3.721 \(\int (d+e x)^m (a+c x^2)^3 \, dx\)

Optimal. Leaf size=223 \[ -\frac{4 c^2 d \left (3 a e^2+5 c d^2\right ) (d+e x)^{m+4}}{e^7 (m+4)}+\frac{3 c^2 \left (a e^2+5 c d^2\right ) (d+e x)^{m+5}}{e^7 (m+5)}+\frac{\left (a e^2+c d^2\right )^3 (d+e x)^{m+1}}{e^7 (m+1)}-\frac{6 c d \left (a e^2+c d^2\right )^2 (d+e x)^{m+2}}{e^7 (m+2)}+\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) (d+e x)^{m+3}}{e^7 (m+3)}-\frac{6 c^3 d (d+e x)^{m+6}}{e^7 (m+6)}+\frac{c^3 (d+e x)^{m+7}}{e^7 (m+7)} \]

[Out]

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

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Rubi [A]  time = 0.133102, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {697} \[ -\frac{4 c^2 d \left (3 a e^2+5 c d^2\right ) (d+e x)^{m+4}}{e^7 (m+4)}+\frac{3 c^2 \left (a e^2+5 c d^2\right ) (d+e x)^{m+5}}{e^7 (m+5)}+\frac{\left (a e^2+c d^2\right )^3 (d+e x)^{m+1}}{e^7 (m+1)}-\frac{6 c d \left (a e^2+c d^2\right )^2 (d+e x)^{m+2}}{e^7 (m+2)}+\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) (d+e x)^{m+3}}{e^7 (m+3)}-\frac{6 c^3 d (d+e x)^{m+6}}{e^7 (m+6)}+\frac{c^3 (d+e x)^{m+7}}{e^7 (m+7)} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(a + c*x^2)^3,x]

[Out]

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

Rule 697

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

Rubi steps

\begin{align*} \int (d+e x)^m \left (a+c x^2\right )^3 \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^3 (d+e x)^m}{e^6}-\frac{6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{1+m}}{e^6}+\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{2+m}}{e^6}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{3+m}}{e^6}+\frac{3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{4+m}}{e^6}-\frac{6 c^3 d (d+e x)^{5+m}}{e^6}+\frac{c^3 (d+e x)^{6+m}}{e^6}\right ) \, dx\\ &=\frac{\left (c d^2+a e^2\right )^3 (d+e x)^{1+m}}{e^7 (1+m)}-\frac{6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{2+m}}{e^7 (2+m)}+\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{3+m}}{e^7 (3+m)}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac{3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{5+m}}{e^7 (5+m)}-\frac{6 c^3 d (d+e x)^{6+m}}{e^7 (6+m)}+\frac{c^3 (d+e x)^{7+m}}{e^7 (7+m)}\\ \end{align*}

Mathematica [A]  time = 0.699113, size = 379, normalized size = 1.7 \[ \frac{(d+e x)^{m+1} \left (\frac{6 \left ((m+6) \left (a e^2+c d^2\right ) \left (4 (m+4) \left (a e^2+c d^2\right ) \left (a e^2 \left (m^2+5 m+6\right )+c \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )-4 c d (m+1) (d+e x) \left (a e^2 \left (m^2+7 m+12\right )+c \left (2 d^2-2 d e (m+2) x+e^2 \left (m^2+5 m+6\right ) x^2\right )\right )+e^4 (m+1) (m+2) (m+3) (m+4) \left (a+c x^2\right )^2\right )-c d (m+1) (d+e x) \left (4 (m+5) \left (a e^2+c d^2\right ) \left (a e^2 \left (m^2+7 m+12\right )+c \left (2 d^2-2 d e (m+2) x+e^2 \left (m^2+5 m+6\right ) x^2\right )\right )-4 c d (m+2) (d+e x) \left (a e^2 \left (m^2+9 m+20\right )+c \left (2 d^2-2 d e (m+3) x+e^2 \left (m^2+7 m+12\right ) x^2\right )\right )+e^4 (m+2) (m+3) (m+4) (m+5) \left (a+c x^2\right )^2\right )\right )}{e^6 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6)}+\left (a+c x^2\right )^3\right )}{e (m+7)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^m*(a + c*x^2)^3,x]

[Out]

((d + e*x)^(1 + m)*((a + c*x^2)^3 + (6*((c*d^2 + a*e^2)*(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)))
- c*d*(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*(2
0 + 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))))/(e*(7 + m))

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Maple [B]  time = 0.049, size = 1140, normalized size = 5.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(c*x^2+a)^3,x)

[Out]

