3.1.9 \(\int \frac {x^8 (d+e x^3)}{a+b x^3+c x^6} \, dx\) [9]

Optimal. Leaf size=132 \[ \frac {(c d-b e) x^3}{3 c^2}+\frac {e x^6}{6 c}-\frac {\left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{\sqrt {b^2-4 a c}}\right )}{3 c^3 \sqrt {b^2-4 a c}}-\frac {\left (b c d-b^2 e+a c e\right ) \log \left (a+b x^3+c x^6\right )}{6 c^3} \]

[Out]

1/3*(-b*e+c*d)*x^3/c^2+1/6*e*x^6/c-1/6*(a*c*e-b^2*e+b*c*d)*ln(c*x^6+b*x^3+a)/c^3-1/3*(3*a*b*c*e-2*a*c^2*d-b^3*
e+b^2*c*d)*arctanh((2*c*x^3+b)/(-4*a*c+b^2)^(1/2))/c^3/(-4*a*c+b^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.15, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1488, 814, 648, 632, 212, 642} \begin {gather*} -\frac {\left (a c e+b^2 (-e)+b c d\right ) \log \left (a+b x^3+c x^6\right )}{6 c^3}-\frac {\left (3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{\sqrt {b^2-4 a c}}\right )}{3 c^3 \sqrt {b^2-4 a c}}+\frac {x^3 (c d-b e)}{3 c^2}+\frac {e x^6}{6 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^8*(d + e*x^3))/(a + b*x^3 + c*x^6),x]

[Out]

((c*d - b*e)*x^3)/(3*c^2) + (e*x^6)/(6*c) - ((b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)*ArcTanh[(b + 2*c*x^3)/S
qrt[b^2 - 4*a*c]])/(3*c^3*Sqrt[b^2 - 4*a*c]) - ((b*c*d - b^2*e + a*c*e)*Log[a + b*x^3 + c*x^6])/(6*c^3)

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 1488

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>
Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a,
b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^8 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2 (d+e x)}{a+b x+c x^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {c d-b e}{c^2}+\frac {e x}{c}-\frac {a (c d-b e)+\left (b c d-b^2 e+a c e\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^3\right )\\ &=\frac {(c d-b e) x^3}{3 c^2}+\frac {e x^6}{6 c}-\frac {\text {Subst}\left (\int \frac {a (c d-b e)+\left (b c d-b^2 e+a c e\right ) x}{a+b x+c x^2} \, dx,x,x^3\right )}{3 c^2}\\ &=\frac {(c d-b e) x^3}{3 c^2}+\frac {e x^6}{6 c}-\frac {\left (b c d-b^2 e+a c e\right ) \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^3\right )}{6 c^3}+\frac {\left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^3\right )}{6 c^3}\\ &=\frac {(c d-b e) x^3}{3 c^2}+\frac {e x^6}{6 c}-\frac {\left (b c d-b^2 e+a c e\right ) \log \left (a+b x^3+c x^6\right )}{6 c^3}-\frac {\left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^3\right )}{3 c^3}\\ &=\frac {(c d-b e) x^3}{3 c^2}+\frac {e x^6}{6 c}-\frac {\left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{\sqrt {b^2-4 a c}}\right )}{3 c^3 \sqrt {b^2-4 a c}}-\frac {\left (b c d-b^2 e+a c e\right ) \log \left (a+b x^3+c x^6\right )}{6 c^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.05, size = 126, normalized size = 0.95 \begin {gather*} \frac {2 c (c d-b e) x^3+c^2 e x^6+\frac {2 \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \tan ^{-1}\left (\frac {b+2 c x^3}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+\left (-b c d+b^2 e-a c e\right ) \log \left (a+b x^3+c x^6\right )}{6 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^8*(d + e*x^3))/(a + b*x^3 + c*x^6),x]

[Out]

(2*c*(c*d - b*e)*x^3 + c^2*e*x^6 + (2*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)*ArcTan[(b + 2*c*x^3)/Sqrt[-b^2
+ 4*a*c]])/Sqrt[-b^2 + 4*a*c] + (-(b*c*d) + b^2*e - a*c*e)*Log[a + b*x^3 + c*x^6])/(6*c^3)

