3.1.63 \(\int \frac {d+e x^2+f x^4}{x (a+b x^2+c x^4)^2} \, dx\)

Optimal. Leaf size=166 \[ \frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (4 a^2 c e-2 a b (a f+3 c d)+b^3 d\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac {d \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac {d \log (x)}{a^2}+\frac {x^2 (a b f-2 a c e+b c d)-a b e-2 a (c d-a f)+b^2 d}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

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Rubi [A]  time = 0.39, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {1663, 1646, 800, 634, 618, 206, 628} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (4 a^2 c e-2 a b (a f+3 c d)+b^3 d\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac {d \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac {d \log (x)}{a^2}+\frac {x^2 (a b f-2 a c e+b c d)-a b e-2 a (c d-a f)+b^2 d}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2 + f*x^4)/(x*(a + b*x^2 + c*x^4)^2),x]

[Out]

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

Rule 206

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

Rule 618

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 628

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 634

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 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 1646

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1663

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)
*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && Inte
gerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {d+e x^2+f x^4}{x \left (a+b x^2+c x^4\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {d+e x+f x^2}{x \left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {b^2 d-a b e-2 a (c d-a f)+(b c d-2 a c e+a b f) x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\operatorname {Subst}\left (\int \frac {-\left (\left (\frac {b^2}{a}-4 c\right ) d\right )-\frac {(b c d-2 a c e+a b f) x}{a}}{x \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=\frac {b^2 d-a b e-2 a (c d-a f)+(b c d-2 a c e+a b f) x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\operatorname {Subst}\left (\int \left (\frac {\left (-b^2+4 a c\right ) d}{a^2 x}+\frac {b^3 d+2 a^2 c e-a b (5 c d+a f)+c \left (b^2-4 a c\right ) d x}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=\frac {b^2 d-a b e-2 a (c d-a f)+(b c d-2 a c e+a b f) x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {d \log (x)}{a^2}-\frac {\operatorname {Subst}\left (\int \frac {b^3 d+2 a^2 c e-a b (5 c d+a f)+c \left (b^2-4 a c\right ) d x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^2 \left (b^2-4 a c\right )}\\ &=\frac {b^2 d-a b e-2 a (c d-a f)+(b c d-2 a c e+a b f) x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {d \log (x)}{a^2}-\frac {d \operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2}-\frac {\left (b^3 d+4 a^2 c e-2 a b (3 c d+a f)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2 \left (b^2-4 a c\right )}\\ &=\frac {b^2 d-a b e-2 a (c d-a f)+(b c d-2 a c e+a b f) x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {d \log (x)}{a^2}-\frac {d \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac {\left (b^3 d+4 a^2 c e-2 a b (3 c d+a f)\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^2 \left (b^2-4 a c\right )}\\ &=\frac {b^2 d-a b e-2 a (c d-a f)+(b c d-2 a c e+a b f) x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (b^3 d+4 a^2 c e-2 a b (3 c d+a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}+\frac {d \log (x)}{a^2}-\frac {d \log \left (a+b x^2+c x^4\right )}{4 a^2}\\ \end {align*}

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Mathematica [A]  time = 0.45, size = 268, normalized size = 1.61 \begin {gather*} -\frac {-\frac {2 a \left (b \left (-a e+a f x^2+c d x^2\right )+2 a \left (a f-c \left (d+e x^2\right )\right )+b^2 d\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\log \left (-\sqrt {b^2-4 a c}+b+2 c x^2\right ) \left (4 a c \left (a e-d \sqrt {b^2-4 a c}\right )+b^2 d \sqrt {b^2-4 a c}-2 a b (a f+3 c d)+b^3 d\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\log \left (\sqrt {b^2-4 a c}+b+2 c x^2\right ) \left (-4 a c \left (d \sqrt {b^2-4 a c}+a e\right )+b^2 d \sqrt {b^2-4 a c}+2 a b (a f+3 c d)+b^3 (-d)\right )}{\left (b^2-4 a c\right )^{3/2}}-4 d \log (x)}{4 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2 + f*x^4)/(x*(a + b*x^2 + c*x^4)^2),x]

[Out]

