3.9.85 \(\int \frac {x^2}{(a+b x^2+c x^4)^3} \, dx\) [885]

Optimal. Leaf size=311 \[ -\frac {x \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x \left (b \left (b^2+8 a c\right )+c \left (b^2+20 a c\right ) x^2\right )}{8 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\sqrt {c} \left (b^2+20 a c+\frac {b \left (b^2-52 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (b^2+20 a c-\frac {b \left (b^2-52 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}} \]

[Out]

-1/4*x*(2*c*x^2+b)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+1/8*x*(b*(8*a*c+b^2)+c*(20*a*c+b^2)*x^2)/a/(-4*a*c+b^2)^2/(c
*x^4+b*x^2+a)+1/16*arctan(x*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(b^2+20*a*c+b*(-52*a*c+b^2)/
(-4*a*c+b^2)^(1/2))/a/(-4*a*c+b^2)^2*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/16*arctan(x*2^(1/2)*c^(1/2)/(b+(-4
*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(b^2+20*a*c-b*(-52*a*c+b^2)/(-4*a*c+b^2)^(1/2))/a/(-4*a*c+b^2)^2*2^(1/2)/(b+(-
4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]
time = 0.49, antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1133, 1192, 1180, 211} \begin {gather*} \frac {\sqrt {c} \left (\frac {b \left (b^2-52 a c\right )}{\sqrt {b^2-4 a c}}+20 a c+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (-\frac {b \left (b^2-52 a c\right )}{\sqrt {b^2-4 a c}}+20 a c+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{8 \sqrt {2} a \left (b^2-4 a c\right )^2 \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {x \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x \left (c x^2 \left (20 a c+b^2\right )+b \left (8 a c+b^2\right )\right )}{8 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x^2 + c*x^4)^3,x]

[Out]

-1/4*(x*(b + 2*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x*(b*(b^2 + 8*a*c) + c*(b^2 + 20*a*c)*x^2))/(8
*a*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(b^2 + 20*a*c + (b*(b^2 - 52*a*c))/Sqrt[b^2 - 4*a*c])*ArcTa
n[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a*(b^2 - 4*a*c)^2*Sqrt[b - Sqrt[b^2 - 4*a*c]])
+ (Sqrt[c]*(b^2 + 20*a*c - (b*(b^2 - 52*a*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2
- 4*a*c]]])/(8*Sqrt[2]*a*(b^2 - 4*a*c)^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 1133

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d*(d*x)^(m - 1)*(b + 2*c*x
^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*(p + 1)*(b^2 - 4*a*c))), x] - Dist[d^2/(2*(p + 1)*(b^2 - 4*a*c)), Int[(d*x
)^(m - 2)*(b*(m - 1) + 2*c*(m + 4*p + 5)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

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 1192

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*(a*b*e - d*(b^2 - 2*a
*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {x^2}{\left (a+b x^2+c x^4\right )^3} \, dx &=-\frac {x \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\int \frac {b-10 c x^2}{\left (a+b x^2+c x^4\right )^2} \, dx}{4 \left (b^2-4 a c\right )}\\ &=-\frac {x \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x \left (b \left (b^2+8 a c\right )+c \left (b^2+20 a c\right ) x^2\right )}{8 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\int \frac {-b \left (b^2-16 a c\right )-c \left (b^2+20 a c\right ) x^2}{a+b x^2+c x^4} \, dx}{8 a \left (b^2-4 a c\right )^2}\\ &=-\frac {x \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x \left (b \left (b^2+8 a c\right )+c \left (b^2+20 a c\right ) x^2\right )}{8 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (c \left (b^2+20 a c-\frac {b \left (b^2-52 a c\right )}{\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}{16 a \left (b^2-4 a c\right )^2}+\frac {\left (c \left (b^2+20 a c+\frac {b \left (b^2-52 a c\right )}{\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}{16 a \left (b^2-4 a c\right )^2}\\ &=-\frac {x \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x \left (b \left (b^2+8 a c\right )+c \left (b^2+20 a c\right ) x^2\right )}{8 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\sqrt {c} \left (b^2+20 a c+\frac {b \left (b^2-52 a c\right )}{\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 )}{8 \sqrt {2} a \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (b^2+20 a c-\frac {b \left (b^2-52 a c\right )}{\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 )}{8 \sqrt {2} a \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [A]
time = 0.55, size = 334, normalized size = 1.07 \begin {gather*} \frac {1}{16} \left (-\frac {4 x \left (b+2 c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 x \left (b^3+8 a b c+b^2 c x^2+20 a c^2 x^2\right )}{a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \sqrt {c} \left (b^3-52 a b c+b^2 \sqrt {b^2-4 a c}+20 a c \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 )}{a \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-b^3+52 a b c+b^2 \sqrt {b^2-4 a c}+20 a c \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 )}{a \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x^2 + c*x^4)^3,x]

