3.4.16 \(\int \frac {1}{x (a+b x^4+c x^8)} \, dx\) [316]

Optimal. Leaf size=69 \[ \frac {b \tanh ^{-1}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{4 a \sqrt {b^2-4 a c}}+\frac {\log (x)}{a}-\frac {\log \left (a+b x^4+c x^8\right )}{8 a} \]

[Out]

ln(x)/a-1/8*ln(c*x^8+b*x^4+a)/a+1/4*b*arctanh((2*c*x^4+b)/(-4*a*c+b^2)^(1/2))/a/(-4*a*c+b^2)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1371, 719, 29, 648, 632, 212, 642} \begin {gather*} \frac {b \tanh ^{-1}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{4 a \sqrt {b^2-4 a c}}-\frac {\log \left (a+b x^4+c x^8\right )}{8 a}+\frac {\log (x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x^4 + c*x^8)),x]

[Out]

(b*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]])/(4*a*Sqrt[b^2 - 4*a*c]) + Log[x]/a - Log[a + b*x^4 + c*x^8]/(8*a)

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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 719

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

Rule 1371

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x \left (a+b x^4+c x^8\right )} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{x \left (a+b x+c x^2\right )} \, dx,x,x^4\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^4\right )}{4 a}+\frac {\text {Subst}\left (\int \frac {-b-c x}{a+b x+c x^2} \, dx,x,x^4\right )}{4 a}\\ &=\frac {\log (x)}{a}-\frac {\text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^4\right )}{8 a}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^4\right )}{8 a}\\ &=\frac {\log (x)}{a}-\frac {\log \left (a+b x^4+c x^8\right )}{8 a}+\frac {b \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^4\right )}{4 a}\\ &=\frac {b \tanh ^{-1}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{4 a \sqrt {b^2-4 a c}}+\frac {\log (x)}{a}-\frac {\log \left (a+b x^4+c x^8\right )}{8 a}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.02, size = 66, normalized size = 0.96 \begin {gather*} \frac {\log (x)}{a}-\frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b \log (x-\text {$\#$1})+c \log (x-\text {$\#$1}) \text {$\#$1}^4}{b+2 c \text {$\#$1}^4}\&\right ]}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x^4 + c*x^8)),x]

[Out]

Log[x]/a - RootSum[a + b*#1^4 + c*#1^8 & , (b*Log[x - #1] + c*Log[x - #1]*#1^4)/(b + 2*c*#1^4) & ]/(4*a)

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Maple [A]
time = 0.04, size = 66, normalized size = 0.96

method	result	size
default	\(-\frac {\frac {\ln \left (c \,x^{8}+b \,x^{4}+a \right )}{4}+\frac {b \arctan \left (\frac {2 c \,x^{4}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}}}{2 a}+\frac {\ln \left (x \right )}{a}\)	\(66\)
risch	\(\frac {\ln \left (x \right )}{a}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (4 a^{2} c -a \,b^{2}\right ) \textit {\_Z}^{2}+\left (4 a c -b^{2}\right ) \textit {\_Z} +c \right )}{\sum }\textit {\_R} \ln \left (\left (\left (18 a c -5 b^{2}\right ) \textit {\_R} +9 c \right ) x^{4}-a b \textit {\_R} +4 b \right )\right )}{4}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(c*x^8+b*x^4+a),x,method=_RETURNVERBOSE)

[Out]

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

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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(1/x/(c*x^8+b*x^4+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 = 0.39, size = 223, normalized size = 3.23 \begin {gather*} \left [\frac {\sqrt {b^{2} - 4 \, a c} b \log \left (\frac {2 \, c^{2} x^{8} + 2 \, b c x^{4} + b^{2} - 2 \, a c + {\left (2 \, c x^{4} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{8} + b x^{4} + a}\right ) - {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{8} + b x^{4} + a\right ) + 8 \, {\left (b^{2} - 4 \, a c\right )} \log \left (x\right )}{8 \, {\left (a b^{2} - 4 \, a^{2} c\right )}}, \frac {2 \, \sqrt {-b^{2} + 4 \, a c} b \arctan \left (-\frac {{\left (2 \, c x^{4} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{8} + b x^{4} + a\right ) + 8 \, {\left (b^{2} - 4 \, a c\right )} \log \left (x\right )}{8 \, {\left (a b^{2} - 4 \, a^{2} c\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c*x^8+b*x^4+a),x, algorithm="fricas")

[Out]

