3.1.68 \(\int \frac {x^6 (d+e x^2+f x^4)}{(a+b x^2+c x^4)^2} \, dx\) [68]

Optimal. Leaf size=550 \[ \frac {(c e-2 b f) x}{c^3}+\frac {f x^3}{3 c^2}+\frac {x \left (a \left (b^2 c e-2 a c^2 e-b^3 f-b c (c d-3 a f)\right )+\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)\right ) x^2\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (3 b^3 c e-13 a b c^2 e-5 b^4 f-b^2 c (c d-24 a f)+2 a c^2 (3 c d-7 a f)-\frac {3 b^4 c e-19 a b^2 c^2 e+20 a^2 c^3 e-5 b^5 f-b^3 c (c d-34 a f)+4 a b c^2 (2 c d-13 a f)}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} c^{7/2} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (3 b^3 c e-13 a b c^2 e-5 b^4 f-b^2 c (c d-24 a f)+2 a c^2 (3 c d-7 a f)+\frac {3 b^4 c e-19 a b^2 c^2 e+20 a^2 c^3 e-5 b^5 f-b^3 c (c d-34 a f)+4 a b c^2 (2 c d-13 a f)}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} c^{7/2} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}} \]

[Out]

(-2*b*f+c*e)*x/c^3+1/3*f*x^3/c^2+1/2*x*(a*(b^2*c*e-2*a*c^2*e-b^3*f-b*c*(-3*a*f+c*d))+(b^3*c*e-3*a*b*c^2*e-b^4*
f-b^2*c*(-4*a*f+c*d)+2*a*c^2*(-a*f+c*d))*x^2)/c^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))*(3*b^3*c*e-13*a*b*c^2*e-5*b^4*f-b^2*c*(-24*a*f+c*d)+2*a*c^2*(-7*a*f+3*c*d)+(-3*b^4
*c*e+19*a*b^2*c^2*e-20*a^2*c^3*e+5*b^5*f+b^3*c*(-34*a*f+c*d)-4*a*b*c^2*(-13*a*f+2*c*d))/(-4*a*c+b^2)^(1/2))/c^
(7/2)/(-4*a*c+b^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))*(3*b^3*c*e-13*a*b*c^2*e-5*b^4*f-b^2*c*(-24*a*f+c*d)+2*a*c^2*(-7*a*f+3*c*d)+(3*b^4*c*e-19*a*b^2*c^2*e+20*a
^2*c^3*e-5*b^5*f-b^3*c*(-34*a*f+c*d)+4*a*b*c^2*(-13*a*f+2*c*d))/(-4*a*c+b^2)^(1/2))/c^(7/2)/(-4*a*c+b^2)*2^(1/
2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]
time = 9.87, antiderivative size = 550, 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 = {1682, 1690, 1180, 211} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (-\frac {20 a^2 c^3 e-b^3 c (c d-34 a f)-19 a b^2 c^2 e+4 a b c^2 (2 c d-13 a f)-5 b^5 f+3 b^4 c e}{\sqrt {b^2-4 a c}}-b^2 c (c d-24 a f)-13 a b c^2 e+2 a c^2 (3 c d-7 a f)-5 b^4 f+3 b^3 c e\right )}{2 \sqrt {2} c^{7/2} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (\frac {20 a^2 c^3 e-b^3 c (c d-34 a f)-19 a b^2 c^2 e+4 a b c^2 (2 c d-13 a f)-5 b^5 f+3 b^4 c e}{\sqrt {b^2-4 a c}}-b^2 c (c d-24 a f)-13 a b c^2 e+2 a c^2 (3 c d-7 a f)-5 b^4 f+3 b^3 c e\right )}{2 \sqrt {2} c^{7/2} \left (b^2-4 a c\right ) \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {x \left (a \left (-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )+x^2 \left (-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^4 (-f)+b^3 c e\right )\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {x (c e-2 b f)}{c^3}+\frac {f x^3}{3 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^6*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x]

[Out]

