3.2.29 \(\int \frac {d+e x^2+f x^4+g x^6}{x^2 (a+b x^2+c x^4)^2} \, dx\) [129]

Optimal. Leaf size=460 \[ -\frac {d}{a^2 x}-\frac {x \left (a \left (\frac {b^3 d}{a}-b (3 c d+b e)+a (2 c e+b f)-2 a^2 g\right )+\left (b^2 c d-2 a c (c d-a f)-a b (c e+a g)\right ) x^2\right )}{2 a^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (3 b^2 c d-2 a c (5 c d-a f)-a b (c e+a g)+\frac {3 b^3 c d-4 a b c (4 c d+a f)-a b^2 (c e-a g)+4 a^2 c (3 c e+a g)}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (3 b^2 c d-2 a c (5 c d-a f)-a b (c e+a g)-\frac {3 b^3 c d-4 a b c (4 c d+a f)-a b^2 (c e-a g)+4 a^2 c (3 c e+a g)}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}} \]

[Out]

-d/a^2/x-1/2*x*(a*(b^3*d/a-b*(b*e+3*c*d)+a*(b*f+2*c*e)-2*a^2*g)+(b^2*c*d-2*a*c*(-a*f+c*d)-a*b*(a*g+c*e))*x^2)/
a^2/(-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^2*c*d-2*a*c*(
-a*f+5*c*d)-a*b*(a*g+c*e)+(3*b^3*c*d-4*a*b*c*(a*f+4*c*d)-a*b^2*(-a*g+c*e)+4*a^2*c*(a*g+3*c*e))/(-4*a*c+b^2)^(1
/2))/a^2/(-4*a*c+b^2)*2^(1/2)/c^(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^2*c*d-2*a*c*(-a*f+5*c*d)-a*b*(a*g+c*e)+(-3*b^3*c*d+4*a*b*c*(a*f+4*c*d)+a*b^2*(-a*g+c*e)-4
*a^2*c*(a*g+3*c*e))/(-4*a*c+b^2)^(1/2))/a^2/(-4*a*c+b^2)*2^(1/2)/c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]
time = 1.81, antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1683, 1678, 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 {4 a^2 c (a g+3 c e)-a b^2 (c e-a g)-4 a b c (a f+4 c d)+3 b^3 c d}{\sqrt {b^2-4 a c}}-a b (a g+c e)-2 a c (5 c d-a f)+3 b^2 c d\right )}{2 \sqrt {2} a^2 \sqrt {c} \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 {4 a^2 c (a g+3 c e)-a b^2 (c e-a g)-4 a b c (a f+4 c d)+3 b^3 c d}{\sqrt {b^2-4 a c}}-a b (a g+c e)-2 a c (5 c d-a f)+3 b^2 c d\right )}{2 \sqrt {2} a^2 \sqrt {c} \left (b^2-4 a c\right ) \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {x \left (a \left (-2 a^2 g+\frac {b^3 d}{a}+a (b f+2 c e)-b (b e+3 c d)\right )+x^2 \left (-a b (a g+c e)-2 a c (c d-a f)+b^2 c d\right )\right )}{2 a^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {d}{a^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(d/(a^2*x)) - (x*(a*((b^3*d)/a - b*(3*c*d + b*e) + a*(2*c*e + b*f) - 2*a^2*g) + (b^2*c*d - 2*a*c*(c*d - a*f)
- a*b*(c*e + a*g))*x^2))/(2*a^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((3*b^2*c*d - 2*a*c*(5*c*d - a*f) - a*b*(
c*e + a*g) + (3*b^3*c*d - 4*a*b*c*(4*c*d + a*f) - a*b^2*(c*e - a*g) + 4*a^2*c*(3*c*e + a*g))/Sqrt[b^2 - 4*a*c]
)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^2*Sqrt[c]*(b^2 - 4*a*c)*Sqrt[b - Sqrt[
b^2 - 4*a*c]]) - ((3*b^2*c*d - 2*a*c*(5*c*d - a*f) - a*b*(c*e + a*g) - (3*b^3*c*d - 4*a*b*c*(4*c*d + a*f) - a*
b^2*(c*e - a*g) + 4*a^2*c*(3*c*e + a*g))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a
