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

Optimal. Leaf size=237 \[ \frac {x^{2 n+1} (e x)^m \left (A \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a B c (a d+b c)\right )}{m+2 n+1}+\frac {x^{3 n+1} (e x)^m \left (a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{m+3 n+1}+\frac {a^2 A c^2 (e x)^{m+1}}{e (m+1)}+\frac {a c x^{n+1} (e x)^m (2 A (a d+b c)+a B c)}{m+n+1}+\frac {b d x^{4 n+1} (e x)^m (2 a B d+A b d+2 b B c)}{m+4 n+1}+\frac {b^2 B d^2 x^{5 n+1} (e x)^m}{m+5 n+1} \]

[Out]

a*c*(a*B*c+2*A*(a*d+b*c))*x^(1+n)*(e*x)^m/(1+m+n)+(2*a*B*c*(a*d+b*c)+A*(a^2*d^2+4*a*b*c*d+b^2*c^2))*x^(1+2*n)*
(e*x)^m/(1+m+2*n)+(a^2*B*d^2+2*a*b*d*(A*d+2*B*c)+b^2*c*(2*A*d+B*c))*x^(1+3*n)*(e*x)^m/(1+m+3*n)+b*d*(A*b*d+2*B
*a*d+2*B*b*c)*x^(1+4*n)*(e*x)^m/(1+m+4*n)+b^2*B*d^2*x^(1+5*n)*(e*x)^m/(1+m+5*n)+a^2*A*c^2*(e*x)^(1+m)/e/(1+m)

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Rubi [A]  time = 0.31, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {570, 20, 30} \[ \frac {x^{2 n+1} (e x)^m \left (A \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a B c (a d+b c)\right )}{m+2 n+1}+\frac {x^{3 n+1} (e x)^m \left (a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{m+3 n+1}+\frac {a^2 A c^2 (e x)^{m+1}}{e (m+1)}+\frac {a c x^{n+1} (e x)^m (2 A (a d+b c)+a B c)}{m+n+1}+\frac {b d x^{4 n+1} (e x)^m (2 a B d+A b d+2 b B c)}{m+4 n+1}+\frac {b^2 B d^2 x^{5 n+1} (e x)^m}{m+5 n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*c*(a*B*c + 2*A*(b*c + a*d))*x^(1 + n)*(e*x)^m)/(1 + m + n) + ((2*a*B*c*(b*c + a*d) + A*(b^2*c^2 + 4*a*b*c*d
+ a^2*d^2))*x^(1 + 2*n)*(e*x)^m)/(1 + m + 2*n) + ((a^2*B*d^2 + 2*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))*x
^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (b*d*(2*b*B*c + A*b*d + 2*a*B*d)*x^(1 + 4*n)*(e*x)^m)/(1 + m + 4*n) + (b^2
*B*d^2*x^(1 + 5*n)*(e*x)^m)/(1 + m + 5*n) + (a^2*A*c^2*(e*x)^(1 + m))/(e*(1 + m))

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]

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]

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 )^2 \, dx &=\int \left (a^2 A c^2 (e x)^m+a c (a B c+2 A (b c+a d)) x^n (e x)^m+\left (2 a B c (b c+a d)+A \left (b^2 c^2+4 a b c d+a^2 d^2\right )\right ) x^{2 n} (e x)^m+\left (a^2 B d^2+2 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) x^{3 n} (e x)^m+b d (2 b B c+A b d+2 a B d) x^{4 n} (e x)^m+b^2 B d^2 x^{5 n} (e x)^m\right ) \, dx\\ &=\frac {a^2 A c^2 (e x)^{1+m}}{e (1+m)}+\left (b^2 B d^2\right ) \int x^{5 n} (e x)^m \, dx+(b d (2 b B c+A b d+2 a B d)) \int x^{4 n} (e x)^m \, dx+(a c (a B c+2 A (b c+a d))) \int x^n (e x)^m \, dx+\left (a^2 B d^2+2 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) \int x^{3 n} (e x)^m \, dx+\left (2 a B c (b c+a d)+A \left (b^2 c^2+4 a b c d+a^2 d^2\right )\right ) \int x^{2 n} (e x)^m \, dx\\ &=\frac {a^2 A c^2 (e x)^{1+m}}{e (1+m)}+\left (b^2 B d^2 x^{-m} (e x)^m\right ) \int x^{m+5 n} \, dx+\left (b d (2 b B c+A b d+2 a B d) x^{-m} (e x)^m\right ) \int x^{m+4 n} \, dx+\left (a c (a B c+2 A (b c+a d)) x^{-m} (e x)^m\right ) \int x^{m+n} \, dx+\left (\left (a^2 B d^2+2 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) x^{-m} (e x)^m\right ) \int x^{m+3 n} \, dx+\left (\left (2 a B c (b c+a d)+A \left (b^2 c^2+4 a b c d+a^2 d^2\right )\right ) x^{-m} (e x)^m\right ) \int x^{m+2 n} \, dx\\ &=\frac {a c (a B c+2 A (b c+a d)) x^{1+n} (e x)^m}{1+m+n}+\frac {\left (2 a B c (b c+a d)+A \left (b^2 c^2+4 a b c d+a^2 d^2\right )\right ) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {\left (a^2 B d^2+2 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) x^{1+3 n} (e x)^m}{1+m+3 n}+\frac {b d (2 b B c+A b d+2 a B d) x^{1+4 n} (e x)^m}{1+m+4 n}+\frac {b^2 B d^2 x^{1+5 n} (e x)^m}{1+m+5 n}+\frac {a^2 A c^2 (e x)^{1+m}}{e (1+m)}\\ \end {align*}

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Mathematica [A]  time = 0.60, size = 199, normalized size = 0.84 \[ x (e x)^m \left (\frac {x^{2 n} \left (A \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a B c (a d+b c)\right )}{m+2 n+1}+\frac {x^{3 n} \left (a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{m+3 n+1}+\frac {a^2 A c^2}{m+1}+\frac {b d x^{4 n} (2 a B d+A b d+2 b B c)}{m+4 n+1}+\frac {a c x^n (2 A (a d+b c)+a B c)}{m+n+1}+\frac {b^2 B d^2 x^{5 n}}{m+5 n+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(e*x)^m*((a^2*A*c^2)/(1 + m) + (a*c*(a*B*c + 2*A*(b*c + a*d))*x^n)/(1 + m + n) + ((2*a*B*c*(b*c + a*d) + A*(
b^2*c^2 + 4*a*b*c*d + a^2*d^2))*x^(2*n))/(1 + m + 2*n) + ((a^2*B*d^2 + 2*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*
A*d))*x^(3*n))/(1 + m + 3*n) + (b*d*(2*b*B*c + A*b*d + 2*a*B*d)*x^(4*n))/(1 + m + 4*n) + (b^2*B*d^2*x^(5*n))/(
1 + m + 5*n))

