3.3181 \(\int (a+b x)^3 (A+B x) (d+e x)^m \, dx\)

Optimal. Leaf size=186 \[ -\frac {b^2 (d+e x)^{m+4} (-3 a B e-A b e+4 b B d)}{e^5 (m+4)}+\frac {(b d-a e)^3 (B d-A e) (d+e x)^{m+1}}{e^5 (m+1)}-\frac {(b d-a e)^2 (d+e x)^{m+2} (-a B e-3 A b e+4 b B d)}{e^5 (m+2)}+\frac {3 b (b d-a e) (d+e x)^{m+3} (-a B e-A b e+2 b B d)}{e^5 (m+3)}+\frac {b^3 B (d+e x)^{m+5}}{e^5 (m+5)} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ -\frac {b^2 (d+e x)^{m+4} (-3 a B e-A b e+4 b B d)}{e^5 (m+4)}+\frac {(b d-a e)^3 (B d-A e) (d+e x)^{m+1}}{e^5 (m+1)}-\frac {(b d-a e)^2 (d+e x)^{m+2} (-a B e-3 A b e+4 b B d)}{e^5 (m+2)}+\frac {3 b (b d-a e) (d+e x)^{m+3} (-a B e-A b e+2 b B d)}{e^5 (m+3)}+\frac {b^3 B (d+e x)^{m+5}}{e^5 (m+5)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3*(A + B*x)*(d + e*x)^m,x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]  time = 0.18, size = 165, normalized size = 0.89 \[ \frac {(d+e x)^{m+1} \left (-\frac {b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)}{m+4}+\frac {3 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)}{m+3}-\frac {(d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{m+2}+\frac {(b d-a e)^3 (B d-A e)}{m+1}+\frac {b^3 B (d+e x)^4}{m+5}\right )}{e^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3*(A + B*x)*(d + e*x)^m,x]

[Out]

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

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fricas [B]  time = 1.02, size = 1282, normalized size = 6.89 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^m,x, algorithm="fricas")

[Out]

(A*a^3*d*e^4*m^4 + 24*B*b^3*d^5 + 120*A*a^3*d*e^4 - 30*(3*B*a*b^2 + A*b^3)*d^4*e + 120*(B*a^2*b + A*a*b^2)*d^3
*e^2 - 60*(B*a^3 + 3*A*a^2*b)*d^2*e^3 + (B*b^3*e^5*m^4 + 10*B*b^3*e^5*m^3 + 35*B*b^3*e^5*m^2 + 50*B*b^3*e^5*m
+ 24*B*b^3*e^5)*x^5 + (30*(3*B*a*b^2 + A*b^3)*e^5 + (B*b^3*d*e^4 + (3*B*a*b^2 + A*b^3)*e^5)*m^4 + (6*B*b^3*d*e
^4 + 11*(3*B*a*b^2 + A*b^3)*e^5)*m^3 + (11*B*b^3*d*e^4 + 41*(3*B*a*b^2 + A*b^3)*e^5)*m^2 + (6*B*b^3*d*e^4 + 61
*(3*B*a*b^2 + A*b^3)*e^5)*m)*x^4 + (14*A*a^3*d*e^4 - (B*a^3 + 3*A*a^2*b)*d^2*e^3)*m^3 + (120*(B*a^2*b + A*a*b^
2)*e^5 + ((3*B*a*b^2 + A*b^3)*d*e^4 + 3*(B*a^2*b + A*a*b^2)*e^5)*m^4 - 4*(B*b^3*d^2*e^3 - 2*(3*B*a*b^2 + A*b^3
)*d*e^4 - 9*(B*a^2*b + A*a*b^2)*e^5)*m^3 - (12*B*b^3*d^2*e^3 - 17*(3*B*a*b^2 + A*b^3)*d*e^4 - 147*(B*a^2*b + A
*a*b^2)*e^5)*m^2 - 2*(4*B*b^3*d^2*e^3 - 5*(3*B*a*b^2 + A*b^3)*d*e^4 - 117*(B*a^2*b + A*a*b^2)*e^5)*m)*x^3 + (7
1*A*a^3*d*e^4 + 6*(B*a^2*b + A*a*b^2)*d^3*e^2 - 12*(B*a^3 + 3*A*a^2*b)*d^2*e^3)*m^2 + (60*(B*a^3 + 3*A*a^2*b)*
e^5 + (3*(B*a^2*b + A*a*b^2)*d*e^4 + (B*a^3 + 3*A*a^2*b)*e^5)*m^4 - (3*(3*B*a*b^2 + A*b^3)*d^2*e^3 - 30*(B*a^2
*b + A*a*b^2)*d*e^4 - 13*(B*a^3 + 3*A*a^2*b)*e^5)*m^3 + (12*B*b^3*d^3*e^2 - 18*(3*B*a*b^2 + A*b^3)*d^2*e^3 + 8
7*(B*a^2*b + A*a*b^2)*d*e^4 + 59*(B*a^3 + 3*A*a^2*b)*e^5)*m^2 + (12*B*b^3*d^3*e^2 - 15*(3*B*a*b^2 + A*b^3)*d^2
*e^3 + 60*(B*a^2*b + A*a*b^2)*d*e^4 + 107*(B*a^3 + 3*A*a^2*b)*e^5)*m)*x^2 + (154*A*a^3*d*e^4 - 6*(3*B*a*b^2 +
A*b^3)*d^4*e + 54*(B*a^2*b + A*a*b^2)*d^3*e^2 - 47*(B*a^3 + 3*A*a^2*b)*d^2*e^3)*m + (120*A*a^3*e^5 + (A*a^3*e^
5 + (B*a^3 + 3*A*a^2*b)*d*e^4)*m^4 + 2*(7*A*a^3*e^5 - 3*(B*a^2*b + A*a*b^2)*d^2*e^3 + 6*(B*a^3 + 3*A*a^2*b)*d*
e^4)*m^3 + (71*A*a^3*e^5 + 6*(3*B*a*b^2 + A*b^3)*d^3*e^2 - 54*(B*a^2*b + A*a*b^2)*d^2*e^3 + 47*(B*a^3 + 3*A*a^
2*b)*d*e^4)*m^2 - 2*(12*B*b^3*d^4*e - 77*A*a^3*e^5 - 15*(3*B*a*b^2 + A*b^3)*d^3*e^2 + 60*(B*a^2*b + A*a*b^2)*d
^2*e^3 - 30*(B*a^3 + 3*A*a^2*b)*d*e^4)*m)*x)*(e*x + d)^m/(e^5*m^5 + 15*e^5*m^4 + 85*e^5*m^3 + 225*e^5*m^2 + 27
4*e^5*m + 120*e^5)

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giac [B]  time = 1.26, size = 2524, normalized size = 13.57 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^m,x, algorithm="giac")

[Out]

