3.1.67 \(\int \frac {d+e x^2+f x^4}{x^5 (a+b x^2+c x^4)^2} \, dx\) [67]

Optimal. Leaf size=329 \[ -\frac {d}{4 a^2 x^4}+\frac {2 b d-a e}{2 a^3 x^2}+\frac {b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (3 b^5 d-2 a b^4 e+12 a^2 b^2 c e-12 a^3 c^2 e+6 a^2 b c (5 c d-a f)-a b^3 (20 c d-a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{3/2}}+\frac {\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \log (x)}{a^4}-\frac {\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4} \]

[Out]

-1/4*d/a^2/x^4+1/2*(-a*e+2*b*d)/a^3/x^2+1/2*(b^4*d-a*b^3*e+3*a^2*b*c*e+2*a^2*c*(-a*f+c*d)-a*b^2*(-a*f+4*c*d)+c
*(b^3*d-a*b^2*e+2*a^2*c*e-a*b*(-a*f+3*c*d))*x^2)/a^3/(-4*a*c+b^2)/(c*x^4+b*x^2+a)+1/2*(3*b^5*d-2*a*b^4*e+12*a^
2*b^2*c*e-12*a^3*c^2*e+6*a^2*b*c*(-a*f+5*c*d)-a*b^3*(-a*f+20*c*d))*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/a^4
/(-4*a*c+b^2)^(3/2)+(3*b^2*d-2*a*b*e-a*(-a*f+2*c*d))*ln(x)/a^4-1/4*(3*b^2*d-2*a*b*e-a*(-a*f+2*c*d))*ln(c*x^4+b
*x^2+a)/a^4

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Rubi [A]
time = 0.78, antiderivative size = 329, 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 = {1677, 1660, 1642, 648, 632, 212, 642} \begin {gather*} -\frac {\log \left (a+b x^2+c x^4\right ) \left (-2 a b e-a (2 c d-a f)+3 b^2 d\right )}{4 a^4}+\frac {\log (x) \left (-2 a b e-a (2 c d-a f)+3 b^2 d\right )}{a^4}+\frac {2 b d-a e}{2 a^3 x^2}-\frac {d}{4 a^2 x^4}+\frac {c x^2 \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )+3 a^2 b c e+2 a^2 c (c d-a f)-a b^3 e-a b^2 (4 c d-a f)+b^4 d}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-12 a^3 c^2 e+12 a^2 b^2 c e+6 a^2 b c (5 c d-a f)-2 a b^4 e-a b^3 (20 c d-a f)+3 b^5 d\right )}{2 a^4 \left (b^2-4 a c\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2 + f*x^4)/(x^5*(a + b*x^2 + c*x^4)^2),x]

[Out]

-1/4*d/(a^2*x^4) + (2*b*d - a*e)/(2*a^3*x^2) + (b^4*d - a*b^3*e + 3*a^2*b*c*e + 2*a^2*c*(c*d - a*f) - a*b^2*(4
*c*d - a*f) + c*(b^3*d - a*b^2*e + 2*a^2*c*e - a*b*(3*c*d - a*f))*x^2)/(2*a^3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4
)) + ((3*b^5*d - 2*a*b^4*e + 12*a^2*b^2*c*e - 12*a^3*c^2*e + 6*a^2*b*c*(5*c*d - a*f) - a*b^3*(20*c*d - a*f))*A
rcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^4*(b^2 - 4*a*c)^(3/2)) + ((3*b^2*d - 2*a*b*e - a*(2*c*d - a*f))*
Log[x])/a^4 - ((3*b^2*d - 2*a*b*e - a*(2*c*d - a*f))*Log[a + b*x^2 + c*x^4])/(4*a^4)

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 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 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1660

