3.2 \(\int (e x)^m (a+b x^n)^2 (A+B x^n) (c+d x^n) \, dx\)

Optimal. Leaf size=160 \[ \frac{a^2 A c (e x)^{m+1}}{e (m+1)}+\frac{a x^{n+1} (e x)^m (a A d+a B c+2 A b c)}{m+n+1}+\frac{x^{2 n+1} (e x)^m (A b (2 a d+b c)+a B (a d+2 b c))}{m+2 n+1}+\frac{b x^{3 n+1} (e x)^m (2 a B d+A b d+b B c)}{m+3 n+1}+\frac{b^2 B d x^{4 n+1} (e x)^m}{m+4 n+1} \]

[Out]

(a*(2*A*b*c + a*B*c + a*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + ((a*B*(2*b*c + a*d) + A*b*(b*c + 2*a*d))*x^(1 +
2*n)*(e*x)^m)/(1 + m + 2*n) + (b*(b*B*c + A*b*d + 2*a*B*d)*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (b^2*B*d*x^(1
+ 4*n)*(e*x)^m)/(1 + m + 4*n) + (a^2*A*c*(e*x)^(1 + m))/(e*(1 + m))

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Rubi [A]  time = 0.176407, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {570, 20, 30} \[ \frac{a^2 A c (e x)^{m+1}}{e (m+1)}+\frac{a x^{n+1} (e x)^m (a A d+a B c+2 A b c)}{m+n+1}+\frac{x^{2 n+1} (e x)^m (A b (2 a d+b c)+a B (a d+2 b c))}{m+2 n+1}+\frac{b x^{3 n+1} (e x)^m (2 a B d+A b d+b B c)}{m+3 n+1}+\frac{b^2 B d x^{4 n+1} (e x)^m}{m+4 n+1} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(a + b*x^n)^2*(A + B*x^n)*(c + d*x^n),x]

[Out]

(a*(2*A*b*c + a*B*c + a*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + ((a*B*(2*b*c + a*d) + A*b*(b*c + 2*a*d))*x^(1 +
2*n)*(e*x)^m)/(1 + m + 2*n) + (b*(b*B*c + A*b*d + 2*a*B*d)*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (b^2*B*d*x^(1
+ 4*n)*(e*x)^m)/(1 + m + 4*n) + (a^2*A*c*(e*x)^(1 + m))/(e*(1 + m))

Rule 570

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int (e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx &=\int \left (a^2 A c (e x)^m+a (2 A b c+a B c+a A d) x^n (e x)^m+(a B (2 b c+a d)+A b (b c+2 a d)) x^{2 n} (e x)^m+b (b B c+A b d+2 a B d) x^{3 n} (e x)^m+b^2 B d x^{4 n} (e x)^m\right ) \, dx\\ &=\frac{a^2 A c (e x)^{1+m}}{e (1+m)}+\left (b^2 B d\right ) \int x^{4 n} (e x)^m \, dx+(a (2 A b c+a B c+a A d)) \int x^n (e x)^m \, dx+(b (b B c+A b d+2 a B d)) \int x^{3 n} (e x)^m \, dx+(a B (2 b c+a d)+A b (b c+2 a d)) \int x^{2 n} (e x)^m \, dx\\ &=\frac{a^2 A c (e x)^{1+m}}{e (1+m)}+\left (b^2 B d x^{-m} (e x)^m\right ) \int x^{m+4 n} \, dx+\left (a (2 A b c+a B c+a A d) x^{-m} (e x)^m\right ) \int x^{m+n} \, dx+\left (b (b B c+A b d+2 a B d) x^{-m} (e x)^m\right ) \int x^{m+3 n} \, dx+\left ((a B (2 b c+a d)+A b (b c+2 a d)) x^{-m} (e x)^m\right ) \int x^{m+2 n} \, dx\\ &=\frac{a (2 A b c+a B c+a A d) x^{1+n} (e x)^m}{1+m+n}+\frac{(a B (2 b c+a d)+A b (b c+2 a d)) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac{b (b B c+A b d+2 a B d) x^{1+3 n} (e x)^m}{1+m+3 n}+\frac{b^2 B d x^{1+4 n} (e x)^m}{1+m+4 n}+\frac{a^2 A c (e x)^{1+m}}{e (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.310725, size = 129, normalized size = 0.81 \[ x (e x)^m \left (\frac{a^2 A c}{m+1}+\frac{a x^n (a A d+a B c+2 A b c)}{m+n+1}+\frac{x^{2 n} (A b (2 a d+b c)+a B (a d+2 b c))}{m+2 n+1}+\frac{b x^{3 n} (2 a B d+A b d+b B c)}{m+3 n+1}+\frac{b^2 B d x^{4 n}}{m+4 n+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*(a + b*x^n)^2*(A + B*x^n)*(c + d*x^n),x]

