\(\int \frac {(d+e x)^{3/2}}{x (a+b x+c x^2)} \, dx\) [538]

Optimal result

Integrand size = 25, antiderivative size = 340 \[ \int \frac {(d+e x)^{3/2}}{x \left (a+b x+c x^2\right )} \, dx=-\frac {2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a}-\frac {\sqrt {2} \left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d-4 a e\right )-b \left (c d^2+a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{a \sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {2} \left (a \sqrt {b^2-4 a c} e^2-c d \left (\sqrt {b^2-4 a c} d+4 a e\right )+b \left (c d^2+a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{a \sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

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

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {911, 1301, 212, 1180, 214} \[ \int \frac {(d+e x)^{3/2}}{x \left (a+b x+c x^2\right )} \, dx=-\frac {\sqrt {2} \left (-c d \left (d \sqrt {b^2-4 a c}-4 a e\right )+a e^2 \sqrt {b^2-4 a c}-b \left (a e^2+c d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{a \sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\sqrt {2} \left (-c d \left (d \sqrt {b^2-4 a c}+4 a e\right )+a e^2 \sqrt {b^2-4 a c}+b \left (a e^2+c d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{a \sqrt {c} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a} \]

[In]

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

[Out]

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

Rule 212

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

Rule 214

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

Rule 911

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 1180

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 1301

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.30 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.09 \[ \int \frac {(d+e x)^{3/2}}{x \left (a+b x+c x^2\right )} \, dx=-\frac {\frac {\sqrt {2} \left (-a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d+4 i a e\right )-i b \left (c d^2+a e^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {c} \sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\sqrt {2} \left (-a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d-4 i a e\right )+i b \left (c d^2+a e^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {c} \sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}+2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(2 e^{2} \left (\frac {4 c \left (\frac {\left (a b \,e^{3}-4 a c d \,e^{2}+b c \,d^{2} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c \,d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-a b \,e^{3}+4 a c d \,e^{2}-b c \,d^{2} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c \,d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a \,e^{2}}-\frac {d^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{a \,e^{2}}\right )\)	\(370\)
default	\(2 e^{2} \left (\frac {4 c \left (\frac {\left (a b \,e^{3}-4 a c d \,e^{2}+b c \,d^{2} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c \,d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-a b \,e^{3}+4 a c d \,e^{2}-b c \,d^{2} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c \,d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a \,e^{2}}-\frac {d^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{a \,e^{2}}\right )\)	\(370\)
pseudoelliptic	\(\frac {-\sqrt {2}\, \left (\left (e^{2} a -c \,d^{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-a b \,e^{3}+4 a c d \,e^{2}-b c \,d^{2} e \right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (\sqrt {2}\, \left (\left (e^{2} a -c \,d^{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (a b \,e^{2}-4 a c d e +b c \,d^{2}\right )\right ) \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )-2 \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, d^{\frac {3}{2}}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, a}\)	\(395\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2581 vs. \(2 (286) = 572\).

Time = 4.77 (sec) , antiderivative size = 5167, normalized size of antiderivative = 15.20 \[ \int \frac {(d+e x)^{3/2}}{x \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

\[ \int \frac {(d+e x)^{3/2}}{x \left (a+b x+c x^2\right )} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{x \left (a + b x + c x^{2}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(d+e x)^{3/2}}{x \left (a+b x+c x^2\right )} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x + a\right )} x} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 833 vs. \(2 (286) = 572\).

