3.2147 \(\int (a+b x) (d+e x)^m (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=175 \[ -\frac{10 b^2 (b d-a e)^3 (d+e x)^{m+3}}{e^6 (m+3)}+\frac{10 b^3 (b d-a e)^2 (d+e x)^{m+4}}{e^6 (m+4)}-\frac{5 b^4 (b d-a e) (d+e x)^{m+5}}{e^6 (m+5)}-\frac{(b d-a e)^5 (d+e x)^{m+1}}{e^6 (m+1)}+\frac{5 b (b d-a e)^4 (d+e x)^{m+2}}{e^6 (m+2)}+\frac{b^5 (d+e x)^{m+6}}{e^6 (m+6)} \]

[Out]

-(((b*d - a*e)^5*(d + e*x)^(1 + m))/(e^6*(1 + m))) + (5*b*(b*d - a*e)^4*(d + e*x)^(2 + m))/(e^6*(2 + m)) - (10
*b^2*(b*d - a*e)^3*(d + e*x)^(3 + m))/(e^6*(3 + m)) + (10*b^3*(b*d - a*e)^2*(d + e*x)^(4 + m))/(e^6*(4 + m)) -
(5*b^4*(b*d - a*e)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (b^5*(d + e*x)^(6 + m))/(e^6*(6 + m))

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Rubi [A]  time = 0.085458, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ -\frac{10 b^2 (b d-a e)^3 (d+e x)^{m+3}}{e^6 (m+3)}+\frac{10 b^3 (b d-a e)^2 (d+e x)^{m+4}}{e^6 (m+4)}-\frac{5 b^4 (b d-a e) (d+e x)^{m+5}}{e^6 (m+5)}-\frac{(b d-a e)^5 (d+e x)^{m+1}}{e^6 (m+1)}+\frac{5 b (b d-a e)^4 (d+e x)^{m+2}}{e^6 (m+2)}+\frac{b^5 (d+e x)^{m+6}}{e^6 (m+6)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((b*d - a*e)^5*(d + e*x)^(1 + m))/(e^6*(1 + m))) + (5*b*(b*d - a*e)^4*(d + e*x)^(2 + m))/(e^6*(2 + m)) - (10
*b^2*(b*d - a*e)^3*(d + e*x)^(3 + m))/(e^6*(3 + m)) + (10*b^3*(b*d - a*e)^2*(d + e*x)^(4 + m))/(e^6*(4 + m)) -
(5*b^4*(b*d - a*e)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (b^5*(d + e*x)^(6 + m))/(e^6*(6 + m))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^5 (d+e x)^m \, dx\\ &=\int \left (\frac{(-b d+a e)^5 (d+e x)^m}{e^5}+\frac{5 b (b d-a e)^4 (d+e x)^{1+m}}{e^5}-\frac{10 b^2 (b d-a e)^3 (d+e x)^{2+m}}{e^5}+\frac{10 b^3 (b d-a e)^2 (d+e x)^{3+m}}{e^5}-\frac{5 b^4 (b d-a e) (d+e x)^{4+m}}{e^5}+\frac{b^5 (d+e x)^{5+m}}{e^5}\right ) \, dx\\ &=-\frac{(b d-a e)^5 (d+e x)^{1+m}}{e^6 (1+m)}+\frac{5 b (b d-a e)^4 (d+e x)^{2+m}}{e^6 (2+m)}-\frac{10 b^2 (b d-a e)^3 (d+e x)^{3+m}}{e^6 (3+m)}+\frac{10 b^3 (b d-a e)^2 (d+e x)^{4+m}}{e^6 (4+m)}-\frac{5 b^4 (b d-a e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac{b^5 (d+e x)^{6+m}}{e^6 (6+m)}\\ \end{align*}

Mathematica [A]  time = 0.148806, size = 149, normalized size = 0.85 \[ \frac{(d+e x)^{m+1} \left (-\frac{10 b^2 (d+e x)^2 (b d-a e)^3}{m+3}+\frac{10 b^3 (d+e x)^3 (b d-a e)^2}{m+4}-\frac{5 b^4 (d+e x)^4 (b d-a e)}{m+5}+\frac{5 b (d+e x) (b d-a e)^4}{m+2}-\frac{(b d-a e)^5}{m+1}+\frac{b^5 (d+e x)^5}{m+6}\right )}{e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*(-((b*d - a*e)^5/(1 + m)) + (5*b*(b*d - a*e)^4*(d + e*x))/(2 + m) - (10*b^2*(b*d - a*e)^3*(
d + e*x)^2)/(3 + m) + (10*b^3*(b*d - a*e)^2*(d + e*x)^3)/(4 + m) - (5*b^4*(b*d - a*e)*(d + e*x)^4)/(5 + m) + (
b^5*(d + e*x)^5)/(6 + m)))/e^6

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Maple [B]  time = 0.012, size = 1345, normalized size = 7.7 \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*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

(e*x+d)^(1+m)*(b^5*e^5*m^5*x^5+5*a*b^4*e^5*m^5*x^4+15*b^5*e^5*m^4*x^5+10*a^2*b^3*e^5*m^5*x^3+80*a*b^4*e^5*m^4*
x^4-5*b^5*d*e^4*m^4*x^4+85*b^5*e^5*m^3*x^5+10*a^3*b^2*e^5*m^5*x^2+170*a^2*b^3*e^5*m^4*x^3-20*a*b^4*d*e^4*m^4*x
^3+475*a*b^4*e^5*m^3*x^4-50*b^5*d*e^4*m^3*x^4+225*b^5*e^5*m^2*x^5+5*a^4*b*e^5*m^5*x+180*a^3*b^2*e^5*m^4*x^2-30
*a^2*b^3*d*e^4*m^4*x^2+1070*a^2*b^3*e^5*m^3*x^3-240*a*b^4*d*e^4*m^3*x^3+1300*a*b^4*e^5*m^2*x^4+20*b^5*d^2*e^3*
m^3*x^3-175*b^5*d*e^4*m^2*x^4+274*b^5*e^5*m*x^5+a^5*e^5*m^5+95*a^4*b*e^5*m^4*x-20*a^3*b^2*d*e^4*m^4*x+1210*a^3
*b^2*e^5*m^3*x^2-420*a^2*b^3*d*e^4*m^3*x^2+3070*a^2*b^3*e^5*m^2*x^3+60*a*b^4*d^2*e^3*m^3*x^2-940*a*b^4*d*e^4*m