*c]]])/(2*Sqrt[2]*a^2*Sqrt[c]*(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 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+g x^6}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx &=-\frac {x \left (a \left (\frac {b^3 d}{a}-b (3 c d+b e)+a (2 c e+b f)-2 a^2 g\right )+\left (b^2 c d-2 a c (c d-a f)-a b (c e+a g)\right ) x^2\right )}{2 a^2 \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+\left (\frac {b^3 d}{a}-b (5 c d+b e)+a (6 c e-b f)+2 a^2 g\right ) x^2+\left (\frac {b^2 c d}{a}-c (2 c d+b e)+a (2 c f-b g)\right ) x^4}{x^2 \left (a+b x^2+c x^4\right )} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=-\frac {x \left (a \left (\frac {b^3 d}{a}-b (3 c d+b e)+a (2 c e+b f)-2 a^2 g\right )+\left (b^2 c d-2 a c (c d-a f)-a b (c e+a g)\right ) x^2\right )}{2 a^2 \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^2}+\frac {3 b^3 d-a b^2 e-a b (13 c d+a f)+2 a^2 (3 c e+a g)+\left (3 b^2 c d-2 a c (5 c d-a f)-a b (c e+a g)\right ) x^2}{a \left (a+b x^2+c x^4\right )}\right ) \, dx}{2 a \left (b^2-4 a c\right )}\\ &=-\frac {d}{a^2 x}-\frac {x \left (a \left (\frac {b^3 d}{a}-b (3 c d+b e)+a (2 c e+b f)-2 a^2 g\right )+\left (b^2 c d-2 a c (c d-a f)-a b (c e+a g)\right ) x^2\right )}{2 a^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {3 b^3 d-a b^2 e-a b (13 c d+a f)+2 a^2 (3 c e+a g)+\left (3 b^2 c d-2 a c (5 c d-a f)-a b (c e+a g)\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a^2 \left (b^2-4 a c\right )}\\ &=-\frac {d}{a^2 x}-\frac {x \left (a \left (\frac {b^3 d}{a}-b (3 c d+b e)+a (2 c e+b f)-2 a^2 g\right )+\left (b^2 c d-2 a c (c d-a f)-a b (c e+a g)\right ) x^2\right )}{2 a^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (3 b^2 c d-2 a c (5 c d-a f)-a b (c e+a g)-\frac {3 b^3 c d-4 a b c (4 c d+a f)-a b^2 (c e-a g)+4 a^2 c (3 c e+a g)}{\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 a^2 \left (b^2-4 a c\right )}-\frac {\left (3 b^2 c d-2 a c (5 c d-a f)-a b (c e+a g)+\frac {3 b^3 c d-4 a b c (4 c d+a f)-a b^2 (c e-a g)+4 a^2 c (3 c e+a g)}{\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 a^2 \left (b^2-4 a c\right )}\\ &=-\frac {d}{a^2 x}-\frac {x \left (a \left (\frac {b^3 d}{a}-b (3 c d+b e)+a (2 c e+b f)-2 a^2 g\right )+\left (b^2 c d-2 a c (c d-a f)-a b (c e+a g)\right ) x^2\right )}{2 a^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (3 b^2 c d-2 a c (5 c d-a f)-a b (c e+a g)+\frac {3 b^3 c d-4 a b c (4 c d+a f)-a b^2 (c e-a g)+4 a^2 c (3 c e+a g)}{\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} a^2 \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (3 b^2 c d-2 a c (5 c d-a f)-a b (c e+a g)-\frac {3 b^3 c d-4 a b c (4 c d+a f)-a b^2 (c e-a g)+4 a^2 c (3 c e+a g)}{\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} a^2 \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [A]