((c*e - 2*b*f)*x)/c^3 + (f*x^3)/(3*c^2) + (x*(a*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f)) + (b^3*c*e -
3*a*b*c^2*e - b^4*f - b^2*c*(c*d - 4*a*f) + 2*a*c^2*(c*d - a*f))*x^2))/(2*c^3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^
4)) - ((3*b^3*c*e - 13*a*b*c^2*e - 5*b^4*f - b^2*c*(c*d - 24*a*f) + 2*a*c^2*(3*c*d - 7*a*f) - (3*b^4*c*e - 19*
a*b^2*c^2*e + 20*a^2*c^3*e - 5*b^5*f - b^3*c*(c*d - 34*a*f) + 4*a*b*c^2*(2*c*d - 13*a*f))/Sqrt[b^2 - 4*a*c])*A
rcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*c^(7/2)*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4
*a*c]]) - ((3*b^3*c*e - 13*a*b*c^2*e - 5*b^4*f - b^2*c*(c*d - 24*a*f) + 2*a*c^2*(3*c*d - 7*a*f) + (3*b^4*c*e -
19*a*b^2*c^2*e + 20*a^2*c^3*e - 5*b^5*f - b^3*c*(c*d - 34*a*f) + 4*a*b*c^2*(2*c*d - 13*a*f))/Sqrt[b^2 - 4*a*c
])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*c^(7/2)*(b^2 - 4*a*c)*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 1682

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[(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] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p +
7)*(b*d - 2*a*e)*x^2, 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] && IGtQ[m/2, 0]

Rule 1690

Int[(Pq_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^2 + c*x^4), x], x
] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1

Rubi steps

\begin {align*} \int \frac {x^6 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx &=\frac {x \left (a \left (b^2 c e-2 a c^2 e-b^3 f-b c (c d-3 a f)\right )+\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)\right ) x^2\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {\frac {a^2 \left (b^2 c e-2 a c^2 e-b^3 f-b c (c d-3 a f)\right )}{c^3}+\frac {a \left (b^3 c e-5 a b c^2 e-b^4 f-b^2 c (c d-6 a f)+6 a c^2 (c d-a f)\right ) x^2}{c^3}-\frac {2 a \left (b^2-4 a c\right ) (c e-b f) x^4}{c^2}+2 a \left (4 a-\frac {b^2}{c}\right ) f x^6}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac {x \left (a \left (b^2 c e-2 a c^2 e-b^3 f-b c (c d-3 a f)\right )+\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)\right ) x^2\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \left (-\frac {2 a \left (b^2-4 a c\right ) (c e-2 b f)}{c^3}-\frac {2 a \left (b^2-4 a c\right ) f x^2}{c^2}-\frac {-a^2 \left (3 b^2 c e-10 a c^2 e-5 b^3 f-b c (c d-19 a f)\right )-a \left (3 b^3 c e-13 a b c^2 e-5 b^4 f-b^2 c (c d-24 a f)+2 a c^2 (3 c d-7 a f)\right ) x^2}{c^3 \left (a+b x^2+c x^4\right )}\right ) \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac {(c e-2 b f) x}{c^3}+\frac {f x^3}{3 c^2}+\frac {x \left (a \left (b^2 c e-2 a c^2 e-b^3 f-b c (c d-3 a f)\right )+\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)\right ) x^2\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\int \frac {-a^2 \left (3 b^2 c e-10 a c^2 e-5 b^3 f-b c (c d-19 a f)\right )-a \left (3 b^3 c e-13 a b c^2 e-5 b^4 f-b^2 c (c d-24 a f)+2 a c^2 (3 c d-7 a f)\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a c^3 \left (b^2-4 a c\right )}\\ &=\frac {(c e-2 b f) x}{c^3}+\frac {f x^3}{3 c^2}+\frac {x \left (a \left (b^2 c e-2 a c^2 e-b^3 f-b c (c d-3 a f)\right )+\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)\right ) x^2\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (3 b^3 c e-13 a b c^2 e-5 b^4 f-b^2 c (c d-24 a f)+2 a c^2 (3 c d-7 a f)-\frac {3 b^4 c e-19 a b^2 c^2 e+20 a^2 c^3 e-5 b^5 f-b^3 c (c d-34 a f)+4 a b c^2 (2 c d-13 a f)}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{4 c^3 \left (b^2-4 a c\right )}-\frac {\left (3 b^3 c e-13 a b c^2 e-5 b^4 f-b^2 c (c d-24 a f)+2 a c^2 (3 c d-7 a f)+\frac {3 b^4 c e-19 a b^2 c^2 e+20 a^2 c^3 e-5 b^5 f-b^3 c (c d-34 a f)+4 a b c^2 (2 c d-13 a f)}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{4 c^3 \left (b^2-4 a c\right )}\\ &=\frac {(c e-2 b f) x}{c^3}+\frac {f x^3}{3 c^2}+\frac {x \left (a \left (b^2 c e-2 a c^2 e-b^3 f-b c (c d-3 a f)\right )+\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)\right ) x^2\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (3 b^3 c e-13 a b c^2 e-5 b^4 f-b^2 c (c d-24 a f)+2 a c^2 (3 c d-7 a f)-\frac {3 b^4 c e-19 a b^2 c^2 e+20 a^2 c^3 e-5 b^5 f-b^3 c (c d-34 a f)+4 a b c^2 (2 c d-13 a f)}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} c^{7/2} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (3 b^3 c e-13 a b c^2 e-5 b^4 f-b^2 c (c d-24 a f)+2 a c^2 (3 c d-7 a f)+\frac {3 b^4 c e-19 a b^2 c^2 e+20 a^2 c^3 e-5 b^5 f-b^3 c (c d-34 a f)+4 a b c^2 (2 c d-13 a f)}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} c^{7/2} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [A]
time = 1.30, size = 648, normalized size = 1.18 \begin {gather*} \frac {12 \sqrt {c} (c e-2 b f) x+4 c^{3/2} f x^3-\frac {6 \sqrt {c} x \left (b^2 \left (c^2 d-b c e+b^2 f\right ) x^2+a^2 c \left (-3 b f+2 c \left (e+f x^2\right )\right )+a \left (b^3 f-2 c^3 d x^2+b c^2 \left (d+3 e x^2\right )-b^2 c \left (e+4 f 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} \left (-5 b^5 f+a b c^2 \left (8 c d+13 \sqrt {b^2-4 a c} e-52 a f\right )-b^3 c \left (c d+3 \sqrt {b^2-4 a c} e-34 a f\right )+b^4 \left (3 c e+5 \sqrt {b^2-4 a c} f\right )+b^2 c \left (c \sqrt {b^2-4 a c} d-19 a c e-24 a \sqrt {b^2-4 a c} f\right )+2 a c^2 \left (-3 c \sqrt {b^2-4 a c} d+10 a c e+7 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} \left (5 b^5 f+b^3 c \left (c d-3 \sqrt {b^2-4 a c} e-34 a f\right )+a b c^2 \left (-8 c d+13 \sqrt {b^2-4 a c} e+52 a f\right )+b^4 \left (-3 c e+5 \sqrt {b^2-4 a c} f\right )+b^2 c \left (c \sqrt {b^2-4 a c} d+19 a c e-24 a \sqrt {b^2-4 a c} f\right )-2 a c^2 \left (3 c \sqrt {b^2-4 a c} d+10 a c e-7 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 c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^6*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x]