((x*e + d)^m*B*b^3*m^4*x^5*e^5 + (x*e + d)^m*B*b^3*d*m^4*x^4*e^4 + 3*(x*e + d)^m*B*a*b^2*m^4*x^4*e^5 + (x*e +
d)^m*A*b^3*m^4*x^4*e^5 + 10*(x*e + d)^m*B*b^3*m^3*x^5*e^5 + 3*(x*e + d)^m*B*a*b^2*d*m^4*x^3*e^4 + (x*e + d)^m*
A*b^3*d*m^4*x^3*e^4 + 6*(x*e + d)^m*B*b^3*d*m^3*x^4*e^4 - 4*(x*e + d)^m*B*b^3*d^2*m^3*x^3*e^3 + 3*(x*e + d)^m*
B*a^2*b*m^4*x^3*e^5 + 3*(x*e + d)^m*A*a*b^2*m^4*x^3*e^5 + 33*(x*e + d)^m*B*a*b^2*m^3*x^4*e^5 + 11*(x*e + d)^m*
A*b^3*m^3*x^4*e^5 + 35*(x*e + d)^m*B*b^3*m^2*x^5*e^5 + 3*(x*e + d)^m*B*a^2*b*d*m^4*x^2*e^4 + 3*(x*e + d)^m*A*a
*b^2*d*m^4*x^2*e^4 + 24*(x*e + d)^m*B*a*b^2*d*m^3*x^3*e^4 + 8*(x*e + d)^m*A*b^3*d*m^3*x^3*e^4 + 11*(x*e + d)^m
*B*b^3*d*m^2*x^4*e^4 - 9*(x*e + d)^m*B*a*b^2*d^2*m^3*x^2*e^3 - 3*(x*e + d)^m*A*b^3*d^2*m^3*x^2*e^3 - 12*(x*e +
d)^m*B*b^3*d^2*m^2*x^3*e^3 + 12*(x*e + d)^m*B*b^3*d^3*m^2*x^2*e^2 + (x*e + d)^m*B*a^3*m^4*x^2*e^5 + 3*(x*e +
d)^m*A*a^2*b*m^4*x^2*e^5 + 36*(x*e + d)^m*B*a^2*b*m^3*x^3*e^5 + 36*(x*e + d)^m*A*a*b^2*m^3*x^3*e^5 + 123*(x*e
+ d)^m*B*a*b^2*m^2*x^4*e^5 + 41*(x*e + d)^m*A*b^3*m^2*x^4*e^5 + 50*(x*e + d)^m*B*b^3*m*x^5*e^5 + (x*e + d)^m*B
*a^3*d*m^4*x*e^4 + 3*(x*e + d)^m*A*a^2*b*d*m^4*x*e^4 + 30*(x*e + d)^m*B*a^2*b*d*m^3*x^2*e^4 + 30*(x*e + d)^m*A
*a*b^2*d*m^3*x^2*e^4 + 51*(x*e + d)^m*B*a*b^2*d*m^2*x^3*e^4 + 17*(x*e + d)^m*A*b^3*d*m^2*x^3*e^4 + 6*(x*e + d)
^m*B*b^3*d*m*x^4*e^4 - 6*(x*e + d)^m*B*a^2*b*d^2*m^3*x*e^3 - 6*(x*e + d)^m*A*a*b^2*d^2*m^3*x*e^3 - 54*(x*e + d
)^m*B*a*b^2*d^2*m^2*x^2*e^3 - 18*(x*e + d)^m*A*b^3*d^2*m^2*x^2*e^3 - 8*(x*e + d)^m*B*b^3*d^2*m*x^3*e^3 + 18*(x
*e + d)^m*B*a*b^2*d^3*m^2*x*e^2 + 6*(x*e + d)^m*A*b^3*d^3*m^2*x*e^2 + 12*(x*e + d)^m*B*b^3*d^3*m*x^2*e^2 - 24*
(x*e + d)^m*B*b^3*d^4*m*x*e + (x*e + d)^m*A*a^3*m^4*x*e^5 + 13*(x*e + d)^m*B*a^3*m^3*x^2*e^5 + 39*(x*e + d)^m*
A*a^2*b*m^3*x^2*e^5 + 147*(x*e + d)^m*B*a^2*b*m^2*x^3*e^5 + 147*(x*e + d)^m*A*a*b^2*m^2*x^3*e^5 + 183*(x*e + d
)^m*B*a*b^2*m*x^4*e^5 + 61*(x*e + d)^m*A*b^3*m*x^4*e^5 + 24*(x*e + d)^m*B*b^3*x^5*e^5 + (x*e + d)^m*A*a^3*d*m^
4*e^4 + 12*(x*e + d)^m*B*a^3*d*m^3*x*e^4 + 36*(x*e + d)^m*A*a^2*b*d*m^3*x*e^4 + 87*(x*e + d)^m*B*a^2*b*d*m^2*x
^2*e^4 + 87*(x*e + d)^m*A*a*b^2*d*m^2*x^2*e^4 + 30*(x*e + d)^m*B*a*b^2*d*m*x^3*e^4 + 10*(x*e + d)^m*A*b^3*d*m*
x^3*e^4 - (x*e + d)^m*B*a^3*d^2*m^3*e^3 - 3*(x*e + d)^m*A*a^2*b*d^2*m^3*e^3 - 54*(x*e + d)^m*B*a^2*b*d^2*m^2*x
*e^3 - 54*(x*e + d)^m*A*a*b^2*d^2*m^2*x*e^3 - 45*(x*e + d)^m*B*a*b^2*d^2*m*x^2*e^3 - 15*(x*e + d)^m*A*b^3*d^2*
m*x^2*e^3 + 6*(x*e + d)^m*B*a^2*b*d^3*m^2*e^2 + 6*(x*e + d)^m*A*a*b^2*d^3*m^2*e^2 + 90*(x*e + d)^m*B*a*b^2*d^3
*m*x*e^2 + 30*(x*e + d)^m*A*b^3*d^3*m*x*e^2 - 18*(x*e + d)^m*B*a*b^2*d^4*m*e - 6*(x*e + d)^m*A*b^3*d^4*m*e + 2
4*(x*e + d)^m*B*b^3*d^5 + 14*(x*e + d)^m*A*a^3*m^3*x*e^5 + 59*(x*e + d)^m*B*a^3*m^2*x^2*e^5 + 177*(x*e + d)^m*
A*a^2*b*m^2*x^2*e^5 + 234*(x*e + d)^m*B*a^2*b*m*x^3*e^5 + 234*(x*e + d)^m*A*a*b^2*m*x^3*e^5 + 90*(x*e + d)^m*B
*a*b^2*x^4*e^5 + 30*(x*e + d)^m*A*b^3*x^4*e^5 + 14*(x*e + d)^m*A*a^3*d*m^3*e^4 + 47*(x*e + d)^m*B*a^3*d*m^2*x*
e^4 + 141*(x*e + d)^m*A*a^2*b*d*m^2*x*e^4 + 60*(x*e + d)^m*B*a^2*b*d*m*x^2*e^4 + 60*(x*e + d)^m*A*a*b^2*d*m*x^
2*e^4 - 12*(x*e + d)^m*B*a^3*d^2*m^2*e^3 - 36*(x*e + d)^m*A*a^2*b*d^2*m^2*e^3 - 120*(x*e + d)^m*B*a^2*b*d^2*m*
x*e^3 - 120*(x*e + d)^m*A*a*b^2*d^2*m*x*e^3 + 54*(x*e + d)^m*B*a^2*b*d^3*m*e^2 + 54*(x*e + d)^m*A*a*b^2*d^3*m*
e^2 - 90*(x*e + d)^m*B*a*b^2*d^4*e - 30*(x*e + d)^m*A*b^3*d^4*e + 71*(x*e + d)^m*A*a^3*m^2*x*e^5 + 107*(x*e +
d)^m*B*a^3*m*x^2*e^5 + 321*(x*e + d)^m*A*a^2*b*m*x^2*e^5 + 120*(x*e + d)^m*B*a^2*b*x^3*e^5 + 120*(x*e + d)^m*A
*a*b^2*x^3*e^5 + 71*(x*e + d)^m*A*a^3*d*m^2*e^4 + 60*(x*e + d)^m*B*a^3*d*m*x*e^4 + 180*(x*e + d)^m*A*a^2*b*d*m
*x*e^4 - 47*(x*e + d)^m*B*a^3*d^2*m*e^3 - 141*(x*e + d)^m*A*a^2*b*d^2*m*e^3 + 120*(x*e + d)^m*B*a^2*b*d^3*e^2
+ 120*(x*e + d)^m*A*a*b^2*d^3*e^2 + 154*(x*e + d)^m*A*a^3*m*x*e^5 + 60*(x*e + d)^m*B*a^3*x^2*e^5 + 180*(x*e +
d)^m*A*a^2*b*x^2*e^5 + 154*(x*e + d)^m*A*a^3*d*m*e^4 - 60*(x*e + d)^m*B*a^3*d^2*e^3 - 180*(x*e + d)^m*A*a^2*b*
d^2*e^3 + 120*(x*e + d)^m*A*a^3*x*e^5 + 120*(x*e + d)^m*A*a^3*d*e^4)/(m^5*e^5 + 15*m^4*e^5 + 85*m^3*e^5 + 225*
m^2*e^5 + 274*m*e^5 + 120*e^5)

