3.3.98 \(\int \frac {x^3}{(d+e x^2) (a+b x^2+c x^4)} \, dx\) [298]

Optimal. Leaf size=132 \[ \frac {(b d-2 a e) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}-\frac {d \log \left (d+e x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}+\frac {d \log \left (a+b x^2+c x^4\right )}{4 \left (c d^2-b d e+a e^2\right )} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 132, 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, 814, 648, 632, 212, 642} \begin {gather*} \frac {(b d-2 a e) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac {d \log \left (d+e x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}+\frac {d \log \left (a+b x^2+c x^4\right )}{4 \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*d - 2*a*e)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)) - (d*Lo
g[d + e*x^2])/(2*(c*d^2 - b*d*e + a*e^2)) + (d*Log[a + b*x^2 + c*x^4])/(4*(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 814

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

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]

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 114, normalized size = 0.86 \begin {gather*} \frac {2 (b d-2 a e) \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )+\sqrt {-b^2+4 a c} d \left (2 \log \left (d+e x^2\right )-\log \left (a+b x^2+c x^4\right )\right )}{4 \sqrt {-b^2+4 a c} \left (-c d^2+e (b d-a e)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.16, size = 112, normalized size = 0.85

method	result	size
default	\(\frac {\frac {d \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2}+\frac {2 \left (a e -\frac {b d}{2}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 a \,e^{2}-2 d e b +2 c \,d^{2}}-\frac {d \ln \left (e \,x^{2}+d \right )}{2 \left (a \,e^{2}-d e b +c \,d^{2}\right )}\)	\(112\)
risch	\(-\frac {d \ln \left (e \,x^{2}+d \right )}{2 \left (a \,e^{2}-d e b +c \,d^{2}\right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (4 a^{2} c \,e^{2}-a \,b^{2} e^{2}-4 a b c d e +4 a \,c^{2} d^{2}+b^{3} d e -b^{2} c \,d^{2}\right ) \textit {\_Z}^{2}+\left (-4 a c d +b^{2} d \right ) \textit {\_Z} +a \right )}{\sum }\textit {\_R} \ln \left (\left (\left (4 e^{3} c \,a^{2}-a \,b^{2} e^{3}+3 a b c d \,e^{2}-4 a \,c^{2} d^{2} e -e^{2} d \,b^{3}+2 b^{2} c \,d^{2} e -b \,c^{2} d^{3}\right ) \textit {\_R}^{2}-3 \textit {\_R} a c d e +a e +b d \right ) x^{2}+\left (6 a^{2} c d \,e^{2}-2 a \,b^{2} d \,e^{2}+2 a b c \,d^{2} e -2 a \,c^{2} d^{3}\right ) \textit {\_R}^{2}+\left (-a b d e -a c \,d^{2}\right ) \textit {\_R} +a d \right )\right )}{2}\)	\(260\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/(a*e^2-b*d*e+c*d^2)*(1/2*d*ln(c*x^4+b*x^2+a)+2*(a*e-1/2*b*d)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b
^2)^(1/2)))-1/2*d*ln(e*x^2+d)/(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^3/(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 [A]
time = 11.05, size = 325, normalized size = 2.46 \begin {gather*} \left [\frac {{\left (b^{2} - 4 \, a c\right )} d \log \left (c x^{4} + b x^{2} + a\right ) - 2 \, {\left (b^{2} - 4 \, a c\right )} d \log \left (x^{2} e + d\right ) - \sqrt {b^{2} - 4 \, a c} {\left (b d - 2 \, a e\right )} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c - {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right )}{4 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} - {\left (b^{3} - 4 \, a b c\right )} d e + {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )}}, \frac {{\left (b^{2} - 4 \, a c\right )} d \log \left (c x^{4} + b x^{2} + a\right ) - 2 \, {\left (b^{2} - 4 \, a c\right )} d \log \left (x^{2} e + d\right ) + 2 \, \sqrt {-b^{2} + 4 \, a c} {\left (b d - 2 \, a e\right )} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{4 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} - {\left (b^{3} - 4 \, a b c\right )} d e + {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [A]
time = 5.53, size = 133, normalized size = 1.01 \begin {gather*} -\frac {d e \log \left ({\left | x^{2} e + d \right |}\right )}{2 \, {\left (c d^{2} e - b d e^{2} + a e^{3}\right )}} + \frac {d \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (c d^{2} - b d e + a e^{2}\right )}} - \frac {{\left (b d - 2 \, a e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 9.75, size = 2500, normalized size = 18.94 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((d + e*x^2)*(a + b*x^2 + c*x^4)),x)