Time = 0.33 (sec) , antiderivative size = 833, normalized size of antiderivative = 2.45 \[ \int \frac {(d+e x)^{3/2}}{x \left (a+b x+c x^2\right )} \, dx=\frac {2 \, d^{2} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{a \sqrt {-d}} - \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} - {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )} a^{2} e^{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 | e \right |} - {\left (2 \, a^{2} b c^{2} d^{3} e + 6 \, a^{3} b c d e^{3} - a^{3} b^{2} e^{4} - {\left (a^{2} b^{2} c + 8 \, a^{3} c^{2}\right )} d^{2} e^{2}\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 {e x + 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 )}{4 \, {\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 |} {\left | e \right |}} + \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} - {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )} a^{2} e^{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 | e \right |} - {\left (2 \, a^{2} b c^{2} d^{3} e + 6 \, a^{3} b c d e^{3} - a^{3} b^{2} e^{4} - {\left (a^{2} b^{2} c + 8 \, a^{3} c^{2}\right )} d^{2} e^{2}\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 {e x + 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 )}{4 \, {\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 |} {\left | e \right |}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 18.02 (sec) , antiderivative size = 20897, normalized size of antiderivative = 61.46 \[ \int \frac {(d+e x)^{3/2}}{x \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan(((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3
*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^
3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((((b^4
*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 +
4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 1
2*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((d + e*x)^(1/2)*((
b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^
2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2)
+ 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(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^1
0 + 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) - 384*a^3*c^5*d^4*e^8 - 384*a^
4*c^4*d^2*e^10 + 96*a^2*b^2*c^4*d^4*e^8 - 128*a^2*b^3*c^3*d^3*e^9 + 32*a^2*b^4*c^2*d^2*e^10 - 32*a^3*b^2*c^3*d
^2*e^10 + 128*a^4*b*c^3*d*e^11 + 512*a^3*b*c^4*d^3*e^9 - 32*a^3*b^3*c^2*d*e^11) + (d + e*x)^(1/2)*(32*a^3*b^3*
c*e^13 - 128*a^4*b*c^2*e^13 + 704*a^4*c^3*d*e^12 - 576*a^2*c^5*d^5*e^8 + 896*a^3*c^4*d^3*e^10 - 64*b^4*c^3*d^5
*e^8 + 64*b^5*c^2*d^4*e^9 + 192*a^2*b^2*c^3*d^3*e^10 + 448*a^2*b^3*c^2*d^2*e^11 - 64*a^2*b^4*c*d*e^12 + 384*a*
b^2*c^4*d^5*e^8 - 320*a*b^3*c^3*d^4*e^9 - 128*a*b^4*c^2*d^3*e^10 + 384*a^2*b*c^4*d^4*e^9 - 1664*a^3*b*c^3*d^2*
e^11 + 64*a^3*b^2*c^2*d*e^12))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) -
6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*
c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*
b^2*c^2)))^(1/2) + 96*a*c^5*d^7*e^8 + 32*a^4*c^2*d*e^14 - 672*a^2*c^4*d^5*e^10 - 736*a^3*c^3*d^3*e^12 - 32*b^2
*c^4*d^7*e^8 - 32*b^3*c^3*d^6*e^9 + 64*b^4*c^2*d^5*e^10 - 96*a^2*b^2*c^2*d^3*e^12 + 256*a*b*c^4*d^6*e^9 - 32*a
^3*b^2*c*d*e^14 - 288*a*b^2*c^3*d^5*e^10 - 160*a*b^3*c^2*d^4*e^11 + 1280*a^2*b*c^3*d^4*e^11 + 32*a^2*b^3*c*d^2
*e^13 + 128*a^3*b*c^2*d^2*e^13) + (d + e*x)^(1/2)*(32*a^4*c*e^16 + 96*c^5*d^8*e^8 - 256*a*c^4*d^6*e^10 - 256*b
*c^4*d^7*e^9 + 64*b^4*c*d^4*e^12 + 256*a^2*c^3*d^4*e^12 + 128*a^3*c^2*d^2*e^14 + 384*b^2*c^3*d^6*e^10 - 256*b^
3*c^2*d^5*e^11 - 128*a^3*b*c*d*e^15 - 128*a*b^3*c*d^3*e^13 + 256*a*b^2*c^2*d^4*e^12 - 384*a^2*b*c^2*d^3*e^13 +
192*a^2*b^2*c*d^2*e^14))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b
^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2
*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c
^2)))^(1/2)*1i + (((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d
^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*
a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1
/2)*((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*
c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3
)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((d + e*
