3.1.4 \(\int \frac {A+B x}{(a+b x+c x^2) (d+f x^2)} \, dx\) [4]

Optimal. Leaf size=274 \[ \frac {\sqrt {f} (b B d-A c d+a A f) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {d}}\right )}{\sqrt {d} \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}-\frac {\left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}+\frac {(B c d+A b f-a B f) \log \left (a+b x+c x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}-\frac {(B c d+A b f-a B f) \log \left (d+f x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )} \]

[Out]

1/2*(A*b*f-B*a*f+B*c*d)*ln(c*x^2+b*x+a)/(c^2*d^2-2*a*c*d*f+f*(a^2*f+b^2*d))-1/2*(A*b*f-B*a*f+B*c*d)*ln(f*x^2+d
)/(c^2*d^2-2*a*c*d*f+f*(a^2*f+b^2*d))-(A*b^2*f+2*A*c*(-a*f+c*d)-b*B*(a*f+c*d))*arctanh((2*c*x+b)/(-4*a*c+b^2)^
(1/2))/(c^2*d^2-2*a*c*d*f+f*(a^2*f+b^2*d))/(-4*a*c+b^2)^(1/2)+(A*a*f-A*c*d+B*b*d)*arctan(x*f^(1/2)/d^(1/2))*f^
(1/2)/(c^2*d^2-2*a*c*d*f+f*(a^2*f+b^2*d))/d^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1037, 648, 632, 212, 642, 649, 211, 266} \begin {gather*} \frac {\sqrt {f} \text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {d}}\right ) (a A f-A c d+b B d)}{\sqrt {d} \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )}+\frac {\log \left (a+b x+c x^2\right ) (-a B f+A b f+B c d)}{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )}-\frac {\log \left (d+f x^2\right ) (-a B f+A b f+B c d)}{2 \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )}-\frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )}{\sqrt {b^2-4 a c} \left (f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((a + b*x + c*x^2)*(d + f*x^2)),x]

[Out]

(Sqrt[f]*(b*B*d - A*c*d + a*A*f)*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/(Sqrt[d]*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f
))) - ((A*b^2*f + 2*A*c*(c*d - a*f) - b*B*(c*d + a*f))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a
*c]*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))) + ((B*c*d + A*b*f - a*B*f)*Log[a + b*x + c*x^2])/(2*(c^2*d^2 -
2*a*c*d*f + f*(b^2*d + a^2*f))) - ((B*c*d + A*b*f - a*B*f)*Log[d + f*x^2])/(2*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d
+ a^2*f)))

Rule 211

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

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 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 649

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

Rule 1037

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (f_.)*(x_)^2)), x_Symbol] :> With[{q = Si
mplify[c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2]}, Dist[1/q, Int[Simp[g*c^2*d + g*b^2*f - a*b*h*f - a*g*c*f + c
*(h*c*d + g*b*f - a*h*f)*x, x]/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[Simp[b*h*d*f - g*c*d*f + a*g*f^2 - f*
(h*c*d + g*b*f - a*h*f)*x, x]/(d + f*x^2), x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[b^2
- 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {A+B x}{\left (a+b x+c x^2\right ) \left (d+f x^2\right )} \, dx &=\frac {\int \frac {-a b B f+A \left (c^2 d+b^2 f-a c f\right )+c (B c d+A b f-a B f) x}{a+b x+c x^2} \, dx}{c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )}+\frac {\int \frac {f (b B d-A c d+a A f)-f (B c d+A b f-a B f) x}{d+f x^2} \, dx}{c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )}\\ &=\frac {(f (b B d-A c d+a A f)) \int \frac {1}{d+f x^2} \, dx}{c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )}+\frac {(B c d+A b f-a B f) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}-\frac {(f (B c d+A b f-a B f)) \int \frac {x}{d+f x^2} \, dx}{c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )}+\frac {\left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}\\ &=\frac {\sqrt {f} (b B d-A c d+a A f) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {d}}\right )}{\sqrt {d} \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}+\frac {(B c d+A b f-a B f) \log \left (a+b x+c x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}-\frac {(B c d+A b f-a B f) \log \left (d+f x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}-\frac {\left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )}\\ &=\frac {\sqrt {f} (b B d-A c d+a A f) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {d}}\right )}{\sqrt {d} \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}-\frac {\left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}+\frac {(B c d+A b f-a B f) \log \left (a+b x+c x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}-\frac {(B c d+A b f-a B f) \log \left (d+f x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )}\\ \end {align*}

