*e^13 - a^8*b^5*c^2*e^13 + 7*a^9*b^3*c^3*e^13 + 1152*a^8*c^7*d^5*e^8

+ 512*a^9*c^6*d^3*e^10 + 128*a^4*b^8*c^3*d^5*e^8 - 128*a^4*b^9*c^2*d^4*e^9 - 1152*a^5*b^6*c^4*d^5*e^8 + 1088*a

^5*b^7*c^3*d^4*e^9 + 192*a^5*b^8*c^2*d^3*e^10 + 3520*a^6*b^4*c^5*d^5*e^8 - 2816*a^6*b^5*c^4*d^4*e^9 - 1728*a^6

*b^6*c^3*d^3*e^10 - 64*a^6*b^7*c^2*d^2*e^11 - 4096*a^7*b^2*c^6*d^5*e^8 + 1792*a^7*b^3*c^5*d^4*e^9 + 4944*a^7*b

^4*c^4*d^3*e^10 + 568*a^7*b^5*c^3*d^2*e^11 - 4512*a^8*b^2*c^5*d^3*e^10 - 1536*a^8*b^3*c^4*d^2*e^11 - 8*a^7*b^6

*c^2*d*e^12 + 896*a^8*b*c^6*d^4*e^9 + 57*a^8*b^4*c^3*d*e^12 + 1152*a^9*b*c^5*d^2*e^11 - 102*a^9*b^2*c^4*d*e^12

))/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d + b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b

^2*c^3*d - 25*a^3*b^3*c^2*e - a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d - a*b^4*e*(-(4*a*c - b^2)^3)^(

1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e - 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3

)^(1/2) + 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) + (4*a^9*c^5

*e^14 - a^6*b^6*c^2*e^14 + 7*a^7*b^4*c^3*e^14 - 13*a^8*b^2*c^4*e^14 - 192*a^6*c^8*d^6*e^8 - 192*a^7*c^7*d^4*e^

10 + 4*a^8*c^6*d^2*e^12 - 128*a^2*b^8*c^4*d^6*e^8 + 96*a^2*b^9*c^3*d^5*e^9 + 32*a^2*b^10*c^2*d^4*e^10 + 960*a^

3*b^6*c^5*d^6*e^8 - 512*a^3*b^7*c^4*d^5*e^9 - 552*a^3*b^8*c^3*d^4*e^10 - 56*a^3*b^9*c^2*d^3*e^11 - 2176*a^4*b^

4*c^6*d^6*e^8 + 224*a^4*b^5*c^5*d^5*e^9 + 2688*a^4*b^6*c^4*d^4*e^10 + 672*a^4*b^7*c^3*d^3*e^11 + 24*a^4*b^8*c^

2*d^2*e^12 + 1600*a^5*b^2*c^7*d^6*e^8 + 1408*a^5*b^3*c^6*d^5*e^9 - 4536*a^5*b^4*c^5*d^4*e^10 - 2616*a^5*b^5*c^

4*d^3*e^11 - 209*a^5*b^6*c^3*d^2*e^12 + 2336*a^6*b^2*c^6*d^4*e^10 + 3648*a^6*b^3*c^5*d^3*e^11 + 559*a^6*b^4*c^

4*d^2*e^12 - 429*a^7*b^2*c^5*d^2*e^12 - 132*a^8*b*c^5*d*e^13 + a^5*b^7*c^2*d*e^13 - 1088*a^6*b*c^7*d^5*e^9 - 2

3*a^6*b^5*c^3*d*e^13 - 1408*a^7*b*c^6*d^3*e^11 + 109*a^7*b^3*c^4*d*e^13)/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d +

b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e - a^3*c^2*e*

(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d - a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e -

4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*c*e*(-(4*a*c - b^2)^

3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) + ((d + e*x)^(1/2)*(a^6*b^2*c^5*e^14 - 2*a^7*c^6*e^1