-1/4*((-2*a*(b^2*d + b*(-(a*e) + c*d*x^2 + a*f*x^2) + 2*a*(a*f - c*(d + e*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 +
c*x^4)) - 4*d*Log[x] + ((b^3*d + b^2*Sqrt[b^2 - 4*a*c]*d + 4*a*c*(-(Sqrt[b^2 - 4*a*c]*d) + a*e) - 2*a*b*(3*c*d
+ a*f))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((-(b^3*d) + b^2*Sqrt[b^2 - 4*a*c]*d - 4*
a*c*(Sqrt[b^2 - 4*a*c]*d + a*e) + 2*a*b*(3*c*d + a*f))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/
2))/a^2

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x^2+f x^4}{x \left (a+b x^2+c x^4\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x^2 + f*x^4)/(x*(a + b*x^2 + c*x^4)^2),x]

[Out]

IntegrateAlgebraic[(d + e*x^2 + f*x^4)/(x*(a + b*x^2 + c*x^4)^2), x]

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fricas [B]  time = 3.33, size = 1103, normalized size = 6.64

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/4*(2*((a*b^3*c - 4*a^2*b*c^2)*d - 2*(a^2*b^2*c - 4*a^3*c^2)*e + (a^2*b^3 - 4*a^3*b*c)*f)*x^2 + (4*a^3*c*e -
2*a^3*b*f + (4*a^2*c^2*e - 2*a^2*b*c*f + (b^3*c - 6*a*b*c^2)*d)*x^4 + (4*a^2*b*c*e - 2*a^2*b^2*f + (b^4 - 6*a
*b^2*c)*d)*x^2 + (a*b^3 - 6*a^2*b*c)*d)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^2
+ b)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)) + 2*(a*b^4 - 6*a^2*b^2*c + 8*a^3*c^2)*d - 2*(a^2*b^3 - 4*a^3*b*c)
*e + 4*(a^3*b^2 - 4*a^4*c)*f - ((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*x^4 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d*
x^2 + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*d)*log(c*x^4 + b*x^2 + a) + 4*((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*x
^4 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d*x^2 + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*d)*log(x))/(a^3*b^4 - 8*a^4*b
^2*c + 16*a^5*c^2 + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*x^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^2)
, 1/4*(2*((a*b^3*c - 4*a^2*b*c^2)*d - 2*(a^2*b^2*c - 4*a^3*c^2)*e + (a^2*b^3 - 4*a^3*b*c)*f)*x^2 + 2*(4*a^3*c*
e - 2*a^3*b*f + (4*a^2*c^2*e - 2*a^2*b*c*f + (b^3*c - 6*a*b*c^2)*d)*x^4 + (4*a^2*b*c*e - 2*a^2*b^2*f + (b^4 -
6*a*b^2*c)*d)*x^2 + (a*b^3 - 6*a^2*b*c)*d)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 -
4*a*c)) + 2*(a*b^4 - 6*a^2*b^2*c + 8*a^3*c^2)*d - 2*(a^2*b^3 - 4*a^3*b*c)*e + 4*(a^3*b^2 - 4*a^4*c)*f - ((b^4*
c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*x^4 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d*x^2 + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c
^2)*d)*log(c*x^4 + b*x^2 + a) + 4*((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*x^4 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)
*d*x^2 + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*d)*log(x))/(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 + (a^2*b^4*c - 8*a^
3*b^2*c^2 + 16*a^4*c^3)*x^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^2)]

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giac [A]  time = 2.00, size = 227, normalized size = 1.37 \begin {gather*} -\frac {{\left (b^{3} d - 6 \, a b c d - 2 \, a^{2} b f + 4 \, a^{2} c e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {d \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{2}} + \frac {d \log \left (x^{2}\right )}{2 \, a^{2}} + \frac {b^{2} c d x^{4} - 4 \, a c^{2} d x^{4} + b^{3} d x^{2} - 2 \, a b c d x^{2} + 2 \, a^{2} b f x^{2} - 4 \, a^{2} c x^{2} e + 3 \, a b^{2} d - 8 \, a^{2} c d + 4 \, a^{3} f - 2 \, a^{2} b e}{4 \, {\left (c x^{4} + b x^{2} + a\right )} {\left (a^{2} b^{2} - 4 \, a^{3} c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.02, size = 462, normalized size = 2.78 \begin {gather*} -\frac {b c d \,x^{2}}{2 \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right ) a}-\frac {b f \,x^{2}}{2 \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right )}+\frac {c e \,x^{2}}{\left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right )}-\frac {3 b c d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a}+\frac {b^{3} d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \left (4 a c -b^{2}\right )^{\frac {3}{2}} a^{2}}-\frac {b f \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}+\frac {2 c e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}-\frac {a f}{\left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right )}-\frac {b^{2} d}{2 \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right ) a}-\frac {c d \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{\left (4 a c -b^{2}\right ) a}+\frac {b^{2} d \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 \left (4 a c -b^{2}\right ) a^{2}}+\frac {b e}{2 \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right )}+\frac {c d}{\left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right )}+\frac {d \ln \relax (x )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^4+e*x^2+d)/x/(c*x^4+b*x^2+a)^2,x)