(e*x+d)^(1+m)*(c^3*e^6*m^6*x^6+21*c^3*e^6*m^5*x^6+3*a*c^2*e^6*m^6*x^4-6*c^3*d*e^5*m^5*x^5+175*c^3*e^6*m^4*x^6+
69*a*c^2*e^6*m^5*x^4-90*c^3*d*e^5*m^4*x^5+735*c^3*e^6*m^3*x^6+3*a^2*c*e^6*m^6*x^2-12*a*c^2*d*e^5*m^5*x^3+621*a
*c^2*e^6*m^4*x^4+30*c^3*d^2*e^4*m^4*x^4-510*c^3*d*e^5*m^3*x^5+1624*c^3*e^6*m^2*x^6+75*a^2*c*e^6*m^5*x^2-228*a*
c^2*d*e^5*m^4*x^3+2775*a*c^2*e^6*m^3*x^4+300*c^3*d^2*e^4*m^3*x^4-1350*c^3*d*e^5*m^2*x^5+1764*c^3*e^6*m*x^6+a^3
*e^6*m^6-6*a^2*c*d*e^5*m^5*x+741*a^2*c*e^6*m^4*x^2+36*a*c^2*d^2*e^4*m^4*x^2-1572*a*c^2*d*e^5*m^3*x^3+6432*a*c^
2*e^6*m^2*x^4-120*c^3*d^3*e^3*m^3*x^3+1050*c^3*d^2*e^4*m^2*x^4-1644*c^3*d*e^5*m*x^5+720*c^3*e^6*x^6+27*a^3*e^6
*m^5-138*a^2*c*d*e^5*m^4*x+3657*a^2*c*e^6*m^3*x^2+576*a*c^2*d^2*e^4*m^3*x^2-4812*a*c^2*d*e^5*m^2*x^3+7236*a*c^
2*e^6*m*x^4-720*c^3*d^3*e^3*m^2*x^3+1500*c^3*d^2*e^4*m*x^4-720*c^3*d*e^5*x^5+295*a^3*e^6*m^4+6*a^2*c*d^2*e^4*m
^4-1206*a^2*c*d*e^5*m^3*x+9336*a^2*c*e^6*m^2*x^2-72*a*c^2*d^3*e^3*m^3*x+2988*a*c^2*d^2*e^4*m^2*x^2-6480*a*c^2*
d*e^5*m*x^3+3024*a*c^2*e^6*x^4+360*c^3*d^4*e^2*m^2*x^2-1320*c^3*d^3*e^3*m*x^3+720*c^3*d^2*e^4*x^4+1665*a^3*e^6
*m^3+132*a^2*c*d^2*e^4*m^3-4902*a^2*c*d*e^5*m^2*x+11388*a^2*c*e^6*m*x^2-1008*a*c^2*d^3*e^3*m^2*x+5472*a*c^2*d^
2*e^4*m*x^2-3024*a*c^2*d*e^5*x^3+1080*c^3*d^4*e^2*m*x^2-720*c^3*d^3*e^3*x^3+5104*a^3*e^6*m^2+1074*a^2*c*d^2*e^
4*m^2-8868*a^2*c*d*e^5*m*x+5040*a^2*c*e^6*x^2+72*a*c^2*d^4*e^2*m^2-3960*a*c^2*d^3*e^3*m*x+3024*a*c^2*d^2*e^4*x
^2-720*c^3*d^5*e*m*x+720*c^3*d^4*e^2*x^2+8028*a^3*e^6*m+3828*a^2*c*d^2*e^4*m-5040*a^2*c*d*e^5*x+936*a*c^2*d^4*
e^2*m-3024*a*c^2*d^3*e^3*x-720*c^3*d^5*e*x+5040*a^3*e^6+5040*a^2*c*d^2*e^4+3024*a*c^2*d^4*e^2+720*c^3*d^6)/e^7
/(m^7+28*m^6+322*m^5+1960*m^4+6769*m^3+13132*m^2+13068*m+5040)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.2828, size = 2692, normalized size = 12.07 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*x^2+a)^3,x, algorithm="fricas")

[Out]

(a^3*d*e^6*m^6 + 27*a^3*d*e^6*m^5 + 720*c^3*d^7 + 3024*a*c^2*d^5*e^2 + 5040*a^2*c*d^3*e^4 + 5040*a^3*d*e^6 + (
c^3*e^7*m^6 + 21*c^3*e^7*m^5 + 175*c^3*e^7*m^4 + 735*c^3*e^7*m^3 + 1624*c^3*e^7*m^2 + 1764*c^3*e^7*m + 720*c^3
*e^7)*x^7 + (c^3*d*e^6*m^6 + 15*c^3*d*e^6*m^5 + 85*c^3*d*e^6*m^4 + 225*c^3*d*e^6*m^3 + 274*c^3*d*e^6*m^2 + 120
*c^3*d*e^6*m)*x^6 + 3*(a*c^2*e^7*m^6 + 1008*a*c^2*e^7 - (2*c^3*d^2*e^5 - 23*a*c^2*e^7)*m^5 - (20*c^3*d^2*e^5 -
207*a*c^2*e^7)*m^4 - 5*(14*c^3*d^2*e^5 - 185*a*c^2*e^7)*m^3 - 4*(25*c^3*d^2*e^5 - 536*a*c^2*e^7)*m^2 - 12*(4*
c^3*d^2*e^5 - 201*a*c^2*e^7)*m)*x^5 + (6*a^2*c*d^3*e^4 + 295*a^3*d*e^6)*m^4 + 3*(a*c^2*d*e^6*m^6 + 19*a*c^2*d*
e^6*m^5 + (10*c^3*d^3*e^4 + 131*a*c^2*d*e^6)*m^4 + (60*c^3*d^3*e^4 + 401*a*c^2*d*e^6)*m^3 + 10*(11*c^3*d^3*e^4
+ 54*a*c^2*d*e^6)*m^2 + 12*(5*c^3*d^3*e^4 + 21*a*c^2*d*e^6)*m)*x^4 + 3*(44*a^2*c*d^3*e^4 + 555*a^3*d*e^6)*m^3
+ 3*(a^2*c*e^7*m^6 + 1680*a^2*c*e^7 - (4*a*c^2*d^2*e^5 - 25*a^2*c*e^7)*m^5 - (64*a*c^2*d^2*e^5 - 247*a^2*c*e^
7)*m^4 - (40*c^3*d^4*e^3 + 332*a*c^2*d^2*e^5 - 1219*a^2*c*e^7)*m^3 - 8*(15*c^3*d^4*e^3 + 76*a*c^2*d^2*e^5 - 38
9*a^2*c*e^7)*m^2 - 4*(20*c^3*d^4*e^3 + 84*a*c^2*d^2*e^5 - 949*a^2*c*e^7)*m)*x^3 + 2*(36*a*c^2*d^5*e^2 + 537*a^
2*c*d^3*e^4 + 2552*a^3*d*e^6)*m^2 + 3*(a^2*c*d*e^6*m^6 + 23*a^2*c*d*e^6*m^5 + 3*(4*a*c^2*d^3*e^4 + 67*a^2*c*d*
e^6)*m^4 + (168*a*c^2*d^3*e^4 + 817*a^2*c*d*e^6)*m^3 + 2*(60*c^3*d^5*e^2 + 330*a*c^2*d^3*e^4 + 739*a^2*c*d*e^6
)*m^2 + 24*(5*c^3*d^5*e^2 + 21*a*c^2*d^3*e^4 + 35*a^2*c*d*e^6)*m)*x^2 + 12*(78*a*c^2*d^5*e^2 + 319*a^2*c*d^3*e
^4 + 669*a^3*d*e^6)*m + (a^3*e^7*m^6 + 5040*a^3*e^7 - 3*(2*a^2*c*d^2*e^5 - 9*a^3*e^7)*m^5 - (132*a^2*c*d^2*e^5
- 295*a^3*e^7)*m^4 - 3*(24*a*c^2*d^4*e^3 + 358*a^2*c*d^2*e^5 - 555*a^3*e^7)*m^3 - 4*(234*a*c^2*d^4*e^3 + 957*
a^2*c*d^2*e^5 - 1276*a^3*e^7)*m^2 - 36*(20*c^3*d^6*e + 84*a*c^2*d^4*e^3 + 140*a^2*c*d^2*e^5 - 223*a^3*e^7)*m)*
x)*(e*x + d)^m/(e^7*m^7 + 28*e^7*m^6 + 322*e^7*m^5 + 1960*e^7*m^4 + 6769*e^7*m^3 + 13132*e^7*m^2 + 13068*e^7*m
+ 5040*e^7)

