3.1.73 \(\int \frac {d+e x^2+f x^4}{x^4 (a+b x^2+c x^4)^2} \, dx\) [73]

Optimal. Leaf size=575 \[ -\frac {d}{3 a^2 x^3}+\frac {2 b d-a e}{a^3 x}+\frac {x \left (a^2 \left (\frac {b^4 d}{a^2}+2 c^2 d+3 b c e-\frac {b^2 (4 c d+b e)}{a}+b^2 f-2 a c f\right )+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {c} \left (5 b^4 d+b^3 \left (5 \sqrt {b^2-4 a c} d-3 a e\right )+2 a^2 c \left (14 c d+5 \sqrt {b^2-4 a c} e-6 a f\right )-a b^2 \left (29 c d+3 \sqrt {b^2-4 a c} e-a f\right )-a b \left (19 c \sqrt {b^2-4 a c} d-16 a c e-a \sqrt {b^2-4 a c} f\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (5 b^4 d-b^3 \left (5 \sqrt {b^2-4 a c} d+3 a e\right )+2 a^2 c \left (14 c d-5 \sqrt {b^2-4 a c} e-6 a f\right )-a b^2 \left (29 c d-3 \sqrt {b^2-4 a c} e-a f\right )+a b \left (19 c \sqrt {b^2-4 a c} d+16 a c e-a \sqrt {b^2-4 a c} f\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}} \]

[Out]

-1/3*d/a^2/x^3+(-a*e+2*b*d)/a^3/x+1/2*x*(a^2*(b^4*d/a^2+2*c^2*d+3*b*c*e-b^2*(b*e+4*c*d)/a+b^2*f-2*a*c*f)+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)+1/4*arctan(x*2^(1/2)*c^(1/2)/(b-
(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(5*b^4*d+b^3*(-3*a*e+5*d*(-4*a*c+b^2)^(1/2))-a*b^2*(29*c*d-a*f+3*e*(-4*a*c+
b^2)^(1/2))+2*a^2*c*(14*c*d-6*a*f+5*e*(-4*a*c+b^2)^(1/2))-a*b*(-16*a*c*e+19*c*d*(-4*a*c+b^2)^(1/2)-a*f*(-4*a*c
+b^2)^(1/2)))/a^3/(-4*a*c+b^2)^(3/2)*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-1/4*arctan(x*2^(1/2)*c^(1/2)/(b+(-4*
a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(5*b^4*d-b^3*(3*a*e+5*d*(-4*a*c+b^2)^(1/2))+2*a^2*c*(14*c*d-6*a*f-5*e*(-4*a*c+b
^2)^(1/2))-a*b^2*(29*c*d-a*f-3*e*(-4*a*c+b^2)^(1/2))+a*b*(16*a*c*e+19*c*d*(-4*a*c+b^2)^(1/2)-a*f*(-4*a*c+b^2)^
(1/2)))/a^3/(-4*a*c+b^2)^(3/2)*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]
time = 6.51, antiderivative size = 575, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1683, 1678, 1180, 211} \begin {gather*} \frac {2 b d-a e}{a^3 x}-\frac {d}{3 a^2 x^3}+\frac {\sqrt {c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (2 a^2 c \left (5 e \sqrt {b^2-4 a c}-6 a f+14 c d\right )-a b^2 \left (3 e \sqrt {b^2-4 a c}-a f+29 c d\right )-a b \left (19 c d \sqrt {b^2-4 a c}-a f \sqrt {b^2-4 a c}-16 a c e\right )+b^3 \left (5 d \sqrt {b^2-4 a c}-3 a e\right )+5 b^4 d\right )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (2 a^2 c \left (-5 e \sqrt {b^2-4 a c}-6 a f+14 c d\right )-a b^2 \left (-3 e \sqrt {b^2-4 a c}-a f+29 c d\right )+a b \left (19 c d \sqrt {b^2-4 a c}-a f \sqrt {b^2-4 a c}+16 a c e\right )-b^3 \left (5 d \sqrt {b^2-4 a c}+3 a e\right )+5 b^4 d\right )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {x \left (a^2 \left (\frac {b^4 d}{a^2}-\frac {b^2 (b e+4 c d)}{a}-2 a c f+b^2 f+3 b c e+2 c^2 d\right )+c x^2 \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )\right )}{2 a^3 \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^4*(a + b*x^2 + c*x^4)^2),x]

[Out]