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fricas [B]  time = 0.80, size = 3515, normalized size = 14.83 \[ \text {result too large to display} \]

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)^2,x, algorithm="fricas")

[Out]

((B*b^2*d^2*m^5 + 5*B*b^2*d^2*m^4 + 10*B*b^2*d^2*m^3 + 10*B*b^2*d^2*m^2 + 5*B*b^2*d^2*m + B*b^2*d^2 + 24*(B*b^
2*d^2*m + B*b^2*d^2)*n^4 + 50*(B*b^2*d^2*m^2 + 2*B*b^2*d^2*m + B*b^2*d^2)*n^3 + 35*(B*b^2*d^2*m^3 + 3*B*b^2*d^
2*m^2 + 3*B*b^2*d^2*m + B*b^2*d^2)*n^2 + 10*(B*b^2*d^2*m^4 + 4*B*b^2*d^2*m^3 + 6*B*b^2*d^2*m^2 + 4*B*b^2*d^2*m
+ B*b^2*d^2)*n)*x*x^(5*n)*e^(m*log(e) + m*log(x)) + ((2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^5 + 2*B*b^2*c*d
+ 5*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^4 + 30*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2 + (2*B*b^2*c*d + (2*B*
a*b + A*b^2)*d^2)*m)*n^4 + 10*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^3 + 61*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*
d^2 + (2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^2 + 2*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m)*n^3 + (2*B*a*b +
A*b^2)*d^2 + 10*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^2 + 41*(2*B*b^2*c*d + (2*B*b^2*c*d + (2*B*a*b + A*b^2)
*d^2)*m^3 + (2*B*a*b + A*b^2)*d^2 + 3*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^2 + 3*(2*B*b^2*c*d + (2*B*a*b +
A*b^2)*d^2)*m)*n^2 + 5*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m + 11*(2*B*b^2*c*d + (2*B*b^2*c*d + (2*B*a*b + A
*b^2)*d^2)*m^4 + 4*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^3 + (2*B*a*b + A*b^2)*d^2 + 6*(2*B*b^2*c*d + (2*B*a
*b + A*b^2)*d^2)*m^2 + 4*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m)*n)*x*x^(4*n)*e^(m*log(e) + m*log(x)) + ((B*b
^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^5 + B*b^2*c^2 + 5*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)
*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^4 + 40*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2 + (B*b^2*c
^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m)*n^4 + 10*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^
2 + 2*A*a*b)*d^2)*m^3 + 78*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2 + (B*b^2*c^2 + 2*(2*B*
a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^2 + 2*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2
)*m)*n^3 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2 + 10*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2
+ 2*A*a*b)*d^2)*m^2 + 49*(B*b^2*c^2 + (B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^3 + 2*(2
*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2 + 3*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*
m^2 + 3*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m)*n^2 + 5*(B*b^2*c^2 + 2*(2*B*a*b + A*b
^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m + 12*(B*b^2*c^2 + (B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*
d^2)*m^4 + 4*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^3 + 2*(2*B*a*b + A*b^2)*c*d + (B*
a^2 + 2*A*a*b)*d^2 + 6*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^2 + 4*(B*b^2*c^2 + 2*(2
*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m)*n)*x*x^(3*n)*e^(m*log(e) + m*log(x)) + ((A*a^2*d^2 + (2*B*a*b
+ A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^5 + A*a^2*d^2 + 5*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*
A*a*b)*c*d)*m^4 + 60*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d + (A*a^2*d^2 + (2*B*a*b + A*
b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m)*n^4 + 10*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*
m^3 + 107*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d + (A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 +
2*(B*a^2 + 2*A*a*b)*c*d)*m^2 + 2*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m)*n^3 + (2*B*a
*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d + 10*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m
^2 + 59*(A*a^2*d^2 + (A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^3 + (2*B*a*b + A*b^2)*c^2
+ 2*(B*a^2 + 2*A*a*b)*c*d + 3*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^2 + 3*(A*a^2*d^
2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m)*n^2 + 5*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2
+ 2*A*a*b)*c*d)*m + 13*(A*a^2*d^2 + (A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^4 + 4*(A*a
^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^3 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*
d + 6*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^2 + 4*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2
+ 2*(B*a^2 + 2*A*a*b)*c*d)*m)*n)*x*x^(2*n)*e^(m*log(e) + m*log(x)) + ((2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m
^5 + 2*A*a^2*c*d + 5*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^4 + 120*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2 + (2
*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m)*n^4 + 10*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^3 + 154*(2*A*a^2*c*d +
(B*a^2 + 2*A*a*b)*c^2 + (2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^2 + 2*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m
)*n^3 + (B*a^2 + 2*A*a*b)*c^2 + 10*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^2 + 71*(2*A*a^2*c*d + (2*A*a^2*c*d
+ (B*a^2 + 2*A*a*b)*c^2)*m^3 + (B*a^2 + 2*A*a*b)*c^2 + 3*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^2 + 3*(2*A*a^
2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m)*n^2 + 5*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m + 14*(2*A*a^2*c*d + (2*A*a^2
*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^4 + 4*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^3 + (B*a^2 + 2*A*a*b)*c^2 + 6*(2
*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^2 + 4*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m)*n)*x*x^n*e^(m*log(e) + m*
log(x)) + (A*a^2*c^2*m^5 + 120*A*a^2*c^2*n^5 + 5*A*a^2*c^2*m^4 + 10*A*a^2*c^2*m^3 + 10*A*a^2*c^2*m^2 + 5*A*a^2
*c^2*m + A*a^2*c^2 + 274*(A*a^2*c^2*m + A*a^2*c^2)*n^4 + 225*(A*a^2*c^2*m^2 + 2*A*a^2*c^2*m + A*a^2*c^2)*n^3 +
85*(A*a^2*c^2*m^3 + 3*A*a^2*c^2*m^2 + 3*A*a^2*c^2*m + A*a^2*c^2)*n^2 + 15*(A*a^2*c^2*m^4 + 4*A*a^2*c^2*m^3 +
6*A*a^2*c^2*m^2 + 4*A*a^2*c^2*m + A*a^2*c^2)*n)*x*e^(m*log(e) + m*log(x)))/(m^6 + 120*(m + 1)*n^5 + 6*m^5 + 27
4*(m^2 + 2*m + 1)*n^4 + 15*m^4 + 225*(m^3 + 3*m^2 + 3*m + 1)*n^3 + 20*m^3 + 85*(m^4 + 4*m^3 + 6*m^2 + 4*m + 1)
*n^2 + 15*m^2 + 15*(m^5 + 5*m^4 + 10*m^3 + 10*m^2 + 5*m + 1)*n + 6*m + 1)