[Out]

d*ln(x)/a^2-1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*b*f+c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*e-1/2/a/(c*x^4+b*x^2+a)/
(4*a*c-b^2)*x^2*b*c*d-a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*f+1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b*e+1/(c*x^4+b*x^2+a)/(4
*a*c-b^2)*c*d-1/2/a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b^2*d-1/a/(4*a*c-b^2)*c*ln(c*x^4+b*x^2+a)*d+1/4/a^2/(4*a*c-b^2
)*ln(c*x^4+b*x^2+a)*b^2*d-1/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b*f+2/(4*a*c-b^2)^(3/2)*ar
ctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*c*e-3/a/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b*c*d+1/2/
a^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^3*d

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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((f*x^4+e*x^2+d)/x/(c*x^4+b*x^2+a)^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*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 11.85, size = 8706, normalized size = 52.45

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2 + f*x^4)/(x*(a + b*x^2 + c*x^4)^2),x)

[Out]

(d*log(x))/a^2 - ((b^2*d + 2*a^2*f - a*b*e - 2*a*c*d)/(2*a*(4*a*c - b^2)) + (x^2*(a*b*f - 2*a*c*e + b*c*d))/(2
*a*(4*a*c - b^2)))/(a + b*x^2 + c*x^4) - (log((((d + a^2*(-(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d)^2/(a^4*
(4*a*c - b^2)^3))^(1/2))*(((d + a^2*(-(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d)^2/(a^4*(4*a*c - b^2)^3))^(1/
2))*((2*c^2*x^2*(20*a^2*c^2*e + 4*a*b^3*f - b^3*c*d + 10*a*b*c^2*d - 8*a*b^2*c*e - 10*a^2*b*c*f))/(a*(4*a*c -
b^2)) + (b*c^2*(d + a^2*(-(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d)^2/(a^4*(4*a*c - b^2)^3))^(1/2))*(a*b + 3
*b^2*x^2 - 10*a*c*x^2))/a^2 - (4*b*c^2*(b^3*d - a^2*b*f + 2*a^2*c*e - 5*a*b*c*d))/(a*(4*a*c - b^2))))/(4*a^2)
+ (c^2*(a^3*b^2*f^2 - 4*b^4*c*d^2 + 4*a^3*c^2*e^2 + 17*a*b^2*c^2*d^2 - 4*a*b^4*d*f - 36*a^2*b*c^2*d*e + 18*a^2
*b^2*c*d*f + 8*a*b^3*c*d*e - 4*a^3*b*c*e*f))/(a^2*(4*a*c - b^2)^2) - (c^2*x^2*(a^2*b^3*f^2 + 6*b^3*c^2*d^2 + 4
*a^2*b*c^2*e^2 - 20*a*b*c^3*d^2 + 40*a^2*c^3*d*e - 14*a*b^2*c^2*d*e - 20*a^2*b*c^2*d*f - 4*a^2*b^2*c*e*f + 7*a
*b^3*c*d*f))/(a^2*(4*a*c - b^2)^2)))/(4*a^2) - (c^2*x^2*(a*b*f - 2*a*c*e + b*c*d)^3)/(a^3*(4*a*c - b^2)^3) + (
c^2*d*(a*b*f - 2*a*c*e + b*c*d)^2)/(a^3*(4*a*c - b^2)^2))*(((d - a^2*(-(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*
c*d)^2/(a^4*(4*a*c - b^2)^3))^(1/2))*(((d - a^2*(-(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d)^2/(a^4*(4*a*c -
b^2)^3))^(1/2))*((2*c^2*x^2*(20*a^2*c^2*e + 4*a*b^3*f - b^3*c*d + 10*a*b*c^2*d - 8*a*b^2*c*e - 10*a^2*b*c*f))/
(a*(4*a*c - b^2)) + (b*c^2*(d - a^2*(-(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d)^2/(a^4*(4*a*c - b^2)^3))^(1/