[1/8*(sqrt(b^2 - 4*a*c)*b*log((2*c^2*x^8 + 2*b*c*x^4 + b^2 - 2*a*c + (2*c*x^4 + b)*sqrt(b^2 - 4*a*c))/(c*x^8 +
b*x^4 + a)) - (b^2 - 4*a*c)*log(c*x^8 + b*x^4 + a) + 8*(b^2 - 4*a*c)*log(x))/(a*b^2 - 4*a^2*c), 1/8*(2*sqrt(-
b^2 + 4*a*c)*b*arctan(-(2*c*x^4 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (b^2 - 4*a*c)*log(c*x^8 + b*x^4 + a)
+ 8*(b^2 - 4*a*c)*log(x))/(a*b^2 - 4*a^2*c)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (60) = 120\).
time = 98.46, size = 253, normalized size = 3.67 \begin {gather*} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{8 a \left (4 a c - b^{2}\right )} - \frac {1}{8 a}\right ) \log {\left (x^{4} + \frac {- 16 a^{2} c \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{8 a \left (4 a c - b^{2}\right )} - \frac {1}{8 a}\right ) + 4 a b^{2} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{8 a \left (4 a c - b^{2}\right )} - \frac {1}{8 a}\right ) - 2 a c + b^{2}}{b c} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}}}{8 a \left (4 a c - b^{2}\right )} - \frac {1}{8 a}\right ) \log {\left (x^{4} + \frac {- 16 a^{2} c \left (\frac {b \sqrt {- 4 a c + b^{2}}}{8 a \left (4 a c - b^{2}\right )} - \frac {1}{8 a}\right ) + 4 a b^{2} \left (\frac {b \sqrt {- 4 a c + b^{2}}}{8 a \left (4 a c - b^{2}\right )} - \frac {1}{8 a}\right ) - 2 a c + b^{2}}{b c} \right )} + \frac {\log {\left (x \right )}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c*x**8+b*x**4+a),x)

[Out]

(-b*sqrt(-4*a*c + b**2)/(8*a*(4*a*c - b**2)) - 1/(8*a))*log(x**4 + (-16*a**2*c*(-b*sqrt(-4*a*c + b**2)/(8*a*(4
*a*c - b**2)) - 1/(8*a)) + 4*a*b**2*(-b*sqrt(-4*a*c + b**2)/(8*a*(4*a*c - b**2)) - 1/(8*a)) - 2*a*c + b**2)/(b
*c)) + (b*sqrt(-4*a*c + b**2)/(8*a*(4*a*c - b**2)) - 1/(8*a))*log(x**4 + (-16*a**2*c*(b*sqrt(-4*a*c + b**2)/(8
*a*(4*a*c - b**2)) - 1/(8*a)) + 4*a*b**2*(b*sqrt(-4*a*c + b**2)/(8*a*(4*a*c - b**2)) - 1/(8*a)) - 2*a*c + b**2
)/(b*c)) + log(x)/a

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Giac [A]
time = 7.26, size = 68, normalized size = 0.99 \begin {gather*} -\frac {b \arctan \left (\frac {2 \, c x^{4} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt {-b^{2} + 4 \, a c} a} - \frac {\log \left (c x^{8} + b x^{4} + a\right )}{8 \, a} + \frac {\log \left (x^{4}\right )}{4 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c*x^8+b*x^4+a),x, algorithm="giac")

[Out]

-1/4*b*arctan((2*c*x^4 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a) - 1/8*log(c*x^8 + b*x^4 + a)/a + 1/4*lo
g(x^4)/a