time = 1.51, size = 529, normalized size = 1.15 \begin {gather*} -\frac {\frac {4 d}{x}-\frac {2 x \left (-b^3 d+b^2 \left (a e-c d x^2\right )+a b \left (3 c d-a f+c e x^2+a g x^2\right )+2 a \left (a^2 g+c^2 d x^2-a c \left (e+f x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \left (3 b^3 c d+b^2 \left (3 c \sqrt {b^2-4 a c} d-a c e+a^2 g\right )+2 a c \left (-5 c \sqrt {b^2-4 a c} d+6 a c e+a \sqrt {b^2-4 a c} f+2 a^2 g\right )-a b \left (16 c^2 d+c \sqrt {b^2-4 a c} e+4 a c f+a \sqrt {b^2-4 a c} g\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (-3 b^3 c d+b^2 \left (3 c \sqrt {b^2-4 a c} d+a c e-a^2 g\right )-2 a c \left (5 c \sqrt {b^2-4 a c} d+6 a c e-a \sqrt {b^2-4 a c} f+2 a^2 g\right )+a b \left (16 c^2 d-c \sqrt {b^2-4 a c} e+4 a c f-a \sqrt {b^2-4 a c} g\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{4 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((4*d)/x - (2*x*(-(b^3*d) + b^2*(a*e - c*d*x^2) + a*b*(3*c*d - a*f + c*e*x^2 + a*g*x^2) + 2*a*(a^2*g + c^
2*d*x^2 - a*c*(e + f*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(3*b^3*c*d + b^2*(3*c*Sqrt[b^2 - 4
*a*c]*d - a*c*e + a^2*g) + 2*a*c*(-5*c*Sqrt[b^2 - 4*a*c]*d + 6*a*c*e + a*Sqrt[b^2 - 4*a*c]*f + 2*a^2*g) - a*b*
(16*c^2*d + c*Sqrt[b^2 - 4*a*c]*e + 4*a*c*f + a*Sqrt[b^2 - 4*a*c]*g))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt
[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(-3*b^3*c*d + b^2*(3*c*S
qrt[b^2 - 4*a*c]*d + a*c*e - a^2*g) - 2*a*c*(5*c*Sqrt[b^2 - 4*a*c]*d + 6*a*c*e - a*Sqrt[b^2 - 4*a*c]*f + 2*a^2
*g) + a*b*(16*c^2*d - c*Sqrt[b^2 - 4*a*c]*e + 4*a*c*f - a*Sqrt[b^2 - 4*a*c]*g))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqr
t[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/a^2

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Maple [A]
time = 0.14, size = 550, normalized size = 1.20

method	result	size
default	\(\frac {\frac {-\frac {\left (a^{2} b g -2 a^{2} c f +a b c e +2 a \,c^{2} d -b^{2} c d \right ) x^{3}}{2 \left (4 a c -b^{2}\right )}-\frac {\left (2 a^{3} g -a^{2} b f -2 a^{2} c e +a \,b^{2} e +3 a b c d -b^{3} d \right ) x}{2 \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {2 c \left (-\frac {\left (-a^{2} b g \sqrt {-4 a c +b^{2}}+2 a^{2} c f \sqrt {-4 a c +b^{2}}-a b c e \sqrt {-4 a c +b^{2}}-10 a \,c^{2} d \sqrt {-4 a c +b^{2}}+3 b^{2} c d \sqrt {-4 a c +b^{2}}+4 a^{3} g c +a^{2} b^{2} g -4 a^{2} b c f +12 a^{2} c^{2} e -a \,b^{2} c e -16 a b \,c^{2} d +3 b^{3} c d \right ) \sqrt {2}\, \arctanh \left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, c \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-a^{2} b g \sqrt {-4 a c +b^{2}}+2 a^{2} c f \sqrt {-4 a c +b^{2}}-a b c e \sqrt {-4 a c +b^{2}}-10 a \,c^{2} d \sqrt {-4 a c +b^{2}}+3 b^{2} c d \sqrt {-4 a c +b^{2}}-4 a^{3} g c -a^{2} b^{2} g +4 a^{2} b c f -12 a^{2} c^{2} e +a \,b^{2} c e +16 a b \,c^{2} d -3 b^{3} c d \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, c \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}}{a^{2}}-\frac {d}{a^{2} x}\)	\(550\)