[Out]

(12*Sqrt[c]*(c*e - 2*b*f)*x + 4*c^(3/2)*f*x^3 - (6*Sqrt[c]*x*(b^2*(c^2*d - b*c*e + b^2*f)*x^2 + a^2*c*(-3*b*f
+ 2*c*(e + f*x^2)) + a*(b^3*f - 2*c^3*d*x^2 + b*c^2*(d + 3*e*x^2) - b^2*c*(e + 4*f*x^2))))/((b^2 - 4*a*c)*(a +
b*x^2 + c*x^4)) + (3*Sqrt[2]*(-5*b^5*f + a*b*c^2*(8*c*d + 13*Sqrt[b^2 - 4*a*c]*e - 52*a*f) - b^3*c*(c*d + 3*S
qrt[b^2 - 4*a*c]*e - 34*a*f) + b^4*(3*c*e + 5*Sqrt[b^2 - 4*a*c]*f) + b^2*c*(c*Sqrt[b^2 - 4*a*c]*d - 19*a*c*e -
24*a*Sqrt[b^2 - 4*a*c]*f) + 2*a*c^2*(-3*c*Sqrt[b^2 - 4*a*c]*d + 10*a*c*e + 7*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
]*(5*b^5*f + b^3*c*(c*d - 3*Sqrt[b^2 - 4*a*c]*e - 34*a*f) + a*b*c^2*(-8*c*d + 13*Sqrt[b^2 - 4*a*c]*e + 52*a*f)
+ b^4*(-3*c*e + 5*Sqrt[b^2 - 4*a*c]*f) + b^2*c*(c*Sqrt[b^2 - 4*a*c]*d + 19*a*c*e - 24*a*Sqrt[b^2 - 4*a*c]*f)
- 2*a*c^2*(3*c*Sqrt[b^2 - 4*a*c]*d + 10*a*c*e - 7*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]]))/(12*c^(7/2))