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 1677

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^5 \left (a+b x^2+c x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {d+e x+f x^2}{x^3 \left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\text {Subst}\left (\int \frac {-\left (\frac {b^2}{a}-4 c\right ) d+\frac {\left (b^2-4 a c\right ) (b d-a e) x}{a^2}-\frac {\left (b^2-4 a c\right ) \left (b^2 d-a b e-a (c d-a f)\right ) x^2}{a^3}-\frac {c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^3}{a^3}}{x^3 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=\frac {b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\text {Subst}\left (\int \left (\frac {\left (-b^2+4 a c\right ) d}{a^2 x^3}+\frac {\left (-b^2+4 a c\right ) (-2 b d+a e)}{a^3 x^2}+\frac {\left (b^2-4 a c\right ) \left (-3 b^2 d+2 a b e+a (2 c d-a f)\right )}{a^4 x}+\frac {3 b^5 d-2 a b^4 e+10 a^2 b^2 c e-6 a^3 c^2 e+a^2 b c (19 c d-5 a f)-a b^3 (17 c d-a f)+c \left (b^2-4 a c\right ) \left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) x}{a^4 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=-\frac {d}{4 a^2 x^4}+\frac {2 b d-a e}{2 a^3 x^2}+\frac {b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \log (x)}{a^4}-\frac {\text {Subst}\left (\int \frac {3 b^5 d-2 a b^4 e+10 a^2 b^2 c e-6 a^3 c^2 e+a^2 b c (19 c d-5 a f)-a b^3 (17 c d-a f)+c \left (b^2-4 a c\right ) \left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^4 \left (b^2-4 a c\right )}\\ &=-\frac {d}{4 a^2 x^4}+\frac {2 b d-a e}{2 a^3 x^2}+\frac {b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \log (x)}{a^4}-\frac {\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^4}-\frac {\left (3 b^5 d-2 a b^4 e+12 a^2 b^2 c e-12 a^3 c^2 e+6 a^2 b c (5 c d-a f)-a b^3 (20 c d-a f)\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^4 \left (b^2-4 a c\right )}\\ &=-\frac {d}{4 a^2 x^4}+\frac {2 b d-a e}{2 a^3 x^2}+\frac {b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \log (x)}{a^4}-\frac {\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}+\frac {\left (3 b^5 d-2 a b^4 e+12 a^2 b^2 c e-12 a^3 c^2 e+6 a^2 b c (5 c d-a f)-a b^3 (20 c d-a f)\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^4 \left (b^2-4 a c\right )}\\ &=-\frac {d}{4 a^2 x^4}+\frac {2 b d-a e}{2 a^3 x^2}+\frac {b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (3 b^5 d-2 a b^4 e+12 a^2 b^2 c e-12 a^3 c^2 e+6 a^2 b c (5 c d-a f)-a b^3 (20 c d-a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{3/2}}+\frac {\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \log (x)}{a^4}-\frac {\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}\\ \end {align*}

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Mathematica [A]
time = 0.78, size = 592, normalized size = 1.80 \begin {gather*} -\frac {\frac {a^2 d}{x^4}+\frac {2 a (-2 b d+a e)}{x^2}+\frac {2 a \left (-b^4 d+b^3 \left (a e-c d x^2\right )+a b^2 \left (4 c d-a f+c e x^2\right )-a b c \left (3 a e-3 c d x^2+a f x^2\right )+2 a^2 c \left (a f-c \left (d+e x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-4 \left (3 b^2 d-2 a b e+a (-2 c d+a f)\right ) \log (x)+\frac {\left (3 b^5 d+b^4 \left (3 \sqrt {b^2-4 a c} d-2 a e\right )+2 a^2 b c \left (15 c d+4 \sqrt {b^2-4 a c} e-3 a f\right )+a b^3 \left (-20 c d-2 \sqrt {b^2-4 a c} e+a f\right )-4 a^2 c \left (-2 c \sqrt {b^2-4 a c} d+3 a c e+a \sqrt {b^2-4 a c} f\right )+a b^2 \left (-14 c \sqrt {b^2-4 a c} d+12 a c e+a \sqrt {b^2-4 a c} f\right )\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (-3 b^5 d+b^4 \left (3 \sqrt {b^2-4 a c} d+2 a e\right )-a b^3 \left (-20 c d+2 \sqrt {b^2-4 a c} e+a f\right )+2 a^2 b c \left (-15 c d+4 \sqrt {b^2-4 a c} e+3 a f\right )+4 a^2 c \left (2 c \sqrt {b^2-4 a c} d+3 a c e-a \sqrt {b^2-4 a c} f\right )+a b^2 \left (-2 c \left (7 \sqrt {b^2-4 a c} d+6 a e\right )+a \sqrt {b^2-4 a c} f\right )\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}}{4 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2 + f*x^4)/(x^5*(a + b*x^2 + c*x^4)^2),x]

[Out]

-1/4*((a^2*d)/x^4 + (2*a*(-2*b*d + a*e))/x^2 + (2*a*(-(b^4*d) + b^3*(a*e - c*d*x^2) + a*b^2*(4*c*d - a*f + c*e
*x^2) - a*b*c*(3*a*e - 3*c*d*x^2 + a*f*x^2) + 2*a^2*c*(a*f - c*(d + e*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^
4)) - 4*(3*b^2*d - 2*a*b*e + a*(-2*c*d + a*f))*Log[x] + ((3*b^5*d + b^4*(3*Sqrt[b^2 - 4*a*c]*d - 2*a*e) + 2*a^
2*b*c*(15*c*d + 4*Sqrt[b^2 - 4*a*c]*e - 3*a*f) + a*b^3*(-20*c*d - 2*Sqrt[b^2 - 4*a*c]*e + a*f) - 4*a^2*c*(-2*c
*Sqrt[b^2 - 4*a*c]*d + 3*a*c*e + a*Sqrt[b^2 - 4*a*c]*f) + a*b^2*(-14*c*Sqrt[b^2 - 4*a*c]*d + 12*a*c*e + a*Sqrt
[b^2 - 4*a*c]*f))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((-3*b^5*d + b^4*(3*Sqrt[b^2 - 4
*a*c]*d + 2*a*e) - a*b^3*(-20*c*d + 2*Sqrt[b^2 - 4*a*c]*e + a*f) + 2*a^2*b*c*(-15*c*d + 4*Sqrt[b^2 - 4*a*c]*e
+ 3*a*f) + 4*a^2*c*(2*c*Sqrt[b^2 - 4*a*c]*d + 3*a*c*e - a*Sqrt[b^2 - 4*a*c]*f) + a*b^2*(-2*c*(7*Sqrt[b^2 - 4*a
*c]*d + 6*a*e) + a*Sqrt[b^2 - 4*a*c]*f))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/a^4

