3.3.95 \(\int \frac {x^9}{(d+e x^2) (a+b x^2+c x^4)} \, dx\) [295]

Optimal. Leaf size=230 \[ -\frac {(c d+b e) x^2}{2 c^2 e^2}+\frac {x^4}{4 c e}-\frac {\left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-a b^3 e+3 a^2 b c e\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}+\frac {d^4 \log \left (d+e x^2\right )}{2 e^3 \left (c d^2-b d e+a e^2\right )}-\frac {\left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3 \left (c d^2-b d e+a e^2\right )} \]

[Out]

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

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Rubi [A]
time = 0.33, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1265, 1642, 648, 632, 212, 642} \begin {gather*} -\frac {\left (a^2 c e-a b^2 e-2 a b c d+b^3 d\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3 \left (a e^2-b d e+c d^2\right )}-\frac {\left (3 a^2 b c e+2 a^2 c^2 d-a b^3 e-4 a b^2 c d+b^4 d\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac {d^4 \log \left (d+e x^2\right )}{2 e^3 \left (a e^2-b d e+c d^2\right )}-\frac {x^2 (b e+c d)}{2 c^2 e^2}+\frac {x^4}{4 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^9/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {x^9}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^4}{(d+e x) \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {-c d-b e}{c^2 e^2}+\frac {x}{c e}+\frac {d^4}{e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {-a \left (b^2 d-a c d-a b e\right )-\left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) x}{c^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {(c d+b e) x^2}{2 c^2 e^2}+\frac {x^4}{4 c e}+\frac {d^4 \log \left (d+e x^2\right )}{2 e^3 \left (c d^2-b d e+a e^2\right )}+\frac {\text {Subst}\left (\int \frac {-a \left (b^2 d-a c d-a b e\right )-\left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {(c d+b e) x^2}{2 c^2 e^2}+\frac {x^4}{4 c e}+\frac {d^4 \log \left (d+e x^2\right )}{2 e^3 \left (c d^2-b d e+a e^2\right )}-\frac {\left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3 \left (c d^2-b d e+a e^2\right )}+\frac {\left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-a b^3 e+3 a^2 b c e\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {(c d+b e) x^2}{2 c^2 e^2}+\frac {x^4}{4 c e}+\frac {d^4 \log \left (d+e x^2\right )}{2 e^3 \left (c d^2-b d e+a e^2\right )}-\frac {\left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3 \left (c d^2-b d e+a e^2\right )}-\frac {\left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-a b^3 e+3 a^2 b c e\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^3 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {(c d+b e) x^2}{2 c^2 e^2}+\frac {x^4}{4 c e}-\frac {\left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-a b^3 e+3 a^2 b c e\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}+\frac {d^4 \log \left (d+e x^2\right )}{2 e^3 \left (c d^2-b d e+a e^2\right )}-\frac {\left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3 \left (c d^2-b d e+a e^2\right )}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 228, normalized size = 0.99 \begin {gather*} \frac {1}{4} \left (-\frac {2 (c d+b e) x^2}{c^2 e^2}+\frac {x^4}{c e}-\frac {2 \left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-a b^3 e+3 a^2 b c e\right ) \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{c^3 \sqrt {-b^2+4 a c} \left (-c d^2+e (b d-a e)\right )}+\frac {2 d^4 \log \left (d+e x^2\right )}{e^3 \left (c d^2+e (-b d+a e)\right )}+\frac {\left (-b^3 d+2 a b c d+a b^2 e-a^2 c e\right ) \log \left (a+b x^2+c x^4\right )}{c^3 \left (c d^2+e (-b d+a e)\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^9/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]

[Out]

((-2*(c*d + b*e)*x^2)/(c^2*e^2) + x^4/(c*e) - (2*(b^4*d - 4*a*b^2*c*d + 2*a^2*c^2*d - a*b^3*e + 3*a^2*b*c*e)*A
rcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(c^3*Sqrt[-b^2 + 4*a*c]*(-(c*d^2) + e*(b*d - a*e))) + (2*d^4*Log[d +
e*x^2])/(e^3*(c*d^2 + e*(-(b*d) + a*e))) + ((-(b^3*d) + 2*a*b*c*d + a*b^2*e - a^2*c*e)*Log[a + b*x^2 + c*x^4])
/(c^3*(c*d^2 + e*(-(b*d) + a*e))))/4

