3.1488 \(\int (A+B x) (d+e x)^m (a+c x^2)^2 \, dx\)

Optimal. Leaf size=234 \[ -\frac{\left (a e^2+c d^2\right )^2 (B d-A e) (d+e x)^{m+1}}{e^6 (m+1)}+\frac{\left (a e^2+c d^2\right ) (d+e x)^{m+2} \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6 (m+2)}-\frac{2 c (d+e x)^{m+3} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 (m+3)}+\frac{2 c (d+e x)^{m+4} \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (m+4)}-\frac{c^2 (5 B d-A e) (d+e x)^{m+5}}{e^6 (m+5)}+\frac{B c^2 (d+e x)^{m+6}}{e^6 (m+6)} \]

[Out]

-(((B*d - A*e)*(c*d^2 + a*e^2)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) + ((c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e +
a*B*e^2)*(d + e*x)^(2 + m))/(e^6*(2 + m)) - (2*c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^
(3 + m))/(e^6*(3 + m)) + (2*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(4 + m))/(e^6*(4 + m)) - (c^2*(5*B*d
- A*e)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (B*c^2*(d + e*x)^(6 + m))/(e^6*(6 + m))

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Rubi [A]  time = 0.136468, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ -\frac{\left (a e^2+c d^2\right )^2 (B d-A e) (d+e x)^{m+1}}{e^6 (m+1)}+\frac{\left (a e^2+c d^2\right ) (d+e x)^{m+2} \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6 (m+2)}-\frac{2 c (d+e x)^{m+3} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 (m+3)}+\frac{2 c (d+e x)^{m+4} \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (m+4)}-\frac{c^2 (5 B d-A e) (d+e x)^{m+5}}{e^6 (m+5)}+\frac{B c^2 (d+e x)^{m+6}}{e^6 (m+6)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((B*d - A*e)*(c*d^2 + a*e^2)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) + ((c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e +
a*B*e^2)*(d + e*x)^(2 + m))/(e^6*(2 + m)) - (2*c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^
(3 + m))/(e^6*(3 + m)) + (2*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(4 + m))/(e^6*(4 + m)) - (c^2*(5*B*d
- A*e)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (B*c^2*(d + e*x)^(6 + m))/(e^6*(6 + m))

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (A+B x) (d+e x)^m \left (a+c x^2\right )^2 \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^2 (d+e x)^m}{e^5}+\frac{\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^{1+m}}{e^5}+\frac{2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right ) (d+e x)^{2+m}}{e^5}-\frac{2 c \left (-5 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^{3+m}}{e^5}+\frac{c^2 (-5 B d+A e) (d+e x)^{4+m}}{e^5}+\frac{B c^2 (d+e x)^{5+m}}{e^5}\right ) \, dx\\ &=-\frac{(B d-A e) \left (c d^2+a e^2\right )^2 (d+e x)^{1+m}}{e^6 (1+m)}+\frac{\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^{2+m}}{e^6 (2+m)}-\frac{2 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^{3+m}}{e^6 (3+m)}+\frac{2 c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{4+m}}{e^6 (4+m)}-\frac{c^2 (5 B d-A e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac{B c^2 (d+e x)^{6+m}}{e^6 (6+m)}\\ \end{align*}

Mathematica [A]  time = 0.551043, size = 355, normalized size = 1.52 \[ \frac{(d+e x)^{m+1} \left (B (m+1) (d+e x) \left (4 (m+5) \left (a e^2+c d^2\right ) \left (a e^2 \left (m^2+7 m+12\right )+c \left (2 d^2-2 d e (m+2) x+e^2 \left (m^2+5 m+6\right ) x^2\right )\right )-4 c d (m+2) (d+e x) \left (a e^2 \left (m^2+9 m+20\right )+c \left (2 d^2-2 d e (m+3) x+e^2 \left (m^2+7 m+12\right ) x^2\right )\right )+e^4 (m+2) (m+3) (m+4) (m+5) \left (a+c x^2\right )^2\right )-(m+6) (B d-A e) \left (4 (m+4) \left (a e^2+c d^2\right ) \left (a e^2 \left (m^2+5 m+6\right )+c \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )-4 c d (m+1) (d+e x) \left (a e^2 \left (m^2+7 m+12\right )+c \left (2 d^2-2 d e (m+2) x+e^2 \left (m^2+5 m+6\right ) x^2\right )\right )+e^4 (m+1) (m+2) (m+3) (m+4) \left (a+c x^2\right )^2\right )\right )}{e^6 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*(-((B*d - A*e)*(6 + m)*(e^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(a + c*x^2)^2 + 4*(c*d^2 + a*e^
2)*(4 + m)*(a*e^2*(6 + 5*m + m^2) + c*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2)) - 4*c*d*(1 + m)*(d
+ e*x)*(a*e^2*(12 + 7*m + m^2) + c*(2*d^2 - 2*d*e*(2 + m)*x + e^2*(6 + 5*m + m^2)*x^2)))) + B*(1 + m)*(d + e*x
)*(e^4*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(a + c*x^2)^2 + 4*(c*d^2 + a*e^2)*(5 + m)*(a*e^2*(12 + 7*m + m^2) + c*(
2*d^2 - 2*d*e*(2 + m)*x + e^2*(6 + 5*m + m^2)*x^2)) - 4*c*d*(2 + m)*(d + e*x)*(a*e^2*(20 + 9*m + m^2) + c*(2*d
^2 - 2*d*e*(3 + m)*x + e^2*(12 + 7*m + m^2)*x^2)))))/(e^6*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m))

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Maple [B]  time = 0.012, size = 1271, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x+d)^(1+m)*(B*c^2*e^5*m^5*x^5+A*c^2*e^5*m^5*x^4+15*B*c^2*e^5*m^4*x^5+16*A*c^2*e^5*m^4*x^4+2*B*a*c*e^5*m^5*x