-1/3*d/(a^2*x^3) + (2*b*d - a*e)/(a^3*x) + (x*(a^2*((b^4*d)/a^2 + 2*c^2*d + 3*b*c*e - (b^2*(4*c*d + b*e))/a +
b^2*f - 2*a*c*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)) + (Sqrt[c]*(5*b^4*d + b^3*(5*Sqrt[b^2 - 4*a*c]*d - 3*a*e) + 2*a^2*c*(14*c*d + 5*Sqrt[b^2 - 4*a*c]*e -
6*a*f) - a*b^2*(29*c*d + 3*Sqrt[b^2 - 4*a*c]*e - a*f) - a*b*(19*c*Sqrt[b^2 - 4*a*c]*d - 16*a*c*e - a*Sqrt[b^2
- 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a*c)^(3/2)*Sqrt
[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[c]*(5*b^4*d - b^3*(5*Sqrt[b^2 - 4*a*c]*d + 3*a*e) + 2*a^2*c*(14*c*d - 5*Sqrt[
b^2 - 4*a*c]*e - 6*a*f) - a*b^2*(29*c*d - 3*Sqrt[b^2 - 4*a*c]*e - a*f) + a*b*(19*c*Sqrt[b^2 - 4*a*c]*d + 16*a*
c*e - a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4
*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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 1180

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

Rule 1678

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x
)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]

Rule 1683

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainde
r[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, S
imp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))
), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[x^m*(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[(2*a*(p + 1)*(b^2
- 4*a*c)*PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x])/x^m + (b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)
/x^m + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - m), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[
Pq, x^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && ILtQ[m/2, 0]