x)^(1/2)*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a
^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2
)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(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) + 384*a^3*c^5*d^4*e
^8 + 384*a^4*c^4*d^2*e^10 - 96*a^2*b^2*c^4*d^4*e^8 + 128*a^2*b^3*c^3*d^3*e^9 - 32*a^2*b^4*c^2*d^2*e^10 + 32*a^
3*b^2*c^3*d^2*e^10 - 128*a^4*b*c^3*d*e^11 - 512*a^3*b*c^4*d^3*e^9 + 32*a^3*b^3*c^2*d*e^11) + (d + e*x)^(1/2)*(
32*a^3*b^3*c*e^13 - 128*a^4*b*c^2*e^13 + 704*a^4*c^3*d*e^12 - 576*a^2*c^5*d^5*e^8 + 896*a^3*c^4*d^3*e^10 - 64*
b^4*c^3*d^5*e^8 + 64*b^5*c^2*d^4*e^9 + 192*a^2*b^2*c^3*d^3*e^10 + 448*a^2*b^3*c^2*d^2*e^11 - 64*a^2*b^4*c*d*e^
12 + 384*a*b^2*c^4*d^5*e^8 - 320*a*b^3*c^3*d^4*e^9 - 128*a*b^4*c^2*d^3*e^10 + 384*a^2*b*c^4*d^4*e^9 - 1664*a^3
*b*c^3*d^2*e^11 + 64*a^3*b^2*c^2*d*e^12))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^
3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d
^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4
*c - 8*a^3*b^2*c^2)))^(1/2) - 96*a*c^5*d^7*e^8 - 32*a^4*c^2*d*e^14 + 672*a^2*c^4*d^5*e^10 + 736*a^3*c^3*d^3*e^
12 + 32*b^2*c^4*d^7*e^8 + 32*b^3*c^3*d^6*e^9 - 64*b^4*c^2*d^5*e^10 + 96*a^2*b^2*c^2*d^3*e^12 - 256*a*b*c^4*d^6
*e^9 + 32*a^3*b^2*c*d*e^14 + 288*a*b^2*c^3*d^5*e^10 + 160*a*b^3*c^2*d^4*e^11 - 1280*a^2*b*c^3*d^4*e^11 - 32*a^
2*b^3*c*d^2*e^13 - 128*a^3*b*c^2*d^2*e^13) + (d + e*x)^(1/2)*(32*a^4*c*e^16 + 96*c^5*d^8*e^8 - 256*a*c^4*d^6*e
^10 - 256*b*c^4*d^7*e^9 + 64*b^4*c*d^4*e^12 + 256*a^2*c^3*d^4*e^12 + 128*a^3*c^2*d^2*e^14 + 384*b^2*c^3*d^6*e^
10 - 256*b^3*c^2*d^5*e^11 - 128*a^3*b*c*d*e^15 - 128*a*b^3*c*d^3*e^13 + 256*a*b^2*c^2*d^4*e^12 - 384*a^2*b*c^2
*d^3*e^13 + 192*a^2*b^2*c*d^2*e^14))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1
/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e
- 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c -
8*a^3*b^2*c^2)))^(1/2)*1i)/((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*
a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*
d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^
2*c^2)))^(1/2)*((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^
3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a
*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/
2)*((d + e*x)^(1/2)*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2
*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(
4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(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) + 384*a^
3*c^5*d^4*e^8 + 384*a^4*c^4*d^2*e^10 - 96*a^2*b^2*c^4*d^4*e^8 + 128*a^2*b^3*c^3*d^3*e^9 - 32*a^2*b^4*c^2*d^2*e
^10 + 32*a^3*b^2*c^3*d^2*e^10 - 128*a^4*b*c^3*d*e^11 - 512*a^3*b*c^4*d^3*e^9 + 32*a^3*b^3*c^2*d*e^11) + (d + e
*x)^(1/2)*(32*a^3*b^3*c*e^13 - 128*a^4*b*c^2*e^13 + 704*a^4*c^3*d*e^12 - 576*a^2*c^5*d^5*e^8 + 896*a^3*c^4*d^3
*e^10 - 64*b^4*c^3*d^5*e^8 + 64*b^5*c^2*d^4*e^9 + 192*a^2*b^2*c^3*d^3*e^10 + 448*a^2*b^3*c^2*d^2*e^11 - 64*a^2
*b^4*c*d*e^12 + 384*a*b^2*c^4*d^5*e^8 - 320*a*b^3*c^3*d^4*e^9 - 128*a*b^4*c^2*d^3*e^10 + 384*a^2*b*c^4*d^4*e^9
- 1664*a^3*b*c^3*d^2*e^11 + 64*a^3*b^2*c^2*d*e^12))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*
a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) -
3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^
3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) - 96*a*c^5*d^7*e^8 - 32*a^4*c^2*d*e^14 + 672*a^2*c^4*d^5*e^10 + 736*a^3
*c^3*d^3*e^12 + 32*b^2*c^4*d^7*e^8 + 32*b^3*c^3*d^6*e^9 - 64*b^4*c^2*d^5*e^10 + 96*a^2*b^2*c^2*d^3*e^12 - 256*
a*b*c^4*d^6*e^9 + 32*a^3*b^2*c*d*e^14 + 288*a*b^2*c^3*d^5*e^10 + 160*a*b^3*c^2*d^4*e^11 - 1280*a^2*b*c^3*d^4*e
^11 - 32*a^2*b^3*c*d^2*e^13 - 128*a^3*b*c^2*d^2*e^13) + (d + e*x)^(1/2)*(32*a^4*c*e^16 + 96*c^5*d^8*e^8 - 256*
a*c^4*d^6*e^10 - 256*b*c^4*d^7*e^9 + 64*b^4*c*d^4*e^12 + 256*a^2*c^3*d^4*e^12 + 128*a^3*c^2*d^2*e^14 + 384*b^2
*c^3*d^6*e^10 - 256*b^3*c^2*d^5*e^11 - 128*a^3*b*c*d*e^15 - 128*a*b^3*c*d^3*e^13 + 256*a*b^2*c^2*d^4*e^12 - 38
4*a^2*b*c^2*d^3*e^13 + 192*a^2*b^2*c*d^2*e^14))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c -
b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b
^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a