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Sympy [A]  time = 18.1348, size = 15781, normalized size = 70.77 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*(c*x**2+a)**3,x)

[Out]

Piecewise((d**m*(a**3*x + a**2*c*x**3 + 3*a*c**2*x**5/5 + c**3*x**7/7), Eq(e, 0)), (-10*a**3*d**4*e**6/(60*d**
10*e**7 + 360*d**9*e**8*x + 900*d**8*e**9*x**2 + 1200*d**7*e**10*x**3 + 900*d**6*e**11*x**4 + 360*d**5*e**12*x
**5 + 60*d**4*e**13*x**6) + 60*a**2*c*d**3*e**7*x**3/(60*d**10*e**7 + 360*d**9*e**8*x + 900*d**8*e**9*x**2 + 1
200*d**7*e**10*x**3 + 900*d**6*e**11*x**4 + 360*d**5*e**12*x**5 + 60*d**4*e**13*x**6) + 45*a**2*c*d**2*e**8*x*
*4/(60*d**10*e**7 + 360*d**9*e**8*x + 900*d**8*e**9*x**2 + 1200*d**7*e**10*x**3 + 900*d**6*e**11*x**4 + 360*d*
*5*e**12*x**5 + 60*d**4*e**13*x**6) + 18*a**2*c*d*e**9*x**5/(60*d**10*e**7 + 360*d**9*e**8*x + 900*d**8*e**9*x
**2 + 1200*d**7*e**10*x**3 + 900*d**6*e**11*x**4 + 360*d**5*e**12*x**5 + 60*d**4*e**13*x**6) + 3*a**2*c*e**10*
x**6/(60*d**10*e**7 + 360*d**9*e**8*x + 900*d**8*e**9*x**2 + 1200*d**7*e**10*x**3 + 900*d**6*e**11*x**4 + 360*
d**5*e**12*x**5 + 60*d**4*e**13*x**6) + 36*a*c**2*d**3*e**7*x**5/(60*d**10*e**7 + 360*d**9*e**8*x + 900*d**8*e
**9*x**2 + 1200*d**7*e**10*x**3 + 900*d**6*e**11*x**4 + 360*d**5*e**12*x**5 + 60*d**4*e**13*x**6) + 6*a*c**2*d
**2*e**8*x**6/(60*d**10*e**7 + 360*d**9*e**8*x + 900*d**8*e**9*x**2 + 1200*d**7*e**10*x**3 + 900*d**6*e**11*x*
*4 + 360*d**5*e**12*x**5 + 60*d**4*e**13*x**6) + 60*c**3*d**10*log(d/e + x)/(60*d**10*e**7 + 360*d**9*e**8*x +
900*d**8*e**9*x**2 + 1200*d**7*e**10*x**3 + 900*d**6*e**11*x**4 + 360*d**5*e**12*x**5 + 60*d**4*e**13*x**6) +
22*c**3*d**10/(60*d**10*e**7 + 360*d**9*e**8*x + 900*d**8*e**9*x**2 + 1200*d**7*e**10*x**3 + 900*d**6*e**11*x
**4 + 360*d**5*e**12*x**5 + 60*d**4*e**13*x**6) + 360*c**3*d**9*e*x*log(d/e + x)/(60*d**10*e**7 + 360*d**9*e**
8*x + 900*d**8*e**9*x**2 + 1200*d**7*e**10*x**3 + 900*d**6*e**11*x**4 + 360*d**5*e**12*x**5 + 60*d**4*e**13*x*
*6) + 72*c**3*d**9*e*x/(60*d**10*e**7 + 360*d**9*e**8*x + 900*d**8*e**9*x**2 + 1200*d**7*e**10*x**3 + 900*d**6
*e**11*x**4 + 360*d**5*e**12*x**5 + 60*d**4*e**13*x**6) + 900*c**3*d**8*e**2*x**2*log(d/e + x)/(60*d**10*e**7
+ 360*d**9*e**8*x + 900*d**8*e**9*x**2 + 1200*d**7*e**10*x**3 + 900*d**6*e**11*x**4 + 360*d**5*e**12*x**5 + 60
*d**4*e**13*x**6) + 1200*c**3*d**7*e**3*x**3*log(d/e + x)/(60*d**10*e**7 + 360*d**9*e**8*x + 900*d**8*e**9*x**
2 + 1200*d**7*e**10*x**3 + 900*d**6*e**11*x**4 + 360*d**5*e**12*x**5 + 60*d**4*e**13*x**6) - 300*c**3*d**7*e**
3*x**3/(60*d**10*e**7 + 360*d**9*e**8*x + 900*d**8*e**9*x**2 + 1200*d**7*e**10*x**3 + 900*d**6*e**11*x**4 + 36
0*d**5*e**12*x**5 + 60*d**4*e**13*x**6) + 900*c**3*d**6*e**4*x**4*log(d/e + x)/(60*d**10*e**7 + 360*d**9*e**8*
x + 900*d**8*e**9*x**2 + 1200*d**7*e**10*x**3 + 900*d**6*e**11*x**4 + 360*d**5*e**12*x**5 + 60*d**4*e**13*x**6
) - 525*c**3*d**6*e**4*x**4/(60*d**10*e**7 + 360*d**9*e**8*x + 900*d**8*e**9*x**2 + 1200*d**7*e**10*x**3 + 900
*d**6*e**11*x**4 + 360*d**5*e**12*x**5 + 60*d**4*e**13*x**6) + 360*c**3*d**5*e**5*x**5*log(d/e + x)/(60*d**10*
e**7 + 360*d**9*e**8*x + 900*d**8*e**9*x**2 + 1200*d**7*e**10*x**3 + 900*d**6*e**11*x**4 + 360*d**5*e**12*x**5
+ 60*d**4*e**13*x**6) - 390*c**3*d**5*e**5*x**5/(60*d**10*e**7 + 360*d**9*e**8*x + 900*d**8*e**9*x**2 + 1200*
d**7*e**10*x**3 + 900*d**6*e**11*x**4 + 360*d**5*e**12*x**5 + 60*d**4*e**13*x**6) + 60*c**3*d**4*e**6*x**6*log
(d/e + x)/(60*d**10*e**7 + 360*d**9*e**8*x + 900*d**8*e**9*x**2 + 1200*d**7*e**10*x**3 + 900*d**6*e**11*x**4 +
360*d**5*e**12*x**5 + 60*d**4*e**13*x**6) - 125*c**3*d**4*e**6*x**6/(60*d**10*e**7 + 360*d**9*e**8*x + 900*d*
*8*e**9*x**2 + 1200*d**7*e**10*x**3 + 900*d**6*e**11*x**4 + 360*d**5*e**12*x**5 + 60*d**4*e**13*x**6), Eq(m, -
7)), (-2*a**3*d**3*e**6/(10*d**8*e**7 + 50*d**7*e**8*x + 100*d**6*e**9*x**2 + 100*d**5*e**10*x**3 + 50*d**4*e*
*11*x**4 + 10*d**3*e**12*x**5) + 10*a**2*c*d**2*e**7*x**3/(10*d**8*e**7 + 50*d**7*e**8*x + 100*d**6*e**9*x**2
+ 100*d**5*e**10*x**3 + 50*d**4*e**11*x**4 + 10*d**3*e**12*x**5) + 5*a**2*c*d*e**8*x**4/(10*d**8*e**7 + 50*d**
7*e**8*x + 100*d**6*e**9*x**2 + 100*d**5*e**10*x**3 + 50*d**4*e**11*x**4 + 10*d**3*e**12*x**5) + a**2*c*e**9*x
**5/(10*d**8*e**7 + 50*d**7*e**8*x + 100*d**6*e**9*x**2 + 100*d**5*e**10*x**3 + 50*d**4*e**11*x**4 + 10*d**3*e
**12*x**5) + 6*a*c**2*d**2*e**7*x**5/(10*d**8*e**7 + 50*d**7*e**8*x + 100*d**6*e**9*x**2 + 100*d**5*e**10*x**3
+ 50*d**4*e**11*x**4 + 10*d**3*e**12*x**5) - 60*c**3*d**9*log(d/e + x)/(10*d**8*e**7 + 50*d**7*e**8*x + 100*d
**6*e**9*x**2 + 100*d**5*e**10*x**3 + 50*d**4*e**11*x**4 + 10*d**3*e**12*x**5) - 27*c**3*d**9/(10*d**8*e**7 +
50*d**7*e**8*x + 100*d**6*e**9*x**2 + 100*d**5*e**10*x**3 + 50*d**4*e**11*x**4 + 10*d**3*e**12*x**5) - 300*c**
3*d**8*e*x*log(d/e + x)/(10*d**8*e**7 + 50*d**7*e**8*x + 100*d**6*e**9*x**2 + 100*d**5*e**10*x**3 + 50*d**4*e*
*11*x**4 + 10*d**3*e**12*x**5) - 75*c**3*d**8*e*x/(10*d**8*e**7 + 50*d**7*e**8*x + 100*d**6*e**9*x**2 + 100*d*
*5*e**10*x**3 + 50*d**4*e**11*x**4 + 10*d**3*e**12*x**5) - 600*c**3*d**7*e**2*x**2*log(d/e + x)/(10*d**8*e**7
