3.1.5 \(\int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx\) [5]

Optimal. Leaf size=122 \[ -\frac {C \sqrt {1-d^2 x^2}}{d^2 f}-\frac {(C e-B f) \sin ^{-1}(d x)}{d f^2}+\frac {\left (C e^2-B e f+A f^2\right ) \tan ^{-1}\left (\frac {f+d^2 e x}{\sqrt {d^2 e^2-f^2} \sqrt {1-d^2 x^2}}\right )}{f^2 \sqrt {d^2 e^2-f^2}} \]

[Out]

-(-B*f+C*e)*arcsin(d*x)/d/f^2+(A*f^2-B*e*f+C*e^2)*arctan((d^2*e*x+f)/(d^2*e^2-f^2)^(1/2)/(-d^2*x^2+1)^(1/2))/f
^2/(d^2*e^2-f^2)^(1/2)-C*(-d^2*x^2+1)^(1/2)/d^2/f

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Rubi [A]
time = 0.20, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {1623, 1668, 858, 222, 739, 210} \begin {gather*} \frac {\left (A f^2-B e f+C e^2\right ) \text {ArcTan}\left (\frac {d^2 e x+f}{\sqrt {1-d^2 x^2} \sqrt {d^2 e^2-f^2}}\right )}{f^2 \sqrt {d^2 e^2-f^2}}-\frac {\text {ArcSin}(d x) (C e-B f)}{d f^2}-\frac {C \sqrt {1-d^2 x^2}}{d^2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)),x]

[Out]

-((C*Sqrt[1 - d^2*x^2])/(d^2*f)) - ((C*e - B*f)*ArcSin[d*x])/(d*f^2) + ((C*e^2 - B*e*f + A*f^2)*ArcTan[(f + d^
2*e*x)/(Sqrt[d^2*e^2 - f^2]*Sqrt[1 - d^2*x^2])])/(f^2*Sqrt[d^2*e^2 - f^2])

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 222

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

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 1623

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

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*} \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx &=\int \frac {A+B x+C x^2}{(e+f x) \sqrt {1-d^2 x^2}} \, dx\\ &=-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}-\frac {\int \frac {-A d^2 f^2+d^2 f (C e-B f) x}{(e+f x) \sqrt {1-d^2 x^2}} \, dx}{d^2 f^2}\\ &=-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}-\frac {(C e-B f) \int \frac {1}{\sqrt {1-d^2 x^2}} \, dx}{f^2}+\frac {\left (C e^2-B e f+A f^2\right ) \int \frac {1}{(e+f x) \sqrt {1-d^2 x^2}} \, dx}{f^2}\\ &=-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}-\frac {(C e-B f) \sin ^{-1}(d x)}{d f^2}-\frac {\left (C e^2-B e f+A f^2\right ) \text {Subst}\left (\int \frac {1}{-d^2 e^2+f^2-x^2} \, dx,x,\frac {f+d^2 e x}{\sqrt {1-d^2 x^2}}\right )}{f^2}\\ &=-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}-\frac {(C e-B f) \sin ^{-1}(d x)}{d f^2}+\frac {\left (C e^2-B e f+A f^2\right ) \tan ^{-1}\left (\frac {f+d^2 e x}{\sqrt {d^2 e^2-f^2} \sqrt {1-d^2 x^2}}\right )}{f^2 \sqrt {d^2 e^2-f^2}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(559\) vs. \(2(122)=244\).
time = 2.21, size = 559, normalized size = 4.58 \begin {gather*} \frac {-\frac {C f^3 \sqrt {1-d^2 x^2}}{d^2}+\frac {\left (C e^2+f (-B e+A f)\right ) \sqrt {2 d^2 e^2-f^2-2 d e \sqrt {d^2 e^2-f^2}} \left (d^2 e^2-f^2+d e \sqrt {d^2 e^2-f^2}\right ) \tan ^{-1}\left (\frac {f \left (\sqrt {-d^2} x-\sqrt {1-d^2 x^2}\right )}{\sqrt {2 d^2 e^2-f^2-2 d e \sqrt {d^2 e^2-f^2}}}\right )}{(d e-f) (d e+f)}-\frac {\left (C e^2+f (-B e+A f)\right ) \left (-d^2 e^2+f^2+d e \sqrt {d^2 e^2-f^2}\right ) \sqrt {2 d^2 e^2-f^2+2 d e \sqrt {d^2 e^2-f^2}} \tan ^{-1}\left (\frac {f \left (\sqrt {-d^2} x-\sqrt {1-d^2 x^2}\right )}{\sqrt {2 d^2 e^2-f^2+2 d e \sqrt {d^2 e^2-f^2}}}\right )}{(d e-f) (d e+f)}+\frac {d f^2 \sqrt {-d^2 e^2+f^2} \left (C e^2+f (-B e+A f)\right ) \tan ^{-1}\left (\frac {-\sqrt {-d^2} f^2 x \sqrt {1-d^2 x^2}+d^2 \left (e^2-f^2 x^2\right )}{d e \sqrt {-d^2 e^2+f^2}}\right )}{\sqrt {-d^2} (d e-f) (d e+f)}+\frac {f^2 (C e-B f) \log \left (-\sqrt {-d^2} x+\sqrt {1-d^2 x^2}\right )}{\sqrt {-d^2}}}{f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)),x]

