\(\int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx\) [31]

Optimal result

Integrand size = 40, antiderivative size = 278 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=-\frac {C \left (a^2-b^2 x^2\right )}{b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {(C e-B f) \sqrt {a^2 c-b^2 c x^2} \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{b \sqrt {c} f^2 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (C e^2-B e f+A f^2\right ) \sqrt {a^2 c-b^2 c x^2} \arctan \left (\frac {\sqrt {c} \left (a^2 f+b^2 e x\right )}{\sqrt {b^2 e^2-a^2 f^2} \sqrt {a^2 c-b^2 c x^2}}\right )}{\sqrt {c} f^2 \sqrt {b^2 e^2-a^2 f^2} \sqrt {a+b x} \sqrt {a c-b c x}} \]

[Out]

-C*(-b^2*x^2+a^2)/b^2/f/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-(-B*f+C*e)*arctan(b*x*c^(1/2)/(-b^2*c*x^2+a^2*c)^(1/2
))*(-b^2*c*x^2+a^2*c)^(1/2)/b/f^2/c^(1/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)+(A*f^2-B*e*f+C*e^2)*arctan((b^2*e*x
+a^2*f)*c^(1/2)/(-a^2*f^2+b^2*e^2)^(1/2)/(-b^2*c*x^2+a^2*c)^(1/2))*(-b^2*c*x^2+a^2*c)^(1/2)/f^2/c^(1/2)/(-a^2*
f^2+b^2*e^2)^(1/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1624, 1668, 858, 223, 209, 739, 210} \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=\frac {\sqrt {a^2 c-b^2 c x^2} \left (A f^2-B e f+C e^2\right ) \arctan \left (\frac {\sqrt {c} \left (a^2 f+b^2 e x\right )}{\sqrt {a^2 c-b^2 c x^2} \sqrt {b^2 e^2-a^2 f^2}}\right )}{\sqrt {c} f^2 \sqrt {a+b x} \sqrt {a c-b c x} \sqrt {b^2 e^2-a^2 f^2}}-\frac {\sqrt {a^2 c-b^2 c x^2} (C e-B f) \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{b \sqrt {c} f^2 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C \left (a^2-b^2 x^2\right )}{b^2 f \sqrt {a+b x} \sqrt {a c-b c x}} \]

[In]

Int[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)),x]

[Out]

-((C*(a^2 - b^2*x^2))/(b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])) - ((C*e - B*f)*Sqrt[a^2*c - b^2*c*x^2]*ArcTan[(
b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(b*Sqrt[c]*f^2*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + ((C*e^2 - B*e*f + A*f
^2)*Sqrt[a^2*c - b^2*c*x^2]*ArcTan[(Sqrt[c]*(a^2*f + b^2*e*x))/(Sqrt[b^2*e^2 - a^2*f^2]*Sqrt[a^2*c - b^2*c*x^2
])])/(Sqrt[c]*f^2*Sqrt[b^2*e^2 - a^2*f^2]*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])

Rule 209

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

Rule 210

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1624

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[(a
+ b*x)^FracPart[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]), Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1668

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2 c-b^2 c x^2} \int \frac {A+B x+C x^2}{(e+f x) \sqrt {a^2 c-b^2 c x^2}} \, dx}{\sqrt {a+b x} \sqrt {a c-b c x}} \\ & = -\frac {C \left (a^2-b^2 x^2\right )}{b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\sqrt {a^2 c-b^2 c x^2} \int \frac {-A b^2 c f^2+b^2 c f (C e-B f) x}{(e+f x) \sqrt {a^2 c-b^2 c x^2}} \, dx}{b^2 c f^2 \sqrt {a+b x} \sqrt {a c-b c x}} \\ & = -\frac {C \left (a^2-b^2 x^2\right )}{b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left ((C e-B f) \sqrt {a^2 c-b^2 c x^2}\right ) \int \frac {1}{\sqrt {a^2 c-b^2 c x^2}} \, dx}{f^2 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (\left (C e^2-B e f+A f^2\right ) \sqrt {a^2 c-b^2 c x^2}\right ) \int \frac {1}{(e+f x) \sqrt {a^2 c-b^2 c x^2}} \, dx}{f^2 \sqrt {a+b x} \sqrt {a c-b c x}} \\ & = -\frac {C \left (a^2-b^2 x^2\right )}{b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left ((C e-B f) \sqrt {a^2 c-b^2 c x^2}\right ) \text {Subst}\left (\int \frac {1}{1+b^2 c x^2} \, dx,x,\frac {x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{f^2 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\left (\left (C e^2-B e f+A f^2\right ) \sqrt {a^2 c-b^2 c x^2}\right ) \text {Subst}\left (\int \frac {1}{-b^2 c e^2+a^2 c f^2-x^2} \, dx,x,\frac {a^2 c f+b^2 c e x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{f^2 \sqrt {a+b x} \sqrt {a c-b c x}} \\ & = -\frac {C \left (a^2-b^2 x^2\right )}{b^2 f \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {(C e-B f) \sqrt {a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{b \sqrt {c} f^2 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {\left (C e^2-B e f+A f^2\right ) \sqrt {a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac {\sqrt {c} \left (a^2 f+b^2 e x\right )}{\sqrt {b^2 e^2-a^2 f^2} \sqrt {a^2 c-b^2 c x^2}}\right )}{\sqrt {c} f^2 \sqrt {b^2 e^2-a^2 f^2} \sqrt {a+b x} \sqrt {a c-b c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.64 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=\frac {\frac {C f (-a+b x) \sqrt {a+b x}}{b^2}-\frac {2 (C e-B f) \sqrt {a-b x} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{b}+\frac {2 \left (C e^2+f (-B e+A f)\right ) \sqrt {a-b x} \arctan \left (\frac {\sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a-b x}}\right )}{\sqrt {b e-a f} \sqrt {b e+a f}}}{f^2 \sqrt {c (a-b x)}} \]