+ 50*d**7*e**8*x + 100*d**6*e**9*x**2 + 100*d**5*e**10*x**3 + 50*d**4*e**11*x**4 + 10*d**3*e**12*x**5) - 600*c
**3*d**6*e**3*x**3*log(d/e + x)/(10*d**8*e**7 + 50*d**7*e**8*x + 100*d**6*e**9*x**2 + 100*d**5*e**10*x**3 + 50
*d**4*e**11*x**4 + 10*d**3*e**12*x**5) + 200*c**3*d**6*e**3*x**3/(10*d**8*e**7 + 50*d**7*e**8*x + 100*d**6*e**
9*x**2 + 100*d**5*e**10*x**3 + 50*d**4*e**11*x**4 + 10*d**3*e**12*x**5) - 300*c**3*d**5*e**4*x**4*log(d/e + x)
/(10*d**8*e**7 + 50*d**7*e**8*x + 100*d**6*e**9*x**2 + 100*d**5*e**10*x**3 + 50*d**4*e**11*x**4 + 10*d**3*e**1
2*x**5) + 250*c**3*d**5*e**4*x**4/(10*d**8*e**7 + 50*d**7*e**8*x + 100*d**6*e**9*x**2 + 100*d**5*e**10*x**3 +
50*d**4*e**11*x**4 + 10*d**3*e**12*x**5) - 60*c**3*d**4*e**5*x**5*log(d/e + x)/(10*d**8*e**7 + 50*d**7*e**8*x
+ 100*d**6*e**9*x**2 + 100*d**5*e**10*x**3 + 50*d**4*e**11*x**4 + 10*d**3*e**12*x**5) + 110*c**3*d**4*e**5*x**
5/(10*d**8*e**7 + 50*d**7*e**8*x + 100*d**6*e**9*x**2 + 100*d**5*e**10*x**3 + 50*d**4*e**11*x**4 + 10*d**3*e**
12*x**5) + 10*c**3*d**3*e**6*x**6/(10*d**8*e**7 + 50*d**7*e**8*x + 100*d**6*e**9*x**2 + 100*d**5*e**10*x**3 +
50*d**4*e**11*x**4 + 10*d**3*e**12*x**5), Eq(m, -6)), (-a**3*d**2*e**6/(4*d**6*e**7 + 16*d**5*e**8*x + 24*d**4
*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) + 4*a**2*c*d*e**7*x**3/(4*d**6*e**7 + 16*d**5*e**8*x + 24
*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) + a**2*c*e**8*x**4/(4*d**6*e**7 + 16*d**5*e**8*x + 2
4*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) + 12*a*c**2*d**6*e**2*log(d/e + x)/(4*d**6*e**7 + 1
6*d**5*e**8*x + 24*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) + 7*a*c**2*d**6*e**2/(4*d**6*e**7
+ 16*d**5*e**8*x + 24*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) + 48*a*c**2*d**5*e**3*x*log(d/e
+ x)/(4*d**6*e**7 + 16*d**5*e**8*x + 24*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) + 16*a*c**2*
d**5*e**3*x/(4*d**6*e**7 + 16*d**5*e**8*x + 24*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) + 72*a
*c**2*d**4*e**4*x**2*log(d/e + x)/(4*d**6*e**7 + 16*d**5*e**8*x + 24*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d
**2*e**11*x**4) + 48*a*c**2*d**3*e**5*x**3*log(d/e + x)/(4*d**6*e**7 + 16*d**5*e**8*x + 24*d**4*e**9*x**2 + 16
*d**3*e**10*x**3 + 4*d**2*e**11*x**4) - 24*a*c**2*d**3*e**5*x**3/(4*d**6*e**7 + 16*d**5*e**8*x + 24*d**4*e**9*
x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) + 12*a*c**2*d**2*e**6*x**4*log(d/e + x)/(4*d**6*e**7 + 16*d**5*
e**8*x + 24*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) - 18*a*c**2*d**2*e**6*x**4/(4*d**6*e**7 +
16*d**5*e**8*x + 24*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) + 60*c**3*d**8*log(d/e + x)/(4*d
**6*e**7 + 16*d**5*e**8*x + 24*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) + 35*c**3*d**8/(4*d**6
*e**7 + 16*d**5*e**8*x + 24*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) + 240*c**3*d**7*e*x*log(d
/e + x)/(4*d**6*e**7 + 16*d**5*e**8*x + 24*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) + 80*c**3*
d**7*e*x/(4*d**6*e**7 + 16*d**5*e**8*x + 24*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) + 360*c**
3*d**6*e**2*x**2*log(d/e + x)/(4*d**6*e**7 + 16*d**5*e**8*x + 24*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*
e**11*x**4) + 240*c**3*d**5*e**3*x**3*log(d/e + x)/(4*d**6*e**7 + 16*d**5*e**8*x + 24*d**4*e**9*x**2 + 16*d**3
*e**10*x**3 + 4*d**2*e**11*x**4) - 120*c**3*d**5*e**3*x**3/(4*d**6*e**7 + 16*d**5*e**8*x + 24*d**4*e**9*x**2 +
16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) + 60*c**3*d**4*e**4*x**4*log(d/e + x)/(4*d**6*e**7 + 16*d**5*e**8*x +
24*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) - 90*c**3*d**4*e**4*x**4/(4*d**6*e**7 + 16*d**5*e
**8*x + 24*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) - 12*c**3*d**3*e**5*x**5/(4*d**6*e**7 + 16
*d**5*e**8*x + 24*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4) + 2*c**3*d**2*e**6*x**6/(4*d**6*e**
7 + 16*d**5*e**8*x + 24*d**4*e**9*x**2 + 16*d**3*e**10*x**3 + 4*d**2*e**11*x**4), Eq(m, -5)), (-a**3*d*e**6/(3
*d**4*e**7 + 9*d**3*e**8*x + 9*d**2*e**9*x**2 + 3*d*e**10*x**3) + 3*a**2*c*e**7*x**3/(3*d**4*e**7 + 9*d**3*e**
8*x + 9*d**2*e**9*x**2 + 3*d*e**10*x**3) - 36*a*c**2*d**5*e**2*log(d/e + x)/(3*d**4*e**7 + 9*d**3*e**8*x + 9*d
**2*e**9*x**2 + 3*d*e**10*x**3) - 30*a*c**2*d**5*e**2/(3*d**4*e**7 + 9*d**3*e**8*x + 9*d**2*e**9*x**2 + 3*d*e*
*10*x**3) - 108*a*c**2*d**4*e**3*x*log(d/e + x)/(3*d**4*e**7 + 9*d**3*e**8*x + 9*d**2*e**9*x**2 + 3*d*e**10*x*
*3) - 54*a*c**2*d**4*e**3*x/(3*d**4*e**7 + 9*d**3*e**8*x + 9*d**2*e**9*x**2 + 3*d*e**10*x**3) - 108*a*c**2*d**
3*e**4*x**2*log(d/e + x)/(3*d**4*e**7 + 9*d**3*e**8*x + 9*d**2*e**9*x**2 + 3*d*e**10*x**3) - 36*a*c**2*d**2*e*
*5*x**3*log(d/e + x)/(3*d**4*e**7 + 9*d**3*e**8*x + 9*d**2*e**9*x**2 + 3*d*e**10*x**3) + 36*a*c**2*d**2*e**5*x
**3/(3*d**4*e**7 + 9*d**3*e**8*x + 9*d**2*e**9*x**2 + 3*d*e**10*x**3) + 9*a*c**2*d*e**6*x**4/(3*d**4*e**7 + 9*
d**3*e**8*x + 9*d**2*e**9*x**2 + 3*d*e**10*x**3) - 60*c**3*d**7*log(d/e + x)/(3*d**4*e**7 + 9*d**3*e**8*x + 9*
d**2*e**9*x**2 + 3*d*e**10*x**3) - 50*c**3*d**7/(3*d**4*e**7 + 9*d**3*e**8*x + 9*d**2*e**9*x**2 + 3*d*e**10*x*