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Mathematica [A]
time = 0.26, size = 212, normalized size = 0.77 \begin {gather*} \frac {2 \sqrt {-b^2+4 a c} \sqrt {f} (b B d-A c d+a A f) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {d}}\right )+\sqrt {d} \left (2 \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )+\sqrt {-b^2+4 a c} (B c d+A b f-a B f) \left (-\log \left (d+f x^2\right )+\log (a+x (b+c x))\right )\right )}{2 \sqrt {-b^2+4 a c} \sqrt {d} \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((a + b*x + c*x^2)*(d + f*x^2)),x]

[Out]

(2*Sqrt[-b^2 + 4*a*c]*Sqrt[f]*(b*B*d - A*c*d + a*A*f)*ArcTan[(Sqrt[f]*x)/Sqrt[d]] + Sqrt[d]*(2*(A*b^2*f + 2*A*
c*(c*d - a*f) - b*B*(c*d + a*f))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*(B*c*d + A*b*f -
a*B*f)*(-Log[d + f*x^2] + Log[a + x*(b + c*x)])))/(2*Sqrt[-b^2 + 4*a*c]*Sqrt[d]*(c^2*d^2 - 2*a*c*d*f + f*(b^2*
d + a^2*f)))

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Maple [A]
time = 0.31, size = 239, normalized size = 0.87

method	result	size
default	\(\frac {f \left (\frac {\left (-A b f +B a f -B c d \right ) \ln \left (f \,x^{2}+d \right )}{2 f}+\frac {\left (A a f -A c d +B b d \right ) \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f}}\right )}{a^{2} f^{2}-2 a c d f +b^{2} d f +c^{2} d^{2}}+\frac {\frac {\left (A b c f -B a c f +B \,c^{2} d \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-A a c f +A \,b^{2} f +A \,c^{2} d -B a b f -\frac {\left (A b c f -B a c f +B \,c^{2} d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{a^{2} f^{2}-2 a c d f +b^{2} d f +c^{2} d^{2}}\)	\(239\)
risch	\(\text {Expression too large to display}\)	\(239016\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(c*x^2+b*x+a)/(f*x^2+d),x,method=_RETURNVERBOSE)

[Out]

f/(a^2*f^2-2*a*c*d*f+b^2*d*f+c^2*d^2)*(1/2*(-A*b*f+B*a*f-B*c*d)/f*ln(f*x^2+d)+(A*a*f-A*c*d+B*b*d)/(d*f)^(1/2)*
arctan(f*x/(d*f)^(1/2)))+1/(a^2*f^2-2*a*c*d*f+b^2*d*f+c^2*d^2)*(1/2*(A*b*c*f-B*a*c*f+B*c^2*d)/c*ln(c*x^2+b*x+a
)+2*(-A*a*c*f+A*b^2*f+A*c^2*d-B*a*b*f-1/2*(A*b*c*f-B*a*c*f+B*c^2*d)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4
*a*c-b^2)^(1/2)))

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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((B*x+A)/(c*x^2+b*x+a)/(f*x^2+d),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*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x**2+b*x+a)/(f*x**2+d),x)

[Out]

Timed out

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Giac [A]
time = 5.05, size = 266, normalized size = 0.97 \begin {gather*} \frac {{\left (B c d - B a f + A b f\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (c^{2} d^{2} + b^{2} d f - 2 \, a c d f + a^{2} f^{2}\right )}} - \frac {{\left (B c d - B a f + A b f\right )} \log \left (f x^{2} + d\right )}{2 \, {\left (c^{2} d^{2} + b^{2} d f - 2 \, a c d f + a^{2} f^{2}\right )}} + \frac {{\left (B b d f - A c d f + A a f^{2}\right )} \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{{\left (c^{2} d^{2} + b^{2} d f - 2 \, a c d f + a^{2} f^{2}\right )} \sqrt {d f}} - \frac {{\left (B b c d - 2 \, A c^{2} d + B a b f - A b^{2} f + 2 \, A a c f\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (c^{2} d^{2} + b^{2} d f - 2 \, a c d f + a^{2} f^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(B*c*d - B*a*f + A*b*f)*log(c*x^2 + b*x + a)/(c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2) - 1/2*(B*c*d - B*a*
f + A*b*f)*log(f*x^2 + d)/(c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2) + (B*b*d*f - A*c*d*f + A*a*f^2)*arctan(f*x
/sqrt(d*f))/((c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2)*sqrt(d*f)) - (B*b*c*d - 2*A*c^2*d + B*a*b*f - A*b^2*f +
2*A*a*c*f)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2)*sqrt(-b^2 + 4*a*
c))

