### 3.434 $$\int (d+e x)^m (c d x+c e x^2)^3 \, dx$$

Optimal. Leaf size=95 $-\frac{c^3 d^3 (d+e x)^{m+4}}{e^4 (m+4)}+\frac{3 c^3 d^2 (d+e x)^{m+5}}{e^4 (m+5)}-\frac{3 c^3 d (d+e x)^{m+6}}{e^4 (m+6)}+\frac{c^3 (d+e x)^{m+7}}{e^4 (m+7)}$

[Out]

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

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Rubi [A]  time = 0.0735214, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {626, 12, 43} $-\frac{c^3 d^3 (d+e x)^{m+4}}{e^4 (m+4)}+\frac{3 c^3 d^2 (d+e x)^{m+5}}{e^4 (m+5)}-\frac{3 c^3 d (d+e x)^{m+6}}{e^4 (m+6)}+\frac{c^3 (d+e x)^{m+7}}{e^4 (m+7)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

Mathematica [A]  time = 0.0592135, size = 70, normalized size = 0.74 $\frac{c^3 (d+e x)^{m+4} \left (\frac{3 d^2 (d+e x)}{m+5}-\frac{d^3}{m+4}-\frac{3 d (d+e x)^2}{m+6}+\frac{(d+e x)^3}{m+7}\right )}{e^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 129, normalized size = 1.4 \begin{align*} -{\frac{{c}^{3} \left ( ex+d \right ) ^{4+m} \left ( -{e}^{3}{m}^{3}{x}^{3}-15\,{e}^{3}{m}^{2}{x}^{3}+3\,d{e}^{2}{m}^{2}{x}^{2}-74\,{e}^{3}m{x}^{3}+27\,d{e}^{2}m{x}^{2}-120\,{x}^{3}{e}^{3}-6\,{d}^{2}emx+60\,d{x}^{2}{e}^{2}-24\,{d}^{2}xe+6\,{d}^{3} \right ) }{{e}^{4} \left ({m}^{4}+22\,{m}^{3}+179\,{m}^{2}+638\,m+840 \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^3*(e*x+d)^(4+m)*(-e^3*m^3*x^3-15*e^3*m^2*x^3+3*d*e^2*m^2*x^2-74*e^3*m*x^3+27*d*e^2*m*x^2-120*e^3*x^3-6*d^2*
e*m*x+60*d*e^2*x^2-24*d^2*e*x+6*d^3)/e^4/(m^4+22*m^3+179*m^2+638*m+840)

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Maxima [B]  time = 1.37479, size = 910, normalized size = 9.58 \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*e*x^2+c*d*x)^3,x, algorithm="maxima")

[Out]

((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*
d^4)*(e*x + d)^m*c^3*d^3/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 3*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^
5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2
- 24*d^4*e*m*x + 24*d^5)*(e*x + d)^m*c^3*d^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^4) + 3*((m^5 +
15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*e^5*x^5 - 5*(m^4
+ 6*m^3 + 11*m^2 + 6*m)*d^2*e^4*x^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*e^3*x^3 - 60*(m^2 + m)*d^4*e^2*x^2 + 120*d^5
*e*m*x - 120*d^6)*(e*x + d)^m*c^3*d/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^4) + ((m^6
+ 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^7*x^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2
+ 120*m)*d*e^6*x^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^2*e^5*x^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d
^3*e^4*x^4 - 120*(m^3 + 3*m^2 + 2*m)*d^4*e^3*x^3 + 360*(m^2 + m)*d^5*e^2*x^2 - 720*d^6*e*m*x + 720*d^7)*(e*x +
d)^m*c^3/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)*e^4)