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Maple [A]
time = 0.10, size = 682, normalized size = 1.24

method	result	size
risch	\(\frac {f \,x^{3}}{3 c^{2}}-\frac {2 b f x}{c^{3}}+\frac {e x}{c^{2}}+\frac {\frac {\left (2 a^{2} c^{2} f -4 a \,b^{2} c f +3 a b \,c^{2} e -2 c^{3} a d +b^{4} f -b^{3} c e +b^{2} c^{2} d \right ) x^{3}}{8 a c -2 b^{2}}-\frac {a \left (3 a b c f -2 a \,c^{2} e -b^{3} f +b^{2} c e -b \,c^{2} d \right ) x}{2 \left (4 a c -b^{2}\right )}}{c^{3} \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (-\frac {\left (14 a^{2} c^{2} f -24 a \,b^{2} c f +13 a b \,c^{2} e -6 c^{3} a d +5 b^{4} f -3 b^{3} c e +b^{2} c^{2} d \right ) \textit {\_R}^{2}}{4 a c -b^{2}}+\frac {a \left (19 a b c f -10 a \,c^{2} e -5 b^{3} f +3 b^{2} c e -b \,c^{2} d \right )}{4 a c -b^{2}}\right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}}{4 c^{3}}\)	\(323\)
default	\(-\frac {-\frac {1}{3} c \,x^{3} f +2 b f x -c e x}{c^{3}}+\frac {\frac {\frac {\left (2 a^{2} c^{2} f -4 a \,b^{2} c f +3 a b \,c^{2} e -2 c^{3} a d +b^{4} f -b^{3} c e +b^{2} c^{2} d \right ) x^{3}}{8 a c -2 b^{2}}-\frac {a \left (3 a b c f -2 a \,c^{2} e -b^{3} f +b^{2} c e -b \,c^{2} d \right ) x}{2 \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {2 c \left (-\frac {\left (-14 a^{2} c^{2} f \sqrt {-4 a c +b^{2}}+24 a \,b^{2} c f \sqrt {-4 a c +b^{2}}-13 a b \,c^{2} e \sqrt {-4 a c +b^{2}}+6 c^{3} a d \sqrt {-4 a c +b^{2}}-5 b^{4} f \sqrt {-4 a c +b^{2}}+3 b^{3} c e \sqrt {-4 a c +b^{2}}-b^{2} c^{2} d \sqrt {-4 a c +b^{2}}+52 a^{2} b \,c^{2} f -20 a^{2} c^{3} e -34 a \,b^{3} c f +19 a \,b^{2} c^{2} e -8 a b \,c^{3} d +5 b^{5} f -3 b^{4} c e +b^{3} c^{2} d \right ) \sqrt {2}\, \arctanh \left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, c \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-14 a^{2} c^{2} f \sqrt {-4 a c +b^{2}}+24 a \,b^{2} c f \sqrt {-4 a c +b^{2}}-13 a b \,c^{2} e \sqrt {-4 a c +b^{2}}+6 c^{3} a d \sqrt {-4 a c +b^{2}}-5 b^{4} f \sqrt {-4 a c +b^{2}}+3 b^{3} c e \sqrt {-4 a c +b^{2}}-b^{2} c^{2} d \sqrt {-4 a c +b^{2}}-52 a^{2} b \,c^{2} f +20 a^{2} c^{3} e +34 a \,b^{3} c f -19 a \,b^{2} c^{2} e +8 a b \,c^{3} d -5 b^{5} f +3 b^{4} c e -b^{3} c^{2} d \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, c \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}}{c^{3}}\)	\(682\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

-1/c^3*(-1/3*c*x^3*f+2*b*f*x-c*e*x)+1/c^3*((1/2*(2*a^2*c^2*f-4*a*b^2*c*f+3*a*b*c^2*e-2*a*c^3*d+b^4*f-b^3*c*e+b
^2*c^2*d)/(4*a*c-b^2)*x^3-1/2*a*(3*a*b*c*f-2*a*c^2*e-b^3*f+b^2*c*e-b*c^2*d)/(4*a*c-b^2)*x)/(c*x^4+b*x^2+a)+2/(
4*a*c-b^2)*c*(-1/8*(-14*a^2*c^2*f*(-4*a*c+b^2)^(1/2)+24*a*b^2*c*f*(-4*a*c+b^2)^(1/2)-13*a*b*c^2*e*(-4*a*c+b^2)
^(1/2)+6*c^3*a*d*(-4*a*c+b^2)^(1/2)-5*b^4*f*(-4*a*c+b^2)^(1/2)+3*b^3*c*e*(-4*a*c+b^2)^(1/2)-b^2*c^2*d*(-4*a*c+
b^2)^(1/2)+52*a^2*b*c^2*f-20*a^2*c^3*e-34*a*b^3*c*f+19*a*b^2*c^2*e-8*a*b*c^3*d+5*b^5*f-3*b^4*c*e+b^3*c^2*d)/(-
4*a*c+b^2)^(1/2)/c*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*(-14*a^2*c^2*f*(-4*a*c+b^2)^(1/2)+24*a*b^2*c*f*(-4*a*c+b^2)^(1/2)-13*a*b*c^2*e*(-4*a*c+b^2)^(1/2)+6*
c^3*a*d*(-4*a*c+b^2)^(1/2)-5*b^4*f*(-4*a*c+b^2)^(1/2)+3*b^3*c*e*(-4*a*c+b^2)^(1/2)-b^2*c^2*d*(-4*a*c+b^2)^(1/2
)-52*a^2*b*c^2*f+20*a^2*c^3*e+34*a*b^3*c*f-19*a*b^2*c^2*e+8*a*b*c^3*d-5*b^5*f+3*b^4*c*e-b^3*c^2*d)/(-4*a*c+b^2
)^(1/2)/c*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))))