2))*(a*b + 3*b^2*x^2 - 10*a*c*x^2))/a^2 - (4*b*c^2*(b^3*d - a^2*b*f + 2*a^2*c*e - 5*a*b*c*d))/(a*(4*a*c - b^2)
)))/(4*a^2) + (c^2*(a^3*b^2*f^2 - 4*b^4*c*d^2 + 4*a^3*c^2*e^2 + 17*a*b^2*c^2*d^2 - 4*a*b^4*d*f - 36*a^2*b*c^2*
d*e + 18*a^2*b^2*c*d*f + 8*a*b^3*c*d*e - 4*a^3*b*c*e*f))/(a^2*(4*a*c - b^2)^2) - (c^2*x^2*(a^2*b^3*f^2 + 6*b^3
*c^2*d^2 + 4*a^2*b*c^2*e^2 - 20*a*b*c^3*d^2 + 40*a^2*c^3*d*e - 14*a*b^2*c^2*d*e - 20*a^2*b*c^2*d*f - 4*a^2*b^2
*c*e*f + 7*a*b^3*c*d*f))/(a^2*(4*a*c - b^2)^2)))/(4*a^2) - (c^2*x^2*(a*b*f - 2*a*c*e + b*c*d)^3)/(a^3*(4*a*c -
b^2)^3) + (c^2*d*(a*b*f - 2*a*c*e + b*c*d)^2)/(a^3*(4*a*c - b^2)^2)))*(2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c
^2*d - 24*a*b^4*c*d))/(2*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)) - (atan((x^2*((((b^3*c^5*
d^3 - 8*a^3*c^5*e^3 + a^3*b^3*c^2*f^3 - 6*a*b^2*c^5*d^2*e + 12*a^2*b*c^5*d*e^2 + 3*a*b^3*c^4*d^2*f + 12*a^3*b*
c^4*e^2*f + 3*a^2*b^3*c^3*d*f^2 - 6*a^3*b^2*c^3*e*f^2 - 12*a^2*b^2*c^4*d*e*f)/(a^3*b^6 - 64*a^6*c^3 - 12*a^4*b
^4*c + 48*a^5*b^2*c^2) - (((6*a*b^5*c^4*d^2 + 80*a^3*b*c^6*d^2 - 16*a^4*b*c^5*e^2 - 44*a^2*b^3*c^5*d^2 + 4*a^3
*b^3*c^4*e^2 + a^3*b^5*c^2*f^2 - 4*a^4*b^3*c^3*f^2 - 160*a^4*c^6*d*e + 80*a^4*b*c^5*d*f - 14*a^2*b^4*c^4*d*e +
96*a^3*b^2*c^5*d*e + 7*a^2*b^5*c^3*d*f - 48*a^3*b^3*c^4*d*f - 4*a^3*b^4*c^3*e*f + 16*a^4*b^2*c^4*e*f)/(a^3*b^
6 - 64*a^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2*c^2) + (((640*a^6*c^6*e - 2*a^2*b^7*c^3*d + 36*a^3*b^5*c^4*d - 192*
a^4*b^3*c^5*d - 16*a^3*b^6*c^3*e + 168*a^4*b^4*c^4*e - 576*a^5*b^2*c^5*e + 8*a^3*b^7*c^2*f - 84*a^4*b^5*c^3*f
+ 288*a^5*b^3*c^4*f + 320*a^5*b*c^6*d - 320*a^6*b*c^5*f)/(a^3*b^6 - 64*a^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2*c^2
) - ((2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^2*d - 24*a*b^4*c*d)*(2560*a^7*b*c^6 + 12*a^3*b^9*c^2 - 184*a^4*b^
7*c^3 + 1056*a^5*b^5*c^4 - 2688*a^6*b^3*c^5))/(2*(a^3*b^6 - 64*a^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2*c^2)*(4*a^2
*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)))*(2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^2*d - 24*a*b^4*
c*d))/(2*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)))*(2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^
2*d - 24*a*b^4*c*d))/(2*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)) + (((((640*a^6*c^6*e - 2*a
^2*b^7*c^3*d + 36*a^3*b^5*c^4*d - 192*a^4*b^3*c^5*d - 16*a^3*b^6*c^3*e + 168*a^4*b^4*c^4*e - 576*a^5*b^2*c^5*e
+ 8*a^3*b^7*c^2*f - 84*a^4*b^5*c^3*f + 288*a^5*b^3*c^4*f + 320*a^5*b*c^6*d - 320*a^6*b*c^5*f)/(a^3*b^6 - 64*a
^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2*c^2) - ((2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^2*d - 24*a*b^4*c*d)*(2560*a
