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3.4.4 \(\int \frac {x^6}{(d+e x^2) (a+b x^2+c x^4)} \, dx\) [304]

Optimal. Leaf size=323 \[ \frac {x}{c e}+\frac {\left (b^2 d-a c d-a b e-\frac {b^3 d-3 a b c d-a b^2 e+2 a^2 c e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}+\frac {\left (b^2 d-a c d-a b e+\frac {b^3 d-3 a b c d-a b^2 e+2 a^2 c e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2} \left (c d^2-b d e+a e^2\right )} \]

[Out]

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

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Rubi [A]
time = 0.94, antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1301, 211, 1180} \begin {gather*} \frac {\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-3 a b c d+b^3 d}{\sqrt {b^2-4 a c}}-a b e-a c d+b^2 d\right )}{\sqrt {2} c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} \left (a e^2-b d e+c d^2\right )}+\frac {\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-3 a b c d+b^3 d}{\sqrt {b^2-4 a c}}-a b e-a c d+b^2 d\right )}{\sqrt {2} c^{3/2} \sqrt {\sqrt {b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )}-\frac {d^{5/2} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac {x}{c e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 1301

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]

Rubi steps

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

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Mathematica [A]
time = 0.34, size = 385, normalized size = 1.19 \begin {gather*} \frac {x}{c e}+\frac {\left (b^3 d-b^2 \left (\sqrt {b^2-4 a c} d+a e\right )+a c \left (\sqrt {b^2-4 a c} d+2 a e\right )+a b \left (-3 c d+\sqrt {b^2-4 a c} e\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}} \left (-c d^2+e (b d-a e)\right )}+\frac {\left (b^3 d+b^2 \left (\sqrt {b^2-4 a c} d-a e\right )+a c \left (-\sqrt {b^2-4 a c} d+2 a e\right )-a b \left (3 c d+\sqrt {b^2-4 a c} e\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2+e (-b d+a e)\right )}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2} \left (c d^2-b d e+a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(c*e) + ((b^3*d - b^2*(Sqrt[b^2 - 4*a*c]*d + a*e) + a*c*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) + a*b*(-3*c*d + Sqrt[b
^2 - 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sq
rt[b - Sqrt[b^2 - 4*a*c]]*(-(c*d^2) + e*(b*d - a*e))) + ((b^3*d + b^2*(Sqrt[b^2 - 4*a*c]*d - a*e) + a*c*(-(Sqr
t[b^2 - 4*a*c]*d) + 2*a*e) - a*b*(3*c*d + Sqrt[b^2 - 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 -
 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 + e*(-(b*d) + a*e))) - (d^(5/
2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(e^(3/2)*(c*d^2 - b*d*e + a*e^2))

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Maple [A]
time = 0.21, size = 331, normalized size = 1.02