^3-5*B*c^2*d*e^4*m^4*x^4+85*B*c^2*e^5*m^3*x^5+2*A*a*c*e^5*m^5*x^2-4*A*c^2*d*e^4*m^4*x^3+95*A*c^2*e^5*m^3*x^4+3
4*B*a*c*e^5*m^4*x^3-50*B*c^2*d*e^4*m^3*x^4+225*B*c^2*e^5*m^2*x^5+36*A*a*c*e^5*m^4*x^2-48*A*c^2*d*e^4*m^3*x^3+2
60*A*c^2*e^5*m^2*x^4+B*a^2*e^5*m^5*x-6*B*a*c*d*e^4*m^4*x^2+214*B*a*c*e^5*m^3*x^3+20*B*c^2*d^2*e^3*m^3*x^3-175*
B*c^2*d*e^4*m^2*x^4+274*B*c^2*e^5*m*x^5+A*a^2*e^5*m^5-4*A*a*c*d*e^4*m^4*x+242*A*a*c*e^5*m^3*x^2+12*A*c^2*d^2*e
^3*m^3*x^2-188*A*c^2*d*e^4*m^2*x^3+324*A*c^2*e^5*m*x^4+19*B*a^2*e^5*m^4*x-84*B*a*c*d*e^4*m^3*x^2+614*B*a*c*e^5
*m^2*x^3+120*B*c^2*d^2*e^3*m^2*x^3-250*B*c^2*d*e^4*m*x^4+120*B*c^2*e^5*x^5+20*A*a^2*e^5*m^4-64*A*a*c*d*e^4*m^3
*x+744*A*a*c*e^5*m^2*x^2+108*A*c^2*d^2*e^3*m^2*x^2-288*A*c^2*d*e^4*m*x^3+144*A*c^2*e^5*x^4-B*a^2*d*e^4*m^4+137
*B*a^2*e^5*m^3*x+12*B*a*c*d^2*e^3*m^3*x-390*B*a*c*d*e^4*m^2*x^2+792*B*a*c*e^5*m*x^3-60*B*c^2*d^3*e^2*m^2*x^2+2
20*B*c^2*d^2*e^3*m*x^3-120*B*c^2*d*e^4*x^4+155*A*a^2*e^5*m^3+4*A*a*c*d^2*e^3*m^3-356*A*a*c*d*e^4*m^2*x+1016*A*
a*c*e^5*m*x^2-24*A*c^2*d^3*e^2*m^2*x+240*A*c^2*d^2*e^3*m*x^2-144*A*c^2*d*e^4*x^3-18*B*a^2*d*e^4*m^3+461*B*a^2*
e^5*m^2*x+144*B*a*c*d^2*e^3*m^2*x-672*B*a*c*d*e^4*m*x^2+360*B*a*c*e^5*x^3-180*B*c^2*d^3*e^2*m*x^2+120*B*c^2*d^
2*e^3*x^3+580*A*a^2*e^5*m^2+60*A*a*c*d^2*e^3*m^2-776*A*a*c*d*e^4*m*x+480*A*a*c*e^5*x^2-168*A*c^2*d^3*e^2*m*x+1
44*A*c^2*d^2*e^3*x^2-119*B*a^2*d*e^4*m^2+702*B*a^2*e^5*m*x-12*B*a*c*d^3*e^2*m^2+492*B*a*c*d^2*e^3*m*x-360*B*a*
c*d*e^4*x^2+120*B*c^2*d^4*e*m*x-120*B*c^2*d^3*e^2*x^2+1044*A*a^2*e^5*m+296*A*a*c*d^2*e^3*m-480*A*a*c*d*e^4*x+2
4*A*c^2*d^4*e*m-144*A*c^2*d^3*e^2*x-342*B*a^2*d*e^4*m+360*B*a^2*e^5*x-132*B*a*c*d^3*e^2*m+360*B*a*c*d^2*e^3*x+
120*B*c^2*d^4*e*x+720*A*a^2*e^5+480*A*a*c*d^2*e^3+144*A*c^2*d^4*e-360*B*a^2*d*e^4-360*B*a*c*d^3*e^2-120*B*c^2*
d^5)/e^6/(m^6+21*m^5+175*m^4+735*m^3+1624*m^2+1764*m+720)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.03555, size = 3032, normalized size = 12.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(A*a^2*d*e^5*m^5 - 120*B*c^2*d^6 + 144*A*c^2*d^5*e - 360*B*a*c*d^4*e^2 + 480*A*a*c*d^3*e^3 - 360*B*a^2*d^2*e^4
+ 720*A*a^2*d*e^5 + (B*c^2*e^6*m^5 + 15*B*c^2*e^6*m^4 + 85*B*c^2*e^6*m^3 + 225*B*c^2*e^6*m^2 + 274*B*c^2*e^6*
m + 120*B*c^2*e^6)*x^6 + (144*A*c^2*e^6 + (B*c^2*d*e^5 + A*c^2*e^6)*m^5 + 2*(5*B*c^2*d*e^5 + 8*A*c^2*e^6)*m^4
+ 5*(7*B*c^2*d*e^5 + 19*A*c^2*e^6)*m^3 + 10*(5*B*c^2*d*e^5 + 26*A*c^2*e^6)*m^2 + 12*(2*B*c^2*d*e^5 + 27*A*c^2*
e^6)*m)*x^5 - (B*a^2*d^2*e^4 - 20*A*a^2*d*e^5)*m^4 + (360*B*a*c*e^6 + (A*c^2*d*e^5 + 2*B*a*c*e^6)*m^5 - (5*B*c
^2*d^2*e^4 - 12*A*c^2*d*e^5 - 34*B*a*c*e^6)*m^4 - (30*B*c^2*d^2*e^4 - 47*A*c^2*d*e^5 - 214*B*a*c*e^6)*m^3 - (5
5*B*c^2*d^2*e^4 - 72*A*c^2*d*e^5 - 614*B*a*c*e^6)*m^2 - 6*(5*B*c^2*d^2*e^4 - 6*A*c^2*d*e^5 - 132*B*a*c*e^6)*m)
*x^4 + (4*A*a*c*d^3*e^3 - 18*B*a^2*d^2*e^4 + 155*A*a^2*d*e^5)*m^3 + 2*(240*A*a*c*e^6 + (B*a*c*d*e^5 + A*a*c*e^
6)*m^5 - 2*(A*c^2*d^2*e^4 - 7*B*a*c*d*e^5 - 9*A*a*c*e^6)*m^4 + (10*B*c^2*d^3*e^3 - 18*A*c^2*d^2*e^4 + 65*B*a*c
*d*e^5 + 121*A*a*c*e^6)*m^3 + 2*(15*B*c^2*d^3*e^3 - 20*A*c^2*d^2*e^4 + 56*B*a*c*d*e^5 + 186*A*a*c*e^6)*m^2 + 4
*(5*B*c^2*d^3*e^3 - 6*A*c^2*d^2*e^4 + 15*B*a*c*d*e^5 + 127*A*a*c*e^6)*m)*x^3 - (12*B*a*c*d^4*e^2 - 60*A*a*c*d^
3*e^3 + 119*B*a^2*d^2*e^4 - 580*A*a^2*d*e^5)*m^2 + (360*B*a^2*e^6 + (2*A*a*c*d*e^5 + B*a^2*e^6)*m^5 - (6*B*a*c
*d^2*e^4 - 32*A*a*c*d*e^5 - 19*B*a^2*e^6)*m^4 + (12*A*c^2*d^3*e^3 - 72*B*a*c*d^2*e^4 + 178*A*a*c*d*e^5 + 137*B
*a^2*e^6)*m^3 - (60*B*c^2*d^4*e^2 - 84*A*c^2*d^3*e^3 + 246*B*a*c*d^2*e^4 - 388*A*a*c*d*e^5 - 461*B*a^2*e^6)*m^
2 - 6*(10*B*c^2*d^4*e^2 - 12*A*c^2*d^3*e^3 + 30*B*a*c*d^2*e^4 - 40*A*a*c*d*e^5 - 117*B*a^2*e^6)*m)*x^2 + 2*(12
*A*c^2*d^5*e - 66*B*a*c*d^4*e^2 + 148*A*a*c*d^3*e^3 - 171*B*a^2*d^2*e^4 + 522*A*a^2*d*e^5)*m + (720*A*a^2*e^6
+ (B*a^2*d*e^5 + A*a^2*e^6)*m^5 - 2*(2*A*a*c*d^2*e^4 - 9*B*a^2*d*e^5 - 10*A*a^2*e^6)*m^4 + (12*B*a*c*d^3*e^3 -
60*A*a*c*d^2*e^4 + 119*B*a^2*d*e^5 + 155*A*a^2*e^6)*m^3 - 2*(12*A*c^2*d^4*e^2 - 66*B*a*c*d^3*e^3 + 148*A*a*c*
d^2*e^4 - 171*B*a^2*d*e^5 - 290*A*a^2*e^6)*m^2 + 12*(10*B*c^2*d^5*e - 12*A*c^2*d^4*e^2 + 30*B*a*c*d^3*e^3 - 40
*A*a*c*d^2*e^4 + 30*B*a^2*d*e^5 + 87*A*a^2*e^6)*m)*x)*(e*x + d)^m/(e^6*m^6 + 21*e^6*m^5 + 175*e^6*m^4 + 735*e^
6*m^3 + 1624*e^6*m^2 + 1764*e^6*m + 720*e^6)

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Sympy [A]  time = 23.1765, size = 16038, normalized size = 68.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((d**m*(A*a**2*x + 2*A*a*c*x**3/3 + A*c**2*x**5/5 + B*a**2*x**2/2 + B*a*c*x**4/2 + B*c**2*x**6/6), Eq
(e, 0)), (-12*A*a**2*d**2*e**5/(60*d**7*e**6 + 300*d**6*e**7*x + 600*d**5*e**8*x**2 + 600*d**4*e**9*x**3 + 300
*d**3*e**10*x**4 + 60*d**2*e**11*x**5) - 4*A*a*c*d**4*e**3/(60*d**7*e**6 + 300*d**6*e**7*x + 600*d**5*e**8*x**
2 + 600*d**4*e**9*x**3 + 300*d**3*e**10*x**4 + 60*d**2*e**11*x**5) - 20*A*a*c*d**3*e**4*x/(60*d**7*e**6 + 300*