[Out]

x*(e*x)^m*((a^2*A*c)/(1 + m) + (a*(2*A*b*c + a*B*c + a*A*d)*x^n)/(1 + m + n) + ((a*B*(2*b*c + a*d) + A*b*(b*c
+ 2*a*d))*x^(2*n))/(1 + m + 2*n) + (b*(b*B*c + A*b*d + 2*a*B*d)*x^(3*n))/(1 + m + 3*n) + (b^2*B*d*x^(4*n))/(1
+ m + 4*n))

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Maple [C]  time = 0.075, size = 2410, normalized size = 15.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(a+b*x^n)^2*(A+B*x^n)*(c+d*x^n),x)

[Out]

x*(26*A*a^2*d*m^2*n^2*x^n+24*A*a^2*d*m*n^3*x^n+2*A*a*b*c*m^4*x^n+19*B*a^2*d*m^2*n^2*(x^n)^2+12*B*a^2*d*m*n^3*(
x^n)^2+2*B*a*b*c*m^4*(x^n)^2+8*B*a*b*d*m^3*(x^n)^3+16*B*a*b*d*n^3*(x^n)^3+21*B*b^2*c*m^2*n*(x^n)^3+12*A*b^2*c*
n^3*(x^n)^2+6*B*a^2*c*m^2*x^n+26*B*a^2*c*n^2*x^n+4*B*a^2*d*(x^n)^2*m+8*B*a^2*d*(x^n)^2*n+2*B*a*b*d*(x^n)^3+4*A
*a^2*d*x^n*m+9*A*a^2*d*x^n*n+2*A*a*b*d*(x^n)^2+24*B*a^2*c*m*n^3*x^n+24*B*a^2*d*m^2*n*(x^n)^2+38*B*a^2*d*m*n^2*
(x^n)^2+6*A*b^2*d*m^2*(x^n)^3+14*A*b^2*d*n^2*(x^n)^3+B*a^2*c*m^4*x^n+4*B*a^2*c*x^n*m+9*B*a^2*c*x^n*n+2*B*a*b*c
*(x^n)^2+76*B*a*b*c*m*n^2*(x^n)^2+42*B*a*b*d*m*n*(x^n)^3+54*A*a*b*c*m^2*n*x^n+104*A*a*b*c*m*n^2*x^n+18*A*a*b*c
*m^3*n*x^n+52*A*a*b*c*m^2*n^2*x^n+48*A*a*b*c*m*n^3*x^n+48*A*a*b*d*m^2*n*(x^n)^2+76*A*a*b*d*m*n^2*(x^n)^2+48*B*
a*b*c*m^2*n*(x^n)^2+B*a^2*d*(x^n)^2+A*a^2*d*x^n+B*a^2*c*x^n+26*A*a^2*d*n^2*x^n+4*A*b^2*c*(x^n)^2*m+6*B*a^2*d*m
^2*(x^n)^2+19*B*a^2*d*n^2*(x^n)^2+b^2*B*d*(x^n)^4+B*b^2*c*(x^n)^3+(x^n)^3*A*b^2*d+(x^n)^2*A*b^2*c+24*B*a^2*d*m