________________________________________________________________________________________

giac [B]  time = 2.01, size = 8103, normalized size = 34.19 \[ \text {result too large to display} \]

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)^2,x, algorithm="giac")

[Out]

(B*b^2*d^2*m^5*x*x^m*x^(5*n)*e^m + 10*B*b^2*d^2*m^4*n*x*x^m*x^(5*n)*e^m + 35*B*b^2*d^2*m^3*n^2*x*x^m*x^(5*n)*e
^m + 50*B*b^2*d^2*m^2*n^3*x*x^m*x^(5*n)*e^m + 24*B*b^2*d^2*m*n^4*x*x^m*x^(5*n)*e^m + 2*B*b^2*c*d*m^5*x*x^m*x^(
4*n)*e^m + 2*B*a*b*d^2*m^5*x*x^m*x^(4*n)*e^m + A*b^2*d^2*m^5*x*x^m*x^(4*n)*e^m + 22*B*b^2*c*d*m^4*n*x*x^m*x^(4
*n)*e^m + 22*B*a*b*d^2*m^4*n*x*x^m*x^(4*n)*e^m + 11*A*b^2*d^2*m^4*n*x*x^m*x^(4*n)*e^m + 82*B*b^2*c*d*m^3*n^2*x
*x^m*x^(4*n)*e^m + 82*B*a*b*d^2*m^3*n^2*x*x^m*x^(4*n)*e^m + 41*A*b^2*d^2*m^3*n^2*x*x^m*x^(4*n)*e^m + 122*B*b^2
*c*d*m^2*n^3*x*x^m*x^(4*n)*e^m + 122*B*a*b*d^2*m^2*n^3*x*x^m*x^(4*n)*e^m + 61*A*b^2*d^2*m^2*n^3*x*x^m*x^(4*n)*
e^m + 60*B*b^2*c*d*m*n^4*x*x^m*x^(4*n)*e^m + 60*B*a*b*d^2*m*n^4*x*x^m*x^(4*n)*e^m + 30*A*b^2*d^2*m*n^4*x*x^m*x
^(4*n)*e^m + B*b^2*c^2*m^5*x*x^m*x^(3*n)*e^m + 4*B*a*b*c*d*m^5*x*x^m*x^(3*n)*e^m + 2*A*b^2*c*d*m^5*x*x^m*x^(3*
n)*e^m + B*a^2*d^2*m^5*x*x^m*x^(3*n)*e^m + 2*A*a*b*d^2*m^5*x*x^m*x^(3*n)*e^m + 12*B*b^2*c^2*m^4*n*x*x^m*x^(3*n
)*e^m + 48*B*a*b*c*d*m^4*n*x*x^m*x^(3*n)*e^m + 24*A*b^2*c*d*m^4*n*x*x^m*x^(3*n)*e^m + 12*B*a^2*d^2*m^4*n*x*x^m
*x^(3*n)*e^m + 24*A*a*b*d^2*m^4*n*x*x^m*x^(3*n)*e^m + 49*B*b^2*c^2*m^3*n^2*x*x^m*x^(3*n)*e^m + 196*B*a*b*c*d*m
^3*n^2*x*x^m*x^(3*n)*e^m + 98*A*b^2*c*d*m^3*n^2*x*x^m*x^(3*n)*e^m + 49*B*a^2*d^2*m^3*n^2*x*x^m*x^(3*n)*e^m + 9
8*A*a*b*d^2*m^3*n^2*x*x^m*x^(3*n)*e^m + 78*B*b^2*c^2*m^2*n^3*x*x^m*x^(3*n)*e^m + 312*B*a*b*c*d*m^2*n^3*x*x^m*x
^(3*n)*e^m + 156*A*b^2*c*d*m^2*n^3*x*x^m*x^(3*n)*e^m + 78*B*a^2*d^2*m^2*n^3*x*x^m*x^(3*n)*e^m + 156*A*a*b*d^2*
m^2*n^3*x*x^m*x^(3*n)*e^m + 40*B*b^2*c^2*m*n^4*x*x^m*x^(3*n)*e^m + 160*B*a*b*c*d*m*n^4*x*x^m*x^(3*n)*e^m + 80*
A*b^2*c*d*m*n^4*x*x^m*x^(3*n)*e^m + 40*B*a^2*d^2*m*n^4*x*x^m*x^(3*n)*e^m + 80*A*a*b*d^2*m*n^4*x*x^m*x^(3*n)*e^
m + 2*B*a*b*c^2*m^5*x*x^m*x^(2*n)*e^m + A*b^2*c^2*m^5*x*x^m*x^(2*n)*e^m + 2*B*a^2*c*d*m^5*x*x^m*x^(2*n)*e^m +
4*A*a*b*c*d*m^5*x*x^m*x^(2*n)*e^m + A*a^2*d^2*m^5*x*x^m*x^(2*n)*e^m + 26*B*a*b*c^2*m^4*n*x*x^m*x^(2*n)*e^m + 1
3*A*b^2*c^2*m^4*n*x*x^m*x^(2*n)*e^m + 26*B*a^2*c*d*m^4*n*x*x^m*x^(2*n)*e^m + 52*A*a*b*c*d*m^4*n*x*x^m*x^(2*n)*
e^m + 13*A*a^2*d^2*m^4*n*x*x^m*x^(2*n)*e^m + 118*B*a*b*c^2*m^3*n^2*x*x^m*x^(2*n)*e^m + 59*A*b^2*c^2*m^3*n^2*x*
x^m*x^(2*n)*e^m + 118*B*a^2*c*d*m^3*n^2*x*x^m*x^(2*n)*e^m + 236*A*a*b*c*d*m^3*n^2*x*x^m*x^(2*n)*e^m + 59*A*a^2
*d^2*m^3*n^2*x*x^m*x^(2*n)*e^m + 214*B*a*b*c^2*m^2*n^3*x*x^m*x^(2*n)*e^m + 107*A*b^2*c^2*m^2*n^3*x*x^m*x^(2*n)
*e^m + 214*B*a^2*c*d*m^2*n^3*x*x^m*x^(2*n)*e^m + 428*A*a*b*c*d*m^2*n^3*x*x^m*x^(2*n)*e^m + 107*A*a^2*d^2*m^2*n
^3*x*x^m*x^(2*n)*e^m + 120*B*a*b*c^2*m*n^4*x*x^m*x^(2*n)*e^m + 60*A*b^2*c^2*m*n^4*x*x^m*x^(2*n)*e^m + 120*B*a^
2*c*d*m*n^4*x*x^m*x^(2*n)*e^m + 240*A*a*b*c*d*m*n^4*x*x^m*x^(2*n)*e^m + 60*A*a^2*d^2*m*n^4*x*x^m*x^(2*n)*e^m +
B*a^2*c^2*m^5*x*x^m*x^n*e^m + 2*A*a*b*c^2*m^5*x*x^m*x^n*e^m + 2*A*a^2*c*d*m^5*x*x^m*x^n*e^m + 14*B*a^2*c^2*m^
4*n*x*x^m*x^n*e^m + 28*A*a*b*c^2*m^4*n*x*x^m*x^n*e^m + 28*A*a^2*c*d*m^4*n*x*x^m*x^n*e^m + 71*B*a^2*c^2*m^3*n^2
*x*x^m*x^n*e^m + 142*A*a*b*c^2*m^3*n^2*x*x^m*x^n*e^m + 142*A*a^2*c*d*m^3*n^2*x*x^m*x^n*e^m + 154*B*a^2*c^2*m^2
*n^3*x*x^m*x^n*e^m + 308*A*a*b*c^2*m^2*n^3*x*x^m*x^n*e^m + 308*A*a^2*c*d*m^2*n^3*x*x^m*x^n*e^m + 120*B*a^2*c^2