^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) - (((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1
/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e
- 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c -
8*a^3*b^2*c^2)))^(1/2)*((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^
2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*
e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^
2)))^(1/2)*((d + e*x)^(1/2)*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a
*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d
^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(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)
- 384*a^3*c^5*d^4*e^8 - 384*a^4*c^4*d^2*e^10 + 96*a^2*b^2*c^4*d^4*e^8 - 128*a^2*b^3*c^3*d^3*e^9 + 32*a^2*b^4*c
^2*d^2*e^10 - 32*a^3*b^2*c^3*d^2*e^10 + 128*a^4*b*c^3*d*e^11 + 512*a^3*b*c^4*d^3*e^9 - 32*a^3*b^3*c^2*d*e^11)
+ (d + e*x)^(1/2)*(32*a^3*b^3*c*e^13 - 128*a^4*b*c^2*e^13 + 704*a^4*c^3*d*e^12 - 576*a^2*c^5*d^5*e^8 + 896*a^3
*c^4*d^3*e^10 - 64*b^4*c^3*d^5*e^8 + 64*b^5*c^2*d^4*e^9 + 192*a^2*b^2*c^3*d^3*e^10 + 448*a^2*b^3*c^2*d^2*e^11
- 64*a^2*b^4*c*d*e^12 + 384*a*b^2*c^4*d^5*e^8 - 320*a*b^3*c^3*d^4*e^9 - 128*a*b^4*c^2*d^3*e^10 + 384*a^2*b*c^4
*d^4*e^9 - 1664*a^3*b*c^3*d^2*e^11 + 64*a^3*b^2*c^2*d*e^12))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e
^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^
(1/2) - 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(1
6*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) + 96*a*c^5*d^7*e^8 + 32*a^4*c^2*d*e^14 - 672*a^2*c^4*d^5*e^10 -
736*a^3*c^3*d^3*e^12 - 32*b^2*c^4*d^7*e^8 - 32*b^3*c^3*d^6*e^9 + 64*b^4*c^2*d^5*e^10 - 96*a^2*b^2*c^2*d^3*e^1
2 + 256*a*b*c^4*d^6*e^9 - 32*a^3*b^2*c*d*e^14 - 288*a*b^2*c^3*d^5*e^10 - 160*a*b^3*c^2*d^4*e^11 + 1280*a^2*b*c
^3*d^4*e^11 + 32*a^2*b^3*c*d^2*e^13 + 128*a^3*b*c^2*d^2*e^13) + (d + e*x)^(1/2)*(32*a^4*c*e^16 + 96*c^5*d^8*e^
8 - 256*a*c^4*d^6*e^10 - 256*b*c^4*d^7*e^9 + 64*b^4*c*d^4*e^12 + 256*a^2*c^3*d^4*e^12 + 128*a^3*c^2*d^2*e^14 +
384*b^2*c^3*d^6*e^10 - 256*b^3*c^2*d^5*e^11 - 128*a^3*b*c*d*e^15 - 128*a*b^3*c*d^3*e^13 + 256*a*b^2*c^2*d^4*e
^12 - 384*a^2*b*c^2*d^3*e^13 + 192*a^2*b^2*c*d^2*e^14))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-
(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2)
- 3*a*b^3*c*d^2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4
*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) + 192*c^4*d^8*e^10 + 448*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 512*b*
c^3*d^7*e^11 - 128*b^3*c*d^5*e^13 + 320*a^2*c^2*d^4*e^14 + 448*b^2*c^2*d^6*e^12 - 768*a*b*c^2*d^5*e^13 + 320*a
*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 + a^2*e^3*(-(4*a*c - b^2)^3
)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 + b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^
2*e - 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*
c - 8*a^3*b^2*c^2)))^(1/2)*2i + atan(((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^
(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*
e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c
- 8*a^3*b^2*c^2)))^(1/2)*((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*
b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^
2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*
c^2)))^(1/2)*((d + e*x)^(1/2)*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6
*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c
*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(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
) - 384*a^3*c^5*d^4*e^8 - 384*a^4*c^4*d^2*e^10 + 96*a^2*b^2*c^4*d^4*e^8 - 128*a^2*b^3*c^3*d^3*e^9 + 32*a^2*b^4
*c^2*d^2*e^10 - 32*a^3*b^2*c^3*d^2*e^10 + 128*a^4*b*c^3*d*e^11 + 512*a^3*b*c^4*d^3*e^9 - 32*a^3*b^3*c^2*d*e^11
) + (d + e*x)^(1/2)*(32*a^3*b^3*c*e^13 - 128*a^4*b*c^2*e^13 + 704*a^4*c^3*d*e^12 - 576*a^2*c^5*d^5*e^8 + 896*a
^3*c^4*d^3*e^10 - 64*b^4*c^3*d^5*e^8 + 64*b^5*c^2*d^4*e^9 + 192*a^2*b^2*c^3*d^3*e^10 + 448*a^2*b^3*c^2*d^2*e^1
1 - 64*a^2*b^4*c*d*e^12 + 384*a*b^2*c^4*d^5*e^8 - 320*a*b^3*c^3*d^4*e^9 - 128*a*b^4*c^2*d^3*e^10 + 384*a^2*b*c
^4*d^4*e^9 - 1664*a^3*b*c^3*d^2*e^11 + 64*a^3*b^2*c^2*d*e^12))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2
*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3