[Out]

((-4*x*(b + 2*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (2*x*(b^3 + 8*a*b*c + b^2*c*x^2 + 20*a*c^2*x^2))
/(a*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(b^3 - 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*S
qrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqr
t[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-b^3 + 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqrt[b^2 - 4*a*c])*ArcTa
n[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/16

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(835\) vs. \(2(267)=534\).
time = 0.16, size = 836, normalized size = 2.69

method	result	size
risch	\(\frac {\frac {c^{2} \left (20 a c +b^{2}\right ) x^{7}}{8 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c b \left (14 a c +b^{2}\right ) x^{5}}{4 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (36 a^{2} c^{2}+5 a \,b^{2} c +b^{4}\right ) x^{3}}{8 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {b \left (16 a c -b^{2}\right ) x}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4} c +\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\frac {c \left (20 a c +b^{2}\right ) \textit {\_R}^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {b \left (16 a c -b^{2}\right )}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} b}}{16 a}\)	\(289\)
default	\(64 c^{3} \left (-\frac {\frac {-\frac {\left (320 \sqrt {-4 a c +b^{2}}\, a^{3} c^{3}-144 \sqrt {-4 a c +b^{2}}\, a^{2} b^{2} c^{2}+12 \sqrt {-4 a c +b^{2}}\, a \,b^{4} c +\sqrt {-4 a c +b^{2}}\, b^{6}+64 a^{3} b \,c^{3}-48 a^{2} b^{3} c^{2}+12 a \,b^{5} c -b^{7}\right ) x^{3}}{16 a \,c^{2}}+\frac {\left (-96 \sqrt {-4 a c +b^{2}}\, a^{2} b \,c^{2}+48 \sqrt {-4 a c +b^{2}}\, a \,b^{3} c -6 \sqrt {-4 a c +b^{2}}\, b^{5}+448 a^{3} c^{3}-336 a^{2} b^{2} c^{2}+84 a \,b^{4} c -7 b^{6}\right ) x}{8 c^{2}}}{\left (x^{2}+\frac {\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}\right )^{2}}-\frac {\left (320 \sqrt {-4 a c +b^{2}}\, a^{3} c^{3}-144 \sqrt {-4 a c +b^{2}}\, a^{2} b^{2} c^{2}+12 \sqrt {-4 a c +b^{2}}\, a \,b^{4} c +\sqrt {-4 a c +b^{2}}\, b^{6}+832 a^{3} b \,c^{3}-432 a^{2} b^{3} c^{2}+60 a \,b^{5} c -b^{7}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a c \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{64 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} c \left (4 a c -b^{2}\right )^{2}}+\frac {\frac {-\frac {\left (-320 \sqrt {-4 a c +b^{2}}\, a^{3} c^{3}+144 \sqrt {-4 a c +b^{2}}\, a^{2} b^{2} c^{2}-12 \sqrt {-4 a c +b^{2}}\, a \,b^{4} c -\sqrt {-4 a c +b^{2}}\, b^{6}+64 a^{3} b \,c^{3}-48 a^{2} b^{3} c^{2}+12 a \,b^{5} c -b^{7}\right ) x^{3}}{16 a \,c^{2}}+\frac {\left (96 \sqrt {-4 a c +b^{2}}\, a^{2} b \,c^{2}-48 \sqrt {-4 a c +b^{2}}\, a \,b^{3} c +6 \sqrt {-4 a c +b^{2}}\, b^{5}+448 a^{3} c^{3}-336 a^{2} b^{2} c^{2}+84 a \,b^{4} c -7 b^{6}\right ) x}{8 c^{2}}}{\left (x^{2}+\frac {b}{2 c}-\frac {\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2}}-\frac {\left (320 \sqrt {-4 a c +b^{2}}\, a^{3} c^{3}-144 \sqrt {-4 a c +b^{2}}\, a^{2} b^{2} c^{2}+12 \sqrt {-4 a c +b^{2}}\, a \,b^{4} c +\sqrt {-4 a c +b^{2}}\, b^{6}-832 a^{3} b \,c^{3}+432 a^{2} b^{3} c^{2}-60 a \,b^{5} c +b^{7}\right ) \sqrt {2}\, \arctanh \left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a c \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{64 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} c \left (4 a c -b^{2}\right )^{2}}\right )\)	\(836\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(c*x^4+b*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