[Out]

(log(76*d^3*e^3*(b^2 - 4*a*c)^(9/2) - 64*a^3*b^6*e^6 - 4608*a^3*c^6*d^6 + 512*a^6*c^3*e^6 - 320*a*b^4*c^4*d^6
+ 512*a^4*b^4*c*e^6 - 64*a*b^8*d^2*e^4 - 128*a^2*b^7*d*e^5 + 32*a^3*b^3*e^6*(b^2 - 4*a*c)^(3/2) - 48*b^3*c^3*d
^6*(b^2 - 4*a*c)^(3/2) - 68*b^2*d^3*e^3*(b^2 - 4*a*c)^(7/2) - 28*b^4*d^3*e^3*(b^2 - 4*a*c)^(5/2) + 20*b^6*d^3*
e^3*(b^2 - 4*a*c)^(3/2) + 4*a^2*e^6*x^2*(b^2 - 4*a*c)^(7/2) + 144*c^4*d^6*x^2*(b^2 - 4*a*c)^(5/2) + 39*d^2*e^4
*x^2*(b^2 - 4*a*c)^(9/2) + 2432*a^2*b^2*c^5*d^6 - 1152*a^5*b^2*c^2*e^6 + 40448*a^4*c^5*d^4*e^2 - 19968*a^5*c^4
*d^2*e^4 - 64*a^2*b^7*e^6*x^2 - 64*b^5*c^4*d^6*x^2 - 64*b^9*d^2*e^4*x^2 + 32*a^3*b*e^6*(b^2 - 4*a*c)^(5/2) + 4
8*b*c^3*d^6*(b^2 - 4*a*c)^(5/2) + 40*a^2*d*e^5*(b^2 - 4*a*c)^(7/2) + 168*c^2*d^5*e*(b^2 - 4*a*c)^(7/2) + 40*a^
2*b^2*e^6*x^2*(b^2 - 4*a*c)^(5/2) + 20*a^2*b^4*e^6*x^2*(b^2 - 4*a*c)^(3/2) - 80*b^2*c^4*d^6*x^2*(b^2 - 4*a*c)^
(3/2) + 155*b^2*d^2*e^4*x^2*(b^2 - 4*a*c)^(7/2) - 155*b^4*d^2*e^4*x^2*(b^2 - 4*a*c)^(5/2) + 25*b^6*d^2*e^4*x^2
*(b^2 - 4*a*c)^(3/2) + 316*c^2*d^4*e^2*x^2*(b^2 - 4*a*c)^(7/2) + 5120*a^2*b^4*c^3*d^4*e^2 - 4096*a^2*b^5*c^2*d
^3*e^3 - 24448*a^3*b^2*c^4*d^4*e^2 + 21760*a^3*b^3*c^3*d^3*e^3 - 9920*a^3*b^4*c^2*d^2*e^4 + 26240*a^4*b^2*c^3*
d^2*e^4 - 1600*a^4*b^3*c^2*e^6*x^2 + 38912*a^4*c^5*d^3*e^3*x^2 - 384*b^7*c^2*d^4*e^2*x^2 + 212*a*b*d^2*e^4*(b^
2 - 4*a*c)^(7/2) - 176*b*c*d^4*e^2*(b^2 - 4*a*c)^(7/2) + 256*a*b^5*c^3*d^5*e + 256*a*b^7*c*d^3*e^3 + 2560*a^3*
b*c^5*d^5*e + 1664*a^3*b^5*c*d*e^5 + 8704*a^5*b*c^3*d*e^5 - 128*a*b^8*d*e^5*x^2 - 168*a*b^3*d^2*e^4*(b^2 - 4*a
*c)^(5/2) + 20*a*b^5*d^2*e^4*(b^2 - 4*a*c)^(3/2) + 144*a^2*b^2*d*e^5*(b^2 - 4*a*c)^(5/2) - 56*a^2*b^4*d*e^5*(b