method result size
default \(\frac {x}{c e}+\frac {-\frac {\left (-a b e \sqrt {-4 a c +b^{2}}-\sqrt {-4 a c +b^{2}}\, a c d +\sqrt {-4 a c +b^{2}}\, b^{2} d -2 a^{2} c e +a \,b^{2} e +3 a b c d -b^{3} d \right ) \sqrt {2}\, \arctanh \left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-a b e \sqrt {-4 a c +b^{2}}-\sqrt {-4 a c +b^{2}}\, a c d +\sqrt {-4 a c +b^{2}}\, b^{2} d +2 a^{2} c e -a \,b^{2} e -3 a b c d +b^{3} d \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{a \,e^{2}-d e b +c \,d^{2}}-\frac {d^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e \left (a \,e^{2}-d e b +c \,d^{2}\right ) \sqrt {d e}}\) \(331\)
risch \(\text {Expression too large to display}\) \(5281\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c/e+4/(a*e^2-b*d*e+c*d^2)*(-1/8*(-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
-2*a^2*c*e+a*b^2*e+3*a*b*c*d-b^3*d)/c/(-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*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+2*a^2*c*e-a*b^2*e-3*a*b*c*d+b^3*d)/c/(-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/e*d^3/(a*e^2-b*d*e+c*d^2)/(d*e)^(1/2)*arctan(e*x/(d*
e)^(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/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 9909 vs. \(2 (284) = 568\).
time = 25.98, size = 19849, normalized size = 61.45 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(-d*e^(-1))*c*d^2*log((x^2*e - 2*sqrt(-d*e^(-1))*x*e - d)/(x^2*e + d)) + 2*c*d^2*x - 2*b*d*x*e + 2*a
*x*e^2 + sqrt(1/2)*(c^2*d^2*e - b*c*d*e^2 + a*c*e^3)*sqrt(-((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d^2 - 2*(a*b^4 - 4
*a^2*b^2*c + 2*a^3*c^2)*d*e + (a^2*b^3 - 3*a^3*b*c)*e^2 + ((b^2*c^5 - 4*a*c^6)*d^4 - 2*(b^3*c^4 - 4*a*b*c^5)*d
^3*e + (b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d^2*e^2 - 2*(a*b^3*c^3 - 4*a^2*b*c^4)*d*e^3 + (a^2*b^2*c^3 - 4*a^3*
c^4)*e^4)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^4 - 4*(a*b^7 - 5*a^2*b^5*c + 7*
a^3*b^3*c^2 - 2*a^4*b*c^3)*d^3*e + 2*(3*a^2*b^6 - 12*a^3*b^4*c + 12*a^4*b^2*c^2 - a^5*c^3)*d^2*e^2 - 4*(a^3*b^
5 - 3*a^4*b^3*c + 2*a^5*b*c^2)*d*e^3 + (a^4*b^4 - 2*a^5*b^2*c + a^6*c^2)*e^4)/((b^2*c^10 - 4*a*c^11)*d^8 - 4*(