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Maple [A]
time = 0.26, size = 216, normalized size = 0.94

method	result	size
default	\(\frac {\left (-c e \,x^{2}+e b +c d \right )^{2}}{4 c^{3} e^{3}}+\frac {\frac {\left (-a^{2} c e +a \,b^{2} e +2 a b c d -b^{3} d \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (a^{2} b e +a^{2} c d -a \,b^{2} d -\frac {\left (-a^{2} c e +a \,b^{2} e +2 a b c d -b^{3} d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 \left (a \,e^{2}-d e b +c \,d^{2}\right ) c^{2}}+\frac {d^{4} \ln \left (e \,x^{2}+d \right )}{2 e^{3} \left (a \,e^{2}-d e b +c \,d^{2}\right )}\)	\(216\)
risch	\(\text {Expression too large to display}\)	\(29566\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(e*x^2+d)/(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [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^9/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out

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Giac [A]
time = 4.17, size = 236, normalized size = 1.03 \begin {gather*} \frac {d^{4} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \, {\left (c d^{2} e^{3} - b d e^{4} + a e^{5}\right )}} - \frac {{\left (b^{3} d - 2 \, a b c d - a b^{2} e + a^{2} c e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (c^{4} d^{2} - b c^{3} d e + a c^{3} e^{2}\right )}} + \frac {{\left (b^{4} d - 4 \, a b^{2} c d + 2 \, a^{2} c^{2} d - a b^{3} e + 3 \, a^{2} b c e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (c^{4} d^{2} - b c^{3} d e + a c^{3} e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (c x^{4} e - 2 \, c d x^{2} - 2 \, b x^{2} e\right )} e^{\left (-2\right )}}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 69.94, size = 2500, normalized size = 10.87 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/((d + e*x^2)*(a + b*x^2 + c*x^4)),x)

[Out]