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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(x^6*(f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 18909 vs. \(2 (506) = 1012\).
time = 75.86, size = 18909, normalized size = 34.38 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/12*(4*(b^2*c^2 - 4*a*c^3)*f*x^7 + 4*(3*(b^2*c^2 - 4*a*c^3)*e - 5*(b^3*c - 4*a*b*c^2)*f)*x^5 - 2*(3*(b^2*c^2
- 2*a*c^3)*d - 3*(3*b^3*c - 11*a*b*c^2)*e + (15*b^4 - 62*a*b^2*c + 14*a^2*c^2)*f)*x^3 + 3*sqrt(1/2)*(a*b^2*c^3
- 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^4 + (b^3*c^3 - 4*a*b*c^4)*x^2)*sqrt(-((b^5*c^4 - 15*a*b^3*c^5 + 60*a^2*b*
c^6)*d^2 - 2*(3*b^6*c^3 - 40*a*b^4*c^4 + 150*a^2*b^2*c^5 - 120*a^3*c^6)*d*e + (9*b^7*c^2 - 105*a*b^5*c^3 + 385
*a^2*b^3*c^4 - 420*a^3*b*c^5)*e^2 + (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)*f^2 + 2*((5*b^7*c^2 - 69*a*b^5*c^3 + 285*a^2*b^3*c^4 - 340*a^3*b*c^5)*d - (15*b^8*c - 182*a*b^6*c^2 + 735*
a^2*b^4*c^3 - 1050*a^3*b^2*c^4 + 280*a^4*c^5)*e)*f + (b^6*c^7 - 12*a*b^4*c^8 + 48*a^2*b^2*c^9 - 64*a^3*c^10)*s
qrt(((b^4*c^8 - 18*a*b^2*c^9 + 81*a^2*c^10)*d^4 - 4*(3*b^5*c^7 - 49*a*b^3*c^8 + 198*a^2*b*c^9)*d^3*e + 6*(9*b^
6*c^6 - 132*a*b^4*c^7 + 484*a^2*b^2*c^8 - 75*a^3*c^9)*d^2*e^2 - 4*(27*b^7*c^5 - 351*a*b^5*c^6 + 1197*a^2*b^3*c
^7 - 550*a ...

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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(x**6*(f*x**4+e*x**2+d)/(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. 8957 vs. \(2 (517) = 1034\).
time = 7.93, size = 8957, normalized size = 16.29 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