b^3*c^9 - 4*a*b*c^10)*d^7*e + 2*(3*b^4*c^8 - 10*a*b^2*c^9 - 8*a^2*c^10)*d^6*e^2 - 4*(b^5*c^7 - a*b^3*c^8 - 12*
a^2*b*c^9)*d^5*e^3 + (b^6*c^6 + 8*a*b^4*c^7 - 42*a^2*b^2*c^8 - 24*a^3*c^9)*d^4*e^4 - 4*(a*b^5*c^6 - a^2*b^3*c^
7 - 12*a^3*b*c^8)*d^3*e^5 + 2*(3*a^2*b^4*c^6 - 10*a^3*b^2*c^7 - 8*a^4*c^8)*d^2*e^6 - 4*(a^3*b^3*c^6 - 4*a^4*b*
c^7)*d*e^7 + (a^4*b^2*c^6 - 4*a^5*c^7)*e^8)))/((b^2*c^5 - 4*a*c^6)*d^4 - 2*(b^3*c^4 - 4*a*b*c^5)*d^3*e + (b^4*
c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d^2*e^2 - 2*(a*b^3*c^3 - 4*a^2*b*c^4)*d*e^3 + (a^2*b^2*c^3 - 4*a^3*c^4)*e^4))*l
og(-2*(a^2*b^4 - 3*a^3*b^2*c + a^4*c^2)*d^2*x + 4*(a^3*b^3 - 2*a^4*b*c)*d*x*e - 2*(a^4*b^2 - a^5*c)*x*e^2 + sq
rt(1/2)*((b^7 - 7*a*b^5*c + 13*a^2*b^3*c^2 - 4*a^3*b*c^3)*d^3 - (3*a*b^6 - 19*a^2*b^4*c + 29*a^3*b^2*c^2 - 4*a
^4*c^3)*d^2*e + (3*a^2*b^5 - 17*a^3*b^3*c + 20*a^4*b*c^2)*d*e^2 - (a^3*b^4 - 5*a^4*b^2*c + 4*a^5*c^2)*e^3 - ((
b^4*c^5 - 6*a*b^2*c^6 + 8*a^2*c^7)*d^5 - (2*b^5*c^4 - 11*a*b^3*c^5 + 12*a^2*b*c^6)*d^4*e + (b^6*c^3 - 2*a*b^4*
c^4 - 12*a^2*b^2*c^5 + 16*a^3*c^6)*d^3*e^2 - (3*a*b^5*c^3 - 14*a^2*b^3*c^4 + 8*a^3*b*c^5)*d^2*e^3 + (3*a^2*b^4
*c^3 - 14*a^3*b^2*c^4 + 8*a^4*c^5)*d*e^4 - (a^3*b^3*c^3 - 4*a^4*b*c^4)*e^5)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^
4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*d^4 - 4*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d^3*e + 2*(3*a^2*
b^6 - 12*a^3*b^4*c + 12*a^4*b^2*c^2 - a^5*c^3)*d^2*e^2 - 4*(a^3*b^5 - 3*a^4*b^3*c + 2*a^5*b*c^2)*d*e^3 + (a^4*
b^4 - 2*a^5*b^2*c + a^6*c^2)*e^4)/((b^2*c^10 - 4*a*c^11)*d^8 - 4*(b^3*c^9 - 4*a*b*c^10)*d^7*e + 2*(3*b^4*c^8 -
 10*a*b^2*c^9 - 8*a^2*c^10)*d^6*e^2 - 4*(b^5*c^7 - a*b^3*c^8 - 12*a^2*b*c^9)*d^5*e^3 + (b^6*c^6 + 8*a*b^4*c^7
- 42*a^2*b^2*c^8 - 24*a^3*c^9)*d^4*e^4 - 4*(a*b^5*c^6 - a^2*b^3*c^7 - 12*a^3*b*c^8)*d^3*e^5 + 2*(3*a^2*b^4*c^6
 - 10*a^3*b^2*c^7 - 8*a^4*c^8)*d^2*e^6 - 4*(a^3*b^3*c^6 - 4*a^4*b*c^7)*d*e^7 + (a^4*b^2*c^6 - 4*a^5*c^7)*e^8))
)*sqrt(-((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d^2 - 2*(a*b^4 - 4*a^2*b^2*c + 2*a^3*c^2)*d*e + (a^2*b^3 - 3*a^3*b*c)
*e^2 + ((b^2*c^5 - 4*a*c^6)*d^4 - 2*(b^3*c^4 - 4*a*b*c^5)*d^3*e + (b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d^2*e^2