[Out]

(-((C*f^3*Sqrt[1 - d^2*x^2])/d^2) + ((C*e^2 + f*(-(B*e) + A*f))*Sqrt[2*d^2*e^2 - f^2 - 2*d*e*Sqrt[d^2*e^2 - f^
2]]*(d^2*e^2 - f^2 + d*e*Sqrt[d^2*e^2 - f^2])*ArcTan[(f*(Sqrt[-d^2]*x - Sqrt[1 - d^2*x^2]))/Sqrt[2*d^2*e^2 - f
^2 - 2*d*e*Sqrt[d^2*e^2 - f^2]]])/((d*e - f)*(d*e + f)) - ((C*e^2 + f*(-(B*e) + A*f))*(-(d^2*e^2) + f^2 + d*e*
Sqrt[d^2*e^2 - f^2])*Sqrt[2*d^2*e^2 - f^2 + 2*d*e*Sqrt[d^2*e^2 - f^2]]*ArcTan[(f*(Sqrt[-d^2]*x - Sqrt[1 - d^2*
x^2]))/Sqrt[2*d^2*e^2 - f^2 + 2*d*e*Sqrt[d^2*e^2 - f^2]]])/((d*e - f)*(d*e + f)) + (d*f^2*Sqrt[-(d^2*e^2) + f^
2]*(C*e^2 + f*(-(B*e) + A*f))*ArcTan[(-(Sqrt[-d^2]*f^2*x*Sqrt[1 - d^2*x^2]) + d^2*(e^2 - f^2*x^2))/(d*e*Sqrt[-
(d^2*e^2) + f^2])])/(Sqrt[-d^2]*(d*e - f)*(d*e + f)) + (f^2*(C*e - B*f)*Log[-(Sqrt[-d^2]*x) + Sqrt[1 - d^2*x^2
]])/Sqrt[-d^2])/f^4

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.14, size = 373, normalized size = 3.06

