\(\int \frac {1}{(d+e x) (a+b (d+e x)^2+c (d+e x)^4)} \, dx\) [617]
Optimal result
Integrand size = 30, antiderivative size = 94 \[
\int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx=\frac {b \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a \sqrt {b^2-4 a c} e}+\frac {\log (d+e x)}{a e}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a e}
\]
[Out]
ln(e*x+d)/a/e-1/4*ln(a+b*(e*x+d)^2+c*(e*x+d)^4)/a/e+1/2*b*arctanh((b+2*c*(e*x+d)^2)/(-4*a*c+b^2)^(1/2))/a/e/(-
4*a*c+b^2)^(1/2)
Rubi [A] (verified)
Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of
steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1156, 1128, 719, 29, 648, 632,
212, 642} \[
\int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx=\frac {b \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a e \sqrt {b^2-4 a c}}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a e}+\frac {\log (d+e x)}{a e}
\]
[In]
Int[1/((d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]
[Out]
(b*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/(2*a*Sqrt[b^2 - 4*a*c]*e) + Log[d + e*x]/(a*e) - Log[a +
b*(d + e*x)^2 + c*(d + e*x)^4]/(4*a*e)
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 1128
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
Rule 1156
Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]
Rubi steps \begin{align*}
\text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x^2+c x^4\right )} \, dx,x,d+e x\right )}{e} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x+c x^2\right )} \, dx,x,(d+e x)^2\right )}{2 e} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,(d+e x)^2\right )}{2 a e}+\frac {\text {Subst}\left (\int \frac {-b-c x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 a e} \\ & = \frac {\log (d+e x)}{a e}-\frac {\text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a e}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a e} \\ & = \frac {\log (d+e x)}{a e}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a e}+\frac {b \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{2 a e} \\ & = \frac {b \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a \sqrt {b^2-4 a c} e}+\frac {\log (d+e x)}{a e}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a e} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.36
\[
\int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx=\frac {4 \sqrt {b^2-4 a c} \log (d+e x)-\left (b+\sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )+\left (b-\sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{4 a \sqrt {b^2-4 a c} e}
\]
[In]
Integrate[1/((d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]
[Out]
(4*Sqrt[b^2 - 4*a*c]*Log[d + e*x] - (b + Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2] + (b
- Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(4*a*Sqrt[b^2 - 4*a*c]*e)
Maple [C] (verified)
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.63 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.65
| | |
| method | result | size |
| | |
| risch |
\(\frac {\ln \left (e x +d \right )}{a e}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4 a^{2} c \,e^{2}-a \,b^{2} e^{2}\right ) \textit {\_Z}^{2}+\left (4 a c e -b^{2} e \right ) \textit {\_Z} +c \right )}{\sum }\textit {\_R} \ln \left (\left (\left (10 e^{3} a c -3 b^{2} e^{3}\right ) \textit {\_R} +5 c \,e^{2}\right ) x^{2}+\left (\left (20 a c d \,e^{2}-6 b^{2} d \,e^{2}\right ) \textit {\_R} +10 d c e \right ) x +\left (10 a c \,d^{2} e -3 b^{2} d^{2} e -a b e \right ) \textit {\_R} +5 c \,d^{2}+2 b \right )\right )}{2}\) |