)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*
(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) + 96*a*c^5*d^7*e^8 + 32*a^4*c^2*d*e^14 - 672*a^2*c^4*d^5*e^10
- 736*a^3*c^3*d^3*e^12 - 32*b^2*c^4*d^7*e^8 - 32*b^3*c^3*d^6*e^9 + 64*b^4*c^2*d^5*e^10 - 96*a^2*b^2*c^2*d^3*e
^12 + 256*a*b*c^4*d^6*e^9 - 32*a^3*b^2*c*d*e^14 - 288*a*b^2*c^3*d^5*e^10 - 160*a*b^3*c^2*d^4*e^11 + 1280*a^2*b
*c^3*d^4*e^11 + 32*a^2*b^3*c*d^2*e^13 + 128*a^3*b*c^2*d^2*e^13) + (d + e*x)^(1/2)*(32*a^4*c*e^16 + 96*c^5*d^8*
e^8 - 256*a*c^4*d^6*e^10 - 256*b*c^4*d^7*e^9 + 64*b^4*c*d^4*e^12 + 256*a^2*c^3*d^4*e^12 + 128*a^3*c^2*d^2*e^14
+ 384*b^2*c^3*d^6*e^10 - 256*b^3*c^2*d^5*e^11 - 128*a^3*b*c*d*e^15 - 128*a*b^3*c*d^3*e^13 + 256*a*b^2*c^2*d^4
*e^12 - 384*a^2*b*c^2*d^3*e^13 + 192*a^2*b^2*c*d^2*e^14))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*
(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/
2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a
^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*1i + (((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c
- b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a
*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 +
a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(
1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e
+ 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c -
8*a^3*b^2*c^2)))^(1/2)*((d + e*x)^(1/2)*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3
)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^
2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(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) + 384*a^3*c^5*d^4*e^8 + 384*a^4*c^4*d^2*e^10 - 96*a^2*b^2*c^4*d^4*e^8 + 128*a^2*b^3*c^3*d^3*e^9 -
32*a^2*b^4*c^2*d^2*e^10 + 32*a^3*b^2*c^3*d^2*e^10 - 128*a^4*b*c^3*d*e^11 - 512*a^3*b*c^4*d^3*e^9 + 32*a^3*b^3
*c^2*d*e^11) + (d + e*x)^(1/2)*(32*a^3*b^3*c*e^13 - 128*a^4*b*c^2*e^13 + 704*a^4*c^3*d*e^12 - 576*a^2*c^5*d^5*
e^8 + 896*a^3*c^4*d^3*e^10 - 64*b^4*c^3*d^5*e^8 + 64*b^5*c^2*d^4*e^9 + 192*a^2*b^2*c^3*d^3*e^10 + 448*a^2*b^3*
c^2*d^2*e^11 - 64*a^2*b^4*c*d*e^12 + 384*a*b^2*c^4*d^5*e^8 - 320*a*b^3*c^3*d^4*e^9 - 128*a*b^4*c^2*d^3*e^10 +
384*a^2*b*c^4*d^4*e^9 - 1664*a^3*b*c^3*d^2*e^11 + 64*a^3*b^2*c^2*d*e^12))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^
3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a
*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c
*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) - 96*a*c^5*d^7*e^8 - 32*a^4*c^2*d*e^14 + 672*a^2*c
^4*d^5*e^10 + 736*a^3*c^3*d^3*e^12 + 32*b^2*c^4*d^7*e^8 + 32*b^3*c^3*d^6*e^9 - 64*b^4*c^2*d^5*e^10 + 96*a^2*b^
2*c^2*d^3*e^12 - 256*a*b*c^4*d^6*e^9 + 32*a^3*b^2*c*d*e^14 + 288*a*b^2*c^3*d^5*e^10 + 160*a*b^3*c^2*d^4*e^11 -
1280*a^2*b*c^3*d^4*e^11 - 32*a^2*b^3*c*d^2*e^13 - 128*a^3*b*c^2*d^2*e^13) + (d + e*x)^(1/2)*(32*a^4*c*e^16 +
96*c^5*d^8*e^8 - 256*a*c^4*d^6*e^10 - 256*b*c^4*d^7*e^9 + 64*b^4*c*d^4*e^12 + 256*a^2*c^3*d^4*e^12 + 128*a^3*c
^2*d^2*e^14 + 384*b^2*c^3*d^6*e^10 - 256*b^3*c^2*d^5*e^11 - 128*a^3*b*c*d*e^15 - 128*a*b^3*c*d^3*e^13 + 256*a*
b^2*c^2*d^4*e^12 - 384*a^2*b*c^2*d^3*e^13 + 192*a^2*b^2*c*d^2*e^14))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3
- a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c -
b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^
2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*1i)/((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e
^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^
(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(1
6*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c
- b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*
b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 +
a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((d + e*x)^(1/2)*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a
*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3
*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(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) + 384*a^3*c^5*d^4*e^8 + 384*a^4*c^4*d^2*e^10 - 96*a^2*b^2*c^4*d^4*e^8 + 128*a^2*b^3*c^
3*d^3*e^9 - 32*a^2*b^4*c^2*d^2*e^10 + 32*a^3*b^2*c^3*d^2*e^10 - 128*a^4*b*c^3*d*e^11 - 512*a^3*b*c^4*d^3*e^9 +