- 2*(a*b^3*c^3 - 4*a^2*b*c^4)*d*e^3 + (a^2*b^2*c^3 - 4*a^3*c^4)*e^4)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 -
 6*a^3*b^2*c^3 + a^4*c^4)*d^4 - 4*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d^3*e + 2*(3*a^2*b^6 - 1
2*a^3*b^4*c + 12*a^4*b^2*c^2 - a^5*c^3)*d^2*e^2 - 4*(a^3*b^5 - 3*a^4*b^3*c + 2*a^5*b*c^2)*d*e^3 + (a^4*b^4 - 2
*a^5*b^2*c + a^6*c^2)*e^4)/((b^2*c^10 - 4*a*c^11)*d^8 - 4*(b^3*c^9 - 4*a*b*c^10)*d^7*e + 2*(3*b^4*c^8 - 10*a*b
^2*c^9 - 8*a^2*c^10)*d^6*e^2 - 4*(b^5*c^7 - a*b^3*c^8 - 12*a^2*b*c^9)*d^5*e^3 + (b^6*c^6 + 8*a*b^4*c^7 - 42*a^
2*b^2*c^8 - 24*a^3*c^9)*d^4*e^4 - 4*(a*b^5*c^6 - a^2*b^3*c^7 - 12*a^3*b*c^8)*d^3*e^5 + 2*(3*a^2*b^4*c^6 - 10*a
^3*b^2*c^7 - 8*a^4*c^8)*d^2*e^6 - 4*(a^3*b^3*c^6 - 4*a^4*b*c^7)*d*e^7 + (a^4*b^2*c^6 - 4*a^5*c^7)*e^8)))/((b^2
*c^5 - 4*a*c^6)*d^4 - 2*(b^3*c^4 - 4*a*b*c^5)*d^3*e + (b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d^2*e^2 - 2*(a*b^3*c
^3 - 4*a^2*b*c^4)*d*e^3 + (a^2*b^2*c^3 - 4*a^3*c^4)*e^4))) - sqrt(1/2)*(c^2*d^2*e - b*c*d*e^2 + a*c*e^3)*sqrt(
-((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d^2 - 2*(a*b^4 - 4*a^2*b^2*c + 2*a^3*c^2)*d*e + (a^2*b^3 - 3*a^3*b*c)*e^2 +
((b^2*c^5 - 4*a*c^6)*d^4 - 2*(b^3*c^4 - 4*a*b*c^5)*d^3*e + (b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d^2*e^2 - 2*(a*
b^3*c^3 - 4*a^2*b*c^4)*d*e^3 + (a^2*b^2*c^3 - 4*a^3*c^4)*e^4)*sqrt(((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*
b^2*c^3 + a^4*c^4)*d^4 - 4*(a*b^7 - 5*a^2*b^5*c + 7*a^3*b^3*c^2 - 2*a^4*b*c^3)*d^3*e + 2*(3*a^2*b^6 - 12*a^3*b
^4*c + 12*a^4*b^2*c^2 - a^5*c^3)*d^2*e^2 - 4*(a^3*b^5 - 3*a^4*b^3*c + 2*a^5*b*c^2)*d*e^3 + (a^4*b^4 - 2*a^5*b^
2*c + a^6*c^2)*e^4)/((b^2*c^10 - 4*a*c^11)*d^8 - 4*(b^3*c^9 - 4*a*b*c^10)*d^7*e + 2*(3*b^4*c^8 - 10*a*b^2*c^9
- 8*a^2*c^10)*d^6*e^2 - 4*(b^5*c^7 - a*b^3*c^8 - 12*a^2*b*c^9)*d^5*e^3 + (b^6*c^6 + 8*a*b^4*c^7 - 42*a^2*b^2*c
^8 - 24*a^3*c^9)*d^4*e^4 - 4*(a*b^5*c^6 - a^2*b^3*c^7 - 12*a^3*b*c^8)*d^3*e^5 + 2*(3*a^2*b^4*c^6 - 10*a^3*b^2*
c^7 - 8*a^4*c^8)*d^2*e^6 - 4*(a^3*b^3*c^6 - 4*a^4*b*c^7)*d*e^7 + (a^4*b^2*c^6 - 4*a^5*c^7)*e^8)))/((b^2*c^5 -
4*a*c^6)*d^4 - 2*(b^3*c^4 - 4*a*b*c^5)*d^3*e + (b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d^2*e^2 - 2*(a*b^3*c^3 - 4*
a^2*b*c^4)*d*e^3 + (a^2*b^2*c^3 - 4*a^3*c^4)*e^...