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maple [B]  time = 0.01, size = 1270, normalized size = 6.83 \[ \frac {\left (B \,b^{3} e^{4} m^{4} x^{4}+A \,b^{3} e^{4} m^{4} x^{3}+3 B a \,b^{2} e^{4} m^{4} x^{3}+10 B \,b^{3} e^{4} m^{3} x^{4}+3 A a \,b^{2} e^{4} m^{4} x^{2}+11 A \,b^{3} e^{4} m^{3} x^{3}+3 B \,a^{2} b \,e^{4} m^{4} x^{2}+33 B a \,b^{2} e^{4} m^{3} x^{3}-4 B \,b^{3} d \,e^{3} m^{3} x^{3}+35 B \,b^{3} e^{4} m^{2} x^{4}+3 A \,a^{2} b \,e^{4} m^{4} x +36 A a \,b^{2} e^{4} m^{3} x^{2}-3 A \,b^{3} d \,e^{3} m^{3} x^{2}+41 A \,b^{3} e^{4} m^{2} x^{3}+B \,a^{3} e^{4} m^{4} x +36 B \,a^{2} b \,e^{4} m^{3} x^{2}-9 B a \,b^{2} d \,e^{3} m^{3} x^{2}+123 B a \,b^{2} e^{4} m^{2} x^{3}-24 B \,b^{3} d \,e^{3} m^{2} x^{3}+50 B \,b^{3} e^{4} m \,x^{4}+A \,a^{3} e^{4} m^{4}+39 A \,a^{2} b \,e^{4} m^{3} x -6 A a \,b^{2} d \,e^{3} m^{3} x +147 A a \,b^{2} e^{4} m^{2} x^{2}-24 A \,b^{3} d \,e^{3} m^{2} x^{2}+61 A \,b^{3} e^{4} m \,x^{3}+13 B \,a^{3} e^{4} m^{3} x -6 B \,a^{2} b d \,e^{3} m^{3} x +147 B \,a^{2} b \,e^{4} m^{2} x^{2}-72 B a \,b^{2} d \,e^{3} m^{2} x^{2}+183 B a \,b^{2} e^{4} m \,x^{3}+12 B \,b^{3} d^{2} e^{2} m^{2} x^{2}-44 B \,b^{3} d \,e^{3} m \,x^{3}+24 B \,b^{3} x^{4} e^{4}+14 A \,a^{3} e^{4} m^{3}-3 A \,a^{2} b d \,e^{3} m^{3}+177 A \,a^{2} b \,e^{4} m^{2} x -60 A a \,b^{2} d \,e^{3} m^{2} x +234 A a \,b^{2} e^{4} m \,x^{2}+6 A \,b^{3} d^{2} e^{2} m^{2} x -51 A \,b^{3} d \,e^{3} m \,x^{2}+30 A \,b^{3} e^{4} x^{3}-B \,a^{3} d \,e^{3} m^{3}+59 B \,a^{3} e^{4} m^{2} x -60 B \,a^{2} b d \,e^{3} m^{2} x +234 B \,a^{2} b \,e^{4} m \,x^{2}+18 B a \,b^{2} d^{2} e^{2} m^{2} x -153 B a \,b^{2} d \,e^{3} m \,x^{2}+90 B a \,b^{2} e^{4} x^{3}+36 B \,b^{3} d^{2} e^{2} m \,x^{2}-24 B \,b^{3} d \,e^{3} x^{3}+71 A \,a^{3} e^{4} m^{2}-36 A \,a^{2} b d \,e^{3} m^{2}+321 A \,a^{2} b \,e^{4} m x +6 A a \,b^{2} d^{2} e^{2} m^{2}-174 A a \,b^{2} d \,e^{3} m x +120 A a \,b^{2} e^{4} x^{2}+36 A \,b^{3} d^{2} e^{2} m x -30 A \,b^{3} d \,e^{3} x^{2}-12 B \,a^{3} d \,e^{3} m^{2}+107 B \,a^{3} e^{4} m x +6 B \,a^{2} b \,d^{2} e^{2} m^{2}-174 B \,a^{2} b d \,e^{3} m x +120 B \,a^{2} b \,e^{4} x^{2}+108 B a \,b^{2} d^{2} e^{2} m x -90 B a \,b^{2} d \,e^{3} x^{2}-24 B \,b^{3} d^{3} e m x +24 B \,b^{3} d^{2} e^{2} x^{2}+154 A \,a^{3} e^{4} m -141 A \,a^{2} b d \,e^{3} m +180 A \,a^{2} b \,e^{4} x +54 A a \,b^{2} d^{2} e^{2} m -120 A a \,b^{2} d \,e^{3} x -6 A \,b^{3} d^{3} e m +30 A \,b^{3} d^{2} e^{2} x -47 B \,a^{3} d \,e^{3} m +60 B \,a^{3} e^{4} x +54 B \,a^{2} b \,d^{2} e^{2} m -120 B \,a^{2} b d \,e^{3} x -18 B a \,b^{2} d^{3} e m +90 B a \,b^{2} d^{2} e^{2} x -24 B \,b^{3} d^{3} e x +120 A \,a^{3} e^{4}-180 A \,a^{2} b d \,e^{3}+120 A a \,b^{2} d^{2} e^{2}-30 A \,b^{3} d^{3} e -60 B \,a^{3} d \,e^{3}+120 B \,a^{2} b \,d^{2} e^{2}-90 B a \,b^{2} d^{3} e +24 B \,b^{3} d^{4}\right ) \left (e x +d \right )^{m +1}}{\left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right ) e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)*(e*x+d)^m,x)