64*c^3*(-1/64/(-4*a*c+b^2)^(5/2)/c/(4*a*c-b^2)^2*((-1/16/a/c^2*(320*(-4*a*c+b^2)^(1/2)*a^3*c^3-144*(-4*a*c+b^2
)^(1/2)*a^2*b^2*c^2+12*(-4*a*c+b^2)^(1/2)*a*b^4*c+(-4*a*c+b^2)^(1/2)*b^6+64*a^3*b*c^3-48*a^2*b^3*c^2+12*a*b^5*
c-b^7)*x^3+1/8/c^2*(-96*(-4*a*c+b^2)^(1/2)*a^2*b*c^2+48*(-4*a*c+b^2)^(1/2)*a*b^3*c-6*(-4*a*c+b^2)^(1/2)*b^5+44
8*a^3*c^3-336*a^2*b^2*c^2+84*a*b^4*c-7*b^6)*x)/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2-1/16*(320*(-4*a*c+b^2)
^(1/2)*a^3*c^3-144*(-4*a*c+b^2)^(1/2)*a^2*b^2*c^2+12*(-4*a*c+b^2)^(1/2)*a*b^4*c+(-4*a*c+b^2)^(1/2)*b^6+832*a^3
*b*c^3-432*a^2*b^3*c^2+60*a*b^5*c-b^7)/a/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)))+1/64/(-4*a*c+b^2)^(5/2)/c/(4*a*c-b^2)^2*((-1/16/a/c^2*(-320*(-4*a*c+b^2)^(1/2)*a^3
*c^3+144*(-4*a*c+b^2)^(1/2)*a^2*b^2*c^2-12*(-4*a*c+b^2)^(1/2)*a*b^4*c-(-4*a*c+b^2)^(1/2)*b^6+64*a^3*b*c^3-48*a
^2*b^3*c^2+12*a*b^5*c-b^7)*x^3+1/8/c^2*(96*(-4*a*c+b^2)^(1/2)*a^2*b*c^2-48*(-4*a*c+b^2)^(1/2)*a*b^3*c+6*(-4*a*
c+b^2)^(1/2)*b^5+448*a^3*c^3-336*a^2*b^2*c^2+84*a*b^4*c-7*b^6)*x)/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2-1/1
6*(320*(-4*a*c+b^2)^(1/2)*a^3*c^3-144*(-4*a*c+b^2)^(1/2)*a^2*b^2*c^2+12*(-4*a*c+b^2)^(1/2)*a*b^4*c+(-4*a*c+b^2
)^(1/2)*b^6-832*a^3*b*c^3+432*a^2*b^3*c^2-60*a*b^5*c+b^7)/a/c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan
h(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^2/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")

[Out]