d**6*e**7*x + 600*d**5*e**8*x**2 + 600*d**4*e**9*x**3 + 300*d**3*e**10*x**4 + 60*d**2*e**11*x**5) - 40*A*a*c*d
**2*e**5*x**2/(60*d**7*e**6 + 300*d**6*e**7*x + 600*d**5*e**8*x**2 + 600*d**4*e**9*x**3 + 300*d**3*e**10*x**4
+ 60*d**2*e**11*x**5) + 12*A*c**2*d*e**6*x**5/(60*d**7*e**6 + 300*d**6*e**7*x + 600*d**5*e**8*x**2 + 600*d**4*
e**9*x**3 + 300*d**3*e**10*x**4 + 60*d**2*e**11*x**5) - 3*B*a**2*d**3*e**4/(60*d**7*e**6 + 300*d**6*e**7*x + 6
00*d**5*e**8*x**2 + 600*d**4*e**9*x**3 + 300*d**3*e**10*x**4 + 60*d**2*e**11*x**5) - 15*B*a**2*d**2*e**5*x/(60
*d**7*e**6 + 300*d**6*e**7*x + 600*d**5*e**8*x**2 + 600*d**4*e**9*x**3 + 300*d**3*e**10*x**4 + 60*d**2*e**11*x
**5) + 30*B*a*c*d*e**6*x**4/(60*d**7*e**6 + 300*d**6*e**7*x + 600*d**5*e**8*x**2 + 600*d**4*e**9*x**3 + 300*d*
*3*e**10*x**4 + 60*d**2*e**11*x**5) + 6*B*a*c*e**7*x**5/(60*d**7*e**6 + 300*d**6*e**7*x + 600*d**5*e**8*x**2 +
600*d**4*e**9*x**3 + 300*d**3*e**10*x**4 + 60*d**2*e**11*x**5) + 60*B*c**2*d**7*log(d/e + x)/(60*d**7*e**6 +
300*d**6*e**7*x + 600*d**5*e**8*x**2 + 600*d**4*e**9*x**3 + 300*d**3*e**10*x**4 + 60*d**2*e**11*x**5) + 47*B*c
**2*d**7/(60*d**7*e**6 + 300*d**6*e**7*x + 600*d**5*e**8*x**2 + 600*d**4*e**9*x**3 + 300*d**3*e**10*x**4 + 60*
d**2*e**11*x**5) + 300*B*c**2*d**6*e*x*log(d/e + x)/(60*d**7*e**6 + 300*d**6*e**7*x + 600*d**5*e**8*x**2 + 600
*d**4*e**9*x**3 + 300*d**3*e**10*x**4 + 60*d**2*e**11*x**5) + 175*B*c**2*d**6*e*x/(60*d**7*e**6 + 300*d**6*e**
7*x + 600*d**5*e**8*x**2 + 600*d**4*e**9*x**3 + 300*d**3*e**10*x**4 + 60*d**2*e**11*x**5) + 600*B*c**2*d**5*e*
*2*x**2*log(d/e + x)/(60*d**7*e**6 + 300*d**6*e**7*x + 600*d**5*e**8*x**2 + 600*d**4*e**9*x**3 + 300*d**3*e**1
0*x**4 + 60*d**2*e**11*x**5) + 200*B*c**2*d**5*e**2*x**2/(60*d**7*e**6 + 300*d**6*e**7*x + 600*d**5*e**8*x**2
+ 600*d**4*e**9*x**3 + 300*d**3*e**10*x**4 + 60*d**2*e**11*x**5) + 600*B*c**2*d**4*e**3*x**3*log(d/e + x)/(60*
d**7*e**6 + 300*d**6*e**7*x + 600*d**5*e**8*x**2 + 600*d**4*e**9*x**3 + 300*d**3*e**10*x**4 + 60*d**2*e**11*x*
*5) + 300*B*c**2*d**3*e**4*x**4*log(d/e + x)/(60*d**7*e**6 + 300*d**6*e**7*x + 600*d**5*e**8*x**2 + 600*d**4*e
**9*x**3 + 300*d**3*e**10*x**4 + 60*d**2*e**11*x**5) - 150*B*c**2*d**3*e**4*x**4/(60*d**7*e**6 + 300*d**6*e**7
*x + 600*d**5*e**8*x**2 + 600*d**4*e**9*x**3 + 300*d**3*e**10*x**4 + 60*d**2*e**11*x**5) + 60*B*c**2*d**2*e**5
*x**5*log(d/e + x)/(60*d**7*e**6 + 300*d**6*e**7*x + 600*d**5*e**8*x**2 + 600*d**4*e**9*x**3 + 300*d**3*e**10*
x**4 + 60*d**2*e**11*x**5) - 90*B*c**2*d**2*e**5*x**5/(60*d**7*e**6 + 300*d**6*e**7*x + 600*d**5*e**8*x**2 + 6
00*d**4*e**9*x**3 + 300*d**3*e**10*x**4 + 60*d**2*e**11*x**5), Eq(m, -6)), (-3*A*a**2*d*e**5/(12*d**5*e**6 + 4
8*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) - 2*A*a*c*d**3*e**3/(12*d**5*e**6 + 4
8*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) - 8*A*a*c*d**2*e**4*x/(12*d**5*e**6 +
48*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) - 12*A*a*c*d*e**5*x**2/(12*d**5*e**
6 + 48*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) + 12*A*c**2*d**5*e*log(d/e + x)/
(12*d**5*e**6 + 48*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) + 13*A*c**2*d**5*e/(
12*d**5*e**6 + 48*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) + 48*A*c**2*d**4*e**2
*x*log(d/e + x)/(12*d**5*e**6 + 48*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) + 40
*A*c**2*d**4*e**2*x/(12*d**5*e**6 + 48*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4)
+ 72*A*c**2*d**3*e**3*x**2*log(d/e + x)/(12*d**5*e**6 + 48*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3
+ 12*d*e**10*x**4) + 36*A*c**2*d**3*e**3*x**2/(12*d**5*e**6 + 48*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e*
*9*x**3 + 12*d*e**10*x**4) + 48*A*c**2*d**2*e**4*x**3*log(d/e + x)/(12*d**5*e**6 + 48*d**4*e**7*x + 72*d**3*e*
*8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) + 12*A*c**2*d*e**5*x**4*log(d/e + x)/(12*d**5*e**6 + 48*d**4*e*
*7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) - 12*A*c**2*d*e**5*x**4/(12*d**5*e**6 + 48*d**
4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) - B*a**2*d**2*e**4/(12*d**5*e**6 + 48*d**4
*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) - 4*B*a**2*d*e**5*x/(12*d**5*e**6 + 48*d**4
*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) + 6*B*a*c*e**6*x**4/(12*d**5*e**6 + 48*d**4
*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) - 60*B*c**2*d**6*log(d/e + x)/(12*d**5*e**6
+ 48*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) - 65*B*c**2*d**6/(12*d**5*e**6 +
48*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) - 240*B*c**2*d**5*e*x*log(d/e + x)/(
12*d**5*e**6 + 48*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) - 200*B*c**2*d**5*e*x
/(12*d**5*e**6 + 48*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) - 360*B*c**2*d**4*e
**2*x**2*log(d/e + x)/(12*d**5*e**6 + 48*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4
) - 180*B*c**2*d**4*e**2*x**2/(12*d**5*e**6 + 48*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e*