[Out]

(e*x+d)^(m+1)*(B*b^3*e^4*m^4*x^4+A*b^3*e^4*m^4*x^3+3*B*a*b^2*e^4*m^4*x^3+10*B*b^3*e^4*m^3*x^4+3*A*a*b^2*e^4*m^
4*x^2+11*A*b^3*e^4*m^3*x^3+3*B*a^2*b*e^4*m^4*x^2+33*B*a*b^2*e^4*m^3*x^3-4*B*b^3*d*e^3*m^3*x^3+35*B*b^3*e^4*m^2
*x^4+3*A*a^2*b*e^4*m^4*x+36*A*a*b^2*e^4*m^3*x^2-3*A*b^3*d*e^3*m^3*x^2+41*A*b^3*e^4*m^2*x^3+B*a^3*e^4*m^4*x+36*
B*a^2*b*e^4*m^3*x^2-9*B*a*b^2*d*e^3*m^3*x^2+123*B*a*b^2*e^4*m^2*x^3-24*B*b^3*d*e^3*m^2*x^3+50*B*b^3*e^4*m*x^4+
A*a^3*e^4*m^4+39*A*a^2*b*e^4*m^3*x-6*A*a*b^2*d*e^3*m^3*x+147*A*a*b^2*e^4*m^2*x^2-24*A*b^3*d*e^3*m^2*x^2+61*A*b
^3*e^4*m*x^3+13*B*a^3*e^4*m^3*x-6*B*a^2*b*d*e^3*m^3*x+147*B*a^2*b*e^4*m^2*x^2-72*B*a*b^2*d*e^3*m^2*x^2+183*B*a
*b^2*e^4*m*x^3+12*B*b^3*d^2*e^2*m^2*x^2-44*B*b^3*d*e^3*m*x^3+24*B*b^3*e^4*x^4+14*A*a^3*e^4*m^3-3*A*a^2*b*d*e^3
*m^3+177*A*a^2*b*e^4*m^2*x-60*A*a*b^2*d*e^3*m^2*x+234*A*a*b^2*e^4*m*x^2+6*A*b^3*d^2*e^2*m^2*x-51*A*b^3*d*e^3*m
*x^2+30*A*b^3*e^4*x^3-B*a^3*d*e^3*m^3+59*B*a^3*e^4*m^2*x-60*B*a^2*b*d*e^3*m^2*x+234*B*a^2*b*e^4*m*x^2+18*B*a*b
^2*d^2*e^2*m^2*x-153*B*a*b^2*d*e^3*m*x^2+90*B*a*b^2*e^4*x^3+36*B*b^3*d^2*e^2*m*x^2-24*B*b^3*d*e^3*x^3+71*A*a^3
*e^4*m^2-36*A*a^2*b*d*e^3*m^2+321*A*a^2*b*e^4*m*x+6*A*a*b^2*d^2*e^2*m^2-174*A*a*b^2*d*e^3*m*x+120*A*a*b^2*e^4*
x^2+36*A*b^3*d^2*e^2*m*x-30*A*b^3*d*e^3*x^2-12*B*a^3*d*e^3*m^2+107*B*a^3*e^4*m*x+6*B*a^2*b*d^2*e^2*m^2-174*B*a
^2*b*d*e^3*m*x+120*B*a^2*b*e^4*x^2+108*B*a*b^2*d^2*e^2*m*x-90*B*a*b^2*d*e^3*x^2-24*B*b^3*d^3*e*m*x+24*B*b^3*d^
2*e^2*x^2+154*A*a^3*e^4*m-141*A*a^2*b*d*e^3*m+180*A*a^2*b*e^4*x+54*A*a*b^2*d^2*e^2*m-120*A*a*b^2*d*e^3*x-6*A*b
^3*d^3*e*m+30*A*b^3*d^2*e^2*x-47*B*a^3*d*e^3*m+60*B*a^3*e^4*x+54*B*a^2*b*d^2*e^2*m-120*B*a^2*b*d*e^3*x-18*B*a*
b^2*d^3*e*m+90*B*a*b^2*d^2*e^2*x-24*B*b^3*d^3*e*x+120*A*a^3*e^4-180*A*a^2*b*d*e^3+120*A*a*b^2*d^2*e^2-30*A*b^3
*d^3*e-60*B*a^3*d*e^3+120*B*a^2*b*d^2*e^2-90*B*a*b^2*d^3*e+24*B*b^3*d^4)/e^5/(m^5+15*m^4+85*m^3+225*m^2+274*m+
120)

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maxima [B]  time = 0.56, size = 620, normalized size = 3.33 \[ \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} B a^{3}}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {3 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} A a^{2} b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} A a^{3}}{e {\left (m + 1\right )}} + \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} B a^{2} b}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} A a b^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {3 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} B a b^{2}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} A b^{3}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} B b^{3}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^m,x, algorithm="maxima")

[Out]

(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*B*a^3/((m^2 + 3*m + 2)*e^2) + 3*(e^2*(m + 1)*x^2 + d*e*m*x - d^2
)*(e*x + d)^m*A*a^2*b/((m^2 + 3*m + 2)*e^2) + (e*x + d)^(m + 1)*A*a^3/(e*(m + 1)) + 3*((m^2 + 3*m + 2)*e^3*x^3
+ (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*B*a^2*b/((m^3 + 6*m^2 + 11*m + 6)*e^3) + 3*((m^2 + 3
*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*A*a*b^2/((m^3 + 6*m^2 + 11*m + 6)*e^3
) + 3*((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*
x - 6*d^4)*(e*x + d)^m*B*a*b^2/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 +
(m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*A*b^3/((m^4 + 10*m
^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*
e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d)^m*B*
b^3/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5)