32*a^3*b^3*c^2*d*e^11) + (d + e*x)^(1/2)*(32*a^3*b^3*c*e^13 - 128*a^4*b*c^2*e^13 + 704*a^4*c^3*d*e^12 - 576*a
^2*c^5*d^5*e^8 + 896*a^3*c^4*d^3*e^10 - 64*b^4*c^3*d^5*e^8 + 64*b^5*c^2*d^4*e^9 + 192*a^2*b^2*c^3*d^3*e^10 + 4
48*a^2*b^3*c^2*d^2*e^11 - 64*a^2*b^4*c*d*e^12 + 384*a*b^2*c^4*d^5*e^8 - 320*a*b^3*c^3*d^4*e^9 - 128*a*b^4*c^2*
d^3*e^10 + 384*a^2*b*c^4*d^4*e^9 - 1664*a^3*b*c^3*d^2*e^11 + 64*a^3*b^2*c^2*d*e^12))*((b^4*c*d^3 - a^2*b^3*e^3
+ 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c
*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e +
6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) - 96*a*c^5*d^7*e^8 - 32*a^4*c^2*d*e^14
+ 672*a^2*c^4*d^5*e^10 + 736*a^3*c^3*d^3*e^12 + 32*b^2*c^4*d^7*e^8 + 32*b^3*c^3*d^6*e^9 - 64*b^4*c^2*d^5*e^10
+ 96*a^2*b^2*c^2*d^3*e^12 - 256*a*b*c^4*d^6*e^9 + 32*a^3*b^2*c*d*e^14 + 288*a*b^2*c^3*d^5*e^10 + 160*a*b^3*c^2
*d^4*e^11 - 1280*a^2*b*c^3*d^4*e^11 - 32*a^2*b^3*c*d^2*e^13 - 128*a^3*b*c^2*d^2*e^13) + (d + e*x)^(1/2)*(32*a^
4*c*e^16 + 96*c^5*d^8*e^8 - 256*a*c^4*d^6*e^10 - 256*b*c^4*d^7*e^9 + 64*b^4*c*d^4*e^12 + 256*a^2*c^3*d^4*e^12
+ 128*a^3*c^2*d^2*e^14 + 384*b^2*c^3*d^6*e^10 - 256*b^3*c^2*d^5*e^11 - 128*a^3*b*c*d*e^15 - 128*a*b^3*c*d^3*e^
13 + 256*a*b^2*c^2*d^4*e^12 - 384*a^2*b*c^2*d^3*e^13 + 192*a^2*b^2*c*d^2*e^14))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*
a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*
(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2
*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) - (((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3
- a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c -
b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^
2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(
-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2
) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^
4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*((d + e*x)^(1/2)*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^
3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(
1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(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) - 384*a^3*c^5*d^4*e^8 - 384*a^4*c^4*d^2*e^10 + 96*a^2*b^2*c^4*d^4*e^8 - 128*a^
2*b^3*c^3*d^3*e^9 + 32*a^2*b^4*c^2*d^2*e^10 - 32*a^3*b^2*c^3*d^2*e^10 + 128*a^4*b*c^3*d*e^11 + 512*a^3*b*c^4*d
^3*e^9 - 32*a^3*b^3*c^2*d*e^11) + (d + e*x)^(1/2)*(32*a^3*b^3*c*e^13 - 128*a^4*b*c^2*e^13 + 704*a^4*c^3*d*e^12
- 576*a^2*c^5*d^5*e^8 + 896*a^3*c^4*d^3*e^10 - 64*b^4*c^3*d^5*e^8 + 64*b^5*c^2*d^4*e^9 + 192*a^2*b^2*c^3*d^3*
e^10 + 448*a^2*b^3*c^2*d^2*e^11 - 64*a^2*b^4*c*d*e^12 + 384*a*b^2*c^4*d^5*e^8 - 320*a*b^3*c^3*d^4*e^9 - 128*a*
b^4*c^2*d^3*e^10 + 384*a^2*b*c^4*d^4*e^9 - 1664*a^3*b*c^3*d^2*e^11 + 64*a^3*b^2*c^2*d*e^12))*((b^4*c*d^3 - a^2
*b^3*e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e
^3 - b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*
d^2*e + 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) + 96*a*c^5*d^7*e^8 + 32*a^4*c^2
*d*e^14 - 672*a^2*c^4*d^5*e^10 - 736*a^3*c^3*d^3*e^12 - 32*b^2*c^4*d^7*e^8 - 32*b^3*c^3*d^6*e^9 + 64*b^4*c^2*d
^5*e^10 - 96*a^2*b^2*c^2*d^3*e^12 + 256*a*b*c^4*d^6*e^9 - 32*a^3*b^2*c*d*e^14 - 288*a*b^2*c^3*d^5*e^10 - 160*a
*b^3*c^2*d^4*e^11 + 1280*a^2*b*c^3*d^4*e^11 + 32*a^2*b^3*c*d^2*e^13 + 128*a^3*b*c^2*d^2*e^13) + (d + e*x)^(1/2
)*(32*a^4*c*e^16 + 96*c^5*d^8*e^8 - 256*a*c^4*d^6*e^10 - 256*b*c^4*d^7*e^9 + 64*b^4*c*d^4*e^12 + 256*a^2*c^3*d
^4*e^12 + 128*a^3*c^2*d^2*e^14 + 384*b^2*c^3*d^6*e^10 - 256*b^3*c^2*d^5*e^11 - 128*a^3*b*c*d*e^15 - 128*a*b^3*
c*d^3*e^13 + 256*a*b^2*c^2*d^4*e^12 - 384*a^2*b*c^2*d^3*e^13 + 192*a^2*b^2*c*d^2*e^14))*((b^4*c*d^3 - a^2*b^3*
e^3 + 8*a^2*c^3*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 -
b*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e
+ 6*a^2*b^2*c*d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2) + 192*c^4*d^8*e^10 + 448*a*c^3*d^6*e
^12 + 64*a^3*c*d^2*e^16 - 512*b*c^3*d^7*e^11 - 128*b^3*c*d^5*e^13 + 320*a^2*c^2*d^4*e^14 + 448*b^2*c^2*d^6*e^1
2 - 768*a*b*c^2*d^5*e^13 + 320*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15))*((b^4*c*d^3 - a^2*b^3*e^3 + 8*a^2*c^3
*d^3 - a^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d^3 - 24*a^3*c^2*d*e^2 + 4*a^3*b*c*e^3 - b*c*d^3*(-(4*a*