^2 - 4*a*c)^(3/2) - 272*b^2*c^2*d^5*e*(b^2 - 4*a*c)^(5/2) + 256*b^3*c*d^4*e^2*(b^2 - 4*a*c)^(5/2) + 104*b^4*c^
2*d^5*e*(b^2 - 4*a*c)^(3/2) - 80*b^5*c*d^4*e^2*(b^2 - 4*a*c)^(3/2) - 384*a*b^6*c^2*d^4*e^2 - 1664*a^2*b^3*c^4*
d^5*e + 1408*a^2*b^6*c*d^2*e^4 - 37888*a^4*b*c^4*d^3*e^3 - 6784*a^4*b^3*c^2*d*e^5 + 448*a*b^3*c^5*d^6*x^2 - 76
8*a^2*b*c^6*d^6*x^2 + 576*a^3*b^5*c*e^6*x^2 + 1280*a^5*b*c^3*e^6*x^2 - 21504*a^3*c^6*d^5*e*x^2 - 5120*a^5*c^4*
d*e^5*x^2 + 256*b^6*c^3*d^5*e*x^2 + 256*b^8*c*d^3*e^3*x^2 - 26560*a^2*b^3*c^4*d^4*e^2*x^2 + 25600*a^2*b^4*c^3*
d^3*e^3*x^2 - 11264*a^2*b^5*c^2*d^2*e^4*x^2 - 58880*a^3*b^2*c^4*d^3*e^3*x^2 + 34880*a^3*b^3*c^3*d^2*e^4*x^2 +
80*a*b^3*d*e^5*x^2*(b^2 - 4*a*c)^(5/2) - 40*a*b^5*d*e^5*x^2*(b^2 - 4*a*c)^(3/2) - 448*b*c*d^3*e^3*x^2*(b^2 - 4
*a*c)^(7/2) - 416*b*c^3*d^5*e*x^2*(b^2 - 4*a*c)^(5/2) - 3200*a*b^4*c^4*d^5*e*x^2 + 1472*a*b^7*c*d^2*e^4*x^2 +
1792*a^2*b^6*c*d*e^5*x^2 + 192*b^3*c*d^3*e^3*x^2*(b^2 - 4*a*c)^(5/2) + 160*b^3*c^3*d^5*e*x^2*(b^2 - 4*a*c)^(3/
2) + 5504*a*b^5*c^3*d^4*e^2*x^2 - 4352*a*b^6*c^2*d^3*e^3*x^2 + 14080*a^2*b^2*c^5*d^5*e*x^2 + 42752*a^3*b*c^5*d
^4*e^2*x^2 - 8320*a^3*b^4*c^2*d*e^5*x^2 - 37120*a^4*b*c^4*d^2*e^4*x^2 + 14080*a^4*b^2*c^3*d*e^5*x^2 + 88*a*b*d
*e^5*x^2*(b^2 - 4*a*c)^(7/2) + 168*b^2*c^2*d^4*e^2*x^2*(b^2 - 4*a*c)^(5/2) - 100*b^4*c^2*d^4*e^2*x^2*(b^2 - 4*
a*c)^(3/2))*(d*((b*(b^2 - 4*a*c)^(1/2))/4 - a*c + b^2/4) - (a*e*(b^2 - 4*a*c)^(1/2))/2))/(a*b^2*e^2 - 4*a*c^2*
d^2 - 4*a^2*c*e^2 + b^2*c*d^2 - b^3*d*e + 4*a*b*c*d*e) - (log(76*d^3*e^3*(b^2 - 4*a*c)^(9/2) + 64*a^3*b^6*e^6
+ 4608*a^3*c^6*d^6 - 512*a^6*c^3*e^6 + 320*a*b^4*c^4*d^6 - 512*a^4*b^4*c*e^6 + 64*a*b^8*d^2*e^4 + 128*a^2*b^7*
d*e^5 + 32*a^3*b^3*e^6*(b^2 - 4*a*c)^(3/2) - 48*b^3*c^3*d^6*(b^2 - 4*a*c)^(3/2) - 68*b^2*d^3*e^3*(b^2 - 4*a*c)
