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

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

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Rubi [A]  time = 0.07, antiderivative size = 95, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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)} \end {gather*}

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 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])

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]

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*}

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Mathematica [A]  time = 0.06, size = 70, normalized size = 0.74 \begin {gather*} \frac {c^3 (d+e x)^{m+4} \left (-\frac {d^3}{m+4}+\frac {3 d^2 (d+e x)}{m+5}-\frac {3 d (d+e x)^2}{m+6}+\frac {(d+e x)^3}{m+7}\right )}{e^4} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.43, size = 343, normalized size = 3.61 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.21, size = 528, normalized size = 5.56 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.05, size = 129, normalized size = 1.36 \begin {gather*} -\frac {\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} e m x +60 d \,x^{2} e^{2}-24 d^{2} x e +6 d^{3}\right ) c^{3} \left (e x +d \right )^{m +4}}{\left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right ) e^{4}} \end {gather*}

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)^(m+4)*(-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 = 1.60, size = 674, normalized size = 7.09 \begin {gather*} \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} c^{3} d^{3}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {3 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} c^{3} d^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{4}} + \frac {3 \, {\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{6} x^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d e^{5} x^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{2} e^{4} x^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{3} e^{3} x^{3} - 60 \, {\left (m^{2} + m\right )} d^{4} e^{2} x^{2} + 120 \, d^{5} e m x - 120 \, d^{6}\right )} {\left (e x + d\right )}^{m} c^{3} d}{{\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{4}} + \frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{7} x^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} d e^{6} x^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d^{2} e^{5} x^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{3} e^{4} x^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{4} e^{3} x^{3} + 360 \, {\left (m^{2} + m\right )} d^{5} e^{2} x^{2} - 720 \, d^{6} e m x + 720 \, d^{7}\right )} {\left (e x + d\right )}^{m} c^{3}}{{\left (m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040\right )} e^{4}} \end {gather*}

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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mupad [B]  time = 0.59, size = 333, normalized size = 3.51 \begin {gather*} {\left (d+e\,x\right )}^m\,\left (\frac {c^3\,e^3\,x^7\,\left (m^3+15\,m^2+74\,m+120\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}-\frac {6\,c^3\,d^7}{e^4\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}+\frac {2\,c^3\,d^3\,x^4\,\left (2\,m^3+21\,m^2+79\,m+105\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {6\,c^3\,d^6\,m\,x}{e^3\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}+\frac {c^3\,d\,e^2\,x^6\,\left (4\,m^3+57\,m^2+269\,m+420\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {6\,c^3\,d^2\,e\,x^5\,\left (m^3+13\,m^2+57\,m+84\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}-\frac {3\,c^3\,d^5\,m\,x^2\,\left (m+1\right )}{e^2\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}+\frac {c^3\,d^4\,m\,x^3\,\left (m^2+3\,m+2\right )}{e\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d + e*x)^m*((c^3*e^3*x^7*(74*m + 15*m^2 + m^3 + 120))/(638*m + 179*m^2 + 22*m^3 + m^4 + 840) - (6*c^3*d^7)/(e
^4*(638*m + 179*m^2 + 22*m^3 + m^4 + 840)) + (2*c^3*d^3*x^4*(79*m + 21*m^2 + 2*m^3 + 105))/(638*m + 179*m^2 +
22*m^3 + m^4 + 840) + (6*c^3*d^6*m*x)/(e^3*(638*m + 179*m^2 + 22*m^3 + m^4 + 840)) + (c^3*d*e^2*x^6*(269*m + 5
7*m^2 + 4*m^3 + 420))/(638*m + 179*m^2 + 22*m^3 + m^4 + 840) + (6*c^3*d^2*e*x^5*(57*m + 13*m^2 + m^3 + 84))/(6
38*m + 179*m^2 + 22*m^3 + m^4 + 840) - (3*c^3*d^5*m*x^2*(m + 1))/(e^2*(638*m + 179*m^2 + 22*m^3 + m^4 + 840))
+ (c^3*d^4*m*x^3*(3*m + m^2 + 2))/(e*(638*m + 179*m^2 + 22*m^3 + m^4 + 840)))

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sympy [A]  time = 6.48, size = 2218, normalized size = 23.35

result too large to display

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)), (6*c**3*d**3*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e*
*6*x**2 + 6*e**7*x**3) + 11*c**3*d**3/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*c**3*
d**2*e*x*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 27*c**3*d**2*e*x/(6*d**3
*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*c**3*d*e**2*x**2*log(d/e + x)/(6*d**3*e**4 + 18*d*
*2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*c**3*d*e**2*x**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2
+ 6*e**7*x**3) + 6*c**3*e**3*x**3*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3),
Eq(m, -7)), (-6*c**3*d**3*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 15*c**3*d**3/(2*d**2*e**4 +
4*d*e**5*x + 2*e**6*x**2) - 12*c**3*d**2*e*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 24*c**3*
d**2*e*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*c**3*d*e**2*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x
+ 2*e**6*x**2) - 6*c**3*d*e**2*x**2/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*c**3*e**3*x**3/(2*d**2*e**4 +
4*d*e**5*x + 2*e**6*x**2), Eq(m, -6)), (6*c**3*d**3*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 12*c**3*d**3/(2*d*e*
*4 + 2*e**5*x) + 6*c**3*d**2*e*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 6*c**3*d**2*e*x/(2*d*e**4 + 2*e**5*x) -
3*c**3*d*e**2*x**2/(2*d*e**4 + 2*e**5*x) + c**3*e**3*x**3/(2*d*e**4 + 2*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 + 17
9*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 + 179*e**4*m**2 + 63
8*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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