risch	\(\text {Expression too large to display}\)	\(7011\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^2*((-1/2*(a^2*b*g-2*a^2*c*f+a*b*c*e+2*a*c^2*d-b^2*c*d)/(4*a*c-b^2)*x^3-1/2*(2*a^3*g-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)/(c*x^4+b*x^2+a)+2/(4*a*c-b^2)*c*(-1/8*(-a^2*b*g*(-4*a*c+b^2)^(1/2)+2*a^2*c
*f*(-4*a*c+b^2)^(1/2)-a*b*c*e*(-4*a*c+b^2)^(1/2)-10*a*c^2*d*(-4*a*c+b^2)^(1/2)+3*b^2*c*d*(-4*a*c+b^2)^(1/2)+4*
a^3*g*c+a^2*b^2*g-4*a^2*b*c*f+12*a^2*c^2*e-a*b^2*c*e-16*a*b*c^2*d+3*b^3*c*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*(-a^2*b*g*(-4*a*c+b^2
)^(1/2)+2*a^2*c*f*(-4*a*c+b^2)^(1/2)-a*b*c*e*(-4*a*c+b^2)^(1/2)-10*a*c^2*d*(-4*a*c+b^2)^(1/2)+3*b^2*c*d*(-4*a*
c+b^2)^(1/2)-4*a^3*g*c-a^2*b^2*g+4*a^2*b*c*f-12*a^2*c^2*e+a*b^2*c*e+16*a*b*c^2*d-3*b^3*c*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))))-d/a^2/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((g*x^6+f*x^4+e*x^2+d)/x^2/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

-1/2*((2*a^2*c*f - a^2*b*g - a*b*c*e + (3*b^2*c - 10*a*c^2)*d)*x^4 + (a^2*b*f - 2*a^3*g - a*b^2*e + 2*a^2*c*e
+ (3*b^3 - 11*a*b*c)*d)*x^2 + 2*(a*b^2 - 4*a^2*c)*d)/((a^2*b^2*c - 4*a^3*c^2)*x^5 + (a^2*b^3 - 4*a^3*b*c)*x^3
+ (a^3*b^2 - 4*a^4*c)*x) + 1/2*integrate((a^2*b*f - 2*a^3*g + a*b^2*e - 6*a^2*c*e - (2*a^2*c*f - a^2*b*g - a*b
*c*e + (3*b^2*c - 10*a*c^2)*d)*x^2 - (3*b^3 - 13*a*b*c)*d)/(c*x^4 + b*x^2 + a), x)/(a^2*b^2 - 4*a^3*c)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 23991 vs. \(2 (418) = 836\).
time = 128.53, size = 23991, normalized size = 52.15 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*(a*b*c*e - 2*a^2*c*f + a^2*b*g - (3*b^2*c - 10*a*c^2)*d)*x^4 - 2*(a^2*b*f - 2*a^3*g + (3*b^3 - 11*a*b*c
)*d - (a*b^2 - 2*a^2*c)*e)*x^2 - sqrt(1/2)*((a^2*b^2*c - 4*a^3*c^2)*x^5 + (a^2*b^3 - 4*a^3*b*c)*x^3 + (a^3*b^2
- 4*a^4*c)*x)*sqrt(-((9*b^7*c - 105*a*b^5*c^2 + 385*a^2*b^3*c^3 - 420*a^3*b*c^4)*d^2 - 2*(3*a*b^6*c - 40*a^2*
b^4*c^2 + 150*a^3*b^2*c^3 - 120*a^4*c^4)*d*e + (a^2*b^5*c - 15*a^3*b^3*c^2 + 60*a^4*b*c^3)*e^2 + (a^4*b^3*c +
12*a^5*b*c^2)*f^2 + (a^5*b^3 + 12*a^6*b*c)*g^2 - 2*((3*a^2*b^5*c - 13*a^3*b^3*c^2 - 12*a^4*b*c^3)*d - (a^3*b^4
*c - 6*a^4*b^2*c^2 - 24*a^5*c^3)*e)*f + 2*((9*a^3*b^4*c - 54*a^4*b^2*c^2 + 40*a^5*c^3)*d - (3*a^4*b^3*c - 28*a
^5*b*c^2)*e - 2*(3*a^5*b^2*c + 4*a^6*c^2)*f)*g + (a^5*b^6*c - 12*a^6*b^4*c^2 + 48*a^7*b^2*c^3 - 64*a^8*c^4)*sq
rt((a^8*c^2*f^4 + a^10*g^4 + (81*b^8*c^2 - 918*a*b^6*c^3 + 3051*a^2*b^4*c^4 - 2550*a^3*b^2*c^5 + 625*a^4*c^6)*
d^4 - 4*(27*a*b^7*c^2 - 351*a^2*b^5*c^3 + 1197*a^3*b^3*c^4 - 550*a^4*b*c^5)*d^3*e + 6*(9*a^2*b^6*c^2 - 132*a^3
*b^4*c^3 + ...