Rubi steps

\begin {align*} \int \frac {d+e x^2+f x^4}{x^4 \left (a+b x^2+c x^4\right )^2} \, dx &=\frac {x \left (a^2 \left (\frac {b^4 d}{a^2}+2 c^2 d+3 b c e-\frac {b^2 (4 c d+b e)}{a}+b^2 f-2 a c f\right )+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {-2 \left (b^2-4 a c\right ) d+\frac {2 \left (b^2-4 a c\right ) (b d-a e) x^2}{a}-\left (\frac {b^4 d}{a^2}+6 c^2 d+5 b c e-\frac {b^2 (6 c d+b e)}{a}+b^2 f-6 a c f\right ) x^4-\frac {c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^6}{a^2}}{x^4 \left (a+b x^2+c x^4\right )} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac {x \left (a^2 \left (\frac {b^4 d}{a^2}+2 c^2 d+3 b c e-\frac {b^2 (4 c d+b e)}{a}+b^2 f-2 a c f\right )+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \left (\frac {2 \left (-b^2+4 a c\right ) d}{a x^4}+\frac {2 \left (-b^2+4 a c\right ) (-2 b d+a e)}{a^2 x^2}+\frac {-5 b^4 d+3 a b^3 e-13 a^2 b c e-2 a^2 c (7 c d-3 a f)+a b^2 (24 c d-a f)-c \left (5 b^3 d-3 a b^2 e+10 a^2 c e-a b (19 c d-a f)\right ) x^2}{a^2 \left (a+b x^2+c x^4\right )}\right ) \, dx}{2 a \left (b^2-4 a c\right )}\\ &=-\frac {d}{3 a^2 x^3}+\frac {2 b d-a e}{a^3 x}+\frac {x \left (a^2 \left (\frac {b^4 d}{a^2}+2 c^2 d+3 b c e-\frac {b^2 (4 c d+b e)}{a}+b^2 f-2 a c f\right )+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {-5 b^4 d+3 a b^3 e-13 a^2 b c e-2 a^2 c (7 c d-3 a f)+a b^2 (24 c d-a f)-c \left (5 b^3 d-3 a b^2 e+10 a^2 c e-a b (19 c d-a f)\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a^3 \left (b^2-4 a c\right )}\\ &=-\frac {d}{3 a^2 x^3}+\frac {2 b d-a e}{a^3 x}+\frac {x \left (a^2 \left (\frac {b^4 d}{a^2}+2 c^2 d+3 b c e-\frac {b^2 (4 c d+b e)}{a}+b^2 f-2 a c f\right )+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (c \left (5 b^4 d+b^3 \left (5 \sqrt {b^2-4 a c} d-3 a e\right )+2 a^2 c \left (14 c d+5 \sqrt {b^2-4 a c} e-6 a f\right )-a b^2 \left (29 c d+3 \sqrt {b^2-4 a c} e-a f\right )-a b \left (19 c \sqrt {b^2-4 a c} d-16 a c e-a \sqrt {b^2-4 a c} f\right )\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{4 a^3 \left (b^2-4 a c\right )^{3/2}}-\frac {\left (c \left (5 b^4 d-b^3 \left (5 \sqrt {b^2-4 a c} d+3 a e\right )+2 a^2 c \left (14 c d-5 \sqrt {b^2-4 a c} e-6 a f\right )-a b^2 \left (29 c d-3 \sqrt {b^2-4 a c} e-a f\right )+a b \left (19 c \sqrt {b^2-4 a c} d+16 a c e-a \sqrt {b^2-4 a c} f\right )\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{4 a^3 \left (b^2-4 a c\right )^{3/2}}\\ &=-\frac {d}{3 a^2 x^3}+\frac {2 b d-a e}{a^3 x}+\frac {x \left (a^2 \left (\frac {b^4 d}{a^2}+2 c^2 d+3 b c e-\frac {b^2 (4 c d+b e)}{a}+b^2 f-2 a c f\right )+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {c} \left (5 b^4 d+b^3 \left (5 \sqrt {b^2-4 a c} d-3 a e\right )+2 a^2 c \left (14 c d+5 \sqrt {b^2-4 a c} e-6 a f\right )-a b^2 \left (29 c d+3 \sqrt {b^2-4 a c} e-a f\right )-a b \left (19 c \sqrt {b^2-4 a c} d-16 a c e-a \sqrt {b^2-4 a c} f\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (5 b^4 d-b^3 \left (5 \sqrt {b^2-4 a c} d+3 a e\right )+2 a^2 c \left (14 c d-5 \sqrt {b^2-4 a c} e-6 a f\right )-a b^2 \left (29 c d-3 \sqrt {b^2-4 a c} e-a f\right )+a b \left (19 c \sqrt {b^2-4 a c} d+16 a c e-a \sqrt {b^2-4 a c} f\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [A]
time = 1.12, size = 548, normalized size = 0.95 \begin {gather*} \frac {-\frac {4 a d}{x^3}+\frac {24 b d-12 a e}{x}+\frac {6 x \left (b^4 d+b^3 \left (-a e+c d 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 )+a b^2 \left (a f-c \left (4 d+e x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {3 \sqrt {2} \sqrt {c} \left (5 b^4 d+b^3 \left (5 \sqrt {b^2-4 a c} d-3 a e\right )+2 a^2 c \left (14 c d+5 \sqrt {b^2-4 a c} e-6 a f\right )+a b^2 \left (-29 c d-3 \sqrt {b^2-4 a c} e+a f\right )+a b \left (-19 c \sqrt {b^2-4 a c} d+16 a c e+a \sqrt {b^2-4 a c} f\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {2} \sqrt {c} \left (-5 b^4 d+b^3 \left (5 \sqrt {b^2-4 a c} d+3 a e\right )-a b^2 \left (-29 c d+3 \sqrt {b^2-4 a c} e+a f\right )+2 a^2 c \left (-14 c d+5 \sqrt {b^2-4 a c} e+6 a f\right )+a b \left (-19 c \sqrt {b^2-4 a c} d-16 a c e+a \sqrt {b^2-4 a c} f\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{12 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*a*d)/x^3 + (24*b*d - 12*a*e)/x + (6*x*(b^4*d + b^3*(-(a*e) + c*d*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)) + a*b^2*(a*f - c*(4*d + e*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) +
(3*Sqrt[2]*Sqrt[c]*(5*b^4*d + b^3*(5*Sqrt[b^2 - 4*a*c]*d - 3*a*e) + 2*a^2*c*(14*c*d + 5*Sqrt[b^2 - 4*a*c]*e -
6*a*f) + a*b^2*(-29*c*d - 3*Sqrt[b^2 - 4*a*c]*e + a*f) + a*b*(-19*c*Sqrt[b^2 - 4*a*c]*d + 16*a*c*e + a*Sqrt[b^
2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2
- 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(-5*b^4*d + b^3*(5*Sqrt[b^2 - 4*a*c]*d + 3*a*e) - a*b^2*(-29*c*d + 3*Sqrt[b^2
- 4*a*c]*e + a*f) + 2*a^2*c*(-14*c*d + 5*Sqrt[b^2 - 4*a*c]*e + 6*a*f) + a*b*(-19*c*Sqrt[b^2 - 4*a*c]*d - 16*a
*c*e + a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*S
qrt[b + Sqrt[b^2 - 4*a*c]]))/(12*a^3)

