3.21.84 \(\int (d+e x)^m (a d e+(c d^2+a e^2) x+c d e x^2)^3 \, dx\) [2084]

Optimal. Leaf size=130 \[ -\frac {\left (c d^2-a e^2\right )^3 (d+e x)^{4+m}}{e^4 (4+m)}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{5+m}}{e^4 (5+m)}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{6+m}}{e^4 (6+m)}+\frac {c^3 d^3 (d+e x)^{7+m}}{e^4 (7+m)} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \begin {gather*} -\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{m+6}}{e^4 (m+6)}-\frac {\left (c d^2-a e^2\right )^3 (d+e x)^{m+4}}{e^4 (m+4)}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{m+5}}{e^4 (m+5)}+\frac {c^3 d^3 (d+e x)^{m+7}}{e^4 (m+7)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 45

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 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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 (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx &=\int (a e+c d x)^3 (d+e x)^{3+m} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^3 (d+e x)^{3+m}}{e^3}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{4+m}}{e^3}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{5+m}}{e^3}+\frac {c^3 d^3 (d+e x)^{6+m}}{e^3}\right ) \, dx\\ &=-\frac {\left (c d^2-a e^2\right )^3 (d+e x)^{4+m}}{e^4 (4+m)}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{5+m}}{e^4 (5+m)}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{6+m}}{e^4 (6+m)}+\frac {c^3 d^3 (d+e x)^{7+m}}{e^4 (7+m)}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 114, normalized size = 0.88 \begin {gather*} \frac {(d+e x)^{4+m} \left (-\frac {\left (c d^2-a e^2\right )^3}{4+m}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)}{5+m}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^2}{6+m}+\frac {c^3 d^3 (d+e x)^3}{7+m}\right )}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(435\) vs. \(2(130)=260\).
time = 0.74, size = 436, normalized size = 3.35 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x+d)^(4+m)*(c^3*d^3*e^3*m^3*x^3+3*a*c^2*d^2*e^4*m^3*x^2+15*c^3*d^3*e^3*m^2*x^3+3*a^2*c*d*e^5*m^3*x+48*a*c^2
*d^2*e^4*m^2*x^2-3*c^3*d^4*e^2*m^2*x^2+74*c^3*d^3*e^3*m*x^3+a^3*e^6*m^3+51*a^2*c*d*e^5*m^2*x-6*a*c^2*d^3*e^3*m
^2*x+249*a*c^2*d^2*e^4*m*x^2-27*c^3*d^4*e^2*m*x^2+120*c^3*d^3*e^3*x^3+18*a^3*e^6*m^2-3*a^2*c*d^2*e^4*m^2+282*a
^2*c*d*e^5*m*x-66*a*c^2*d^3*e^3*m*x+420*a*c^2*d^2*e^4*x^2+6*c^3*d^5*e*m*x-60*c^3*d^4*e^2*x^2+107*a^3*e^6*m-39*
a^2*c*d^2*e^4*m+504*a^2*c*d*e^5*x+6*a*c^2*d^4*e^2*m-168*a*c^2*d^3*e^3*x+24*c^3*d^5*e*x+210*a^3*e^6-126*a^2*c*d
^2*e^4+42*a*c^2*d^4*e^2-6*c^3*d^6)/e^4/(m^4+22*m^3+179*m^2+638*m+840)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1812 vs. \(2 (127) = 254\).
time = 0.33, size = 1812, normalized size = 13.94 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((m^3 + 6*m^2 + 11*m + 6)*x^4*e^4 + (m^3 + 3*m^2 + 2*m)*d*x^3*e^3 - 3*(m^2 + m)*d^2*x^2*e^2 + 6*d^3*m*x*e - 6*
d^4)*c^3*d^6*e^(m*log(x*e + d) - 4)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) + 3*((m^2 + 3*m + 2)*x^3*e^3 + (m^2 +
m)*d*x^2*e^2 - 2*d^2*m*x*e + 2*d^3)*a*c^2*d^5*e^(m*log(x*e + d) - 2)/(m^3 + 6*m^2 + 11*m + 6) + 3*((m^4 + 10*m