*10*x**4) - 240*B*c**2*d**3*e**3*x**3*log(d/e + x)/(12*d**5*e**6 + 48*d**4*e**7*x + 72*d**3*e**8*x**2 + 48*d**
2*e**9*x**3 + 12*d*e**10*x**4) - 60*B*c**2*d**2*e**4*x**4*log(d/e + x)/(12*d**5*e**6 + 48*d**4*e**7*x + 72*d**
3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) + 60*B*c**2*d**2*e**4*x**4/(12*d**5*e**6 + 48*d**4*e**7*x +
72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4) + 12*B*c**2*d*e**5*x**5/(12*d**5*e**6 + 48*d**4*e**7
*x + 72*d**3*e**8*x**2 + 48*d**2*e**9*x**3 + 12*d*e**10*x**4), Eq(m, -5)), (-2*A*a**2*e**5/(6*d**3*e**6 + 18*d
**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) - 4*A*a*c*d**2*e**3/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2
+ 6*e**9*x**3) - 12*A*a*c*d*e**4*x/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) - 12*A*a*c*e*
*5*x**2/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) - 24*A*c**2*d**4*e*log(d/e + x)/(6*d**3*
e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) - 44*A*c**2*d**4*e/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*
e**8*x**2 + 6*e**9*x**3) - 72*A*c**2*d**3*e**2*x*log(d/e + x)/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 +
6*e**9*x**3) - 108*A*c**2*d**3*e**2*x/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) - 72*A*c*
*2*d**2*e**3*x**2*log(d/e + x)/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) - 72*A*c**2*d**2*
e**3*x**2/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) - 24*A*c**2*d*e**4*x**3*log(d/e + x)/(
6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) + 6*A*c**2*e**5*x**4/(6*d**3*e**6 + 18*d**2*e**7*
x + 18*d*e**8*x**2 + 6*e**9*x**3) - B*a**2*d*e**4/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3
) - 3*B*a**2*e**5*x/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) + 12*B*a*c*d**3*e**2*log(d/e
+ x)/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) + 22*B*a*c*d**3*e**2/(6*d**3*e**6 + 18*d**
2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) + 36*B*a*c*d**2*e**3*x*log(d/e + x)/(6*d**3*e**6 + 18*d**2*e**7*x + 1
8*d*e**8*x**2 + 6*e**9*x**3) + 54*B*a*c*d**2*e**3*x/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x*
*3) + 36*B*a*c*d*e**4*x**2*log(d/e + x)/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) + 36*B*a
*c*d*e**4*x**2/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) + 12*B*a*c*e**5*x**3*log(d/e + x)
/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) + 60*B*c**2*d**5*log(d/e + x)/(6*d**3*e**6 + 18
*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) + 110*B*c**2*d**5/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2
+ 6*e**9*x**3) + 180*B*c**2*d**4*e*x*log(d/e + x)/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3
) + 270*B*c**2*d**4*e*x/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) + 180*B*c**2*d**3*e**2*x
**2*log(d/e + x)/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) + 180*B*c**2*d**3*e**2*x**2/(6*
d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) + 60*B*c**2*d**2*e**3*x**3*log(d/e + x)/(6*d**3*e**
6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) - 15*B*c**2*d*e**4*x**4/(6*d**3*e**6 + 18*d**2*e**7*x + 18*
d*e**8*x**2 + 6*e**9*x**3) + 3*B*c**2*e**5*x**5/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3),
Eq(m, -4)), (-3*A*a**2*e**5/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 12*A*a*c*d**2*e**3*log(d/e + x)/(6*d*
*2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 18*A*a*c*d**2*e**3/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 24*A*a*c
*d*e**4*x*log(d/e + x)/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 24*A*a*c*d*e**4*x/(6*d**2*e**6 + 12*d*e**7*
x + 6*e**8*x**2) + 12*A*a*c*e**5*x**2*log(d/e + x)/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 36*A*c**2*d**4*
e*log(d/e + x)/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 54*A*c**2*d**4*e/(6*d**2*e**6 + 12*d*e**7*x + 6*e**
8*x**2) + 72*A*c**2*d**3*e**2*x*log(d/e + x)/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 72*A*c**2*d**3*e**2*x
/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 36*A*c**2*d**2*e**3*x**2*log(d/e + x)/(6*d**2*e**6 + 12*d*e**7*x
+ 6*e**8*x**2) - 12*A*c**2*d*e**4*x**3/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 3*A*c**2*e**5*x**4/(6*d**2*
e**6 + 12*d*e**7*x + 6*e**8*x**2) - 3*B*a**2*d*e**4/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 6*B*a**2*e**5*
x/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 36*B*a*c*d**3*e**2*log(d/e + x)/(6*d**2*e**6 + 12*d*e**7*x + 6*e
**8*x**2) - 54*B*a*c*d**3*e**2/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 72*B*a*c*d**2*e**3*x*log(d/e + x)/(
6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 72*B*a*c*d**2*e**3*x/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 36
*B*a*c*d*e**4*x**2*log(d/e + x)/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 12*B*a*c*e**5*x**3/(6*d**2*e**6 +