-1/2*(b^2*c^2*d*x^3 - 2*a*c^3*d*x^3 + b^4*f*x^3 - 4*a*b^2*c*f*x^3 + 2*a^2*c^2*f*x^3 - b^3*c*x^3*e + 3*a*b*c^2*
x^3*e + a*b*c^2*d*x + a*b^3*f*x - 3*a^2*b*c*f*x - a*b^2*c*x*e + 2*a^2*c^2*x*e)/((b^2*c^3 - 4*a*c^4)*(c*x^4 + b
*x^2 + a)) - 1/16*((2*b^4*c^4 - 20*a*b^2*c^5 + 48*a^2*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*
a*c)*c)*b^4*c^2 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 + 2*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^3 - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c
)*a^2*c^4 - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*s
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^4 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^5 - 2
*(b^2 - 4*a*c)*b^2*c^4 + 12*(b^2 - 4*a*c)*a*c^5)*(b^2*c^3 - 4*a*c^4)^2*d + (10*b^6*c^2 - 88*a*b^4*c^3 + 220*a^
2*b^2*c^4 - 112*a^3*c^5 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6 + 44*sqrt(2)*sqrt(b^
2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)
*c)*b^5*c - 110*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 - 48*sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)
*b^4*c^2 + 56*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^3 + 28*sqrt(2)*sqrt(b^2 - 4*a*c)
*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^
2*c^3 - 14*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^4 - 10*(b^2 - 4*a*c)*b^4*c^2 + 48*(
b^2 - 4*a*c)*a*b^2*c^3 - 28*(b^2 - 4*a*c)*a^2*c^4)*(b^2*c^3 - 4*a*c^4)^2*f - (6*b^5*c^3 - 50*a*b^3*c^4 + 104*a
^2*b*c^5 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c + 25*sqrt(2)*sqrt(b^2 - 4*a*c)*sq
rt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2
- 52*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 - 26*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b
*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^3 + 13
*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^4 - 6*(b^2 - 4*a*c)*b^3*c^3 + 26*(b^2 - 4*a*c
)*a*b*c^4)*(b^2*c^3 - 4*a*c^4)^2*e - 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^6 - 8*sqrt(2)*sqrt(b*c
+ sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^7 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^7 - 2*a*b^5*c^7 + 16*s
qrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^8 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^8 + sqr
t(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^8 + 16*a^2*b^3*c^8 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^
2*b*c^9 - 32*a^3*b*c^9 + 2*(b^2 - 4*a*c)*a*b^3*c^7 - 8*(b^2 - 4*a*c)*a^2*b*c^8)*d*abs(b^2*c^3 - 4*a*c^4) - 2*(
5*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^7*c^4 - 59*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c^5 -
10*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6*c^5 - 10*a*b^7*c^5 + 232*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c
)*c)*a^3*b^3*c^6 + 78*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^6 + 5*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*
a*c)*c)*a*b^5*c^6 + 118*a^2*b^5*c^6 - 304*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^7 - 152*sqrt(2)*sqrt
(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^7 - 39*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^7 - 464*a^3*b^3
*c^7 + 76*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^8 + 608*a^4*b*c^8 + 10*(b^2 - 4*a*c)*a*b^5*c^5 - 78*
(b^2 - 4*a*c)*a^2*b^3*c^6 + 152*(b^2 - 4*a*c)*a^3*b*c^7)*f*abs(b^2*c^3 - 4*a*c^4) + 2*(3*sqrt(2)*sqrt(b*c + sq
rt(b^2 - 4*a*c)*c)*a*b^6*c^5 - 34*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^6 - 6*sqrt(2)*sqrt(b*c + s
qrt(b^2 - 4*a*c)*c)*a*b^5*c^6 - 6*a*b^6*c^6 + 128*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^7 + 44*sqr
t(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^7 + 3*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^7 + 68*a^
2*b^4*c^7 - 160*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*c^8 - 80*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a
^3*b*c^8 - 22*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^8 - 256*a^3*b^2*c^8 + 40*sqrt(2)*sqrt(b*c + sq
rt(b^2 - 4*a*c)*c)*a^3*c^9 + 320*a^4*c^9 + 6*(b^2 - 4*a*c)*a*b^4*c^6 - 44*(b^2 - 4*a*c)*a^2*b^2*c^7 + 80*(b^2
- 4*a*c)*a^3*c^8)*abs(b^2*c^3 - 4*a*c^4)*e - (2*b^8*c^10 - 32*a*b^6*c^11 + 160*a^2*b^4*c^12 - 256*a^3*b^2*c^13
- sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^8*c^8 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*a*b^6*c^9 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^7*c^9 - 80*sqr
t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^10 - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*a*b^5*c^10 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6*c^10 + 128*sqr
t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^11 + 64*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^11 + 12*sqrt(2)*...

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Mupad [B]
time = 4.10, size = 2500, normalized size = 4.55 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x)

[Out]