\(155\) |
| default |
\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,e^{4} \textit {\_Z}^{4}+4 c d \,e^{3} \textit {\_Z}^{3}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 b d e \right ) \textit {\_Z} +d^{4} c +b \,d^{2}+a \right )}{\sum }\frac {\left (-e^{3} c \,\textit {\_R}^{3}-3 c d \,e^{2} \textit {\_R}^{2}+e \left (-3 c \,d^{2}-b \right ) \textit {\_R} -d^{3} c -b d \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 c d \,e^{2} \textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 d^{3} c +b e \textit {\_R} +b d}}{2 a e}+\frac {\ln \left (e x +d \right )}{a e}\) |
\(184\) |
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[In]
int(1/(e*x+d)/(a+b*(e*x+d)^2+c*(e*x+d)^4),x,method=_RETURNVERBOSE)
[Out]
ln(e*x+d)/a/e+1/2*sum(_R*ln(((10*a*c*e^3-3*b^2*e^3)*_R+5*c*e^2)*x^2+((20*a*c*d*e^2-6*b^2*d*e^2)*_R+10*d*c*e)*x
+(10*a*c*d^2*e-3*b^2*d^2*e-a*b*e)*_R+5*c*d^2+2*b),_R=RootOf((4*a^2*c*e^2-a*b^2*e^2)*_Z^2+(4*a*c*e-b^2*e)*_Z+c)
)
Fricas [A] (verification not implemented)
none
Time = 0.28 (sec) , antiderivative size = 468, normalized size of antiderivative = 4.98
\[
\int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx=\left [\frac {\sqrt {b^{2} - 4 \, a c} b \log \left (\frac {2 \, c^{2} e^{4} x^{4} + 8 \, c^{2} d e^{3} x^{3} + 2 \, c^{2} d^{4} + 2 \, {\left (6 \, c^{2} d^{2} + b c\right )} e^{2} x^{2} + 2 \, b c d^{2} + 4 \, {\left (2 \, c^{2} d^{3} + b c d\right )} e x + b^{2} - 2 \, a c + {\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a}\right ) - {\left (b^{2} - 4 \, a c\right )} \log \left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a\right ) + 4 \, {\left (b^{2} - 4 \, a c\right )} \log \left (e x + d\right )}{4 \, {\left (a b^{2} - 4 \, a^{2} c\right )} e}, \frac {2 \, \sqrt {-b^{2} + 4 \, a c} b \arctan \left (-\frac {{\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left (b^{2} - 4 \, a c\right )} \log \left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a\right ) + 4 \, {\left (b^{2} - 4 \, a c\right )} \log \left (e x + d\right )}{4 \, {\left (a b^{2} - 4 \, a^{2} c\right )} e}\right ]
\]
[In]
integrate(1/(e*x+d)/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="fricas")
[Out]
[1/4*(sqrt(b^2 - 4*a*c)*b*log((2*c^2*e^4*x^4 + 8*c^2*d*e^3*x^3 + 2*c^2*d^4 + 2*(6*c^2*d^2 + b*c)*e^2*x^2 + 2*b
*c*d^2 + 4*(2*c^2*d^3 + b*c*d)*e*x + b^2 - 2*a*c + (2*c*e^2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqrt(b^2 - 4*a*c))/
(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a)) - (b^2 - 4*a*
c)*log(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a) + 4*(b^2
- 4*a*c)*log(e*x + d))/((a*b^2 - 4*a^2*c)*e), 1/4*(2*sqrt(-b^2 + 4*a*c)*b*arctan(-(2*c*e^2*x^2 + 4*c*d*e*x +
2*c*d^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (b^2 - 4*a*c)*log(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^
2 + b)*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a) + 4*(b^2 - 4*a*c)*log(e*x + d))/((a*b^2 - 4*a^2*c)*e)]
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (78) = 156\).
