3.1.60 \(\int \frac {d+e x^2+f x^4}{x^6 (a+b x^2+c x^4)} \, dx\) [60]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.30, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1678, 1180, 211} \begin {gather*} -\frac {-a b e-a (c d-a f)+b^2 d}{a^3 x}+\frac {b d-a e}{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 (\frac {2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d}{\sqrt {b^2-4 a c}}-a b e-a (c d-a f)+b^2 d\right )}{\sqrt {2} a^3 \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 (-\frac {2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d}{\sqrt {b^2-4 a c}}-a b e-a (c d-a f)+b^2 d\right )}{\sqrt {2} a^3 \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {d}{5 a x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {d+e x^2+f x^4}{x^6 \left (a+b x^2+c x^4\right )} \, dx &=\int \left (\frac {d}{a x^6}+\frac {-b d+a e}{a^2 x^4}+\frac {b^2 d-a b e-a (c d-a f)}{a^3 x^2}+\frac {-b^3 d+a b^2 e-a^2 c e+a b (2 c d-a f)-c \left (b^2 d-a b e-a (c d-a f)\right ) x^2}{a^3 \left (a+b x^2+c x^4\right )}\right ) \, dx\\ &=-\frac {d}{5 a x^5}+\frac {b d-a e}{3 a^2 x^3}-\frac {b^2 d-a b e-a (c d-a f)}{a^3 x}+\frac {\int \frac {-b^3 d+a b^2 e-a^2 c e+a b (2 c d-a f)-c \left (b^2 d-a b e-a (c d-a f)\right ) x^2}{a+b x^2+c x^4} \, dx}{a^3}\\ &=-\frac {d}{5 a x^5}+\frac {b d-a e}{3 a^2 x^3}-\frac {b^2 d-a b e-a (c d-a f)}{a^3 x}-\frac {\left (c \left (b^2 d-a b e-a (c d-a f)-\frac {b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{2 a^3}-\frac {\left (c \left (b^2 d-a b e-a (c d-a f)+\frac {b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{2 a^3}\\ &=-\frac {d}{5 a x^5}+\frac {b d-a e}{3 a^2 x^3}-\frac {b^2 d-a b e-a (c d-a f)}{a^3 x}-\frac {\sqrt {c} \left (b^2 d-a b e-a (c d-a f)+\frac {b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-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 )}{\sqrt {2} a^3 \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (b^2 d-a b e-a (c d-a f)-\frac {b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-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 )}{\sqrt {2} a^3 \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.36, size = 394, normalized size = 1.20 \begin {gather*} \frac {-\frac {6 a^2 d}{x^5}+\frac {10 a (b d-a e)}{x^3}+\frac {30 \left (-b^2 d+a b e+a (c d-a f)\right )}{x}-\frac {15 \sqrt {2} \sqrt {c} \left (b^3 d+b^2 \left (\sqrt {b^2-4 a c} d-a e\right )+a b \left (-3 c d-\sqrt {b^2-4 a c} e+a f\right )+a \left (-c \sqrt {b^2-4 a c} d+2 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 )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {15 \sqrt {2} \sqrt {c} \left (b^3 d-b^2 \left (\sqrt {b^2-4 a c} d+a e\right )+a b \left (-3 c d+\sqrt {b^2-4 a c} e+a f\right )+a \left (c \sqrt {b^2-4 a c} d+2 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 )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{30 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*a^2*d)/x^5 + (10*a*(b*d - a*e))/x^3 + (30*(-(b^2*d) + a*b*e + a*(c*d - a*f)))/x - (15*Sqrt[2]*Sqrt[c]*(b^
3*d + b^2*(Sqrt[b^2 - 4*a*c]*d - a*e) + a*b*(-3*c*d - Sqrt[b^2 - 4*a*c]*e + a*f) + a*(-(c*Sqrt[b^2 - 4*a*c]*d)
+ 2*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]]])/(Sqrt[b^2 - 4*a*
c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (15*Sqrt[2]*Sqrt[c]*(b^3*d - b^2*(Sqrt[b^2 - 4*a*c]*d + a*e) + a*b*(-3*c*d +
Sqrt[b^2 - 4*a*c]*e + a*f) + a*(c*Sqrt[b^2 - 4*a*c]*d + 2*a*c*e - a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqr
t[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(30*a^3)