________________________________________________________________________________________

Maple [A]
time = 0.13, size = 136, normalized size = 1.03

method	result	size
default	\(-\frac {-\frac {1}{2} c e \,x^{6}+b e \,x^{3}-c d \,x^{3}}{3 c^{2}}+\frac {\frac {\left (-a c e +b^{2} e -b c d \right ) \ln \left (c \,x^{6}+b \,x^{3}+a \right )}{2 c}+\frac {2 \left (a b e -a c d -\frac {\left (-a c e +b^{2} e -b c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{3}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{3 c^{2}}\)	\(136\)
risch	\(\text {Expression too large to display}\)	\(2131\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(e*x^3+d)/(c*x^6+b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

-1/3/c^2*(-1/2*c*e*x^6+b*e*x^3-c*d*x^3)+1/3/c^2*(1/2*(-a*c*e+b^2*e-b*c*d)/c*ln(c*x^6+b*x^3+a)+2*(a*b*e-a*c*d-1
/2*(-a*c*e+b^2*e-b*c*d)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^3+b)/(4*a*c-b^2)^(1/2)))

________________________________________________________________________________________

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^8*(e*x^3+d)/(c*x^6+b*x^3+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

________________________________________________________________________________________

Fricas [A]
time = 0.59, size = 440, normalized size = 3.33 \begin {gather*} \left [\frac {2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d x^{3} + \sqrt {b^{2} - 4 \, a c} {\left ({\left (b^{2} c - 2 \, a c^{2}\right )} d - {\left (b^{3} - 3 \, a b c\right )} e\right )} \log \left (\frac {2 \, c^{2} x^{6} + 2 \, b c x^{3} + b^{2} - 2 \, a c - {\left (2 \, c x^{3} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{6} + b x^{3} + a}\right ) + {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{6} - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3}\right )} e - {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}, \frac {2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d x^{3} - 2 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (b^{2} c - 2 \, a c^{2}\right )} d - {\left (b^{3} - 3 \, a b c\right )} e\right )} \arctan \left (-\frac {{\left (2 \, c x^{3} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{6} - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3}\right )} e - {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(e*x^3+d)/(c*x^6+b*x^3+a),x, algorithm="fricas")

[Out]

[1/6*(2*(b^2*c^2 - 4*a*c^3)*d*x^3 + sqrt(b^2 - 4*a*c)*((b^2*c - 2*a*c^2)*d - (b^3 - 3*a*b*c)*e)*log((2*c^2*x^6
+ 2*b*c*x^3 + b^2 - 2*a*c - (2*c*x^3 + b)*sqrt(b^2 - 4*a*c))/(c*x^6 + b*x^3 + a)) + ((b^2*c^2 - 4*a*c^3)*x^6
- 2*(b^3*c - 4*a*b*c^2)*x^3)*e - ((b^3*c - 4*a*b*c^2)*d - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e)*log(c*x^6 + b*x^3 +
a))/(b^2*c^3 - 4*a*c^4), 1/6*(2*(b^2*c^2 - 4*a*c^3)*d*x^3 - 2*sqrt(-b^2 + 4*a*c)*((b^2*c - 2*a*c^2)*d - (b^3
- 3*a*b*c)*e)*arctan(-(2*c*x^3 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + ((b^2*c^2 - 4*a*c^3)*x^6 - 2*(b^3*c -
4*a*b*c^2)*x^3)*e - ((b^3*c - 4*a*b*c^2)*d - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e)*log(c*x^6 + b*x^3 + a))/(b^2*c^3
- 4*a*c^4)]

________________________________________________________________________________________

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**8*(e*x**3+d)/(c*x**6+b*x**3+a),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 3.08, size = 131, normalized size = 0.99 \begin {gather*} \frac {c x^{6} e + 2 \, c d x^{3} - 2 \, b x^{3} e}{6 \, c^{2}} - \frac {{\left (b c d - b^{2} e + a c e\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \, c^{3}} + \frac {{\left (b^{2} c d - 2 \, a c^{2} d - b^{3} e + 3 \, a b c e\right )} \arctan \left (\frac {2 \, c x^{3} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{3 \, \sqrt {-b^{2} + 4 \, a c} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(e*x^3+d)/(c*x^6+b*x^3+a),x, algorithm="giac")

[Out]

