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

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

[Out]

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

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Rubi [A]  time = 0.171506, 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} \[ \frac{c x^{n+1} (e x)^m (2 a A d+a B c+A b c)}{m+n+1}+\frac{x^{2 n+1} (e x)^m (a d (A d+2 B c)+b c (2 A d+B c))}{m+2 n+1}+\frac{d x^{3 n+1} (e x)^m (a B d+A b d+2 b B c)}{m+3 n+1}+\frac{a A c^2 (e x)^{m+1}}{e (m+1)}+\frac{b B d^2 x^{4 n+1} (e x)^m}{m+4 n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.342733, size = 129, normalized size = 0.81 \[ x (e x)^m \left (\frac{c x^n (2 a A d+a B c+A b c)}{m+n+1}+\frac{x^{2 n} (a d (A d+2 B c)+b c (2 A d+B c))}{m+2 n+1}+\frac{d x^{3 n} (a B d+A b d+2 b B c)}{m+3 n+1}+\frac{a A c^2}{m+1}+\frac{b B d^2 x^{4 n}}{m+4 n+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.076, 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)*(A+B*x^n)*(c+d*x^n)^2,x)

[Out]

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

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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)*(A+B*x^n)*(c+d*x^n)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.24824, size = 3268, normalized size = 20.42 \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)*(A+B*x^n)*(c+d*x^n)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.18954, 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)*(A+B*x^n)*(c+d*x^n)^2,x, algorithm="giac")

[Out]

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