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mupad [B]  time = 3.03, size = 1273, normalized size = 6.84 \[ \frac {{\left (d+e\,x\right )}^m\,\left (-B\,a^3\,d^2\,e^3\,m^3-12\,B\,a^3\,d^2\,e^3\,m^2-47\,B\,a^3\,d^2\,e^3\,m-60\,B\,a^3\,d^2\,e^3+A\,a^3\,d\,e^4\,m^4+14\,A\,a^3\,d\,e^4\,m^3+71\,A\,a^3\,d\,e^4\,m^2+154\,A\,a^3\,d\,e^4\,m+120\,A\,a^3\,d\,e^4+6\,B\,a^2\,b\,d^3\,e^2\,m^2+54\,B\,a^2\,b\,d^3\,e^2\,m+120\,B\,a^2\,b\,d^3\,e^2-3\,A\,a^2\,b\,d^2\,e^3\,m^3-36\,A\,a^2\,b\,d^2\,e^3\,m^2-141\,A\,a^2\,b\,d^2\,e^3\,m-180\,A\,a^2\,b\,d^2\,e^3-18\,B\,a\,b^2\,d^4\,e\,m-90\,B\,a\,b^2\,d^4\,e+6\,A\,a\,b^2\,d^3\,e^2\,m^2+54\,A\,a\,b^2\,d^3\,e^2\,m+120\,A\,a\,b^2\,d^3\,e^2+24\,B\,b^3\,d^5-6\,A\,b^3\,d^4\,e\,m-30\,A\,b^3\,d^4\,e\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (B\,a^3\,d\,e^4\,m^4+12\,B\,a^3\,d\,e^4\,m^3+47\,B\,a^3\,d\,e^4\,m^2+60\,B\,a^3\,d\,e^4\,m+A\,a^3\,e^5\,m^4+14\,A\,a^3\,e^5\,m^3+71\,A\,a^3\,e^5\,m^2+154\,A\,a^3\,e^5\,m+120\,A\,a^3\,e^5-6\,B\,a^2\,b\,d^2\,e^3\,m^3-54\,B\,a^2\,b\,d^2\,e^3\,m^2-120\,B\,a^2\,b\,d^2\,e^3\,m+3\,A\,a^2\,b\,d\,e^4\,m^4+36\,A\,a^2\,b\,d\,e^4\,m^3+141\,A\,a^2\,b\,d\,e^4\,m^2+180\,A\,a^2\,b\,d\,e^4\,m+18\,B\,a\,b^2\,d^3\,e^2\,m^2+90\,B\,a\,b^2\,d^3\,e^2\,m-6\,A\,a\,b^2\,d^2\,e^3\,m^3-54\,A\,a\,b^2\,d^2\,e^3\,m^2-120\,A\,a\,b^2\,d^2\,e^3\,m-24\,B\,b^3\,d^4\,e\,m+6\,A\,b^3\,d^3\,e^2\,m^2+30\,A\,b^3\,d^3\,e^2\,m\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (B\,a^3\,e^3\,m^3+12\,B\,a^3\,e^3\,m^2+47\,B\,a^3\,e^3\,m+60\,B\,a^3\,e^3+3\,B\,a^2\,b\,d\,e^2\,m^3+27\,B\,a^2\,b\,d\,e^2\,m^2+60\,B\,a^2\,b\,d\,e^2\,m+3\,A\,a^2\,b\,e^3\,m^3+36\,A\,a^2\,b\,e^3\,m^2+141\,A\,a^2\,b\,e^3\,m+180\,A\,a^2\,b\,e^3-9\,B\,a\,b^2\,d^2\,e\,m^2-45\,B\,a\,b^2\,d^2\,e\,m+3\,A\,a\,b^2\,d\,e^2\,m^3+27\,A\,a\,b^2\,d\,e^2\,m^2+60\,A\,a\,b^2\,d\,e^2\,m+12\,B\,b^3\,d^3\,m-3\,A\,b^3\,d^2\,e\,m^2-15\,A\,b^3\,d^2\,e\,m\right )}{e^3\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {B\,b^3\,x^5\,{\left (d+e\,x\right )}^m\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {b^2\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )\,\left (5\,A\,b\,e+15\,B\,a\,e+A\,b\,e\,m+3\,B\,a\,e\,m+B\,b\,d\,m\right )}{e\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {b\,x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (3\,B\,a^2\,e^2\,m^2+27\,B\,a^2\,e^2\,m+60\,B\,a^2\,e^2+3\,B\,a\,b\,d\,e\,m^2+15\,B\,a\,b\,d\,e\,m+3\,A\,a\,b\,e^2\,m^2+27\,A\,a\,b\,e^2\,m+60\,A\,a\,b\,e^2-4\,B\,b^2\,d^2\,m+A\,b^2\,d\,e\,m^2+5\,A\,b^2\,d\,e\,m\right )}{e^2\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^3*(d + e*x)^m,x)

[Out]

((d + e*x)^m*(24*B*b^3*d^5 + 120*A*a^3*d*e^4 - 30*A*b^3*d^4*e - 60*B*a^3*d^2*e^3 + 120*A*a*b^2*d^3*e^2 - 180*A
*a^2*b*d^2*e^3 + 120*B*a^2*b*d^3*e^2 + 71*A*a^3*d*e^4*m^2 + 14*A*a^3*d*e^4*m^3 + A*a^3*d*e^4*m^4 - 47*B*a^3*d^
2*e^3*m - 12*B*a^3*d^2*e^3*m^2 - B*a^3*d^2*e^3*m^3 - 90*B*a*b^2*d^4*e + 154*A*a^3*d*e^4*m - 6*A*b^3*d^4*e*m +
6*A*a*b^2*d^3*e^2*m^2 - 36*A*a^2*b*d^2*e^3*m^2 - 3*A*a^2*b*d^2*e^3*m^3 + 6*B*a^2*b*d^3*e^2*m^2 - 18*B*a*b^2*d^
4*e*m + 54*A*a*b^2*d^3*e^2*m - 141*A*a^2*b*d^2*e^3*m + 54*B*a^2*b*d^3*e^2*m))/(e^5*(274*m + 225*m^2 + 85*m^3 +
15*m^4 + m^5 + 120)) + (x*(d + e*x)^m*(120*A*a^3*e^5 + 154*A*a^3*e^5*m + 71*A*a^3*e^5*m^2 + 14*A*a^3*e^5*m^3
+ A*a^3*e^5*m^4 + 30*A*b^3*d^3*e^2*m + 47*B*a^3*d*e^4*m^2 + 12*B*a^3*d*e^4*m^3 + B*a^3*d*e^4*m^4 + 6*A*b^3*d^3
*e^2*m^2 + 60*B*a^3*d*e^4*m - 24*B*b^3*d^4*e*m - 54*A*a*b^2*d^2*e^3*m^2 - 6*A*a*b^2*d^2*e^3*m^3 + 18*B*a*b^2*d
^3*e^2*m^2 - 54*B*a^2*b*d^2*e^3*m^2 - 6*B*a^2*b*d^2*e^3*m^3 + 180*A*a^2*b*d*e^4*m - 120*A*a*b^2*d^2*e^3*m + 14
1*A*a^2*b*d*e^4*m^2 + 36*A*a^2*b*d*e^4*m^3 + 3*A*a^2*b*d*e^4*m^4 + 90*B*a*b^2*d^3*e^2*m - 120*B*a^2*b*d^2*e^3*
m))/(e^5*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (x^2*(m + 1)*(d + e*x)^m*(60*B*a^3*e^3 + 180*A*a^2
*b*e^3 + 47*B*a^3*e^3*m + 12*B*b^3*d^3*m + 12*B*a^3*e^3*m^2 + B*a^3*e^3*m^3 + 36*A*a^2*b*e^3*m^2 + 3*A*a^2*b*e
^3*m^3 - 3*A*b^3*d^2*e*m^2 + 141*A*a^2*b*e^3*m - 15*A*b^3*d^2*e*m + 60*A*a*b^2*d*e^2*m - 45*B*a*b^2*d^2*e*m +
60*B*a^2*b*d*e^2*m + 27*A*a*b^2*d*e^2*m^2 + 3*A*a*b^2*d*e^2*m^3 - 9*B*a*b^2*d^2*e*m^2 + 27*B*a^2*b*d*e^2*m^2 +
3*B*a^2*b*d*e^2*m^3))/(e^3*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (B*b^3*x^5*(d + e*x)^m*(50*m +
35*m^2 + 10*m^3 + m^4 + 24))/(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120) + (b^2*x^4*(d + e*x)^m*(11*m + 6*
m^2 + m^3 + 6)*(5*A*b*e + 15*B*a*e + A*b*e*m + 3*B*a*e*m + B*b*d*m))/(e*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m
^5 + 120)) + (b*x^3*(d + e*x)^m*(3*m + m^2 + 2)*(60*B*a^2*e^2 + 60*A*a*b*e^2 + 27*B*a^2*e^2*m - 4*B*b^2*d^2*m
+ 3*B*a^2*e^2*m^2 + 27*A*a*b*e^2*m + 5*A*b^2*d*e*m + 3*A*a*b*e^2*m^2 + A*b^2*d*e*m^2 + 15*B*a*b*d*e*m + 3*B*a*
b*d*e*m^2))/(e^2*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))