[In]

Integrate[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)),x]

[Out]

((C*f*(-a + b*x)*Sqrt[a + b*x])/b^2 - (2*(C*e - B*f)*Sqrt[a - b*x]*ArcTan[Sqrt[a + b*x]/Sqrt[a - b*x]])/b + (2
*(C*e^2 + f*(-(B*e) + A*f))*Sqrt[a - b*x]*ArcTan[(Sqrt[b*e + a*f]*Sqrt[a + b*x])/(Sqrt[b*e - a*f]*Sqrt[a - b*x
])])/(Sqrt[b*e - a*f]*Sqrt[b*e + a*f]))/(f^2*Sqrt[c*(a - b*x)])

Maple [A] (verified)

Time = 1.70 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.08

method	result	size
risch	\(-\frac {C \sqrt {b x +a}\, \left (-b x +a \right )}{f \,b^{2} \sqrt {-c \left (b x -a \right )}}+\frac {\left (\frac {\left (B f -C e \right ) \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right )}{f \sqrt {b^{2} c}}-\frac {\left (A \,f^{2}-B e f +C \,e^{2}\right ) \ln \left (\frac {\frac {2 c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}+\frac {2 b^{2} c e \left (x +\frac {e}{f}\right )}{f}+2 \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}\, \sqrt {-b^{2} c \left (x +\frac {e}{f}\right )^{2}+\frac {2 b^{2} c e \left (x +\frac {e}{f}\right )}{f}+\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}}{x +\frac {e}{f}}\right )}{f^{2} \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}}{f \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(299\)
default	\(\frac {\left (-A \ln \left (\frac {2 b^{2} c e x +2 a^{2} c f +2 \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, f}{f x +e}\right ) b^{2} c \,f^{2} \sqrt {b^{2} c}+B \ln \left (\frac {2 b^{2} c e x +2 a^{2} c f +2 \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, f}{f x +e}\right ) b^{2} c e f \sqrt {b^{2} c}+B \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) b^{2} c \,f^{2} \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}-C \ln \left (\frac {2 b^{2} c e x +2 a^{2} c f +2 \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, f}{f x +e}\right ) b^{2} c \,e^{2} \sqrt {b^{2} c}-C \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) b^{2} c e f \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}-C \,f^{2} \sqrt {b^{2} c}\, \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\right ) \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}{\sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}\, f^{3} \sqrt {b^{2} c}\, b^{2} c \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\)	\(487\)

[In]

int((C*x^2+B*x+A)/(f*x+e)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-C*(b*x+a)^(1/2)*(-b*x+a)/f/b^2/(-c*(b*x-a))^(1/2)+1/f*((B*f-C*e)/f/(b^2*c)^(1/2)*arctan((b^2*c)^(1/2)*x/(-b^2
*c*x^2+a^2*c)^(1/2))-(A*f^2-B*e*f+C*e^2)/f^2/(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*ln((2*c*(a^2*f^2-b^2*e^2)/f^2+2*b
^2*c*e/f*(x+e/f)+2*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*(-b^2*c*(x+e/f)^2+2*b^2*c*e/f*(x+e/f)+c*(a^2*f^2-b^2*e^2)/f
^2)^(1/2))/(x+e/f)))*(-(b*x+a)*c*(b*x-a))^(1/2)/(b*x+a)^(1/2)/(-c*(b*x-a))^(1/2)

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=\text {Timed out} \]

[In]

integrate((C*x^2+B*x+A)/(f*x+e)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=\int \frac {A + B x + C x^{2}}{\sqrt {- c \left (- a + b x\right )} \sqrt {a + b x} \left (e + f x\right )}\, dx \]

[In]

integrate((C*x**2+B*x+A)/(f*x+e)/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)

[Out]

Integral((A + B*x + C*x**2)/(sqrt(-c*(-a + b*x))*sqrt(a + b*x)*(e + f*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((C*x^2+B*x+A)/(f*x+e)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume((4*b^2*c>0)', see `assume?` fo
r more detai

Giac [F(-2)]

Exception generated. \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((C*x^2+B*x+A)/(f*x+e)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 9298, normalized size of antiderivative = 33.45 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=\text {Too large to display} \]

[In]

int((A + B*x + C*x^2)/((e + f*x)*(a*c - b*c*x)^(1/2)*(a + b*x)^(1/2)),x)

[Out]