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Maple [A]
time = 0.13, size = 570, normalized size = 0.99

method	result	size
default	\(\frac {\frac {-\frac {c \left (a^{2} b f +2 a^{2} c e -a \,b^{2} e -3 a b c d +b^{3} d \right ) x^{3}}{2 \left (4 a c -b^{2}\right )}+\frac {\left (2 a^{3} c f -a^{2} b^{2} f -3 a^{2} b c e -2 a^{2} c^{2} d +a \,b^{3} e +4 a \,b^{2} c d -b^{4} d \right ) x}{8 a c -2 b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {2 c \left (-\frac {\left (-a^{2} b f \sqrt {-4 a c +b^{2}}-10 a^{2} c e \sqrt {-4 a c +b^{2}}+3 a \,b^{2} e \sqrt {-4 a c +b^{2}}+19 \sqrt {-4 a c +b^{2}}\, a b c d -5 \sqrt {-4 a c +b^{2}}\, b^{3} d +12 a^{3} c f -a^{2} b^{2} f -16 a^{2} b c e -28 a^{2} c^{2} d +3 a \,b^{3} e +29 a \,b^{2} c d -5 b^{4} d \right ) \sqrt {2}\, \arctanh \left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-a^{2} b f \sqrt {-4 a c +b^{2}}-10 a^{2} c e \sqrt {-4 a c +b^{2}}+3 a \,b^{2} e \sqrt {-4 a c +b^{2}}+19 \sqrt {-4 a c +b^{2}}\, a b c d -5 \sqrt {-4 a c +b^{2}}\, b^{3} d -12 a^{3} c f +a^{2} b^{2} f +16 a^{2} b c e +28 a^{2} c^{2} d -3 a \,b^{3} e -29 a \,b^{2} c d +5 b^{4} d \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}}{a^{3}}-\frac {d}{3 a^{2} x^{3}}-\frac {a e -2 b d}{a^{3} x}\)	\(570\)
risch	\(\text {Expression too large to display}\)	\(3685\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^3*((-1/2*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^3+1/2*(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)*x)/(c*x^4+b*x^2+a)+2/(4*a*c-b^2)*c*(-1/8*(-a^2*b*f*(-4*a*
c+b^2)^(1/2)-10*a^2*c*e*(-4*a*c+b^2)^(1/2)+3*a*b^2*e*(-4*a*c+b^2)^(1/2)+19*(-4*a*c+b^2)^(1/2)*a*b*c*d-5*(-4*a*
c+b^2)^(1/2)*b^3*d+12*a^3*c*f-a^2*b^2*f-16*a^2*b*c*e-28*a^2*c^2*d+3*a*b^3*e+29*a*b^2*c*d-5*b^4*d)/(-4*a*c+b^2)
^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))+1/8*(-
a^2*b*f*(-4*a*c+b^2)^(1/2)-10*a^2*c*e*(-4*a*c+b^2)^(1/2)+3*a*b^2*e*(-4*a*c+b^2)^(1/2)+19*(-4*a*c+b^2)^(1/2)*a*
b*c*d-5*(-4*a*c+b^2)^(1/2)*b^3*d-12*a^3*c*f+a^2*b^2*f+16*a^2*b*c*e+28*a^2*c^2*d-3*a*b^3*e-29*a*b^2*c*d+5*b^4*d
)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1
/2))))-1/3*d/a^2/x^3-(a*e-2*b*d)/a^3/x