*m*n^4*x*x^m*x^n*e^m + 240*A*a*b*c^2*m*n^4*x*x^m*x^n*e^m + 240*A*a^2*c*d*m*n^4*x*x^m*x^n*e^m + A*a^2*c^2*m^5*x
*x^m*e^m + 15*A*a^2*c^2*m^4*n*x*x^m*e^m + 85*A*a^2*c^2*m^3*n^2*x*x^m*e^m + 225*A*a^2*c^2*m^2*n^3*x*x^m*e^m + 2
74*A*a^2*c^2*m*n^4*x*x^m*e^m + 120*A*a^2*c^2*n^5*x*x^m*e^m + 5*B*b^2*d^2*m^4*x*x^m*x^(5*n)*e^m + 40*B*b^2*d^2*
m^3*n*x*x^m*x^(5*n)*e^m + 105*B*b^2*d^2*m^2*n^2*x*x^m*x^(5*n)*e^m + 100*B*b^2*d^2*m*n^3*x*x^m*x^(5*n)*e^m + 24
*B*b^2*d^2*n^4*x*x^m*x^(5*n)*e^m + 10*B*b^2*c*d*m^4*x*x^m*x^(4*n)*e^m + 10*B*a*b*d^2*m^4*x*x^m*x^(4*n)*e^m + 5
*A*b^2*d^2*m^4*x*x^m*x^(4*n)*e^m + 88*B*b^2*c*d*m^3*n*x*x^m*x^(4*n)*e^m + 88*B*a*b*d^2*m^3*n*x*x^m*x^(4*n)*e^m
+ 44*A*b^2*d^2*m^3*n*x*x^m*x^(4*n)*e^m + 246*B*b^2*c*d*m^2*n^2*x*x^m*x^(4*n)*e^m + 246*B*a*b*d^2*m^2*n^2*x*x^
m*x^(4*n)*e^m + 123*A*b^2*d^2*m^2*n^2*x*x^m*x^(4*n)*e^m + 244*B*b^2*c*d*m*n^3*x*x^m*x^(4*n)*e^m + 244*B*a*b*d^
2*m*n^3*x*x^m*x^(4*n)*e^m + 122*A*b^2*d^2*m*n^3*x*x^m*x^(4*n)*e^m + 60*B*b^2*c*d*n^4*x*x^m*x^(4*n)*e^m + 60*B*
a*b*d^2*n^4*x*x^m*x^(4*n)*e^m + 30*A*b^2*d^2*n^4*x*x^m*x^(4*n)*e^m + 5*B*b^2*c^2*m^4*x*x^m*x^(3*n)*e^m + 20*B*
a*b*c*d*m^4*x*x^m*x^(3*n)*e^m + 10*A*b^2*c*d*m^4*x*x^m*x^(3*n)*e^m + 5*B*a^2*d^2*m^4*x*x^m*x^(3*n)*e^m + 10*A*
a*b*d^2*m^4*x*x^m*x^(3*n)*e^m + 48*B*b^2*c^2*m^3*n*x*x^m*x^(3*n)*e^m + 192*B*a*b*c*d*m^3*n*x*x^m*x^(3*n)*e^m +
96*A*b^2*c*d*m^3*n*x*x^m*x^(3*n)*e^m + 48*B*a^2*d^2*m^3*n*x*x^m*x^(3*n)*e^m + 96*A*a*b*d^2*m^3*n*x*x^m*x^(3*n
)*e^m + 147*B*b^2*c^2*m^2*n^2*x*x^m*x^(3*n)*e^m + 588*B*a*b*c*d*m^2*n^2*x*x^m*x^(3*n)*e^m + 294*A*b^2*c*d*m^2*
n^2*x*x^m*x^(3*n)*e^m + 147*B*a^2*d^2*m^2*n^2*x*x^m*x^(3*n)*e^m + 294*A*a*b*d^2*m^2*n^2*x*x^m*x^(3*n)*e^m + 15
6*B*b^2*c^2*m*n^3*x*x^m*x^(3*n)*e^m + 624*B*a*b*c*d*m*n^3*x*x^m*x^(3*n)*e^m + 312*A*b^2*c*d*m*n^3*x*x^m*x^(3*n
)*e^m + 156*B*a^2*d^2*m*n^3*x*x^m*x^(3*n)*e^m + 312*A*a*b*d^2*m*n^3*x*x^m*x^(3*n)*e^m + 40*B*b^2*c^2*n^4*x*x^m
*x^(3*n)*e^m + 160*B*a*b*c*d*n^4*x*x^m*x^(3*n)*e^m + 80*A*b^2*c*d*n^4*x*x^m*x^(3*n)*e^m + 40*B*a^2*d^2*n^4*x*x
^m*x^(3*n)*e^m + 80*A*a*b*d^2*n^4*x*x^m*x^(3*n)*e^m + 10*B*a*b*c^2*m^4*x*x^m*x^(2*n)*e^m + 5*A*b^2*c^2*m^4*x*x
^m*x^(2*n)*e^m + 10*B*a^2*c*d*m^4*x*x^m*x^(2*n)*e^m + 20*A*a*b*c*d*m^4*x*x^m*x^(2*n)*e^m + 5*A*a^2*d^2*m^4*x*x
^m*x^(2*n)*e^m + 104*B*a*b*c^2*m^3*n*x*x^m*x^(2*n)*e^m + 52*A*b^2*c^2*m^3*n*x*x^m*x^(2*n)*e^m + 104*B*a^2*c*d*
m^3*n*x*x^m*x^(2*n)*e^m + 208*A*a*b*c*d*m^3*n*x*x^m*x^(2*n)*e^m + 52*A*a^2*d^2*m^3*n*x*x^m*x^(2*n)*e^m + 354*B
*a*b*c^2*m^2*n^2*x*x^m*x^(2*n)*e^m + 177*A*b^2*c^2*m^2*n^2*x*x^m*x^(2*n)*e^m + 354*B*a^2*c*d*m^2*n^2*x*x^m*x^(
2*n)*e^m + 708*A*a*b*c*d*m^2*n^2*x*x^m*x^(2*n)*e^m + 177*A*a^2*d^2*m^2*n^2*x*x^m*x^(2*n)*e^m + 428*B*a*b*c^2*m
*n^3*x*x^m*x^(2*n)*e^m + 214*A*b^2*c^2*m*n^3*x*x^m*x^(2*n)*e^m + 428*B*a^2*c*d*m*n^3*x*x^m*x^(2*n)*e^m + 856*A
*a*b*c*d*m*n^3*x*x^m*x^(2*n)*e^m + 214*A*a^2*d^2*m*n^3*x*x^m*x^(2*n)*e^m + 120*B*a*b*c^2*n^4*x*x^m*x^(2*n)*e^m
+ 60*A*b^2*c^2*n^4*x*x^m*x^(2*n)*e^m + 120*B*a^2*c*d*n^4*x*x^m*x^(2*n)*e^m + 240*A*a*b*c*d*n^4*x*x^m*x^(2*n)*
e^m + 60*A*a^2*d^2*n^4*x*x^m*x^(2*n)*e^m + 5*B*a^2*c^2*m^4*x*x^m*x^n*e^m + 10*A*a*b*c^2*m^4*x*x^m*x^n*e^m + 10
*A*a^2*c*d*m^4*x*x^m*x^n*e^m + 56*B*a^2*c^2*m^3*n*x*x^m*x^n*e^m + 112*A*a*b*c^2*m^3*n*x*x^m*x^n*e^m + 112*A*a^
2*c*d*m^3*n*x*x^m*x^n*e^m + 213*B*a^2*c^2*m^2*n^2*x*x^m*x^n*e^m + 426*A*a*b*c^2*m^2*n^2*x*x^m*x^n*e^m + 426*A*
a^2*c*d*m^2*n^2*x*x^m*x^n*e^m + 308*B*a^2*c^2*m*n^3*x*x^m*x^n*e^m + 616*A*a*b*c^2*m*n^3*x*x^m*x^n*e^m + 616*A*