(d^4*log(d + e*x^2))/(2*a*e^5 + 2*c*d^2*e^3 - 2*b*d*e^4) + (log((x^2*(a^7*e^7 + b^7*d^7 - 2*a^3*b*c^3*d^7 - a^
4*c^3*d^6*e - 2*a^6*c*d^2*e^5 + 7*a^2*b^3*c^2*d^7 + 3*a^2*b^5*d^5*e^2 + 4*a^3*b^4*d^4*e^3 + 4*a^4*b^3*d^3*e^4
+ 3*a^5*b^2*d^2*e^5 + 2*a^5*c^2*d^4*e^3 - 5*a*b^5*c*d^7 + 2*a*b^6*d^6*e + 2*a^6*b*d*e^6 - 8*a^2*b^4*c*d^6*e -
6*a^5*b*c*d^3*e^4 + 8*a^3*b^2*c^2*d^6*e - 9*a^3*b^3*c*d^5*e^2 + 5*a^4*b*c^2*d^5*e^2 - 9*a^4*b^2*c*d^4*e^3))/(c
^4*e^4) + (a*d*(a^3*e^3 + b^3*d^3 - 2*a*b*c*d^3 + a*b^2*d^2*e + a^2*b*d*e^2 - a^2*c*d^2*e)^2)/(c^4*e^4) + (((x
^2*(4*a^2*c^6*d^8 + 6*a^4*b^4*e^8 + 18*a^6*c^2*e^8 + 6*b^4*c^4*d^8 + 6*b^8*d^4*e^4 - 16*a*b^2*c^5*d^8 - 26*a^5
*b^2*c*e^8 + 8*a*b^7*d^3*e^5 + 8*a^3*b^5*d*e^7 - 2*b^5*c^3*d^7*e - 2*b^7*c*d^5*e^3 + 8*a^2*b^6*d^2*e^6 - 20*a^
3*c^5*d^6*e^2 + 40*a^4*c^4*d^4*e^4 - 36*a^5*c^3*d^2*e^6 + 2*b^6*c^2*d^6*e^2 + 42*a^2*b^2*c^4*d^6*e^2 - 28*a^2*
b^3*c^3*d^5*e^3 + 80*a^2*b^4*c^2*d^4*e^4 - 64*a^3*b^2*c^3*d^4*e^4 + 80*a^3*b^3*c^2*d^3*e^5 + 48*a^4*b^2*c^2*d^
2*e^6 + 18*a*b^3*c^4*d^7*e - 40*a*b^6*c*d^4*e^4 - 26*a^2*b*c^5*d^7*e - 32*a^4*b^3*c*d*e^7 + 12*a^5*b*c^2*d*e^7
- 16*a*b^4*c^3*d^6*e^2 + 10*a*b^5*c^2*d^5*e^3 - 48*a^2*b^5*c*d^3*e^5 + 46*a^3*b*c^4*d^5*e^3 - 40*a^3*b^4*c*d^
2*e^6 - 48*a^4*b*c^3*d^3*e^5))/(c^4*e^4) + (((x^2*(8*a*b^8*e^9 + 8*b*c^8*d^9 + 8*b^9*d*e^8 + 120*a^5*c^4*e^9 -
72*a^2*b^6*c*e^9 - 8*b^2*c^7*d^8*e - 8*b^8*c*d^2*e^7 + 212*a^3*b^4*c^2*e^9 - 240*a^4*b^2*c^3*e^9 - 112*a^2*c^
7*d^6*e^3 + 240*a^3*c^6*d^4*e^5 - 228*a^4*c^5*d^2*e^7 + 4*b^3*c^6*d^7*e^2 - 24*b^4*c^5*d^6*e^3 + 32*b^5*c^4*d^
5*e^4 - 24*b^6*c^3*d^4*e^5 + 4*b^7*c^2*d^3*e^6 + 32*a*c^8*d^8*e - 56*a*b^7*c*d*e^8 - 428*a^2*b^2*c^5*d^4*e^5 +
108*a^2*b^3*c^4*d^3*e^6 - 216*a^2*b^4*c^3*d^2*e^7 + 424*a^3*b^2*c^4*d^2*e^7 - 16*a*b*c^7*d^7*e^2 + 8*a^4*b*c^
4*d*e^8 + 88*a*b^2*c^6*d^6*e^3 - 116*a*b^3*c^5*d^5*e^4 + 188*a*b^4*c^4*d^4*e^5 - 36*a*b^5*c^3*d^3*e^6 + 60*a*b
^6*c^2*d^2*e^7 + 40*a^2*b*c^6*d^5*e^4 + 100*a^2*b^5*c^2*d*e^8 - 72*a^3*b*c^5*d^3*e^6 - 4*a^3*b^3*c^3*d*e^8))/(
c^4*e^4) - (((x^2*(32*a*b^6*c^3*e^10 - 352*a^4*c^6*e^10 + 128*a*c^9*d^6*e^4 + 32*b*c^9*d^7*e^3 + 32*b^7*c^3*d*
e^9 - 256*a^2*b^4*c^4*e^10 + 600*a^3*b^2*c^5*e^10 - 464*a^2*c^8*d^4*e^6 + 592*a^3*c^7*d^2*e^8 - 64*b^2*c^8*d^6
*e^4 + 56*b^3*c^7*d^5*e^5 - 48*b^4*c^6*d^4*e^6 + 56*b^5*c^5*d^3*e^7 - 64*b^6*c^4*d^2*e^8 - 688*a^2*b^2*c^6*d^2
*e^8 - 192*a*b*c^8*d^5*e^5 - 224*a*b^5*c^4*d*e^9 - 72*a^3*b*c^6*d*e^9 + 272*a*b^2*c^7*d^4*e^6 - 200*a*b^3*c^6*
d^3*e^7 + 360*a*b^4*c^5*d^2*e^8 + 136*a^2*b*c^7*d^3*e^7 + 424*a^2*b^3*c^5*d*e^9))/(c^4*e^4) + (32*a*d*(2*b^6*e