*3) - 180*c**3*d**6*e*x*log(d/e + x)/(3*d**4*e**7 + 9*d**3*e**8*x + 9*d**2*e**9*x**2 + 3*d*e**10*x**3) - 90*c*
*3*d**6*e*x/(3*d**4*e**7 + 9*d**3*e**8*x + 9*d**2*e**9*x**2 + 3*d*e**10*x**3) - 180*c**3*d**5*e**2*x**2*log(d/
e + x)/(3*d**4*e**7 + 9*d**3*e**8*x + 9*d**2*e**9*x**2 + 3*d*e**10*x**3) - 60*c**3*d**4*e**3*x**3*log(d/e + x)
/(3*d**4*e**7 + 9*d**3*e**8*x + 9*d**2*e**9*x**2 + 3*d*e**10*x**3) + 60*c**3*d**4*e**3*x**3/(3*d**4*e**7 + 9*d
**3*e**8*x + 9*d**2*e**9*x**2 + 3*d*e**10*x**3) + 15*c**3*d**3*e**4*x**4/(3*d**4*e**7 + 9*d**3*e**8*x + 9*d**2
*e**9*x**2 + 3*d*e**10*x**3) - 3*c**3*d**2*e**5*x**5/(3*d**4*e**7 + 9*d**3*e**8*x + 9*d**2*e**9*x**2 + 3*d*e**
10*x**3) + c**3*d*e**6*x**6/(3*d**4*e**7 + 9*d**3*e**8*x + 9*d**2*e**9*x**2 + 3*d*e**10*x**3), Eq(m, -4)), (-2
*a**3*e**6/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 12*a**2*c*d**2*e**4*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8
*x + 4*e**9*x**2) + 18*a**2*c*d**2*e**4/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 24*a**2*c*d*e**5*x*log(d/e
+ x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 24*a**2*c*d*e**5*x/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) +
12*a**2*c*e**6*x**2*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 72*a*c**2*d**4*e**2*log(d/e + x)/(
4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 108*a*c**2*d**4*e**2/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 144*
a*c**2*d**3*e**3*x*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 144*a*c**2*d**3*e**3*x/(4*d**2*e**7
+ 8*d*e**8*x + 4*e**9*x**2) + 72*a*c**2*d**2*e**4*x**2*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2)
- 24*a*c**2*d*e**5*x**3/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 6*a*c**2*e**6*x**4/(4*d**2*e**7 + 8*d*e**8*
x + 4*e**9*x**2) + 60*c**3*d**6*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 90*c**3*d**6/(4*d**2*e
**7 + 8*d*e**8*x + 4*e**9*x**2) + 120*c**3*d**5*e*x*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 12
0*c**3*d**5*e*x/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 60*c**3*d**4*e**2*x**2*log(d/e + x)/(4*d**2*e**7 +
8*d*e**8*x + 4*e**9*x**2) - 20*c**3*d**3*e**3*x**3/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 5*c**3*d**2*e**4
*x**4/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) - 2*c**3*d*e**5*x**5/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) +
c**3*e**6*x**6/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2), Eq(m, -3)), (-10*a**3*e**6/(10*d*e**7 + 10*e**8*x) -
60*a**2*c*d**2*e**4*log(d/e + x)/(10*d*e**7 + 10*e**8*x) - 60*a**2*c*d**2*e**4/(10*d*e**7 + 10*e**8*x) - 60*a
**2*c*d*e**5*x*log(d/e + x)/(10*d*e**7 + 10*e**8*x) + 30*a**2*c*e**6*x**2/(10*d*e**7 + 10*e**8*x) - 120*a*c**2
*d**4*e**2*log(d/e + x)/(10*d*e**7 + 10*e**8*x) - 120*a*c**2*d**4*e**2/(10*d*e**7 + 10*e**8*x) - 120*a*c**2*d*
*3*e**3*x*log(d/e + x)/(10*d*e**7 + 10*e**8*x) + 60*a*c**2*d**2*e**4*x**2/(10*d*e**7 + 10*e**8*x) - 20*a*c**2*
d*e**5*x**3/(10*d*e**7 + 10*e**8*x) + 10*a*c**2*e**6*x**4/(10*d*e**7 + 10*e**8*x) - 60*c**3*d**6*log(d/e + x)/
(10*d*e**7 + 10*e**8*x) - 60*c**3*d**6/(10*d*e**7 + 10*e**8*x) - 60*c**3*d**5*e*x*log(d/e + x)/(10*d*e**7 + 10
*e**8*x) + 30*c**3*d**4*e**2*x**2/(10*d*e**7 + 10*e**8*x) - 10*c**3*d**3*e**3*x**3/(10*d*e**7 + 10*e**8*x) + 5
*c**3*d**2*e**4*x**4/(10*d*e**7 + 10*e**8*x) - 3*c**3*d*e**5*x**5/(10*d*e**7 + 10*e**8*x) + 2*c**3*e**6*x**6/(
10*d*e**7 + 10*e**8*x), Eq(m, -2)), (a**3*log(d/e + x)/e + 3*a**2*c*d**2*log(d/e + x)/e**3 - 3*a**2*c*d*x/e**2
+ 3*a**2*c*x**2/(2*e) + 3*a*c**2*d**4*log(d/e + x)/e**5 - 3*a*c**2*d**3*x/e**4 + 3*a*c**2*d**2*x**2/(2*e**3)
- a*c**2*d*x**3/e**2 + 3*a*c**2*x**4/(4*e) + c**3*d**6*log(d/e + x)/e**7 - c**3*d**5*x/e**6 + c**3*d**4*x**2/(
2*e**5) - c**3*d**3*x**3/(3*e**4) + c**3*d**2*x**4/(4*e**3) - c**3*d*x**5/(5*e**2) + c**3*x**6/(6*e), Eq(m, -1
)), (a**3*d*e**6*m**6*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3
+ 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 27*a**3*d*e**6*m**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 +
322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 295*a**3*d*e**
6*m**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m
**2 + 13068*e**7*m + 5040*e**7) + 1665*a**3*d*e**6*m**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5
+ 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 5104*a**3*d*e**6*m**2*(d +
e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*
e**7*m + 5040*e**7) + 8028*a**3*d*e**6*m*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m*
*4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 5040*a**3*d*e**6*(d + e*x)**m/(e**7*m**7 +
28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7)
+ a**3*e**7*m**6*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 +
13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 27*a**3*e**7*m**5*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322
*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 295*a**3*e**7*m**
4*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2
+ 13068*e**7*m + 5040*e**7) + 1665*a**3*e**7*m**3*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 +
1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 5104*a**3*e**7*m**2*x*(d + e*x