________________________________________________________________________________________

sympy [A]  time = 14.47, size = 14256, normalized size = 76.65 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)*(e*x+d)**m,x)

[Out]

Piecewise((d**m*(A*a**3*x + 3*A*a**2*b*x**2/2 + A*a*b**2*x**3 + A*b**3*x**4/4 + B*a**3*x**2/2 + B*a**2*b*x**3
+ 3*B*a*b**2*x**4/4 + B*b**3*x**5/5), Eq(e, 0)), (-3*A*a**3*e**4/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7
*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 3*A*a**2*b*d*e**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2
+ 48*d*e**8*x**3 + 12*e**9*x**4) - 12*A*a**2*b*e**4*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*
d*e**8*x**3 + 12*e**9*x**4) - 3*A*a*b**2*d**2*e**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e
**8*x**3 + 12*e**9*x**4) - 12*A*a*b**2*d*e**3*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8
*x**3 + 12*e**9*x**4) - 18*A*a*b**2*e**4*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x
**3 + 12*e**9*x**4) - 3*A*b**3*d**3*e/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12
*e**9*x**4) - 12*A*b**3*d**2*e**2*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e
**9*x**4) - 18*A*b**3*d*e**3*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**
9*x**4) - 12*A*b**3*e**4*x**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x*
*4) - B*a**3*d*e**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 4*B*
a**3*e**4*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 3*B*a**2*b*d
**2*e**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 12*B*a**2*b*d*e
**3*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 18*B*a**2*b*e**4*x
**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 9*B*a*b**2*d**3*e/(1
2*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 36*B*a*b**2*d**2*e**2*x/(1
2*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 54*B*a*b**2*d*e**3*x**2/(1
2*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 36*B*a*b**2*e**4*x**3/(12*
d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 12*B*b**3*d**4*log(d/e + x)/
(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 25*B*b**3*d**4/(12*d**4*
e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 48*B*b**3*d**3*e*x*log(d/e + x)/(
12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 88*B*b**3*d**3*e*x/(12*d*
*4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 72*B*b**3*d**2*e**2*x**2*log(d
/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 108*B*b**3*d**2*
e**2*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 48*B*b**3*d*e*
*3*x**3*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 48*
B*b**3*d*e**3*x**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 12*B*
b**3*e**4*x**4*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4
), Eq(m, -5)), (-2*A*a**3*e**4/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 3*A*a**2*b*d*e*
*3/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 9*A*a**2*b*e**4*x/(6*d**3*e**5 + 18*d**2*e*
*6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 6*A*a*b**2*d**2*e**2/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6
*e**8*x**3) - 18*A*a*b**2*d*e**3*x/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 18*A*a*b**2
*e**4*x**2/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 6*A*b**3*d**3*e*log(d/e + x)/(6*d**
3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 11*A*b**3*d**3*e/(6*d**3*e**5 + 18*d**2*e**6*x + 18*
d*e**7*x**2 + 6*e**8*x**3) + 18*A*b**3*d**2*e**2*x*log(d/e + x)/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2
+ 6*e**8*x**3) + 27*A*b**3*d**2*e**2*x/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 18*A*b
**3*d*e**3*x**2*log(d/e + x)/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 18*A*b**3*d*e**3*
x**2/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 6*A*b**3*e**4*x**3*log(d/e + x)/(6*d**3*e
**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - B*a**3*d*e**3/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7
*x**2 + 6*e**8*x**3) - 3*B*a**3*e**4*x/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 6*B*a**
2*b*d**2*e**2/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 18*B*a**2*b*d*e**3*x/(6*d**3*e**
5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 18*B*a**2*b*e**4*x**2/(6*d**3*e**5 + 18*d**2*e**6*x + 18*
d*e**7*x**2 + 6*e**8*x**3) + 18*B*a*b**2*d**3*e*log(d/e + x)/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 +
6*e**8*x**3) + 33*B*a*b**2*d**3*e/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 54*B*a*b**2*
d**2*e**2*x*log(d/e + x)/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 81*B*a*b**2*d**2*e**2
*x/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 54*B*a*b**2*d*e**3*x**2*log(d/e + x)/(6*d**
3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) + 54*B*a*b**2*d*e**3*x**2/(6*d**3*e**5 + 18*d**2*e**6*
x + 18*d*e**7*x**2 + 6*e**8*x**3) + 18*B*a*b**2*e**4*x**3*log(d/e + x)/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e*
*7*x**2 + 6*e**8*x**3) - 24*B*b**3*d**4*log(d/e + x)/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x
**3) - 44*B*b**3*d**4/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 72*B*b**3*d**3*e*x*log(d
/e + x)/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 108*B*b**3*d**3*e*x/(6*d**3*e**5 + 18*
d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 72*B*b**3*d**2*e**2*x**2*log(d/e + x)/(6*d**3*e**5 + 18*d**2*e**
6*x + 18*d*e**7*x**2 + 6*e**8*x**3) - 72*B*b**3*d**2*e**2*x**2/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2
+ 6*e**8*x**3) - 24*B*b**3*d*e**3*x**3*log(d/e + x)/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x*
*3) + 6*B*b**3*e**4*x**4/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3), Eq(m, -4)), (-A*a**3*e
**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 3*A*a**2*b*d*e**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 6*
A*a**2*b*e**4*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 6*A*a*b**2*d**2*e**2*log(d/e + x)/(2*d**2*e**5 + 4*
d*e**6*x + 2*e**7*x**2) + 9*A*a*b**2*d**2*e**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*A*a*b**2*d*e**3*x
*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*A*a*b**2*d*e**3*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e*
*7*x**2) + 6*A*a*b**2*e**4*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 6*A*b**3*d**3*e*log(d/
e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 9*A*b**3*d**3*e/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 1
2*A*b**3*d**2*e**2*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 12*A*b**3*d**2*e**2*x/(2*d**2*e**
5 + 4*d*e**6*x + 2*e**7*x**2) - 6*A*b**3*d*e**3*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 2
*A*b**3*e**4*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - B*a**3*d*e**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*
x**2) - 2*B*a**3*e**4*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 6*B*a**2*b*d**2*e**2*log(d/e + x)/(2*d**2*e
**5 + 4*d*e**6*x + 2*e**7*x**2) + 9*B*a**2*b*d**2*e**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*B*a**2*b*
d*e**3*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*B*a**2*b*d*e**3*x/(2*d**2*e**5 + 4*d*e**6*
x + 2*e**7*x**2) + 6*B*a**2*b*e**4*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 18*B*a*b**2*d*
*3*e*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 27*B*a*b**2*d**3*e/(2*d**2*e**5 + 4*d*e**6*x + 2*