12*d*e**7*x + 6*e**8*x**2) - 60*B*c**2*d**5*log(d/e + x)/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 90*B*c**2
*d**5/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 120*B*c**2*d**4*e*x*log(d/e + x)/(6*d**2*e**6 + 12*d*e**7*x
+ 6*e**8*x**2) - 120*B*c**2*d**4*e*x/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 60*B*c**2*d**3*e**2*x**2*log(
d/e + x)/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 20*B*c**2*d**2*e**3*x**3/(6*d**2*e**6 + 12*d*e**7*x + 6*e
**8*x**2) - 5*B*c**2*d*e**4*x**4/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 2*B*c**2*e**5*x**5/(6*d**2*e**6 +
12*d*e**7*x + 6*e**8*x**2), Eq(m, -3)), (-12*A*a**2*e**5/(12*d*e**6 + 12*e**7*x) - 48*A*a*c*d**2*e**3*log(d/e
+ x)/(12*d*e**6 + 12*e**7*x) - 48*A*a*c*d**2*e**3/(12*d*e**6 + 12*e**7*x) - 48*A*a*c*d*e**4*x*log(d/e + x)/(1
2*d*e**6 + 12*e**7*x) + 24*A*a*c*e**5*x**2/(12*d*e**6 + 12*e**7*x) - 48*A*c**2*d**4*e*log(d/e + x)/(12*d*e**6
+ 12*e**7*x) - 48*A*c**2*d**4*e/(12*d*e**6 + 12*e**7*x) - 48*A*c**2*d**3*e**2*x*log(d/e + x)/(12*d*e**6 + 12*e
**7*x) + 24*A*c**2*d**2*e**3*x**2/(12*d*e**6 + 12*e**7*x) - 8*A*c**2*d*e**4*x**3/(12*d*e**6 + 12*e**7*x) + 4*A
*c**2*e**5*x**4/(12*d*e**6 + 12*e**7*x) + 12*B*a**2*d*e**4*log(d/e + x)/(12*d*e**6 + 12*e**7*x) + 12*B*a**2*d*
e**4/(12*d*e**6 + 12*e**7*x) + 12*B*a**2*e**5*x*log(d/e + x)/(12*d*e**6 + 12*e**7*x) + 72*B*a*c*d**3*e**2*log(
d/e + x)/(12*d*e**6 + 12*e**7*x) + 72*B*a*c*d**3*e**2/(12*d*e**6 + 12*e**7*x) + 72*B*a*c*d**2*e**3*x*log(d/e +
x)/(12*d*e**6 + 12*e**7*x) - 36*B*a*c*d*e**4*x**2/(12*d*e**6 + 12*e**7*x) + 12*B*a*c*e**5*x**3/(12*d*e**6 + 1
2*e**7*x) + 60*B*c**2*d**5*log(d/e + x)/(12*d*e**6 + 12*e**7*x) + 60*B*c**2*d**5/(12*d*e**6 + 12*e**7*x) + 60*
B*c**2*d**4*e*x*log(d/e + x)/(12*d*e**6 + 12*e**7*x) - 30*B*c**2*d**3*e**2*x**2/(12*d*e**6 + 12*e**7*x) + 10*B
*c**2*d**2*e**3*x**3/(12*d*e**6 + 12*e**7*x) - 5*B*c**2*d*e**4*x**4/(12*d*e**6 + 12*e**7*x) + 3*B*c**2*e**5*x*
*5/(12*d*e**6 + 12*e**7*x), Eq(m, -2)), (A*a**2*log(d/e + x)/e + 2*A*a*c*d**2*log(d/e + x)/e**3 - 2*A*a*c*d*x/
e**2 + A*a*c*x**2/e + A*c**2*d**4*log(d/e + x)/e**5 - A*c**2*d**3*x/e**4 + A*c**2*d**2*x**2/(2*e**3) - A*c**2*
d*x**3/(3*e**2) + A*c**2*x**4/(4*e) - B*a**2*d*log(d/e + x)/e**2 + B*a**2*x/e - 2*B*a*c*d**3*log(d/e + x)/e**4
+ 2*B*a*c*d**2*x/e**3 - B*a*c*d*x**2/e**2 + 2*B*a*c*x**3/(3*e) - B*c**2*d**5*log(d/e + x)/e**6 + B*c**2*d**4*
x/e**5 - B*c**2*d**3*x**2/(2*e**4) + B*c**2*d**2*x**3/(3*e**3) - B*c**2*d*x**4/(4*e**2) + B*c**2*x**5/(5*e), E
q(m, -1)), (A*a**2*d*e**5*m**5*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e
**6*m**2 + 1764*e**6*m + 720*e**6) + 20*A*a**2*d*e**5*m**4*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m
**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 155*A*a**2*d*e**5*m**3*(d + e*x)**m/(e**6*m**
6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 580*A*a**2*d*e**
5*m**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m +
720*e**6) + 1044*A*a**2*d*e**5*m*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 162
4*e**6*m**2 + 1764*e**6*m + 720*e**6) + 720*A*a**2*d*e**5*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m*
*4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + A*a**2*e**6*m**5*x*(d + e*x)**m/(e**6*m**6 + 2
1*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 20*A*a**2*e**6*m**4*x
*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e
**6) + 155*A*a**2*e**6*m**3*x*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e*
*6*m**2 + 1764*e**6*m + 720*e**6) + 580*A*a**2*e**6*m**2*x*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m
**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 1044*A*a**2*e**6*m*x*(d + e*x)**m/(e**6*m**6
+ 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 720*A*a**2*e**6*x*
(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e*
*6) + 4*A*a*c*d**3*e**3*m**3*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**
6*m**2 + 1764*e**6*m + 720*e**6) + 60*A*a*c*d**3*e**3*m**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m
**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 296*A*a*c*d**3*e**3*m*(d + e*x)**m/(e**6*m**6
+ 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 480*A*a*c*d**3*e*
*3*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720
*e**6) - 4*A*a*c*d**2*e**4*m**4*x*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 162
4*e**6*m**2 + 1764*e**6*m + 720*e**6) - 60*A*a*c*d**2*e**4*m**3*x*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175
*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 296*A*a*c*d**2*e**4*m**2*x*(d + e*x)**
m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 480*A
*a*c*d**2*e**4*m*x*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1
764*e**6*m + 720*e**6) + 2*A*a*c*d*e**5*m**5*x**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735
*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 32*A*a*c*d*e**5*m**4*x**2*(d + e*x)**m/(e**6*m**6 + 21