^(7/2) - 28*b^4*d^3*e^3*(b^2 - 4*a*c)^(5/2) + 20*b^6*d^3*e^3*(b^2 - 4*a*c)^(3/2) + 4*a^2*e^6*x^2*(b^2 - 4*a*c)
^(7/2) + 144*c^4*d^6*x^2*(b^2 - 4*a*c)^(5/2) + 39*d^2*e^4*x^2*(b^2 - 4*a*c)^(9/2) - 2432*a^2*b^2*c^5*d^6 + 115
2*a^5*b^2*c^2*e^6 - 40448*a^4*c^5*d^4*e^2 + 19968*a^5*c^4*d^2*e^4 + 64*a^2*b^7*e^6*x^2 + 64*b^5*c^4*d^6*x^2 +
64*b^9*d^2*e^4*x^2 + 32*a^3*b*e^6*(b^2 - 4*a*c)^(5/2) + 48*b*c^3*d^6*(b^2 - 4*a*c)^(5/2) + 40*a^2*d*e^5*(b^2 -
4*a*c)^(7/2) + 168*c^2*d^5*e*(b^2 - 4*a*c)^(7/2) + 40*a^2*b^2*e^6*x^2*(b^2 - 4*a*c)^(5/2) + 20*a^2*b^4*e^6*x^
2*(b^2 - 4*a*c)^(3/2) - 80*b^2*c^4*d^6*x^2*(b^2 - 4*a*c)^(3/2) + 155*b^2*d^2*e^4*x^2*(b^2 - 4*a*c)^(7/2) - 155
*b^4*d^2*e^4*x^2*(b^2 - 4*a*c)^(5/2) + 25*b^6*d^2*e^4*x^2*(b^2 - 4*a*c)^(3/2) + 316*c^2*d^4*e^2*x^2*(b^2 - 4*a
*c)^(7/2) - 5120*a^2*b^4*c^3*d^4*e^2 + 4096*a^2*b^5*c^2*d^3*e^3 + 24448*a^3*b^2*c^4*d^4*e^2 - 21760*a^3*b^3*c^
3*d^3*e^3 + 9920*a^3*b^4*c^2*d^2*e^4 - 26240*a^4*b^2*c^3*d^2*e^4 + 1600*a^4*b^3*c^2*e^6*x^2 - 38912*a^4*c^5*d^
3*e^3*x^2 + 384*b^7*c^2*d^4*e^2*x^2 + 212*a*b*d^2*e^4*(b^2 - 4*a*c)^(7/2) - 176*b*c*d^4*e^2*(b^2 - 4*a*c)^(7/2
) - 256*a*b^5*c^3*d^5*e - 256*a*b^7*c*d^3*e^3 - 2560*a^3*b*c^5*d^5*e - 1664*a^3*b^5*c*d*e^5 - 8704*a^5*b*c^3*d
*e^5 + 128*a*b^8*d*e^5*x^2 - 168*a*b^3*d^2*e^4*(b^2 - 4*a*c)^(5/2) + 20*a*b^5*d^2*e^4*(b^2 - 4*a*c)^(3/2) + 14
4*a^2*b^2*d*e^5*(b^2 - 4*a*c)^(5/2) - 56*a^2*b^4*d*e^5*(b^2 - 4*a*c)^(3/2) - 272*b^2*c^2*d^5*e*(b^2 - 4*a*c)^(
5/2) + 256*b^3*c*d^4*e^2*(b^2 - 4*a*c)^(5/2) + 104*b^4*c^2*d^5*e*(b^2 - 4*a*c)^(3/2) - 80*b^5*c*d^4*e^2*(b^2 -
4*a*c)^(3/2) + 384*a*b^6*c^2*d^4*e^2 + 1664*a^...

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