∫ A+Bx+Cx2 _____________________________________

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

Optimal. Leaf size=278 \[ \frac {\sqrt {a^2 c-b^2 c x^2} \left (A f^2-B e f+C e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \left (a^2 f+b^2 e x\right )}{\sqrt {a^2 c-b^2 c x^2} \sqrt {b^2 e^2-a^2 f^2}}\right )}{\sqrt {c} f^2 \sqrt {a+b x} \sqrt {a c-b c x} \sqrt {b^2 e^2-a^2 f^2}}-\frac {\sqrt {a^2 c-b^2 c x^2} (C e-B f) \tan ^{-1}\left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{b \sqrt {c} f^2 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C \left (a^2-b^2 x^2\right )}{b^2 f \sqrt {a+b x} \sqrt {a c-b c x}} \]

[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)

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Rubi [A] time = 0.49, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1610, 1654, 844, 217, 203, 725, 204} \[ \frac {\sqrt {a^2 c-b^2 c x^2} \left (A f^2-B e f+C e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \left (a^2 f+b^2 e x\right )}{\sqrt {a^2 c-b^2 c x^2} \sqrt {b^2 e^2-a^2 f^2}}\right )}{\sqrt {c} f^2 \sqrt {a+b x} \sqrt {a c-b c x} \sqrt {b^2 e^2-a^2 f^2}}-\frac {\sqrt {a^2 c-b^2 c x^2} (C e-B f) \tan ^{-1}\left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{b \sqrt {c} f^2 \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {C \left (a^2-b^2 x^2\right )}{b^2 f \sqrt {a+b x} \sqrt {a c-b c x}} \]

Antiderivative was successfully veriﬁed.

[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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;

FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /

; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

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 725

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 844

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 1610

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 1654

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 {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx &=\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 ) \operatorname {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 ) \operatorname {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*}

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Mathematica [A] time = 0.77, size = 225, normalized size = 0.81 \[ \frac {\sqrt {a-b x} \left (\frac {2 \left (f (A f-B e)+C e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b x} \sqrt {b e-a f}}{\sqrt {a+b x} \sqrt {-a f-b e}}\right )}{\sqrt {-a f-b e} \sqrt {b e-a f}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {a+b x}}\right ) (a C f-b B f+b C e)}{b^2}+\frac {C f \sqrt {a+b x} \left (-\sqrt {a-b x}-\frac {2 \sqrt {a} \sin ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {\frac {b x}{a}+1}}\right )}{b^2}\right )}{f^2 \sqrt {c (a-b x)}} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[a - b*x]*((C*f*Sqrt[a + b*x]*(-Sqrt[a - b*x] - (2*Sqrt[a]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])])/Sqrt[

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

e) + A*f))*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[a - b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])])/(Sqrt[-(b*e) - a*f]*Sq

rt[b*e - a*f])))/(f^2*Sqrt[c*(a - b*x)])

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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((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

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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((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]

Timed out

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maple [B] time = 0.07, size = 503, normalized size = 1.81 \[ \frac {\left (-\sqrt {b^{2} c}\, A \,b^{2} c \,f^{2} \ln \left (\frac {2 b^{2} c e x +2 a^{2} c f +2 \sqrt {\frac {\left (a^{2} f^{2}-b^{2} e^{2}\right ) c}{f^{2}}}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, f}{f x +e}\right )+\sqrt {b^{2} c}\, B \,b^{2} c e f \ln \left (\frac {2 b^{2} c e x +2 a^{2} c f +2 \sqrt {\frac {\left (a^{2} f^{2}-b^{2} e^{2}\right ) c}{f^{2}}}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, f}{f x +e}\right )+\sqrt {\frac {\left (a^{2} f^{2}-b^{2} e^{2}\right ) c}{f^{2}}}\, B \,b^{2} c \,f^{2} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )-\sqrt {b^{2} c}\, C \,b^{2} c \,e^{2} \ln \left (\frac {2 b^{2} c e x +2 a^{2} c f +2 \sqrt {\frac {\left (a^{2} f^{2}-b^{2} e^{2}\right ) c}{f^{2}}}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, f}{f x +e}\right )-\sqrt {\frac {\left (a^{2} f^{2}-b^{2} e^{2}\right ) c}{f^{2}}}\, C \,b^{2} c e f \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}}\right )-\sqrt {b^{2} c}\, \sqrt {\frac {\left (a^{2} f^{2}-b^{2} e^{2}\right ) c}{f^{2}}}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, C \,f^{2}\right ) \sqrt {b x +a}\, \sqrt {-\left (b x -a \right ) c}}{\sqrt {\frac {\left (a^{2} f^{2}-b^{2} e^{2}\right ) c}{f^{2}}}\, \sqrt {b^{2} c}\, \sqrt {-\left (b^{2} x^{2}-a^{2}\right ) c}\, b^{2} c \,f^{3}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

*(b^2*c)^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2))*(b*x+a)^(1/2)*(-(b*x-a)*c)^(1/2)/(c*(

a^2*f^2-b^2*e^2)/f^2)^(1/2)/f^3/(b^2*c)^(1/2)/b^2/c/(-(b^2*x^2-a^2)*c)^(1/2)

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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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 details)Is (4*b^2*c *(a^2*c-(b^2*c*e^2) /f^2)) /f^2 +(4*b^4*c

^2*e^2)/f^4 zero or nonzero?

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mupad [B] time = 44.56, size = 9298, normalized size = 33.45 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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^