c - b^2)^3)^(1/2) - 3*a*b^3*c*d^2*e + 3*a*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*d^2*e + 6*a^2*b^2*c*
d*e^2)/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))^(1/2)*2i - (2*atanh((64*a^3*c*e^16*(d^3)^(1/2)*(d + e*x)^
(1/2))/(2304*c^4*d^8*e^10 + 1920*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c^3*d^7*e^11 - 192*b^3*c*d^5*e^13
+ 256*a^2*c^2*d^4*e^14 + (576*c^5*d^10*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b^2*c^3*d^8*e^10)/a + (384*b^3*c^
2*d^7*e^11)/a - (128*b^2*c^4*d^10*e^8)/a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*b^4*c^2*d^8*e^10)/a^2 - 1024*a*b
*c^2*d^5*e^13 + 384*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*e^9)/a) + (576*c^5*d^8*e^8*(d^3)
^(1/2)*(d + e*x)^(1/2))/(576*c^5*d^10*e^8 + 2304*a*c^4*d^8*e^10 + 64*a^4*c*d^2*e^16 - 1536*b*c^4*d^9*e^9 + 192
0*a^2*c^3*d^6*e^12 + 256*a^3*c^2*d^4*e^14 + 640*b^2*c^3*d^8*e^10 + 384*b^3*c^2*d^7*e^11 - (128*b^2*c^4*d^10*e^
8)/a + (320*b^3*c^3*d^9*e^9)/a - (192*b^4*c^2*d^8*e^10)/a - 3328*a*b*c^3*d^7*e^11 - 192*a*b^3*c*d^5*e^13 - 256
*a^3*b*c*d^3*e^15 + 640*a*b^2*c^2*d^6*e^12 - 1024*a^2*b*c^2*d^5*e^13 + 384*a^2*b^2*c*d^4*e^14) + (2304*c^4*d^6
*e^10*(d^3)^(1/2)*(d + e*x)^(1/2))/(2304*c^4*d^8*e^10 + 1920*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c^3*d
^7*e^11 - 192*b^3*c*d^5*e^13 + 256*a^2*c^2*d^4*e^14 + (576*c^5*d^10*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b^2*c
^3*d^8*e^10)/a + (384*b^3*c^2*d^7*e^11)/a - (128*b^2*c^4*d^10*e^8)/a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*b^4*
c^2*d^8*e^10)/a^2 - 1024*a*b*c^2*d^5*e^13 + 384*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*e^9)
/a) - (128*b^2*c^4*d^8*e^8*(d^3)^(1/2)*(d + e*x)^(1/2))/(576*a*c^5*d^10*e^8 + 64*a^5*c*d^2*e^16 + 2304*a^2*c^4
*d^8*e^10 + 1920*a^3*c^3*d^6*e^12 + 256*a^4*c^2*d^4*e^14 - 128*b^2*c^4*d^10*e^8 + 320*b^3*c^3*d^9*e^9 - 192*b^
4*c^2*d^8*e^10 + 640*a^2*b^2*c^2*d^6*e^12 - 1536*a*b*c^4*d^9*e^9 - 256*a^4*b*c*d^3*e^15 + 640*a*b^2*c^3*d^8*e^
10 + 384*a*b^3*c^2*d^7*e^11 - 3328*a^2*b*c^3*d^7*e^11 - 192*a^2*b^3*c*d^5*e^13 - 1024*a^3*b*c^2*d^5*e^13 + 384
*a^3*b^2*c*d^4*e^14) + (320*b^3*c^3*d^7*e^9*(d^3)^(1/2)*(d + e*x)^(1/2))/(576*a*c^5*d^10*e^8 + 64*a^5*c*d^2*e^
16 + 2304*a^2*c^4*d^8*e^10 + 1920*a^3*c^3*d^6*e^12 + 256*a^4*c^2*d^4*e^14 - 128*b^2*c^4*d^10*e^8 + 320*b^3*c^3
*d^9*e^9 - 192*b^4*c^2*d^8*e^10 + 640*a^2*b^2*c^2*d^6*e^12 - 1536*a*b*c^4*d^9*e^9 - 256*a^4*b*c*d^3*e^15 + 640
*a*b^2*c^3*d^8*e^10 + 384*a*b^3*c^2*d^7*e^11 - 3328*a^2*b*c^3*d^7*e^11 - 192*a^2*b^3*c*d^5*e^13 - 1024*a^3*b*c
^2*d^5*e^13 + 384*a^3*b^2*c*d^4*e^14) - (192*b^4*c^2*d^6*e^10*(d^3)^(1/2)*(d + e*x)^(1/2))/(576*a*c^5*d^10*e^8
+ 64*a^5*c*d^2*e^16 + 2304*a^2*c^4*d^8*e^10 + 1920*a^3*c^3*d^6*e^12 + 256*a^4*c^2*d^4*e^14 - 128*b^2*c^4*d^10
*e^8 + 320*b^3*c^3*d^9*e^9 - 192*b^4*c^2*d^8*e^10 + 640*a^2*b^2*c^2*d^6*e^12 - 1536*a*b*c^4*d^9*e^9 - 256*a^4*
b*c*d^3*e^15 + 640*a*b^2*c^3*d^8*e^10 + 384*a*b^3*c^2*d^7*e^11 - 3328*a^2*b*c^3*d^7*e^11 - 192*a^2*b^3*c*d^5*e
^13 - 1024*a^3*b*c^2*d^5*e^13 + 384*a^3*b^2*c*d^4*e^14) + (1920*a*c^3*d^4*e^12*(d^3)^(1/2)*(d + e*x)^(1/2))/(2
304*c^4*d^8*e^10 + 1920*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c^3*d^7*e^11 - 192*b^3*c*d^5*e^13 + 256*a^
2*c^2*d^4*e^14 + (576*c^5*d^10*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b^2*c^3*d^8*e^10)/a + (384*b^3*c^2*d^7*e^1
1)/a - (128*b^2*c^4*d^10*e^8)/a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*b^4*c^2*d^8*e^10)/a^2 - 1024*a*b*c^2*d^5*
e^13 + 384*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*e^9)/a) - (3328*b*c^3*d^5*e^11*(d^3)^(1/2
)*(d + e*x)^(1/2))/(2304*c^4*d^8*e^10 + 1920*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c^3*d^7*e^11 - 192*b^
3*c*d^5*e^13 + 256*a^2*c^2*d^4*e^14 + (576*c^5*d^10*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b^2*c^3*d^8*e^10)/a +
(384*b^3*c^2*d^7*e^11)/a - (128*b^2*c^4*d^10*e^8)/a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*b^4*c^2*d^8*e^10)/a^
2 - 1024*a*b*c^2*d^5*e^13 + 384*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*e^9)/a) - (192*b^3*c
*d^3*e^13*(d^3)^(1/2)*(d + e*x)^(1/2))/(2304*c^4*d^8*e^10 + 1920*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c