^2*x^3+1620*a*b^4*e^5*m*x^4+120*b^5*d^2*e^3*m^2*x^3-250*b^5*d*e^4*m*x^4+120*b^5*e^5*x^5+20*a^5*e^5*m^4-5*a^4*b
*d*e^4*m^4+685*a^4*b*e^5*m^3*x-320*a^3*b^2*d*e^4*m^3*x+3720*a^3*b^2*e^5*m^2*x^2+60*a^2*b^3*d^2*e^3*m^3*x-1950*
a^2*b^3*d*e^4*m^2*x^2+3960*a^2*b^3*e^5*m*x^3+540*a*b^4*d^2*e^3*m^2*x^2-1440*a*b^4*d*e^4*m*x^3+720*a*b^4*e^5*x^
4-60*b^5*d^3*e^2*m^2*x^2+220*b^5*d^2*e^3*m*x^3-120*b^5*d*e^4*x^4+155*a^5*e^5*m^3-90*a^4*b*d*e^4*m^3+2305*a^4*b
*e^5*m^2*x+20*a^3*b^2*d^2*e^3*m^3-1780*a^3*b^2*d*e^4*m^2*x+5080*a^3*b^2*e^5*m*x^2+720*a^2*b^3*d^2*e^3*m^2*x-33
60*a^2*b^3*d*e^4*m*x^2+1800*a^2*b^3*e^5*x^3-120*a*b^4*d^3*e^2*m^2*x+1200*a*b^4*d^2*e^3*m*x^2-720*a*b^4*d*e^4*x
^3-180*b^5*d^3*e^2*m*x^2+120*b^5*d^2*e^3*x^3+580*a^5*e^5*m^2-595*a^4*b*d*e^4*m^2+3510*a^4*b*e^5*m*x+300*a^3*b^
2*d^2*e^3*m^2-3880*a^3*b^2*d*e^4*m*x+2400*a^3*b^2*e^5*x^2-60*a^2*b^3*d^3*e^2*m^2+2460*a^2*b^3*d^2*e^3*m*x-1800
*a^2*b^3*d*e^4*x^2-840*a*b^4*d^3*e^2*m*x+720*a*b^4*d^2*e^3*x^2+120*b^5*d^4*e*m*x-120*b^5*d^3*e^2*x^2+1044*a^5*
e^5*m-1710*a^4*b*d*e^4*m+1800*a^4*b*e^5*x+1480*a^3*b^2*d^2*e^3*m-2400*a^3*b^2*d*e^4*x-660*a^2*b^3*d^3*e^2*m+18
00*a^2*b^3*d^2*e^3*x+120*a*b^4*d^4*e*m-720*a*b^4*d^3*e^2*x+120*b^5*d^4*e*x+720*a^5*e^5-1800*a^4*b*d*e^4+2400*a
^3*b^2*d^2*e^3-1800*a^2*b^3*d^3*e^2+720*a*b^4*d^4*e-120*b^5*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*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.14491, size = 3060, normalized size = 17.49 \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*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

(a^5*d*e^5*m^5 - 120*b^5*d^6 + 720*a*b^4*d^5*e - 1800*a^2*b^3*d^4*e^2 + 2400*a^3*b^2*d^3*e^3 - 1800*a^4*b*d^2*
e^4 + 720*a^5*d*e^5 + (b^5*e^6*m^5 + 15*b^5*e^6*m^4 + 85*b^5*e^6*m^3 + 225*b^5*e^6*m^2 + 274*b^5*e^6*m + 120*b
^5*e^6)*x^6 + (720*a*b^4*e^6 + (b^5*d*e^5 + 5*a*b^4*e^6)*m^5 + 10*(b^5*d*e^5 + 8*a*b^4*e^6)*m^4 + 5*(7*b^5*d*e
^5 + 95*a*b^4*e^6)*m^3 + 50*(b^5*d*e^5 + 26*a*b^4*e^6)*m^2 + 12*(2*b^5*d*e^5 + 135*a*b^4*e^6)*m)*x^5 - 5*(a^4*
b*d^2*e^4 - 4*a^5*d*e^5)*m^4 + 5*(360*a^2*b^3*e^6 + (a*b^4*d*e^5 + 2*a^2*b^3*e^6)*m^5 - (b^5*d^2*e^4 - 12*a*b^
4*d*e^5 - 34*a^2*b^3*e^6)*m^4 - (6*b^5*d^2*e^4 - 47*a*b^4*d*e^5 - 214*a^2*b^3*e^6)*m^3 - (11*b^5*d^2*e^4 - 72*
a*b^4*d*e^5 - 614*a^2*b^3*e^6)*m^2 - 6*(b^5*d^2*e^4 - 6*a*b^4*d*e^5 - 132*a^2*b^3*e^6)*m)*x^4 + 5*(4*a^3*b^2*d
^3*e^3 - 18*a^4*b*d^2*e^4 + 31*a^5*d*e^5)*m^3 + 10*(240*a^3*b^2*e^6 + (a^2*b^3*d*e^5 + a^3*b^2*e^6)*m^5 - 2*(a
*b^4*d^2*e^4 - 7*a^2*b^3*d*e^5 - 9*a^3*b^2*e^6)*m^4 + (2*b^5*d^3*e^3 - 18*a*b^4*d^2*e^4 + 65*a^2*b^3*d*e^5 + 1
21*a^3*b^2*e^6)*m^3 + 2*(3*b^5*d^3*e^3 - 20*a*b^4*d^2*e^4 + 56*a^2*b^3*d*e^5 + 186*a^3*b^2*e^6)*m^2 + 4*(b^5*d
^3*e^3 - 6*a*b^4*d^2*e^4 + 15*a^2*b^3*d*e^5 + 127*a^3*b^2*e^6)*m)*x^3 - 5*(12*a^2*b^3*d^4*e^2 - 60*a^3*b^2*d^3
*e^3 + 119*a^4*b*d^2*e^4 - 116*a^5*d*e^5)*m^2 + 5*(360*a^4*b*e^6 + (2*a^3*b^2*d*e^5 + a^4*b*e^6)*m^5 - (6*a^2*
b^3*d^2*e^4 - 32*a^3*b^2*d*e^5 - 19*a^4*b*e^6)*m^4 + (12*a*b^4*d^3*e^3 - 72*a^2*b^3*d^2*e^4 + 178*a^3*b^2*d*e^
5 + 137*a^4*b*e^6)*m^3 - (12*b^5*d^4*e^2 - 84*a*b^4*d^3*e^3 + 246*a^2*b^3*d^2*e^4 - 388*a^3*b^2*d*e^5 - 461*a^
4*b*e^6)*m^2 - 6*(2*b^5*d^4*e^2 - 12*a*b^4*d^3*e^3 + 30*a^2*b^3*d^2*e^4 - 40*a^3*b^2*d*e^5 - 117*a^4*b*e^6)*m)
*x^2 + 2*(60*a*b^4*d^5*e - 330*a^2*b^3*d^4*e^2 + 740*a^3*b^2*d^3*e^3 - 855*a^4*b*d^2*e^4 + 522*a^5*d*e^5)*m +
(720*a^5*e^6 + (5*a^4*b*d*e^5 + a^5*e^6)*m^5 - 10*(2*a^3*b^2*d^2*e^4 - 9*a^4*b*d*e^5 - 2*a^5*e^6)*m^4 + 5*(12*
a^2*b^3*d^3*e^3 - 60*a^3*b^2*d^2*e^4 + 119*a^4*b*d*e^5 + 31*a^5*e^6)*m^3 - 10*(12*a*b^4*d^4*e^2 - 66*a^2*b^3*d
^3*e^3 + 148*a^3*b^2*d^2*e^4 - 171*a^4*b*d*e^5 - 58*a^5*e^6)*m^2 + 12*(10*b^5*d^5*e - 60*a*b^4*d^4*e^2 + 150*a