^6 + 2*c^6*d^6 - 15*a^3*c^3*e^6 - 10*a*c^5*d^4*e^2 + 29*a^2*b^2*c^2*e^6 + 17*a^2*c^4*d^2*e^4 + 3*b^2*c^4*d^4*e
^2 - b^3*c^3*d^3*e^3 + 3*b^4*c^2*d^2*e^4 - 14*a*b^4*c*e^6 - 2*b*c^5*d^5*e - 2*b^5*c*d*e^5 + 2*a*b*c^4*d^3*e^3
+ 6*a*b^3*c^2*d*e^5 + a^2*b*c^3*d*e^5 - 13*a*b^2*c^3*d^2*e^4))/(c*e) - (8*e^2*(b^2*e^2 + c^2*d^2 - 3*a*c*e^2 -
b*c*d*e)*(b^5*d + b^4*d*(b^2 - 4*a*c)^(1/2) - 4*a^3*c^2*e - a*b^4*e - 6*a*b^3*c*d - a*b^3*e*(b^2 - 4*a*c)^(1/
2) + 8*a^2*b*c^2*d + 5*a^2*b^2*c*e + 2*a^2*c^2*d*(b^2 - 4*a*c)^(1/2) - 4*a*b^2*c*d*(b^2 - 4*a*c)^(1/2) + 3*a^2
*b*c*e*(b^2 - 4*a*c)^(1/2))*(2*a*c^2*d^3 + a*b^2*e^3*x^2 + b*c^2*d^3*x^2 - 4*a^2*c*e^3*x^2 + b^3*d*e^2*x^2 + 2
*a*b^2*d*e^2 - 6*a^2*c*d*e^2 + 4*a*c^2*d^2*e*x^2 - 2*b^2*c*d^2*e*x^2 - 2*a*b*c*d^2*e - 3*a*b*c*d*e^2*x^2))/(c*
(4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)))*(b^5*d + b^4*d*(b^2 - 4*a*c)^(1/2) - 4*a^3*c^2*e - a*b^4*e - 6*a*b^3*c
*d - a*b^3*e*(b^2 - 4*a*c)^(1/2) + 8*a^2*b*c^2*d + 5*a^2*b^2*c*e + 2*a^2*c^2*d*(b^2 - 4*a*c)^(1/2) - 4*a*b^2*c
*d*(b^2 - 4*a*c)^(1/2) + 3*a^2*b*c*e*(b^2 - 4*a*c)^(1/2)))/(4*c^3*(4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)) + (4*
a*d*(4*b^8*e^8 + 4*c^8*d^8 + 37*a^4*c^4*e^8 - 16*a*c^7*d^6*e^2 + 84*a^2*b^4*c^2*e^8 - 84*a^3*b^2*c^3*e^8 + 40*
a^2*c^6*d^4*e^4 - 56*a^3*c^5*d^2*e^6 + 4*b^2*c^6*d^6*e^2 - 4*b^3*c^5*d^5*e^3 + 13*b^4*c^4*d^4*e^4 - 4*b^5*c^3*
d^3*e^5 + 4*b^6*c^2*d^2*e^6 - 32*a*b^6*c*e^8 + 98*a^2*b^2*c^4*d^2*e^6 - 8*a*b^5*c^2*d*e^7 - 4*a^3*b*c^4*d*e^7
- 52*a*b^2*c^5*d^4*e^4 + 20*a*b^3*c^4*d^3*e^5 - 36*a*b^4*c^3*d^2*e^6 - 16*a^2*b*c^5*d^3*e^5 + 28*a^2*b^3*c^3*d
*e^7))/(c^4*e^4))*(b^5*d + b^4*d*(b^2 - 4*a*c)^(1/2) - 4*a^3*c^2*e - a*b^4*e - 6*a*b^3*c*d - a*b^3*e*(b^2 - 4*
a*c)^(1/2) + 8*a^2*b*c^2*d + 5*a^2*b^2*c*e + 2*a^2*c^2*d*(b^2 - 4*a*c)^(1/2) - 4*a*b^2*c*d*(b^2 - 4*a*c)^(1/2)
+ 3*a^2*b*c*e*(b^2 - 4*a*c)^(1/2)))/(4*c^3*(4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)) + (4*a*d*(2*a^3*b^4*e^7 + 5
*a^5*c^2*e^7 + 2*b^3*c^4*d^7 + 2*b^7*d^3*e^4 - 8*a^4*b^2*c*e^7 + 2*a*b^6*d^2*e^5 + 2*a^2*b^5*d*e^6 - 2*a^2*c^5
*d^6*e + 6*a^3*c^4*d^4*e^3 - 9*a^4*c^3*d^2*e^5 + b^5*c^2*d^5*e^2 - 4*a*b*c^5*d^7 - a^2*b^2*c^3*d^4*e^3 + 20*a^
2*b^3*c^2*d^3*e^4 + 12*a^3*b^2*c^2*d^2*e^5 + 2*a*b^2*c^4*d^6*e - 12*a*b^5*c*d^3*e^4 - 8*a^3*b^3*c*d*e^6 + 3*a^
4*b*c^2*d*e^6 - 6*a*b^3*c^3*d^5*e^2 - a*b^4*c^2*d^4*e^3 + 10*a^2*b*c^4*d^5*e^2 - 10*a^2*b^4*c*d^2*e^5 - 12*a^3
*b*c^3*d^3*e^4))/(c^4*e^4))*(b^5*d + b^4*d*(b^2 - 4*a*c)^(1/2) - 4*a^3*c^2*e - a*b^4*e - 6*a*b^3*c*d - a*b^3*e
*(b^2 - 4*a*c)^(1/2) + 8*a^2*b*c^2*d + 5*a^2*b^...

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