(B*a*e*atan(((B*a*e*((4096*(32*B^3*a^(17/2)*c^3*e*f^2*(a*c)^(5/2) + 24*B^3*a^(15/2)*b^2*c^4*e^3*(a*c)^(3/2)))/
(a^6*b^8*e^6) - (4096*(32*B^3*a^(17/2)*c^2*e*f^2*(a*c)^(5/2) - 96*B^3*a^(15/2)*b^2*c^3*e^3*(a*c)^(3/2))*((a*c
- b*c*x)^(1/2) - (a*c)^(1/2))^2)/(a^6*b^8*e^6*((a + b*x)^(1/2) - a^(1/2))^2) - (B*a*e*((4096*(16*B^2*a^12*c^6*
f^4 + 9*B^2*a^8*b^4*c^6*e^4))/(a^6*b^8*e^6) + (B*a*e*((4096*(24*B*a^(17/2)*b^2*c^4*e*f^4*(a*c)^(5/2) - 30*B*a^
(15/2)*b^4*c^5*e^3*f^2*(a*c)^(3/2)))/(a^6*b^8*e^6) + (16384*(20*B*a^12*c^6*f^5 - 22*B*a^10*b^2*c^6*e^2*f^3)*((
a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(a^6*b^7*e^6*((a + b*x)^(1/2) - a^(1/2))) + (B*a*e*((4096*(9*a^8*b^6*c^7*e^
4*f^2 - 7*a^10*b^4*c^7*e^2*f^4))/(a^6*b^8*e^6) + (4096*(9*a^8*b^6*c^6*e^4*f^2 - 11*a^10*b^4*c^6*e^2*f^4)*((a*c
- b*c*x)^(1/2) - (a*c)^(1/2))^2)/(a^6*b^8*e^6*((a + b*x)^(1/2) - a^(1/2))^2) - (16384*(5*a^(17/2)*b^2*c^4*e*f
^5*(a*c)^(5/2) - 6*a^(15/2)*b^4*c^5*e^3*f^3*(a*c)^(3/2))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(a^6*b^7*e^6*((a
+ b*x)^(1/2) - a^(1/2)))))/(f*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)) + (4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^
2*(96*B*a^(17/2)*b^2*c^3*e*f^4*(a*c)^(5/2) - 90*B*a^(15/2)*b^4*c^4*e^3*f^2*(a*c)^(3/2)))/(a^6*b^8*e^6*((a + b*
x)^(1/2) - a^(1/2))^2)))/(f*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)) + (16384*(8*B^2*a^(17/2)*c^3*e*f^3*(a*c)^(5/2)
+ 3*B^2*a^(15/2)*b^2*c^4*e^3*f*(a*c)^(3/2))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(a^6*b^7*e^6*((a + b*x)^(1/2)
- a^(1/2))) + (4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2*(9*B^2*a^8*b^4*c^5*e^4 - 144*B^2*a^12*c^5*f^4 + 128
*B^2*a^10*b^2*c^5*e^2*f^2))/(a^6*b^8*e^6*((a + b*x)^(1/2) - a^(1/2))^2)))/(f*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)
) + (458752*B^3*a^4*c^5*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(b^7*e^4*((a + b*x)^(1/2) - a^(1/2))))*1i)/(f*(
a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)) + (B*a*e*((4096*(32*B^3*a^(17/2)*c^3*e*f^2*(a*c)^(5/2) + 24*B^3*a^(15/2)*b^2
*c^4*e^3*(a*c)^(3/2)))/(a^6*b^8*e^6) - (4096*(32*B^3*a^(17/2)*c^2*e*f^2*(a*c)^(5/2) - 96*B^3*a^(15/2)*b^2*c^3*
e^3*(a*c)^(3/2))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(a^6*b^8*e^6*((a + b*x)^(1/2) - a^(1/2))^2) + (B*a*e*(
(4096*(16*B^2*a^12*c^6*f^4 + 9*B^2*a^8*b^4*c^6*e^4))/(a^6*b^8*e^6) - (B*a*e*((4096*(24*B*a^(17/2)*b^2*c^4*e*f^
4*(a*c)^(5/2) - 30*B*a^(15/2)*b^4*c^5*e^3*f^2*(a*c)^(3/2)))/(a^6*b^8*e^6) + (16384*(20*B*a^12*c^6*f^5 - 22*B*a
^10*b^2*c^6*e^2*f^3)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(a^6*b^7*e^6*((a + b*x)^(1/2) - a^(1/2))) - (B*a*e*(
(4096*(9*a^8*b^6*c^7*e^4*f^2 - 7*a^10*b^4*c^7*e^2*f^4))/(a^6*b^8*e^6) + (4096*(9*a^8*b^6*c^6*e^4*f^2 - 11*a^10
*b^4*c^6*e^2*f^4)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(a^6*b^8*e^6*((a + b*x)^(1/2) - a^(1/2))^2) - (16384*
(5*a^(17/2)*b^2*c^4*e*f^5*(a*c)^(5/2) - 6*a^(15/2)*b^4*c^5*e^3*f^3*(a*c)^(3/2))*((a*c - b*c*x)^(1/2) - (a*c)^(