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 11030 vs. \(2 (284) = 568\).
time = 9.92, size = 11030, normalized size = 34.15 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d^(5/2)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/(c*d^2*e - b*d*e^2 + a*e^3) - 1/8*((2*b^6*c^6 - 14*a*b^4*c^7 + 24*
a^2*b^2*c^8 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^6*c^4 + 7*sqrt(2)*sqrt(b^2 - 4*a*c)*
sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4*c^5 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^5*c^
5 - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^6 - 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr
t(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4*c^6 + 3
*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^7 - 2*(b^2 - 4*a*c)*b^4*c^6 + 6*(b^2 - 4*a*
c)*a*b^2*c^7)*d^5 - (4*b^7*c^5 - 26*a*b^5*c^6 + 36*a^2*b^3*c^7 + 16*a^3*b*c^8 - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sq
rt(b*c - sqrt(b^2 - 4*a*c)*c)*b^7*c^3 + 13*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^5*c^4
 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^6*c^4 - 18*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c
 - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^5 - 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4*c^5 -
 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^5*c^5 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c -
sqrt(b^2 - 4*a*c)*c)*a^3*b*c^6 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^6 + 5*s
qrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^6 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sq
rt(b^2 - 4*a*c)*c)*a^2*b*c^7 - 4*(b^2 - 4*a*c)*b^5*c^5 + 10*(b^2 - 4*a*c)*a*b^3*c^6 + 4*(b^2 - 4*a*c)*a^2*b*c^
7)*d^4*e - 2*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^5*c^3 - 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^
2*b^3*c^4 - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4*c^4 + 2*a*b^5*c^4 + 16*sqrt(2)*sqrt(b*c - sqrt(b^2
 - 4*a*c)*c)*a^3*b*c^5 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^5 + sqrt(2)*sqrt(b*c - sqrt(b^2 -
 4*a*c)*c)*a*b^3*c^5 - 16*a^2*b^3*c^5 - 4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^6 + 32*a^3*b*c^6 - 2
*(b^2 - 4*a*c)*a*b^3*c^4 + 8*(b^2 - 4*a*c)*a^2*b*c^5)*d^3*abs(-c^2*d^2 + b*c*d*e - a*c*e^2) + (2*b^8*c^4 - 6*a
*b^6*c^5 - 28*a^2*b^4*c^6 + 80*a^3*b^2*c^7 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^8*c^2
 + 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^6*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c - sqrt(b^2 - 4*a*c)*c)*b^7*c^3 + 14*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^4 +
2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^5*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - s
qrt(b^2 - 4*a*c)*c)*b^6*c^4 - 40*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^5 - 20*sq
rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqr
t(b^2 - 4*a*c)*c)*a*b^4*c^5 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^6 - 2*(b^
2 - 4*a*c)*b^6*c^4 - 2*(b^2 - 4*a*c)*a*b^4*c^5 + 20*(b^2 - 4*a*c)*a^2*b^2*c^6)*d^3*e^2 + 2*(sqrt(2)*sqrt(b*c -
 sqrt(b^2 - 4*a*c)*c)*a*b^6*c^2 - 7*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^3 - 2*sqrt(2)*sqrt(b*c -
 sqrt(b^2 - 4*a*c)*c)*a*b^5*c^3 + 2*a*b^6*c^3 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^4 + 6*sqrt
(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^4 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4*c^4 - 14*a^2*b
^4*c^4 + 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*c^5 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b*
c^5 - 3*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^5 + 16*a^3*b^2*c^5 - 4*sqrt(2)*sqrt(b*c - sqrt(b^2 -
 4*a*c)*c)*a^3*c^6 + 32*a^4*c^6 - 2*(b^2 - 4*a*c)*a*b^4*c^3 + 6*(b^2 - 4*a*c)*a^2*b^2*c^4 + 8*(b^2 - 4*a*c)*a^
3*c^5)*d^2*abs(-c^2*d^2 + b*c*d*e - a*c*e^2)*e - (2*b^6*c^2 - 18*a*b^4*c^3 + 48*a^2*b^2*c^4 - 32*a^3*c^5 - sqr
t(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)*sq
rt(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)*(c^2*d^2 - b*c*d*e + a*c*e^2)^2*d - (6*a*b^7*c^4 - 36*a^2*b^5*c^5 + 40*a^3*b^3*c^6 + 32*a^4*b*c^7 -
3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^7*c^2 + 18*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c
- sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c^3 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^6*c^3 - 2
0*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^4 - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c - sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^4 - 3*sqrt(2...