*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 178*A*a*c*d*e**5*m**3*
x**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 7
20*e**6) + 388*A*a*c*d*e**5*m**2*x**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 +
1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 240*A*a*c*d*e**5*m*x**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 1
75*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 2*A*a*c*e**6*m**5*x**3*(d + e*x)**m/
(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 36*A*a*
c*e**6*m**4*x**3*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 176
4*e**6*m + 720*e**6) + 242*A*a*c*e**6*m**3*x**3*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e
**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 744*A*a*c*e**6*m**2*x**3*(d + e*x)**m/(e**6*m**6 + 21*e*
*6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 1016*A*a*c*e**6*m*x**3*(d
+ e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6
) + 480*A*a*c*e**6*x**3*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**
2 + 1764*e**6*m + 720*e**6) + 24*A*c**2*d**5*e*m*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*
e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 144*A*c**2*d**5*e*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**
5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 24*A*c**2*d**4*e**2*m**2*x*(d +
e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6)
- 144*A*c**2*d**4*e**2*m*x*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*
m**2 + 1764*e**6*m + 720*e**6) + 12*A*c**2*d**3*e**3*m**3*x**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e*
*6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 84*A*c**2*d**3*e**3*m**2*x**2*(d + e*x)**
m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 72*A*
c**2*d**3*e**3*m*x**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2
+ 1764*e**6*m + 720*e**6) - 4*A*c**2*d**2*e**4*m**4*x**3*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**
4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 36*A*c**2*d**2*e**4*m**3*x**3*(d + e*x)**m/(e**
6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 80*A*c**2*d
**2*e**4*m**2*x**3*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1
764*e**6*m + 720*e**6) - 48*A*c**2*d**2*e**4*m*x**3*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 7
35*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + A*c**2*d*e**5*m**5*x**4*(d + e*x)**m/(e**6*m**6 + 21
*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 12*A*c**2*d*e**5*m**4*
x**4*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 7
20*e**6) + 47*A*c**2*d*e**5*m**3*x**4*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 +
1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 72*A*c**2*d*e**5*m**2*x**4*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5
+ 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 36*A*c**2*d*e**5*m*x**4*(d + e*x)
**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + A*c
**2*e**6*m**5*x**5*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1
764*e**6*m + 720*e**6) + 16*A*c**2*e**6*m**4*x**5*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735
*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 95*A*c**2*e**6*m**3*x**5*(d + e*x)**m/(e**6*m**6 + 21*
e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 260*A*c**2*e**6*m**2*x*
*5*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720
*e**6) + 324*A*c**2*e**6*m*x**5*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*
e**6*m**2 + 1764*e**6*m + 720*e**6) + 144*A*c**2*e**6*x**5*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m
**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - B*a**2*d**2*e**4*m**4*(d + e*x)**m/(e**6*m**6
+ 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 18*B*a**2*d**2*e*
*4*m**3*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m
+ 720*e**6) - 119*B*a**2*d**2*e**4*m**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3
+ 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 342*B*a**2*d**2*e**4*m*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 +
175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 360*B*a**2*d**2*e**4*(d + e*x)**m/(
e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + B*a**2*d
*e**5*m**5*x*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e*
*6*m + 720*e**6) + 18*B*a**2*d*e**5*m**4*x*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m
**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 119*B*a**2*d*e**5*m**3*x*(d + e*x)**m/(e**6*m**6 + 21*e**6*m*
*5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 342*B*a**2*d*e**5*m**2*x*(d +
e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) +
360*B*a**2*d*e**5*m*x*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2
+ 1764*e**6*m + 720*e**6) + B*a**2*e**6*m**5*x**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 73
5*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 19*B*a**2*e**6*m**4*x**2*(d + e*x)**m/(e**6*m**6 + 21