4 + 192*a^4*c^9*d^6*e^8 + 32*a^5*c^8*d^4*e^10 + 34*a^6*c^7*d^2*e^12 + 64*b^8*c^5*d^6*e^8 + 704*a^2*b^4*c^7*d^6

*e^8 + 960*a^2*b^5*c^6*d^5*e^9 + 192*a^2*b^6*c^5*d^4*e^10 - 512*a^3*b^2*c^8*d^6*e^8 - 1280*a^3*b^3*c^7*d^5*e^9

- 752*a^3*b^4*c^6*d^4*e^10 - 56*a^3*b^5*c^5*d^3*e^11 + 704*a^4*b^2*c^7*d^4*e^10 + 128*a^4*b^3*c^6*d^3*e^11 -

15*a^4*b^4*c^5*d^2*e^12 + 60*a^5*b^2*c^6*d^2*e^12 - 10*a^6*b*c^6*d*e^13 - 384*a*b^6*c^6*d^6*e^8 - 192*a*b^7*c^

5*d^5*e^9 + 384*a^4*b*c^8*d^5*e^9 - 144*a^5*b*c^7*d^3*e^11 + 6*a^5*b^3*c^5*d*e^13))/(2*a^8*d^2))*((b^8*d + 8*a

^4*c^4*d + b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e -

a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d - a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4

*b*c^3*e - 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*c*e*(-(4*

a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2)*1i)/((((((128*a^12*c^4*d*e^12 + 768*a^10*

c^6*d^5*e^8 + 896*a^11*c^5*d^3*e^10 + 128*a^8*b^4*c^4*d^5*e^8 - 96*a^8*b^5*c^3*d^4*e^9 - 32*a^8*b^6*c^2*d^3*e^

10 - 704*a^9*b^2*c^5*d^5*e^8 + 448*a^9*b^3*c^4*d^4*e^9 + 392*a^9*b^4*c^3*d^3*e^10 + 24*a^9*b^5*c^2*d^2*e^11 -

1280*a^10*b^2*c^4*d^3*e^10 - 192*a^10*b^3*c^3*d^2*e^11 - 256*a^10*b*c^5*d^4*e^9 + 8*a^10*b^4*c^2*d*e^12 + 384*

a^11*b*c^4*d^2*e^11 - 64*a^11*b^2*c^3*d*e^12)/(2*a^8*d^2) - ((d + e*x)^(1/2)*((b^8*d + 8*a^4*c^4*d + b^5*d*(-(

4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e - a^3*c^2*e*(-(4*a*c

- b^2)^3)^(1/2) - 10*a*b^6*c*d - a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e - 4*a*b^3*c

*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))

/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2)*(1536*a^12*c^5*d^4*e^8 + 1024*a^13*c^4*d^2*e^10 + 128*a^10*b^

4*c^3*d^4*e^8 - 128*a^10*b^5*c^2*d^3*e^9 - 896*a^11*b^2*c^4*d^4*e^8 + 960*a^11*b^3*c^3*d^3*e^9 + 64*a^11*b^4*c

^2*d^2*e^10 - 512*a^12*b^2*c^3*d^2*e^10 - 1792*a^12*b*c^4*d^3*e^9))/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d + b^5*d

*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e - a^3*c^2*e*(-(4*

a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d - a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e - 4*a*b

^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1

/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) + ((d + e*x)^(1/2)*(8*a^10*c^5*d*e^12 - 12*a^10*b*c^4*e^1

3 - a^8*b^5*c^2*e^13 + 7*a^9*b^3*c^3*e^13 + 1152*a^8*c^7*d^5*e^8 + 512*a^9*c^6*d^3*e^10 + 128*a^4*b^8*c^3*d^5*

e^8 - 128*a^4*b^9*c^2*d^4*e^9 - 1152*a^5*b^6*c^4*d^5*e^8 + 1088*a^5*b^7*c^3*d^4*e^9 + 192*a^5*b^8*c^2*d^3*e^10