^3 + 35*m^2 + 50*m + 24)*x^5*e^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*x^4*e^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*x^3*e^3
+ 12*(m^2 + m)*d^3*x^2*e^2 - 24*d^4*m*x*e + 24*d^5)*c^3*d^5*e^(m*log(x*e + d) - 4)/(m^5 + 15*m^4 + 85*m^3 + 22
5*m^2 + 274*m + 120) + 3*((m + 1)*x^2*e^2 + d*m*x*e - d^2)*(x*e + d)^m*a^2*c*d^4/(m^2 + 3*m + 2) + 9*((m^3 + 6
*m^2 + 11*m + 6)*x^4*e^4 + (m^3 + 3*m^2 + 2*m)*d*x^3*e^3 - 3*(m^2 + m)*d^2*x^2*e^2 + 6*d^3*m*x*e - 6*d^4)*a*c^
2*d^4*e^(m*log(x*e + d) - 2)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) + 3*((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m
+ 120)*x^6*e^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*x^5*e^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^2*x^4*e
^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*x^3*e^3 - 60*(m^2 + m)*d^4*x^2*e^2 + 120*d^5*m*x*e - 120*d^6)*c^3*d^4*e^(m*log
(x*e + d) - 4)/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720) + 9*((m^2 + 3*m + 2)*x^3*e^3 + (m^
2 + m)*d*x^2*e^2 - 2*d^2*m*x*e + 2*d^3)*(x*e + d)^m*a^2*c*d^3/(m^3 + 6*m^2 + 11*m + 6) + (x*e + d)^(m + 1)*a^3
*d^3*e^2/(m + 1) + 9*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*x^5*e^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*x^4*e^4 - 4
*(m^3 + 3*m^2 + 2*m)*d^2*x^3*e^3 + 12*(m^2 + m)*d^3*x^2*e^2 - 24*d^4*m*x*e + 24*d^5)*a*c^2*d^3*e^(m*log(x*e +
d) - 2)/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120) + ((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764
*m + 720)*x^7*e^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*d*x^6*e^6 - 6*(m^5 + 10*m^4 + 35*m^3 +
50*m^2 + 24*m)*d^2*x^5*e^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^3*x^4*e^4 - 120*(m^3 + 3*m^2 + 2*m)*d^4*x^3*e^
3 + 360*(m^2 + m)*d^5*x^2*e^2 - 720*d^6*m*x*e + 720*d^7)*c^3*d^3*e^(m*log(x*e + d) - 4)/(m^7 + 28*m^6 + 322*m^
5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040) + 9*((m^3 + 6*m^2 + 11*m + 6)*x^4*e^4 + (m^3 + 3*m^2 + 2
*m)*d*x^3*e^3 - 3*(m^2 + m)*d^2*x^2*e^2 + 6*d^3*m*x*e - 6*d^4)*(x*e + d)^m*a^2*c*d^2/(m^4 + 10*m^3 + 35*m^2 +
50*m + 24) + 3*((m + 1)*x^2*e^2 + d*m*x*e - d^2)*a^3*d^2*e^(m*log(x*e + d) + 2)/(m^2 + 3*m + 2) + 3*((m^5 + 15
*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*x^6*e^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*x^5*e^5 - 5*(m^4 +
6*m^3 + 11*m^2 + 6*m)*d^2*x^4*e^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*x^3*e^3 - 60*(m^2 + m)*d^4*x^2*e^2 + 120*d^5*m*
x*e - 120*d^6)*a*c^2*d^2*e^(m*log(x*e + d) - 2)/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720) +
3*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*x^5*e^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*x^4*e^4 - 4*(m^3 + 3*m^2 + 2*
m)*d^2*x^3*e^3 + 12*(m^2 + m)*d^3*x^2*e^2 - 24*d^4*m*x*e + 24*d^5)*(x*e + d)^m*a^2*c*d/(m^5 + 15*m^4 + 85*m^3
+ 225*m^2 + 274*m + 120) + 3*((m^2 + 3*m + 2)*x^3*e^3 + (m^2 + m)*d*x^2*e^2 - 2*d^2*m*x*e + 2*d^3)*a^3*d*e^(m*
log(x*e + d) + 2)/(m^3 + 6*m^2 + 11*m + 6) + ((m^3 + 6*m^2 + 11*m + 6)*x^4*e^4 + (m^3 + 3*m^2 + 2*m)*d*x^3*e^3
- 3*(m^2 + m)*d^2*x^2*e^2 + 6*d^3*m*x*e - 6*d^4)*a^3*e^(m*log(x*e + d) + 2)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 2
4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1045 vs. \(2 (127) = 254\).