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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((g*x**6+f*x**4+e*x**2+d)/x**2/(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. 9176 vs. \(2 (425) = 850\).
time = 9.08, size = 9176, normalized size = 19.95 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*b^2*c*d*x^4 - 10*a*c^2*d*x^4 + 2*a^2*c*f*x^4 - a^2*b*g*x^4 - a*b*c*x^4*e + 3*b^3*d*x^2 - 11*a*b*c*d*x^
2 + a^2*b*f*x^2 - 2*a^3*g*x^2 - a*b^2*x^2*e + 2*a^2*c*x^2*e + 2*a*b^2*d - 8*a^2*c*d)/((c*x^5 + b*x^3 + a*x)*(a
^2*b^2 - 4*a^3*c)) + 1/16*((6*b^4*c^3 - 44*a*b^2*c^4 + 80*a^2*c^5 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqr
t(b^2 - 4*a*c)*c)*b^4*c + 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 6*sqrt(2)*s
qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 - 40*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*a^2*c^3 - 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 3*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^3 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c
)*a*c^4 - 6*(b^2 - 4*a*c)*b^2*c^3 + 20*(b^2 - 4*a*c)*a*c^4)*(a^2*b^2 - 4*a^3*c)^2*d + 2*(2*a^2*b^2*c^3 - 8*a^3
*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(
b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - s
qrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^3 - 2*(b^2 - 4*a*c)*a^2*c^3)*(a^2*b^2 - 4*a^3*c
)^2*f - (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*s
qrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(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^2*b^2 - 4*a^3*c)^2*g - (2*a*b^3*c^3 - 8*a^2*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq
rt(b^2 - 4*a*c)*c)*a*b^3*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 + 2*sqrt(2)
*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*a*b*c^3 - 2*(b^2 - 4*a*c)*a*b*c^3)*(a^2*b^2 - 4*a^3*c)^2*e - 2*(3*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a
*c)*c)*a^2*b^7*c - 37*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c^2 - 6*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*
a*c)*c)*a^2*b^6*c^2 - 6*a^2*b^7*c^2 + 152*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^3 + 50*sqrt(2)*sqr
t(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^3 + 3*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c^3 + 74*a^3*b^5*
c^3 - 208*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b*c^4 - 104*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*
b^2*c^4 - 25*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^4 - 304*a^4*b^3*c^4 + 52*sqrt(2)*sqrt(b*c + sqr
t(b^2 - 4*a*c)*c)*a^4*b*c^5 + 416*a^5*b*c^5 + 6*(b^2 - 4*a*c)*a^2*b^5*c^2 - 50*(b^2 - 4*a*c)*a^3*b^3*c^3 + 104