1/8*((b^2*c^2 + 20*a*c^3)*x^7 + 2*(b^3*c + 14*a*b*c^2)*x^5 + (b^4 + 5*a*b^2*c + 36*a^2*c^2)*x^3 - (a*b^3 - 16*
a^2*b*c)*x)/((a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*x^8 + a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 + 2*(a*b^5*c -
8*a^2*b^3*c^2 + 16*a^3*b*c^3)*x^6 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)*x^4 + 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4
*b*c^2)*x^2) + 1/8*integrate((b^3 - 16*a*b*c + (b^2*c + 20*a*c^2)*x^2)/(c*x^4 + b*x^2 + a), x)/(a*b^4 - 8*a^2*
b^2*c + 16*a^3*c^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3777 vs. \(2 (267) = 534\).
time = 0.57, size = 3777, normalized size = 12.14 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/16*(2*(b^2*c^2 + 20*a*c^3)*x^7 + 4*(b^3*c + 14*a*b*c^2)*x^5 + 2*(b^4 + 5*a*b^2*c + 36*a^2*c^2)*x^3 + sqrt(1/
2)*((a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*x^8 + a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 + 2*(a*b^5*c - 8*a^2*b^3
*c^2 + 16*a^3*b*c^3)*x^6 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)*x^4 + 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x
^2)*sqrt(-(b^7 - 35*a*b^5*c + 280*a^2*b^3*c^2 + 1680*a^3*b*c^3 + (a^3*b^10 - 20*a^4*b^8*c + 160*a^5*b^6*c^2 -
640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4 - 1024*a^8*c^5)*sqrt((b^4 - 50*a*b^2*c + 625*a^2*c^2)/(a^6*b^10 - 20*a^7*b^
8*c + 160*a^8*b^6*c^2 - 640*a^9*b^4*c^3 + 1280*a^10*b^2*c^4 - 1024*a^11*c^5)))/(a^3*b^10 - 20*a^4*b^8*c + 160*
a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4 - 1024*a^8*c^5))*log((35*b^6*c^2 - 1491*a*b^4*c^3 + 15000*a^2
*b^2*c^4 + 10000*a^3*c^5)*x + 1/2*sqrt(1/2)*(b^11 - 53*a*b^9*c + 940*a^2*b^7*c^2 - 6832*a^3*b^5*c^3 + 21824*a^
4*b^3*c^4 - 25600*a^5*b*c^5 - (a^3*b^14 - 38*a^4*b^12*c + 480*a^5*b^10*c^2 - 2720*a^6*b^8*c^3 + 6400*a^7*b^6*c
^4 + 1536*a^8*b^4*c^5 - 32768*a^9*b^2*c^6 + 40960*a^10*c^7)*sqrt((b^4 - 50*a*b^2*c + 625*a^2*c^2)/(a^6*b^10 -
20*a^7*b^8*c + 160*a^8*b^6*c^2 - 640*a^9*b^4*c^3 + 1280*a^10*b^2*c^4 - 1024*a^11*c^5)))*sqrt(-(b^7 - 35*a*b^5*
c + 280*a^2*b^3*c^2 + 1680*a^3*b*c^3 + (a^3*b^10 - 20*a^4*b^8*c + 160*a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280*a^7
*b^2*c^4 - 1024*a^8*c^5)*sqrt((b^4 - 50*a*b^2*c + 625*a^2*c^2)/(a^6*b^10 - 20*a^7*b^8*c + 160*a^8*b^6*c^2 - 64
0*a^9*b^4*c^3 + 1280*a^10*b^2*c^4 - 1024*a^11*c^5)))/(a^3*b^10 - 20*a^4*b^8*c + 160*a^5*b^6*c^2 - 640*a^6*b^4*
c^3 + 1280*a^7*b^2*c^4 - 1024*a^8*c^5))) - sqrt(1/2)*((a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*x^8 + a^3*b^4 -
8*a^4*b^2*c + 16*a^5*c^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*x^6 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3
)*x^4 + 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^2)*sqrt(-(b^7 - 35*a*b^5*c + 280*a^2*b^3*c^2 + 1680*a^3*b*c
^3 + (a^3*b^10 - 20*a^4*b^8*c + 160*a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4 - 1024*a^8*c^5)*sqrt((b^4
- 50*a*b^2*c + 625*a^2*c^2)/(a^6*b^10 - 20*a^7*b^8*c + 160*a^8*b^6*c^2 - 640*a^9*b^4*c^3 + 1280*a^10*b^2*c^4