x*(e/c^2 - (2*b*f)/c^3) + ((x^3*(b^4*f + b^2*c^2*d + 2*a^2*c^2*f - 2*a*c^3*d - b^3*c*e + 3*a*b*c^2*e - 4*a*b^2
*c*f))/(2*(4*a*c - b^2)) + (x*(2*a^2*c^2*e + a*b^3*f + a*b*c^2*d - a*b^2*c*e - 3*a^2*b*c*f))/(2*(4*a*c - b^2))
)/(a*c^3 + c^4*x^4 + b*c^3*x^2) - atan(((((10240*a^5*c^9*e + 192*a^2*b^5*c^7*d - 768*a^3*b^3*c^8*d - 736*a^2*b
^6*c^6*e + 4224*a^3*b^4*c^7*e - 10752*a^4*b^2*c^8*e + 1264*a^2*b^7*c^5*f - 7488*a^3*b^5*c^6*f + 19712*a^4*b^3*
c^7*f - 16*a*b^7*c^6*d + 1024*a^4*b*c^9*d + 48*a*b^8*c^5*e - 80*a*b^9*c^4*f - 19456*a^5*b*c^8*f)/(8*(64*a^3*c^
8 - b^6*c^5 + 12*a*b^4*c^6 - 48*a^2*b^2*c^7)) - (x*(-(25*b^15*f^2 + b^11*c^4*d^2 + 9*b^13*c^2*e^2 + 25*b^6*f^2
*(-(4*a*c - b^2)^9)^(1/2) - 27*a*b^9*c^5*d^2 - 3840*a^5*b*c^9*d^2 - 9*a*c^5*d^2*(-(4*a*c - b^2)^9)^(1/2) - 213
*a*b^11*c^3*e^2 + 26880*a^6*b*c^8*e^2 - 80640*a^7*b*c^7*f^2 - 30*b^14*c*e*f + 288*a^2*b^7*c^6*d^2 - 1504*a^3*b
^5*c^7*d^2 + 3840*a^4*b^3*c^8*d^2 + 2077*a^2*b^9*c^4*e^2 - 10656*a^3*b^7*c^5*e^2 + 30240*a^4*b^5*c^6*e^2 - 448
00*a^5*b^3*c^7*e^2 + 25*a^2*c^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + b^2*c^4*d^2*(-(4*a*c - b^2)^9)^(1/2) + 6366*a^2
*b^11*c^2*f^2 - 35767*a^3*b^9*c^3*f^2 + 116928*a^4*b^7*c^4*f^2 - 219744*a^5*b^5*c^5*f^2 + 215040*a^6*b^3*c^6*f
^2 - 49*a^3*c^3*f^2*(-(4*a*c - b^2)^9)^(1/2) + 9*b^4*c^2*e^2*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c*f^2 - 153
60*a^6*c^9*d*e - 6*b^12*c^3*d*e + 35840*a^7*c^8*e*f + 10*b^13*c^2*d*f + 152*a*b^10*c^4*d*e - 258*a*b^11*c^3*d*
f + 43520*a^6*b*c^8*d*f + 724*a*b^12*c^2*e*f - 30*b^5*c*e*f*(-(4*a*c - b^2)^9)^(1/2) + 246*a^2*b^2*c^2*f^2*(-(
4*a*c - b^2)^9)^(1/2) - 165*a*b^4*c*f^2*(-(4*a*c - b^2)^9)^(1/2) - 1548*a^2*b^8*c^5*d*e + 8064*a^3*b^6*c^6*d*e
- 22400*a^4*b^4*c^7*d*e + 30720*a^5*b^2*c^8*d*e + 2706*a^2*b^9*c^4*d*f - 14784*a^3*b^7*c^5*d*f + 44352*a^4*b^
5*c^6*d*f - 69120*a^5*b^3*c^7*d*f + 42*a^2*c^4*d*f*(-(4*a*c - b^2)^9)^(1/2) - 6*b^3*c^3*d*e*(-(4*a*c - b^2)^9)
^(1/2) - 7278*a^2*b^10*c^3*e*f + 39132*a^3*b^8*c^4*e*f - 119616*a^4*b^6*c^5*e*f + 201600*a^5*b^4*c^6*e*f - 161
280*a^6*b^2*c^7*e*f + 10*b^4*c^2*d*f*(-(4*a*c - b^2)^9)^(1/2) - 51*a*b^2*c^3*e^2*(-(4*a*c - b^2)^9)^(1/2) + 44
*a*b*c^4*d*e*(-(4*a*c - b^2)^9)^(1/2) - 78*a*b^2*c^3*d*f*(-(4*a*c - b^2)^9)^(1/2) + 184*a*b^3*c^2*e*f*(-(4*a*c
- b^2)^9)^(1/2) - 186*a^2*b*c^3*e*f*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^6*c^13 + b^12*c^7 - 24*a*b^10*c^8 +
240*a^2*b^8*c^9 - 1280*a^3*b^6*c^10 + 3840*a^4*b^4*c^11 - 6144*a^5*b^2*c^12)))^(1/2)*(16*b^7*c^7 - 192*a*b^5*
c^8 - 1024*a^3*b*c^10 + 768*a^2*b^3*c^9))/(2*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))*(-(25*b^15*f^2 + b^11*c^4*