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Mupad [B]
time = 2.18, size = 1690, normalized size = 24.49 \begin {gather*} \frac {\ln \left (x\right )}{a}+\frac {\ln \left (c\,x^8+b\,x^4+a\right )\,\left (16\,a\,c-4\,b^2\right )}{2\,\left (16\,a\,b^2-64\,a^2\,c\right )}-\frac {b\,\mathrm {atan}\left (\frac {4\,{\left (4\,a\,c-b^2\right )}^2\,\left (-18\,a^3\,c^3+61\,a^2\,b^2\,c^2-34\,a\,b^4\,c+5\,b^6\right )\,\left (\frac {b^9\,c^4}{128\,a^4\,{\left (4\,a\,c-b^2\right )}^{5/2}}+\frac {2\,b^5\,c^4\,{\left (16\,a\,c-4\,b^2\right )}^4}{{\left (16\,a\,b^2-64\,a^2\,c\right )}^4\,\sqrt {4\,a\,c-b^2}}-\frac {b\,{\left (16\,a\,c-4\,b^2\right )}^3\,\left (256\,b^4\,c^4-\frac {128\,a\,b^4\,c^4\,\left (16\,a\,c-4\,b^2\right )}{16\,a\,b^2-64\,a^2\,c}\right )}{16\,a\,{\left (16\,a\,b^2-64\,a^2\,c\right )}^3\,\sqrt {4\,a\,c-b^2}}+\frac {b^3\,\left (16\,a\,c-4\,b^2\right )\,\left (256\,b^4\,c^4-\frac {128\,a\,b^4\,c^4\,\left (16\,a\,c-4\,b^2\right )}{16\,a\,b^2-64\,a^2\,c}\right )}{256\,a^3\,\left (16\,a\,b^2-64\,a^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^{3/2}}-\frac {3\,b^7\,c^4\,{\left (16\,a\,c-4\,b^2\right )}^2}{4\,a^2\,{\left (16\,a\,b^2-64\,a^2\,c\right )}^2\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )}{b^4\,c^8\,\left (81\,a\,c-20\,b^2\right )}+\frac {128\,a^5\,x^4\,\left (\frac {\left (23\,a^2\,b\,c^2-24\,a\,b^3\,c+5\,b^5\right )\,\left (\frac {\left (576\,b^3\,c^5-\frac {\left (1280\,b^5\,c^4-4608\,a\,b^3\,c^5\right )\,\left (16\,a\,c-4\,b^2\right )}{2\,\left (16\,a\,b^2-64\,a^2\,c\right )}\right )\,{\left (16\,a\,c-4\,b^2\right )}^4}{16\,{\left (16\,a\,b^2-64\,a^2\,c\right )}^4}+\frac {b^4\,\left (576\,b^3\,c^5-\frac {\left (1280\,b^5\,c^4-4608\,a\,b^3\,c^5\right )\,\left (16\,a\,c-4\,b^2\right )}{2\,\left (16\,a\,b^2-64\,a^2\,c\right )}\right )}{4096\,a^4\,{\left (4\,a\,c-b^2\right )}^2}+\frac {b^2\,\left (1280\,b^5\,c^4-4608\,a\,b^3\,c^5\right )\,{\left (16\,a\,c-4\,b^2\right )}^3}{128\,a^2\,{\left (16\,a\,b^2-64\,a^2\,c\right )}^3\,\left (4\,a\,c-b^2\right )}-\frac {3\,b^2\,\left (576\,b^3\,c^5-\frac {\left (1280\,b^5\,c^4-4608\,a\,b^3\,c^5\right )\,\left (16\,a\,c-4\,b^2\right )}{2\,\left (16\,a\,b^2-64\,a^2\,c\right )}\right )\,{\left (16\,a\,c-4\,b^2\right )}^2}{128\,a^2\,{\left (16\,a\,b^2-64\,a^2\,c\right )}^2\,\left (4\,a\,c-b^2\right )}-\frac {b^4\,\left (1280\,b^5\,c^4-4608\,a\,b^3\,c^5\right )\,\left (16\,a\,c-4\,b^2\right )}{2048\,a^4\,\left (16\,a\,b^2-64\,a^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}\right )}{32\,a^5\,c^4\,\left (81\,a\,c-20\,b^2\right )}+\frac {\left (-18\,a^3\,c^3+61\,a^2\,b^2\,c^2-34\,a\,b^4\,c+5\,b^6\right )\,\left (\frac {b^5\,\left (1280\,b^5\,c^4-4608\,a\,b^3\,c^5\right )}{32768\,a^5\,{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {3\,b^3\,\left (1280\,b^5\,c^4-4608\,a\,b^3\,c^5\right )\,{\left (16\,a\,c-4\,b^2\right )}^2}{1024\,a^3\,{\left (16\,a\,b^2-64\,a^2\,c\right )}^2\,{\left (4\,a\,c-b^2\right )}^{3/2}}+\frac {b\,\left (1280\,b^5\,c^4-4608\,a\,b^3\,c^5\right )\,{\left (16\,a\,c-4\,b^2\right )}^4}{128\,a\,{\left (16\,a\,b^2-64\,a^2\,c\right )}^4\,\sqrt {4\,a\,c-b^2}}-\frac {b\,\left (576\,b^3\,c^5-\frac {\left (1280\,b^5\,c^4-4608\,a\,b^3\,c^5\right )\,\left (16\,a\,c-4\,b^2\right )}{2\,\left (16\,a\,b^2-64\,a^2\,c\right )}\right )\,{\left (16\,a\,c-4\,b^2\right )}^3}{16\,a\,{\left (16\,a\,b^2-64\,a^2\,c\right )}^3\,\sqrt {4\,a\,c-b^2}}+\frac {b^3\,\left (576\,b^3\,c^5-\frac {\left (1280\,b^5\,c^4-4608\,a\,b^3\,c^5\right )\,\left (16\,a\,c-4\,b^2\right )}{2\,\left (16\,a\,b^2-64\,a^2\,c\right )}\right )\,\left (16\,a\,c-4\,b^2\right )}{256\,a^3\,\left (16\,a\,b^2-64\,a^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )}{32\,a^5\,c^4\,\sqrt {4\,a\,c-b^2}\,\left (81\,a\,c-20\,b^2\right )}\right )\,{\left (4\,a\,c-b^2\right )}^{5/2}}{b^4\,c^4}+\frac {4\,{\left (4\,a\,c-b^2\right )}^{5/2}\,\left (23\,a^2\,b\,c^2-24\,a\,b^3\,c+5\,b^5\right )\,\left (\frac {{\left (16\,a\,c-4\,b^2\right )}^4\,\left (256\,b^4\,c^4-\frac {128\,a\,b^4\,c^4\,\left (16\,a\,c-4\,b^2\right )}{16\,a\,b^2-64\,a^2\,c}\right )}{16\,{\left (16\,a\,b^2-64\,a^2\,c\right )}^4}+\frac {b^4\,\left (256\,b^4\,c^4-\frac {128\,a\,b^4\,c^4\,\left (16\,a\,c-4\,b^2\right )}{16\,a\,b^2-64\,a^2\,c}\right )}{4096\,a^4\,{\left (4\,a\,c-b^2\right )}^2}-\frac {b^8\,c^4\,\left (16\,a\,c-4\,b^2\right )}{8\,a^3\,\left (16\,a\,b^2-64\,a^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}+\frac {2\,b^6\,c^4\,{\left (16\,a\,c-4\,b^2\right )}^3}{a\,{\left (16\,a\,b^2-64\,a^2\,c\right )}^3\,\left (4\,a\,c-b^2\right )}-\frac {3\,b^2\,{\left (16\,a\,c-4\,b^2\right )}^2\,\left (256\,b^4\,c^4-\frac {128\,a\,b^4\,c^4\,\left (16\,a\,c-4\,b^2\right )}{16\,a\,b^2-64\,a^2\,c}\right )}{128\,a^2\,{\left (16\,a\,b^2-64\,a^2\,c\right )}^2\,\left (4\,a\,c-b^2\right )}\right )}{b^4\,c^8\,\left (81\,a\,c-20\,b^2\right )}\right )}{4\,a\,\sqrt {4\,a\,c-b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x^4 + c*x^8)),x)