- 1024*a^11*c^5)))/(a^3*b^10 - 20*a^4*b^8*c + 160*a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4 - 1024*a^8*
c^5))*log((35*b^6*c^2 - 1491*a*b^4*c^3 + 15000*a^2*b^2*c^4 + 10000*a^3*c^5)*x - 1/2*sqrt(1/2)*(b^11 - 53*a*b^9
*c + 940*a^2*b^7*c^2 - 6832*a^3*b^5*c^3 + 21824*a^4*b^3*c^4 - 25600*a^5*b*c^5 - (a^3*b^14 - 38*a^4*b^12*c + 48
0*a^5*b^10*c^2 - 2720*a^6*b^8*c^3 + 6400*a^7*b^6*c^4 + 1536*a^8*b^4*c^5 - 32768*a^9*b^2*c^6 + 40960*a^10*c^7)*
sqrt((b^4 - 50*a*b^2*c + 625*a^2*c^2)/(a^6*b^10 - 20*a^7*b^8*c + 160*a^8*b^6*c^2 - 640*a^9*b^4*c^3 + 1280*a^10
*b^2*c^4 - 1024*a^11*c^5)))*sqrt(-(b^7 - 35*a*b^5*c + 280*a^2*b^3*c^2 + 1680*a^3*b*c^3 + (a^3*b^10 - 20*a^4*b^
8*c + 160*a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4 - 1024*a^8*c^5)*sqrt((b^4 - 50*a*b^2*c + 625*a^2*c^
2)/(a^6*b^10 - 20*a^7*b^8*c + 160*a^8*b^6*c^2 - 640*a^9*b^4*c^3 + 1280*a^10*b^2*c^4 - 1024*a^11*c^5)))/(a^3*b^
10 - 20*a^4*b^8*c + 160*a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4 - 1024*a^8*c^5))) + sqrt(1/2)*((a*b^4
*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*x^8 + a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*
a^3*b*c^3)*x^6 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)*x^4 + 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^2)*sqrt(-
(b^7 - 35*a*b^5*c + 280*a^2*b^3*c^2 + 1680*a^3*b*c^3 - (a^3*b^10 - 20*a^4*b^8*c + 160*a^5*b^6*c^2 - 640*a^6*b^
4*c^3 + 1280*a^7*b^2*c^4 - 1024*a^8*c^5)*sqrt((b^4 - 50*a*b^2*c + 625*a^2*c^2)/(a^6*b^10 - 20*a^7*b^8*c + 160*
a^8*b^6*c^2 - 640*a^9*b^4*c^3 + 1280*a^10*b^2*c^4 - 1024*a^11*c^5)))/(a^3*b^10 - 20*a^4*b^8*c + 160*a^5*b^6*c^
2 - 640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4 - 1024*a^8*c^5))*log((35*b^6*c^2 - 1491*a*b^4*c^3 + 15000*a^2*b^2*c^4 +
10000*a^3*c^5)*x + 1/2*sqrt(1/2)*(b^11 - 53*a*b^9*c + 940*a^2*b^7*c^2 - 6832*a^3*b^5*c^3 + 21824*a^4*b^3*c^4
- 25600*a^5*b*c^5 + (a^3*b^14 - 38*a^4*b^12*c + 480*a^5*b^10*c^2 - 2720*a^6*b^8*c^3 + 6400*a^7*b^6*c^4 + 1536*
a^8*b^4*c^5 - 32768*a^9*b^2*c^6 + 40960*a^10*c^7)*sqrt((b^4 - 50*a*b^2*c + 625*a^2*c^2)/(a^6*b^10 - 20*a^7*b^8
*c + 160*a^8*b^6*c^2 - 640*a^9*b^4*c^3 + 1280*a^10*b^2*c^4 - 1024*a^11*c^5)))*sqrt(-(b^7 - 35*a*b^5*c + 280*a^
2*b^3*c^2 + 1680*a^3*b*c^3 - (a^3*b^10 - 20*a^4*b^8*c + 160*a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4 -
1024*a^8*c^5)*sqrt((b^4 - 50*a*b^2*c + 625*a^2*c^2)/(a^6*b^10 - 20*a^7*b^8*c + 160*a^8*b^6*c^2 - 640*a^9*b^4*
c^3 + 1280*a^10*b^2*c^4 - 1024*a^11*c^5)))/(a^3*b^10 - 20*a^4*b^8*c + 160*a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280
*a^7*b^2*c^4 - 1024*a^8*c^5))) - sqrt(1/2)*((a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*x^8 + a^3*b^4 - 8*a^4*b^2
*c + 16*a^5*c^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*x^6 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)*x^4 + 2*
(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^2)*sqrt(-(b^7 - 35*a*b^5*c + 280*a^2*b^3*c^2 + 1680*a^3*b*c^3 - (a^3*
b^10 - 20*a^4*b^8*c + 160*a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4 - 1024*a^8*c^5)*sqrt((b^4 - 50*a*b^
2*c + 625*a^2*c^2)/(a^6*b^10 - 20*a^7*b^8*c + 1...