1/2)))/(a^6*b^7*e^6*((a + b*x)^(1/2) - a^(1/2)))))/(f*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)) + (4096*((a*c - b*c*x
)^(1/2) - (a*c)^(1/2))^2*(96*B*a^(17/2)*b^2*c^3*e*f^4*(a*c)^(5/2) - 90*B*a^(15/2)*b^4*c^4*e^3*f^2*(a*c)^(3/2))
)/(a^6*b^8*e^6*((a + b*x)^(1/2) - a^(1/2))^2)))/(f*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)) + (16384*(8*B^2*a^(17/2)
*c^3*e*f^3*(a*c)^(5/2) + 3*B^2*a^(15/2)*b^2*c^4*e^3*f*(a*c)^(3/2))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(a^6*b
^7*e^6*((a + b*x)^(1/2) - a^(1/2))) + (4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2*(9*B^2*a^8*b^4*c^5*e^4 - 144
*B^2*a^12*c^5*f^4 + 128*B^2*a^10*b^2*c^5*e^2*f^2))/(a^6*b^8*e^6*((a + b*x)^(1/2) - a^(1/2))^2)))/(f*(a^4*c*f^2
- a^2*b^2*c*e^2)^(1/2)) + (458752*B^3*a^4*c^5*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(b^7*e^4*((a + b*x)^(1/2
) - a^(1/2))))*1i)/(f*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)))/((131072*B^4*a^4*c^5)/(b^8*e^4) - (B*a*e*((4096*(32*
B^3*a^(17/2)*c^3*e*f^2*(a*c)^(5/2) + 24*B^3*a^(15/2)*b^2*c^4*e^3*(a*c)^(3/2)))/(a^6*b^8*e^6) - (4096*(32*B^3*a
^(17/2)*c^2*e*f^2*(a*c)^(5/2) - 96*B^3*a^(15/2)*b^2*c^3*e^3*(a*c)^(3/2))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2
)/(a^6*b^8*e^6*((a + b*x)^(1/2) - a^(1/2))^2) - (B*a*e*((4096*(16*B^2*a^12*c^6*f^4 + 9*B^2*a^8*b^4*c^6*e^4))/(
a^6*b^8*e^6) + (B*a*e*((4096*(24*B*a^(17/2)*b^2*c^4*e*f^4*(a*c)^(5/2) - 30*B*a^(15/2)*b^4*c^5*e^3*f^2*(a*c)^(3
/2)))/(a^6*b^8*e^6) + (16384*(20*B*a^12*c^6*f^5 - 22*B*a^10*b^2*c^6*e^2*f^3)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2
)))/(a^6*b^7*e^6*((a + b*x)^(1/2) - a^(1/2))) + (B*a*e*((4096*(9*a^8*b^6*c^7*e^4*f^2 - 7*a^10*b^4*c^7*e^2*f^4)
)/(a^6*b^8*e^6) + (4096*(9*a^8*b^6*c^6*e^4*f^2 - 11*a^10*b^4*c^6*e^2*f^4)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^
2)/(a^6*b^8*e^6*((a + b*x)^(1/2) - a^(1/2))^2) - (16384*(5*a^(17/2)*b^2*c^4*e*f^5*(a*c)^(5/2) - 6*a^(15/2)*b^4
*c^5*e^3*f^3*(a*c)^(3/2))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(a^6*b^7*e^6*((a + b*x)^(1/2) - a^(1/2)))))/(f*
(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)) + (4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2*(96*B*a^(17/2)*b^2*c^3*e*f^4*
(a*c)^(5/2) - 90*B*a^(15/2)*b^4*c^4*e^3*f^2*(a*c)^(3/2)))/(a^6*b^8*e^6*((a + b*x)^(1/2) - a^(1/2))^2)))/(f*(a^
4*c*f^2 - a^2*b^2*c*e^2)^(1/2)) + (16384*(8*B^2*a^(17/2)*c^3*e*f^3*(a*c)^(5/2) + 3*B^2*a^(15/2)*b^2*c^4*e^3*f*
(a*c)^(3/2))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(a^6*b^7*e^6*((a + b*x)^(1/2) - a^(1/2))) + (4096*((a*c - b*
c*x)^(1/2) - (a*c)^(1/2))^2*(9*B^2*a^8*b^4*c^5*e^4 - 144*B^2*a^12*c^5*f^4 + 128*B^2*a^10*b^2*c^5*e^2*f^2))/(a^
6*b^8*e^6*((a + b*x)^(1/2) - a^(1/2))^2)))/(f*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)) + (458752*B^3*a^4*c^5*f*((a*c
- b*c*x)^(1/2) - (a*c)^(1/2)))/(b^7*e^4*((a + b*x)^(1/2) - a^(1/2)))))/(f*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2))
+ (B*a*e*((4096*(32*B^3*a^(17/2)*c^3*e*f^2*(a*c)^(5/2) + 24*B^3*a^(15/2)*b^2*c^4*e^3*(a*c)^(3/2)))/(a^6*b^8*e^