Time = 13.83 (sec) , antiderivative size = 320, normalized size of antiderivative = 3.40
\[
\int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx=\left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a^{2} c e \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) + 2 a b^{2} e \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) - 2 a c + b^{2} + b c d^{2}}{b c e^{2}} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a^{2} c e \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) + 2 a b^{2} e \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) - 2 a c + b^{2} + b c d^{2}}{b c e^{2}} \right )} + \frac {\log {\left (\frac {d}{e} + x \right )}}{a e}
\]
[In]
integrate(1/(e*x+d)/(a+b*(e*x+d)**2+c*(e*x+d)**4),x)
[Out]
(-b*sqrt(-4*a*c + b**2)/(4*a*e*(4*a*c - b**2)) - 1/(4*a*e))*log(2*d*x/e + x**2 + (-8*a**2*c*e*(-b*sqrt(-4*a*c
+ b**2)/(4*a*e*(4*a*c - b**2)) - 1/(4*a*e)) + 2*a*b**2*e*(-b*sqrt(-4*a*c + b**2)/(4*a*e*(4*a*c - b**2)) - 1/(4
*a*e)) - 2*a*c + b**2 + b*c*d**2)/(b*c*e**2)) + (b*sqrt(-4*a*c + b**2)/(4*a*e*(4*a*c - b**2)) - 1/(4*a*e))*log
(2*d*x/e + x**2 + (-8*a**2*c*e*(b*sqrt(-4*a*c + b**2)/(4*a*e*(4*a*c - b**2)) - 1/(4*a*e)) + 2*a*b**2*e*(b*sqrt
(-4*a*c + b**2)/(4*a*e*(4*a*c - b**2)) - 1/(4*a*e)) - 2*a*c + b**2 + b*c*d**2)/(b*c*e**2)) + log(d/e + x)/(a*e
)
Maxima [F]
\[
\int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx=\int { \frac {1}{{\left ({\left (e x + d\right )}^{4} c + {\left (e x + d\right )}^{2} b + a\right )} {\left (e x + d\right )}} \,d x }
\]
[In]
integrate(1/(e*x+d)/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="maxima")
[Out]
-integrate((c*e^3*x^3 + 3*c*d*e^2*x^2 + c*d^3 + (3*c*d^2 + b)*e*x + b*d)/(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 +
(6*c*d^2 + b)*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a), x)/a + log(e*x + d)/(a*e)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (86) = 172\).
Time = 0.36 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.98
\[
\int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx=-\frac {\log \left ({\left | c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + 6 \, c d^{2} e^{2} x^{2} + 4 \, c d^{3} e x + c d^{4} + b e^{2} x^{2} + 2 \, b d e x + b d^{2} + a \right |}\right )}{4 \, a e} + \frac {\log \left ({\left | e x + d \right |}\right )}{a e} - \frac {\frac {a b c e^{3} \log \left ({\left | b e^{2} x^{2} + \sqrt {b^{2} - 4 \, a c} e^{2} x^{2} + 2 \, b