*n*(x^n)^2+12*B*a*b*c*m^2*(x^n)^2+38*B*a*b*c*n^2*(x^n)^2+8*B*a*b*d*(x^n)^3*m+14*B*a*b*d*(x^n)^3*n+4*B*a^2*c*m^
3*x^n+16*B*a*b*d*m*n^3*(x^n)^3+14*B*a*b*d*m^3*n*(x^n)^3+28*B*a*b*d*m^2*n^2*(x^n)^3+35*A*a^2*c*n^2+24*A*a^2*c*n
^4+A*a^2*c*m^4+4*A*a^2*c*m^3+50*A*a^2*c*n^3+6*A*a^2*c*m^2+30*A*a^2*c*m*n+a^2*A*c+10*A*a^2*c*m^3*n+35*A*a^2*c*m
^2*n^2+50*A*a^2*c*m*n^3+30*A*a^2*c*m^2*n+70*A*a^2*c*m*n^2+7*B*b^2*c*(x^n)^3*n+6*A*a^2*d*m^2*x^n+28*B*b^2*c*m*n
^2*(x^n)^3+18*B*b^2*d*m*n*(x^n)^4+9*A*a^2*d*m^3*n*x^n+B*b^2*d*m^4*(x^n)^4+A*b^2*d*m^4*(x^n)^3+38*A*a*b*d*n^2*(
x^n)^2+24*A*b^2*c*m*n*(x^n)^2+27*B*a^2*c*m^2*n*x^n+52*B*a^2*c*m*n^2*x^n+8*A*a*b*d*m^3*(x^n)^2+24*A*a*b*d*n^3*(
x^n)^2+24*A*b^2*c*m^2*n*(x^n)^2+38*A*b^2*c*m*n^2*(x^n)^2+21*A*b^2*d*m*n*(x^n)^3+9*B*a^2*c*m^3*n*x^n+26*B*a^2*c
*m^2*n^2*x^n+22*B*b^2*d*m*n^2*(x^n)^4+2*A*a*b*d*m^4*(x^n)^2+4*m*b^2*B*d*(x^n)^4+6*b^2*B*d*(x^n)^4*n+4*A*a^2*d*
m^3*x^n+24*A*a^2*d*n^3*x^n+6*A*b^2*c*m^2*(x^n)^2+19*A*b^2*c*n^2*(x^n)^2+4*A*b^2*d*(x^n)^3*m+7*A*b^2*d*(x^n)^3*
n+48*A*a*b*d*m*n*(x^n)^2+16*B*a*b*c*m^3*n*(x^n)^2+38*B*a*b*c*m^2*n^2*(x^n)^2+24*B*a*b*c*m*n^3*(x^n)^2+42*B*a*b
*d*m^2*n*(x^n)^3+56*B*a*b*d*m*n^2*(x^n)^3+8*A*b^2*d*n^3*(x^n)^3+B*a^2*d*m^4*(x^n)^2+4*a^2*A*c*m+10*a^2*A*c*n+4
*B*b^2*c*m^3*(x^n)^3+8*B*b^2*c*n^3*(x^n)^3+6*B*b^2*d*m^2*(x^n)^4+11*B*b^2*d*n^2*(x^n)^4+A*a^2*d*m^4*x^n+4*A*b^
2*c*m^3*(x^n)^2+24*B*a^2*c*n^3*x^n+8*A*b^2*c*(x^n)^2*n+4*B*a^2*d*m^3*(x^n)^2+12*B*a^2*d*n^3*(x^n)^2+6*B*b^2*c*
m^2*(x^n)^3+14*B*b^2*c*n^2*(x^n)^3+B*b^2*c*m^4*(x^n)^3+4*B*b^2*d*m^3*(x^n)^4+6*B*b^2*d*n^3*(x^n)^4+A*b^2*c*m^4