________________________________________________________________________________________

Maple [A]
time = 0.08, size = 360, normalized size = 1.09

method	result	size
default	\(\frac {4 c \left (-\frac {\left (-a^{2} f \sqrt {-4 a c +b^{2}}+a b e \sqrt {-4 a c +b^{2}}+\sqrt {-4 a c +b^{2}}\, a c d -\sqrt {-4 a c +b^{2}}\, b^{2} d -a^{2} b f -2 a^{2} c e +a \,b^{2} e +3 a b c d -b^{3} d \right ) \sqrt {2}\, \arctanh \left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-a^{2} f \sqrt {-4 a c +b^{2}}+a b e \sqrt {-4 a c +b^{2}}+\sqrt {-4 a c +b^{2}}\, a c d -\sqrt {-4 a c +b^{2}}\, b^{2} d +a^{2} b f +2 a^{2} c e -a \,b^{2} e -3 a b c d +b^{3} d \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a^{3}}-\frac {d}{5 a \,x^{5}}-\frac {a e -b d}{3 a^{2} x^{3}}-\frac {a^{2} f -a b e -a c d +b^{2} d}{a^{3} x}\)	\(360\)
risch	\(\frac {-\frac {\left (a^{2} f -a b e -a c d +b^{2} d \right ) x^{4}}{a^{3}}-\frac {\left (a e -b d \right ) x^{2}}{3 a^{2}}-\frac {d}{5 a}}{x^{5}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (-4 a \,c^{6} d^{3} f +2 b^{2} c^{5} d^{3} f -2 a^{2} b \,c^{4} e^{3} f -4 a^{2} c^{5} d \,e^{2} f -2 b^{3} c^{4} d^{2} e f -4 a^{3} c^{4} d \,f^{3}+2 a^{3} c^{4} e^{2} f^{2}+6 a^{2} c^{5} d^{2} f^{2}+b^{4} c^{3} d^{2} f^{2}+b^{2} c^{5} d^{2} e^{2}-2 b \,c^{6} d^{3} e +\left (16 c^{2} a^{9}-8 b^{2} c \,a^{8}+b^{4} a^{7}\right ) \textit {\_Z}^{4}+\left (12 a^{6} b \,c^{2} f^{2}+16 a^{6} c^{3} e f -7 a^{5} b^{3} c \,f^{2}-36 a^{5} b^{2} c^{2} e f -40 a^{5} b \,c^{3} d f -20 a^{5} b \,c^{3} e^{2}-16 a^{5} c^{4} d e +a^{4} b^{5} f^{2}+16 a^{4} b^{4} c e f +50 a^{4} b^{3} c^{2} d f +25 a^{4} b^{3} c^{2} e^{2}+76 a^{4} b^{2} c^{3} d e +28 a^{4} b \,c^{4} d^{2}-2 a^{3} b^{6} e f -18 a^{3} b^{5} c d f -9 a^{3} b^{5} c \,e^{2}-66 a^{3} b^{4} c^{2} d e -63 a^{3} b^{3} c^{3} d^{2}+2 a^{2} b^{7} d f +a^{2} b^{7} e^{2}+20 a^{2} b^{6} c d e +42 a^{2} b^{5} c^{2} d^{2}-2 a \,b^{8} d e -11 a \,b^{7} c \,d^{2}+b^{9} d^{2}\right ) \textit {\_Z}^{2}+4 a \,b^{2} c^{4} d \,e^{2} f +2 a b \,c^{5} d^{2} e f +2 a \,c^{6} d^{2} e^{2}+c^{7} d^{4}+2 a^{2} b \,c^{4} d e \,f^{2}-2 a \,b^{3} c^{3} d e \,f^{2}+a^{2} c^{5} e^{4}+a^{4} c^{3} f^{4}-2 a^{3} b \,c^{3} e \,f^{3}+2 a^{2} b^{2} c^{3} d \,f^{3}+a^{2} b^{2} c^{3} e^{2} f^{2}-4 a \,b^{2} c^{4} d^{2} f^{2}-2 a b \,c^{5} d \,e^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-8 a \,c^{6} d^{3} f +4 b^{2} c^{5} d^{3} f -4 a^{2} b \,c^{4} e^{3} f -8 a^{2} c^{5} d \,e^{2} f -4 b^{3} c^{4} d^{2} e f -8 a^{3} c^{4} d \,f^{3}+4 a^{3} c^{4} e^{2} f^{2}+12 a^{2} c^{5} d^{2} f^{2}+2 b^{4} c^{3} d^{2} f^{2}+2 b^{2} c^{5} d^{2} e^{2}-4 b \,c^{6} d^{3} e +\left (40 c^{2} a^{9}-22 b^{2} c \,a^{8}+3 b^{4} a^{7}\right ) \textit {\_R}^{4}+\left (25 a^{6} b \,c^{2} f^{2}+36 a^{6} c^{3} e f -14 a^{5} b^{3} c \,f^{2}-74 a^{5} b^{2} c^{2} e f -86 a^{5} b \,c^{3} d f -43 a^{5} b \,c^{3} e^{2}-36 a^{5} c^{4} d e +2 a^{4} b^{5} f^{2}+32 a^{4} b^{4} c e f +102 a^{4} b^{3} c^{2} d f +51 a^{4} b^{3} c^{2} e^{2}+160 a^{4} b^{2} c^{3} d e +61 a^{4} b \,c^{4} d^{2}-4 a^{3} b^{6} e f -36 a^{3} b^{5} c d f -18 a^{3} b^{5} c \,e^{2}-134 a^{3} b^{4} c^{2} d e -131 a^{3} b^{3} c^{3} d^{2}+4 a^{2} b^{7} d f +2 a^{2} b^{7} e^{2}+40 a^{2} b^{6} c d e +85 a^{2} b^{5} c^{2} d^{2}-4 a \,b^{8} d e -22 a \,b^{7} c \,d^{2}+2 b^{9} d^{2}\right ) \textit {\_R}^{2}+8 a \,b^{2} c^{4} d \,e^{2} f +4 a b \,c^{5} d^{2} e f +4 a \,c^{6} d^{2} e^{2}+2 c^{7} d^{4}+4 a^{2} b \,c^{4} d e \,f^{2}-4 a \,b^{3} c^{3} d e \,f^{2}+2 a^{2} c^{5} e^{4}+2 a^{4} c^{3} f^{4}-4 a^{3} b \,c^{3} e \,f^{3}+4 a^{2} b^{2} c^{3} d \,f^{3}+2 a^{2} b^{2} c^{3} e^{2} f^{2}-8 a \,b^{2} c^{4} d^{2} f^{2}-4 a b \,c^{5} d \,e^{3}\right ) x +\left (4 a^{8} c^{2} f -5 a^{7} b^{2} c f -8 a^{7} b \,c^{2} e -4 a^{7} c^{3} d +a^{6} b^{4} f +6 a^{6} b^{3} c e +13 a^{6} b^{2} c^{2} d -a^{5} b^{5} e -7 a^{5} b^{4} c d +a^{4} b^{6} d \right ) \textit {\_R}^{3}+\left (a^{5} c^{3} e \,f^{2}-a^{4} b \,c^{3} d \,f^{2}-a^{4} b \,c^{3} e^{2} f -2 a^{4} c^{4} d e f +a^{4} c^{4} e^{3}+2 a^{3} b^{2} c^{3} d e f +2 a^{3} b \,c^{4} d^{2} f -2 a^{3} b \,c^{4} d \,e^{2}+a^{3} c^{5} d^{2} e -a^{2} b^{3} c^{3} d^{2} f +a^{2} b^{2} c^{4} d^{2} e -a^{2} b \,c^{5} d^{3}\right ) \textit {\_R} \right )\right )}{2}\)	\(1569\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/a^3*c*(-1/8*(-a^2*f*(-4*a*c+b^2)^(1/2)+a*b*e*(-4*a*c+b^2)^(1/2)+(-4*a*c+b^2)^(1/2)*a*c*d-(-4*a*c+b^2)^(1/2)*
b^2*d-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)^(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*f*(-4*a*c+b^2)^(1/2)+a*b*e*(-4*a*c+b^2)^(1/2)
+(-4*a*c+b^2)^(1/2)*a*c*d-(-4*a*c+b^2)^(1/2)*b^2*d+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)^(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)))-1/5*d/a/x^5-
1/3*(a*e-b*d)/a^2/x^3-(a^2*f-a*b*e-a*c*d+b^2*d)/a^3/x