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Mupad [B]
time = 38.32, size = 2500, normalized size = 9.12 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((d + f*x^2)*(a + b*x + c*x^2)),x)

[Out]

(log(B^3*c^2*f^2*x + (((B*c*d^2)/2 + (A*b*d*f)/2 - (B*a*d*f)/2 - (A*a*f*(-d*f)^(1/2))/2 + (A*c*d*(-d*f)^(1/2))
/2 - (B*b*d*(-d*f)^(1/2))/2)*((((B*c*d^2)/2 + (A*b*d*f)/2 - (B*a*d*f)/2 - (A*a*f*(-d*f)^(1/2))/2 + (A*c*d*(-d*
f)^(1/2))/2 - (B*b*d*(-d*f)^(1/2))/2)*(4*A*a^2*c^2*f^4 + 4*A*c^4*d^2*f^2 - c*f^2*x*(3*A*b^3*f^2 + 4*B*c^3*d^2
- B*a*b^2*f^2 + 4*B*a^2*c*f^2 - 12*A*a*b*c*f^2 + 12*A*b*c^2*d*f - 8*B*a*c^2*d*f - 3*B*b^2*c*d*f) - 3*A*b^2*c^2
*d*f^3 - 4*B*b*c^3*d^2*f^2 - A*a*b^2*c*f^4 - 8*A*a*c^3*d*f^3 + B*b^3*c*d*f^3 + (2*c*f^2*((B*c*d^2)/2 + (A*b*d*
f)/2 - (B*a*d*f)/2 - (A*a*f*(-d*f)^(1/2))/2 + (A*c*d*(-d*f)^(1/2))/2 - (B*b*d*(-d*f)^(1/2))/2)*(2*b*c^3*d^3 +
4*c^4*d^3*x - a^2*b^2*f^3*x + 4*a*b^3*d*f^2 - 2*b^3*c*d^2*f + 4*a^3*c*f^3*x + 3*b^4*d*f^2*x + 12*a*b*c^2*d^2*f
- 14*a^2*b*c*d*f^2 - 4*a*c^3*d^2*f*x - 4*a^2*c^2*d*f^2*x + 3*b^2*c^2*d^2*f*x - 10*a*b^2*c*d*f^2*x))/(d*(a^2*f
^2 + c^2*d^2 + b^2*d*f - 2*a*c*d*f)) + 4*B*a*b*c^2*d*f^3))/(d*(a^2*f^2 + c^2*d^2 + b^2*d*f - 2*a*c*d*f)) - c*f
^2*x*(2*B^2*c^2*d - 4*A^2*c^2*f - B^2*b^2*f + 2*B^2*a*c*f + 2*A*B*b*c*f) + A^2*b*c^2*f^3 + B^2*b*c^2*d*f^2 - 4
*A*B*a*c^2*f^3 + A*B*b^2*c*f^3 - 4*A*B*c^3*d*f^2))/(d*(a^2*f^2 + c^2*d^2 + b^2*d*f - 2*a*c*d*f)) + A*B^2*c^2*f
^2)*(f*((B*a*d)/2 - (A*b*d)/2 + (A*a*(-d*f)^(1/2))/2) - (B*c*d^2)/2 - (A*c*d*(-d*f)^(1/2))/2 + (B*b*d*(-d*f)^(
1/2))/2))/(c^2*d^3 + a^2*d*f^2 + b^2*d^2*f - 2*a*c*d^2*f) - (log(B^3*c^2*f^2*x + (((B*c*d^2)/2 + (A*b*d*f)/2 -
(B*a*d*f)/2 + (A*a*f*(-d*f)^(1/2))/2 - (A*c*d*(-d*f)^(1/2))/2 + (B*b*d*(-d*f)^(1/2))/2)*((((B*c*d^2)/2 + (A*b