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*(a^2*b*c*f - 3*a*b^2*c*e + 10*a^2*c^2*e + (5*b^3*c - 19*a*b*c^2)*d)*x^6 - (9*a*b^3*e - 33*a^2*b*c*e - (
15*b^4 - 62*a*b^2*c + 14*a^2*c^2)*d - 3*(a^2*b^2 - 2*a^3*c)*f)*x^4 - 2*(3*a^2*b^2*e - 12*a^3*c*e - 5*(a*b^3 -
4*a^2*b*c)*d)*x^2 - 2*(a^2*b^2 - 4*a^3*c)*d)/((a^3*b^2*c - 4*a^4*c^2)*x^7 + (a^3*b^3 - 4*a^4*b*c)*x^5 + (a^4*b
^2 - 4*a^5*c)*x^3) + 1/2*integrate(-(3*a*b^3*e - 13*a^2*b*c*e - (a^2*b*c*f - 3*a*b^2*c*e + 10*a^2*c^2*e + (5*b
^3*c - 19*a*b*c^2)*d)*x^2 - (5*b^4 - 24*a*b^2*c + 14*a^2*c^2)*d - (a^2*b^2 - 6*a^3*c)*f)/(c*x^4 + b*x^2 + a),
x)/(a^3*b^2 - 4*a^4*c)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 19333 vs. \(2 (510) = 1020\).
time = 84.86, size = 19333, normalized size = 33.62 \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^4/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/12*(6*(a^2*b*c*f + (5*b^3*c - 19*a*b*c^2)*d - (3*a*b^2*c - 10*a^2*c^2)*e)*x^6 + 2*((15*b^4 - 62*a*b^2*c + 14
*a^2*c^2)*d - 3*(3*a*b^3 - 11*a^2*b*c)*e + 3*(a^2*b^2 - 2*a^3*c)*f)*x^4 + 4*(5*(a*b^3 - 4*a^2*b*c)*d - 3*(a^2*
b^2 - 4*a^3*c)*e)*x^2 + 3*sqrt(1/2)*((a^3*b^2*c - 4*a^4*c^2)*x^7 + (a^3*b^3 - 4*a^4*b*c)*x^5 + (a^4*b^2 - 4*a^
5*c)*x^3)*sqrt(-((25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4)*d^2 - 2*(15*a*b
^8 - 182*a^2*b^6*c + 735*a^3*b^4*c^2 - 1050*a^4*b^2*c^3 + 280*a^5*c^4)*d*e + (9*a^2*b^7 - 105*a^3*b^5*c + 385*
a^4*b^3*c^2 - 420*a^5*b*c^3)*e^2 + (a^4*b^5 - 15*a^5*b^3*c + 60*a^6*b*c^2)*f^2 + 2*((5*a^2*b^7 - 69*a^3*b^5*c
+ 285*a^4*b^3*c^2 - 340*a^5*b*c^3)*d - (3*a^3*b^6 - 40*a^4*b^4*c + 150*a^5*b^2*c^2 - 120*a^6*c^3)*e)*f + (a^7*
b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt(((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630
*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)*d^4 - 4*(375*a*b^11 - 4775*a^2*b^9*c + 21
195*a^3*b^ ...

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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**4/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 8660 vs. \(2 (523) = 1046\).
time = 8.75, size = 8660, normalized size = 15.06 \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^4/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