^2*b^3*d^3*e^3 - 200*a^3*b^2*d^2*e^4 + 150*a^4*b*d*e^5 + 87*a^5*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 = 15.9207, size = 16058, normalized size = 91.76 \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*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Piecewise((d**m*(a**5*x + 5*a**4*b*x**2/2 + 10*a**3*b**2*x**3/3 + 5*a**2*b**3*x**4/2 + a*b**4*x**5 + b**5*x**6
/6), Eq(e, 0)), (-12*a**5*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) - 15*a**4*b*d**3*e**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) - 75*a**4*b*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) - 20
*a**3*b**2*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**1
0*x**4 + 60*d**2*e**11*x**5) - 100*a**3*b**2*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) - 200*a**3*b**2*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) + 150*
a**2*b**3*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) + 30*a**2*b**3*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*a*b**4*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) + 60*b**5*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**5*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**5*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**5*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**5*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**10*x**4 + 60*d**2*e**11*x**5) + 200*b**5*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**5*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**5*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**5*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**
5*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**5*d**2*e**5*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), Eq(m, -6)), (-3*a**5*d*e**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) - 5*a**4*b*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) - 20*a**4*b*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) - 10*a**3*b**2*d**3*e**3/(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**3*b**2*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) - 60*a**3*b**2*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) + 30*a**2
*b**3*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
*a*b**4*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**1
0*x**4) + 65*a*b**4*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) + 240*a*b**4*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) + 200*a*b**4*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) + 360*a*b**4*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) + 180*a*b**4*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) + 240*a*b**4*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) + 60*a*b**4*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) - 60*a*b**4*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) - 60*b**5
*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**5*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**5*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**5*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**5*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**5*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**5*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**5*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**5*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**5*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**5*e**5/(6*d**3
*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) - 5*a**4*b*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) - 15*a**4*b*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) - 20