*(b^2 - 4*a*c)*a^4*b*c^4)*d*abs(a^2*b^2 - 4*a^3*c) + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^5*c - 8*
sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^3*c^2 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^4*c^2 -
2*a^4*b^5*c^2 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b*c^3 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*
c)*a^5*b^2*c^3 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^3 + 16*a^5*b^3*c^3 - 4*sqrt(2)*sqrt(b*c + s
qrt(b^2 - 4*a*c)*c)*a^5*b*c^4 - 32*a^6*b*c^4 + 2*(b^2 - 4*a*c)*a^4*b^3*c^2 - 8*(b^2 - 4*a*c)*a^5*b*c^3)*f*abs(
a^2*b^2 - 4*a^3*c) - 4*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^4*c - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*
a*c)*c)*a^6*b^2*c^2 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^3*c^2 - 2*a^5*b^4*c^2 + 16*sqrt(2)*sqrt(
b*c + sqrt(b^2 - 4*a*c)*c)*a^7*c^3 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b*c^3 + sqrt(2)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*a^5*b^2*c^3 + 16*a^6*b^2*c^3 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*c^4 - 32*a^7
*c^4 + 2*(b^2 - 4*a*c)*a^5*b^2*c^2 - 8*(b^2 - 4*a*c)*a^6*c^3)*g*abs(a^2*b^2 - 4*a^3*c) + 2*(sqrt(2)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*a^3*b^6*c - 14*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^4*c^2 - 2*sqrt(2)*sqrt(b*c
+ sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c^2 - 2*a^3*b^6*c^2 + 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^2*c^3 +
20*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^3 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^3 +
28*a^4*b^4*c^3 - 96*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*c^4 - 48*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)
*c)*a^5*b*c^4 - 10*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^4 - 128*a^5*b^2*c^4 + 24*sqrt(2)*sqrt(b*c
+ sqrt(b^2 - 4*a*c)*c)*a^5*c^5 + 192*a^6*c^5 + 2*(b^2 - 4*a*c)*a^3*b^4*c^2 - 20*(b^2 - 4*a*c)*a^4*b^2*c^3 + 4
8*(b^2 - 4*a*c)*a^5*c^4)*abs(a^2*b^2 - 4*a^3*c)*e + (6*a^4*b^8*c^3 - 80*a^5*b^6*c^4 + 352*a^6*b^4*c^5 - 512*a^
7*b^2*c^6 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^8*c + 40*sqrt(2)*sqrt(b^2 - 4*a*
c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^6*c^2 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a
^4*b^7*c^2 - 176*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^3 - 56*sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^5*c^3 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)
*c)*a^4*b^6*c^3 + 256*sqrt(2)*sqrt(b^2 - 4*a*c)...

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Mupad [B]