1/2*(b^3*c*d*x^3 - 3*a*b*c^2*d*x^3 + a^2*b*c*f*x^3 - a*b^2*c*x^3*e + 2*a^2*c^2*x^3*e + b^4*d*x - 4*a*b^2*c*d*x
+ 2*a^2*c^2*d*x + a^2*b^2*f*x - 2*a^3*c*f*x - a*b^3*x*e + 3*a^2*b*c*x*e)/((a^3*b^2 - 4*a^4*c)*(c*x^4 + b*x^2
+ a)) + 1/16*((10*b^5*c^2 - 78*a*b^3*c^3 + 152*a^2*b*c^4 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4
*a*c)*c)*b^5 + 39*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 10*sqrt(2)*sqrt(b^2 - 4*
a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c - 76*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*
b*c^2 - 38*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*s
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 19*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3
- 10*(b^2 - 4*a*c)*b^3*c^2 + 38*(b^2 - 4*a*c)*a*b*c^3)*(a^3*b^2 - 4*a^4*c)^2*d + (2*a^2*b^3*c^2 - 8*a^3*b*c^3
- sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + s
qrt(b^2 - 4*a*c)*c)*a^3*b*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c - sqrt(2)*
sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - 2*(b^2 - 4*a*c)*a^2*b*c^2)*(a^3*b^2 - 4*a^4*c)^2
*f - (6*a*b^4*c^2 - 44*a^2*b^2*c^3 + 80*a^3*c^4 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*
a*b^4 + 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*s
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c - 40*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^2
- 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^3 - 6*
(b^2 - 4*a*c)*a*b^2*c^2 + 20*(b^2 - 4*a*c)*a^2*c^3)*(a^3*b^2 - 4*a^4*c)^2*e + 2*(5*sqrt(2)*sqrt(b*c + sqrt(b^2
- 4*a*c)*c)*a^3*b^8 - 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^6*c - 10*sqrt(2)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*a^3*b^7*c - 10*a^3*b^8*c + 286*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^4*c^2 + 88*sqrt(2)*sqr
t(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^5*c^2 + 5*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^6*c^2 + 128*a^4*b^6
*c^2 - 496*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^2*c^3 - 220*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a
^5*b^3*c^3 - 44*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^4*c^3 - 572*a^5*b^4*c^3 + 224*sqrt(2)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*a^7*c^4 + 112*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b*c^4 + 110*sqrt(2)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*a^5*b^2*c^4 + 992*a^6*b^2*c^4 - 56*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*c^5 - 448
*a^7*c^5 + 10*(b^2 - 4*a*c)*a^3*b^6*c - 88*(b^2 - 4*a*c)*a^4*b^4*c^2 + 220*(b^2 - 4*a*c)*a^5*b^2*c^3 - 112*(b^
2 - 4*a*c)*a^6*c^4)*d*abs(a^3*b^2 - 4*a^4*c) + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^6 - 14*sqrt(2)
*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^5*c - 2*a^5*b^6*c
+ 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^2*c^2 + 20*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^3
*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^4*c^2 + 28*a^6*b^4*c^2 - 96*sqrt(2)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*a^8*c^3 - 48*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b*c^3 - 10*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4
*a*c)*c)*a^6*b^2*c^3 - 128*a^7*b^2*c^3 + 24*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*c^4 + 192*a^8*c^4 + 2*
(b^2 - 4*a*c)*a^5*b^4*c - 20*(b^2 - 4*a*c)*a^6*b^2*c^2 + 48*(b^2 - 4*a*c)*a^7*c^3)*f*abs(a^3*b^2 - 4*a^4*c) -
2*(3*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^7 - 37*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^5*c -
6*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^6*c - 6*a^4*b^7*c + 152*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c
)*a^6*b^3*c^2 + 50*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^4*c^2 + 3*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c
)*c)*a^4*b^5*c^2 + 74*a^5*b^5*c^2 - 208*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b*c^3 - 104*sqrt(2)*sqrt(b
*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^2*c^3 - 25*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^3*c^3 - 304*a^6*b^3*c
^3 + 52*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b*c^4 + 416*a^7*b*c^4 + 6*(b^2 - 4*a*c)*a^4*b^5*c - 50*(b^
2 - 4*a*c)*a^5*b^3*c^2 + 104*(b^2 - 4*a*c)*a^6*b*c^3)*abs(a^3*b^2 - 4*a^4*c)*e + (10*a^6*b^9*c^2 - 138*a^7*b^7
*c^3 + 680*a^8*b^5*c^4 - 1376*a^9*b^3*c^5 + 896*a^10*b*c^6 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*a^6*b^9 + 69*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^7*c + 10*sqrt(2)*sqrt(
b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^8*c - 340*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4
*a*c)*c)*a^8*b^5*c^2 - 98*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^6*c^2 - 5*sqrt(2)*sq
rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^7*c^2 + 688*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^
2 - 4*a*c)*c)*a^9*b^3*c^3 + 288*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^4*c^3 + 49*sqr
t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*...

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Mupad [B]
time = 7.37, size = 2500, normalized size = 4.35 \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^4*(a + b*x^2 + c*x^4)^2),x)

[Out]