d e x + 2 \, \sqrt {b^{2} - 4 \, a c} d e x + b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} + 2 \, a \right |}\right )}{\sqrt {b^{2} - 4 \, a c}} - \frac {a b c e^{3} \log \left ({\left | -b e^{2} x^{2} + \sqrt {b^{2} - 4 \, a c} e^{2} x^{2} - 2 \, b d e x + 2 \, \sqrt {b^{2} - 4 \, a c} d e x - b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} - 2 \, a \right |}\right )}{\sqrt {b^{2} - 4 \, a c}}}{4 \, a^{2} c e^{4}}
\]
[In]
integrate(1/(e*x+d)/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="giac")
[Out]
-1/4*log(abs(c*e^4*x^4 + 4*c*d*e^3*x^3 + 6*c*d^2*e^2*x^2 + 4*c*d^3*e*x + c*d^4 + b*e^2*x^2 + 2*b*d*e*x + b*d^2
+ a))/(a*e) + log(abs(e*x + d))/(a*e) - 1/4*(a*b*c*e^3*log(abs(b*e^2*x^2 + sqrt(b^2 - 4*a*c)*e^2*x^2 + 2*b*d*
e*x + 2*sqrt(b^2 - 4*a*c)*d*e*x + b*d^2 + sqrt(b^2 - 4*a*c)*d^2 + 2*a))/sqrt(b^2 - 4*a*c) - a*b*c*e^3*log(abs(
-b*e^2*x^2 + sqrt(b^2 - 4*a*c)*e^2*x^2 - 2*b*d*e*x + 2*sqrt(b^2 - 4*a*c)*d*e*x - b*d^2 + sqrt(b^2 - 4*a*c)*d^2
- 2*a))/sqrt(b^2 - 4*a*c))/(a^2*c*e^4)
Mupad [B] (verification not implemented)
Time = 9.56 (sec) , antiderivative size = 2173, normalized size of antiderivative = 23.12
\[
\int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx=\text {Too large to display}
\]
[In]
int(1/((d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x)
[Out]
log(d + e*x)/(a*e) - (log(a + b*d^2 + c*d^4 + b*e^2*x^2 + c*e^4*x^4 + 2*b*d*e*x + 6*c*d^2*e^2*x^2 + 4*c*d^3*e*
x + 4*c*d*e^3*x^3)*(2*b^2*e - 8*a*c*e))/(2*(4*a*b^2*e^2 - 16*a^2*c*e^2)) - (b*atan((16*a^3*x^2*(((3*b^3 - 8*a*
b*c)*((b^2*(10*b*c^3*e^18 + ((2*b^2*e - 8*a*c*e)*(12*b^3*c^2*e^19 - 40*a*b*c^3*e^19))/(2*(4*a*b^2*e^2 - 16*a^2
*c*e^2))))/(16*a^2*e^2*(4*a*c - b^2)) - ((2*b^2*e - 8*a*c*e)^2*(10*b*c^3*e^18 + ((2*b^2*e - 8*a*c*e)*(12*b^3*c
^2*e^19 - 40*a*b*c^3*e^19))/(2*(4*a*b^2*e^2 - 16*a^2*c*e^2))))/(4*(4*a*b^2*e^2 - 16*a^2*c*e^2)^2) + (b^2*(2*b^
2*e - 8*a*c*e)*(12*b^3*c^2*e^19 - 40*a*b*c^3*e^19))/(16*a^2*e^2*(4*a*c - b^2)*(4*a*b^2*e^2 - 16*a^2*c*e^2))))/
(8*a^3*c^2*(25*a*c - 6*b^2)) - (((b*(2*b^2*e - 8*a*c*e)^2*(12*b^3*c^2*e^19 - 40*a*b*c^3*e^19))/(16*a*e*(4*a*c
- b^2)^(1/2)*(4*a*b^2*e^2 - 16*a^2*c*e^2)^2) - (b^3*(12*b^3*c^2*e^19 - 40*a*b*c^3*e^19))/(64*a^3*e^3*(4*a*c -
b^2)^(3/2)) + (b*(2*b^2*e - 8*a*c*e)*(10*b*c^3*e^18 + ((2*b^2*e - 8*a*c*e)*(12*b^3*c^2*e^19 - 40*a*b*c^3*e^19)
)/(2*(4*a*b^2*e^2 - 16*a^2*c*e^2))))/(4*a*e*(4*a*c - b^2)^(1/2)*(4*a*b^2*e^2 - 16*a^2*c*e^2)))*(3*b^4 + 10*a^2