time = 7.76, size = 2500, normalized size = 5.43 \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 + g*x^6)/(x^2*(a + b*x^2 + c*x^4)^2),x)

[Out]

atan(((((213*a*b^11*c^2*d^2 - a^5*b^9*g^2 - a^5*g^2*(-(4*a*c - b^2)^9)^(1/2) - 9*b^13*c*d^2 - 26880*a^6*b*c^7*
d^2 - a^2*b^11*c*e^2 + 3840*a^7*b*c^6*e^2 + 9*b^4*c*d^2*(-(4*a*c - b^2)^9)^(1/2) - a^4*b^9*c*f^2 + 768*a^8*b*c
^5*f^2 + a^4*c*f^2*(-(4*a*c - b^2)^9)^(1/2) + 768*a^9*b*c^4*g^2 - 2077*a^2*b^9*c^3*d^2 + 10656*a^3*b^7*c^4*d^2
- 30240*a^4*b^5*c^5*d^2 + 44800*a^5*b^3*c^6*d^2 + 25*a^2*c^3*d^2*(-(4*a*c - b^2)^9)^(1/2) + 27*a^3*b^9*c^2*e^
2 - 288*a^4*b^7*c^3*e^2 + 1504*a^5*b^5*c^4*e^2 - 3840*a^6*b^3*c^5*e^2 - 9*a^3*c^2*e^2*(-(4*a*c - b^2)^9)^(1/2)
+ 96*a^6*b^5*c^3*f^2 - 512*a^7*b^3*c^4*f^2 + 96*a^7*b^5*c^2*g^2 - 512*a^8*b^3*c^3*g^2 + 15360*a^7*c^7*d*e + 5
120*a^8*c^6*d*g - 3072*a^8*c^6*e*f - 1024*a^9*c^5*f*g + 6*a^2*b^11*c*d*f + 1536*a^7*b*c^6*d*f - 18*a^3*b^10*c*
d*g - 2*a^3*b^10*c*e*f + 6*a^4*b^9*c*e*g + 3584*a^8*b*c^5*e*g - 6*a^4*c*e*g*(-(4*a*c - b^2)^9)^(1/2) + 12*a^5*
b^8*c*f*g - 152*a^2*b^10*c^2*d*e + 1548*a^3*b^8*c^3*d*e - 8064*a^4*b^6*c^4*d*e + 22400*a^5*b^4*c^5*d*e - 30720
*a^6*b^2*c^6*d*e - 98*a^3*b^9*c^2*d*f + 576*a^4*b^7*c^3*d*f - 1344*a^5*b^5*c^4*d*f + 512*a^6*b^3*c^5*d*f - 10*
a^3*c^2*d*f*(-(4*a*c - b^2)^9)^(1/2) + 324*a^4*b^8*c^2*d*g + 36*a^4*b^8*c^2*e*f - 2240*a^5*b^6*c^3*d*g - 192*a
^5*b^6*c^3*e*f + 7296*a^6*b^4*c^4*d*g + 128*a^6*b^4*c^4*e*f - 10752*a^7*b^2*c^5*d*g + 1536*a^7*b^2*c^5*e*f - 1
28*a^5*b^7*c^2*e*g + 960*a^6*b^5*c^3*e*g - 3072*a^7*b^3*c^4*e*g - 128*a^6*b^6*c^2*f*g + 384*a^7*b^4*c^3*f*g +
6*a*b^12*c*d*e - 51*a*b^2*c^2*d^2*(-(4*a*c - b^2)^9)^(1/2) + a^2*b^2*c*e^2*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^3*
c*d*e*(-(4*a*c - b^2)^9)^(1/2) + 18*a^3*b*c*d*g*(-(4*a*c - b^2)^9)^(1/2) + 2*a^3*b*c*e*f*(-(4*a*c - b^2)^9)^(1
/2) + 44*a^2*b*c^2*d*e*(-(4*a*c - b^2)^9)^(1/2) - 6*a^2*b^2*c*d*f*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^11*c^7
+ a^5*b^12*c - 24*a^6*b^10*c^2 + 240*a^7*b^8*c^3 - 1280*a^8*b^6*c^4 + 3840*a^9*b^4*c^5 - 6144*a^10*b^2*c^6)))
^(1/2)*(x*((213*a*b^11*c^2*d^2 - a^5*b^9*g^2 - a^5*g^2*(-(4*a*c - b^2)^9)^(1/2) - 9*b^13*c*d^2 - 26880*a^6*b*c
^7*d^2 - a^2*b^11*c*e^2 + 3840*a^7*b*c^6*e^2 + 9*b^4*c*d^2*(-(4*a*c - b^2)^9)^(1/2) - a^4*b^9*c*f^2 + 768*a^8*
b*c^5*f^2 + a^4*c*f^2*(-(4*a*c - b^2)^9)^(1/2) + 768*a^9*b*c^4*g^2 - 2077*a^2*b^9*c^3*d^2 + 10656*a^3*b^7*c^4*
d^2 - 30240*a^4*b^5*c^5*d^2 + 44800*a^5*b^3*c^6*d^2 + 25*a^2*c^3*d^2*(-(4*a*c - b^2)^9)^(1/2) + 27*a^3*b^9*c^2
*e^2 - 288*a^4*b^7*c^3*e^2 + 1504*a^5*b^5*c^4*e^2 - 3840*a^6*b^3*c^5*e^2 - 9*a^3*c^2*e^2*(-(4*a*c - b^2)^9)^(1