e**7*x**2) - 36*B*a*b**2*d**2*e**2*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 36*B*a*b**2*d**2*
e**2*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 18*B*a*b**2*d*e**3*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6
*x + 2*e**7*x**2) + 6*B*a*b**2*e**4*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*B*b**3*d**4*log(d/e + x
)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 18*B*b**3*d**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 24*B*b*
*3*d**3*e*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 24*B*b**3*d**3*e*x/(2*d**2*e**5 + 4*d*e**6
*x + 2*e**7*x**2) + 12*B*b**3*d**2*e**2*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 4*B*b**3*
d*e**3*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + B*b**3*e**4*x**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**
2), Eq(m, -3)), (-6*A*a**3*e**4/(6*d*e**5 + 6*e**6*x) + 18*A*a**2*b*d*e**3*log(d/e + x)/(6*d*e**5 + 6*e**6*x)
+ 18*A*a**2*b*d*e**3/(6*d*e**5 + 6*e**6*x) + 18*A*a**2*b*e**4*x*log(d/e + x)/(6*d*e**5 + 6*e**6*x) - 36*A*a*b*
*2*d**2*e**2*log(d/e + x)/(6*d*e**5 + 6*e**6*x) - 36*A*a*b**2*d**2*e**2/(6*d*e**5 + 6*e**6*x) - 36*A*a*b**2*d*
e**3*x*log(d/e + x)/(6*d*e**5 + 6*e**6*x) + 18*A*a*b**2*e**4*x**2/(6*d*e**5 + 6*e**6*x) + 18*A*b**3*d**3*e*log
(d/e + x)/(6*d*e**5 + 6*e**6*x) + 18*A*b**3*d**3*e/(6*d*e**5 + 6*e**6*x) + 18*A*b**3*d**2*e**2*x*log(d/e + x)/
(6*d*e**5 + 6*e**6*x) - 9*A*b**3*d*e**3*x**2/(6*d*e**5 + 6*e**6*x) + 3*A*b**3*e**4*x**3/(6*d*e**5 + 6*e**6*x)
+ 6*B*a**3*d*e**3*log(d/e + x)/(6*d*e**5 + 6*e**6*x) + 6*B*a**3*d*e**3/(6*d*e**5 + 6*e**6*x) + 6*B*a**3*e**4*x
*log(d/e + x)/(6*d*e**5 + 6*e**6*x) - 36*B*a**2*b*d**2*e**2*log(d/e + x)/(6*d*e**5 + 6*e**6*x) - 36*B*a**2*b*d
**2*e**2/(6*d*e**5 + 6*e**6*x) - 36*B*a**2*b*d*e**3*x*log(d/e + x)/(6*d*e**5 + 6*e**6*x) + 18*B*a**2*b*e**4*x*
*2/(6*d*e**5 + 6*e**6*x) + 54*B*a*b**2*d**3*e*log(d/e + x)/(6*d*e**5 + 6*e**6*x) + 54*B*a*b**2*d**3*e/(6*d*e**
5 + 6*e**6*x) + 54*B*a*b**2*d**2*e**2*x*log(d/e + x)/(6*d*e**5 + 6*e**6*x) - 27*B*a*b**2*d*e**3*x**2/(6*d*e**5
+ 6*e**6*x) + 9*B*a*b**2*e**4*x**3/(6*d*e**5 + 6*e**6*x) - 24*B*b**3*d**4*log(d/e + x)/(6*d*e**5 + 6*e**6*x)
- 24*B*b**3*d**4/(6*d*e**5 + 6*e**6*x) - 24*B*b**3*d**3*e*x*log(d/e + x)/(6*d*e**5 + 6*e**6*x) + 12*B*b**3*d**
2*e**2*x**2/(6*d*e**5 + 6*e**6*x) - 4*B*b**3*d*e**3*x**3/(6*d*e**5 + 6*e**6*x) + 2*B*b**3*e**4*x**4/(6*d*e**5
+ 6*e**6*x), Eq(m, -2)), (A*a**3*log(d/e + x)/e - 3*A*a**2*b*d*log(d/e + x)/e**2 + 3*A*a**2*b*x/e + 3*A*a*b**2
*d**2*log(d/e + x)/e**3 - 3*A*a*b**2*d*x/e**2 + 3*A*a*b**2*x**2/(2*e) - A*b**3*d**3*log(d/e + x)/e**4 + A*b**3
*d**2*x/e**3 - A*b**3*d*x**2/(2*e**2) + A*b**3*x**3/(3*e) - B*a**3*d*log(d/e + x)/e**2 + B*a**3*x/e + 3*B*a**2
*b*d**2*log(d/e + x)/e**3 - 3*B*a**2*b*d*x/e**2 + 3*B*a**2*b*x**2/(2*e) - 3*B*a*b**2*d**3*log(d/e + x)/e**4 +
3*B*a*b**2*d**2*x/e**3 - 3*B*a*b**2*d*x**2/(2*e**2) + B*a*b**2*x**3/e + B*b**3*d**4*log(d/e + x)/e**5 - B*b**3
*d**3*x/e**4 + B*b**3*d**2*x**2/(2*e**3) - B*b**3*d*x**3/(3*e**2) + B*b**3*x**4/(4*e), Eq(m, -1)), (A*a**3*d*e
**4*m**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 14*A
*a**3*d*e**4*m**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**
5) + 71*A*a**3*d*e**4*m**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m
+ 120*e**5) + 154*A*a**3*d*e**4*m*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*
e**5*m + 120*e**5) + 120*A*a**3*d*e**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 +
274*e**5*m + 120*e**5) + A*a**3*e**5*m**4*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*
m**2 + 274*e**5*m + 120*e**5) + 14*A*a**3*e**5*m**3*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 +
225*e**5*m**2 + 274*e**5*m + 120*e**5) + 71*A*a**3*e**5*m**2*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**
5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 154*A*a**3*e**5*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 +
85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 120*A*a**3*e**5*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m
**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 3*A*a**2*b*d**2*e**3*m**3*(d + e*x)**m/(e**5*m**
5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 36*A*a**2*b*d**2*e**3*m**2*(d + e*x
)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 141*A*a**2*b*d**2*e**
3*m*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 180*A*a**
2*b*d**2*e**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) +
3*A*a**2*b*d*e**4*m**4*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m +
120*e**5) + 36*A*a**2*b*d*e**4*m**3*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 +
274*e**5*m + 120*e**5) + 141*A*a**2*b*d*e**4*m**2*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 2
25*e**5*m**2 + 274*e**5*m + 120*e**5) + 180*A*a**2*b*d*e**4*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e*
*5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 3*A*a**2*b*e**5*m**4*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5
*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 39*A*a**2*b*e**5*m**3*x**2*(d + e*x)**m/(e**5*
m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 177*A*a**2*b*e**5*m**2*x**2*(d +
e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 321*A*a**2*b*e**5
*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 180*A
*a**2*b*e**5*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**
5) + 6*A*a*b**2*d**3*e**2*m**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**
5*m + 120*e**5) + 54*A*a*b**2*d**3*e**2*m*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**
2 + 274*e**5*m + 120*e**5) + 120*A*a*b**2*d**3*e**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 22
5*e**5*m**2 + 274*e**5*m + 120*e**5) - 6*A*a*b**2*d**2*e**3*m**3*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85
*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 54*A*a*b**2*d**2*e**3*m**2*x*(d + e*x)**m/(e**5*m**5 + 1
5*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 120*A*a*b**2*d**2*e**3*m*x*(d + e*x)**m/
(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 3*A*a*b**2*d*e**4*m**4*x**
2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 30*A*a*b**2
*d*e**4*m**3*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**
5) + 87*A*a*b**2*d*e**4*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*
e**5*m + 120*e**5) + 60*A*a*b**2*d*e**4*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**
5*m**2 + 274*e**5*m + 120*e**5) + 3*A*a*b**2*e**5*m**4*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m
**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 36*A*a*b**2*e**5*m**3*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m*
*4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 147*A*a*b**2*e**5*m**2*x**3*(d + e*x)**m/(e**5*m*
*5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 234*A*a*b**2*e**5*m*x**3*(d + e*x)
**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 120*A*a*b**2*e**5*x**3
*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 6*A*b**3*d**
4*e*m*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 30*A*b*
*3*d**4*e*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 6*A
*b**3*d**3*e**2*m**2*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 12
0*e**5) + 30*A*b**3*d**3*e**2*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*
e**5*m + 120*e**5) - 3*A*b**3*d**2*e**3*m**3*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*