*c^2 - 14*a*b^2*c))/(8*a^3*c^2*(4*a*c - b^2)^(1/2)*(25*a*c - 6*b^2)))*(4*a*c - b^2)^(3/2))/(b^2*c^2*e^14) + (2
*(3*b^3 - 8*a*b*c)*(4*a*c - b^2)^(3/2)*((b^2*(((2*b^2*e - 8*a*c*e)*(4*a*b^2*c^2*e^17 + 12*b^3*c^2*d^2*e^17 - 4
0*a*b*c^3*d^2*e^17))/(2*(4*a*b^2*e^2 - 16*a^2*c*e^2)) + 4*b^2*c^2*e^16 + 10*b*c^3*d^2*e^16))/(16*a^2*e^2*(4*a*
c - b^2)) - ((2*b^2*e - 8*a*c*e)^2*(((2*b^2*e - 8*a*c*e)*(4*a*b^2*c^2*e^17 + 12*b^3*c^2*d^2*e^17 - 40*a*b*c^3*
d^2*e^17))/(2*(4*a*b^2*e^2 - 16*a^2*c*e^2)) + 4*b^2*c^2*e^16 + 10*b*c^3*d^2*e^16))/(4*(4*a*b^2*e^2 - 16*a^2*c*
e^2)^2) + (b^2*(2*b^2*e - 8*a*c*e)*(4*a*b^2*c^2*e^17 + 12*b^3*c^2*d^2*e^17 - 40*a*b*c^3*d^2*e^17))/(16*a^2*e^2
*(4*a*c - b^2)*(4*a*b^2*e^2 - 16*a^2*c*e^2))))/(b^2*c^4*e^14*(25*a*c - 6*b^2)) - (2*(4*a*c - b^2)*(3*b^4 + 10*
a^2*c^2 - 14*a*b^2*c)*((b*(2*b^2*e - 8*a*c*e)*(((2*b^2*e - 8*a*c*e)*(4*a*b^2*c^2*e^17 + 12*b^3*c^2*d^2*e^17 -
40*a*b*c^3*d^2*e^17))/(2*(4*a*b^2*e^2 - 16*a^2*c*e^2)) + 4*b^2*c^2*e^16 + 10*b*c^3*d^2*e^16))/(4*a*e*(4*a*c -
b^2)^(1/2)*(4*a*b^2*e^2 - 16*a^2*c*e^2)) - (b^3*(4*a*b^2*c^2*e^17 + 12*b^3*c^2*d^2*e^17 - 40*a*b*c^3*d^2*e^17)
)/(64*a^3*e^3*(4*a*c - b^2)^(3/2)) + (b*(2*b^2*e - 8*a*c*e)^2*(4*a*b^2*c^2*e^17 + 12*b^3*c^2*d^2*e^17 - 40*a*b
*c^3*d^2*e^17))/(16*a*e*(4*a*c - b^2)^(1/2)*(4*a*b^2*e^2 - 16*a^2*c*e^2)^2)))/(b^2*c^4*e^14*(25*a*c - 6*b^2))
+ (16*a^3*x*(((3*b^3 - 8*a*b*c)*((b^2*(((2*b^2*e - 8*a*c*e)*(24*b^3*c^2*d*e^18 - 80*a*b*c^3*d*e^18))/(2*(4*a*b
^2*e^2 - 16*a^2*c*e^2)) + 20*b*c^3*d*e^17))/(16*a^2*e^2*(4*a*c - b^2)) - ((2*b^2*e - 8*a*c*e)^2*(((2*b^2*e - 8
*a*c*e)*(24*b^3*c^2*d*e^18 - 80*a*b*c^3*d*e^18))/(2*(4*a*b^2*e^2 - 16*a^2*c*e^2)) + 20*b*c^3*d*e^17))/(4*(4*a*
b^2*e^2 - 16*a^2*c*e^2)^2) + (b^2*(2*b^2*e - 8*a*c*e)*(24*b^3*c^2*d*e^18 - 80*a*b*c^3*d*e^18))/(16*a^2*e^2*(4*
a*c - b^2)*(4*a*b^2*e^2 - 16*a^2*c*e^2))))/(8*a^3*c^2*(25*a*c - 6*b^2)) - ((3*b^4 + 10*a^2*c^2 - 14*a*b^2*c)*(
(b*(2*b^2*e - 8*a*c*e)*(((2*b^2*e - 8*a*c*e)*(24*b^3*c^2*d*e^18 - 80*a*b*c^3*d*e^18))/(2*(4*a*b^2*e^2 - 16*a^2
*c*e^2)) + 20*b*c^3*d*e^17))/(4*a*e*(4*a*c - b^2)^(1/2)*(4*a*b^2*e^2 - 16*a^2*c*e^2)) - (b^3*(24*b^3*c^2*d*e^1
8 - 80*a*b*c^3*d*e^18))/(64*a^3*e^3*(4*a*c - b^2)^(3/2)) + (b*(2*b^2*e - 8*a*c*e)^2*(24*b^3*c^2*d*e^18 - 80*a*
b*c^3*d*e^18))/(16*a*e*(4*a*c - b^2)^(1/2)*(4*a*b^2*e^2 - 16*a^2*c*e^2)^2)))/(8*a^3*c^2*(4*a*c - b^2)^(1/2)*(2
5*a*c - 6*b^2)))*(4*a*c - b^2)^(3/2))/(b^2*c^2*e^14)))/(2*a*e*(4*a*c - b^2)^(1/2))