*a**3*b**2*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) - 60*a**3*b**2*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) - 60*a**3*b**2*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) + 60*a**2*b**3*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) + 110*a**2*b**3*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) + 180*a**2*b**3*d**2*e**3*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) + 2
70*a**2*b**3*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) + 180*a**2*b**3*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) + 180*a**2*b**3*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) + 60*a**2*b**3*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) - 120*a*b**4*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) - 220*a*b**4*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) - 360*a*b**4*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) - 5
40*a*b**4*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) - 360*a*b**4*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) - 360*a*b**4*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) - 120*a*b**4*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) + 30*a*b**4*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) + 60*b**5*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**5*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**5*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**5*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**5*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**5*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**5*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**5*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**5*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**5*e**5/(6*d**2*e**6 + 12*d*e**7*x + 6*e*
*8*x**2) - 15*a**4*b*d*e**4/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 30*a**4*b*e**5*x/(6*d**2*e**6 + 12*d*e
**7*x + 6*e**8*x**2) + 60*a**3*b**2*d**2*e**3*log(d/e + x)/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 90*a**3
*b**2*d**2*e**3/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 120*a**3*b**2*d*e**4*x*log(d/e + x)/(6*d**2*e**6 +
12*d*e**7*x + 6*e**8*x**2) + 120*a**3*b**2*d*e**4*x/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 60*a**3*b**2*
e**5*x**2*log(d/e + x)/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 180*a**2*b**3*d**3*e**2*log(d/e + x)/(6*d**
2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 270*a**2*b**3*d**3*e**2/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 360*
a**2*b**3*d**2*e**3*x*log(d/e + x)/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 360*a**2*b**3*d**2*e**3*x/(6*d*
*2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 180*a**2*b**3*d*e**4*x**2*log(d/e + x)/(6*d**2*e**6 + 12*d*e**7*x + 6*e
**8*x**2) + 60*a**2*b**3*e**5*x**3/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 180*a*b**4*d**4*e*log(d/e + x)/
(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 270*a*b**4*d**4*e/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 360*
a*b**4*d**3*e**2*x*log(d/e + x)/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 360*a*b**4*d**3*e**2*x/(6*d**2*e**
6 + 12*d*e**7*x + 6*e**8*x**2) + 180*a*b**4*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) - 60*a*b**4*d*e**4*x**3/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 15*a*b**4*e**5*x**4/(6*d**2*e**6 + 12*
d*e**7*x + 6*e**8*x**2) - 60*b**5*d**5*log(d/e + x)/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 90*b**5*d**5/(
6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 120*b**5*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**5*d**4*e*x/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 60*b**5*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**5*d**2*e**3*x**3/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) - 5*b