time = 2.79, size = 1045, normalized size = 8.04 \begin {gather*} \frac {{\left (6 \, c^{3} d^{9} m x e - 6 \, c^{3} d^{10} + {\left (a^{3} m^{3} + 18 \, a^{3} m^{2} + 107 \, a^{3} m + 210 \, a^{3}\right )} x^{4} e^{10} + {\left (3 \, {\left (a^{2} c d m^{3} + 17 \, a^{2} c d m^{2} + 94 \, a^{2} c d m + 168 \, a^{2} c d\right )} x^{5} + 4 \, {\left (a^{3} d m^{3} + 18 \, a^{3} d m^{2} + 107 \, a^{3} d m + 210 \, a^{3} d\right )} x^{3}\right )} e^{9} + 3 \, {\left ({\left (a c^{2} d^{2} m^{3} + 16 \, a c^{2} d^{2} m^{2} + 83 \, a c^{2} d^{2} m + 140 \, a c^{2} d^{2}\right )} x^{6} + {\left (4 \, a^{2} c d^{2} m^{3} + 67 \, a^{2} c d^{2} m^{2} + 363 \, a^{2} c d^{2} m + 630 \, a^{2} c d^{2}\right )} x^{4} + 2 \, {\left (a^{3} d^{2} m^{3} + 18 \, a^{3} d^{2} m^{2} + 107 \, a^{3} d^{2} m + 210 \, a^{3} d^{2}\right )} x^{2}\right )} e^{8} + {\left ({\left (c^{3} d^{3} m^{3} + 15 \, c^{3} d^{3} m^{2} + 74 \, c^{3} d^{3} m + 120 \, c^{3} d^{3}\right )} x^{7} + 6 \, {\left (2 \, a c^{2} d^{3} m^{3} + 31 \, a c^{2} d^{3} m^{2} + 155 \, a c^{2} d^{3} m + 252 \, a c^{2} d^{3}\right )} x^{5} + 6 \, {\left (3 \, a^{2} c d^{3} m^{3} + 49 \, a^{2} c d^{3} m^{2} + 256 \, a^{2} c d^{3} m + 420 \, a^{2} c d^{3}\right )} x^{3} + 4 \, {\left (a^{3} d^{3} m^{3} + 18 \, a^{3} d^{3} m^{2} + 107 \, a^{3} d^{3} m + 210 \, a^{3} d^{3}\right )} x\right )} e^{7} + {\left (a^{3} d^{4} m^{3} + 18 \, a^{3} d^{4} m^{2} + 107 \, a^{3} d^{4} m + 210 \, a^{3} d^{4} + {\left (4 \, c^{3} d^{4} m^{3} + 57 \, c^{3} d^{4} m^{2} + 269 \, c^{3} d^{4} m + 420 \, c^{3} d^{4}\right )} x^{6} + 6 \, {\left (3 \, a c^{2} d^{4} m^{3} + 44 \, a c^{2} d^{4} m^{2} + 206 \, a c^{2} d^{4} m + 315 \, a c^{2} d^{4}\right )} x^{4} + 6 \, {\left (2 \, a^{2} c d^{4} m^{3} + 31 \, a^{2} c d^{4} m^{2} + 149 \, a^{2} c d^{4} m + 210 \, a^{2} c d^{4}\right )} x^{2}\right )} e^{6} + 3 \, {\left (2 \, {\left (c^{3} d^{5} m^{3} + 13 \, c^{3} d^{5} m^{2} + 57 \, c^{3} d^{5} m + 84 \, c^{3} d^{5}\right )} x^{5} + 4 \, {\left (a c^{2} d^{5} m^{3} + 13 \, a c^{2} d^{5} m^{2} + 52 \, a c^{2} d^{5} m + 70 \, a c^{2} d^{5}\right )} x^{3} + {\left (a^{2} c d^{5} m^{3} + 13 \, a^{2} c d^{5} m^{2} + 42 \, a^{2} c d^{5} m\right )} x\right )} e^{5} - {\left (3 \, a^{2} c d^{6} m^{2} + 39 \, a^{2} c d^{6} m + 126 \, a^{2} c d^{6} - 2 \, {\left (2 \, c^{3} d^{6} m^{3} + 21 \, c^{3} d^{6} m^{2} + 79 \, c^{3} d^{6} m + 105 \, c^{3} d^{6}\right )} x^{4} - 3 \, {\left (a c^{2} d^{6} m^{3} + 8 \, a c^{2} d^{6} m^{2} + 7 \, a c^{2} d^{6} m\right )} x^{2}\right )} e^{4} + {\left ({\left (c^{3} d^{7} m^{3} + 3 \, c^{3} d^{7} m^{2} + 2 \, c^{3} d^{7} m\right )} x^{3} - 6 \, {\left (a c^{2} d^{7} m^{2} + 7 \, a c^{2} d^{7} m\right )} x\right )} e^{3} + 3 \, {\left (2 \, a c^{2} d^{8} m + 14 \, a c^{2} d^{8} - {\left (c^{3} d^{8} m^{2} + c^{3} d^{8} m\right )} x^{2}\right )} e^{2}\right )} {\left (x e + d\right )}^{m} e^{\left (-4\right )}}{m^{4} + 22 \, m^{3} + 179 \, m^{2} + 638 \, m + 840} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*c^3*d^9*m*x*e - 6*c^3*d^10 + (a^3*m^3 + 18*a^3*m^2 + 107*a^3*m + 210*a^3)*x^4*e^10 + (3*(a^2*c*d*m^3 + 17*a
^2*c*d*m^2 + 94*a^2*c*d*m + 168*a^2*c*d)*x^5 + 4*(a^3*d*m^3 + 18*a^3*d*m^2 + 107*a^3*d*m + 210*a^3*d)*x^3)*e^9
+ 3*((a*c^2*d^2*m^3 + 16*a*c^2*d^2*m^2 + 83*a*c^2*d^2*m + 140*a*c^2*d^2)*x^6 + (4*a^2*c*d^2*m^3 + 67*a^2*c*d^
2*m^2 + 363*a^2*c*d^2*m + 630*a^2*c*d^2)*x^4 + 2*(a^3*d^2*m^3 + 18*a^3*d^2*m^2 + 107*a^3*d^2*m + 210*a^3*d^2)*
x^2)*e^8 + ((c^3*d^3*m^3 + 15*c^3*d^3*m^2 + 74*c^3*d^3*m + 120*c^3*d^3)*x^7 + 6*(2*a*c^2*d^3*m^3 + 31*a*c^2*d^
3*m^2 + 155*a*c^2*d^3*m + 252*a*c^2*d^3)*x^5 + 6*(3*a^2*c*d^3*m^3 + 49*a^2*c*d^3*m^2 + 256*a^2*c*d^3*m + 420*a
^2*c*d^3)*x^3 + 4*(a^3*d^3*m^3 + 18*a^3*d^3*m^2 + 107*a^3*d^3*m + 210*a^3*d^3)*x)*e^7 + (a^3*d^4*m^3 + 18*a^3*
d^4*m^2 + 107*a^3*d^4*m + 210*a^3*d^4 + (4*c^3*d^4*m^3 + 57*c^3*d^4*m^2 + 269*c^3*d^4*m + 420*c^3*d^4)*x^6 + 6
*(3*a*c^2*d^4*m^3 + 44*a*c^2*d^4*m^2 + 206*a*c^2*d^4*m + 315*a*c^2*d^4)*x^4 + 6*(2*a^2*c*d^4*m^3 + 31*a^2*c*d^
4*m^2 + 149*a^2*c*d^4*m + 210*a^2*c*d^4)*x^2)*e^6 + 3*(2*(c^3*d^5*m^3 + 13*c^3*d^5*m^2 + 57*c^3*d^5*m + 84*c^3
*d^5)*x^5 + 4*(a*c^2*d^5*m^3 + 13*a*c^2*d^5*m^2 + 52*a*c^2*d^5*m + 70*a*c^2*d^5)*x^3 + (a^2*c*d^5*m^3 + 13*a^2
*c*d^5*m^2 + 42*a^2*c*d^5*m)*x)*e^5 - (3*a^2*c*d^6*m^2 + 39*a^2*c*d^6*m + 126*a^2*c*d^6 - 2*(2*c^3*d^6*m^3 + 2
1*c^3*d^6*m^2 + 79*c^3*d^6*m + 105*c^3*d^6)*x^4 - 3*(a*c^2*d^6*m^3 + 8*a*c^2*d^6*m^2 + 7*a*c^2*d^6*m)*x^2)*e^4
+ ((c^3*d^7*m^3 + 3*c^3*d^7*m^2 + 2*c^3*d^7*m)*x^3 - 6*(a*c^2*d^7*m^2 + 7*a*c^2*d^7*m)*x)*e^3 + 3*(2*a*c^2*d^
8*m + 14*a*c^2*d^8 - (c^3*d^8*m^2 + c^3*d^8*m)*x^2)*e^2)*(x*e + d)^m*e^(-4)/(m^4 + 22*m^3 + 179*m^2 + 638*m +
840)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 7164 vs. \(2 (114) = 228\).