*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 137*B*a**2*e**6*m**3*x
**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 72
0*e**6) + 461*B*a**2*e**6*m**2*x**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1
624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 702*B*a**2*e**6*m*x**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*
e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 360*B*a**2*e**6*x**2*(d + e*x)**m/(e**6
*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 12*B*a*c*d**
4*e**2*m**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**
6*m + 720*e**6) - 132*B*a*c*d**4*e**2*m*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3
+ 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 360*B*a*c*d**4*e**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175
*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 12*B*a*c*d**3*e**3*m**3*x*(d + e*x)**m
/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 132*B*
a*c*d**3*e**3*m**2*x*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 +
1764*e**6*m + 720*e**6) + 360*B*a*c*d**3*e**3*m*x*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 73
5*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 6*B*a*c*d**2*e**4*m**4*x**2*(d + e*x)**m/(e**6*m**6 +
21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 72*B*a*c*d**2*e**4*
m**3*x**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*
m + 720*e**6) - 246*B*a*c*d**2*e**4*m**2*x**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**
6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 180*B*a*c*d**2*e**4*m*x**2*(d + e*x)**m/(e**6*m**6 + 21*e*
*6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 2*B*a*c*d*e**5*m**5*x**3*
(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e*
*6) + 28*B*a*c*d*e**5*m**4*x**3*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*
e**6*m**2 + 1764*e**6*m + 720*e**6) + 130*B*a*c*d*e**5*m**3*x**3*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*
e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 224*B*a*c*d*e**5*m**2*x**3*(d + e*x)**m
/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 120*B*
a*c*d*e**5*m*x**3*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 17
64*e**6*m + 720*e**6) + 2*B*a*c*e**6*m**5*x**4*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e*
*6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 34*B*a*c*e**6*m**4*x**4*(d + e*x)**m/(e**6*m**6 + 21*e**6
*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 214*B*a*c*e**6*m**3*x**4*(d
+ e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6
) + 614*B*a*c*e**6*m**2*x**4*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**
6*m**2 + 1764*e**6*m + 720*e**6) + 792*B*a*c*e**6*m*x**4*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**
4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 360*B*a*c*e**6*x**4*(d + e*x)**m/(e**6*m**6 + 2
1*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 120*B*c**2*d**6*(d +
e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) +
120*B*c**2*d**5*e*m*x*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2
+ 1764*e**6*m + 720*e**6) - 60*B*c**2*d**4*e**2*m**2*x**2*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m
**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 60*B*c**2*d**4*e**2*m*x**2*(d + e*x)**m/(e**6
*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 20*B*c**2*d*
*3*e**3*m**3*x**3*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 17
64*e**6*m + 720*e**6) + 60*B*c**2*d**3*e**3*m**2*x**3*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 +
735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 40*B*c**2*d**3*e**3*m*x**3*(d + e*x)**m/(e**6*m**6
+ 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 5*B*c**2*d**2*e**
4*m**4*x**4*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**
6*m + 720*e**6) - 30*B*c**2*d**2*e**4*m**3*x**4*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e
**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 55*B*c**2*d**2*e**4*m**2*x**4*(d + e*x)**m/(e**6*m**6 +
21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) - 30*B*c**2*d**2*e**4*
m*x**4*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m +
720*e**6) + B*c**2*d*e**5*m**5*x**5*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 +
1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 10*B*c**2*d*e**5*m**4*x**5*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 +
175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 35*B*c**2*d*e**5*m**3*x**5*(d + e*
x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 5
0*B*c**2*d*e**5*m**2*x**5*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m
**2 + 1764*e**6*m + 720*e**6) + 24*B*c**2*d*e**5*m*x**5*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4
+ 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + B*c**2*e**6*m**5*x**6*(d + e*x)**m/(e**6*m**6 +
21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 15*B*c**2*e**6*m**4*