________________________________________________________________________________________

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

[Out]

-integrate((a^2*b*f - a*b^2*e + a^2*c*e + (a^2*c*f - a*b*c*e + (b^2*c - a*c^2)*d)*x^2 + (b^3 - 2*a*b*c)*d)/(c*
x^4 + b*x^2 + a), x)/a^3 - 1/15*(15*(a^2*f - a*b*e + (b^2 - a*c)*d)*x^4 + 3*a^2*d - 5*(a*b*d - a^2*e)*x^2)/(a^
3*x^5)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 15830 vs. \(2 (289) = 578\).
time = 42.36, size = 15830, normalized size = 48.12 \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^6/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

-1/30*(15*sqrt(1/2)*a^3*x^5*sqrt(-((b^7 - 7*a*b^5*c + 14*a^2*b^3*c^2 - 7*a^3*b*c^3)*d^2 - 2*(a*b^6 - 6*a^2*b^4
*c + 9*a^3*b^2*c^2 - 2*a^4*c^3)*d*e + (a^2*b^5 - 5*a^3*b^3*c + 5*a^4*b*c^2)*e^2 + (a^4*b^3 - 3*a^5*b*c)*f^2 +
2*((a^2*b^5 - 5*a^3*b^3*c + 5*a^4*b*c^2)*d - (a^3*b^4 - 4*a^4*b^2*c + 2*a^5*c^2)*e)*f + (a^7*b^2 - 4*a^8*c)*sq
rt(((b^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12*a^5*b^2*c^5 + a^6*c^6)*d^4 - 4
*(a*b^11 - 9*a^2*b^9*c + 29*a^3*b^7*c^2 - 40*a^4*b^5*c^3 + 22*a^5*b^3*c^4 - 3*a^6*b*c^5)*d^3*e + 2*(3*a^2*b^10
- 24*a^3*b^8*c + 66*a^4*b^6*c^2 - 72*a^5*b^4*c^3 + 27*a^6*b^2*c^4 - a^7*c^5)*d^2*e^2 - 4*(a^3*b^9 - 7*a^4*b^7
*c + 16*a^5*b^5*c^2 - 13*a^6*b^3*c^3 + 3*a^7*b*c^4)*d*e^3 + (a^4*b^8 - 6*a^5*b^6*c + 11*a^6*b^4*c^2 - 6*a^7*b^
2*c^3 + a^8*c^4)*e^4 + (a^8*b^4 - 2*a^9*b^2*c + a^10*c^2)*f^4 + 4*((a^6*b^6 - 4*a^7*b^4*c + 4*a^8*b^2*c^2 - a^
9*c^3)*d - (a^7*b^5 - 3*a^8*b^3*c + 2*a^9*b*c^2)*e)*f^3 + 2*((3*a^4*b^8 - 18*a^5*b^6*c + 33*a^6*b^4*c^2 - 19*a
^7*b^2*c^3 ...