time = 2.79, size = 7164, normalized size = 55.11 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((c**3*d**6*d**m*x**4/4, Eq(e, 0)), (-2*a**3*e**6/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*
e**7*x**3) - 3*a**2*c*d**2*e**4/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 9*a**2*c*d*e**
5*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*a*c**2*d**4*e**2/(6*d**3*e**4 + 18*d**2*
e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 18*a*c**2*d**3*e**3*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2
+ 6*e**7*x**3) - 18*a*c**2*d**2*e**4*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*d**6*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**6/(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**5*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**5*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**4*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**4*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*d**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)), (-a**3*e**6/(2*d**2*e**4 +
4*d*e**5*x + 2*e**6*x**2) - 3*a**2*c*d**2*e**4/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*a**2*c*d*e**5*x/(
2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 6*a*c**2*d**4*e**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x
**2) + 9*a*c**2*d**4*e**2/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 12*a*c**2*d**3*e**3*x*log(d/e + x)/(2*d**
2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 12*a*c**2*d**3*e**3*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 6*a*c**2
*d**2*e**4*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**6*log(d/e + x)/(2*d**2*e**4
+ 4*d*e**5*x + 2*e**6*x**2) - 15*c**3*d**6/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 12*c**3*d**5*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**5*e*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6
*c**3*d**4*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**4*e**2*x**2/(2*d**2*e**
4 + 4*d*e**5*x + 2*e**6*x**2) + 2*c**3*d**3*e**3*x**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2), Eq(m, -6)), (-
2*a**3*e**6/(2*d*e**4 + 2*e**5*x) + 6*a**2*c*d**2*e**4*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 6*a**2*c*d**2*e**4
/(2*d*e**4 + 2*e**5*x) + 6*a**2*c*d*e**5*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 12*a*c**2*d**4*e**2*log(d/e +
x)/(2*d*e**4 + 2*e**5*x) - 30*a*c**2*d**4*e**2/(2*d*e**4 + 2*e**5*x) - 12*a*c**2*d**3*e**3*x*log(d/e + x)/(2*d
*e**4 + 2*e**5*x) - 18*a*c**2*d**3*e**3*x/(2*d*e**4 + 2*e**5*x) + 6*a*c**2*d**2*e**4*x**2/(2*d*e**4 + 2*e**5*x
) + 6*c**3*d**6*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 12*c**3*d**6/(2*d*e**4 + 2*e**5*x) + 6*c**3*d**5*e*x*log(
d/e + x)/(2*d*e**4 + 2*e**5*x) + 6*c**3*d**5*e*x/(2*d*e**4 + 2*e**5*x) - 3*c**3*d**4*e**2*x**2/(2*d*e**4 + 2*e
**5*x) + c**3*d**3*e**3*x**3/(2*d*e**4 + 2*e**5*x), Eq(m, -5)), (a**3*e**2*log(d/e + x) - 3*a**2*c*d**2*log(d/
e + x) + 3*a**2*c*d*e*x + 3*a*c**2*d**4*log(d/e + x)/e**2 - 3*a*c**2*d**3*x/e + 3*a*c**2*d**2*x**2/2 - c**3*d*
*6*log(d/e + x)/e**4 + c**3*d**5*x/e**3 - c**3*d**4*x**2/(2*e**2) + c**3*d**3*x**3/(3*e), Eq(m, -4)), (a**3*d*