*d*f)/2 - (B*a*d*f)/2 + (A*a*f*(-d*f)^(1/2))/2 - (A*c*d*(-d*f)^(1/2))/2 + (B*b*d*(-d*f)^(1/2))/2)*(4*A*a^2*c^2
*f^4 + 4*A*c^4*d^2*f^2 - c*f^2*x*(3*A*b^3*f^2 + 4*B*c^3*d^2 - B*a*b^2*f^2 + 4*B*a^2*c*f^2 - 12*A*a*b*c*f^2 + 1
2*A*b*c^2*d*f - 8*B*a*c^2*d*f - 3*B*b^2*c*d*f) - 3*A*b^2*c^2*d*f^3 - 4*B*b*c^3*d^2*f^2 - A*a*b^2*c*f^4 - 8*A*a
*c^3*d*f^3 + B*b^3*c*d*f^3 + (2*c*f^2*((B*c*d^2)/2 + (A*b*d*f)/2 - (B*a*d*f)/2 + (A*a*f*(-d*f)^(1/2))/2 - (A*c
*d*(-d*f)^(1/2))/2 + (B*b*d*(-d*f)^(1/2))/2)*(2*b*c^3*d^3 + 4*c^4*d^3*x - a^2*b^2*f^3*x + 4*a*b^3*d*f^2 - 2*b^
3*c*d^2*f + 4*a^3*c*f^3*x + 3*b^4*d*f^2*x + 12*a*b*c^2*d^2*f - 14*a^2*b*c*d*f^2 - 4*a*c^3*d^2*f*x - 4*a^2*c^2*
d*f^2*x + 3*b^2*c^2*d^2*f*x - 10*a*b^2*c*d*f^2*x))/(d*(a^2*f^2 + c^2*d^2 + b^2*d*f - 2*a*c*d*f)) + 4*B*a*b*c^2
*d*f^3))/(d*(a^2*f^2 + c^2*d^2 + b^2*d*f - 2*a*c*d*f)) - c*f^2*x*(2*B^2*c^2*d - 4*A^2*c^2*f - B^2*b^2*f + 2*B^
2*a*c*f + 2*A*B*b*c*f) + A^2*b*c^2*f^3 + B^2*b*c^2*d*f^2 - 4*A*B*a*c^2*f^3 + A*B*b^2*c*f^3 - 4*A*B*c^3*d*f^2))
/(d*(a^2*f^2 + c^2*d^2 + b^2*d*f - 2*a*c*d*f)) + A*B^2*c^2*f^2)*(f*((A*b*d)/2 - (B*a*d)/2 + (A*a*(-d*f)^(1/2))
/2) + (B*c*d^2)/2 - (A*c*d*(-d*f)^(1/2))/2 + (B*b*d*(-d*f)^(1/2))/2))/(c^2*d^3 + a^2*d*f^2 + b^2*d^2*f - 2*a*c
*d^2*f) - (log(B^3*c^2*f^2*x + A*B^2*c^2*f^2 - ((((A*f*(b^2 - 4*a*c)^(3/2) + 2*A*b*f*(4*a*c - b^2) - 2*B*a*f*(
4*a*c - b^2) + 2*B*c*d*(4*a*c - b^2) + 4*A*c^2*d*(b^2 - 4*a*c)^(1/2) + A*b^2*f*(b^2 - 4*a*c)^(1/2) - 2*B*a*b*f
*(b^2 - 4*a*c)^(1/2) - 2*B*b*c*d*(b^2 - 4*a*c)^(1/2))*(c*f^2*x*(3*A*b^3*f^2 + 4*B*c^3*d^2 - B*a*b^2*f^2 + 4*B*
a^2*c*f^2 - 12*A*a*b*c*f^2 + 12*A*b*c^2*d*f - 8*B*a*c^2*d*f - 3*B*b^2*c*d*f) - 4*A*c^4*d^2*f^2 - 4*A*a^2*c^2*f
^4 + 3*A*b^2*c^2*d*f^3 + 4*B*b*c^3*d^2*f^2 + A*a*b^2*c*f^4 + 8*A*a*c^3*d*f^3 - B*b^3*c*d*f^3 - 4*B*a*b*c^2*d*f