[Out]

log(x)/a + (log(a + b*x^4 + c*x^8)*(16*a*c - 4*b^2))/(2*(16*a*b^2 - 64*a^2*c)) - (b*atan((4*(4*a*c - b^2)^2*(5
*b^6 - 18*a^3*c^3 + 61*a^2*b^2*c^2 - 34*a*b^4*c)*((b^9*c^4)/(128*a^4*(4*a*c - b^2)^(5/2)) + (2*b^5*c^4*(16*a*c
- 4*b^2)^4)/((16*a*b^2 - 64*a^2*c)^4*(4*a*c - b^2)^(1/2)) - (b*(16*a*c - 4*b^2)^3*(256*b^4*c^4 - (128*a*b^4*c
^4*(16*a*c - 4*b^2))/(16*a*b^2 - 64*a^2*c)))/(16*a*(16*a*b^2 - 64*a^2*c)^3*(4*a*c - b^2)^(1/2)) + (b^3*(16*a*c
- 4*b^2)*(256*b^4*c^4 - (128*a*b^4*c^4*(16*a*c - 4*b^2))/(16*a*b^2 - 64*a^2*c)))/(256*a^3*(16*a*b^2 - 64*a^2*
c)*(4*a*c - b^2)^(3/2)) - (3*b^7*c^4*(16*a*c - 4*b^2)^2)/(4*a^2*(16*a*b^2 - 64*a^2*c)^2*(4*a*c - b^2)^(3/2))))
/(b^4*c^8*(81*a*c - 20*b^2)) + (128*a^5*x^4*(((5*b^5 + 23*a^2*b*c^2 - 24*a*b^3*c)*(((576*b^3*c^5 - ((1280*b^5*
c^4 - 4608*a*b^3*c^5)*(16*a*c - 4*b^2))/(2*(16*a*b^2 - 64*a^2*c)))*(16*a*c - 4*b^2)^4)/(16*(16*a*b^2 - 64*a^2*
c)^4) + (b^4*(576*b^3*c^5 - ((1280*b^5*c^4 - 4608*a*b^3*c^5)*(16*a*c - 4*b^2))/(2*(16*a*b^2 - 64*a^2*c))))/(40
96*a^4*(4*a*c - b^2)^2) + (b^2*(1280*b^5*c^4 - 4608*a*b^3*c^5)*(16*a*c - 4*b^2)^3)/(128*a^2*(16*a*b^2 - 64*a^2
*c)^3*(4*a*c - b^2)) - (3*b^2*(576*b^3*c^5 - ((1280*b^5*c^4 - 4608*a*b^3*c^5)*(16*a*c - 4*b^2))/(2*(16*a*b^2 -
64*a^2*c)))*(16*a*c - 4*b^2)^2)/(128*a^2*(16*a*b^2 - 64*a^2*c)^2*(4*a*c - b^2)) - (b^4*(1280*b^5*c^4 - 4608*a
*b^3*c^5)*(16*a*c - 4*b^2))/(2048*a^4*(16*a*b^2 - 64*a^2*c)*(4*a*c - b^2)^2)))/(32*a^5*c^4*(81*a*c - 20*b^2))
+ ((5*b^6 - 18*a^3*c^3 + 61*a^2*b^2*c^2 - 34*a*b^4*c)*((b^5*(1280*b^5*c^4 - 4608*a*b^3*c^5))/(32768*a^5*(4*a*c
- b^2)^(5/2)) - (3*b^3*(1280*b^5*c^4 - 4608*a*b^3*c^5)*(16*a*c - 4*b^2)^2)/(1024*a^3*(16*a*b^2 - 64*a^2*c)^2*
(4*a*c - b^2)^(3/2)) + (b*(1280*b^5*c^4 - 4608*a*b^3*c^5)*(16*a*c - 4*b^2)^4)/(128*a*(16*a*b^2 - 64*a^2*c)^4*(
4*a*c - b^2)^(1/2)) - (b*(576*b^3*c^5 - ((1280*b^5*c^4 - 4608*a*b^3*c^5)*(16*a*c - 4*b^2))/(2*(16*a*b^2 - 64*a
^2*c)))*(16*a*c - 4*b^2)^3)/(16*a*(16*a*b^2 - 64*a^2*c)^3*(4*a*c - b^2)^(1/2)) + (b^3*(576*b^3*c^5 - ((1280*b^
5*c^4 - 4608*a*b^3*c^5)*(16*a*c - 4*b^2))/(2*(16*a*b^2 - 64*a^2*c)))*(16*a*c - 4*b^2))/(256*a^3*(16*a*b^2 - 64
*a^2*c)*(4*a*c - b^2)^(3/2))))/(32*a^5*c^4*(4*a*c - b^2)^(1/2)*(81*a*c - 20*b^2)))*(4*a*c - b^2)^(5/2))/(b^4*c
^4) + (4*(4*a*c - b^2)^(5/2)*(5*b^5 + 23*a^2*b*c^2 - 24*a*b^3*c)*(((16*a*c - 4*b^2)^4*(256*b^4*c^4 - (128*a*b^
4*c^4*(16*a*c - 4*b^2))/(16*a*b^2 - 64*a^2*c)))/(16*(16*a*b^2 - 64*a^2*c)^4) + (b^4*(256*b^4*c^4 - (128*a*b^4*
c^4*(16*a*c - 4*b^2))/(16*a*b^2 - 64*a^2*c)))/(4096*a^4*(4*a*c - b^2)^2) - (b^8*c^4*(16*a*c - 4*b^2))/(8*a^3*(
16*a*b^2 - 64*a^2*c)*(4*a*c - b^2)^2) + (2*b^6*c^4*(16*a*c - 4*b^2)^3)/(a*(16*a*b^2 - 64*a^2*c)^3*(4*a*c - b^2
)) - (3*b^2*(16*a*c - 4*b^2)^2*(256*b^4*c^4 - (128*a*b^4*c^4*(16*a*c - 4*b^2))/(16*a*b^2 - 64*a^2*c)))/(128*a^
2*(16*a*b^2 - 64*a^2*c)^2*(4*a*c - b^2))))/(b^4*c^8*(81*a*c - 20*b^2))))/(4*a*(4*a*c - b^2)^(1/2))

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