*(x^n)^2+4*A*b^2*d*m^3*(x^n)^3+8*A*b^2*c*m^3*n*(x^n)^2+19*A*b^2*c*m^2*n^2*(x^n)^2+12*A*b^2*c*m*n^3*(x^n)^2+21*
A*b^2*d*m^2*n*(x^n)^3+28*A*b^2*d*m*n^2*(x^n)^3+8*B*a^2*d*m^3*n*(x^n)^2+6*B*b^2*d*m^3*n*(x^n)^4+11*B*b^2*d*m^2*
n^2*(x^n)^4+6*B*b^2*d*m*n^3*(x^n)^4+7*A*b^2*d*m^3*n*(x^n)^3+14*A*b^2*d*m^2*n^2*(x^n)^3+8*A*b^2*d*m*n^3*(x^n)^3
+2*B*a*b*d*m^4*(x^n)^3+7*B*b^2*c*m^3*n*(x^n)^3+14*B*b^2*c*m^2*n^2*(x^n)^3+8*B*b^2*c*m*n^3*(x^n)^3+18*B*b^2*d*m
^2*n*(x^n)^4+2*A*a*b*c*x^n+48*B*a*b*c*m*n*(x^n)^2+54*A*a*b*c*m*n*x^n+27*A*a^2*d*m*n*x^n+12*A*a*b*c*m^2*x^n+52*
A*a*b*c*n^2*x^n+8*A*a*b*d*(x^n)^2*m+16*A*a*b*d*(x^n)^2*n+27*B*a^2*c*m*n*x^n+8*B*a*b*c*(x^n)^2*m+16*B*a*b*c*(x^
n)^2*n+8*A*a*b*c*x^n*m+18*A*a*b*c*x^n*n+8*B*a*b*c*m^3*(x^n)^2+24*B*a*b*c*n^3*(x^n)^2+12*B*a*b*d*m^2*(x^n)^3+28
*B*a*b*d*n^2*(x^n)^3+21*B*b^2*c*m*n*(x^n)^3+27*A*a^2*d*m^2*n*x^n+52*A*a^2*d*m*n^2*x^n+8*A*a*b*c*m^3*x^n+48*A*a
*b*c*n^3*x^n+12*A*a*b*d*m^2*(x^n)^2+16*A*a*b*d*m^3*n*(x^n)^2+38*A*a*b*d*m^2*n^2*(x^n)^2+24*A*a*b*d*m*n^3*(x^n)
^2+4*B*b^2*c*(x^n)^3*m)/(1+m)/(m+n+1)/(1+m+2*n)/(1+m+3*n)/(1+m+4*n)*exp(1/2*m*(-I*Pi*csgn(I*e*x)^3+I*Pi*csgn(I
*e*x)^2*csgn(I*e)+I*Pi*csgn(I*e*x)^2*csgn(I*x)-I*Pi*csgn(I*e*x)*csgn(I*e)*csgn(I*x)+2*ln(e)+2*ln(x)))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(a+b*x^n)^2*(A+B*x^n)*(c+d*x^n),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.21185, size = 3420, normalized size = 21.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(a+b*x^n)^2*(A+B*x^n)*(c+d*x^n),x, algorithm="fricas")