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 6718 vs. \(2 (297) = 594\).
time = 5.53, size = 6718, normalized size = 20.42 \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^6/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

-1/8*((2*b^6*c^2 - 18*a*b^4*c^3 + 48*a^2*b^2*c^4 - 32*a^3*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2
- 4*a*c)*c)*b^6 + 9*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c + 2*sqrt(2)*sqrt(b^2 - 4
*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2
*b^2*c^2 - 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*
sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^3
+ 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^4 - 2*(
b^2 - 4*a*c)*b^4*c^2 + 10*(b^2 - 4*a*c)*a*b^2*c^3 - 8*(b^2 - 4*a*c)*a^2*c^4)*a^2*d + (2*a^2*b^4*c^2 - 16*a^3*b
^2*c^3 + 32*a^4*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4 + 8*sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c
)*a^2*b^3*c - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*c^2 - 8*sqrt(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^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^3 - 2*(b^2 - 4*a*c)*a^2*b^2*c^2 + 8
*(b^2 - 4*a*c)*a^3*c^3)*a^2*f - (2*a*b^5*c^2 - 16*a^2*b^3*c^3 + 32*a^3*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(
b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c + 2*s
qrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqr
t(b^2 - 4*a*c)*c)*a^3*b*c^2 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 - sqrt(2
)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^
2 - 4*a*c)*c)*a^2*b*c^3 - 2*(b^2 - 4*a*c)*a*b^3*c^2 + 8*(b^2 - 4*a*c)*a^2*b*c^3)*a^2*e + 2*(sqrt(2)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*a*b^7 - 10*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c - 2*sqrt(2)*sqrt(b*c + sqrt
(b^2 - 4*a*c)*c)*a*b^6*c - 2*a*b^7*c + 32*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^2 + 12*sqrt(2)*sqr
t(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 + 20*a^2*b^5*c^2
- 32*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^3 - 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^
3 - 6*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^3 - 64*a^3*b^3*c^3 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4
*a*c)*c)*a^3*b*c^4 + 64*a^4*b*c^4 + 2*(b^2 - 4*a*c)*a*b^5*c - 12*(b^2 - 4*a*c)*a^2*b^3*c^2 + 16*(b^2 - 4*a*c)*
a^3*b*c^3)*d*abs(a) + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^5 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a
*c)*c)*a^4*b^3*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c - 2*a^3*b^5*c + 16*sqrt(2)*sqrt(b*c + s
qrt(b^2 - 4*a*c)*c)*a^5*b*c^2 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^2 + sqrt(2)*sqrt(b*c + sqr
t(b^2 - 4*a*c)*c)*a^3*b^3*c^2 + 16*a^4*b^3*c^2 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^3 - 32*a^5*
b*c^3 + 2*(b^2 - 4*a*c)*a^3*b^3*c - 8*(b^2 - 4*a*c)*a^4*b*c^2)*f*abs(a) - 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a
*c)*c)*a^2*b^6 - 9*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*
c)*a^2*b^5*c - 2*a^2*b^6*c + 24*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^2 + 10*sqrt(2)*sqrt(b*c + sq
rt(b^2 - 4*a*c)*c)*a^3*b^3*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 + 18*a^3*b^4*c^2 - 16*sqr
t(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*c^3 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^3 - 5*sqrt(2)
*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^3 - 48*a^4*b^2*c^3 + 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*
c^4 + 32*a^5*c^4 + 2*(b^2 - 4*a*c)*a^2*b^4*c - 10*(b^2 - 4*a*c)*a^3*b^2*c^2 + 8*(b^2 - 4*a*c)*a^4*c^3)*abs(a)*
e + (2*a^2*b^6*c^2 - 14*a^3*b^4*c^3 + 24*a^4*b^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*
c)*a^2*b^6 + 7*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c + 2*sqrt(2)*sqrt(b^2 - 4*a*
c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^
4*b^2*c^2 - 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c
)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 + 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^
3*b^2*c^3 - 2*(b^2 - 4*a*c)*a^2*b^4*c^2 + 6*(b^2 - 4*a*c)*a^3*b^2*c^3)*d + (2*a^4*b^4*c^2 - 8*a^5*b^2*c^3 - sq
rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^4 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(
b^2 - 4*a*c)*c)*a^5*b^2*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c - sqrt(2)*sq
rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^2 - 2*(b^2 - 4*a*c)*a^4*b^2*c^2)*f - (2*a^3*b^5*c^2
- 12*a^4*b^3*c^3 + 16*a^5*b*c^4 - sqrt(2)*sqrt(...