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Fricas [B]  time = 2.06555, size = 699, normalized size = 7.36 \begin{align*} \frac{{\left (6 \, c^{3} d^{6} e m x - 6 \, c^{3} d^{7} +{\left (c^{3} e^{7} m^{3} + 15 \, c^{3} e^{7} m^{2} + 74 \, c^{3} e^{7} m + 120 \, c^{3} e^{7}\right )} x^{7} +{\left (4 \, c^{3} d e^{6} m^{3} + 57 \, c^{3} d e^{6} m^{2} + 269 \, c^{3} d e^{6} m + 420 \, c^{3} d e^{6}\right )} x^{6} + 6 \,{\left (c^{3} d^{2} e^{5} m^{3} + 13 \, c^{3} d^{2} e^{5} m^{2} + 57 \, c^{3} d^{2} e^{5} m + 84 \, c^{3} d^{2} e^{5}\right )} x^{5} + 2 \,{\left (2 \, c^{3} d^{3} e^{4} m^{3} + 21 \, c^{3} d^{3} e^{4} m^{2} + 79 \, c^{3} d^{3} e^{4} m + 105 \, c^{3} d^{3} e^{4}\right )} x^{4} +{\left (c^{3} d^{4} e^{3} m^{3} + 3 \, c^{3} d^{4} e^{3} m^{2} + 2 \, c^{3} d^{4} e^{3} m\right )} x^{3} - 3 \,{\left (c^{3} d^{5} e^{2} m^{2} + c^{3} d^{5} e^{2} m\right )} x^{2}\right )}{\left (e x + d\right )}^{m}}{e^{4} m^{4} + 22 \, e^{4} m^{3} + 179 \, e^{4} m^{2} + 638 \, e^{4} m + 840 \, e^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*c^3*d^6*e*m*x - 6*c^3*d^7 + (c^3*e^7*m^3 + 15*c^3*e^7*m^2 + 74*c^3*e^7*m + 120*c^3*e^7)*x^7 + (4*c^3*d*e^6*
m^3 + 57*c^3*d*e^6*m^2 + 269*c^3*d*e^6*m + 420*c^3*d*e^6)*x^6 + 6*(c^3*d^2*e^5*m^3 + 13*c^3*d^2*e^5*m^2 + 57*c
^3*d^2*e^5*m + 84*c^3*d^2*e^5)*x^5 + 2*(2*c^3*d^3*e^4*m^3 + 21*c^3*d^3*e^4*m^2 + 79*c^3*d^3*e^4*m + 105*c^3*d^
3*e^4)*x^4 + (c^3*d^4*e^3*m^3 + 3*c^3*d^4*e^3*m^2 + 2*c^3*d^4*e^3*m)*x^3 - 3*(c^3*d^5*e^2*m^2 + c^3*d^5*e^2*m)
*x^2)*(e*x + d)^m/(e^4*m^4 + 22*e^4*m^3 + 179*e^4*m^2 + 638*e^4*m + 840*e^4)

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Sympy [A]  time = 8.11399, size = 2270, normalized size = 23.89 \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*e*x**2+c*d*x)**3,x)

[Out]