^7*b*c^6 + 12*a^3*b^9*c^2 - 184*a^4*b^7*c^3 + 1056*a^5*b^5*c^4 - 2688*a^6*b^3*c^5))/(2*(a^3*b^6 - 64*a^6*c^3 -
12*a^4*b^4*c + 48*a^5*b^2*c^2)*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)))*(b^3*d - 2*a^2*b*
f + 4*a^2*c*e - 6*a*b*c*d))/(4*a^2*(4*a*c - b^2)^(3/2)) - ((2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^2*d - 24*a*
b^4*c*d)*(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d)*(2560*a^7*b*c^6 + 12*a^3*b^9*c^2 - 184*a^4*b^7*c^3 + 1056
*a^5*b^5*c^4 - 2688*a^6*b^3*c^5))/(8*a^2*(4*a*c - b^2)^(3/2)*(a^3*b^6 - 64*a^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2
*c^2)*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)))*(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d)
)/(4*a^2*(4*a*c - b^2)^(3/2)) - ((2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^2*d - 24*a*b^4*c*d)*(b^3*d - 2*a^2*b*
f + 4*a^2*c*e - 6*a*b*c*d)^2*(2560*a^7*b*c^6 + 12*a^3*b^9*c^2 - 184*a^4*b^7*c^3 + 1056*a^5*b^5*c^4 - 2688*a^6*
b^3*c^5))/(32*a^4*(4*a*c - b^2)^3*(a^3*b^6 - 64*a^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2*c^2)*(4*a^2*b^6 - 256*a^5*
c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)))*(3*b^5*d - a^2*b^3*f - 2*a^3*c^2*e - 21*a*b^3*c*d + a^3*b*c*f + 33*a^2
*b*c^2*d + 2*a^2*b^2*c*e))/(8*a^3*c^2*(4*a*c - b^2)^3*(400*a^3*c^3*d^2 - 6*b^6*d^2 + a^4*b^2*f^2 + 4*a^4*c^2*e
^2 - 291*a^2*b^2*c^2*d^2 + 72*a*b^4*c*d^2 - a^2*b^4*d*f + 2*a^2*b^3*c*d*e - 12*a^3*b*c^2*d*e + 6*a^3*b^2*c*d*f
- 4*a^4*b*c*e*f)) + (((((((640*a^6*c^6*e - 2*a^2*b^7*c^3*d + 36*a^3*b^5*c^4*d - 192*a^4*b^3*c^5*d - 16*a^3*b^
6*c^3*e + 168*a^4*b^4*c^4*e - 576*a^5*b^2*c^5*e + 8*a^3*b^7*c^2*f - 84*a^4*b^5*c^3*f + 288*a^5*b^3*c^4*f + 320
*a^5*b*c^6*d - 320*a^6*b*c^5*f)/(a^3*b^6 - 64*a^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2*c^2) - ((2*b^6*d - 128*a^3*c
^3*d + 96*a^2*b^2*c^2*d - 24*a*b^4*c*d)*(2560*a^7*b*c^6 + 12*a^3*b^9*c^2 - 184*a^4*b^7*c^3 + 1056*a^5*b^5*c^4
- 2688*a^6*b^3*c^5))/(2*(a^3*b^6 - 64*a^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2*c^2)*(4*a^2*b^6 - 256*a^5*c^3 - 48*a
^3*b^4*c + 192*a^4*b^2*c^2)))*(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d))/(4*a^2*(4*a*c - b^2)^(3/2)) - ((2*b
^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^2*d - 24*a*b^4*c*d)*(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d)*(2560*a^7*
b*c^6 + 12*a^3*b^9*c^2 - 184*a^4*b^7*c^3 + 1056*a^5*b^5*c^4 - 2688*a^6*b^3*c^5))/(8*a^2*(4*a*c - b^2)^(3/2)*(a
^3*b^6 - 64*a^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2*c^2)*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2
)))*(2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^2*d - 24*a*b^4*c*d))/(2*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c +