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Mupad [B]
time = 6.45, size = 2500, normalized size = 7.74 \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)*(a + b*x^2 + c*x^4)),x)

[Out]

x/(c*e) - atan(((((((64*a^5*c^4*d*e^8 + 64*a^3*c^6*d^5*e^4 + 128*a^4*c^5*d^3*e^6 - 144*a^2*b^2*c^5*d^5*e^4 + 6
4*a^2*b^3*c^4*d^4*e^5 + 16*a^2*b^4*c^3*d^3*e^6 - 96*a^3*b^2*c^4*d^3*e^6 + 16*a^3*b^3*c^3*d^2*e^7 - 16*a*b^3*c^
5*d^6*e^3 + 32*a*b^4*c^4*d^5*e^4 - 16*a*b^5*c^3*d^4*e^5 + 64*a^2*b*c^6*d^6*e^3 - 64*a^4*b*c^4*d^2*e^7 - 16*a^4
*b^2*c^3*d*e^8)/(c*e) - (2*x*(-(b^7*d^2 + a^2*b^5*e^2 - b^4*d^2*(-(4*a*c - b^2)^3)^(1/2) - 20*a^3*b*c^3*d^2 -
7*a^3*b^3*c*e^2 + 12*a^4*b*c^2*e^2 + a^3*c*e^2*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b^6*d*e + 25*a^2*b^3*c^2*d^2 - a
^2*b^2*e^2*(-(4*a*c - b^2)^3)^(1/2) - a^2*c^2*d^2*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*d^2 + 16*a^4*c^3*d*e +
2*a*b^3*d*e*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b^4*c*d*e + 3*a*b^2*c*d^2*(-(4*a*c - b^2)^3)^(1/2) - 36*a^3*b^2*
c^2*d*e - 4*a^2*b*c*d*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^7*d^4 + 16*a^4*c^5*e^4 + b^4*c^5*d^4 - 8*a*b^2*
c^6*d^4 - 2*b^5*c^4*d^3*e + a^2*b^4*c^3*e^4 - 8*a^3*b^2*c^4*e^4 + 32*a^3*c^6*d^2*e^2 + b^6*c^3*d^2*e^2 + 16*a*
b^3*c^5*d^3*e - 2*a*b^5*c^3*d*e^3 - 32*a^2*b*c^6*d^3*e - 32*a^3*b*c^5*d*e^3 - 6*a*b^4*c^4*d^2*e^2 + 16*a^2*b^3
*c^4*d*e^3)))^(1/2)*(128*a^4*b^2*c^4*e^10 - 16*a^3*b^4*c^3*e^10 - 256*a^5*c^5*e^10 + 256*a^2*c^8*d^6*e^4 + 256
*a^3*c^7*d^4*e^6 - 256*a^4*c^6*d^2*e^8 - 16*b^3*c^7*d^7*e^3 + 64*b^4*c^6*d^6*e^4 - 96*b^5*c^5*d^5*e^5 + 64*b^6
*c^4*d^4*e^6 - 16*b^7*c^3*d^3*e^7 + 256*a^2*b^2*c^6*d^4*e^6 + 144*a^2*b^3*c^5*d^3*e^7 - 96*a^2*b^4*c^4*d^2*e^8
 + 192*a^3*b^2*c^5*d^2*e^8 + 64*a*b*c^8*d^7*e^3 + 320*a^4*b*c^5*d*e^9 - 320*a*b^2*c^7*d^6*e^4 + 528*a*b^3*c^6*
d^5*e^5 - 336*a*b^4*c^5*d^4*e^6 + 48*a*b^5*c^4*d^3*e^7 + 16*a*b^6*c^3*d^2*e^8 - 576*a^2*b*c^7*d^5*e^5 + 16*a^2
*b^5*c^3*d*e^9 - 320*a^3*b*c^6*d^3*e^7 - 144*a^3*b^3*c^4*d*e^9))/(c*e))*(-(b^7*d^2 + a^2*b^5*e^2 - b^4*d^2*(-(
4*a*c - b^2)^3)^(1/2) - 20*a^3*b*c^3*d^2 - 7*a^3*b^3*c*e^2 + 12*a^4*b*c^2*e^2 + a^3*c*e^2*(-(4*a*c - b^2)^3)^(
1/2) - 2*a*b^6*d*e + 25*a^2*b^3*c^2*d^2 - a^2*b^2*e^2*(-(4*a*c - b^2)^3)^(1/2) - a^2*c^2*d^2*(-(4*a*c - b^2)^3
)^(1/2) - 9*a*b^5*c*d^2 + 16*a^4*c^3*d*e + 2*a*b^3*d*e*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b^4*c*d*e + 3*a*b^2*c
*d^2*(-(4*a*c - b^2)^3)^(1/2) - 36*a^3*b^2*c^2*d*e - 4*a^2*b*c*d*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^7*d^
4 + 16*a^4*c^5*e^4 + b^4*c^5*d^4 - 8*a*b^2*c^6*d^4 - 2*b^5*c^4*d^3*e + a^2*b^4*c^3*e^4 - 8*a^3*b^2*c^4*e^4 + 3
2*a^3*c^6*d^2*e^2 + b^6*c^3*d^2*e^2 + 16*a*b^3*c^5*d^3*e - 2*a*b^5*c^3*d*e^3 - 32*a^2*b*c^6*d^3*e - 32*a^3*b*c