method	result	size
default	\(\frac {\left (-A \,\mathrm {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d^{2} f^{2}+B \,\mathrm {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d^{2} e f -C \,\mathrm {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} e x +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d^{2} e^{2}+B \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) d \,f^{2} \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}-C \,\mathrm {csgn}\left (d \right ) f^{2} \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}-C \arctan \left (\frac {\mathrm {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) d e f \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\right ) \sqrt {-d x +1}\, \sqrt {d x +1}\, \mathrm {csgn}\left (d \right )}{\sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f^{3} \sqrt {-d^{2} x^{2}+1}\, d^{2}}\)	\(373\)
risch	\(\frac {C \sqrt {d x +1}\, \left (d x -1\right ) \sqrt {\left (-d x +1\right ) \left (d x +1\right )}}{f \,d^{2} \sqrt {-\left (d x +1\right ) \left (d x -1\right )}\, \sqrt {-d x +1}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right ) B}{f \sqrt {d^{2}}}-\frac {\arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right ) C e}{f^{2} \sqrt {d^{2}}}-\frac {\ln \left (\frac {-\frac {2 \left (d^{2} e^{2}-f^{2}\right )}{f^{2}}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}+2 \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, \sqrt {-d^{2} \left (x +\frac {e}{f}\right )^{2}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}{x +\frac {e}{f}}\right ) A}{f \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}+\frac {\ln \left (\frac {-\frac {2 \left (d^{2} e^{2}-f^{2}\right )}{f^{2}}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}+2 \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, \sqrt {-d^{2} \left (x +\frac {e}{f}\right )^{2}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}{x +\frac {e}{f}}\right ) B e}{f^{2} \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}-\frac {\ln \left (\frac {-\frac {2 \left (d^{2} e^{2}-f^{2}\right )}{f^{2}}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}+2 \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, \sqrt {-d^{2} \left (x +\frac {e}{f}\right )^{2}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}{x +\frac {e}{f}}\right ) C \,e^{2}}{f^{3} \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}\right ) \sqrt {\left (-d x +1\right ) \left (d x +1\right )}}{\sqrt {-d x +1}\, \sqrt {d x +1}}\)	\(589\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-A*csgn(d)*ln(2*(d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))*d^2*f^2+B*csgn(d)*ln(2*(
d^2*e*x+(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))*d^2*e*f-C*csgn(d)*ln(2*(d^2*e*x+(-d^2*x^2+
1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)*f+f)/(f*x+e))*d^2*e^2+B*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*d*f^2*(-(d^
2*e^2-f^2)/f^2)^(1/2)-C*csgn(d)*f^2*(-d^2*x^2+1)^(1/2)*(-(d^2*e^2-f^2)/f^2)^(1/2)-C*arctan(csgn(d)*d*x/(-d^2*x
^2+1)^(1/2))*d*e*f*(-(d^2*e^2-f^2)/f^2)^(1/2))*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*csgn(d)/(-(d^2*e^2-f^2)/f^2)^(1/2)
/f^3/(-d^2*x^2+1)^(1/2)/d^2

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(f*x+e)/(-d*x+1)^(1/2)/(d*x+1)^(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(f-%e*d>0)', see `assume?` for
more details