________________________________________________________________________________________

Mupad [B]
time = 6.25, 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^6*(a + b*x^2 + c*x^4)),x)

[Out]

atan(((x*(4*a^13*c^5*e^2 - 4*a^12*c^6*d^2 - 4*a^14*c^4*f^2 + 2*a^9*b^6*c^3*d^2 - 12*a^10*b^4*c^4*d^2 + 18*a^11
*b^2*c^5*d^2 + 2*a^11*b^4*c^3*e^2 - 8*a^12*b^2*c^4*e^2 + 2*a^13*b^2*c^3*f^2 + 8*a^13*c^5*d*f - 20*a^12*b*c^5*d
*e + 12*a^13*b*c^4*e*f - 4*a^10*b^5*c^3*d*e + 20*a^11*b^3*c^4*d*e + 4*a^11*b^4*c^3*d*f - 16*a^12*b^2*c^4*d*f -
4*a^12*b^3*c^3*e*f) - (-(b^9*d^2 + a^2*b^7*e^2 + b^6*d^2*(-(4*a*c - b^2)^3)^(1/2) + a^4*b^5*f^2 + 28*a^4*b*c^
4*d^2 - 9*a^3*b^5*c*e^2 - 20*a^5*b*c^3*e^2 - 7*a^5*b^3*c*f^2 + 12*a^6*b*c^2*f^2 - a^5*c*f^2*(-(4*a*c - b^2)^3)
^(1/2) - 2*a*b^8*d*e + 42*a^2*b^5*c^2*d^2 - 63*a^3*b^3*c^3*d^2 + a^2*b^4*e^2*(-(4*a*c - b^2)^3)^(1/2) - a^3*c^
3*d^2*(-(4*a*c - b^2)^3)^(1/2) + 25*a^4*b^3*c^2*e^2 + a^4*b^2*f^2*(-(4*a*c - b^2)^3)^(1/2) + a^4*c^2*e^2*(-(4*
a*c - b^2)^3)^(1/2) - 11*a*b^7*c*d^2 + 2*a^2*b^7*d*f - 16*a^5*c^4*d*e - 2*a^3*b^6*e*f + 16*a^6*c^3*e*f - 2*a*b
^5*d*e*(-(4*a*c - b^2)^3)^(1/2) + 20*a^2*b^6*c*d*e - 18*a^3*b^5*c*d*f - 40*a^5*b*c^3*d*f + 16*a^4*b^4*c*e*f +
6*a^2*b^2*c^2*d^2*(-(4*a*c - b^2)^3)^(1/2) - 5*a*b^4*c*d^2*(-(4*a*c - b^2)^3)^(1/2) - 66*a^3*b^4*c^2*d*e + 76*
a^4*b^2*c^3*d*e + 2*a^2*b^4*d*f*(-(4*a*c - b^2)^3)^(1/2) + 50*a^4*b^3*c^2*d*f - 2*a^3*b^3*e*f*(-(4*a*c - b^2)^
3)^(1/2) + 2*a^4*c^2*d*f*(-(4*a*c - b^2)^3)^(1/2) - 36*a^5*b^2*c^2*e*f - 3*a^3*b^2*c*e^2*(-(4*a*c - b^2)^3)^(1
/2) + 4*a^4*b*c*e*f*(-(4*a*c - b^2)^3)^(1/2) + 8*a^2*b^3*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a^3*b*c^2*d*e*(-(4
*a*c - b^2)^3)^(1/2) - 6*a^3*b^2*c*d*f*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^7*b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/
2)*(x*(32*a^16*b*c^3 - 8*a^15*b^3*c^2)*(-(b^9*d^2 + a^2*b^7*e^2 + b^6*d^2*(-(4*a*c - b^2)^3)^(1/2) + a^4*b^5*f