*4*e**6*m**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) + 18*a**3*d**4*e*
*6*m**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) + 107*a**3*d**4*e**6*m
*(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*a**3*d**4*e**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) + 4*a**3*d**3*e**7*m**3*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) + 72*a**3*d**3*e**7*m**2*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) + 428*a**3*d**3*e**7*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) + 840*a**3*d**3*e**7*x*(d + e*x)**m/(e**4*m**4 + 2
2*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 6*a**3*d**2*e**8*m**3*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) + 108*a**3*d**2*e**8*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 2
2*e**4*m**3 + 179*e**4*m**2 + 638*e**4*m + 840*e**4) + 642*a**3*d**2*e**8*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) + 1260*a**3*d**2*e**8*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) + 4*a**3*d*e**9*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) + 72*a**3*d*e**9*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) + 428*a**3*d*e**9*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) + 840*a**3*d*e**9*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) + a**3*e**10*m**3*x**4*(d + e*x)**m/(e**4*m**4 + 22*e**4*m**3 + 179*e**4*m**2 + 6
38*e**4*m + 840*e**4) + 18*a**3*e**10*m**2*x**4...

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1997 vs. \(2 (127) = 254\).
time = 1.24, size = 1997, normalized size = 15.36 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)^m*c^3*d^3*m^3*x^7*e^7 + 4*(x*e + d)^m*c^3*d^4*m^3*x^6*e^6 + 6*(x*e + d)^m*c^3*d^5*m^3*x^5*e^5 + 4*(
x*e + d)^m*c^3*d^6*m^3*x^4*e^4 + (x*e + d)^m*c^3*d^7*m^3*x^3*e^3 + 15*(x*e + d)^m*c^3*d^3*m^2*x^7*e^7 + 57*(x*
e + d)^m*c^3*d^4*m^2*x^6*e^6 + 78*(x*e + d)^m*c^3*d^5*m^2*x^5*e^5 + 42*(x*e + d)^m*c^3*d^6*m^2*x^4*e^4 + 3*(x*
e + d)^m*c^3*d^7*m^2*x^3*e^3 - 3*(x*e + d)^m*c^3*d^8*m^2*x^2*e^2 + 3*(x*e + d)^m*a*c^2*d^2*m^3*x^6*e^8 + 12*(x
*e + d)^m*a*c^2*d^3*m^3*x^5*e^7 + 74*(x*e + d)^m*c^3*d^3*m*x^7*e^7 + 18*(x*e + d)^m*a*c^2*d^4*m^3*x^4*e^6 + 26
9*(x*e + d)^m*c^3*d^4*m*x^6*e^6 + 12*(x*e + d)^m*a*c^2*d^5*m^3*x^3*e^5 + 342*(x*e + d)^m*c^3*d^5*m*x^5*e^5 + 3
*(x*e + d)^m*a*c^2*d^6*m^3*x^2*e^4 + 158*(x*e + d)^m*c^3*d^6*m*x^4*e^4 + 2*(x*e + d)^m*c^3*d^7*m*x^3*e^3 - 3*(
x*e + d)^m*c^3*d^8*m*x^2*e^2 + 6*(x*e + d)^m*c^3*d^9*m*x*e + 48*(x*e + d)^m*a*c^2*d^2*m^2*x^6*e^8 + 186*(x*e +
d)^m*a*c^2*d^3*m^2*x^5*e^7 + 120*(x*e + d)^m*c^3*d^3*x^7*e^7 + 264*(x*e + d)^m*a*c^2*d^4*m^2*x^4*e^6 + 420*(x
*e + d)^m*c^3*d^4*x^6*e^6 + 156*(x*e + d)^m*a*c^2*d^5*m^2*x^3*e^5 + 504*(x*e + d)^m*c^3*d^5*x^5*e^5 + 24*(x*e
+ d)^m*a*c^2*d^6*m^2*x^2*e^4 + 210*(x*e + d)^m*c^3*d^6*x^4*e^4 - 6*(x*e + d)^m*a*c^2*d^7*m^2*x*e^3 - 6*(x*e +
d)^m*c^3*d^10 + 3*(x*e + d)^m*a^2*c*d*m^3*x^5*e^9 + 12*(x*e + d)^m*a^2*c*d^2*m^3*x^4*e^8 + 249*(x*e + d)^m*a*c
^2*d^2*m*x^6*e^8 + 18*(x*e + d)^m*a^2*c*d^3*m^3*x^3*e^7 + 930*(x*e + d)^m*a*c^2*d^3*m*x^5*e^7 + 12*(x*e + d)^m