[Out]

((B*b^2*d*m^4 + 4*B*b^2*d*m^3 + 6*B*b^2*d*m^2 + 4*B*b^2*d*m + B*b^2*d + 6*(B*b^2*d*m + B*b^2*d)*n^3 + 11*(B*b^
2*d*m^2 + 2*B*b^2*d*m + B*b^2*d)*n^2 + 6*(B*b^2*d*m^3 + 3*B*b^2*d*m^2 + 3*B*b^2*d*m + B*b^2*d)*n)*x*x^(4*n)*e^
(m*log(e) + m*log(x)) + ((B*b^2*c + (2*B*a*b + A*b^2)*d)*m^4 + B*b^2*c + 4*(B*b^2*c + (2*B*a*b + A*b^2)*d)*m^3
+ 8*(B*b^2*c + (2*B*a*b + A*b^2)*d + (B*b^2*c + (2*B*a*b + A*b^2)*d)*m)*n^3 + 6*(B*b^2*c + (2*B*a*b + A*b^2)*
d)*m^2 + 14*(B*b^2*c + (B*b^2*c + (2*B*a*b + A*b^2)*d)*m^2 + (2*B*a*b + A*b^2)*d + 2*(B*b^2*c + (2*B*a*b + A*b
^2)*d)*m)*n^2 + (2*B*a*b + A*b^2)*d + 4*(B*b^2*c + (2*B*a*b + A*b^2)*d)*m + 7*(B*b^2*c + (B*b^2*c + (2*B*a*b +
A*b^2)*d)*m^3 + 3*(B*b^2*c + (2*B*a*b + A*b^2)*d)*m^2 + (2*B*a*b + A*b^2)*d + 3*(B*b^2*c + (2*B*a*b + A*b^2)*
d)*m)*n)*x*x^(3*n)*e^(m*log(e) + m*log(x)) + (((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m^4 + 4*((2*B*a*b +
A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m^3 + 12*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d + ((2*B*a*b + A*b^2)*c + (
B*a^2 + 2*A*a*b)*d)*m)*n^3 + 6*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m^2 + 19*(((2*B*a*b + A*b^2)*c + (B
*a^2 + 2*A*a*b)*d)*m^2 + (2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d + 2*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b
)*d)*m)*n^2 + (2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d + 4*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m + 8*
(((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m^3 + 3*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m^2 + (2*B*a*
b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d + 3*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m)*n)*x*x^(2*n)*e^(m*log(e)
+ m*log(x)) + ((A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m^4 + A*a^2*d + 4*(A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m^3 + 24*(A*
a^2*d + (B*a^2 + 2*A*a*b)*c + (A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m)*n^3 + 6*(A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m^2 +
26*(A*a^2*d + (A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m^2 + (B*a^2 + 2*A*a*b)*c + 2*(A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m
)*n^2 + (B*a^2 + 2*A*a*b)*c + 4*(A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m + 9*(A*a^2*d + (A*a^2*d + (B*a^2 + 2*A*a*b)*
c)*m^3 + 3*(A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m^2 + (B*a^2 + 2*A*a*b)*c + 3*(A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m)*n)
*x*x^n*e^(m*log(e) + m*log(x)) + (A*a^2*c*m^4 + 24*A*a^2*c*n^4 + 4*A*a^2*c*m^3 + 6*A*a^2*c*m^2 + 4*A*a^2*c*m +
A*a^2*c + 50*(A*a^2*c*m + A*a^2*c)*n^3 + 35*(A*a^2*c*m^2 + 2*A*a^2*c*m + A*a^2*c)*n^2 + 10*(A*a^2*c*m^3 + 3*A
*a^2*c*m^2 + 3*A*a^2*c*m + A*a^2*c)*n)*x*e^(m*log(e) + m*log(x)))/(m^5 + 24*(m + 1)*n^4 + 5*m^4 + 50*(m^2 + 2*
m + 1)*n^3 + 10*m^3 + 35*(m^3 + 3*m^2 + 3*m + 1)*n^2 + 10*m^2 + 10*(m^4 + 4*m^3 + 6*m^2 + 4*m + 1)*n + 5*m + 1
)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(a+b*x**n)**2*(A+B*x**n)*(c+d*x**n),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.12811, size = 4610, normalized size = 28.81 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(a+b*x^n)^2*(A+B*x^n)*(c+d*x^n),x, algorithm="giac")

[Out]