x**6*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 7
20*e**6) + 85*B*c**2*e**6*m**3*x**6*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1
624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 225*B*c**2*e**6*m**2*x**6*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 1
75*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 274*B*c**2*e**6*m*x**6*(d + e*x)**m/
(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 120*B*c
**2*e**6*x**6*(d + e*x)**m/(e**6*m**6 + 21*e**6*m**5 + 175*e**6*m**4 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e
**6*m + 720*e**6), True))

________________________________________________________________________________________

Giac [B]  time = 1.1882, size = 3287, normalized size = 14.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)^m*B*c^2*m^5*x^6*e^6 + (x*e + d)^m*B*c^2*d*m^5*x^5*e^5 + (x*e + d)^m*A*c^2*m^5*x^5*e^6 + 15*(x*e + d
)^m*B*c^2*m^4*x^6*e^6 + (x*e + d)^m*A*c^2*d*m^5*x^4*e^5 + 10*(x*e + d)^m*B*c^2*d*m^4*x^5*e^5 - 5*(x*e + d)^m*B
*c^2*d^2*m^4*x^4*e^4 + 2*(x*e + d)^m*B*a*c*m^5*x^4*e^6 + 16*(x*e + d)^m*A*c^2*m^4*x^5*e^6 + 85*(x*e + d)^m*B*c
^2*m^3*x^6*e^6 + 2*(x*e + d)^m*B*a*c*d*m^5*x^3*e^5 + 12*(x*e + d)^m*A*c^2*d*m^4*x^4*e^5 + 35*(x*e + d)^m*B*c^2
*d*m^3*x^5*e^5 - 4*(x*e + d)^m*A*c^2*d^2*m^4*x^3*e^4 - 30*(x*e + d)^m*B*c^2*d^2*m^3*x^4*e^4 + 20*(x*e + d)^m*B
*c^2*d^3*m^3*x^3*e^3 + 2*(x*e + d)^m*A*a*c*m^5*x^3*e^6 + 34*(x*e + d)^m*B*a*c*m^4*x^4*e^6 + 95*(x*e + d)^m*A*c
^2*m^3*x^5*e^6 + 225*(x*e + d)^m*B*c^2*m^2*x^6*e^6 + 2*(x*e + d)^m*A*a*c*d*m^5*x^2*e^5 + 28*(x*e + d)^m*B*a*c*
d*m^4*x^3*e^5 + 47*(x*e + d)^m*A*c^2*d*m^3*x^4*e^5 + 50*(x*e + d)^m*B*c^2*d*m^2*x^5*e^5 - 6*(x*e + d)^m*B*a*c*
d^2*m^4*x^2*e^4 - 36*(x*e + d)^m*A*c^2*d^2*m^3*x^3*e^4 - 55*(x*e + d)^m*B*c^2*d^2*m^2*x^4*e^4 + 12*(x*e + d)^m
*A*c^2*d^3*m^3*x^2*e^3 + 60*(x*e + d)^m*B*c^2*d^3*m^2*x^3*e^3 - 60*(x*e + d)^m*B*c^2*d^4*m^2*x^2*e^2 + (x*e +
d)^m*B*a^2*m^5*x^2*e^6 + 36*(x*e + d)^m*A*a*c*m^4*x^3*e^6 + 214*(x*e + d)^m*B*a*c*m^3*x^4*e^6 + 260*(x*e + d)^
m*A*c^2*m^2*x^5*e^6 + 274*(x*e + d)^m*B*c^2*m*x^6*e^6 + (x*e + d)^m*B*a^2*d*m^5*x*e^5 + 32*(x*e + d)^m*A*a*c*d
*m^4*x^2*e^5 + 130*(x*e + d)^m*B*a*c*d*m^3*x^3*e^5 + 72*(x*e + d)^m*A*c^2*d*m^2*x^4*e^5 + 24*(x*e + d)^m*B*c^2
*d*m*x^5*e^5 - 4*(x*e + d)^m*A*a*c*d^2*m^4*x*e^4 - 72*(x*e + d)^m*B*a*c*d^2*m^3*x^2*e^4 - 80*(x*e + d)^m*A*c^2
*d^2*m^2*x^3*e^4 - 30*(x*e + d)^m*B*c^2*d^2*m*x^4*e^4 + 12*(x*e + d)^m*B*a*c*d^3*m^3*x*e^3 + 84*(x*e + d)^m*A*
c^2*d^3*m^2*x^2*e^3 + 40*(x*e + d)^m*B*c^2*d^3*m*x^3*e^3 - 24*(x*e + d)^m*A*c^2*d^4*m^2*x*e^2 - 60*(x*e + d)^m
*B*c^2*d^4*m*x^2*e^2 + 120*(x*e + d)^m*B*c^2*d^5*m*x*e + (x*e + d)^m*A*a^2*m^5*x*e^6 + 19*(x*e + d)^m*B*a^2*m^
4*x^2*e^6 + 242*(x*e + d)^m*A*a*c*m^3*x^3*e^6 + 614*(x*e + d)^m*B*a*c*m^2*x^4*e^6 + 324*(x*e + d)^m*A*c^2*m*x^
5*e^6 + 120*(x*e + d)^m*B*c^2*x^6*e^6 + (x*e + d)^m*A*a^2*d*m^5*e^5 + 18*(x*e + d)^m*B*a^2*d*m^4*x*e^5 + 178*(
x*e + d)^m*A*a*c*d*m^3*x^2*e^5 + 224*(x*e + d)^m*B*a*c*d*m^2*x^3*e^5 + 36*(x*e + d)^m*A*c^2*d*m*x^4*e^5 - (x*e
+ d)^m*B*a^2*d^2*m^4*e^4 - 60*(x*e + d)^m*A*a*c*d^2*m^3*x*e^4 - 246*(x*e + d)^m*B*a*c*d^2*m^2*x^2*e^4 - 48*(x
*e + d)^m*A*c^2*d^2*m*x^3*e^4 + 4*(x*e + d)^m*A*a*c*d^3*m^3*e^3 + 132*(x*e + d)^m*B*a*c*d^3*m^2*x*e^3 + 72*(x*
e + d)^m*A*c^2*d^3*m*x^2*e^3 - 12*(x*e + d)^m*B*a*c*d^4*m^2*e^2 - 144*(x*e + d)^m*A*c^2*d^4*m*x*e^2 + 24*(x*e
+ d)^m*A*c^2*d^5*m*e - 120*(x*e + d)^m*B*c^2*d^6 + 20*(x*e + d)^m*A*a^2*m^4*x*e^6 + 137*(x*e + d)^m*B*a^2*m^3*
x^2*e^6 + 744*(x*e + d)^m*A*a*c*m^2*x^3*e^6 + 792*(x*e + d)^m*B*a*c*m*x^4*e^6 + 144*(x*e + d)^m*A*c^2*x^5*e^6
+ 20*(x*e + d)^m*A*a^2*d*m^4*e^5 + 119*(x*e + d)^m*B*a^2*d*m^3*x*e^5 + 388*(x*e + d)^m*A*a*c*d*m^2*x^2*e^5 + 1
20*(x*e + d)^m*B*a*c*d*m*x^3*e^5 - 18*(x*e + d)^m*B*a^2*d^2*m^3*e^4 - 296*(x*e + d)^m*A*a*c*d^2*m^2*x*e^4 - 18
0*(x*e + d)^m*B*a*c*d^2*m*x^2*e^4 + 60*(x*e + d)^m*A*a*c*d^3*m^2*e^3 + 360*(x*e + d)^m*B*a*c*d^3*m*x*e^3 - 132
*(x*e + d)^m*B*a*c*d^4*m*e^2 + 144*(x*e + d)^m*A*c^2*d^5*e + 155*(x*e + d)^m*A*a^2*m^3*x*e^6 + 461*(x*e + d)^m
*B*a^2*m^2*x^2*e^6 + 1016*(x*e + d)^m*A*a*c*m*x^3*e^6 + 360*(x*e + d)^m*B*a*c*x^4*e^6 + 155*(x*e + d)^m*A*a^2*
d*m^3*e^5 + 342*(x*e + d)^m*B*a^2*d*m^2*x*e^5 + 240*(x*e + d)^m*A*a*c*d*m*x^2*e^5 - 119*(x*e + d)^m*B*a^2*d^2*
m^2*e^4 - 480*(x*e + d)^m*A*a*c*d^2*m*x*e^4 + 296*(x*e + d)^m*A*a*c*d^3*m*e^3 - 360*(x*e + d)^m*B*a*c*d^4*e^2
+ 580*(x*e + d)^m*A*a^2*m^2*x*e^6 + 702*(x*e + d)^m*B*a^2*m*x^2*e^6 + 480*(x*e + d)^m*A*a*c*x^3*e^6 + 580*(x*e
+ d)^m*A*a^2*d*m^2*e^5 + 360*(x*e + d)^m*B*a^2*d*m*x*e^5 - 342*(x*e + d)^m*B*a^2*d^2*m*e^4 + 480*(x*e + d)^m*
A*a*c*d^3*e^3 + 1044*(x*e + d)^m*A*a^2*m*x*e^6 + 360*(x*e + d)^m*B*a^2*x^2*e^6 + 1044*(x*e + d)^m*A*a^2*d*m*e^
5 - 360*(x*e + d)^m*B*a^2*d^2*e^4 + 720*(x*e + d)^m*A*a^2*x*e^6 + 720*(x*e + d)^m*A*a^2*d*e^5)/(m^6*e^6 + 21*m
^5*e^6 + 175*m^4*e^6 + 735*m^3*e^6 + 1624*m^2*e^6 + 1764*m*e^6 + 720*e^6)