6) - (4096*(32*B^3*a^(17/2)*c^2*e*f^2*(a*c)^(5/2) - 96*B^3*a^(15/2)*b^2*c^3*e^3*(a*c)^(3/2))*((a*c - b*c*x)^(1
/2) - (a*c)^(1/2))^2)/(a^6*b^8*e^6*((a + b*x)^(1/2) - a^(1/2))^2) + (B*a*e*((4096*(16*B^2*a^12*c^6*f^4 + 9*B^2
*a^8*b^4*c^6*e^4))/(a^6*b^8*e^6) - (B*a*e*((4096*(24*B*a^(17/2)*b^2*c^4*e*f^4*(a*c)^(5/2) - 30*B*a^(15/2)*b^4*
c^5*e^3*f^2*(a*c)^(3/2)))/(a^6*b^8*e^6) + (16384*(20*B*a^12*c^6*f^5 - 22*B*a^10*b^2*c^6*e^2*f^3)*((a*c - b*c*x
)^(1/2) - (a*c)^(1/2)))/(a^6*b^7*e^6*((a + b*x)^(1/2) - a^(1/2))) - (B*a*e*((4096*(9*a^8*b^6*c^7*e^4*f^2 - 7*a
^10*b^4*c^7*e^2*f^4))/(a^6*b^8*e^6) + (4096*(9*a^8*b^6*c^6*e^4*f^2 - 11*a^10*b^4*c^6*e^2*f^4)*((a*c - b*c*x)^(
1/2) - (a*c)^(1/2))^2)/(a^6*b^8*e^6*((a + b*x)^(1/2) - a^(1/2))^2) - (16384*(5*a^(17/2)*b^2*c^4*e*f^5*(a*c)^(5
/2) - 6*a^(15/2)*b^4*c^5*e^3*f^3*(a*c)^(3/2))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(a^6*b^7*e^6*((a + b*x)^(1/
2) - a^(1/2)))))/(f*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)) + (4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2*(96*B*a^(
17/2)*b^2*c^3*e*f^4*(a*c)^(5/2) - 90*B*a^(15/2)*b^4*c^4*e^3*f^2*(a*c)^(3/2)))/(a^6*b^8*e^6*((a + b*x)^(1/2) -
a^(1/2))^2)))/(f*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)) + (16384*(8*B^2*a^(17/2)*c^3*e*f^3*(a*c)^(5/2) + 3*B^2*a^(
15/2)*b^2*c^4*e^3*f*(a*c)^(3/2))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(a^6*b^7*e^6*((a + b*x)^(1/2) - a^(1/2))
) + (4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2*(9*B^2*a^8*b^4*c^5*e^4 - 144*B^2*a^12*c^5*f^4 + 128*B^2*a^10*b
^2*c^5*e^2*f^2))/(a^6*b^8*e^6*((a + b*x)^(1/2) - a^(1/2))^2)))/(f*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)) + (458752
*B^3*a^4*c^5*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(b^7*e^4*((a + b*x)^(1/2) - a^(1/2)))))/(f*(a^4*c*f^2 - a^
2*b^2*c*e^2)^(1/2)) + (917504*B^4*a^4*c^4*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(b^8*e^4*((a + b*x)^(1/2) - a
^(1/2))^2)))*2i)/(f*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)) - (C*e^2*atan(((C*e^2*((4096*(32*C^3*a^(5/2)*c^3*e^2*f^
3*(a*c)^(5/2) + 24*C^3*a^(3/2)*b^2*c^4*e^4*f*(a*c)^(3/2)))/(b^8*e^4*f^4) + (C*e^2*((4096*(16*C^2*a^6*c^6*f^6 +
9*C^2*a^2*b^4*c^6*e^4*f^2))/(b^8*e^4*f^4) - (C*e^2*((4096*(24*C*a^(5/2)*b^2*c^4*f^7*(a*c)^(5/2) - 30*C*a^(3/2
)*b^4*c^5*e^2*f^5*(a*c)^(3/2)))/(b^8*e^4*f^4) + (C*e^2*((4096*(7*a^4*b^4*c^7*f^8 - 9*a^2*b^6*c^7*e^2*f^6))/(b^
8*e^4*f^4) + (16384*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))*(5*a^(5/2)*b^2*c^4*f^7*(a*c)^(5/2) - 6*a^(3/2)*b^4*c^5
*e^2*f^5*(a*c)^(3/2)))/(b^7*e^5*f^2*((a + b*x)^(1/2) - a^(1/2))) + (4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2
*(11*a^4*b^4*c^6*f^8 - 9*a^2*b^6*c^6*e^2*f^6))/(b^8*e^4*f^4*((a + b*x)^(1/2) - a^(1/2))^2)))/(f^2*(a^2*c*f^2 -
b^2*c*e^2)^(1/2)) + (16384*(20*C*a^6*c^6*f^6 - 22*C*a^4*b^2*c^6*e^2*f^4)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))
/(b^7*e^5*f^2*((a + b*x)^(1/2) - a^(1/2))) + (4096*(96*C*a^(5/2)*b^2*c^3*f^7*(a*c)^(5/2) - 90*C*a^(3/2)*b^4*c^