1/6*(c*x^6*e + 2*c*d*x^3 - 2*b*x^3*e)/c^2 - 1/6*(b*c*d - b^2*e + a*c*e)*log(c*x^6 + b*x^3 + a)/c^3 + 1/3*(b^2*
c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)*arctan((2*c*x^3 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^3)

________________________________________________________________________________________

Mupad [B]
time = 2.40, 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^8*(d + e*x^3))/(a + b*x^3 + c*x^6),x)

[Out]

x^3*(d/(3*c) - (b*e)/(3*c^2)) + (e*x^6)/(6*c) - (log(a + b*x^3 + c*x^6)*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d +
12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(36*a*c^4 - 9*b^2*c^3)) - (atan((4*c^6*(4*a*c - b^2)^(3/2)*(x^3*((b*((b^5*c^3
*d^3 - b^8*e^3 - 2*a*b^3*c^4*d^3 + a^2*b*c^5*d^3 + a^3*c^5*d^2*e - 3*b^6*c^2*d^2*e - 8*a^2*b^4*c^2*e^3 + 4*a^3
*b^2*c^3*e^3 + 5*a*b^6*c*e^3 + 3*b^7*c*d*e^2 + 9*a*b^4*c^3*d^2*e - 12*a*b^5*c^2*d*e^2 - 4*a^3*b*c^4*d*e^2 - 7*
a^2*b^2*c^4*d^2*e + 14*a^2*b^3*c^3*d*e^2)/c^6 - (((6*a^2*c^7*d^2 + 12*b^4*c^5*d^2 + 12*b^6*c^3*e^2 - 18*a*b^2*
c^6*d^2 - 42*a*b^4*c^4*e^2 + 36*a^2*b^2*c^5*e^2 - 24*b^5*c^4*d*e + 60*a*b^3*c^5*d*e - 30*a^2*b*c^6*d*e)/c^6 -
(((45*b^3*c^7*d - 45*b^4*c^6*e - 36*a*b*c^8*d + 81*a*b^2*c^7*e)/c^6 - (27*b^2*c^3*(3*b^4*e + 12*a^2*c^2*e - 3*
b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(36*a*c^4 - 9*b^2*c^3))*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c
^2*d - 15*a*b^2*c*e))/(2*(36*a*c^4 - 9*b^2*c^3)))*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^
2*c*e))/(2*(36*a*c^4 - 9*b^2*c^3)) - (((((45*b^3*c^7*d - 45*b^4*c^6*e - 36*a*b*c^8*d + 81*a*b^2*c^7*e)/c^6 - (
27*b^2*c^3*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(36*a*c^4 - 9*b^2*c^3))*(b^3*e
+ 2*a*c^2*d - b^2*c*d - 3*a*b*c*e))/(6*c^3*(4*a*c - b^2)^(1/2)) - (9*b^2*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*
c*e)*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(4*a*c - b^2)^(1/2)*(36*a*c^4 - 9*
b^2*c^3)))*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*c*e))/(6*c^3*(4*a*c - b^2)^(1/2)) + (3*b^2*(b^3*e + 2*a*c^2*d
- b^2*c*d - 3*a*b*c*e)^2*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(4*c^3*(4*a*c - b
^2)*(36*a*c^4 - 9*b^2*c^3))))/(4*a^2*c) - ((2*a*c - b^2)*((((((45*b^3*c^7*d - 45*b^4*c^6*e - 36*a*b*c^8*d + 81
*a*b^2*c^7*e)/c^6 - (27*b^2*c^3*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(36*a*c^4
- 9*b^2*c^3))*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*c*e))/(6*c^3*(4*a*c - b^2)^(1/2)) - (9*b^2*(b^3*e + 2*a*c^2
*d - b^2*c*d - 3*a*b*c*e)*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(4*a*c - b^2)
^(1/2)*(36*a*c^4 - 9*b^2*c^3)))*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(36*a*c
^4 - 9*b^2*c^3)) - (((6*a^2*c^7*d^2 + 12*b^4*c^5*d^2 + 12*b^6*c^3*e^2 - 18*a*b^2*c^6*d^2 - 42*a*b^4*c^4*e^2 +