atan(((x*(204800*a^17*c^9*e^2 - 401408*a^16*c^10*d^2 - 73728*a^18*c^8*f^2 + 400*a^9*b^14*c^3*d^2 - 9440*a^10*b
^12*c^4*d^2 + 92816*a^11*b^10*c^5*d^2 - 488096*a^12*b^8*c^6*d^2 + 1458688*a^13*b^6*c^7*d^2 - 2401280*a^14*b^4*
c^8*d^2 + 1871872*a^15*b^2*c^9*d^2 + 144*a^11*b^12*c^3*e^2 - 3264*a^12*b^10*c^4*e^2 + 30112*a^13*b^8*c^5*e^2 -
143360*a^14*b^6*c^6*e^2 + 365568*a^15*b^4*c^7*e^2 - 458752*a^16*b^2*c^8*e^2 + 16*a^13*b^10*c^3*f^2 - 416*a^14
*b^8*c^4*f^2 + 4608*a^15*b^6*c^5*f^2 - 25600*a^16*b^4*c^6*f^2 + 69632*a^17*b^2*c^7*f^2 + 344064*a^17*c^9*d*f -
1236992*a^16*b*c^9*d*e + 237568*a^17*b*c^8*e*f - 480*a^10*b^13*c^3*d*e + 11104*a^11*b^11*c^4*d*e - 105824*a^1
2*b^9*c^5*d*e + 530432*a^13*b^7*c^6*d*e - 1469440*a^14*b^5*c^7*d*e + 2121728*a^15*b^3*c^8*d*e + 160*a^11*b^12*
c^3*d*f - 3968*a^12*b^10*c^4*d*f + 39488*a^13*b^8*c^5*d*f - 200704*a^14*b^6*c^6*d*f + 542720*a^15*b^4*c^7*d*f
- 720896*a^16*b^2*c^8*d*f - 96*a^12*b^11*c^3*e*f + 2336*a^13*b^9*c^4*e*f - 22528*a^14*b^7*c^5*e*f + 107520*a^1
5*b^5*c^6*e*f - 253952*a^16*b^3*c^7*e*f) + (-(25*b^15*d^2 + 9*a^2*b^13*e^2 + 25*b^6*d^2*(-(4*a*c - b^2)^9)^(1/
2) + a^4*b^11*f^2 - 80640*a^7*b*c^7*d^2 - 213*a^3*b^11*c*e^2 + 26880*a^8*b*c^6*e^2 - 27*a^5*b^9*c*f^2 - 3840*a
^9*b*c^5*f^2 - 9*a^5*c*f^2*(-(4*a*c - b^2)^9)^(1/2) - 30*a*b^14*d*e + 6366*a^2*b^11*c^2*d^2 - 35767*a^3*b^9*c^
3*d^2 + 116928*a^4*b^7*c^4*d^2 - 219744*a^5*b^5*c^5*d^2 + 215040*a^6*b^3*c^6*d^2 + 9*a^2*b^4*e^2*(-(4*a*c - b^
2)^9)^(1/2) - 49*a^3*c^3*d^2*(-(4*a*c - b^2)^9)^(1/2) + 2077*a^4*b^9*c^2*e^2 - 10656*a^5*b^7*c^3*e^2 + 30240*a
^6*b^5*c^4*e^2 - 44800*a^7*b^3*c^5*e^2 + a^4*b^2*f^2*(-(4*a*c - b^2)^9)^(1/2) + 25*a^4*c^2*e^2*(-(4*a*c - b^2)
^9)^(1/2) + 288*a^6*b^7*c^2*f^2 - 1504*a^7*b^5*c^3*f^2 + 3840*a^8*b^3*c^4*f^2 - 615*a*b^13*c*d^2 + 10*a^2*b^13
*d*f + 35840*a^8*c^7*d*e - 6*a^3*b^12*e*f - 15360*a^9*c^6*e*f - 30*a*b^5*d*e*(-(4*a*c - b^2)^9)^(1/2) + 724*a^
2*b^12*c*d*e - 258*a^3*b^11*c*d*f + 43520*a^8*b*c^6*d*f + 152*a^4*b^10*c*e*f + 246*a^2*b^2*c^2*d^2*(-(4*a*c -
b^2)^9)^(1/2) - 165*a*b^4*c*d^2*(-(4*a*c - b^2)^9)^(1/2) - 7278*a^3*b^10*c^2*d*e + 39132*a^4*b^8*c^3*d*e - 119
616*a^5*b^6*c^4*d*e + 201600*a^6*b^4*c^5*d*e - 161280*a^7*b^2*c^6*d*e + 10*a^2*b^4*d*f*(-(4*a*c - b^2)^9)^(1/2
) + 2706*a^4*b^9*c^2*d*f - 14784*a^5*b^7*c^3*d*f + 44352*a^6*b^5*c^4*d*f - 69120*a^7*b^3*c^5*d*f - 6*a^3*b^3*e
*f*(-(4*a*c - b^2)^9)^(1/2) + 42*a^4*c^2*d*f*(-(4*a*c - b^2)^9)^(1/2) - 1548*a^5*b^8*c^2*e*f + 8064*a^6*b^6*c^
3*e*f - 22400*a^7*b^4*c^4*e*f + 30720*a^8*b^2*c^5*e*f - 51*a^3*b^2*c*e^2*(-(4*a*c - b^2)^9)^(1/2) + 44*a^4*b*c