4*e^2*f^5*(a*c)^(3/2))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(b^8*e^4*f^4*((a + b*x)^(1/2) - a^(1/2))^2)))/(f
^2*(a^2*c*f^2 - b^2*c*e^2)^(1/2)) + (4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2*(9*C^2*a^2*b^4*c^5*e^4*f^2 - 1
44*C^2*a^6*c^5*f^6 + 128*C^2*a^4*b^2*c^5*e^2*f^4))/(b^8*e^4*f^4*((a + b*x)^(1/2) - a^(1/2))^2) + (16384*((a*c
- b*c*x)^(1/2) - (a*c)^(1/2))*(8*C^2*a^(5/2)*c^3*e^2*f^3*(a*c)^(5/2) + 3*C^2*a^(3/2)*b^2*c^4*e^4*f*(a*c)^(3/2)
))/(b^7*e^5*f^2*((a + b*x)^(1/2) - a^(1/2)))))/(f^2*(a^2*c*f^2 - b^2*c*e^2)^(1/2)) - (4096*((a*c - b*c*x)^(1/2
) - (a*c)^(1/2))^2*(32*C^3*a^(5/2)*c^2*e^2*f^3*(a*c)^(5/2) - 96*C^3*a^(3/2)*b^2*c^3*e^4*f*(a*c)^(3/2)))/(b^8*e
^4*f^4*((a + b*x)^(1/2) - a^(1/2))^2) + (458752*C^3*a^4*c^5*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(b^7*e*f^2*((
a + b*x)^(1/2) - a^(1/2))))*1i)/(f^2*(a^2*c*f^2 - b^2*c*e^2)^(1/2)) + (C*e^2*((4096*(32*C^3*a^(5/2)*c^3*e^2*f^
3*(a*c)^(5/2) + 24*C^3*a^(3/2)*b^2*c^4*e^4*f*(a*c)^(3/2)))/(b^8*e^4*f^4) - (C*e^2*((4096*(16*C^2*a^6*c^6*f^6 +
9*C^2*a^2*b^4*c^6*e^4*f^2))/(b^8*e^4*f^4) + (C*e^2*((4096*(24*C*a^(5/2)*b^2*c^4*f^7*(a*c)^(5/2) - 30*C*a^(3/2
)*b^4*c^5*e^2*f^5*(a*c)^(3/2)))/(b^8*e^4*f^4) - (C*e^2*((4096*(7*a^4*b^4*c^7*f^8 - 9*a^2*b^6*c^7*e^2*f^6))/(b^
8*e^4*f^4) + (16384*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))*(5*a^(5/2)*b^2*c^4*f^7*(a*c)^(5/2) - 6*a^(3/2)*b^4*c^5
*e^2*f^5*(a*c)^(3/2)))/(b^7*e^5*f^2*((a + b*x)^(1/2) - a^(1/2))) + (4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2
*(11*a^4*b^4*c^6*f^8 - 9*a^2*b^6*c^6*e^2*f^6))/(b^8*e^4*f^4*((a + b*x)^(1/2) - a^(1/2))^2)))/(f^2*(a^2*c*f^2 -
b^2*c*e^2)^(1/2)) + (16384*(20*C*a^6*c^6*f^6 - 22*C*a^4*b^2*c^6*e^2*f^4)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))
/(b^7*e^5*f^2*((a + b*x)^(1/2) - a^(1/2))) + (4096*(96*C*a^(5/2)*b^2*c^3*f^7*(a*c)^(5/2) - 90*C*a^(3/2)*b^4*c^
4*e^2*f^5*(a*c)^(3/2))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(b^8*e^4*f^4*((a + b*x)^(1/2) - a^(1/2))^2)))/(f
^2*(a^2*c*f^2 - b^2*c*e^2)^(1/2)) + (4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2*(9*C^2*a^2*b^4*c^5*e^4*f^2 - 1
44*C^2*a^6*c^5*f^6 + 128*C^2*a^4*b^2*c^5*e^2*f^4))/(b^8*e^4*f^4*((a + b*x)^(1/2) - a^(1/2))^2) + (16384*((a*c
- b*c*x)^(1/2) - (a*c)^(1/2))*(8*C^2*a^(5/2)*c^3*e^2*f^3*(a*c)^(5/2) + 3*C^2*a^(3/2)*b^2*c^4*e^4*f*(a*c)^(3/2)
))/(b^7*e^5*f^2*((a + b*x)^(1/2) - a^(1/2)))))/(f^2*(a^2*c*f^2 - b^2*c*e^2)^(1/2)) - (4096*((a*c - b*c*x)^(1/2
) - (a*c)^(1/2))^2*(32*C^3*a^(5/2)*c^2*e^2*f^3*(a*c)^(5/2) - 96*C^3*a^(3/2)*b^2*c^3*e^4*f*(a*c)^(3/2)))/(b^8*e
^4*f^4*((a + b*x)^(1/2) - a^(1/2))^2) + (458752*C^3*a^4*c^5*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(b^7*e*f^2*((
a + b*x)^(1/2) - a^(1/2))))*1i)/(f^2*(a^2*c*f^2 - b^2*c*e^2)^(1/2)))/((131072*C^4*a^4*c^5)/(b^8*f^4) + (C*e^2*
((4096*(32*C^3*a^(5/2)*c^3*e^2*f^3*(a*c)^(5/2) + 24*C^3*a^(3/2)*b^2*c^4*e^4*f*(a*c)^(3/2)))/(b^8*e^4*f^4) + (C
*e^2*((4096*(16*C^2*a^6*c^6*f^6 + 9*C^2*a^2*b^4*c^6*e^4*f^2))/(b^8*e^4*f^4) - (C*e^2*((4096*(24*C*a^(5/2)*b^2*