**5*d*e**4*x**4/(6*d**2*e**6 + 12*d*e**7*x + 6*e**8*x**2) + 2*b**5*e**5*x**5/(6*d**2*e**6 + 12*d*e**7*x + 6*e*
*8*x**2), Eq(m, -3)), (-12*a**5*e**5/(12*d*e**6 + 12*e**7*x) + 60*a**4*b*d*e**4*log(d/e + x)/(12*d*e**6 + 12*e
**7*x) + 60*a**4*b*d*e**4/(12*d*e**6 + 12*e**7*x) + 60*a**4*b*e**5*x*log(d/e + x)/(12*d*e**6 + 12*e**7*x) - 24
0*a**3*b**2*d**2*e**3*log(d/e + x)/(12*d*e**6 + 12*e**7*x) - 240*a**3*b**2*d**2*e**3/(12*d*e**6 + 12*e**7*x) -
240*a**3*b**2*d*e**4*x*log(d/e + x)/(12*d*e**6 + 12*e**7*x) + 120*a**3*b**2*e**5*x**2/(12*d*e**6 + 12*e**7*x)
+ 360*a**2*b**3*d**3*e**2*log(d/e + x)/(12*d*e**6 + 12*e**7*x) + 360*a**2*b**3*d**3*e**2/(12*d*e**6 + 12*e**7
*x) + 360*a**2*b**3*d**2*e**3*x*log(d/e + x)/(12*d*e**6 + 12*e**7*x) - 180*a**2*b**3*d*e**4*x**2/(12*d*e**6 +
12*e**7*x) + 60*a**2*b**3*e**5*x**3/(12*d*e**6 + 12*e**7*x) - 240*a*b**4*d**4*e*log(d/e + x)/(12*d*e**6 + 12*e
**7*x) - 240*a*b**4*d**4*e/(12*d*e**6 + 12*e**7*x) - 240*a*b**4*d**3*e**2*x*log(d/e + x)/(12*d*e**6 + 12*e**7*
x) + 120*a*b**4*d**2*e**3*x**2/(12*d*e**6 + 12*e**7*x) - 40*a*b**4*d*e**4*x**3/(12*d*e**6 + 12*e**7*x) + 20*a*
b**4*e**5*x**4/(12*d*e**6 + 12*e**7*x) + 60*b**5*d**5*log(d/e + x)/(12*d*e**6 + 12*e**7*x) + 60*b**5*d**5/(12*
d*e**6 + 12*e**7*x) + 60*b**5*d**4*e*x*log(d/e + x)/(12*d*e**6 + 12*e**7*x) - 30*b**5*d**3*e**2*x**2/(12*d*e**
6 + 12*e**7*x) + 10*b**5*d**2*e**3*x**3/(12*d*e**6 + 12*e**7*x) - 5*b**5*d*e**4*x**4/(12*d*e**6 + 12*e**7*x) +
3*b**5*e**5*x**5/(12*d*e**6 + 12*e**7*x), Eq(m, -2)), (a**5*log(d/e + x)/e - 5*a**4*b*d*log(d/e + x)/e**2 + 5
*a**4*b*x/e + 10*a**3*b**2*d**2*log(d/e + x)/e**3 - 10*a**3*b**2*d*x/e**2 + 5*a**3*b**2*x**2/e - 10*a**2*b**3*
d**3*log(d/e + x)/e**4 + 10*a**2*b**3*d**2*x/e**3 - 5*a**2*b**3*d*x**2/e**2 + 10*a**2*b**3*x**3/(3*e) + 5*a*b*
*4*d**4*log(d/e + x)/e**5 - 5*a*b**4*d**3*x/e**4 + 5*a*b**4*d**2*x**2/(2*e**3) - 5*a*b**4*d*x**3/(3*e**2) + 5*
a*b**4*x**4/(4*e) - b**5*d**5*log(d/e + x)/e**6 + b**5*d**4*x/e**5 - b**5*d**3*x**2/(2*e**4) + b**5*d**2*x**3/
(3*e**3) - b**5*d*x**4/(4*e**2) + b**5*x**5/(5*e), Eq(m, -1)), (a**5*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**5*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**5*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**5*d*e**5*m**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) + 1044*a**5*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 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 720*a**5*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**5*e
**6*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) + 20*a**5*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**5*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**5*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**5*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**5*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) - 5*a**4*b*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) - 90*a**4*b*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) - 595*a**4*b*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) - 1710*a**4*b*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) - 1800*a**4*b*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) + 5*a**4*b*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) + 90*a**
4*b*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 + 17
64*e**6*m + 720*e**6) + 595*a**4*b*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) + 1710*a**4*b*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) + 1800*a**4*b*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) + 5*a**4*b*e**6*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) + 95*a**4*b*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) + 685*a**4*b*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 + 720*e**6) + 2305*a**4*b