Piecewise((c**3*d**3*d**m*x**4/4, Eq(e, 0)), (30*c**3*d**3*log(d/e + x)/(30*d**3*e**4 + 90*d**2*e**5*x + 90*d*
e**6*x**2 + 30*e**7*x**3) + 11*c**3*d**3/(30*d**3*e**4 + 90*d**2*e**5*x + 90*d*e**6*x**2 + 30*e**7*x**3) + 90*
c**3*d**2*e*x*log(d/e + x)/(30*d**3*e**4 + 90*d**2*e**5*x + 90*d*e**6*x**2 + 30*e**7*x**3) + 3*c**3*d**2*e*x/(
30*d**3*e**4 + 90*d**2*e**5*x + 90*d*e**6*x**2 + 30*e**7*x**3) + 90*c**3*d*e**2*x**2*log(d/e + x)/(30*d**3*e**
4 + 90*d**2*e**5*x + 90*d*e**6*x**2 + 30*e**7*x**3) - 42*c**3*d*e**2*x**2/(30*d**3*e**4 + 90*d**2*e**5*x + 90*
d*e**6*x**2 + 30*e**7*x**3) + 30*c**3*e**3*x**3*log(d/e + x)/(30*d**3*e**4 + 90*d**2*e**5*x + 90*d*e**6*x**2 +
30*e**7*x**3) - 44*c**3*e**3*x**3/(30*d**3*e**4 + 90*d**2*e**5*x + 90*d*e**6*x**2 + 30*e**7*x**3), Eq(m, -7))
, (-60*c**3*d**3*log(d/e + x)/(20*d**2*e**4 + 40*d*e**5*x + 20*e**6*x**2) - 27*c**3*d**3/(20*d**2*e**4 + 40*d*
e**5*x + 20*e**6*x**2) - 120*c**3*d**2*e*x*log(d/e + x)/(20*d**2*e**4 + 40*d*e**5*x + 20*e**6*x**2) + 6*c**3*d
**2*e*x/(20*d**2*e**4 + 40*d*e**5*x + 20*e**6*x**2) - 60*c**3*d*e**2*x**2*log(d/e + x)/(20*d**2*e**4 + 40*d*e*
*5*x + 20*e**6*x**2) + 63*c**3*d*e**2*x**2/(20*d**2*e**4 + 40*d*e**5*x + 20*e**6*x**2) + 20*c**3*e**3*x**3/(20
*d**2*e**4 + 40*d*e**5*x + 20*e**6*x**2), Eq(m, -6)), (12*c**3*d**3*log(d/e + x)/(4*d*e**4 + 4*e**5*x) + 7*c**
3*d**3/(4*d*e**4 + 4*e**5*x) + 12*c**3*d**2*e*x*log(d/e + x)/(4*d*e**4 + 4*e**5*x) - 5*c**3*d**2*e*x/(4*d*e**4
+ 4*e**5*x) - 6*c**3*d*e**2*x**2/(4*d*e**4 + 4*e**5*x) + 2*c**3*e**3*x**3/(4*d*e**4 + 4*e**5*x), Eq(m, -5)),
(-c**3*d**3*log(d/e + x)/e**4 + c**3*d**2*x/e**3 - c**3*d*x**2/(2*e**2) + c**3*x**3/(3*e), Eq(m, -4)), (-6*c**
3*d**7*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 6*c**3*d**6*e*m*x*(d
+ e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) - 3*c**3*d**5*e**2*m**2*x**2*(d +
e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) - 3*c**3*d**5*e**2*m*x**2*(d + e*x
)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + c**3*d**4*e**3*m**3*x**3*(d + e*x)**
m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 3*c**3*d**4*e**3*m**2*x**3*(d + e*x)**m
/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 2*c**3*d**4*e**3*m*x**3*(d + e*x)**m/(e*
*4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 4*c**3*d**3*e**4*m**3*x**4*(d + e*x)**m/(e**
4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 42*c**3*d**3*e**4*m**2*x**4*(d + e*x)**m/(e**
4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 158*c**3*d**3*e**4*m*x**4*(d + e*x)**m/(e**4*
m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 210*c**3*d**3*e**4*x**4*(d + e*x)**m/(e**4*m**4
+ 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 6*c**3*d**2*e**5*m**3*x**5*(d + e*x)**m/(e**4*m**4
+ 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 78*c**3*d**2*e**5*m**2*x**5*(d + e*x)**m/(e**4*m**4
+ 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 342*c**3*d**2*e**5*m*x**5*(d + e*x)**m/(e**4*m**4 +
22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 504*c**3*d**2*e**5*x**5*(d + e*x)**m/(e**4*m**4 + 22*e
**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 4*c**3*d*e**6*m**3*x**6*(d + e*x)**m/(e**4*m**4 + 22*e**4*
m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 57*c**3*d*e**6*m**2*x**6*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**
3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 269*c**3*d*e**6*m*x**6*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 1
79*e**4*m**2 + 638*e**4*m + 840*e**4) + 420*c**3*d*e**6*x**6*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4
*m**2 + 638*e**4*m + 840*e**4) + c**3*e**7*m**3*x**7*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 +
638*e**4*m + 840*e**4) + 15*c**3*e**7*m**2*x**7*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e
**4*m + 840*e**4) + 74*c**3*e**7*m*x**7*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m +
840*e**4) + 120*c**3*e**7*x**7*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4)
, True))