c^4*f^7*(a*c)^(5/2) - 30*C*a^(3/2)*b^4*c^5*e^2*f^5*(a*c)^(3/2)))/(b^8*e^4*f^4) + (C*e^2*((4096*(7*a^4*b^4*c^7*
f^8 - 9*a^2*b^6*c^7*e^2*f^6))/(b^8*e^4*f^4) + (16384*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))*(5*a^(5/2)*b^2*c^4*f^
7*(a*c)^(5/2) - 6*a^(3/2)*b^4*c^5*e^2*f^5*(a*c)^(3/2)))/(b^7*e^5*f^2*((a + b*x)^(1/2) - a^(1/2))) + (4096*((a*
c - b*c*x)^(1/2) - (a*c)^(1/2))^2*(11*a^4*b^4*c^6*f^8 - 9*a^2*b^6*c^6*e^2*f^6))/(b^8*e^4*f^4*((a + b*x)^(1/2)
- a^(1/2))^2)))/(f^2*(a^2*c*f^2 - b^2*c*e^2)^(1/2)) + (16384*(20*C*a^6*c^6*f^6 - 22*C*a^4*b^2*c^6*e^2*f^4)*((a
*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(b^7*e^5*f^2*((a + b*x)^(1/2) - a^(1/2))) + (4096*(96*C*a^(5/2)*b^2*c^3*f^7*
(a*c)^(5/2) - 90*C*a^(3/2)*b^4*c^4*e^2*f^5*(a*c)^(3/2))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(b^8*e^4*f^4*((
a + b*x)^(1/2) - a^(1/2))^2)))/(f^2*(a^2*c*f^2 - b^2*c*e^2)^(1/2)) + (4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))
^2*(9*C^2*a^2*b^4*c^5*e^4*f^2 - 144*C^2*a^6*c^5*f^6 + 128*C^2*a^4*b^2*c^5*e^2*f^4))/(b^8*e^4*f^4*((a + b*x)^(1
/2) - a^(1/2))^2) + (16384*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))*(8*C^2*a^(5/2)*c^3*e^2*f^3*(a*c)^(5/2) + 3*C^2*
a^(3/2)*b^2*c^4*e^4*f*(a*c)^(3/2)))/(b^7*e^5*f^2*((a + b*x)^(1/2) - a^(1/2)))))/(f^2*(a^2*c*f^2 - b^2*c*e^2)^(
1/2)) - (4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2*(32*C^3*a^(5/2)*c^2*e^2*f^3*(a*c)^(5/2) - 96*C^3*a^(3/2)*b
^2*c^3*e^4*f*(a*c)^(3/2)))/(b^8*e^4*f^4*((a + b*x)^(1/2) - a^(1/2))^2) + (458752*C^3*a^4*c^5*((a*c - b*c*x)^(1
/2) - (a*c)^(1/2)))/(b^7*e*f^2*((a + b*x)^(1/2) - a^(1/2)))))/(f^2*(a^2*c*f^2 - b^2*c*e^2)^(1/2)) - (C*e^2*((4
096*(32*C^3*a^(5/2)*c^3*e^2*f^3*(a*c)^(5/2) + 24*C^3*a^(3/2)*b^2*c^4*e^4*f*(a*c)^(3/2)))/(b^8*e^4*f^4) - (C*e^
2*((4096*(16*C^2*a^6*c^6*f^6 + 9*C^2*a^2*b^4*c^6*e^4*f^2))/(b^8*e^4*f^4) + (C*e^2*((4096*(24*C*a^(5/2)*b^2*c^4
*f^7*(a*c)^(5/2) - 30*C*a^(3/2)*b^4*c^5*e^2*f^5*(a*c)^(3/2)))/(b^8*e^4*f^4) - (C*e^2*((4096*(7*a^4*b^4*c^7*f^8
- 9*a^2*b^6*c^7*e^2*f^6))/(b^8*e^4*f^4) + (16384*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))*(5*a^(5/2)*b^2*c^4*f^7*(
a*c)^(5/2) - 6*a^(3/2)*b^4*c^5*e^2*f^5*(a*c)^(3/2)))/(b^7*e^5*f^2*((a + b*x)^(1/2) - a^(1/2))) + (4096*((a*c -
b*c*x)^(1/2) - (a*c)^(1/2))^2*(11*a^4*b^4*c^6*f^8 - 9*a^2*b^6*c^6*e^2*f^6))/(b^8*e^4*f^4*((a + b*x)^(1/2) - a
^(1/2))^2)))/(f^2*(a^2*c*f^2 - b^2*c*e^2)^(1/2)) + (16384*(20*C*a^6*c^6*f^6 - 22*C*a^4*b^2*c^6*e^2*f^4)*((a*c
- b*c*x)^(1/2) - (a*c)^(1/2)))/(b^7*e^5*f^2*((a + b*x)^(1/2) - a^(1/2))) + (4096*(96*C*a^(5/2)*b^2*c^3*f^7*(a*
c)^(5/2) - 90*C*a^(3/2)*b^4*c^4*e^2*f^5*(a*c)^(3/2))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(b^8*e^4*f^4*((a +
b*x)^(1/2) - a^(1/2))^2)))/(f^2*(a^2*c*f^2 - b^2*c*e^2)^(1/2)) + (4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2*
(9*C^2*a^2*b^4*c^5*e^4*f^2 - 144*C^2*a^6*c^5*f^6 + 128*C^2*a^4*b^2*c^5*e^2*f^4))/(b^8*e^4*f^4*((a + b*x)^(1/2)
- a^(1/2))^2) + (16384*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))*(8*C^2*a^(5/2)*c^3*e^2*f^3*(a*c)^(5/2) + 3*C^2*a^(