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Maple [A]
time = 0.11, size = 466, normalized size = 1.42

method	result	size
default	\(-\frac {\frac {\frac {a c \left (a^{2} b f +2 a^{2} c e -a \,b^{2} e -3 a b c d +b^{3} d \right ) x^{2}}{4 a c -b^{2}}-\frac {a \left (2 a^{3} c f -a^{2} b^{2} f -3 a^{2} b c e -2 a^{2} c^{2} d +a \,b^{3} e +4 a \,b^{2} c d -b^{4} d \right )}{4 a c -b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {\frac {\left (4 a^{3} c^{2} f -a^{2} b^{2} c f -8 a^{2} b \,c^{2} e -8 a^{2} c^{3} d +2 a \,b^{3} c e +14 a \,b^{2} c^{2} d -3 b^{4} c d \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (5 a^{3} b c f +6 e \,a^{3} c^{2}-a^{2} b^{3} f -10 a^{2} b^{2} c e -19 a^{2} b \,c^{2} d +2 a \,b^{4} e +17 a \,b^{3} c d -3 b^{5} d -\frac {\left (4 a^{3} c^{2} f -a^{2} b^{2} c f -8 a^{2} b \,c^{2} e -8 a^{2} c^{3} d +2 a \,b^{3} c e +14 a \,b^{2} c^{2} d -3 b^{4} c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{2 a^{4}}-\frac {d}{4 a^{2} x^{4}}-\frac {a e -2 b d}{2 a^{3} x^{2}}+\frac {\left (a^{2} f -2 a b e -2 a c d +3 b^{2} d \right ) \ln \left (x \right )}{a^{4}}\)	\(466\)
risch	\(\frac {-\frac {c \left (a^{2} b f +6 a^{2} c e -2 a \,b^{2} e -11 a b c d +3 b^{3} d \right ) x^{6}}{2 a^{3} \left (4 a c -b^{2}\right )}+\frac {\left (4 a^{3} c f -2 a^{2} b^{2} f -14 a^{2} b c e -8 a^{2} c^{2} d +4 a \,b^{3} e +25 a \,b^{2} c d -6 b^{4} d \right ) x^{4}}{4 a^{3} \left (4 a c -b^{2}\right )}-\frac {\left (2 a e -3 b d \right ) x^{2}}{4 a^{2}}-\frac {d}{4 a}}{x^{4} \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {\ln \left (x \right ) f}{a^{2}}-\frac {2 \ln \left (x \right ) b e}{a^{3}}-\frac {2 \ln \left (x \right ) c d}{a^{3}}+\frac {3 \ln \left (x \right ) b^{2} d}{a^{4}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (64 a^{7} c^{3}-48 a^{6} b^{2} c^{2}+12 b^{4} a^{5} c -a^{4} b^{6}\right ) \textit {\_Z}^{2}+\left (64 a^{5} c^{3} f -48 b^{2} c^{2} a^{4} f -128 a^{4} b \,c^{3} e -128 a^{4} c^{4} d +12 c \,b^{4} a^{3} f +96 a^{3} b^{3} c^{2} e +288 a^{3} b^{2} c^{3} d -b^{6} a^{2} f -24 a^{2} b^{5} c e -168 a^{2} b^{4} c^{2} d +2 a \,b^{7} e +38 a \,b^{6} c d -3 b^{8} d \right ) \textit {\_Z} +16 a^{3} c^{3} f^{2}-3 a^{2} b^{2} c^{2} f^{2}-28 a^{2} b \,c^{3} e f -64 a^{2} c^{4} d f +36 a^{2} c^{4} e^{2}+6 a \,b^{3} c^{2} e f +54 a \,b^{2} c^{3} d f -8 a \,b^{2} c^{3} e^{2}-52 a b \,c^{4} d e +64 a \,c^{5} d^{2}-9 b^{4} c^{2} d f +12 b^{3} c^{3} d e -15 b^{2} c^{4} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (160 c^{3} a^{9}-128 c^{2} b^{2} a^{8}+34 c \,b^{4} a^{7}-3 a^{6} b^{6}\right ) \textit {\_R}^{2}+\left (80 a^{7} c^{3} f -36 a^{6} b^{2} c^{2} f -136 e \,c^{3} b \,a^{6}-160 d \,c^{4} a^{6}+4 a^{5} b^{4} c f +66 e \,c^{2} b^{3} a^{5}+276 d \,c^{3} b^{2} a^{5}-8 e c \,b^{5} a^{4}-107 d \,c^{2} b^{4} a^{4}+12 d c \,b^{6} a^{3}\right ) \textit {\_R} +2 a^{4} b^{2} c^{2} f^{2}+24 a^{4} b \,c^{3} e f +72 e^{2} c^{4} a^{4}-8 a^{3} b^{3} c^{2} e f -44 a^{3} b^{2} c^{3} d f -48 e^{2} c^{3} b^{2} a^{3}-264 e d \,c^{4} b \,a^{3}+12 a^{2} b^{4} c^{2} d f +8 e^{2} c^{2} b^{4} a^{2}+160 e d \,c^{3} b^{3} a^{2}+242 d^{2} c^{4} b^{2} a^{2}-24 e d \,c^{2} b^{5} a -132 d^{2} c^{3} b^{4} a +18 d^{2} c^{2} b^{6}\right ) x^{2}+\left (-16 a^{9} b \,c^{2}+8 a^{8} b^{3} c -a^{7} b^{5}\right ) \textit {\_R}^{2}+\left (36 a^{7} b \,c^{2} f +24 a^{7} c^{3} e -17 a^{6} b^{3} c f -78 a^{6} b^{2} c^{2} e -108 a^{6} b \,c^{3} d +2 a^{5} b^{5} f +34 a^{5} b^{4} c e +151 a^{5} b^{3} c^{2} d -4 a^{4} b^{6} e -55 a^{4} b^{5} c d +6 a^{3} b^{7} d \right ) \textit {\_R} -8 a^{5} b \,c^{2} f^{2}-48 a^{5} c^{3} e f +2 a^{4} b^{3} c \,f^{2}+44 a^{4} b^{2} c^{2} e f +104 a^{4} b \,c^{3} d f +96 a^{4} b \,c^{3} e^{2}+96 a^{4} c^{4} d e -8 a^{3} b^{4} c e f -74 a^{3} b^{3} c^{2} d f -56 a^{3} b^{3} c^{2} e^{2}-376 a^{3} b^{2} c^{3} d e -176 a^{3} b \,c^{4} d^{2}+12 a^{2} b^{5} c d f +8 a^{2} b^{5} c \,e^{2}+184 a^{2} b^{4} c^{2} d e +356 a^{2} b^{3} c^{3} d^{2}-24 a \,b^{6} c d e -150 a \,b^{5} c^{2} d^{2}+18 b^{7} c \,d^{2}\right )\right )}{2}\)	\(1227\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^4+e*x^2+d)/x^5/(c*x^4+b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