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Fricas [A]
time = 8.24, size = 457, normalized size = 3.75 \begin {gather*} \left [-\frac {{\left (A d^{2} f^{2} - B d^{2} f e + C d^{2} e^{2}\right )} \sqrt {-d^{2} e^{2} + f^{2}} \log \left (\frac {d^{2} f x e - {\left (d^{2} e^{2} - f^{2} + \sqrt {-d^{2} e^{2} + f^{2}} f\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + f^{2} - {\left (d^{2} x e + f\right )} \sqrt {-d^{2} e^{2} + f^{2}}}{f x + e}\right ) + {\left (C d^{2} f e^{2} - C f^{3}\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 2 \, {\left (B d^{3} f e^{2} - B d f^{3} - C d^{3} e^{3} + C d f^{2} e\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{d^{4} f^{2} e^{2} - d^{2} f^{4}}, -\frac {2 \, {\left (A d^{2} f^{2} - B d^{2} f e + C d^{2} e^{2}\right )} \sqrt {d^{2} e^{2} - f^{2}} \arctan \left (-\frac {\sqrt {d^{2} e^{2} - f^{2}} {\left (f x - \sqrt {d x + 1} \sqrt {-d x + 1} e + e\right )}}{d^{2} x e^{2} - f^{2} x}\right ) + {\left (C d^{2} f e^{2} - C f^{3}\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 2 \, {\left (B d^{3} f e^{2} - B d f^{3} - C d^{3} e^{3} + C d f^{2} e\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{d^{4} f^{2} e^{2} - d^{2} f^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-((A*d^2*f^2 - B*d^2*f*e + C*d^2*e^2)*sqrt(-d^2*e^2 + f^2)*log((d^2*f*x*e - (d^2*e^2 - f^2 + sqrt(-d^2*e^2 +
f^2)*f)*sqrt(d*x + 1)*sqrt(-d*x + 1) + f^2 - (d^2*x*e + f)*sqrt(-d^2*e^2 + f^2))/(f*x + e)) + (C*d^2*f*e^2 - C
*f^3)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 2*(B*d^3*f*e^2 - B*d*f^3 - C*d^3*e^3 + C*d*f^2*e)*arctan((sqrt(d*x + 1)*s
qrt(-d*x + 1) - 1)/(d*x)))/(d^4*f^2*e^2 - d^2*f^4), -(2*(A*d^2*f^2 - B*d^2*f*e + C*d^2*e^2)*sqrt(d^2*e^2 - f^2
)*arctan(-sqrt(d^2*e^2 - f^2)*(f*x - sqrt(d*x + 1)*sqrt(-d*x + 1)*e + e)/(d^2*x*e^2 - f^2*x)) + (C*d^2*f*e^2 -
C*f^3)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 2*(B*d^3*f*e^2 - B*d*f^3 - C*d^3*e^3 + C*d*f^2*e)*arctan((sqrt(d*x + 1)
*sqrt(-d*x + 1) - 1)/(d*x)))/(d^4*f^2*e^2 - d^2*f^4)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x + C x^{2}}{\left (e + f x\right ) \sqrt {- d x + 1} \sqrt {d x + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((A + B*x + C*x**2)/((e + f*x)*sqrt(-d*x + 1)*sqrt(d*x + 1)), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Undef/Unsigned Inf encountered in limitLimit: Max order reached or unable to make series expansion Error: B
ad Argument