+ 3520*a^6*b^4*c^5*d^5*e^8 - 2816*a^6*b^5*c^4*d^4*e^9 - 1728*a^6*b^6*c^3*d^3*e^10 - 64*a^6*b^7*c^2*d^2*e^11 -

4096*a^7*b^2*c^6*d^5*e^8 + 1792*a^7*b^3*c^5*d^4*e^9 + 4944*a^7*b^4*c^4*d^3*e^10 + 568*a^7*b^5*c^3*d^2*e^11 -

4512*a^8*b^2*c^5*d^3*e^10 - 1536*a^8*b^3*c^4*d^2*e^11 - 8*a^7*b^6*c^2*d*e^12 + 896*a^8*b*c^6*d^4*e^9 + 57*a^8*

b^4*c^3*d*e^12 + 1152*a^9*b*c^5*d^2*e^11 - 102*a^9*b^2*c^4*d*e^12))/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d + b^5*d

*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e - a^3*c^2*e*(-(4*

a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d - a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e - 4*a*b

^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1

/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2) + (4*a^9*c^5*e^14 - a^6*b^6*c^2*e^14 + 7*a^7*b^4*c^3*e^14

- 13*a^8*b^2*c^4*e^14 - 192*a^6*c^8*d^6*e^8 - 192*a^7*c^7*d^4*e^10 + 4*a^8*c^6*d^2*e^12 - 128*a^2*b^8*c^4*d^6

*e^8 + 96*a^2*b^9*c^3*d^5*e^9 + 32*a^2*b^10*c^2*d^4*e^10 + 960*a^3*b^6*c^5*d^6*e^8 - 512*a^3*b^7*c^4*d^5*e^9 -

552*a^3*b^8*c^3*d^4*e^10 - 56*a^3*b^9*c^2*d^3*e^11 - 2176*a^4*b^4*c^6*d^6*e^8 + 224*a^4*b^5*c^5*d^5*e^9 + 268

8*a^4*b^6*c^4*d^4*e^10 + 672*a^4*b^7*c^3*d^3*e^11 + 24*a^4*b^8*c^2*d^2*e^12 + 1600*a^5*b^2*c^7*d^6*e^8 + 1408*

a^5*b^3*c^6*d^5*e^9 - 4536*a^5*b^4*c^5*d^4*e^10 - 2616*a^5*b^5*c^4*d^3*e^11 - 209*a^5*b^6*c^3*d^2*e^12 + 2336*

a^6*b^2*c^6*d^4*e^10 + 3648*a^6*b^3*c^5*d^3*e^11 + 559*a^6*b^4*c^4*d^2*e^12 - 429*a^7*b^2*c^5*d^2*e^12 - 132*a

^8*b*c^5*d*e^13 + a^5*b^7*c^2*d*e^13 - 1088*a^6*b*c^7*d^5*e^9 - 23*a^6*b^5*c^3*d*e^13 - 1408*a^7*b*c^6*d^3*e^1

1 + 109*a^7*b^3*c^4*d*e^13)/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d + b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33

*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e - a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d - a*b

^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e - 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*

b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^

2*c)))^(1/2) - ((d + e*x)^(1/2)*(a^6*b^2*c^5*e^14 - 2*a^7*c^6*e^14 + 192*a^4*c^9*d^6*e^8 + 32*a^5*c^8*d^4*e^10

+ 34*a^6*c^7*d^2*e^12 + 64*b^8*c^5*d^6*e^8 + 704*a^2*b^4*c^7*d^6*e^8 + 960*a^2*b^5*c^6*d^5*e^9 + 192*a^2*b^6*

c^5*d^4*e^10 - 512*a^3*b^2*c^8*d^6*e^8 - 1280*a^3*b^3*c^7*d^5*e^9 - 752*a^3*b^4*c^6*d^4*e^10 - 56*a^3*b^5*c^5*

d^3*e^11 + 704*a^4*b^2*c^7*d^4*e^10 + 128*a^4*b^3*c^6*d^3*e^11 - 15*a^4*b^4*c^5*d^2*e^12 + 60*a^5*b^2*c^6*d^2*

e^12 - 10*a^6*b*c^6*d*e^13 - 384*a*b^6*c^6*d^6*e^8 - 192*a*b^7*c^5*d^5*e^9 + 384*a^4*b*c^8*d^5*e^9 - 144*a^5*b