-1/2/a^4*((a*c*(a^2*b*f+2*a^2*c*e-a*b^2*e-3*a*b*c*d+b^3*d)/(4*a*c-b^2)*x^2-a*(2*a^3*c*f-a^2*b^2*f-3*a^2*b*c*e-
2*a^2*c^2*d+a*b^3*e+4*a*b^2*c*d-b^4*d)/(4*a*c-b^2))/(c*x^4+b*x^2+a)+1/(4*a*c-b^2)*(1/2*(4*a^3*c^2*f-a^2*b^2*c*
f-8*a^2*b*c^2*e-8*a^2*c^3*d+2*a*b^3*c*e+14*a*b^2*c^2*d-3*b^4*c*d)/c*ln(c*x^4+b*x^2+a)+2*(5*a^3*b*c*f+6*e*a^3*c
^2-a^2*b^3*f-10*a^2*b^2*c*e-19*a^2*b*c^2*d+2*a*b^4*e+17*a*b^3*c*d-3*b^5*d-1/2*(4*a^3*c^2*f-a^2*b^2*c*f-8*a^2*b
*c^2*e-8*a^2*c^3*d+2*a*b^3*c*e+14*a*b^2*c^2*d-3*b^4*c*d)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)
^(1/2))))-1/4*d/a^2/x^4-1/2*(a*e-2*b*d)/a^3/x^2+(a^2*f-2*a*b*e-2*a*c*d+3*b^2*d)/a^4*ln(x)