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Mupad [B]
time = 25.80, size = 2500, normalized size = 20.49 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*C*e*atan((37748736*C^5*d^4*e^10*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/2) - 1)*(37748736*C^5*d^4*e^10 + 6710
8864*C^5*e^6*f^4 - 100663296*C^5*d^2*e^8*f^2)) + (67108864*C^5*e^6*f^4*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/2
) - 1)*(37748736*C^5*d^4*e^10 + 67108864*C^5*e^6*f^4 - 100663296*C^5*d^2*e^8*f^2)) - (100663296*C^5*d^2*e^8*f^
2*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/2) - 1)*(37748736*C^5*d^4*e^10 + 67108864*C^5*e^6*f^4 - 100663296*C^5*
d^2*e^8*f^2))))/(d*f^2) - (4*B*atan((67108864*B^5*e*f^4*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/2) - 1)*(6710886
4*B^5*e*f^4 + 37748736*B^5*d^4*e^5 - 100663296*B^5*d^2*e^3*f^2)) + (37748736*B^5*d^4*e^5*((1 - d*x)^(1/2) - 1)
)/(((d*x + 1)^(1/2) - 1)*(67108864*B^5*e*f^4 + 37748736*B^5*d^4*e^5 - 100663296*B^5*d^2*e^3*f^2)) - (100663296
*B^5*d^2*e^3*f^2*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/2) - 1)*(67108864*B^5*e*f^4 + 37748736*B^5*d^4*e^5 - 10
0663296*B^5*d^2*e^3*f^2))))/(d*f) - (8*C*((1 - d*x)^(1/2) - 1)^2)/(f*((d*x + 1)^(1/2) - 1)^2*(d^2 + (2*d^2*((1
- d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (d^2*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4)) - (A*a
tan((f^2*1i - d^2*e^2*1i - (f^2*((1 - d*x)^(1/2) - 1)^2*1i)/((d*x + 1)^(1/2) - 1)^2 + (d^2*e^2*((1 - d*x)^(1/2
) - 1)^2*1i)/((d*x + 1)^(1/2) - 1)^2)/(f*(f + d*e)^(1/2)*(f - d*e)^(1/2) - (f*((1 - d*x)^(1/2) - 1)^2*(f + d*e
)^(1/2)*(f - d*e)^(1/2))/((d*x + 1)^(1/2) - 1)^2 + (2*d*e*((1 - d*x)^(1/2) - 1)*(f + d*e)^(1/2)*(f - d*e)^(1/2
))/((d*x + 1)^(1/2) - 1)))*2i)/((f + d*e)^(1/2)*(f - d*e)^(1/2)) - (C*e^2*atan(((C*e^2*((4096*(32*C^3*e^5*f^3
+ 24*C^3*d^2*e^7*f))/(d*f^4) - (4096*((1 - d*x)^(1/2) - 1)^2*(32*C^3*e^5*f^3 - 96*C^3*d^2*e^7*f))/(d*f^4*((d*x
+ 1)^(1/2) - 1)^2) + (458752*C^3*e^6*((1 - d*x)^(1/2) - 1))/(f^2*((d*x + 1)^(1/2) - 1)) + (C*e^2*((4096*(16*C
^2*e^3*f^6 + 9*C^2*d^4*e^7*f^2))/(d*f^4) + (16384*((1 - d*x)^(1/2) - 1)*(8*C^2*e^4*f^3 + 3*C^2*d^2*e^6*f))/(f^
2*((d*x + 1)^(1/2) - 1)) + (4096*((1 - d*x)^(1/2) - 1)^2*(128*C^2*d^2*e^5*f^4 - 144*C^2*e^3*f^6 + 9*C^2*d^4*e^
7*f^2))/(d*f^4*((d*x + 1)^(1/2) - 1)^2) - (C*e^2*((4096*(24*C*d^2*e^3*f^7 - 30*C*d^4*e^5*f^5))/(d*f^4) + (1638
4*((1 - d*x)^(1/2) - 1)*(20*C*e^2*f^6 - 22*C*d^2*e^4*f^4))/(f^2*((d*x + 1)^(1/2) - 1)) + (4096*(96*C*d^2*e^3*f
^7 - 90*C*d^4*e^5*f^5)*((1 - d*x)^(1/2) - 1)^2)/(d*f^4*((d*x + 1)^(1/2) - 1)^2) + (C*e^2*((4096*(7*d^4*e^3*f^8
- 9*d^6*e^5*f^6))/(d*f^4) + (16384*((1 - d*x)^(1/2) - 1)*(5*d^2*e^2*f^7 - 6*d^4*e^4*f^5))/(f^2*((d*x + 1)^(1/