^5*d*e^3 - 6*a*b^4*c^4*d^2*e^2 + 16*a^2*b^3*c^4*d*e^3)))^(1/2) + (2*x*(4*a^3*b^5*e^8 + 4*b^3*c^5*d^8 + 4*b^8*d
^3*e^5 - 28*a^4*b^3*c*e^8 + 48*a^5*b*c^2*e^8 - 4*a*b^7*d^2*e^6 - 4*a^2*b^6*d*e^7 - 64*a^2*c^6*d^7*e + 56*a^5*c
^3*d*e^7 - 8*b^4*c^4*d^7*e - 8*b^7*c*d^4*e^4 - 8*a^3*c^5*d^5*e^3 - 16*a^4*c^4*d^3*e^5 + 4*b^5*c^3*d^6*e^2 + 4*
b^6*c^2*d^5*e^3 - 16*a*b*c^6*d^8 + 36*a^2*b^2*c^4*d^5*e^3 - 72*a^2*b^3*c^3*d^4*e^4 - 12*a^2*b^4*c^2*d^3*e^5 +
64*a^3*b^2*c^3*d^3*e^5 + 28*a^3*b^3*c^2*d^2*e^6 + 48*a*b^2*c^5*d^7*e - 16*a*b^6*c*d^3*e^5 + 40*a^3*b^4*c*d*e^7
 - 28*a*b^3*c^4*d^6*e^2 - 24*a*b^4*c^3*d^5*e^3 + 48*a*b^5*c^2*d^4*e^4 + 48*a^2*b*c^5*d^6*e^2 + 12*a^2*b^5*c*d^
2*e^6 + 16*a^3*b*c^4*d^4*e^4 - 64*a^4*b*c^3*d^2*e^6 - 108*a^4*b^2*c^2*d*e^7))/(c*e))*(-(b^7*d^2 + a^2*b^5*e^2
- b^4*d^2*(-(4*a*c - b^2)^3)^(1/2) - 20*a^3*b*c^3*d^2 - 7*a^3*b^3*c*e^2 + 12*a^4*b*c^2*e^2 + a^3*c*e^2*(-(4*a*
c - b^2)^3)^(1/2) - 2*a*b^6*d*e + 25*a^2*b^3*c^2*d^2 - a^2*b^2*e^2*(-(4*a*c - b^2)^3)^(1/2) - a^2*c^2*d^2*(-(4
*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*d^2 + 16*a^4*c^3*d*e + 2*a*b^3*d*e*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b^4*c*d*
e + 3*a*b^2*c*d^2*(-(4*a*c - b^2)^3)^(1/2) - 36*a^3*b^2*c^2*d*e - 4*a^2*b*c*d*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(
16*a^2*c^7*d^4 + 16*a^4*c^5*e^4 + b^4*c^5*d^4 - 8*a*b^2*c^6*d^4 - 2*b^5*c^4*d^3*e + a^2*b^4*c^3*e^4 - 8*a^3*b^
2*c^4*e^4 + 32*a^3*c^6*d^2*e^2 + b^6*c^3*d^2*e^2 + 16*a*b^3*c^5*d^3*e - 2*a*b^5*c^3*d*e^3 - 32*a^2*b*c^6*d^3*e
 - 32*a^3*b*c^5*d*e^3 - 6*a*b^4*c^4*d^2*e^2 + 16*a^2*b^3*c^4*d*e^3)))^(1/2) - (4*a*b^3*c^3*d^7 - 16*a^2*b*c^4*
d^7 + 4*a*b^6*d^4*e^3 + 4*a^4*b^3*d*e^6 + 48*a^3*c^4*d^6*e - 4*a^2*b^5*d^3*e^4 - 4*a^3*b^4*d^2*e^5 - 60*a^4*c^
3*d^4*e^3 + 4*a^5*c^2*d^2*e^5 - 8*a^5*b*c*d*e^6 - 32*a^2*b^3*c^2*d^5*e^2 + 92*a^3*b^2*c^2*d^4*e^3 + 4*a*b^4*c^
2*d^6*e + 4*a*b^5*c*d^5*e^2 - 28*a^2*b^2*c^3*d^6*e - 36*a^2*b^4*c*d^4*e^3 + 64*a^3*b*c^3*d^5*e^2 + 36*a^3*b^3*
c*d^3*e^4 - 60*a^4*b*c^2*d^3*e^4 + 4*a^4*b^2*c*d^2*e^5)/(c*e))*(-(b^7*d^2 + a^2*b^5*e^2 - b^4*d^2*(-(4*a*c - b
^2)^3)^(1/2) - 20*a^3*b*c^3*d^2 - 7*a^3*b^3*c*e^2 + 12*a^4*b*c^2*e^2 + a^3*c*e^2*(-(4*a*c - b^2)^3)^(1/2) - 2*
a*b^6*d*e + 25*a^2*b^3*c^2*d^2 - a^2*b^2*e^2*(-(4*a*c - b^2)^3)^(1/2) - a^2*c^2*d^2*(-(4*a*c - b^2)^3)^(1/2) -
 9*a*b^5*c*d^2 + 16*a^4*c^3*d*e + 2*a*b^3*d*e*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b^4*c*d*e + 3*a*b^2*c*d^2*(-(4
*a*c - b^2)^3)^(1/2) - 36*a^3*b^2*c^2*d*e - 4*a^2*b*c*d*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^7*d^4 + 16*a^
4*c^5*e^4 + b^4*c^5*d^4 - 8*a*b^2*c^6*d^4 - 2*b^5*c^4*d^3*e + a^2*b^4*c^3*e^4 - 8*a^3*b^2*c^4*e^4 + 32*a^3*c^6
*d^2*e^2 + b^6*c^3*d^2*e^2 + 16*a*b^3*c^5*d^3*e...

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