)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**
7*m + 5040*e**7) + 8028*a**3*e**7*m*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4
+ 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 5040*a**3*e**7*x*(d + e*x)**m/(e**7*m**7 + 28
*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 6
*a**2*c*d**3*e**4*m**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**
3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 132*a**2*c*d**3*e**4*m**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*
m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 1074*a*
*2*c*d**3*e**4*m**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 +
13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3828*a**2*c*d**3*e**4*m*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6
+ 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 5040*a**2*c*
d**3*e**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**
7*m**2 + 13068*e**7*m + 5040*e**7) - 6*a**2*c*d**2*e**5*m**5*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e*
*7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 132*a**2*c*d**2*e**5
*m**4*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*
m**2 + 13068*e**7*m + 5040*e**7) - 1074*a**2*c*d**2*e**5*m**3*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e
**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 3828*a**2*c*d**2*e*
*5*m**2*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**
7*m**2 + 13068*e**7*m + 5040*e**7) - 5040*a**2*c*d**2*e**5*m*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e*
*7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3*a**2*c*d*e**6*m**6
*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m*
*2 + 13068*e**7*m + 5040*e**7) + 69*a**2*c*d*e**6*m**5*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*
m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 603*a**2*c*d*e**6*m**4*
x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**
2 + 13068*e**7*m + 5040*e**7) + 2451*a**2*c*d*e**6*m**3*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7
*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 4434*a**2*c*d*e**6*m**
2*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m
**2 + 13068*e**7*m + 5040*e**7) + 2520*a**2*c*d*e**6*m*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*
m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3*a**2*c*e**7*m**6*x**3
*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 +
13068*e**7*m + 5040*e**7) + 75*a**2*c*e**7*m**5*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 +
1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 741*a**2*c*e**7*m**4*x**3*(d +
e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068
*e**7*m + 5040*e**7) + 3657*a**2*c*e**7*m**3*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 196
0*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 9336*a**2*c*e**7*m**2*x**3*(d + e
*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e
**7*m + 5040*e**7) + 11388*a**2*c*e**7*m*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e*
*7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 5040*a**2*c*e**7*x**3*(d + e*x)**m/(e
**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5
040*e**7) + 72*a*c**2*d**5*e**2*m**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 +
6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 936*a*c**2*d**5*e**2*m*(d + e*x)**m/(e**7*m**7
+ 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7
) + 3024*a*c**2*d**5*e**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*
m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 72*a*c**2*d**4*e**3*m**3*x*(d + e*x)**m/(e**7*m**7 + 28*e
**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 936
*a*c**2*d**4*e**3*m**2*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m
**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 3024*a*c**2*d**4*e**3*m*x*(d + e*x)**m/(e**7*m**7 + 28*e**
7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 36*a*
c**2*d**3*e**4*m**4*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m
**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 504*a*c**2*d**3*e**4*m**3*x**2*(d + e*x)**m/(e**7*m**7 + 2
8*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) +
1980*a*c**2*d**3*e**4*m**2*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769
*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 1512*a*c**2*d**3*e**4*m*x**2*(d + e*x)**m/(e**7*m**
7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**
7) - 12*a*c**2*d**2*e**5*m**5*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6
769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 192*a*c**2*d**2*e**5*m**4*x**3*(d + e*x)**m/(e**
7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 504
0*e**7) - 996*a*c**2*d**2*e**5*m**3*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m*
*4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 1824*a*c**2*d**2*e**5*m**2*x**3*(d + e*x)*
*m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*
m + 5040*e**7) - 1008*a*c**2*d**2*e**5*m*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e*
*7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3*a*c**2*d*e**6*m**6*x**4*(d + e*x)**