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Giac [B]  time = 1.25299, size = 713, normalized size = 7.51 \begin{align*} \frac{{\left (x e + d\right )}^{m} c^{3} m^{3} x^{7} e^{7} + 4 \,{\left (x e + d\right )}^{m} c^{3} d m^{3} x^{6} e^{6} + 6 \,{\left (x e + d\right )}^{m} c^{3} d^{2} m^{3} x^{5} e^{5} + 4 \,{\left (x e + d\right )}^{m} c^{3} d^{3} m^{3} x^{4} e^{4} +{\left (x e + d\right )}^{m} c^{3} d^{4} m^{3} x^{3} e^{3} + 15 \,{\left (x e + d\right )}^{m} c^{3} m^{2} x^{7} e^{7} + 57 \,{\left (x e + d\right )}^{m} c^{3} d m^{2} x^{6} e^{6} + 78 \,{\left (x e + d\right )}^{m} c^{3} d^{2} m^{2} x^{5} e^{5} + 42 \,{\left (x e + d\right )}^{m} c^{3} d^{3} m^{2} x^{4} e^{4} + 3 \,{\left (x e + d\right )}^{m} c^{3} d^{4} m^{2} x^{3} e^{3} - 3 \,{\left (x e + d\right )}^{m} c^{3} d^{5} m^{2} x^{2} e^{2} + 74 \,{\left (x e + d\right )}^{m} c^{3} m x^{7} e^{7} + 269 \,{\left (x e + d\right )}^{m} c^{3} d m x^{6} e^{6} + 342 \,{\left (x e + d\right )}^{m} c^{3} d^{2} m x^{5} e^{5} + 158 \,{\left (x e + d\right )}^{m} c^{3} d^{3} m x^{4} e^{4} + 2 \,{\left (x e + d\right )}^{m} c^{3} d^{4} m x^{3} e^{3} - 3 \,{\left (x e + d\right )}^{m} c^{3} d^{5} m x^{2} e^{2} + 6 \,{\left (x e + d\right )}^{m} c^{3} d^{6} m x e + 120 \,{\left (x e + d\right )}^{m} c^{3} x^{7} e^{7} + 420 \,{\left (x e + d\right )}^{m} c^{3} d x^{6} e^{6} + 504 \,{\left (x e + d\right )}^{m} c^{3} d^{2} x^{5} e^{5} + 210 \,{\left (x e + d\right )}^{m} c^{3} d^{3} x^{4} e^{4} - 6 \,{\left (x e + d\right )}^{m} c^{3} d^{7}}{m^{4} e^{4} + 22 \, m^{3} e^{4} + 179 \, m^{2} e^{4} + 638 \, m e^{4} + 840 \, e^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)^m*c^3*m^3*x^7*e^7 + 4*(x*e + d)^m*c^3*d*m^3*x^6*e^6 + 6*(x*e + d)^m*c^3*d^2*m^3*x^5*e^5 + 4*(x*e +
d)^m*c^3*d^3*m^3*x^4*e^4 + (x*e + d)^m*c^3*d^4*m^3*x^3*e^3 + 15*(x*e + d)^m*c^3*m^2*x^7*e^7 + 57*(x*e + d)^m*c
^3*d*m^2*x^6*e^6 + 78*(x*e + d)^m*c^3*d^2*m^2*x^5*e^5 + 42*(x*e + d)^m*c^3*d^3*m^2*x^4*e^4 + 3*(x*e + d)^m*c^3
*d^4*m^2*x^3*e^3 - 3*(x*e + d)^m*c^3*d^5*m^2*x^2*e^2 + 74*(x*e + d)^m*c^3*m*x^7*e^7 + 269*(x*e + d)^m*c^3*d*m*
x^6*e^6 + 342*(x*e + d)^m*c^3*d^2*m*x^5*e^5 + 158*(x*e + d)^m*c^3*d^3*m*x^4*e^4 + 2*(x*e + d)^m*c^3*d^4*m*x^3*
e^3 - 3*(x*e + d)^m*c^3*d^5*m*x^2*e^2 + 6*(x*e + d)^m*c^3*d^6*m*x*e + 120*(x*e + d)^m*c^3*x^7*e^7 + 420*(x*e +
d)^m*c^3*d*x^6*e^6 + 504*(x*e + d)^m*c^3*d^2*x^5*e^5 + 210*(x*e + d)^m*c^3*d^3*x^4*e^4 - 6*(x*e + d)^m*c^3*d^
7)/(m^4*e^4 + 22*m^3*e^4 + 179*m^2*e^4 + 638*m*e^4 + 840*e^4)