^3*d^7*e^11 - 192*b^3*c*d^5*e^13 + 256*a^2*c^2*d^4*e^14 + (576*c^5*d^10*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b
^2*c^3*d^8*e^10)/a + (384*b^3*c^2*d^7*e^11)/a - (128*b^2*c^4*d^10*e^8)/a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*
b^4*c^2*d^8*e^10)/a^2 - 1024*a*b*c^2*d^5*e^13 + 384*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*
e^9)/a) + (640*b^2*c^3*d^6*e^10*(d^3)^(1/2)*(d + e*x)^(1/2))/(576*c^5*d^10*e^8 + 2304*a*c^4*d^8*e^10 + 64*a^4*
c*d^2*e^16 - 1536*b*c^4*d^9*e^9 + 1920*a^2*c^3*d^6*e^12 + 256*a^3*c^2*d^4*e^14 + 640*b^2*c^3*d^8*e^10 + 384*b^
3*c^2*d^7*e^11 - (128*b^2*c^4*d^10*e^8)/a + (320*b^3*c^3*d^9*e^9)/a - (192*b^4*c^2*d^8*e^10)/a - 3328*a*b*c^3*
d^7*e^11 - 192*a*b^3*c*d^5*e^13 - 256*a^3*b*c*d^3*e^15 + 640*a*b^2*c^2*d^6*e^12 - 1024*a^2*b*c^2*d^5*e^13 + 38
4*a^2*b^2*c*d^4*e^14) + (384*b^3*c^2*d^5*e^11*(d^3)^(1/2)*(d + e*x)^(1/2))/(576*c^5*d^10*e^8 + 2304*a*c^4*d^8*
e^10 + 64*a^4*c*d^2*e^16 - 1536*b*c^4*d^9*e^9 + 1920*a^2*c^3*d^6*e^12 + 256*a^3*c^2*d^4*e^14 + 640*b^2*c^3*d^8
*e^10 + 384*b^3*c^2*d^7*e^11 - (128*b^2*c^4*d^10*e^8)/a + (320*b^3*c^3*d^9*e^9)/a - (192*b^4*c^2*d^8*e^10)/a -
3328*a*b*c^3*d^7*e^11 - 192*a*b^3*c*d^5*e^13 - 256*a^3*b*c*d^3*e^15 + 640*a*b^2*c^2*d^6*e^12 - 1024*a^2*b*c^2
*d^5*e^13 + 384*a^2*b^2*c*d^4*e^14) + (256*a^2*c^2*d^2*e^14*(d^3)^(1/2)*(d + e*x)^(1/2))/(2304*c^4*d^8*e^10 +
1920*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c^3*d^7*e^11 - 192*b^3*c*d^5*e^13 + 256*a^2*c^2*d^4*e^14 + (5
76*c^5*d^10*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b^2*c^3*d^8*e^10)/a + (384*b^3*c^2*d^7*e^11)/a - (128*b^2*c^4
*d^10*e^8)/a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*b^4*c^2*d^8*e^10)/a^2 - 1024*a*b*c^2*d^5*e^13 + 384*a*b^2*c*
d^4*e^14 - 256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*e^9)/a) + (640*b^2*c^2*d^4*e^12*(d^3)^(1/2)*(d + e*x)^(1/2))
/(2304*c^4*d^8*e^10 + 1920*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c^3*d^7*e^11 - 192*b^3*c*d^5*e^13 + 256
*a^2*c^2*d^4*e^14 + (576*c^5*d^10*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b^2*c^3*d^8*e^10)/a + (384*b^3*c^2*d^7*
e^11)/a - (128*b^2*c^4*d^10*e^8)/a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*b^4*c^2*d^8*e^10)/a^2 - 1024*a*b*c^2*d
^5*e^13 + 384*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*e^9)/a) - (1536*b*c^4*d^7*e^9*(d^3)^(1
/2)*(d + e*x)^(1/2))/(576*c^5*d^10*e^8 + 2304*a*c^4*d^8*e^10 + 64*a^4*c*d^2*e^16 - 1536*b*c^4*d^9*e^9 + 1920*a
^2*c^3*d^6*e^12 + 256*a^3*c^2*d^4*e^14 + 640*b^2*c^3*d^8*e^10 + 384*b^3*c^2*d^7*e^11 - (128*b^2*c^4*d^10*e^8)/
a + (320*b^3*c^3*d^9*e^9)/a - (192*b^4*c^2*d^8*e^10)/a - 3328*a*b*c^3*d^7*e^11 - 192*a*b^3*c*d^5*e^13 - 256*a^
3*b*c*d^3*e^15 + 640*a*b^2*c^2*d^6*e^12 - 1024*a^2*b*c^2*d^5*e^13 + 384*a^2*b^2*c*d^4*e^14) - (256*a^2*b*c*d*e
^15*(d^3)^(1/2)*(d + e*x)^(1/2))/(2304*c^4*d^8*e^10 + 1920*a*c^3*d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c^3*d^7
*e^11 - 192*b^3*c*d^5*e^13 + 256*a^2*c^2*d^4*e^14 + (576*c^5*d^10*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b^2*c^3
*d^8*e^10)/a + (384*b^3*c^2*d^7*e^11)/a - (128*b^2*c^4*d^10*e^8)/a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*b^4*c^
2*d^8*e^10)/a^2 - 1024*a*b*c^2*d^5*e^13 + 384*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*e^9)/a
) - (1024*a*b*c^2*d^3*e^13*(d^3)^(1/2)*(d + e*x)^(1/2))/(2304*c^4*d^8*e^10 + 1920*a*c^3*d^6*e^12 + 64*a^3*c*d^
2*e^16 - 3328*b*c^3*d^7*e^11 - 192*b^3*c*d^5*e^13 + 256*a^2*c^2*d^4*e^14 + (576*c^5*d^10*e^8)/a + 640*b^2*c^2*
d^6*e^12 + (640*b^2*c^3*d^8*e^10)/a + (384*b^3*c^2*d^7*e^11)/a - (128*b^2*c^4*d^10*e^8)/a^2 + (320*b^3*c^3*d^9
*e^9)/a^2 - (192*b^4*c^2*d^8*e^10)/a^2 - 1024*a*b*c^2*d^5*e^13 + 384*a*b^2*c*d^4*e^14 - 256*a^2*b*c*d^3*e^15 -
(1536*b*c^4*d^9*e^9)/a) + (384*a*b^2*c*d^2*e^14*(d^3)^(1/2)*(d + e*x)^(1/2))/(2304*c^4*d^8*e^10 + 1920*a*c^3*
d^6*e^12 + 64*a^3*c*d^2*e^16 - 3328*b*c^3*d^7*e^11 - 192*b^3*c*d^5*e^13 + 256*a^2*c^2*d^4*e^14 + (576*c^5*d^10
*e^8)/a + 640*b^2*c^2*d^6*e^12 + (640*b^2*c^3*d^8*e^10)/a + (384*b^3*c^2*d^7*e^11)/a - (128*b^2*c^4*d^10*e^8)/
a^2 + (320*b^3*c^3*d^9*e^9)/a^2 - (192*b^4*c^2*d^8*e^10)/a^2 - 1024*a*b*c^2*d^5*e^13 + 384*a*b^2*c*d^4*e^14 -
256*a^2*b*c*d^3*e^15 - (1536*b*c^4*d^9*e^9)/a))*(d^3)^(1/2))/a