e**5*m**2 + 274*e**5*m + 120*e**5) - 18*A*b**3*d**2*e**3*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85
*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 15*A*b**3*d**2*e**3*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*
e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + A*b**3*d*e**4*m**4*x**3*(d + e*x)**m/(e**5
*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 8*A*b**3*d*e**4*m**3*x**3*(d +
e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 17*A*b**3*d*e**4*m
**2*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 10*A
*b**3*d*e**4*m*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e
**5) + A*b**3*e**5*m**4*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*
m + 120*e**5) + 11*A*b**3*e**5*m**3*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2
+ 274*e**5*m + 120*e**5) + 41*A*b**3*e**5*m**2*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 2
25*e**5*m**2 + 274*e**5*m + 120*e**5) + 61*A*b**3*e**5*m*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5
*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 30*A*b**3*e**5*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 +
85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - B*a**3*d**2*e**3*m**3*(d + e*x)**m/(e**5*m**5 + 15*e**
5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 12*B*a**3*d**2*e**3*m**2*(d + e*x)**m/(e**5*m
**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 47*B*a**3*d**2*e**3*m*(d + e*x)**
m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 60*B*a**3*d**2*e**3*(d +
e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + B*a**3*d*e**4*m**
4*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 12*B*a**3
*d*e**4*m**3*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5)
+ 47*B*a**3*d*e**4*m**2*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m +
120*e**5) + 60*B*a**3*d*e**4*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*
e**5*m + 120*e**5) + B*a**3*e**5*m**4*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m*
*2 + 274*e**5*m + 120*e**5) + 13*B*a**3*e**5*m**3*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 +
225*e**5*m**2 + 274*e**5*m + 120*e**5) + 59*B*a**3*e**5*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85
*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 107*B*a**3*e**5*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5
*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 60*B*a**3*e**5*x**2*(d + e*x)**m/(e**5*m**5 +
15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 6*B*a**2*b*d**3*e**2*m**2*(d + e*x)**m/
(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 54*B*a**2*b*d**3*e**2*m*(d
+ e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 120*B*a**2*b*d*
*3*e**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 6*B*a
**2*b*d**2*e**3*m**3*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 12
0*e**5) - 54*B*a**2*b*d**2*e**3*m**2*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 +
274*e**5*m + 120*e**5) - 120*B*a**2*b*d**2*e**3*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 2
25*e**5*m**2 + 274*e**5*m + 120*e**5) + 3*B*a**2*b*d*e**4*m**4*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 8
5*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 30*B*a**2*b*d*e**4*m**3*x**2*(d + e*x)**m/(e**5*m**5 +
15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 87*B*a**2*b*d*e**4*m**2*x**2*(d + e*x)*
*m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 60*B*a**2*b*d*e**4*m*x*
*2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 3*B*a**2*b
*e**5*m**4*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5)
+ 36*B*a**2*b*e**5*m**3*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5
*m + 120*e**5) + 147*B*a**2*b*e**5*m**2*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*
m**2 + 274*e**5*m + 120*e**5) + 234*B*a**2*b*e**5*m*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3
+ 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 120*B*a**2*b*e**5*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85
*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 18*B*a*b**2*d**4*e*m*(d + e*x)**m/(e**5*m**5 + 15*e**5*m
**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 90*B*a*b**2*d**4*e*(d + e*x)**m/(e**5*m**5 + 15*
e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 18*B*a*b**2*d**3*e**2*m**2*x*(d + e*x)**m/
(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 90*B*a*b**2*d**3*e**2*m*x*
(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 9*B*a*b**2*d*
*2*e**3*m**3*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**
5) - 54*B*a*b**2*d**2*e**3*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 2
74*e**5*m + 120*e**5) - 45*B*a*b**2*d**2*e**3*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 2
25*e**5*m**2 + 274*e**5*m + 120*e**5) + 3*B*a*b**2*d*e**4*m**4*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 8
5*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 24*B*a*b**2*d*e**4*m**3*x**3*(d + e*x)**m/(e**5*m**5 +
15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 51*B*a*b**2*d*e**4*m**2*x**3*(d + e*x)*
*m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 30*B*a*b**2*d*e**4*m*x*
*3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 3*B*a*b**2
*e**5*m**4*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5)
+ 33*B*a*b**2*e**5*m**3*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5
*m + 120*e**5) + 123*B*a*b**2*e**5*m**2*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*
m**2 + 274*e**5*m + 120*e**5) + 183*B*a*b**2*e**5*m*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3
+ 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 90*B*a*b**2*e**5*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*
e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 24*B*b**3*d**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 8
5*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 24*B*b**3*d**4*e*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*
m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 12*B*b**3*d**3*e**2*m**2*x**2*(d + e*x)**m/(e**
5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 12*B*b**3*d**3*e**2*m*x**2*(d
+ e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 4*B*b**3*d**2*e*
*3*m**3*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) -
12*B*b**3*d**2*e**3*m**2*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5
*m + 120*e**5) - 8*B*b**3*d**2*e**3*m*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m*
*2 + 274*e**5*m + 120*e**5) + B*b**3*d*e**4*m**4*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 +
225*e**5*m**2 + 274*e**5*m + 120*e**5) + 6*B*b**3*d*e**4*m**3*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85
*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 11*B*b**3*d*e**4*m**2*x**4*(d + e*x)**m/(e**5*m**5 + 15*
e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 6*B*b**3*d*e**4*m*x**4*(d + e*x)**m/(e**5*
m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + B*b**3*e**5*m**4*x**5*(d + e*x)*
*m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 10*B*b**3*e**5*m**3*x**
5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 35*B*b**3*e
**5*m**2*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) +
50*B*b**3*e**5*m*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 12
0*e**5) + 24*B*b**3*e**5*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5
*m + 120*e**5), True))

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