m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m
+ 5040*e**7) + 57*a*c**2*d*e**6*m**5*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*
m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 393*a*c**2*d*e**6*m**4*x**4*(d + e*x)**m
/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m
+ 5040*e**7) + 1203*a*c**2*d*e**6*m**3*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7
*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 1620*a*c**2*d*e**6*m**2*x**4*(d + e*x)*
*m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*
m + 5040*e**7) + 756*a*c**2*d*e**6*m*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m
**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3*a*c**2*e**7*m**6*x**5*(d + e*x)**m/(e**
7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 504
0*e**7) + 69*a*c**2*e**7*m**5*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6
769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 621*a*c**2*e**7*m**4*x**5*(d + e*x)**m/(e**7*m**
7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**
7) + 2775*a*c**2*e**7*m**3*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769
*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 6432*a*c**2*e**7*m**2*x**5*(d + e*x)**m/(e**7*m**7
+ 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7)
+ 7236*a*c**2*e**7*m*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7
*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3024*a*c**2*e**7*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*
m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 720*c**
3*d**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m
**2 + 13068*e**7*m + 5040*e**7) - 720*c**3*d**6*e*m*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 +
1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 360*c**3*d**5*e**2*m**2*x**2*
(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 1
3068*e**7*m + 5040*e**7) + 360*c**3*d**5*e**2*m*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 +
1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 120*c**3*d**4*e**3*m**3*x**3*(
d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13
068*e**7*m + 5040*e**7) - 360*c**3*d**4*e**3*m**2*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5
+ 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 240*c**3*d**4*e**3*m*x**3*(d
+ e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 130
68*e**7*m + 5040*e**7) + 30*c**3*d**3*e**4*m**4*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 +
1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 180*c**3*d**3*e**4*m**3*x**4*(
d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13
068*e**7*m + 5040*e**7) + 330*c**3*d**3*e**4*m**2*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5
+ 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 180*c**3*d**3*e**4*m*x**4*(d
+ e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 130
68*e**7*m + 5040*e**7) - 6*c**3*d**2*e**5*m**5*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1
960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 60*c**3*d**2*e**5*m**4*x**5*(d
+ e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 1306
8*e**7*m + 5040*e**7) - 210*c**3*d**2*e**5*m**3*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 +
1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 300*c**3*d**2*e**5*m**2*x**5*(
d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13
068*e**7*m + 5040*e**7) - 144*c**3*d**2*e**5*m*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1
960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + c**3*d*e**6*m**6*x**6*(d + e*x)
**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7
*m + 5040*e**7) + 15*c**3*d*e**6*m**5*x**6*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*
m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 85*c**3*d*e**6*m**4*x**6*(d + e*x)**m/(e
**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5
040*e**7) + 225*c**3*d*e**6*m**3*x**6*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4
+ 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 274*c**3*d*e**6*m**2*x**6*(d + e*x)**m/(e**7*
m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*
e**7) + 120*c**3*d*e**6*m*x**6*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*
e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + c**3*e**7*m**6*x**7*(d + e*x)**m/(e**7*m**7 + 28*e**
7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 21*c*
*3*e**7*m**5*x**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 1
3132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 175*c**3*e**7*m**4*x**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 +
322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 735*c**3*e**7*
m**3*x**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**
7*m**2 + 13068*e**7*m + 5040*e**7) + 1624*c**3*e**7*m**2*x**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**
7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 1764*c**3*e**7*m*x**7
*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 +
13068*e**7*m + 5040*e**7) + 720*c**3*e**7*x**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e
**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7), True))