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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**2/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4270 vs. \(2 (267) = 534\).
time = 5.26, size = 4270, normalized size = 13.73 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(c*x^4+b*x^2+a)^3,x, algorithm="giac")

[Out]

-1/64*(2*a^2*b^12*c^2 - 136*a^3*b^10*c^3 + 1856*a^4*b^8*c^4 - 10496*a^5*b^6*c^5 + 27136*a^6*b^4*c^6 - 26624*a^
7*b^2*c^7 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^12 + 68*sqrt(2)*sqrt(b^2 - 4*a*c)*
sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^10*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b
^11*c - 928*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^8*c^2 - 128*sqrt(2)*sqrt(b^2 - 4*a
*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^9*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^
2*b^10*c^2 + 5248*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^6*c^3 + 1344*sqrt(2)*sqrt(b^
2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^7*c^3 + 64*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*
a*c)*c)*a^3*b^8*c^3 - 13568*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^4 - 5120*sqrt(
2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^5*c^4 - 672*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq
rt(b^2 - 4*a*c)*c)*a^4*b^6*c^4 + 13312*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^2*c^5 +
6656*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^3*c^5 + 2560*sqrt(2)*sqrt(b^2 - 4*a*c)*s
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^4*c^5 - 3328*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^
6*b^2*c^6 - 2*(b^2 - 4*a*c)*a^2*b^10*c^2 + 128*(b^2 - 4*a*c)*a^3*b^8*c^3 - 1344*(b^2 - 4*a*c)*a^4*b^6*c^4 + 51
20*(b^2 - 4*a*c)*a^5*b^4*c^5 - 6656*(b^2 - 4*a*c)*a^6*b^2*c^6 + (2*b^4*c^2 + 32*a*b^2*c^3 - 160*a^2*c^4 - sqrt
(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 - 16*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)*b^3*c + 80*sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^2 + 40*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)
*a*b*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^2 - 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sq
rt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^3 - 2*(b^2 - 4*a*c)*b^2*c^2 - 40*(b^2 - 4*a*c)*a*c^3)*(a*b^4 - 8*a^2*b^2*c +
16*a^3*c^2)^2 - 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^9 - 28*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)
*a^2*b^7*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^8*c - 2*a*b^9*c + 240*sqrt(2)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*a^3*b^5*c^2 + 48*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^6*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2
- 4*a*c)*c)*a*b^7*c^2 + 56*a^2*b^7*c^2 - 832*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^3 - 288*sqrt(2)
*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^3 - 24*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c^3 - 480*a^
3*b^5*c^3 + 1024*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b*c^4 + 512*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*
c)*a^4*b^2*c^4 + 144*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^4 + 1664*a^4*b^3*c^4 - 256*sqrt(2)*sqrt
(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^5 - 2048*a^5*b*c^5 + 2*(b^2 - 4*a*c)*a*b^7*c - 48*(b^2 - 4*a*c)*a^2*b^5*c^
2 + 288*(b^2 - 4*a*c)*a^3*b^3*c^3 - 512*(b^2 - 4*a*c)*a^4*b*c^4)*abs(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2))*arctan
(2*sqrt(1/2)*x/sqrt((a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + sqrt((a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)^2 - 4*(a^2
*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*(a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)))/(a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^
3)))/((a^3*b^10 - 20*a^4*b^8*c - 2*a^3*b^9*c + 160*a^5*b^6*c^2 + 32*a^4*b^7*c^2 + a^3*b^8*c^2 - 640*a^6*b^4*c^
3 - 192*a^5*b^5*c^3 - 16*a^4*b^6*c^3 + 1280*a^7*b^2*c^4 + 512*a^6*b^3*c^4 + 96*a^5*b^4*c^4 - 1024*a^8*c^5 - 51
2*a^7*b*c^5 - 256*a^6*b^2*c^5 + 256*a^7*c^6)*abs(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*abs(c)) - 1/64*(2*a^2*b^12*
c^2 - 136*a^3*b^10*c^3 + 1856*a^4*b^8*c^4 - 10496*a^5*b^6*c^5 + 27136*a^6*b^4*c^6 - 26624*a^7*b^2*c^7 - sqrt(2
)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^12 + 68*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^
2 - 4*a*c)*c)*a^3*b^10*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^11*c - 928*sqrt(2
)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^8*c^2 - 128*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqr
t(b^2 - 4*a*c)*c)*a^3*b^9*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^10*c^2 + 5248*
sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^6*c^3 + 1344*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c - sqrt(b^2 - 4*a*c)*c)*a^4*b^7*c^3 + 64*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^8*c^
3 - 13568*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^4 - 5120*sqrt(2)*sqrt(b^2 - 4*a*
c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^5*c^4 - 672*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)
*a^4*b^6*c^4 + 13312*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^2*c^5 + 6656*sqrt(2)*sqrt
(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^3*c^5 + 2560*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2
- 4*a*c)*c)*a^5*b^4*c^5 - 3328*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^2*c^6 - 2*(b^2
- 4*a*c)*a^2*b^10*c^2 + 128*(b^2 - 4*a*c)*a^3*...

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Mupad [B]
time = 8.37, size = 2500, normalized size = 8.04 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*x^2 + c*x^4)^3,x)

[Out]