a^2*c*d*m*n^3*x*x^m*x^n*e^m + 120*B*a^2*c^2*n^4*x*x^m*x^n*e^m + 240*A*a*b*c^2*n^4*x*x^m*x^n*e^m + 240*A*a^2*c*
d*n^4*x*x^m*x^n*e^m + 5*A*a^2*c^2*m^4*x*x^m*e^m + 60*A*a^2*c^2*m^3*n*x*x^m*e^m + 255*A*a^2*c^2*m^2*n^2*x*x^m*e
^m + 450*A*a^2*c^2*m*n^3*x*x^m*e^m + 274*A*a^2*c^2*n^4*x*x^m*e^m + 10*B*b^2*d^2*m^3*x*x^m*x^(5*n)*e^m + 60*B*b
^2*d^2*m^2*n*x*x^m*x^(5*n)*e^m + 105*B*b^2*d^2*m*n^2*x*x^m*x^(5*n)*e^m + 50*B*b^2*d^2*n^3*x*x^m*x^(5*n)*e^m +
20*B*b^2*c*d*m^3*x*x^m*x^(4*n)*e^m + 20*B*a*b*d^2*m^3*x*x^m*x^(4*n)*e^m + 10*A*b^2*d^2*m^3*x*x^m*x^(4*n)*e^m +
132*B*b^2*c*d*m^2*n*x*x^m*x^(4*n)*e^m + 132*B*a*b*d^2*m^2*n*x*x^m*x^(4*n)*e^m + 66*A*b^2*d^2*m^2*n*x*x^m*x^(4
*n)*e^m + 246*B*b^2*c*d*m*n^2*x*x^m*x^(4*n)*e^m + 246*B*a*b*d^2*m*n^2*x*x^m*x^(4*n)*e^m + 123*A*b^2*d^2*m*n^2*
x*x^m*x^(4*n)*e^m + 122*B*b^2*c*d*n^3*x*x^m*x^(4*n)*e^m + 122*B*a*b*d^2*n^3*x*x^m*x^(4*n)*e^m + 61*A*b^2*d^2*n
^3*x*x^m*x^(4*n)*e^m + 10*B*b^2*c^2*m^3*x*x^m*x^(3*n)*e^m + 40*B*a*b*c*d*m^3*x*x^m*x^(3*n)*e^m + 20*A*b^2*c*d*
m^3*x*x^m*x^(3*n)*e^m + 10*B*a^2*d^2*m^3*x*x^m*x^(3*n)*e^m + 20*A*a*b*d^2*m^3*x*x^m*x^(3*n)*e^m + 72*B*b^2*c^2
*m^2*n*x*x^m*x^(3*n)*e^m + 288*B*a*b*c*d*m^2*n*x*x^m*x^(3*n)*e^m + 144*A*b^2*c*d*m^2*n*x*x^m*x^(3*n)*e^m + 72*
B*a^2*d^2*m^2*n*x*x^m*x^(3*n)*e^m + 144*A*a*b*d^2*m^2*n*x*x^m*x^(3*n)*e^m + 147*B*b^2*c^2*m*n^2*x*x^m*x^(3*n)*
e^m + 588*B*a*b*c*d*m*n^2*x*x^m*x^(3*n)*e^m + 294*A*b^2*c*d*m*n^2*x*x^m*x^(3*n)*e^m + 147*B*a^2*d^2*m*n^2*x*x^
m*x^(3*n)*e^m + 294*A*a*b*d^2*m*n^2*x*x^m*x^(3*n)*e^m + 78*B*b^2*c^2*n^3*x*x^m*x^(3*n)*e^m + 312*B*a*b*c*d*n^3
*x*x^m*x^(3*n)*e^m + 156*A*b^2*c*d*n^3*x*x^m*x^(3*n)*e^m + 78*B*a^2*d^2*n^3*x*x^m*x^(3*n)*e^m + 156*A*a*b*d^2*
n^3*x*x^m*x^(3*n)*e^m + 20*B*a*b*c^2*m^3*x*x^m*x^(2*n)*e^m + 10*A*b^2*c^2*m^3*x*x^m*x^(2*n)*e^m + 20*B*a^2*c*d
*m^3*x*x^m*x^(2*n)*e^m + 40*A*a*b*c*d*m^3*x*x^m*x^(2*n)*e^m + 10*A*a^2*d^2*m^3*x*x^m*x^(2*n)*e^m + 156*B*a*b*c
^2*m^2*n*x*x^m*x^(2*n)*e^m + 78*A*b^2*c^2*m^2*n*x*x^m*x^(2*n)*e^m + 156*B*a^2*c*d*m^2*n*x*x^m*x^(2*n)*e^m + 31
2*A*a*b*c*d*m^2*n*x*x^m*x^(2*n)*e^m + 78*A*a^2*d^2*m^2*n*x*x^m*x^(2*n)*e^m + 354*B*a*b*c^2*m*n^2*x*x^m*x^(2*n)
*e^m + 177*A*b^2*c^2*m*n^2*x*x^m*x^(2*n)*e^m + 354*B*a^2*c*d*m*n^2*x*x^m*x^(2*n)*e^m + 708*A*a*b*c*d*m*n^2*x*x
^m*x^(2*n)*e^m + 177*A*a^2*d^2*m*n^2*x*x^m*x^(2*n)*e^m + 214*B*a*b*c^2*n^3*x*x^m*x^(2*n)*e^m + 107*A*b^2*c^2*n
^3*x*x^m*x^(2*n)*e^m + 214*B*a^2*c*d*n^3*x*x^m*x^(2*n)*e^m + 428*A*a*b*c*d*n^3*x*x^m*x^(2*n)*e^m + 107*A*a^2*d
^2*n^3*x*x^m*x^(2*n)*e^m + 10*B*a^2*c^2*m^3*x*x^m*x^n*e^m + 20*A*a*b*c^2*m^3*x*x^m*x^n*e^m + 20*A*a^2*c*d*m^3*
x*x^m*x^n*e^m + 84*B*a^2*c^2*m^2*n*x*x^m*x^n*e^m + 168*A*a*b*c^2*m^2*n*x*x^m*x^n*e^m + 168*A*a^2*c*d*m^2*n*x*x
^m*x^n*e^m + 213*B*a^2*c^2*m*n^2*x*x^m*x^n*e^m + 426*A*a*b*c^2*m*n^2*x*x^m*x^n*e^m + 426*A*a^2*c*d*m*n^2*x*x^m
*x^n*e^m + 154*B*a^2*c^2*n^3*x*x^m*x^n*e^m + 308*A*a*b*c^2*n^3*x*x^m*x^n*e^m + 308*A*a^2*c*d*n^3*x*x^m*x^n*e^m
+ 10*A*a^2*c^2*m^3*x*x^m*e^m + 90*A*a^2*c^2*m^2*n*x*x^m*e^m + 255*A*a^2*c^2*m*n^2*x*x^m*e^m + 225*A*a^2*c^2*n
^3*x*x^m*e^m + 10*B*b^2*d^2*m^2*x*x^m*x^(5*n)*e^m + 40*B*b^2*d^2*m*n*x*x^m*x^(5*n)*e^m + 35*B*b^2*d^2*n^2*x*x^
m*x^(5*n)*e^m + 20*B*b^2*c*d*m^2*x*x^m*x^(4*n)*e^m + 20*B*a*b*d^2*m^2*x*x^m*x^(4*n)*e^m + 10*A*b^2*d^2*m^2*x*x
^m*x^(4*n)*e^m + 88*B*b^2*c*d*m*n*x*x^m*x^(4*n)*e^m + 88*B*a*b*d^2*m*n*x*x^m*x^(4*n)*e^m + 44*A*b^2*d^2*m*n*x*
x^m*x^(4*n)*e^m + 82*B*b^2*c*d*n^2*x*x^m*x^(4*n)*e^m + 82*B*a*b*d^2*n^2*x*x^m*x^(4*n)*e^m + 41*A*b^2*d^2*n^2*x
*x^m*x^(4*n)*e^m + 10*B*b^2*c^2*m^2*x*x^m*x^(3*n)*e^m + 40*B*a*b*c*d*m^2*x*x^m*x^(3*n)*e^m + 20*A*b^2*c*d*m^2*