^2 + 28*a^4*b*c^4*d^2 - 9*a^3*b^5*c*e^2 - 20*a^5*b*c^3*e^2 - 7*a^5*b^3*c*f^2 + 12*a^6*b*c^2*f^2 - a^5*c*f^2*(-
(4*a*c - b^2)^3)^(1/2) - 2*a*b^8*d*e + 42*a^2*b^5*c^2*d^2 - 63*a^3*b^3*c^3*d^2 + a^2*b^4*e^2*(-(4*a*c - b^2)^3
)^(1/2) - a^3*c^3*d^2*(-(4*a*c - b^2)^3)^(1/2) + 25*a^4*b^3*c^2*e^2 + a^4*b^2*f^2*(-(4*a*c - b^2)^3)^(1/2) + a
^4*c^2*e^2*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*d^2 + 2*a^2*b^7*d*f - 16*a^5*c^4*d*e - 2*a^3*b^6*e*f + 16*a^6
*c^3*e*f - 2*a*b^5*d*e*(-(4*a*c - b^2)^3)^(1/2) + 20*a^2*b^6*c*d*e - 18*a^3*b^5*c*d*f - 40*a^5*b*c^3*d*f + 16*
a^4*b^4*c*e*f + 6*a^2*b^2*c^2*d^2*(-(4*a*c - b^2)^3)^(1/2) - 5*a*b^4*c*d^2*(-(4*a*c - b^2)^3)^(1/2) - 66*a^3*b
^4*c^2*d*e + 76*a^4*b^2*c^3*d*e + 2*a^2*b^4*d*f*(-(4*a*c - b^2)^3)^(1/2) + 50*a^4*b^3*c^2*d*f - 2*a^3*b^3*e*f*
(-(4*a*c - b^2)^3)^(1/2) + 2*a^4*c^2*d*f*(-(4*a*c - b^2)^3)^(1/2) - 36*a^5*b^2*c^2*e*f - 3*a^3*b^2*c*e^2*(-(4*
a*c - b^2)^3)^(1/2) + 4*a^4*b*c*e*f*(-(4*a*c - b^2)^3)^(1/2) + 8*a^2*b^3*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a^
3*b*c^2*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a^3*b^2*c*d*f*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^7*b^4 + 16*a^9*c^2 - 8*
a^8*b^2*c)))^(1/2) - 16*a^15*c^4*e + 4*a^12*b^5*c^2*d - 24*a^13*b^3*c^3*d - 4*a^13*b^4*c^2*e + 20*a^14*b^2*c^3
*e + 4*a^14*b^3*c^2*f + 32*a^14*b*c^4*d - 16*a^15*b*c^3*f))*(-(b^9*d^2 + a^2*b^7*e^2 + b^6*d^2*(-(4*a*c - b^2)
^3)^(1/2) + a^4*b^5*f^2 + 28*a^4*b*c^4*d^2 - 9*a^3*b^5*c*e^2 - 20*a^5*b*c^3*e^2 - 7*a^5*b^3*c*f^2 + 12*a^6*b*c
^2*f^2 - a^5*c*f^2*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b^8*d*e + 42*a^2*b^5*c^2*d^2 - 63*a^3*b^3*c^3*d^2 + a^2*b^4*
e^2*(-(4*a*c - b^2)^3)^(1/2) - a^3*c^3*d^2*(-(4*a*c - b^2)^3)^(1/2) + 25*a^4*b^3*c^2*e^2 + a^4*b^2*f^2*(-(4*a*