*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 + 1624*e**6*m**2 + 1764
*e**6*m + 720*e**6) + 3510*a**4*b*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) + 1800*a**4*b*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) + 20*a**3*b**2*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
) + 300*a**3*b**2*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) + 1480*a**3*b**2*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) + 2400*a**3*b**2*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) - 20*a**3
*b**2*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 + 1624*e**6*m**2
+ 1764*e**6*m + 720*e**6) - 300*a**3*b**2*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) - 1480*a**3*b**2*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) - 2400*a
**3*b**2*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
+ 1764*e**6*m + 720*e**6) + 10*a**3*b**2*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) + 160*a**3*b**2*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) + 890*a**3
*b**2*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 + 720*e**6) + 1940*a**3*b**2*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) + 1200*a**3*b**2*d*e**5*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) + 10*a**3*
b**2*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) + 180*a**3*b**2*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 + 1764*e**6*m + 720*e**6) + 1210*a**3*b**2*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) + 3720*a**3*b**
2*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 + 176
4*e**6*m + 720*e**6) + 5080*a**3*b**2*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) + 2400*a**3*b**2*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) - 60*a**2*b**3*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 + 7
20*e**6) - 660*a**2*b**3*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) - 1800*a**2*b**3*d**4*e**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) + 60*a**2*b**3*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) +
660*a**2*b**3*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) + 1800*a**2*b**3*d**3*e**3*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) - 30*a**2*b**3*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) -
360*a**2*b**3*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 + 16
24*e**6*m**2 + 1764*e**6*m + 720*e**6) - 1230*a**2*b**3*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) - 900*a**2*b**3*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) + 10*a**2*b**3*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) + 140*a**2*b**3*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) + 650*a**2*b**3*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) + 1120*a**2*b**3*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) + 600*a**2*b**3*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 + 1764*e**6*m + 720*e**6) + 10*a**2*b**3*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) + 170*a**2*b**3*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 + 162
4*e**6*m**2 + 1764*e**6*m + 720*e**6) + 1070*a**2*b**3*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) + 3070*a**2*b**3*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)
+ 3960*a**2*b**3*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) + 1800*a**2*b**3*e**6*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) + 120*a*b**4*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) + 720*a*b**4*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) - 120*a*b**4*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 + 162