/2) + 96*a^6*b^5*c^3*f^2 - 512*a^7*b^3*c^4*f^2 + 96*a^7*b^5*c^2*g^2 - 512*a^8*b^3*c^3*g^2 + 15360*a^7*c^7*d*e
+ 5120*a^8*c^6*d*g - 3072*a^8*c^6*e*f - 1024*a^9*c^5*f*g + 6*a^2*b^11*c*d*f + 1536*a^7*b*c^6*d*f - 18*a^3*b^10
*c*d*g - 2*a^3*b^10*c*e*f + 6*a^4*b^9*c*e*g + 3584*a^8*b*c^5*e*g - 6*a^4*c*e*g*(-(4*a*c - b^2)^9)^(1/2) + 12*a
^5*b^8*c*f*g - 152*a^2*b^10*c^2*d*e + 1548*a^3*b^8*c^3*d*e - 8064*a^4*b^6*c^4*d*e + 22400*a^5*b^4*c^5*d*e - 30
720*a^6*b^2*c^6*d*e - 98*a^3*b^9*c^2*d*f + 576*a^4*b^7*c^3*d*f - 1344*a^5*b^5*c^4*d*f + 512*a^6*b^3*c^5*d*f -
10*a^3*c^2*d*f*(-(4*a*c - b^2)^9)^(1/2) + 324*a^4*b^8*c^2*d*g + 36*a^4*b^8*c^2*e*f - 2240*a^5*b^6*c^3*d*g - 19
2*a^5*b^6*c^3*e*f + 7296*a^6*b^4*c^4*d*g + 128*a^6*b^4*c^4*e*f - 10752*a^7*b^2*c^5*d*g + 1536*a^7*b^2*c^5*e*f
- 128*a^5*b^7*c^2*e*g + 960*a^6*b^5*c^3*e*g - 3072*a^7*b^3*c^4*e*g - 128*a^6*b^6*c^2*f*g + 384*a^7*b^4*c^3*f*g
+ 6*a*b^12*c*d*e - 51*a*b^2*c^2*d^2*(-(4*a*c - b^2)^9)^(1/2) + a^2*b^2*c*e^2*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b
^3*c*d*e*(-(4*a*c - b^2)^9)^(1/2) + 18*a^3*b*c*d*g*(-(4*a*c - b^2)^9)^(1/2) + 2*a^3*b*c*e*f*(-(4*a*c - b^2)^9)
^(1/2) + 44*a^2*b*c^2*d*e*(-(4*a*c - b^2)^9)^(1/2) - 6*a^2*b^2*c*d*f*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^11*
c^7 + a^5*b^12*c - 24*a^6*b^10*c^2 + 240*a^7*b^8*c^3 - 1280*a^8*b^6*c^4 + 3840*a^9*b^4*c^5 - 6144*a^10*b^2*c^6
)))^(1/2)*(1048576*a^16*b*c^8 + 256*a^10*b^13*c^2 - 6144*a^11*b^11*c^3 + 61440*a^12*b^9*c^4 - 327680*a^13*b^7*
c^5 + 983040*a^14*b^5*c^6 - 1572864*a^15*b^3*c^7) - 131072*a^16*c^7*g - 393216*a^15*c^8*e + 192*a^8*b^13*c^2*d
- 4672*a^9*b^11*c^3*d + 47360*a^10*b^9*c^4*d - 256000*a^11*b^7*c^5*d + 778240*a^12*b^5*c^6*d - 1261568*a^13*b
^3*c^7*d - 64*a^9*b^12*c^2*e + 1664*a^10*b^10*c^3*e - 17920*a^11*b^8*c^4*e + 102400*a^12*b^6*c^5*e - 327680*a^
13*b^4*c^6*e + 557056*a^14*b^2*c^7*e - 64*a^10*b^11*c^2*f + 1280*a^11*b^9*c^3*f - 10240*a^12*b^7*c^4*f + 40960
*a^13*b^5*c^5*f - 81920*a^14*b^3*c^6*f + 128*a^11*b^10*c^2*g - 2560*a^12*b^8*c^3*g + 20480*a^13*b^6*c^4*g - 81
920*a^14*b^4*c^5*g + 163840*a^15*b^2*c^6*g + 851968*a^14*b*c^8*d + 65536*a^15*b*c^7*f) + x*(204800*a^12*c^9*d^
2 - 73728*a^13*c^8*e^2 + 8192*a^14*c^7*f^2 - 8192*a^15*c^6*g^2 + 16*a^10*b^10*c*g^2 + 144*a^6*b^12*c^3*d^2 - 3
264*a^7*b^10*c^4*d^2 + 30112*a^8*b^8*c^5*d^2 - 143360*a^9*b^6*c^6*d^2 + 365568*a^10*b^4*c^7*d^2 - 458752*a^11*
b^2*c^8*d^2 + 16*a^8*b^10*c^3*e^2 - 416*a^9*b^8*c^4*e^2 + 4608*a^10*b^6*c^5*e^2 - 25600*a^11*b^4*c^6*e^2 + 696
32*a^12*b^2*c^7*e^2 + 160*a^10*b^8*c^3*f^2 - 2048*a^11*b^6*c^4*f^2 + 9216*a^12*b^4*c^5*f^2 - 16384*a^13*b^2*c^
6*f^2 - 160*a^11*b^8*c^2*g^2 + 512*a^12*b^6*c^3...

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