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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^5/(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 deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1272 vs. \(2 (315) = 630\).
time = 4.31, size = 2567, normalized size = 7.80 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x^5/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/4*(2*((3*a*b^5*c - 23*a^2*b^3*c^2 + 44*a^3*b*c^3)*d - 2*(a^2*b^4*c - 7*a^3*b^2*c^2 + 12*a^4*c^3)*e + (a^3*b
^3*c - 4*a^4*b*c^2)*f)*x^6 + ((6*a*b^6 - 49*a^2*b^4*c + 108*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(2*a^2*b^5 - 15*a^
3*b^3*c + 28*a^4*b*c^2)*e + 2*(a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*f)*x^4 + (3*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*
b*c^2)*d - 2*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*e)*x^2 + (((3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*d - 2*(a*
b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*e + (a^2*b^3*c - 6*a^3*b*c^2)*f)*x^8 + ((3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^
2)*d - 2*(a*b^5 - 6*a^2*b^3*c + 6*a^3*b*c^2)*e + (a^2*b^4 - 6*a^3*b^2*c)*f)*x^6 + ((3*a*b^5 - 20*a^2*b^3*c + 3
0*a^3*b*c^2)*d - 2*(a^2*b^4 - 6*a^3*b^2*c + 6*a^4*c^2)*e + (a^3*b^3 - 6*a^4*b*c)*f)*x^4)*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)) - (a^3*b^4 - 8*a
^4*b^2*c + 16*a^5*c^2)*d - (((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3
*c^2 + 16*a^3*b*c^3)*e + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*f)*x^8 + ((3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c
^2 - 32*a^3*b*c^3)*d - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*f)*
x^6 + ((3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e +
(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*f)*x^4)*log(c*x^4 + b*x^2 + a) + 4*(((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^
2*c^3 - 32*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*e + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3
)*f)*x^8 + ((3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*d - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*
e + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*f)*x^6 + ((3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*d
- 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e + (a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*f)*x^4)*log(x))/((a^4*b^4*
c - 8*a^5*b^2*c^2 + 16*a^6*c^3)*x^8 + (a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^6 + (a^5*b^4 - 8*a^6*b^2*c + 16
*a^7*c^2)*x^4), 1/4*(2*((3*a*b^5*c - 23*a^2*b^3*c^2 + 44*a^3*b*c^3)*d - 2*(a^2*b^4*c - 7*a^3*b^2*c^2 + 12*a^4*
c^3)*e + (a^3*b^3*c - 4*a^4*b*c^2)*f)*x^6 + ((6*a*b^6 - 49*a^2*b^4*c + 108*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(2*
a^2*b^5 - 15*a^3*b^3*c + 28*a^4*b*c^2)*e + 2*(a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*f)*x^4 + (3*(a^2*b^5 - 8*a^3*
b^3*c + 16*a^4*b*c^2)*d - 2*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*e)*x^2 + 2*(((3*b^5*c - 20*a*b^3*c^2 + 30*a^2
*b*c^3)*d - 2*(a*b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*e + (a^2*b^3*c - 6*a^3*b*c^2)*f)*x^8 + ((3*b^6 - 20*a*b^4*
c + 30*a^2*b^2*c^2)*d - 2*(a*b^5 - 6*a^2*b^3*c + 6*a^3*b*c^2)*e + (a^2*b^4 - 6*a^3*b^2*c)*f)*x^6 + ((3*a*b^5 -
20*a^2*b^3*c + 30*a^3*b*c^2)*d - 2*(a^2*b^4 - 6*a^3*b^2*c + 6*a^4*c^2)*e + (a^3*b^3 - 6*a^4*b*c)*f)*x^4)*sqrt
(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*
d - (((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*e
+ (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*f)*x^8 + ((3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*d -
2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*f)*x^6 + ((3*a*b^6 - 26*a^
2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e + (a^3*b^4 - 8*a^4*b^2*c
+ 16*a^5*c^2)*f)*x^4)*log(c*x^4 + b*x^2 + a) + 4*(((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*d -
2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*e + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*f)*x^8 + ((3*b^7 - 26
*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*d - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e + (a^2*b^5 - 8*a^3*b^
3*c + 16*a^4*b*c^2)*f)*x^6 + ((3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(a^2*b^5 - 8*a^3*b^
3*c + 16*a^4*b*c^2)*e + (a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*f)*x^4)*log(x))/((a^4*b^4*c - 8*a^5*b^2*c^2 + 16*
a^6*c^3)*x^8 + (a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^6 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*x^4)]