((b*x*(16*a*c - b^2))/(8*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^3*(b^4 + 36*a^2*c^2 + 5*a*b^2*c))/(8*a*(b^4 + 16
*a^2*c^2 - 8*a*b^2*c)) + (b*x^5*(14*a*c^2 + b^2*c))/(4*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (c*x^7*(20*a*c^2 +
b^2*c))/(8*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^4*(2*a*c + b^2) + a^2 + c^2*x^8 + 2*a*b*x^2 + 2*b*c*x^6) + at
an(((((256*a*b^13*c^2 + 4194304*a^7*b*c^8 - 9216*a^2*b^11*c^3 + 122880*a^3*b^9*c^4 - 819200*a^4*b^7*c^5 + 2949
120*a^5*b^5*c^6 - 5505024*a^6*b^3*c^7)/(512*(a^2*b^12 + 4096*a^8*c^6 - 24*a^3*b^10*c + 240*a^4*b^8*c^2 - 1280*
a^5*b^6*c^3 + 3840*a^6*b^4*c^4 - 6144*a^7*b^2*c^5)) - (x*(-(b^17 + b^2*(-(4*a*c - b^2)^15)^(1/2) - 1720320*a^8
*b*c^8 + 1140*a^2*b^13*c^2 - 10160*a^3*b^11*c^3 + 34880*a^4*b^9*c^4 + 43776*a^5*b^7*c^5 - 680960*a^6*b^5*c^6 +
1863680*a^7*b^3*c^7 - 55*a*b^15*c - 25*a*c*(-(4*a*c - b^2)^15)^(1/2))/(512*(a^3*b^20 + 1048576*a^13*c^10 - 40
*a^4*b^18*c + 720*a^5*b^16*c^2 - 7680*a^6*b^14*c^3 + 53760*a^7*b^12*c^4 - 258048*a^8*b^10*c^5 + 860160*a^9*b^8
*c^6 - 1966080*a^10*b^6*c^7 + 2949120*a^11*b^4*c^8 - 2621440*a^12*b^2*c^9)))^(1/2)*(262144*a^7*b*c^7 - 256*a^2
*b^11*c^2 + 5120*a^3*b^9*c^3 - 40960*a^4*b^7*c^4 + 163840*a^5*b^5*c^5 - 327680*a^6*b^3*c^6))/(32*(a^2*b^8 + 25
6*a^6*c^4 - 16*a^3*b^6*c + 96*a^4*b^4*c^2 - 256*a^5*b^2*c^3)))*(-(b^17 + b^2*(-(4*a*c - b^2)^15)^(1/2) - 17203
20*a^8*b*c^8 + 1140*a^2*b^13*c^2 - 10160*a^3*b^11*c^3 + 34880*a^4*b^9*c^4 + 43776*a^5*b^7*c^5 - 680960*a^6*b^5
*c^6 + 1863680*a^7*b^3*c^7 - 55*a*b^15*c - 25*a*c*(-(4*a*c - b^2)^15)^(1/2))/(512*(a^3*b^20 + 1048576*a^13*c^1
0 - 40*a^4*b^18*c + 720*a^5*b^16*c^2 - 7680*a^6*b^14*c^3 + 53760*a^7*b^12*c^4 - 258048*a^8*b^10*c^5 + 860160*a
^9*b^8*c^6 - 1966080*a^10*b^6*c^7 + 2949120*a^11*b^4*c^8 - 2621440*a^12*b^2*c^9)))^(1/2) - (x*(800*a^3*c^6 - b
^6*c^3 + 34*a*b^4*c^4 - 1472*a^2*b^2*c^5))/(32*(a^2*b^8 + 256*a^6*c^4 - 16*a^3*b^6*c + 96*a^4*b^4*c^2 - 256*a^
5*b^2*c^3)))*(-(b^17 + b^2*(-(4*a*c - b^2)^15)^(1/2) - 1720320*a^8*b*c^8 + 1140*a^2*b^13*c^2 - 10160*a^3*b^11*
c^3 + 34880*a^4*b^9*c^4 + 43776*a^5*b^7*c^5 - 680960*a^6*b^5*c^6 + 1863680*a^7*b^3*c^7 - 55*a*b^15*c - 25*a*c*
(-(4*a*c - b^2)^15)^(1/2))/(512*(a^3*b^20 + 1048576*a^13*c^10 - 40*a^4*b^18*c + 720*a^5*b^16*c^2 - 7680*a^6*b^
14*c^3 + 53760*a^7*b^12*c^4 - 258048*a^8*b^10*c^5 + 860160*a^9*b^8*c^6 - 1966080*a^10*b^6*c^7 + 2949120*a^11*b
^4*c^8 - 2621440*a^12*b^2*c^9)))^(1/2)*1i - (((256*a*b^13*c^2 + 4194304*a^7*b*c^8 - 9216*a^2*b^11*c^3 + 122880
*a^3*b^9*c^4 - 819200*a^4*b^7*c^5 + 2949120*a^5*b^5*c^6 - 5505024*a^6*b^3*c^7)/(512*(a^2*b^12 + 4096*a^8*c^6 -