x*x^m*x^(3*n)*e^m + 10*B*a^2*d^2*m^2*x*x^m*x^(3*n)*e^m + 20*A*a*b*d^2*m^2*x*x^m*x^(3*n)*e^m + 48*B*b^2*c^2*m*n
*x*x^m*x^(3*n)*e^m + 192*B*a*b*c*d*m*n*x*x^m*x^(3*n)*e^m + 96*A*b^2*c*d*m*n*x*x^m*x^(3*n)*e^m + 48*B*a^2*d^2*m
*n*x*x^m*x^(3*n)*e^m + 96*A*a*b*d^2*m*n*x*x^m*x^(3*n)*e^m + 49*B*b^2*c^2*n^2*x*x^m*x^(3*n)*e^m + 196*B*a*b*c*d
*n^2*x*x^m*x^(3*n)*e^m + 98*A*b^2*c*d*n^2*x*x^m*x^(3*n)*e^m + 49*B*a^2*d^2*n^2*x*x^m*x^(3*n)*e^m + 98*A*a*b*d^
2*n^2*x*x^m*x^(3*n)*e^m + 20*B*a*b*c^2*m^2*x*x^m*x^(2*n)*e^m + 10*A*b^2*c^2*m^2*x*x^m*x^(2*n)*e^m + 20*B*a^2*c
*d*m^2*x*x^m*x^(2*n)*e^m + 40*A*a*b*c*d*m^2*x*x^m*x^(2*n)*e^m + 10*A*a^2*d^2*m^2*x*x^m*x^(2*n)*e^m + 104*B*a*b
*c^2*m*n*x*x^m*x^(2*n)*e^m + 52*A*b^2*c^2*m*n*x*x^m*x^(2*n)*e^m + 104*B*a^2*c*d*m*n*x*x^m*x^(2*n)*e^m + 208*A*
a*b*c*d*m*n*x*x^m*x^(2*n)*e^m + 52*A*a^2*d^2*m*n*x*x^m*x^(2*n)*e^m + 118*B*a*b*c^2*n^2*x*x^m*x^(2*n)*e^m + 59*
A*b^2*c^2*n^2*x*x^m*x^(2*n)*e^m + 118*B*a^2*c*d*n^2*x*x^m*x^(2*n)*e^m + 236*A*a*b*c*d*n^2*x*x^m*x^(2*n)*e^m +
59*A*a^2*d^2*n^2*x*x^m*x^(2*n)*e^m + 10*B*a^2*c^2*m^2*x*x^m*x^n*e^m + 20*A*a*b*c^2*m^2*x*x^m*x^n*e^m + 20*A*a^
2*c*d*m^2*x*x^m*x^n*e^m + 56*B*a^2*c^2*m*n*x*x^m*x^n*e^m + 112*A*a*b*c^2*m*n*x*x^m*x^n*e^m + 112*A*a^2*c*d*m*n
*x*x^m*x^n*e^m + 71*B*a^2*c^2*n^2*x*x^m*x^n*e^m + 142*A*a*b*c^2*n^2*x*x^m*x^n*e^m + 142*A*a^2*c*d*n^2*x*x^m*x^
n*e^m + 10*A*a^2*c^2*m^2*x*x^m*e^m + 60*A*a^2*c^2*m*n*x*x^m*e^m + 85*A*a^2*c^2*n^2*x*x^m*e^m + 5*B*b^2*d^2*m*x
*x^m*x^(5*n)*e^m + 10*B*b^2*d^2*n*x*x^m*x^(5*n)*e^m + 10*B*b^2*c*d*m*x*x^m*x^(4*n)*e^m + 10*B*a*b*d^2*m*x*x^m*
x^(4*n)*e^m + 5*A*b^2*d^2*m*x*x^m*x^(4*n)*e^m + 22*B*b^2*c*d*n*x*x^m*x^(4*n)*e^m + 22*B*a*b*d^2*n*x*x^m*x^(4*n
)*e^m + 11*A*b^2*d^2*n*x*x^m*x^(4*n)*e^m + 5*B*b^2*c^2*m*x*x^m*x^(3*n)*e^m + 20*B*a*b*c*d*m*x*x^m*x^(3*n)*e^m
+ 10*A*b^2*c*d*m*x*x^m*x^(3*n)*e^m + 5*B*a^2*d^2*m*x*x^m*x^(3*n)*e^m + 10*A*a*b*d^2*m*x*x^m*x^(3*n)*e^m + 12*B
*b^2*c^2*n*x*x^m*x^(3*n)*e^m + 48*B*a*b*c*d*n*x*x^m*x^(3*n)*e^m + 24*A*b^2*c*d*n*x*x^m*x^(3*n)*e^m + 12*B*a^2*
d^2*n*x*x^m*x^(3*n)*e^m + 24*A*a*b*d^2*n*x*x^m*x^(3*n)*e^m + 10*B*a*b*c^2*m*x*x^m*x^(2*n)*e^m + 5*A*b^2*c^2*m*
x*x^m*x^(2*n)*e^m + 10*B*a^2*c*d*m*x*x^m*x^(2*n)*e^m + 20*A*a*b*c*d*m*x*x^m*x^(2*n)*e^m + 5*A*a^2*d^2*m*x*x^m*
x^(2*n)*e^m + 26*B*a*b*c^2*n*x*x^m*x^(2*n)*e^m + 13*A*b^2*c^2*n*x*x^m*x^(2*n)*e^m + 26*B*a^2*c*d*n*x*x^m*x^(2*
n)*e^m + 52*A*a*b*c*d*n*x*x^m*x^(2*n)*e^m + 13*A*a^2*d^2*n*x*x^m*x^(2*n)*e^m + 5*B*a^2*c^2*m*x*x^m*x^n*e^m + 1
0*A*a*b*c^2*m*x*x^m*x^n*e^m + 10*A*a^2*c*d*m*x*x^m*x^n*e^m + 14*B*a^2*c^2*n*x*x^m*x^n*e^m + 28*A*a*b*c^2*n*x*x
^m*x^n*e^m + 28*A*a^2*c*d*n*x*x^m*x^n*e^m + 5*A*a^2*c^2*m*x*x^m*e^m + 15*A*a^2*c^2*n*x*x^m*e^m + B*b^2*d^2*x*x
^m*x^(5*n)*e^m + 2*B*b^2*c*d*x*x^m*x^(4*n)*e^m + 2*B*a*b*d^2*x*x^m*x^(4*n)*e^m + A*b^2*d^2*x*x^m*x^(4*n)*e^m +
B*b^2*c^2*x*x^m*x^(3*n)*e^m + 4*B*a*b*c*d*x*x^m*x^(3*n)*e^m + 2*A*b^2*c*d*x*x^m*x^(3*n)*e^m + B*a^2*d^2*x*x^m
*x^(3*n)*e^m + 2*A*a*b*d^2*x*x^m*x^(3*n)*e^m + 2*B*a*b*c^2*x*x^m*x^(2*n)*e^m + A*b^2*c^2*x*x^m*x^(2*n)*e^m + 2
*B*a^2*c*d*x*x^m*x^(2*n)*e^m + 4*A*a*b*c*d*x*x^m*x^(2*n)*e^m + A*a^2*d^2*x*x^m*x^(2*n)*e^m + B*a^2*c^2*x*x^m*x
^n*e^m + 2*A*a*b*c^2*x*x^m*x^n*e^m + 2*A*a^2*c*d*x*x^m*x^n*e^m + A*a^2*c^2*x*x^m*e^m)/(m^6 + 15*m^5*n + 85*m^4
*n^2 + 225*m^3*n^3 + 274*m^2*n^4 + 120*m*n^5 + 6*m^5 + 75*m^4*n + 340*m^3*n^2 + 675*m^2*n^3 + 548*m*n^4 + 120*
n^5 + 15*m^4 + 150*m^3*n + 510*m^2*n^2 + 675*m*n^3 + 274*n^4 + 20*m^3 + 150*m^2*n + 340*m*n^2 + 225*n^3 + 15*m
^2 + 75*m*n + 85*n^2 + 6*m + 15*n + 1)