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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((f*x**4+e*x**2+d)/x**5/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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Giac [A]
time = 5.45, size = 535, normalized size = 1.63 \begin {gather*} -\frac {{\left (3 \, b^{5} d - 20 \, a b^{3} c d + 30 \, a^{2} b c^{2} d + a^{2} b^{3} f - 6 \, a^{3} b c f - 2 \, a b^{4} e + 12 \, a^{2} b^{2} c e - 12 \, a^{3} c^{2} e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {3 \, b^{4} c d x^{4} - 14 \, a b^{2} c^{2} d x^{4} + 8 \, a^{2} c^{3} d x^{4} + a^{2} b^{2} c f x^{4} - 4 \, a^{3} c^{2} f x^{4} - 2 \, a b^{3} c x^{4} e + 8 \, a^{2} b c^{2} x^{4} e + 3 \, b^{5} d x^{2} - 12 \, a b^{3} c d x^{2} + 2 \, a^{2} b c^{2} d x^{2} + a^{2} b^{3} f x^{2} - 2 \, a^{3} b c f x^{2} - 2 \, a b^{4} x^{2} e + 6 \, a^{2} b^{2} c x^{2} e + 4 \, a^{3} c^{2} x^{2} e + 5 \, a b^{4} d - 22 \, a^{2} b^{2} c d + 12 \, a^{3} c^{2} d + 3 \, a^{3} b^{2} f - 8 \, a^{4} c f - 4 \, a^{2} b^{3} e + 14 \, a^{3} b c e}{4 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} {\left (c x^{4} + b x^{2} + a\right )}} - \frac {{\left (3 \, b^{2} d - 2 \, a c d + a^{2} f - 2 \, a b e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} + \frac {{\left (3 \, b^{2} d - 2 \, a c d + a^{2} f - 2 \, a b e\right )} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac {9 \, b^{2} d x^{4} - 6 \, a c d x^{4} + 3 \, a^{2} f x^{4} - 6 \, a b x^{4} e - 4 \, a b d x^{2} + 2 \, a^{2} x^{2} e + a^{2} d}{4 \, a^{4} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x^5/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

-1/2*(3*b^5*d - 20*a*b^3*c*d + 30*a^2*b*c^2*d + a^2*b^3*f - 6*a^3*b*c*f - 2*a*b^4*e + 12*a^2*b^2*c*e - 12*a^3*
c^2*e)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((a^4*b^2 - 4*a^5*c)*sqrt(-b^2 + 4*a*c)) + 1/4*(3*b^4*c*d*x^4
- 14*a*b^2*c^2*d*x^4 + 8*a^2*c^3*d*x^4 + a^2*b^2*c*f*x^4 - 4*a^3*c^2*f*x^4 - 2*a*b^3*c*x^4*e + 8*a^2*b*c^2*x^4
*e + 3*b^5*d*x^2 - 12*a*b^3*c*d*x^2 + 2*a^2*b*c^2*d*x^2 + a^2*b^3*f*x^2 - 2*a^3*b*c*f*x^2 - 2*a*b^4*x^2*e + 6*
a^2*b^2*c*x^2*e + 4*a^3*c^2*x^2*e + 5*a*b^4*d - 22*a^2*b^2*c*d + 12*a^3*c^2*d + 3*a^3*b^2*f - 8*a^4*c*f - 4*a^
2*b^3*e + 14*a^3*b*c*e)/((a^4*b^2 - 4*a^5*c)*(c*x^4 + b*x^2 + a)) - 1/4*(3*b^2*d - 2*a*c*d + a^2*f - 2*a*b*e)*
log(c*x^4 + b*x^2 + a)/a^4 + 1/2*(3*b^2*d - 2*a*c*d + a^2*f - 2*a*b*e)*log(x^2)/a^4 - 1/4*(9*b^2*d*x^4 - 6*a*c
*d*x^4 + 3*a^2*f*x^4 - 6*a*b*x^4*e - 4*a*b*d*x^2 + 2*a^2*x^2*e + a^2*d)/(a^4*x^4)

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Mupad [B]
time = 21.02, size = 2500, normalized size = 7.60 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2 + f*x^4)/(x^5*(a + b*x^2 + c*x^4)^2),x)

[Out]