(B*b^2*d*m^4*x*x^m*x^(4*n)*e^m + 6*B*b^2*d*m^3*n*x*x^m*x^(4*n)*e^m + 11*B*b^2*d*m^2*n^2*x*x^m*x^(4*n)*e^m + 6*
B*b^2*d*m*n^3*x*x^m*x^(4*n)*e^m + B*b^2*c*m^4*x*x^m*x^(3*n)*e^m + 2*B*a*b*d*m^4*x*x^m*x^(3*n)*e^m + A*b^2*d*m^
4*x*x^m*x^(3*n)*e^m + 7*B*b^2*c*m^3*n*x*x^m*x^(3*n)*e^m + 14*B*a*b*d*m^3*n*x*x^m*x^(3*n)*e^m + 7*A*b^2*d*m^3*n
*x*x^m*x^(3*n)*e^m + 14*B*b^2*c*m^2*n^2*x*x^m*x^(3*n)*e^m + 28*B*a*b*d*m^2*n^2*x*x^m*x^(3*n)*e^m + 14*A*b^2*d*
m^2*n^2*x*x^m*x^(3*n)*e^m + 8*B*b^2*c*m*n^3*x*x^m*x^(3*n)*e^m + 16*B*a*b*d*m*n^3*x*x^m*x^(3*n)*e^m + 8*A*b^2*d
*m*n^3*x*x^m*x^(3*n)*e^m + 2*B*a*b*c*m^4*x*x^m*x^(2*n)*e^m + A*b^2*c*m^4*x*x^m*x^(2*n)*e^m + B*a^2*d*m^4*x*x^m
*x^(2*n)*e^m + 2*A*a*b*d*m^4*x*x^m*x^(2*n)*e^m + 16*B*a*b*c*m^3*n*x*x^m*x^(2*n)*e^m + 8*A*b^2*c*m^3*n*x*x^m*x^
(2*n)*e^m + 8*B*a^2*d*m^3*n*x*x^m*x^(2*n)*e^m + 16*A*a*b*d*m^3*n*x*x^m*x^(2*n)*e^m + 38*B*a*b*c*m^2*n^2*x*x^m*
x^(2*n)*e^m + 19*A*b^2*c*m^2*n^2*x*x^m*x^(2*n)*e^m + 19*B*a^2*d*m^2*n^2*x*x^m*x^(2*n)*e^m + 38*A*a*b*d*m^2*n^2
*x*x^m*x^(2*n)*e^m + 24*B*a*b*c*m*n^3*x*x^m*x^(2*n)*e^m + 12*A*b^2*c*m*n^3*x*x^m*x^(2*n)*e^m + 12*B*a^2*d*m*n^
3*x*x^m*x^(2*n)*e^m + 24*A*a*b*d*m*n^3*x*x^m*x^(2*n)*e^m + B*a^2*c*m^4*x*x^m*x^n*e^m + 2*A*a*b*c*m^4*x*x^m*x^n
*e^m + A*a^2*d*m^4*x*x^m*x^n*e^m + 9*B*a^2*c*m^3*n*x*x^m*x^n*e^m + 18*A*a*b*c*m^3*n*x*x^m*x^n*e^m + 9*A*a^2*d*
m^3*n*x*x^m*x^n*e^m + 26*B*a^2*c*m^2*n^2*x*x^m*x^n*e^m + 52*A*a*b*c*m^2*n^2*x*x^m*x^n*e^m + 26*A*a^2*d*m^2*n^2
*x*x^m*x^n*e^m + 24*B*a^2*c*m*n^3*x*x^m*x^n*e^m + 48*A*a*b*c*m*n^3*x*x^m*x^n*e^m + 24*A*a^2*d*m*n^3*x*x^m*x^n*
e^m + A*a^2*c*m^4*x*x^m*e^m + 10*A*a^2*c*m^3*n*x*x^m*e^m + 35*A*a^2*c*m^2*n^2*x*x^m*e^m + 50*A*a^2*c*m*n^3*x*x
^m*e^m + 24*A*a^2*c*n^4*x*x^m*e^m + 4*B*b^2*d*m^3*x*x^m*x^(4*n)*e^m + 18*B*b^2*d*m^2*n*x*x^m*x^(4*n)*e^m + 22*
B*b^2*d*m*n^2*x*x^m*x^(4*n)*e^m + 6*B*b^2*d*n^3*x*x^m*x^(4*n)*e^m + 4*B*b^2*c*m^3*x*x^m*x^(3*n)*e^m + 8*B*a*b*