*e*f*(-(4*a*c - b^2)^9)^(1/2) + 184*a^2*b^3*c*d*e*(-(4*a*c - b^2)^9)^(1/2) - 186*a^3*b*c^2*d*e*(-(4*a*c - b^2)
^9)^(1/2) - 78*a^3*b^2*c*d*f*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9
*b^8*c^2 - 1280*a^10*b^6*c^3 + 3840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*(393216*a^20*c^8*f - 917504*a^19
*c^9*d + x*(-(25*b^15*d^2 + 9*a^2*b^13*e^2 + 25*b^6*d^2*(-(4*a*c - b^2)^9)^(1/2) + a^4*b^11*f^2 - 80640*a^7*b*
c^7*d^2 - 213*a^3*b^11*c*e^2 + 26880*a^8*b*c^6*e^2 - 27*a^5*b^9*c*f^2 - 3840*a^9*b*c^5*f^2 - 9*a^5*c*f^2*(-(4*
a*c - b^2)^9)^(1/2) - 30*a*b^14*d*e + 6366*a^2*b^11*c^2*d^2 - 35767*a^3*b^9*c^3*d^2 + 116928*a^4*b^7*c^4*d^2 -
219744*a^5*b^5*c^5*d^2 + 215040*a^6*b^3*c^6*d^2 + 9*a^2*b^4*e^2*(-(4*a*c - b^2)^9)^(1/2) - 49*a^3*c^3*d^2*(-(
4*a*c - b^2)^9)^(1/2) + 2077*a^4*b^9*c^2*e^2 - 10656*a^5*b^7*c^3*e^2 + 30240*a^6*b^5*c^4*e^2 - 44800*a^7*b^3*c
^5*e^2 + a^4*b^2*f^2*(-(4*a*c - b^2)^9)^(1/2) + 25*a^4*c^2*e^2*(-(4*a*c - b^2)^9)^(1/2) + 288*a^6*b^7*c^2*f^2
- 1504*a^7*b^5*c^3*f^2 + 3840*a^8*b^3*c^4*f^2 - 615*a*b^13*c*d^2 + 10*a^2*b^13*d*f + 35840*a^8*c^7*d*e - 6*a^3
*b^12*e*f - 15360*a^9*c^6*e*f - 30*a*b^5*d*e*(-(4*a*c - b^2)^9)^(1/2) + 724*a^2*b^12*c*d*e - 258*a^3*b^11*c*d*
f + 43520*a^8*b*c^6*d*f + 152*a^4*b^10*c*e*f + 246*a^2*b^2*c^2*d^2*(-(4*a*c - b^2)^9)^(1/2) - 165*a*b^4*c*d^2*
(-(4*a*c - b^2)^9)^(1/2) - 7278*a^3*b^10*c^2*d*e + 39132*a^4*b^8*c^3*d*e - 119616*a^5*b^6*c^4*d*e + 201600*a^6
*b^4*c^5*d*e - 161280*a^7*b^2*c^6*d*e + 10*a^2*b^4*d*f*(-(4*a*c - b^2)^9)^(1/2) + 2706*a^4*b^9*c^2*d*f - 14784
*a^5*b^7*c^3*d*f + 44352*a^6*b^5*c^4*d*f - 69120*a^7*b^3*c^5*d*f - 6*a^3*b^3*e*f*(-(4*a*c - b^2)^9)^(1/2) + 42
*a^4*c^2*d*f*(-(4*a*c - b^2)^9)^(1/2) - 1548*a^5*b^8*c^2*e*f + 8064*a^6*b^6*c^3*e*f - 22400*a^7*b^4*c^4*e*f +
30720*a^8*b^2*c^5*e*f - 51*a^3*b^2*c*e^2*(-(4*a*c - b^2)^9)^(1/2) + 44*a^4*b*c*e*f*(-(4*a*c - b^2)^9)^(1/2) +
184*a^2*b^3*c*d*e*(-(4*a*c - b^2)^9)^(1/2) - 186*a^3*b*c^2*d*e*(-(4*a*c - b^2)^9)^(1/2) - 78*a^3*b^2*c*d*f*(-(
4*a*c - b^2)^9)^(1/2))/(32*(a^7*b^12 + 4096*a^13*c^6 - 24*a^8*b^10*c + 240*a^9*b^8*c^2 - 1280*a^10*b^6*c^3 + 3
840*a^11*b^4*c^4 - 6144*a^12*b^2*c^5)))^(1/2)*(1048576*a^21*b*c^8 + 256*a^15*b^13*c^2 - 6144*a^16*b^11*c^3 + 6
1440*a^17*b^9*c^4 - 327680*a^18*b^7*c^5 + 983040*a^19*b^5*c^6 - 1572864*a^20*b^3*c^7) + 320*a^12*b^14*c^2*d -
7936*a^13*b^12*c^3*d + 82816*a^14*b^10*c^4*d - 468480*a^15*b^8*c^5*d + 1536000*a^16*b^6*c^6*d - 2867200*a^17*b
^4*c^7*d + 2719744*a^18*b^2*c^8*d - 192*a^13*b^...

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