(log(x)*(3*b^2*d + a^2*f - 2*a*b*e - 2*a*c*d))/a^4 - (log(((((((4*b*c^2*(3*b^5*d + a^2*b^3*f - 6*a^3*c^2*e - 2
*a*b^4*e - 17*a*b^3*c*d - 5*a^3*b*c*f + 19*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(a^3*(4*a*c - b^2)) - (b*c^2*(a*b +
3*b^2*x^2 - 10*a*c*x^2)*(a^4*(-(3*b^5*d + a^2*b^3*f - 12*a^3*c^2*e - 2*a*b^4*e - 20*a*b^3*c*d - 6*a^3*b*c*f +
30*a^2*b*c^2*d + 12*a^2*b^2*c*e)^2/(a^8*(4*a*c - b^2)^3))^(1/2) + 3*b^2*d + a^2*f - 2*a*b*e - 2*a*c*d))/a^4 +
(2*c^3*x^2*(3*b^5*d + a^2*b^3*f + 60*a^3*c^2*e - 2*a*b^4*e + 4*a*b^3*c*d - 10*a^3*b*c*f - 70*a^2*b*c^2*d - 4*a
^2*b^2*c*e))/(a^3*(4*a*c - b^2)))*(a^4*(-(3*b^5*d + a^2*b^3*f - 12*a^3*c^2*e - 2*a*b^4*e - 20*a*b^3*c*d - 6*a^
3*b*c*f + 30*a^2*b*c^2*d + 12*a^2*b^2*c*e)^2/(a^8*(4*a*c - b^2)^3))^(1/2) + 3*b^2*d + a^2*f - 2*a*b*e - 2*a*c*
d))/(4*a^4) + (c^3*(36*b^8*d^2 + 16*a^2*b^6*e^2 + 4*a^4*b^4*f^2 - 36*a^5*c^3*e^2 - 116*a^3*b^4*c*e^2 - 17*a^5*
b^2*c*f^2 - 48*a*b^7*d*e + 778*a^2*b^4*c^2*d^2 - 473*a^3*b^2*c^3*d^2 + 216*a^4*b^2*c^2*e^2 - 309*a*b^6*c*d^2 +
24*a^2*b^6*d*f - 16*a^3*b^5*e*f + 380*a^2*b^5*c*d*e + 324*a^4*b*c^3*d*e - 154*a^3*b^4*c*d*f + 92*a^4*b^3*c*e*
f - 108*a^5*b*c^2*e*f - 832*a^3*b^3*c^2*d*e + 230*a^4*b^2*c^2*d*f))/(a^6*(4*a*c - b^2)^2) + (c^4*x^2*(54*b^7*d
^2 + 24*a^2*b^5*e^2 + 6*a^4*b^3*f^2 - 440*a^3*b*c^3*d^2 - 164*a^3*b^3*c*e^2 + 276*a^4*b*c^2*e^2 - 72*a*b^6*d*e
+ 1011*a^2*b^3*c^2*d^2 - 441*a*b^5*c*d^2 - 20*a^5*b*c*f^2 + 36*a^2*b^5*d*f + 240*a^4*c^3*d*e - 24*a^3*b^4*e*f
- 120*a^5*c^2*e*f + 540*a^2*b^4*c*d*e - 207*a^3*b^3*c*d*f + 260*a^4*b*c^2*d*f + 122*a^4*b^2*c*e*f - 1072*a^3*
b^2*c^2*d*e))/(a^6*(4*a*c - b^2)^2))*(a^4*(-(3*b^5*d + a^2*b^3*f - 12*a^3*c^2*e - 2*a*b^4*e - 20*a*b^3*c*d - 6
*a^3*b*c*f + 30*a^2*b*c^2*d + 12*a^2*b^2*c*e)^2/(a^8*(4*a*c - b^2)^3))^(1/2) + 3*b^2*d + a^2*f - 2*a*b*e - 2*a
*c*d))/(4*a^4) - (c^4*(3*b^2*d + a^2*f - 2*a*b*e - 2*a*c*d)*(3*b^3*d - 2*a*b^2*e + a^2*b*f + 6*a^2*c*e - 11*a*
b*c*d)^2)/(a^9*(4*a*c - b^2)^2) + (c^5*x^2*(3*b^3*d - 2*a*b^2*e + a^2*b*f + 6*a^2*c*e - 11*a*b*c*d)^3)/(a^9*(4
*a*c - b^2)^3))*((((c^3*(36*b^8*d^2 + 16*a^2*b^6*e^2 + 4*a^4*b^4*f^2 - 36*a^5*c^3*e^2 - 116*a^3*b^4*c*e^2 - 17
*a^5*b^2*c*f^2 - 48*a*b^7*d*e + 778*a^2*b^4*c^2*d^2 - 473*a^3*b^2*c^3*d^2 + 216*a^4*b^2*c^2*e^2 - 309*a*b^6*c*
d^2 + 24*a^2*b^6*d*f - 16*a^3*b^5*e*f + 380*a^2*b^5*c*d*e + 324*a^4*b*c^3*d*e - 154*a^3*b^4*c*d*f + 92*a^4*b^3
*c*e*f - 108*a^5*b*c^2*e*f - 832*a^3*b^3*c^2*d*e + 230*a^4*b^2*c^2*d*f))/(a^6*(4*a*c - b^2)^2) - (((b*c^2*(a*b
+ 3*b^2*x^2 - 10*a*c*x^2)*(a^4*(-(3*b^5*d + a^2*b^3*f - 12*a^3*c^2*e - 2*a*b^4*e - 20*a*b^3*c*d - 6*a^3*b*c*f