*c^7*d^3*e^11 + 6*a^5*b^3*c^5*d*e^13))/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d + b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a

*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e - a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^

6*c*d - a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e - 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/

2) + 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2

- 8*a^7*b^2*c)))^(1/2) + (((((128*a^12*c^4*d*e^12 + 768*a^10*c^6*d^5*e^8 + 896*a^11*c^5*d^3*e^10 + 128*a^8*b^

4*c^4*d^5*e^8 - 96*a^8*b^5*c^3*d^4*e^9 - 32*a^8*b^6*c^2*d^3*e^10 - 704*a^9*b^2*c^5*d^5*e^8 + 448*a^9*b^3*c^4*d

^4*e^9 + 392*a^9*b^4*c^3*d^3*e^10 + 24*a^9*b^5*c^2*d^2*e^11 - 1280*a^10*b^2*c^4*d^3*e^10 - 192*a^10*b^3*c^3*d^

2*e^11 - 256*a^10*b*c^5*d^4*e^9 + 8*a^10*b^4*c^2*d*e^12 + 384*a^11*b*c^4*d^2*e^11 - 64*a^11*b^2*c^3*d*e^12)/(2

*a^8*d^2) + ((d + e*x)^(1/2)*((b^8*d + 8*a^4*c^4*d + b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2

*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e - a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d - a*b^4*e*(-(4*a*

c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e - 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^2*d*(-(4

*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2)

*(1536*a^12*c^5*d^4*e^8 + 1024*a^13*c^4*d^2*e^10 + 128*a^10*b^4*c^3*d^4*e^8 - 128*a^10*b^5*c^2*d^3*e^9 - 896*a

^11*b^2*c^4*d^4*e^8 + 960*a^11*b^3*c^3*d^3*e^9 + 64*a^11*b^4*c^2*d^2*e^10 - 512*a^12*b^2*c^3*d^2*e^10 - 1792*a

^12*b*c^4*d^3*e^9))/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d + b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4

*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e - a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d - a*b^4*e*(-(

4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e - 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^2*d*

(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(

1/2) - ((d + e*x)^(1/2)*(8*a^10*c^5*d*e^12 - 12*a^10*b*c^4*e^13 - a^8*b^5*c^2*e^13 + 7*a^9*b^3*c^3*e^13 + 1152

*a^8*c^7*d^5*e^8 + 512*a^9*c^6*d^3*e^10 + 128*a^4*b^8*c^3*d^5*e^8 - 128*a^4*b^9*c^2*d^4*e^9 - 1152*a^5*b^6*c^4

*d^5*e^8 + 1088*a^5*b^7*c^3*d^4*e^9 + 192*a^5*b^8*c^2*d^3*e^10 + 3520*a^6*b^4*c^5*d^5*e^8 - 2816*a^6*b^5*c^4*d

^4*e^9 - 1728*a^6*b^6*c^3*d^3*e^10 - 64*a^6*b^7*c^2*d^2*e^11 - 4096*a^7*b^2*c^6*d^5*e^8 + 1792*a^7*b^3*c^5*d^4

*e^9 + 4944*a^7*b^4*c^4*d^3*e^10 + 568*a^7*b^5*c^3*d^2*e^11 - 4512*a^8*b^2*c^5*d^3*e^10 - 1536*a^8*b^3*c^4*d^2

*e^11 - 8*a^7*b^6*c^2*d*e^12 + 896*a^8*b*c^6*d^4*e^9 + 57*a^8*b^4*c^3*d*e^12 + 1152*a^9*b*c^5*d^2*e^11 - 102*a

^9*b^2*c^4*d*e^12))/(2*a^8*d^2))*((b^8*d + 8*a^4*c^4*d + b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4