________________________________________________________________________________________

Giac [B]  time = 1.18065, size = 2808, normalized size = 12.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*x^2+a)^3,x, algorithm="giac")

[Out]

((x*e + d)^m*c^3*m^6*x^7*e^7 + (x*e + d)^m*c^3*d*m^6*x^6*e^6 + 21*(x*e + d)^m*c^3*m^5*x^7*e^7 + 15*(x*e + d)^m
*c^3*d*m^5*x^6*e^6 - 6*(x*e + d)^m*c^3*d^2*m^5*x^5*e^5 + 3*(x*e + d)^m*a*c^2*m^6*x^5*e^7 + 175*(x*e + d)^m*c^3
*m^4*x^7*e^7 + 3*(x*e + d)^m*a*c^2*d*m^6*x^4*e^6 + 85*(x*e + d)^m*c^3*d*m^4*x^6*e^6 - 60*(x*e + d)^m*c^3*d^2*m
^4*x^5*e^5 + 30*(x*e + d)^m*c^3*d^3*m^4*x^4*e^4 + 69*(x*e + d)^m*a*c^2*m^5*x^5*e^7 + 735*(x*e + d)^m*c^3*m^3*x
^7*e^7 + 57*(x*e + d)^m*a*c^2*d*m^5*x^4*e^6 + 225*(x*e + d)^m*c^3*d*m^3*x^6*e^6 - 12*(x*e + d)^m*a*c^2*d^2*m^5
*x^3*e^5 - 210*(x*e + d)^m*c^3*d^2*m^3*x^5*e^5 + 180*(x*e + d)^m*c^3*d^3*m^3*x^4*e^4 - 120*(x*e + d)^m*c^3*d^4
*m^3*x^3*e^3 + 3*(x*e + d)^m*a^2*c*m^6*x^3*e^7 + 621*(x*e + d)^m*a*c^2*m^4*x^5*e^7 + 1624*(x*e + d)^m*c^3*m^2*
x^7*e^7 + 3*(x*e + d)^m*a^2*c*d*m^6*x^2*e^6 + 393*(x*e + d)^m*a*c^2*d*m^4*x^4*e^6 + 274*(x*e + d)^m*c^3*d*m^2*
x^6*e^6 - 192*(x*e + d)^m*a*c^2*d^2*m^4*x^3*e^5 - 300*(x*e + d)^m*c^3*d^2*m^2*x^5*e^5 + 36*(x*e + d)^m*a*c^2*d
^3*m^4*x^2*e^4 + 330*(x*e + d)^m*c^3*d^3*m^2*x^4*e^4 - 360*(x*e + d)^m*c^3*d^4*m^2*x^3*e^3 + 360*(x*e + d)^m*c
^3*d^5*m^2*x^2*e^2 + 75*(x*e + d)^m*a^2*c*m^5*x^3*e^7 + 2775*(x*e + d)^m*a*c^2*m^3*x^5*e^7 + 1764*(x*e + d)^m*
c^3*m*x^7*e^7 + 69*(x*e + d)^m*a^2*c*d*m^5*x^2*e^6 + 1203*(x*e + d)^m*a*c^2*d*m^3*x^4*e^6 + 120*(x*e + d)^m*c^
3*d*m*x^6*e^6 - 6*(x*e + d)^m*a^2*c*d^2*m^5*x*e^5 - 996*(x*e + d)^m*a*c^2*d^2*m^3*x^3*e^5 - 144*(x*e + d)^m*c^
3*d^2*m*x^5*e^5 + 504*(x*e + d)^m*a*c^2*d^3*m^3*x^2*e^4 + 180*(x*e + d)^m*c^3*d^3*m*x^4*e^4 - 72*(x*e + d)^m*a
*c^2*d^4*m^3*x*e^3 - 240*(x*e + d)^m*c^3*d^4*m*x^3*e^3 + 360*(x*e + d)^m*c^3*d^5*m*x^2*e^2 - 720*(x*e + d)^m*c
^3*d^6*m*x*e + (x*e + d)^m*a^3*m^6*x*e^7 + 741*(x*e + d)^m*a^2*c*m^4*x^3*e^7 + 6432*(x*e + d)^m*a*c^2*m^2*x^5*
e^7 + 720*(x*e + d)^m*c^3*x^7*e^7 + (x*e + d)^m*a^3*d*m^6*e^6 + 603*(x*e + d)^m*a^2*c*d*m^4*x^2*e^6 + 1620*(x*
e + d)^m*a*c^2*d*m^2*x^4*e^6 - 132*(x*e + d)^m*a^2*c*d^2*m^4*x*e^5 - 1824*(x*e + d)^m*a*c^2*d^2*m^2*x^3*e^5 +
6*(x*e + d)^m*a^2*c*d^3*m^4*e^4 + 1980*(x*e + d)^m*a*c^2*d^3*m^2*x^2*e^4 - 936*(x*e + d)^m*a*c^2*d^4*m^2*x*e^3
+ 72*(x*e + d)^m*a*c^2*d^5*m^2*e^2 + 720*(x*e + d)^m*c^3*d^7 + 27*(x*e + d)^m*a^3*m^5*x*e^7 + 3657*(x*e + d)^
m*a^2*c*m^3*x^3*e^7 + 7236*(x*e + d)^m*a*c^2*m*x^5*e^7 + 27*(x*e + d)^m*a^3*d*m^5*e^6 + 2451*(x*e + d)^m*a^2*c
*d*m^3*x^2*e^6 + 756*(x*e + d)^m*a*c^2*d*m*x^4*e^6 - 1074*(x*e + d)^m*a^2*c*d^2*m^3*x*e^5 - 1008*(x*e + d)^m*a
*c^2*d^2*m*x^3*e^5 + 132*(x*e + d)^m*a^2*c*d^3*m^3*e^4 + 1512*(x*e + d)^m*a*c^2*d^3*m*x^2*e^4 - 3024*(x*e + d)
^m*a*c^2*d^4*m*x*e^3 + 936*(x*e + d)^m*a*c^2*d^5*m*e^2 + 295*(x*e + d)^m*a^3*m^4*x*e^7 + 9336*(x*e + d)^m*a^2*
c*m^2*x^3*e^7 + 3024*(x*e + d)^m*a*c^2*x^5*e^7 + 295*(x*e + d)^m*a^3*d*m^4*e^6 + 4434*(x*e + d)^m*a^2*c*d*m^2*
x^2*e^6 - 3828*(x*e + d)^m*a^2*c*d^2*m^2*x*e^5 + 1074*(x*e + d)^m*a^2*c*d^3*m^2*e^4 + 3024*(x*e + d)^m*a*c^2*d
^5*e^2 + 1665*(x*e + d)^m*a^3*m^3*x*e^7 + 11388*(x*e + d)^m*a^2*c*m*x^3*e^7 + 1665*(x*e + d)^m*a^3*d*m^3*e^6 +
2520*(x*e + d)^m*a^2*c*d*m*x^2*e^6 - 5040*(x*e + d)^m*a^2*c*d^2*m*x*e^5 + 3828*(x*e + d)^m*a^2*c*d^3*m*e^4 +
5104*(x*e + d)^m*a^3*m^2*x*e^7 + 5040*(x*e + d)^m*a^2*c*x^3*e^7 + 5104*(x*e + d)^m*a^3*d*m^2*e^6 + 5040*(x*e +
d)^m*a^2*c*d^3*e^4 + 8028*(x*e + d)^m*a^3*m*x*e^7 + 8028*(x*e + d)^m*a^3*d*m*e^6 + 5040*(x*e + d)^m*a^3*x*e^7
+ 5040*(x*e + d)^m*a^3*d*e^6)/(m^7*e^7 + 28*m^6*e^7 + 322*m^5*e^7 + 1960*m^4*e^7 + 6769*m^3*e^7 + 13132*m^2*e
^7 + 13068*m*e^7 + 5040*e^7)