4*e**6*m**2 + 1764*e**6*m + 720*e**6) - 720*a*b**4*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) + 60*a*b**4*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) + 420
*a*b**4*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) + 360*a*b**4*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) - 20*a*b**4*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) - 180*a*
b**4*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) - 400*a*b**4*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 + 1764*e**6*m + 720*e**6) - 240*a*b**4*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 + 735*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 5*a*b**4
*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 + 17
64*e**6*m + 720*e**6) + 60*a*b**4*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 + 73
5*e**6*m**3 + 1624*e**6*m**2 + 1764*e**6*m + 720*e**6) + 235*a*b**4*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) + 360*a*b**4*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) + 180*a*b**4*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) + 5*a*b**4*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 + 1764*e**6*m + 720*e**6) + 80*a*b**4*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) + 475
*a*b**4*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) + 1300*a*b**4*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) + 1620*a*b**4*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) + 720*a*b**4*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) - 120*b**5*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**5*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**5*d**4*e**2*m**2*x**2*(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) - 60*b**5*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 + 72
0*e**6) + 20*b**5*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 + 1764*e**6*m + 720*e**6) + 60*b**5*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**5*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**5*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**5*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**5*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**5*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**5*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**5*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**5*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) +
50*b**5*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**5*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**5*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**5*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 + 720*e**
6) + 85*b**5*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 + 1624*e**6
*m**2 + 1764*e**6*m + 720*e**6) + 225*b**5*e**6*m**2*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) + 274*b**5*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**5*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 = 1.20169, size = 3409, normalized size = 19.48 \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*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

((x*e + d)^m*b^5*m^5*x^6*e^6 + (x*e + d)^m*b^5*d*m^5*x^5*e^5 + 5*(x*e + d)^m*a*b^4*m^5*x^5*e^6 + 15*(x*e + d)^
m*b^5*m^4*x^6*e^6 + 5*(x*e + d)^m*a*b^4*d*m^5*x^4*e^5 + 10*(x*e + d)^m*b^5*d*m^4*x^5*e^5 - 5*(x*e + d)^m*b^5*d
^2*m^4*x^4*e^4 + 10*(x*e + d)^m*a^2*b^3*m^5*x^4*e^6 + 80*(x*e + d)^m*a*b^4*m^4*x^5*e^6 + 85*(x*e + d)^m*b^5*m^
3*x^6*e^6 + 10*(x*e + d)^m*a^2*b^3*d*m^5*x^3*e^5 + 60*(x*e + d)^m*a*b^4*d*m^4*x^4*e^5 + 35*(x*e + d)^m*b^5*d*m