c - b^2)^3)^(1/2) + a^4*c^2*e^2*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*d^2 + 2*a^2*b^7*d*f - 16*a^5*c^4*d*e - 2
*a^3*b^6*e*f + 16*a^6*c^3*e*f - 2*a*b^5*d*e*(-(4*a*c - b^2)^3)^(1/2) + 20*a^2*b^6*c*d*e - 18*a^3*b^5*c*d*f - 4
0*a^5*b*c^3*d*f + 16*a^4*b^4*c*e*f + 6*a^2*b^2*c^2*d^2*(-(4*a*c - b^2)^3)^(1/2) - 5*a*b^4*c*d^2*(-(4*a*c - b^2
)^3)^(1/2) - 66*a^3*b^4*c^2*d*e + 76*a^4*b^2*c^3*d*e + 2*a^2*b^4*d*f*(-(4*a*c - b^2)^3)^(1/2) + 50*a^4*b^3*c^2
*d*f - 2*a^3*b^3*e*f*(-(4*a*c - b^2)^3)^(1/2) + 2*a^4*c^2*d*f*(-(4*a*c - b^2)^3)^(1/2) - 36*a^5*b^2*c^2*e*f -
3*a^3*b^2*c*e^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a^4*b*c*e*f*(-(4*a*c - b^2)^3)^(1/2) + 8*a^2*b^3*c*d*e*(-(4*a*c -
b^2)^3)^(1/2) - 6*a^3*b*c^2*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a^3*b^2*c*d*f*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^7*
b^4 + 16*a^9*c^2 - 8*a^8*b^2*c)))^(1/2)*1i + (x*(4*a^13*c^5*e^2 - 4*a^12*c^6*d^2 - 4*a^14*c^4*f^2 + 2*a^9*b^6*
c^3*d^2 - 12*a^10*b^4*c^4*d^2 + 18*a^11*b^2*c^5*d^2 + 2*a^11*b^4*c^3*e^2 - 8*a^12*b^2*c^4*e^2 + 2*a^13*b^2*c^3
*f^2 + 8*a^13*c^5*d*f - 20*a^12*b*c^5*d*e + 12*a^13*b*c^4*e*f - 4*a^10*b^5*c^3*d*e + 20*a^11*b^3*c^4*d*e + 4*a
^11*b^4*c^3*d*f - 16*a^12*b^2*c^4*d*f - 4*a^12*b^3*c^3*e*f) - (-(b^9*d^2 + a^2*b^7*e^2 + b^6*d^2*(-(4*a*c - b^
2)^3)^(1/2) + a^4*b^5*f^2 + 28*a^4*b*c^4*d^2 - 9*a^3*b^5*c*e^2 - 20*a^5*b*c^3*e^2 - 7*a^5*b^3*c*f^2 + 12*a^6*b
*c^2*f^2 - a^5*c*f^2*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b^8*d*e + 42*a^2*b^5*c^2*d^2 - 63*a^3*b^3*c^3*d^2 + a^2*b^
4*e^2*(-(4*a*c - b^2)^3)^(1/2) - a^3*c^3*d^2*(-(4*a*c - b^2)^3)^(1/2) + 25*a^4*b^3*c^2*e^2 + a^4*b^2*f^2*(-(4*
a*c - b^2)^3)^(1/2) + a^4*c^2*e^2*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*d^2 + 2*a^2*b^7*d*f - 16*a^5*c^4*d*e -
2*a^3*b^6*e*f + 16*a^6*c^3*e*f - 2*a*b^5*d*e*(...

________________________________________________________________________________________