192*a^4*b^2*c^2)) + (((6*a*b^5*c^4*d^2 + 80*a^3*b*c^6*d^2 - 16*a^4*b*c^5*e^2 - 44*a^2*b^3*c^5*d^2 + 4*a^3*b^3*
c^4*e^2 + a^3*b^5*c^2*f^2 - 4*a^4*b^3*c^3*f^2 - 160*a^4*c^6*d*e + 80*a^4*b*c^5*d*f - 14*a^2*b^4*c^4*d*e + 96*a
^3*b^2*c^5*d*e + 7*a^2*b^5*c^3*d*f - 48*a^3*b^3*c^4*d*f - 4*a^3*b^4*c^3*e*f + 16*a^4*b^2*c^4*e*f)/(a^3*b^6 - 6
4*a^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2*c^2) + (((640*a^6*c^6*e - 2*a^2*b^7*c^3*d + 36*a^3*b^5*c^4*d - 192*a^4*b
^3*c^5*d - 16*a^3*b^6*c^3*e + 168*a^4*b^4*c^4*e - 576*a^5*b^2*c^5*e + 8*a^3*b^7*c^2*f - 84*a^4*b^5*c^3*f + 288
*a^5*b^3*c^4*f + 320*a^5*b*c^6*d - 320*a^6*b*c^5*f)/(a^3*b^6 - 64*a^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2*c^2) - (
(2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^2*d - 24*a*b^4*c*d)*(2560*a^7*b*c^6 + 12*a^3*b^9*c^2 - 184*a^4*b^7*c^3
+ 1056*a^5*b^5*c^4 - 2688*a^6*b^3*c^5))/(2*(a^3*b^6 - 64*a^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2*c^2)*(4*a^2*b^6
- 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)))*(2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^2*d - 24*a*b^4*c*d))
/(2*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)))*(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d))/
(4*a^2*(4*a*c - b^2)^(3/2)) + ((b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d)^3*(2560*a^7*b*c^6 + 12*a^3*b^9*c^2
- 184*a^4*b^7*c^3 + 1056*a^5*b^5*c^4 - 2688*a^6*b^3*c^5))/(64*a^6*(4*a*c - b^2)^(9/2)*(a^3*b^6 - 64*a^6*c^3 -
12*a^4*b^4*c + 48*a^5*b^2*c^2)))*(96*b^6*d - 1280*a^3*c^3*d - 32*a^2*b^4*f + 2208*a^2*b^2*c^2*d - 864*a*b^4*c*
d + 64*a^2*b^3*c*e - 192*a^3*b*c^2*e + 96*a^3*b^2*c*f))/(256*a^3*c^2*(4*a*c - b^2)^(7/2)*(400*a^3*c^3*d^2 - 6*
b^6*d^2 + a^4*b^2*f^2 + 4*a^4*c^2*e^2 - 291*a^2*b^2*c^2*d^2 + 72*a*b^4*c*d^2 - a^2*b^4*d*f + 2*a^2*b^3*c*d*e -
12*a^3*b*c^2*d*e + 6*a^3*b^2*c*d*f - 4*a^4*b*c*e*f)))*(16*a^6*b^6*(4*a*c - b^2)^(9/2) - 1024*a^9*c^3*(4*a*c -
b^2)^(9/2) - 192*a^7*b^4*c*(4*a*c - b^2)^(9/2) + 768*a^8*b^2*c^2*(4*a*c - b^2)^(9/2)))/(16*a^4*c^4*e^2 + b^6*
c^2*d^2 - 12*a*b^4*c^3*d^2 + 36*a^2*b^2*c^4*d^2 + 4*a^4*b^2*c^2*f^2 - 48*a^3*b*c^4*d*e - 16*a^4*b*c^3*e*f + 8*
a^2*b^3*c^3*d*e - 4*a^2*b^4*c^2*d*f + 24*a^3*b^2*c^3*d*f) + ((16*a^6*b^6*(4*a*c - b^2)^(9/2) - 1024*a^9*c^3*(4
*a*c - b^2)^(9/2) - 192*a^7*b^4*c*(4*a*c - b^2)^(9/2) + 768*a^8*b^2*c^2*(4*a*c - b^2)^(9/2))*((b^2*c^4*d^3 + 4
*a^2*c^4*d*e^2 - 4*a*b*c^4*d^2*e + 2*a*b^2*c^3*d^2*f + a^2*b^2*c^2*d*f^2 - 4*a^2*b*c^3*d*e*f)/(a^3*b^4 + 16*a^
5*c^2 - 8*a^4*b^2*c) + (((4*a^4*c^4*e^2 - 4*a*b^4*c^3*d^2 + 17*a^2*b^2*c^4*d^2 + a^4*b^2*c^2*f^2 - 36*a^3*b*c^
4*d*e - 4*a^4*b*c^3*e*f + 8*a^2*b^3*c^3*d*e - 4*a^2*b^4*c^2*d*f + 18*a^3*b^2*c^3*d*f)/(a^3*b^4 + 16*a^5*c^2 -