d^2 + 9*b^13*c^2*e^2 + 25*b^6*f^2*(-(4*a*c - b^2)^9)^(1/2) - 27*a*b^9*c^5*d^2 - 3840*a^5*b*c^9*d^2 - 9*a*c^5*d
^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c^3*e^2 + 26880*a^6*b*c^8*e^2 - 80640*a^7*b*c^7*f^2 - 30*b^14*c*e*f +
288*a^2*b^7*c^6*d^2 - 1504*a^3*b^5*c^7*d^2 + 3840*a^4*b^3*c^8*d^2 + 2077*a^2*b^9*c^4*e^2 - 10656*a^3*b^7*c^5*
e^2 + 30240*a^4*b^5*c^6*e^2 - 44800*a^5*b^3*c^7*e^2 + 25*a^2*c^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + b^2*c^4*d^2*(-
(4*a*c - b^2)^9)^(1/2) + 6366*a^2*b^11*c^2*f^2 - 35767*a^3*b^9*c^3*f^2 + 116928*a^4*b^7*c^4*f^2 - 219744*a^5*b
^5*c^5*f^2 + 215040*a^6*b^3*c^6*f^2 - 49*a^3*c^3*f^2*(-(4*a*c - b^2)^9)^(1/2) + 9*b^4*c^2*e^2*(-(4*a*c - b^2)^
9)^(1/2) - 615*a*b^13*c*f^2 - 15360*a^6*c^9*d*e - 6*b^12*c^3*d*e + 35840*a^7*c^8*e*f + 10*b^13*c^2*d*f + 152*a
*b^10*c^4*d*e - 258*a*b^11*c^3*d*f + 43520*a^6*b*c^8*d*f + 724*a*b^12*c^2*e*f - 30*b^5*c*e*f*(-(4*a*c - b^2)^9
)^(1/2) + 246*a^2*b^2*c^2*f^2*(-(4*a*c - b^2)^9)^(1/2) - 165*a*b^4*c*f^2*(-(4*a*c - b^2)^9)^(1/2) - 1548*a^2*b
^8*c^5*d*e + 8064*a^3*b^6*c^6*d*e - 22400*a^4*b^4*c^7*d*e + 30720*a^5*b^2*c^8*d*e + 2706*a^2*b^9*c^4*d*f - 147
84*a^3*b^7*c^5*d*f + 44352*a^4*b^5*c^6*d*f - 69120*a^5*b^3*c^7*d*f + 42*a^2*c^4*d*f*(-(4*a*c - b^2)^9)^(1/2) -
6*b^3*c^3*d*e*(-(4*a*c - b^2)^9)^(1/2) - 7278*a^2*b^10*c^3*e*f + 39132*a^3*b^8*c^4*e*f - 119616*a^4*b^6*c^5*e
*f + 201600*a^5*b^4*c^6*e*f - 161280*a^6*b^2*c^7*e*f + 10*b^4*c^2*d*f*(-(4*a*c - b^2)^9)^(1/2) - 51*a*b^2*c^3*
e^2*(-(4*a*c - b^2)^9)^(1/2) + 44*a*b*c^4*d*e*(-(4*a*c - b^2)^9)^(1/2) - 78*a*b^2*c^3*d*f*(-(4*a*c - b^2)^9)^(
1/2) + 184*a*b^3*c^2*e*f*(-(4*a*c - b^2)^9)^(1/2) - 186*a^2*b*c^3*e*f*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^6*
c^13 + b^12*c^7 - 24*a*b^10*c^8 + 240*a^2*b^8*c^9 - 1280*a^3*b^6*c^10 + 3840*a^4*b^4*c^11 - 6144*a^5*b^2*c^12)
))^(1/2) - (x*(25*b^10*f^2 - 72*a^3*c^7*d^2 + 200*a^4*c^6*e^2 + b^6*c^4*d^2 - 392*a^5*c^5*f^2 + 9*b^8*c^2*e^2
- 16*a*b^4*c^5*d^2 - 114*a*b^6*c^3*e^2 - 30*b^9*c*e*f + 74*a^2*b^2*c^6*d^2 + 481*a^2*b^4*c^4*e^2 - 718*a^3*b^2
*c^5*e^2 + 1676*a^2*b^6*c^2*f^2 - 3536*a^3*b^4*c^3*f^2 + 2794*a^4*b^2*c^4*f^2 - 340*a*b^8*c*f^2 + 336*a^4*c^6*
d*f - 6*b^7*c^3*d*e + 10*b^8*c^2*d*f + 86*a*b^5*c^4*d*e + 472*a^3*b*c^6*d*e - 148*a*b^6*c^3*d*f + 394*a*b^7*c^
2*e*f - 1768*a^4*b*c^5*e*f - 374*a^2*b^3*c^5*d*e + 698*a^2*b^4*c^4*d*f - 1132*a^3*b^2*c^5*d*f - 1804*a^2*b^5*c
^3*e*f + 3266*a^3*b^3*c^4*e*f))/(2*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))*(-(25*b^15*f^2 + b^11*c^4*d^2 + 9*b^
13*c^2*e^2 + 25*b^6*f^2*(-(4*a*c - b^2)^9)^(1/2...

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