d*m^3*x*x^m*x^(3*n)*e^m + 4*A*b^2*d*m^3*x*x^m*x^(3*n)*e^m + 21*B*b^2*c*m^2*n*x*x^m*x^(3*n)*e^m + 42*B*a*b*d*m^
2*n*x*x^m*x^(3*n)*e^m + 21*A*b^2*d*m^2*n*x*x^m*x^(3*n)*e^m + 28*B*b^2*c*m*n^2*x*x^m*x^(3*n)*e^m + 56*B*a*b*d*m
*n^2*x*x^m*x^(3*n)*e^m + 28*A*b^2*d*m*n^2*x*x^m*x^(3*n)*e^m + 8*B*b^2*c*n^3*x*x^m*x^(3*n)*e^m + 16*B*a*b*d*n^3
*x*x^m*x^(3*n)*e^m + 8*A*b^2*d*n^3*x*x^m*x^(3*n)*e^m + 8*B*a*b*c*m^3*x*x^m*x^(2*n)*e^m + 4*A*b^2*c*m^3*x*x^m*x
^(2*n)*e^m + 4*B*a^2*d*m^3*x*x^m*x^(2*n)*e^m + 8*A*a*b*d*m^3*x*x^m*x^(2*n)*e^m + 48*B*a*b*c*m^2*n*x*x^m*x^(2*n
)*e^m + 24*A*b^2*c*m^2*n*x*x^m*x^(2*n)*e^m + 24*B*a^2*d*m^2*n*x*x^m*x^(2*n)*e^m + 48*A*a*b*d*m^2*n*x*x^m*x^(2*
n)*e^m + 76*B*a*b*c*m*n^2*x*x^m*x^(2*n)*e^m + 38*A*b^2*c*m*n^2*x*x^m*x^(2*n)*e^m + 38*B*a^2*d*m*n^2*x*x^m*x^(2
*n)*e^m + 76*A*a*b*d*m*n^2*x*x^m*x^(2*n)*e^m + 24*B*a*b*c*n^3*x*x^m*x^(2*n)*e^m + 12*A*b^2*c*n^3*x*x^m*x^(2*n)
*e^m + 12*B*a^2*d*n^3*x*x^m*x^(2*n)*e^m + 24*A*a*b*d*n^3*x*x^m*x^(2*n)*e^m + 4*B*a^2*c*m^3*x*x^m*x^n*e^m + 8*A
*a*b*c*m^3*x*x^m*x^n*e^m + 4*A*a^2*d*m^3*x*x^m*x^n*e^m + 27*B*a^2*c*m^2*n*x*x^m*x^n*e^m + 54*A*a*b*c*m^2*n*x*x
^m*x^n*e^m + 27*A*a^2*d*m^2*n*x*x^m*x^n*e^m + 52*B*a^2*c*m*n^2*x*x^m*x^n*e^m + 104*A*a*b*c*m*n^2*x*x^m*x^n*e^m
+ 52*A*a^2*d*m*n^2*x*x^m*x^n*e^m + 24*B*a^2*c*n^3*x*x^m*x^n*e^m + 48*A*a*b*c*n^3*x*x^m*x^n*e^m + 24*A*a^2*d*n
^3*x*x^m*x^n*e^m + 4*A*a^2*c*m^3*x*x^m*e^m + 30*A*a^2*c*m^2*n*x*x^m*e^m + 70*A*a^2*c*m*n^2*x*x^m*e^m + 50*A*a^
2*c*n^3*x*x^m*e^m + 6*B*b^2*d*m^2*x*x^m*x^(4*n)*e^m + 18*B*b^2*d*m*n*x*x^m*x^(4*n)*e^m + 11*B*b^2*d*n^2*x*x^m*
x^(4*n)*e^m + 6*B*b^2*c*m^2*x*x^m*x^(3*n)*e^m + 12*B*a*b*d*m^2*x*x^m*x^(3*n)*e^m + 6*A*b^2*d*m^2*x*x^m*x^(3*n)
*e^m + 21*B*b^2*c*m*n*x*x^m*x^(3*n)*e^m + 42*B*a*b*d*m*n*x*x^m*x^(3*n)*e^m + 21*A*b^2*d*m*n*x*x^m*x^(3*n)*e^m
+ 14*B*b^2*c*n^2*x*x^m*x^(3*n)*e^m + 28*B*a*b*d*n^2*x*x^m*x^(3*n)*e^m + 14*A*b^2*d*n^2*x*x^m*x^(3*n)*e^m + 12*
B*a*b*c*m^2*x*x^m*x^(2*n)*e^m + 6*A*b^2*c*m^2*x*x^m*x^(2*n)*e^m + 6*B*a^2*d*m^2*x*x^m*x^(2*n)*e^m + 12*A*a*b*d