^3 + (c*f^2*(A*f*(b^2 - 4*a*c)^(3/2) + 2*A*b*f*(4*a*c - b^2) - 2*B*a*f*(4*a*c - b^2) + 2*B*c*d*(4*a*c - b^2) +
4*A*c^2*d*(b^2 - 4*a*c)^(1/2) + A*b^2*f*(b^2 - 4*a*c)^(1/2) - 2*B*a*b*f*(b^2 - 4*a*c)^(1/2) - 2*B*b*c*d*(b^2
- 4*a*c)^(1/2))*(2*b*c^3*d^3 + 4*c^4*d^3*x - a^2*b^2*f^3*x + 4*a*b^3*d*f^2 - 2*b^3*c*d^2*f + 4*a^3*c*f^3*x + 3
*b^4*d*f^2*x + 12*a*b*c^2*d^2*f - 14*a^2*b*c*d*f^2 - 4*a*c^3*d^2*f*x - 4*a^2*c^2*d*f^2*x + 3*b^2*c^2*d^2*f*x -
10*a*b^2*c*d*f^2*x))/(2*(4*a*c - b^2)*(a^2*f^2 + c^2*d^2 + b^2*d*f - 2*a*c*d*f))))/(4*(4*a*c - b^2)*(a^2*f^2
+ c^2*d^2 + b^2*d*f - 2*a*c*d*f)) - c*f^2*x*(2*B^2*c^2*d - 4*A^2*c^2*f - B^2*b^2*f + 2*B^2*a*c*f + 2*A*B*b*c*f
) + A^2*b*c^2*f^3 + B^2*b*c^2*d*f^2 - 4*A*B*a*c^2*f^3 + A*B*b^2*c*f^3 - 4*A*B*c^3*d*f^2)*(A*f*(b^2 - 4*a*c)^(3
/2) + 2*A*b*f*(4*a*c - b^2) - 2*B*a*f*(4*a*c - b^2) + 2*B*c*d*(4*a*c - b^2) + 4*A*c^2*d*(b^2 - 4*a*c)^(1/2) +
A*b^2*f*(b^2 - 4*a*c)^(1/2) - 2*B*a*b*f*(b^2 - 4*a*c)^(1/2) - 2*B*b*c*d*(b^2 - 4*a*c)^(1/2)))/(4*(4*a*c - b^2)
*(a^2*f^2 + c^2*d^2 + b^2*d*f - 2*a*c*d*f)))*(A*f*(b^2 - 4*a*c)^(3/2) + 2*A*b*f*(4*a*c - b^2) - 2*B*a*f*(4*a*c
- b^2) + 2*B*c*d*(4*a*c - b^2) + 4*A*c^2*d*(b^2 - 4*a*c)^(1/2) + A*b^2*f*(b^2 - 4*a*c)^(1/2) - 2*B*a*b*f*(b^2
- 4*a*c)^(1/2) - 2*B*b*c*d*(b^2 - 4*a*c)^(1/2)))/(b^2*(4*a^2*f^2 + 4*c^2*d^2 + 4*b^2*d*f - 24*a*c*d*f) - 4*a*
c*(4*a^2*f^2 + 4*c^2*d^2 - 8*a*c*d*f)) + (log(B^3*c^2*f^2*x + A*B^2*c^2*f^2 + ((((A*f*(b^2 - 4*a*c)^(3/2) - 2*
A*b*f*(4*a*c - b^2) + 2*B*a*f*(4*a*c - b^2) - 2*B*c*d*(4*a*c - b^2) + 4*A*c^2*d*(b^2 - 4*a*c)^(1/2) + A*b^2*f*
(b^2 - 4*a*c)^(1/2) - 2*B*a*b*f*(b^2 - 4*a*c)^(1/2) - 2*B*b*c*d*(b^2 - 4*a*c)^(1/2))*(4*A*a^2*c^2*f^4 + 4*A*c^
4*d^2*f^2 - c*f^2*x*(3*A*b^3*f^2 + 4*B*c^3*d^2 ...

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