36*a^2*b^2*c^5*e^2 - 24*b^5*c^4*d*e + 60*a*b^3*c^5*d*e - 30*a^2*b*c^6*d*e)/c^6 - (((45*b^3*c^7*d - 45*b^4*c^6*
e - 36*a*b*c^8*d + 81*a*b^2*c^7*e)/c^6 - (27*b^2*c^3*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a
*b^2*c*e))/(36*a*c^4 - 9*b^2*c^3))*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(36*
a*c^4 - 9*b^2*c^3)))*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*c*e))/(6*c^3*(4*a*c - b^2)^(1/2)) + (b^2*(b^3*e + 2*
a*c^2*d - b^2*c*d - 3*a*b*c*e)^3)/(4*c^6*(4*a*c - b^2)^(3/2))))/(4*a^2*c*(4*a*c - b^2)^(1/2))) - (b*((a*b^7*e^
3 - a*b^4*c^3*d^3 - 4*a^2*b^5*c*e^3 - 2*a^4*b*c^3*e^3 + a^4*c^4*d*e^2 + a^2*b^2*c^4*d^3 + 5*a^3*b^3*c^2*e^3 -
3*a*b^6*c*d*e^2 + 3*a*b^5*c^2*d^2*e + 2*a^3*b*c^4*d^2*e - 6*a^2*b^3*c^3*d^2*e + 9*a^2*b^4*c^2*d*e^2 - 7*a^3*b^
2*c^3*d*e^2)/c^6 + (((15*a*b^3*c^5*d^2 - 12*a^2*b*c^6*d^2 + 15*a*b^5*c^3*e^2 + 27*a^3*b*c^5*e^2 - 42*a^2*b^3*c
^4*e^2 - 12*a^3*c^6*d*e - 30*a*b^4*c^4*d*e + 54*a^2*b^2*c^5*d*e)/c^6 + (((36*a^2*c^8*d - 72*a*b^2*c^7*d + 72*a
*b^3*c^6*e - 108*a^2*b*c^7*e)/c^6 + (54*a*b*c^3*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*
c*e))/(36*a*c^4 - 9*b^2*c^3))*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(36*a*c^4
- 9*b^2*c^3)))*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(36*a*c^4 - 9*b^2*c^3))
- (((((36*a^2*c^8*d - 72*a*b^2*c^7*d + 72*a*b^3*c^6*e - 108*a^2*b*c^7*e)/c^6 + (54*a*b*c^3*(3*b^4*e + 12*a^2*
c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(36*a*c^4 - 9*b^2*c^3))*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b
*c*e))/(6*c^3*(4*a*c - b^2)^(1/2)) + (9*a*b*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*c*e)*(3*b^4*e + 12*a^2*c^2*e
- 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/((4*a*c - b^2)^(1/2)*(36*a*c^4 - 9*b^2*c^3)))*(b^3*e + 2*a*c^2*d -
b^2*c*d - 3*a*b*c*e))/(6*c^3*(4*a*c - b^2)^(1/2)) - (3*a*b*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*c*e)^2*(3*b^4
*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*c^3*(4*a*c - b^2)*(36*a*c^4 - 9*b^2*c^3))))/(
4*a^2*c) + ((2*a*c - b^2)*((((((36*a^2*c^8*d - 72*a*b^2*c^7*d + 72*a*b^3*c^6*e - 108*a^2*b*c^7*e)/c^6 + (54*a*
b*c^3*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(36*a*c^4 - 9*b^2*c^3))*(b^3*e + 2*a
*c^2*d - b^2*c*d - 3*a*b*c*e))/(6*c^3*(4*a*c - b^2)^(1/2)) + (9*a*b*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*c*e)*
(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/((4*a*c - b^2)^(1/2)*(36*a*c^4 - 9*b^2*c^3
)))*(3*b^4*e + 12*a^2*c^2*e - 3*b^3*c*d + 12*a*b*c^2*d - 15*a*b^2*c*e))/(2*(36*a*c^4 - 9*b^2*c^3)) + (((15*a*b
^3*c^5*d^2 - 12*a^2*b*c^6*d^2 + 15*a*b^5*c^3*e^2 + 27*a^3*b*c^5*e^2 - 42*a^2*b^3*c^4*e^2 - 12*a^3*c^6*d*e - 30
*a*b^4*c^4*d*e + 54*a^2*b^2*c^5*d*e)/c^6 + (((3...

________________________________________________________________________________________