8*a^4*b^2*c) + (((4*a^2*b^6*c^2*d - 36*a^3*b^4*c^3*d + 80*a^4*b^2*c^4*d + 8*a^4*b^3*c^3*e - 4*a^4*b^4*c^2*f +
16*a^5*b^2*c^3*f - 32*a^5*b*c^4*e)/(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2*c) + ((4*a^4*b^6*c^2 - 32*a^5*b^4*c^3 + 6
4*a^6*b^2*c^4)*(2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^2*d - 24*a*b^4*c*d))/(2*(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b
^2*c)*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)))*(2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^2*d
- 24*a*b^4*c*d))/(2*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)))*(2*b^6*d - 128*a^3*c^3*d + 9
6*a^2*b^2*c^2*d - 24*a*b^4*c*d))/(2*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)) - (((((4*a^2*b
^6*c^2*d - 36*a^3*b^4*c^3*d + 80*a^4*b^2*c^4*d + 8*a^4*b^3*c^3*e - 4*a^4*b^4*c^2*f + 16*a^5*b^2*c^3*f - 32*a^5
*b*c^4*e)/(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2*c) + ((4*a^4*b^6*c^2 - 32*a^5*b^4*c^3 + 64*a^6*b^2*c^4)*(2*b^6*d -
128*a^3*c^3*d + 96*a^2*b^2*c^2*d - 24*a*b^4*c*d))/(2*(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2*c)*(4*a^2*b^6 - 256*a^
5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)))*(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d))/(4*a^2*(4*a*c - b^2)^(3
/2)) + ((4*a^4*b^6*c^2 - 32*a^5*b^4*c^3 + 64*a^6*b^2*c^4)*(2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^2*d - 24*a*b
^4*c*d)*(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d))/(8*a^2*(4*a*c - b^2)^(3/2)*(a^3*b^4 + 16*a^5*c^2 - 8*a^4*
b^2*c)*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)))*(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d
))/(4*a^2*(4*a*c - b^2)^(3/2)) - ((4*a^4*b^6*c^2 - 32*a^5*b^4*c^3 + 64*a^6*b^2*c^4)*(2*b^6*d - 128*a^3*c^3*d +
96*a^2*b^2*c^2*d - 24*a*b^4*c*d)*(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d)^2)/(32*a^4*(4*a*c - b^2)^3*(a^3*
b^4 + 16*a^5*c^2 - 8*a^4*b^2*c)*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)))*(3*b^5*d - a^2*b^
3*f - 2*a^3*c^2*e - 21*a*b^3*c*d + a^3*b*c*f + 33*a^2*b*c^2*d + 2*a^2*b^2*c*e))/(8*a^3*c^2*(4*a*c - b^2)^3*(16
*a^4*c^4*e^2 + b^6*c^2*d^2 - 12*a*b^4*c^3*d^2 + 36*a^2*b^2*c^4*d^2 + 4*a^4*b^2*c^2*f^2 - 48*a^3*b*c^4*d*e - 16
*a^4*b*c^3*e*f + 8*a^2*b^3*c^3*d*e - 4*a^2*b^4*c^2*d*f + 24*a^3*b^2*c^3*d*f)*(400*a^3*c^3*d^2 - 6*b^6*d^2 + a^
4*b^2*f^2 + 4*a^4*c^2*e^2 - 291*a^2*b^2*c^2*d^2 + 72*a*b^4*c*d^2 - a^2*b^4*d*f + 2*a^2*b^3*c*d*e - 12*a^3*b*c^
2*d*e + 6*a^3*b^2*c*d*f - 4*a^4*b*c*e*f)) - (((((((4*a^2*b^6*c^2*d - 36*a^3*b^4*c^3*d + 80*a^4*b^2*c^4*d + 8*a
^4*b^3*c^3*e - 4*a^4*b^4*c^2*f + 16*a^5*b^2*c^3*f - 32*a^5*b*c^4*e)/(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2*c) + ((4
*a^4*b^6*c^2 - 32*a^5*b^4*c^3 + 64*a^6*b^2*c^4)*(2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^2*d - 24*a*b^4*c*d))/(