+ 30*a^2*b*c^2*d + 12*a^2*b^2*c*e)^2/(a^8*(4*a*c - b^2)^3))^(1/2) - 3*b^2*d - a^2*f + 2*a*b*e + 2*a*c*d))/a^4
+ (4*b*c^2*(3*b^5*d + a^2*b^3*f - 6*a^3*c^2*e - 2*a*b^4*e - 17*a*b^3*c*d - 5*a^3*b*c*f + 19*a^2*b*c^2*d + 10*
a^2*b^2*c*e))/(a^3*(4*a*c - b^2)) + (2*c^3*x^2*(3*b^5*d + a^2*b^3*f + 60*a^3*c^2*e - 2*a*b^4*e + 4*a*b^3*c*d -
10*a^3*b*c*f - 70*a^2*b*c^2*d - 4*a^2*b^2*c*e))/(a^3*(4*a*c - b^2)))*(a^4*(-(3*b^5*d + a^2*b^3*f - 12*a^3*c^2
*e - 2*a*b^4*e - 20*a*b^3*c*d - 6*a^3*b*c*f + 30*a^2*b*c^2*d + 12*a^2*b^2*c*e)^2/(a^8*(4*a*c - b^2)^3))^(1/2)
- 3*b^2*d - a^2*f + 2*a*b*e + 2*a*c*d))/(4*a^4) + (c^4*x^2*(54*b^7*d^2 + 24*a^2*b^5*e^2 + 6*a^4*b^3*f^2 - 440*
a^3*b*c^3*d^2 - 164*a^3*b^3*c*e^2 + 276*a^4*b*c^2*e^2 - 72*a*b^6*d*e + 1011*a^2*b^3*c^2*d^2 - 441*a*b^5*c*d^2
- 20*a^5*b*c*f^2 + 36*a^2*b^5*d*f + 240*a^4*c^3*d*e - 24*a^3*b^4*e*f - 120*a^5*c^2*e*f + 540*a^2*b^4*c*d*e - 2
07*a^3*b^3*c*d*f + 260*a^4*b*c^2*d*f + 122*a^4*b^2*c*e*f - 1072*a^3*b^2*c^2*d*e))/(a^6*(4*a*c - b^2)^2))*(a^4*
(-(3*b^5*d + a^2*b^3*f - 12*a^3*c^2*e - 2*a*b^4*e - 20*a*b^3*c*d - 6*a^3*b*c*f + 30*a^2*b*c^2*d + 12*a^2*b^2*c
*e)^2/(a^8*(4*a*c - b^2)^3))^(1/2) - 3*b^2*d - a^2*f + 2*a*b*e + 2*a*c*d))/(4*a^4) + (c^4*(3*b^2*d + a^2*f - 2
*a*b*e - 2*a*c*d)*(3*b^3*d - 2*a*b^2*e + a^2*b*f + 6*a^2*c*e - 11*a*b*c*d)^2)/(a^9*(4*a*c - b^2)^2) - (c^5*x^2
*(3*b^3*d - 2*a*b^2*e + a^2*b*f + 6*a^2*c*e - 11*a*b*c*d)^3)/(a^9*(4*a*c - b^2)^3)))*(6*b^8*d + 256*a^4*c^4*d
+ 2*a^2*b^6*f - 128*a^5*c^3*f - 4*a*b^7*e + 336*a^2*b^4*c^2*d - 576*a^3*b^2*c^3*d - 192*a^3*b^3*c^2*e + 96*a^4
*b^2*c^2*f - 76*a*b^6*c*d + 48*a^2*b^5*c*e + 256*a^4*b*c^3*e - 24*a^3*b^4*c*f))/(2*(4*a^4*b^6 - 256*a^7*c^3 -
48*a^5*b^4*c + 192*a^6*b^2*c^2)) - (d/(4*a) + (x^2*(2*a*e - 3*b*d))/(4*a^2) + (x^4*(6*b^4*d + 8*a^2*c^2*d + 2*
a^2*b^2*f - 4*a*b^3*e - 4*a^3*c*f - 25*a*b^2*c*d + 14*a^2*b*c*e))/(4*a^3*(4*a*c - b^2)) + (c*x^6*(3*b^3*d - 2*
a*b^2*e + a^2*b*f + 6*a^2*c*e - 11*a*b*c*d))/(2*a^3*(4*a*c - b^2)))/(a*x^4 + b*x^6 + c*x^8) + (atan((x^2*(((((
(1760*a^7*b*c^8*d^2 - 1104*a^8*b*c^7*e^2 + 80*a^9*b*c^6*f^2 + 54*a^3*b^9*c^4*d^2 - 657*a^4*b^7*c^5*d^2 + 2775*
a^5*b^5*c^6*d^2 - 4484*a^6*b^3*c^7*d^2 + 24*a^5*b^7*c^4*e^2 - 260*a^6*b^5*c^5*e^2 + 932*a^7*b^3*c^6*e^2 + 6*a^
7*b^5*c^4*f^2 - 44*a^8*b^3*c^5*f^2 - 960*a^8*c^8*d*e + 480*a^9*c^7*e*f - 1040*a^8*b*c^7*d*f - 72*a^4*b^8*c^4*d
*e + 828*a^5*b^6*c^5*d*e - 3232*a^6*b^4*c^6*d*e + 4528*a^7*b^2*c^7*d*e + 36*a^5*b^7*c^4*d*f - 351*a^6*b^5*c^5*
d*f + 1088*a^7*b^3*c^6*d*f - 24*a^6*b^6*c^4*e*f...

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