3/2)*b^2*c^4*e^4*f*(a*c)^(3/2)))/(b^7*e^5*f^2*((a + b*x)^(1/2) - a^(1/2)))))/(f^2*(a^2*c*f^2 - b^2*c*e^2)^(1/2
)) - (4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2*(32*C^3*a^(5/2)*c^2*e^2*f^3*(a*c)^(5/2) - 96*C^3*a^(3/2)*b^2*
c^3*e^4*f*(a*c)^(3/2)))/(b^8*e^4*f^4*((a + b*x)^(1/2) - a^(1/2))^2) + (458752*C^3*a^4*c^5*((a*c - b*c*x)^(1/2)
- (a*c)^(1/2)))/(b^7*e*f^2*((a + b*x)^(1/2) - a^(1/2)))))/(f^2*(a^2*c*f^2 - b^2*c*e^2)^(1/2)) + (917504*C^4*a
^4*c^4*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(b^8*f^4*((a + b*x)^(1/2) - a^(1/2))^2)))*2i)/(f^2*(a^2*c*f^2 -
b^2*c*e^2)^(1/2)) - (4*B*atan((67108864*B^5*a^16*c^7*f^4*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(((a + b*x)^(1/2
) - a^(1/2))*(67108864*B^5*a^16*c^(15/2)*f^4 + 37748736*B^5*a^12*b^4*c^(15/2)*e^4 - 100663296*B^5*a^14*b^2*c^(
15/2)*e^2*f^2)) + (37748736*B^5*a^12*b^4*c^7*e^4*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(((a + b*x)^(1/2) - a^(1
/2))*(67108864*B^5*a^16*c^(15/2)*f^4 + 37748736*B^5*a^12*b^4*c^(15/2)*e^4 - 100663296*B^5*a^14*b^2*c^(15/2)*e^
2*f^2)) - (100663296*B^5*a^14*b^2*c^7*e^2*f^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(((a + b*x)^(1/2) - a^(1/2)
)*(67108864*B^5*a^16*c^(15/2)*f^4 + 37748736*B^5*a^12*b^4*c^(15/2)*e^4 - 100663296*B^5*a^14*b^2*c^(15/2)*e^2*f
^2))))/(b*c^(1/2)*f) - (A*a*atan((a*c*(a*c - b*c*x)^(1/2)*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)*2i - (a*c)^(3/2)*(
a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)*1i + a*c*(a*c)^(1/2)*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)*1i + b*c*x*(a*c)^(1/2)
*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)*2i - a^(1/2)*c*(a*c)^(1/2)*(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2)*(a + b*x)^(1/2
)*2i)/(2*a^(5/2)*b*c^2*e - 2*a^3*c^2*f*(a + b*x)^(1/2) - 2*a^2*b*c^2*e*(a + b*x)^(1/2) + 2*a^(5/2)*b*c^2*f*x +
2*a^(5/2)*c*f*(a*c - b*c*x)^(1/2)*(a*c)^(1/2) - 2*a^(3/2)*b*c*e*(a*c - b*c*x)^(1/2)*(a*c)^(1/2) + 2*a*b*c*e*(
a*c - b*c*x)^(1/2)*(a*c)^(1/2)*(a + b*x)^(1/2)))*2i)/(a^4*c*f^2 - a^2*b^2*c*e^2)^(1/2) + (4*C*e*atan((67108864
*C^5*a^8*c^7*f^4*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(((a + b*x)^(1/2) - a^(1/2))*(67108864*C^5*a^8*c^(15/2)*
f^4 + 37748736*C^5*a^4*b^4*c^(15/2)*e^4 - 100663296*C^5*a^6*b^2*c^(15/2)*e^2*f^2)) + (37748736*C^5*a^4*b^4*c^7
*e^4*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(((a + b*x)^(1/2) - a^(1/2))*(67108864*C^5*a^8*c^(15/2)*f^4 + 377487
36*C^5*a^4*b^4*c^(15/2)*e^4 - 100663296*C^5*a^6*b^2*c^(15/2)*e^2*f^2)) - (100663296*C^5*a^6*b^2*c^7*e^2*f^2*((
a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(((a + b*x)^(1/2) - a^(1/2))*(67108864*C^5*a^8*c^(15/2)*f^4 + 37748736*C^5*
a^4*b^4*c^(15/2)*e^4 - 100663296*C^5*a^6*b^2*c^(15/2)*e^2*f^2))))/(b*c^(1/2)*f^2) - (8*C*a^(1/2)*(a*c)^(1/2)*(
(a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(b^2*f*((a + b*x)^(1/2) - a^(1/2))^2*(((a*c - b*c*x)^(1/2) - (a*c)^(1/2)
)^4/((a + b*x)^(1/2) - a^(1/2))^4 + c^2 + (2*c*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/
2))^2))