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maple [C]  time = 0.17, size = 5908, normalized size = 24.93 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [B]  time = 0.91, size = 540, normalized size = 2.28 \[ \frac {B b^{2} d^{2} e^{m} x e^{\left (m \log \relax (x) + 5 \, n \log \relax (x)\right )}}{m + 5 \, n + 1} + \frac {2 \, B b^{2} c d e^{m} x e^{\left (m \log \relax (x) + 4 \, n \log \relax (x)\right )}}{m + 4 \, n + 1} + \frac {2 \, B a b d^{2} e^{m} x e^{\left (m \log \relax (x) + 4 \, n \log \relax (x)\right )}}{m + 4 \, n + 1} + \frac {A b^{2} d^{2} e^{m} x e^{\left (m \log \relax (x) + 4 \, n \log \relax (x)\right )}}{m + 4 \, n + 1} + \frac {B b^{2} c^{2} e^{m} x e^{\left (m \log \relax (x) + 3 \, n \log \relax (x)\right )}}{m + 3 \, n + 1} + \frac {4 \, B a b c d e^{m} x e^{\left (m \log \relax (x) + 3 \, n \log \relax (x)\right )}}{m + 3 \, n + 1} + \frac {2 \, A b^{2} c d e^{m} x e^{\left (m \log \relax (x) + 3 \, n \log \relax (x)\right )}}{m + 3 \, n + 1} + \frac {B a^{2} d^{2} e^{m} x e^{\left (m \log \relax (x) + 3 \, n \log \relax (x)\right )}}{m + 3 \, n + 1} + \frac {2 \, A a b d^{2} e^{m} x e^{\left (m \log \relax (x) + 3 \, n \log \relax (x)\right )}}{m + 3 \, n + 1} + \frac {2 \, B a b c^{2} e^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {A b^{2} c^{2} e^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {2 \, B a^{2} c d e^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {4 \, A a b c d e^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {A a^{2} d^{2} e^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {B a^{2} c^{2} e^{m} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{m + n + 1} + \frac {2 \, A a b c^{2} e^{m} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{m + n + 1} + \frac {2 \, A a^{2} c d e^{m} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{m + n + 1} + \frac {\left (e x\right )^{m + 1} A a^{2} c^{2}}{e {\left (m + 1\right )}} \]