*m^2*x*x^m*x^(2*n)*e^m + 48*B*a*b*c*m*n*x*x^m*x^(2*n)*e^m + 24*A*b^2*c*m*n*x*x^m*x^(2*n)*e^m + 24*B*a^2*d*m*n*
x*x^m*x^(2*n)*e^m + 48*A*a*b*d*m*n*x*x^m*x^(2*n)*e^m + 38*B*a*b*c*n^2*x*x^m*x^(2*n)*e^m + 19*A*b^2*c*n^2*x*x^m
*x^(2*n)*e^m + 19*B*a^2*d*n^2*x*x^m*x^(2*n)*e^m + 38*A*a*b*d*n^2*x*x^m*x^(2*n)*e^m + 6*B*a^2*c*m^2*x*x^m*x^n*e
^m + 12*A*a*b*c*m^2*x*x^m*x^n*e^m + 6*A*a^2*d*m^2*x*x^m*x^n*e^m + 27*B*a^2*c*m*n*x*x^m*x^n*e^m + 54*A*a*b*c*m*
n*x*x^m*x^n*e^m + 27*A*a^2*d*m*n*x*x^m*x^n*e^m + 26*B*a^2*c*n^2*x*x^m*x^n*e^m + 52*A*a*b*c*n^2*x*x^m*x^n*e^m +
26*A*a^2*d*n^2*x*x^m*x^n*e^m + 6*A*a^2*c*m^2*x*x^m*e^m + 30*A*a^2*c*m*n*x*x^m*e^m + 35*A*a^2*c*n^2*x*x^m*e^m
+ 4*B*b^2*d*m*x*x^m*x^(4*n)*e^m + 6*B*b^2*d*n*x*x^m*x^(4*n)*e^m + 4*B*b^2*c*m*x*x^m*x^(3*n)*e^m + 8*B*a*b*d*m*
x*x^m*x^(3*n)*e^m + 4*A*b^2*d*m*x*x^m*x^(3*n)*e^m + 7*B*b^2*c*n*x*x^m*x^(3*n)*e^m + 14*B*a*b*d*n*x*x^m*x^(3*n)
*e^m + 7*A*b^2*d*n*x*x^m*x^(3*n)*e^m + 8*B*a*b*c*m*x*x^m*x^(2*n)*e^m + 4*A*b^2*c*m*x*x^m*x^(2*n)*e^m + 4*B*a^2
*d*m*x*x^m*x^(2*n)*e^m + 8*A*a*b*d*m*x*x^m*x^(2*n)*e^m + 16*B*a*b*c*n*x*x^m*x^(2*n)*e^m + 8*A*b^2*c*n*x*x^m*x^
(2*n)*e^m + 8*B*a^2*d*n*x*x^m*x^(2*n)*e^m + 16*A*a*b*d*n*x*x^m*x^(2*n)*e^m + 4*B*a^2*c*m*x*x^m*x^n*e^m + 8*A*a
*b*c*m*x*x^m*x^n*e^m + 4*A*a^2*d*m*x*x^m*x^n*e^m + 9*B*a^2*c*n*x*x^m*x^n*e^m + 18*A*a*b*c*n*x*x^m*x^n*e^m + 9*
A*a^2*d*n*x*x^m*x^n*e^m + 4*A*a^2*c*m*x*x^m*e^m + 10*A*a^2*c*n*x*x^m*e^m + B*b^2*d*x*x^m*x^(4*n)*e^m + B*b^2*c
*x*x^m*x^(3*n)*e^m + 2*B*a*b*d*x*x^m*x^(3*n)*e^m + A*b^2*d*x*x^m*x^(3*n)*e^m + 2*B*a*b*c*x*x^m*x^(2*n)*e^m + A
*b^2*c*x*x^m*x^(2*n)*e^m + B*a^2*d*x*x^m*x^(2*n)*e^m + 2*A*a*b*d*x*x^m*x^(2*n)*e^m + B*a^2*c*x*x^m*x^n*e^m + 2
*A*a*b*c*x*x^m*x^n*e^m + A*a^2*d*x*x^m*x^n*e^m + A*a^2*c*x*x^m*e^m)/(m^5 + 10*m^4*n + 35*m^3*n^2 + 50*m^2*n^3
+ 24*m*n^4 + 5*m^4 + 40*m^3*n + 105*m^2*n^2 + 100*m*n^3 + 24*n^4 + 10*m^3 + 60*m^2*n + 105*m*n^2 + 50*n^3 + 10
*m^2 + 40*m*n + 35*n^2 + 5*m + 10*n + 1)