24*a^3*b^10*c + 240*a^4*b^8*c^2 - 1280*a^5*b^6*c^3 + 3840*a^6*b^4*c^4 - 6144*a^7*b^2*c^5)) + (x*(-(b^17 + b^2
*(-(4*a*c - b^2)^15)^(1/2) - 1720320*a^8*b*c^8 + 1140*a^2*b^13*c^2 - 10160*a^3*b^11*c^3 + 34880*a^4*b^9*c^4 +
43776*a^5*b^7*c^5 - 680960*a^6*b^5*c^6 + 1863680*a^7*b^3*c^7 - 55*a*b^15*c - 25*a*c*(-(4*a*c - b^2)^15)^(1/2))
/(512*(a^3*b^20 + 1048576*a^13*c^10 - 40*a^4*b^18*c + 720*a^5*b^16*c^2 - 7680*a^6*b^14*c^3 + 53760*a^7*b^12*c^
4 - 258048*a^8*b^10*c^5 + 860160*a^9*b^8*c^6 - 1966080*a^10*b^6*c^7 + 2949120*a^11*b^4*c^8 - 2621440*a^12*b^2*
c^9)))^(1/2)*(262144*a^7*b*c^7 - 256*a^2*b^11*c^2 + 5120*a^3*b^9*c^3 - 40960*a^4*b^7*c^4 + 163840*a^5*b^5*c^5
- 327680*a^6*b^3*c^6))/(32*(a^2*b^8 + 256*a^6*c^4 - 16*a^3*b^6*c + 96*a^4*b^4*c^2 - 256*a^5*b^2*c^3)))*(-(b^17
+ b^2*(-(4*a*c - b^2)^15)^(1/2) - 1720320*a^8*b*c^8 + 1140*a^2*b^13*c^2 - 10160*a^3*b^11*c^3 + 34880*a^4*b^9*
c^4 + 43776*a^5*b^7*c^5 - 680960*a^6*b^5*c^6 + 1863680*a^7*b^3*c^7 - 55*a*b^15*c - 25*a*c*(-(4*a*c - b^2)^15)^
(1/2))/(512*(a^3*b^20 + 1048576*a^13*c^10 - 40*a^4*b^18*c + 720*a^5*b^16*c^2 - 7680*a^6*b^14*c^3 + 53760*a^7*b
^12*c^4 - 258048*a^8*b^10*c^5 + 860160*a^9*b^8*c^6 - 1966080*a^10*b^6*c^7 + 2949120*a^11*b^4*c^8 - 2621440*a^1
2*b^2*c^9)))^(1/2) + (x*(800*a^3*c^6 - b^6*c^3 + 34*a*b^4*c^4 - 1472*a^2*b^2*c^5))/(32*(a^2*b^8 + 256*a^6*c^4
- 16*a^3*b^6*c + 96*a^4*b^4*c^2 - 256*a^5*b^2*c^3)))*(-(b^17 + b^2*(-(4*a*c - b^2)^15)^(1/2) - 1720320*a^8*b*c
^8 + 1140*a^2*b^13*c^2 - 10160*a^3*b^11*c^3 + 34880*a^4*b^9*c^4 + 43776*a^5*b^7*c^5 - 680960*a^6*b^5*c^6 + 186
3680*a^7*b^3*c^7 - 55*a*b^15*c - 25*a*c*(-(4*a*c - b^2)^15)^(1/2))/(512*(a^3*b^20 + 1048576*a^13*c^10 - 40*a^4
*b^18*c + 720*a^5*b^16*c^2 - 7680*a^6*b^14*c^3 + 53760*a^7*b^12*c^4 - 258048*a^8*b^10*c^5 + 860160*a^9*b^8*c^6
- 1966080*a^10*b^6*c^7 + 2949120*a^11*b^4*c^8 - 2621440*a^12*b^2*c^9)))^(1/2)*1i)/((((256*a*b^13*c^2 + 419430
4*a^7*b*c^8 - 9216*a^2*b^11*c^3 + 122880*a^3*b^9*c^4 - 819200*a^4*b^7*c^5 + 2949120*a^5*b^5*c^6 - 5505024*a^6*
b^3*c^7)/(512*(a^2*b^12 + 4096*a^8*c^6 - 24*a^3*b^10*c + 240*a^4*b^8*c^2 - 1280*a^5*b^6*c^3 + 3840*a^6*b^4*c^4
- 6144*a^7*b^2*c^5)) - (x*(-(b^17 + b^2*(-(4*a*c - b^2)^15)^(1/2) - 1720320*a^8*b*c^8 + 1140*a^2*b^13*c^2 - 1
0160*a^3*b^11*c^3 + 34880*a^4*b^9*c^4 + 43776*a^5*b^7*c^5 - 680960*a^6*b^5*c^6 + 1863680*a^7*b^3*c^7 - 55*a*b^
15*c - 25*a*c*(-(4*a*c - b^2)^15)^(1/2))/(512*(a^3*b^20 + 1048576*a^13*c^10 - 40*a^4*b^18*c + 720*a^5*b^16*c^2
- 7680*a^6*b^14*c^3 + 53760*a^7*b^12*c^4 - 258...

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