^3*x^5*e^5 - 20*(x*e + d)^m*a*b^4*d^2*m^4*x^3*e^4 - 30*(x*e + d)^m*b^5*d^2*m^3*x^4*e^4 + 20*(x*e + d)^m*b^5*d^
3*m^3*x^3*e^3 + 10*(x*e + d)^m*a^3*b^2*m^5*x^3*e^6 + 170*(x*e + d)^m*a^2*b^3*m^4*x^4*e^6 + 475*(x*e + d)^m*a*b
^4*m^3*x^5*e^6 + 225*(x*e + d)^m*b^5*m^2*x^6*e^6 + 10*(x*e + d)^m*a^3*b^2*d*m^5*x^2*e^5 + 140*(x*e + d)^m*a^2*
b^3*d*m^4*x^3*e^5 + 235*(x*e + d)^m*a*b^4*d*m^3*x^4*e^5 + 50*(x*e + d)^m*b^5*d*m^2*x^5*e^5 - 30*(x*e + d)^m*a^
2*b^3*d^2*m^4*x^2*e^4 - 180*(x*e + d)^m*a*b^4*d^2*m^3*x^3*e^4 - 55*(x*e + d)^m*b^5*d^2*m^2*x^4*e^4 + 60*(x*e +
d)^m*a*b^4*d^3*m^3*x^2*e^3 + 60*(x*e + d)^m*b^5*d^3*m^2*x^3*e^3 - 60*(x*e + d)^m*b^5*d^4*m^2*x^2*e^2 + 5*(x*e
+ d)^m*a^4*b*m^5*x^2*e^6 + 180*(x*e + d)^m*a^3*b^2*m^4*x^3*e^6 + 1070*(x*e + d)^m*a^2*b^3*m^3*x^4*e^6 + 1300*
(x*e + d)^m*a*b^4*m^2*x^5*e^6 + 274*(x*e + d)^m*b^5*m*x^6*e^6 + 5*(x*e + d)^m*a^4*b*d*m^5*x*e^5 + 160*(x*e + d
)^m*a^3*b^2*d*m^4*x^2*e^5 + 650*(x*e + d)^m*a^2*b^3*d*m^3*x^3*e^5 + 360*(x*e + d)^m*a*b^4*d*m^2*x^4*e^5 + 24*(
x*e + d)^m*b^5*d*m*x^5*e^5 - 20*(x*e + d)^m*a^3*b^2*d^2*m^4*x*e^4 - 360*(x*e + d)^m*a^2*b^3*d^2*m^3*x^2*e^4 -
400*(x*e + d)^m*a*b^4*d^2*m^2*x^3*e^4 - 30*(x*e + d)^m*b^5*d^2*m*x^4*e^4 + 60*(x*e + d)^m*a^2*b^3*d^3*m^3*x*e^
3 + 420*(x*e + d)^m*a*b^4*d^3*m^2*x^2*e^3 + 40*(x*e + d)^m*b^5*d^3*m*x^3*e^3 - 120*(x*e + d)^m*a*b^4*d^4*m^2*x
*e^2 - 60*(x*e + d)^m*b^5*d^4*m*x^2*e^2 + 120*(x*e + d)^m*b^5*d^5*m*x*e + (x*e + d)^m*a^5*m^5*x*e^6 + 95*(x*e
+ d)^m*a^4*b*m^4*x^2*e^6 + 1210*(x*e + d)^m*a^3*b^2*m^3*x^3*e^6 + 3070*(x*e + d)^m*a^2*b^3*m^2*x^4*e^6 + 1620*
(x*e + d)^m*a*b^4*m*x^5*e^6 + 120*(x*e + d)^m*b^5*x^6*e^6 + (x*e + d)^m*a^5*d*m^5*e^5 + 90*(x*e + d)^m*a^4*b*d
*m^4*x*e^5 + 890*(x*e + d)^m*a^3*b^2*d*m^3*x^2*e^5 + 1120*(x*e + d)^m*a^2*b^3*d*m^2*x^3*e^5 + 180*(x*e + d)^m*
a*b^4*d*m*x^4*e^5 - 5*(x*e + d)^m*a^4*b*d^2*m^4*e^4 - 300*(x*e + d)^m*a^3*b^2*d^2*m^3*x*e^4 - 1230*(x*e + d)^m
*a^2*b^3*d^2*m^2*x^2*e^4 - 240*(x*e + d)^m*a*b^4*d^2*m*x^3*e^4 + 20*(x*e + d)^m*a^3*b^2*d^3*m^3*e^3 + 660*(x*e
+ d)^m*a^2*b^3*d^3*m^2*x*e^3 + 360*(x*e + d)^m*a*b^4*d^3*m*x^2*e^3 - 60*(x*e + d)^m*a^2*b^3*d^4*m^2*e^2 - 720
*(x*e + d)^m*a*b^4*d^4*m*x*e^2 + 120*(x*e + d)^m*a*b^4*d^5*m*e - 120*(x*e + d)^m*b^5*d^6 + 20*(x*e + d)^m*a^5*
m^4*x*e^6 + 685*(x*e + d)^m*a^4*b*m^3*x^2*e^6 + 3720*(x*e + d)^m*a^3*b^2*m^2*x^3*e^6 + 3960*(x*e + d)^m*a^2*b^
3*m*x^4*e^6 + 720*(x*e + d)^m*a*b^4*x^5*e^6 + 20*(x*e + d)^m*a^5*d*m^4*e^5 + 595*(x*e + d)^m*a^4*b*d*m^3*x*e^5
+ 1940*(x*e + d)^m*a^3*b^2*d*m^2*x^2*e^5 + 600*(x*e + d)^m*a^2*b^3*d*m*x^3*e^5 - 90*(x*e + d)^m*a^4*b*d^2*m^3
*e^4 - 1480*(x*e + d)^m*a^3*b^2*d^2*m^2*x*e^4 - 900*(x*e + d)^m*a^2*b^3*d^2*m*x^2*e^4 + 300*(x*e + d)^m*a^3*b^
2*d^3*m^2*e^3 + 1800*(x*e + d)^m*a^2*b^3*d^3*m*x*e^3 - 660*(x*e + d)^m*a^2*b^3*d^4*m*e^2 + 720*(x*e + d)^m*a*b
^4*d^5*e + 155*(x*e + d)^m*a^5*m^3*x*e^6 + 2305*(x*e + d)^m*a^4*b*m^2*x^2*e^6 + 5080*(x*e + d)^m*a^3*b^2*m*x^3
*e^6 + 1800*(x*e + d)^m*a^2*b^3*x^4*e^6 + 155*(x*e + d)^m*a^5*d*m^3*e^5 + 1710*(x*e + d)^m*a^4*b*d*m^2*x*e^5 +
1200*(x*e + d)^m*a^3*b^2*d*m*x^2*e^5 - 595*(x*e + d)^m*a^4*b*d^2*m^2*e^4 - 2400*(x*e + d)^m*a^3*b^2*d^2*m*x*e
^4 + 1480*(x*e + d)^m*a^3*b^2*d^3*m*e^3 - 1800*(x*e + d)^m*a^2*b^3*d^4*e^2 + 580*(x*e + d)^m*a^5*m^2*x*e^6 + 3
510*(x*e + d)^m*a^4*b*m*x^2*e^6 + 2400*(x*e + d)^m*a^3*b^2*x^3*e^6 + 580*(x*e + d)^m*a^5*d*m^2*e^5 + 1800*(x*e
+ d)^m*a^4*b*d*m*x*e^5 - 1710*(x*e + d)^m*a^4*b*d^2*m*e^4 + 2400*(x*e + d)^m*a^3*b^2*d^3*e^3 + 1044*(x*e + d)
^m*a^5*m*x*e^6 + 1800*(x*e + d)^m*a^4*b*x^2*e^6 + 1044*(x*e + d)^m*a^5*d*m*e^5 - 1800*(x*e + d)^m*a^4*b*d^2*e^
4 + 720*(x*e + d)^m*a^5*x*e^6 + 720*(x*e + d)^m*a^5*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)