2*(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2*c)*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)))*(b^3*d - 2
*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d))/(4*a^2*(4*a*c - b^2)^(3/2)) + ((4*a^4*b^6*c^2 - 32*a^5*b^4*c^3 + 64*a^6*b^2
*c^4)*(2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^2*d - 24*a*b^4*c*d)*(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d))
/(8*a^2*(4*a*c - b^2)^(3/2)*(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2*c)*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192
*a^4*b^2*c^2)))*(2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^2*d - 24*a*b^4*c*d))/(2*(4*a^2*b^6 - 256*a^5*c^3 - 48*
a^3*b^4*c + 192*a^4*b^2*c^2)) + (((4*a^4*c^4*e^2 - 4*a*b^4*c^3*d^2 + 17*a^2*b^2*c^4*d^2 + a^4*b^2*c^2*f^2 - 36
*a^3*b*c^4*d*e - 4*a^4*b*c^3*e*f + 8*a^2*b^3*c^3*d*e - 4*a^2*b^4*c^2*d*f + 18*a^3*b^2*c^3*d*f)/(a^3*b^4 + 16*a
^5*c^2 - 8*a^4*b^2*c) + (((4*a^2*b^6*c^2*d - 36*a^3*b^4*c^3*d + 80*a^4*b^2*c^4*d + 8*a^4*b^3*c^3*e - 4*a^4*b^4
*c^2*f + 16*a^5*b^2*c^3*f - 32*a^5*b*c^4*e)/(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2*c) + ((4*a^4*b^6*c^2 - 32*a^5*b^
4*c^3 + 64*a^6*b^2*c^4)*(2*b^6*d - 128*a^3*c^3*d + 96*a^2*b^2*c^2*d - 24*a*b^4*c*d))/(2*(a^3*b^4 + 16*a^5*c^2
- 8*a^4*b^2*c)*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)))*(2*b^6*d - 128*a^3*c^3*d + 96*a^2*
b^2*c^2*d - 24*a*b^4*c*d))/(2*(4*a^2*b^6 - 256*a^5*c^3 - 48*a^3*b^4*c + 192*a^4*b^2*c^2)))*(b^3*d - 2*a^2*b*f
+ 4*a^2*c*e - 6*a*b*c*d))/(4*a^2*(4*a*c - b^2)^(3/2)) - ((4*a^4*b^6*c^2 - 32*a^5*b^4*c^3 + 64*a^6*b^2*c^4)*(b^
3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d)^3)/(64*a^6*(4*a*c - b^2)^(9/2)*(a^3*b^4 + 16*a^5*c^2 - 8*a^4*b^2*c)))
*(16*a^6*b^6*(4*a*c - b^2)^(9/2) - 1024*a^9*c^3*(4*a*c - b^2)^(9/2) - 192*a^7*b^4*c*(4*a*c - b^2)^(9/2) + 768*
a^8*b^2*c^2*(4*a*c - b^2)^(9/2))*(96*b^6*d - 1280*a^3*c^3*d - 32*a^2*b^4*f + 2208*a^2*b^2*c^2*d - 864*a*b^4*c*
d + 64*a^2*b^3*c*e - 192*a^3*b*c^2*e + 96*a^3*b^2*c*f))/(256*a^3*c^2*(4*a*c - b^2)^(7/2)*(16*a^4*c^4*e^2 + b^6
*c^2*d^2 - 12*a*b^4*c^3*d^2 + 36*a^2*b^2*c^4*d^2 + 4*a^4*b^2*c^2*f^2 - 48*a^3*b*c^4*d*e - 16*a^4*b*c^3*e*f + 8
*a^2*b^3*c^3*d*e - 4*a^2*b^4*c^2*d*f + 24*a^3*b^2*c^3*d*f)*(400*a^3*c^3*d^2 - 6*b^6*d^2 + a^4*b^2*f^2 + 4*a^4*
c^2*e^2 - 291*a^2*b^2*c^2*d^2 + 72*a*b^4*c*d^2 - a^2*b^4*d*f + 2*a^2*b^3*c*d*e - 12*a^3*b*c^2*d*e + 6*a^3*b^2*
c*d*f - 4*a^4*b*c*e*f)))*(b^3*d - 2*a^2*b*f + 4*a^2*c*e - 6*a*b*c*d))/(2*a^2*(4*a*c - b^2)^(3/2))

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sympy [F(-1)]  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((f*x**4+e*x**2+d)/x/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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