2) - 1)) + (4096*((1 - d*x)^(1/2) - 1)^2*(11*d^4*e^3*f^8 - 9*d^6*e^5*f^6))/(d*f^4*((d*x + 1)^(1/2) - 1)^2)))/(
f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2))))/(f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2))))/(f^2*(f + d*e)^(1/2)*(f - d*e)^
(1/2)))*1i)/(f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2)) + (C*e^2*((4096*(32*C^3*e^5*f^3 + 24*C^3*d^2*e^7*f))/(d*f^4)
- (4096*((1 - d*x)^(1/2) - 1)^2*(32*C^3*e^5*f^3 - 96*C^3*d^2*e^7*f))/(d*f^4*((d*x + 1)^(1/2) - 1)^2) + (45875
2*C^3*e^6*((1 - d*x)^(1/2) - 1))/(f^2*((d*x + 1)^(1/2) - 1)) - (C*e^2*((4096*(16*C^2*e^3*f^6 + 9*C^2*d^4*e^7*f
^2))/(d*f^4) + (16384*((1 - d*x)^(1/2) - 1)*(8*C^2*e^4*f^3 + 3*C^2*d^2*e^6*f))/(f^2*((d*x + 1)^(1/2) - 1)) + (
4096*((1 - d*x)^(1/2) - 1)^2*(128*C^2*d^2*e^5*f^4 - 144*C^2*e^3*f^6 + 9*C^2*d^4*e^7*f^2))/(d*f^4*((d*x + 1)^(1
/2) - 1)^2) + (C*e^2*((4096*(24*C*d^2*e^3*f^7 - 30*C*d^4*e^5*f^5))/(d*f^4) + (16384*((1 - d*x)^(1/2) - 1)*(20*
C*e^2*f^6 - 22*C*d^2*e^4*f^4))/(f^2*((d*x + 1)^(1/2) - 1)) + (4096*(96*C*d^2*e^3*f^7 - 90*C*d^4*e^5*f^5)*((1 -
d*x)^(1/2) - 1)^2)/(d*f^4*((d*x + 1)^(1/2) - 1)^2) - (C*e^2*((4096*(7*d^4*e^3*f^8 - 9*d^6*e^5*f^6))/(d*f^4) +
(16384*((1 - d*x)^(1/2) - 1)*(5*d^2*e^2*f^7 - 6*d^4*e^4*f^5))/(f^2*((d*x + 1)^(1/2) - 1)) + (4096*((1 - d*x)^
(1/2) - 1)^2*(11*d^4*e^3*f^8 - 9*d^6*e^5*f^6))/(d*f^4*((d*x + 1)^(1/2) - 1)^2)))/(f^2*(f + d*e)^(1/2)*(f - d*e
)^(1/2))))/(f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2))))/(f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2)))*1i)/(f^2*(f + d*e)^(
1/2)*(f - d*e)^(1/2)))/((131072*C^4*e^7)/(d*f^4) + (C*e^2*((4096*(32*C^3*e^5*f^3 + 24*C^3*d^2*e^7*f))/(d*f^4)
- (4096*((1 - d*x)^(1/2) - 1)^2*(32*C^3*e^5*f^3 - 96*C^3*d^2*e^7*f))/(d*f^4*((d*x + 1)^(1/2) - 1)^2) + (458752
*C^3*e^6*((1 - d*x)^(1/2) - 1))/(f^2*((d*x + 1)^(1/2) - 1)) + (C*e^2*((4096*(16*C^2*e^3*f^6 + 9*C^2*d^4*e^7*f^
2))/(d*f^4) + (16384*((1 - d*x)^(1/2) - 1)*(8*C^2*e^4*f^3 + 3*C^2*d^2*e^6*f))/(f^2*((d*x + 1)^(1/2) - 1)) + (4
096*((1 - d*x)^(1/2) - 1)^2*(128*C^2*d^2*e^5*f^4 - 144*C^2*e^3*f^6 + 9*C^2*d^4*e^7*f^2))/(d*f^4*((d*x + 1)^(1/
2) - 1)^2) - (C*e^2*((4096*(24*C*d^2*e^3*f^7 - 30*C*d^4*e^5*f^5))/(d*f^4) + (16384*((1 - d*x)^(1/2) - 1)*(20*C
*e^2*f^6 - 22*C*d^2*e^4*f^4))/(f^2*((d*x + 1)^(1/2) - 1)) + (4096*(96*C*d^2*e^3*f^7 - 90*C*d^4*e^5*f^5)*((1 -
d*x)^(1/2) - 1)^2)/(d*f^4*((d*x + 1)^(1/2) - 1)^2) + (C*e^2*((4096*(7*d^4*e^3*f^8 - 9*d^6*e^5*f^6))/(d*f^4) +
(16384*((1 - d*x)^(1/2) - 1)*(5*d^2*e^2*f^7 - 6*d^4*e^4*f^5))/(f^2*((d*x + 1)^(1/2) - 1)) + (4096*((1 - d*x)^(
1/2) - 1)^2*(11*d^4*e^3*f^8 - 9*d^6*e^5*f^6))/(d*f^4*((d*x + 1)^(1/2) - 1)^2)))/(f^2*(f + d*e)^(1/2)*(f - d*e)
^(1/2))))/(f^2*(f + d*e)^(1/2)*(f - d*e)^(1/2))...

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