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)^2,x, algorithm="maxima")

[Out]

B*b^2*d^2*e^m*x*e^(m*log(x) + 5*n*log(x))/(m + 5*n + 1) + 2*B*b^2*c*d*e^m*x*e^(m*log(x) + 4*n*log(x))/(m + 4*n
+ 1) + 2*B*a*b*d^2*e^m*x*e^(m*log(x) + 4*n*log(x))/(m + 4*n + 1) + A*b^2*d^2*e^m*x*e^(m*log(x) + 4*n*log(x))/
(m + 4*n + 1) + B*b^2*c^2*e^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + 4*B*a*b*c*d*e^m*x*e^(m*log(x) + 3*n*
log(x))/(m + 3*n + 1) + 2*A*b^2*c*d*e^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + B*a^2*d^2*e^m*x*e^(m*log(x
) + 3*n*log(x))/(m + 3*n + 1) + 2*A*a*b*d^2*e^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + 2*B*a*b*c^2*e^m*x*
e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + A*b^2*c^2*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + 2*B*a^2*c*
d*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + 4*A*a*b*c*d*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) +
A*a^2*d^2*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + B*a^2*c^2*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1)
+ 2*A*a*b*c^2*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + 2*A*a^2*c*d*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1
) + (e*x)^(m + 1)*A*a^2*c^2/(e*(m + 1))

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mupad [B]  time = 5.59, size = 1119, normalized size = 4.72 \[ \frac {x\,x^{2\,n}\,{\left (e\,x\right )}^m\,\left (2\,B\,a^2\,c\,d+A\,a^2\,d^2+2\,B\,a\,b\,c^2+4\,A\,a\,b\,c\,d+A\,b^2\,c^2\right )\,\left (m^4+13\,m^3\,n+4\,m^3+59\,m^2\,n^2+39\,m^2\,n+6\,m^2+107\,m\,n^3+118\,m\,n^2+39\,m\,n+4\,m+60\,n^4+107\,n^3+59\,n^2+13\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {x\,x^{3\,n}\,{\left (e\,x\right )}^m\,\left (B\,a^2\,d^2+4\,B\,a\,b\,c\,d+2\,A\,a\,b\,d^2+B\,b^2\,c^2+2\,A\,b^2\,c\,d\right )\,\left (m^4+12\,m^3\,n+4\,m^3+49\,m^2\,n^2+36\,m^2\,n+6\,m^2+78\,m\,n^3+98\,m\,n^2+36\,m\,n+4\,m+40\,n^4+78\,n^3+49\,n^2+12\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {A\,a^2\,c^2\,x\,{\left (e\,x\right )}^m}{m+1}+\frac {b\,d\,x\,x^{4\,n}\,{\left (e\,x\right )}^m\,\left (A\,b\,d+2\,B\,a\,d+2\,B\,b\,c\right )\,\left (m^4+11\,m^3\,n+4\,m^3+41\,m^2\,n^2+33\,m^2\,n+6\,m^2+61\,m\,n^3+82\,m\,n^2+33\,m\,n+4\,m+30\,n^4+61\,n^3+41\,n^2+11\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {B\,b^2\,d^2\,x\,x^{5\,n}\,{\left (e\,x\right )}^m\,\left (m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {a\,c\,x\,x^n\,{\left (e\,x\right )}^m\,\left (2\,A\,a\,d+2\,A\,b\,c+B\,a\,c\right )\,\left (m^4+14\,m^3\,n+4\,m^3+71\,m^2\,n^2+42\,m^2\,n+6\,m^2+154\,m\,n^3+142\,m\,n^2+42\,m\,n+4\,m+120\,n^4+154\,n^3+71\,n^2+14\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*x^(2*n)*(e*x)^m*(A*a^2*d^2 + A*b^2*c^2 + 2*B*a*b*c^2 + 2*B*a^2*c*d + 4*A*a*b*c*d)*(4*m + 13*n + 39*m*n + 11
8*m*n^2 + 39*m^2*n + 107*m*n^3 + 13*m^3*n + 6*m^2 + 4*m^3 + m^4 + 59*n^2 + 107*n^3 + 60*n^4 + 59*m^2*n^2 + 1))
/(5*m + 15*n + 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 +
5*m^4 + m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1) + (x*x^(3*n)
*(e*x)^m*(B*a^2*d^2 + B*b^2*c^2 + 2*A*a*b*d^2 + 2*A*b^2*c*d + 4*B*a*b*c*d)*(4*m + 12*n + 36*m*n + 98*m*n^2 + 3
6*m^2*n + 78*m*n^3 + 12*m^3*n + 6*m^2 + 4*m^3 + m^4 + 49*n^2 + 78*n^3 + 40*n^4 + 49*m^2*n^2 + 1))/(5*m + 15*n
+ 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5
+ 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1) + (A*a^2*c^2*x*(e*x)^m)/(
m + 1) + (b*d*x*x^(4*n)*(e*x)^m*(A*b*d + 2*B*a*d + 2*B*b*c)*(4*m + 11*n + 33*m*n + 82*m*n^2 + 33*m^2*n + 61*m*
n^3 + 11*m^3*n + 6*m^2 + 4*m^3 + m^4 + 41*n^2 + 61*n^3 + 30*n^4 + 41*m^2*n^2 + 1))/(5*m + 15*n + 60*m*n + 255*
m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*
n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1) + (B*b^2*d^2*x*x^(5*n)*(e*x)^m*(4*m + 10
*n + 30*m*n + 70*m*n^2 + 30*m^2*n + 50*m*n^3 + 10*m^3*n + 6*m^2 + 4*m^3 + m^4 + 35*n^2 + 50*n^3 + 24*n^4 + 35*
m^2*n^2 + 1))/(5*m + 15*n + 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m
^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1)
+ (a*c*x*x^n*(e*x)^m*(2*A*a*d + 2*A*b*c + B*a*c)*(4*m + 14*n + 42*m*n + 142*m*n^2 + 42*m^2*n + 154*m*n^3 + 14
*m^3*n + 6*m^2 + 4*m^3 + m^4 + 71*n^2 + 154*n^3 + 120*n^4 + 71*m^2*n^2 + 1))/(5*m + 15*n + 60*m*n + 255*m*n^2
+ 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*n^3 +
274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**2,x)

[Out]

Timed out

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