*a^2*c*d^4*m^3*x^2*e^6 + 1236*(x*e + d)^m*a*c^2*d^4*m*x^4*e^6 + 3*(x*e + d)^m*a^2*c*d^5*m^3*x*e^5 + 624*(x*e +
d)^m*a*c^2*d^5*m*x^3*e^5 + 21*(x*e + d)^m*a*c^2*d^6*m*x^2*e^4 - 42*(x*e + d)^m*a*c^2*d^7*m*x*e^3 + 6*(x*e + d
)^m*a*c^2*d^8*m*e^2 + 51*(x*e + d)^m*a^2*c*d*m^2*x^5*e^9 + 201*(x*e + d)^m*a^2*c*d^2*m^2*x^4*e^8 + 420*(x*e +
d)^m*a*c^2*d^2*x^6*e^8 + 294*(x*e + d)^m*a^2*c*d^3*m^2*x^3*e^7 + 1512*(x*e + d)^m*a*c^2*d^3*x^5*e^7 + 186*(x*e
+ d)^m*a^2*c*d^4*m^2*x^2*e^6 + 1890*(x*e + d)^m*a*c^2*d^4*x^4*e^6 + 39*(x*e + d)^m*a^2*c*d^5*m^2*x*e^5 + 840*
(x*e + d)^m*a*c^2*d^5*x^3*e^5 - 3*(x*e + d)^m*a^2*c*d^6*m^2*e^4 + 42*(x*e + d)^m*a*c^2*d^8*e^2 + (x*e + d)^m*a
^3*m^3*x^4*e^10 + 4*(x*e + d)^m*a^3*d*m^3*x^3*e^9 + 282*(x*e + d)^m*a^2*c*d*m*x^5*e^9 + 6*(x*e + d)^m*a^3*d^2*
m^3*x^2*e^8 + 1089*(x*e + d)^m*a^2*c*d^2*m*x^4*e^8 + 4*(x*e + d)^m*a^3*d^3*m^3*x*e^7 + 1536*(x*e + d)^m*a^2*c*
d^3*m*x^3*e^7 + (x*e + d)^m*a^3*d^4*m^3*e^6 + 894*(x*e + d)^m*a^2*c*d^4*m*x^2*e^6 + 126*(x*e + d)^m*a^2*c*d^5*
m*x*e^5 - 39*(x*e + d)^m*a^2*c*d^6*m*e^4 + 18*(x*e + d)^m*a^3*m^2*x^4*e^10 + 72*(x*e + d)^m*a^3*d*m^2*x^3*e^9
+ 504*(x*e + d)^m*a^2*c*d*x^5*e^9 + 108*(x*e + d)^m*a^3*d^2*m^2*x^2*e^8 + 1890*(x*e + d)^m*a^2*c*d^2*x^4*e^8 +
72*(x*e + d)^m*a^3*d^3*m^2*x*e^7 + 2520*(x*e + d)^m*a^2*c*d^3*x^3*e^7 + 18*(x*e + d)^m*a^3*d^4*m^2*e^6 + 1260
*(x*e + d)^m*a^2*c*d^4*x^2*e^6 - 126*(x*e + d)^m*a^2*c*d^6*e^4 + 107*(x*e + d)^m*a^3*m*x^4*e^10 + 428*(x*e + d
)^m*a^3*d*m*x^3*e^9 + 642*(x*e + d)^m*a^3*d^2*m*x^2*e^8 + 428*(x*e + d)^m*a^3*d^3*m*x*e^7 + 107*(x*e + d)^m*a^
3*d^4*m*e^6 + 210*(x*e + d)^m*a^3*x^4*e^10 + 840*(x*e + d)^m*a^3*d*x^3*e^9 + 1260*(x*e + d)^m*a^3*d^2*x^2*e^8
+ 840*(x*e + d)^m*a^3*d^3*x*e^7 + 210*(x*e + d)^m*a^3*d^4*e^6)/(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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Mupad [B]
time = 1.52, size = 1202, normalized size = 9.25 \begin {gather*} \frac {d^4\,{\left (d+e\,x\right )}^m\,\left (a^3\,e^6\,m^3+18\,a^3\,e^6\,m^2+107\,a^3\,e^6\,m+210\,a^3\,e^6-3\,a^2\,c\,d^2\,e^4\,m^2-39\,a^2\,c\,d^2\,e^4\,m-126\,a^2\,c\,d^2\,e^4+6\,a\,c^2\,d^4\,e^2\,m+42\,a\,c^2\,d^4\,e^2-6\,c^3\,d^6\right )}{e^4\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}+\frac {x^4\,{\left (d+e\,x\right )}^m\,\left (a^3\,e^{10}\,m^3+18\,a^3\,e^{10}\,m^2+107\,a^3\,e^{10}\,m+210\,a^3\,e^{10}+12\,a^2\,c\,d^2\,e^8\,m^3+201\,a^2\,c\,d^2\,e^8\,m^2+1089\,a^2\,c\,d^2\,e^8\,m+1890\,a^2\,c\,d^2\,e^8+18\,a\,c^2\,d^4\,e^6\,m^3+264\,a\,c^2\,d^4\,e^6\,m^2+1236\,a\,c^2\,d^4\,e^6\,m+1890\,a\,c^2\,d^4\,e^6+4\,c^3\,d^6\,e^4\,m^3+42\,c^3\,d^6\,e^4\,m^2+158\,c^3\,d^6\,e^4\,m+210\,c^3\,d^6\,e^4\right )}{e^4\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}+\frac {3\,d^2\,x^2\,{\left (d+e\,x\right )}^m\,\left (2\,a^3\,e^6\,m^3+36\,a^3\,e^6\,m^2+214\,a^3\,e^6\,m+420\,a^3\,e^6+4\,a^2\,c\,d^2\,e^4\,m^3+62\,a^2\,c\,d^2\,e^4\,m^2+298\,a^2\,c\,d^2\,e^4\,m+420\,a^2\,c\,d^2\,e^4+a\,c^2\,d^4\,e^2\,m^3+8\,a\,c^2\,d^4\,e^2\,m^2+7\,a\,c^2\,d^4\,e^2\,m-c^3\,d^6\,m^2-c^3\,d^6\,m\right )}{e^2\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}+\frac {d^3\,x\,{\left (d+e\,x\right )}^m\,\left (4\,a^3\,e^6\,m^3+72\,a^3\,e^6\,m^2+428\,a^3\,e^6\,m+840\,a^3\,e^6+3\,a^2\,c\,d^2\,e^4\,m^3+39\,a^2\,c\,d^2\,e^4\,m^2+126\,a^2\,c\,d^2\,e^4\,m-6\,a\,c^2\,d^4\,e^2\,m^2-42\,a\,c^2\,d^4\,e^2\,m+6\,c^3\,d^6\,m\right )}{e^3\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}+\frac {d\,x^3\,{\left (d+e\,x\right )}^m\,\left (4\,a^3\,e^6\,m^3+72\,a^3\,e^6\,m^2+428\,a^3\,e^6\,m+840\,a^3\,e^6+18\,a^2\,c\,d^2\,e^4\,m^3+294\,a^2\,c\,d^2\,e^4\,m^2+1536\,a^2\,c\,d^2\,e^4\,m+2520\,a^2\,c\,d^2\,e^4+12\,a\,c^2\,d^4\,e^2\,m^3+156\,a\,c^2\,d^4\,e^2\,m^2+624\,a\,c^2\,d^4\,e^2\,m+840\,a\,c^2\,d^4\,e^2+c^3\,d^6\,m^3+3\,c^3\,d^6\,m^2+2\,c^3\,d^6\,m\right )}{e\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}+\frac {c^3\,d^3\,e^3\,x^7\,{\left (d+e\,x\right )}^m\,\left (m^3+15\,m^2+74\,m+120\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {3\,c\,d\,e\,x^5\,\left (m+4\right )\,{\left (d+e\,x\right )}^m\,\left (a^2\,e^4\,m^2+13\,a^2\,e^4\,m+42\,a^2\,e^4+4\,a\,c\,d^2\,e^2\,m^2+46\,a\,c\,d^2\,e^2\,m+126\,a\,c\,d^2\,e^2+2\,c^2\,d^4\,m^2+18\,c^2\,d^4\,m+42\,c^2\,d^4\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {c^2\,d^2\,e^2\,x^6\,{\left (d+e\,x\right )}^m\,\left (m^2+9\,m+20\right )\,\left (21\,a\,e^2+21\,c\,d^2+3\,a\,e^2\,m+4\,c\,d^2\,m\right )}{m^4+22\,m^3+179\,m^2+638\,m+840} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d^4*(d + e*x)^m*(210*a^3*e^6 - 6*c^3*d^6 + 107*a^3*e^6*m + 18*a^3*e^6*m^2 + a^3*e^6*m^3 + 42*a*c^2*d^4*e^2 -
126*a^2*c*d^2*e^4 + 6*a*c^2*d^4*e^2*m - 39*a^2*c*d^2*e^4*m - 3*a^2*c*d^2*e^4*m^2))/(e^4*(638*m + 179*m^2 + 22*
m^3 + m^4 + 840)) + (x^4*(d + e*x)^m*(210*a^3*e^10 + 107*a^3*e^10*m + 210*c^3*d^6*e^4 + 18*a^3*e^10*m^2 + a^3*
e^10*m^3 + 1890*a*c^2*d^4*e^6 + 1890*a^2*c*d^2*e^8 + 158*c^3*d^6*e^4*m + 42*c^3*d^6*e^4*m^2 + 4*c^3*d^6*e^4*m^
3 + 1236*a*c^2*d^4*e^6*m + 1089*a^2*c*d^2*e^8*m + 264*a*c^2*d^4*e^6*m^2 + 201*a^2*c*d^2*e^8*m^2 + 18*a*c^2*d^4
*e^6*m^3 + 12*a^2*c*d^2*e^8*m^3))/(e^4*(638*m + 179*m^2 + 22*m^3 + m^4 + 840)) + (3*d^2*x^2*(d + e*x)^m*(420*a
^3*e^6 + 214*a^3*e^6*m - c^3*d^6*m + 36*a^3*e^6*m^2 + 2*a^3*e^6*m^3 - c^3*d^6*m^2 + 420*a^2*c*d^2*e^4 + 7*a*c^
2*d^4*e^2*m + 298*a^2*c*d^2*e^4*m + 8*a*c^2*d^4*e^2*m^2 + 62*a^2*c*d^2*e^4*m^2 + a*c^2*d^4*e^2*m^3 + 4*a^2*c*d
^2*e^4*m^3))/(e^2*(638*m + 179*m^2 + 22*m^3 + m^4 + 840)) + (d^3*x*(d + e*x)^m*(840*a^3*e^6 + 428*a^3*e^6*m +
6*c^3*d^6*m + 72*a^3*e^6*m^2 + 4*a^3*e^6*m^3 - 42*a*c^2*d^4*e^2*m + 126*a^2*c*d^2*e^4*m - 6*a*c^2*d^4*e^2*m^2
+ 39*a^2*c*d^2*e^4*m^2 + 3*a^2*c*d^2*e^4*m^3))/(e^3*(638*m + 179*m^2 + 22*m^3 + m^4 + 840)) + (d*x^3*(d + e*x)
^m*(840*a^3*e^6 + 428*a^3*e^6*m + 2*c^3*d^6*m + 72*a^3*e^6*m^2 + 4*a^3*e^6*m^3 + 3*c^3*d^6*m^2 + c^3*d^6*m^3 +
840*a*c^2*d^4*e^2 + 2520*a^2*c*d^2*e^4 + 624*a*c^2*d^4*e^2*m + 1536*a^2*c*d^2*e^4*m + 156*a*c^2*d^4*e^2*m^2 +
294*a^2*c*d^2*e^4*m^2 + 12*a*c^2*d^4*e^2*m^3 + 18*a^2*c*d^2*e^4*m^3))/(e*(638*m + 179*m^2 + 22*m^3 + m^4 + 84
0)) + (c^3*d^3*e^3*x^7*(d + e*x)^m*(74*m + 15*m^2 + m^3 + 120))/(638*m + 179*m^2 + 22*m^3 + m^4 + 840) + (3*c*
d*e*x^5*(m + 4)*(d + e*x)^m*(42*a^2*e^4 + 42*c^2*d^4 + 13*a^2*e^4*m + 18*c^2*d^4*m + a^2*e^4*m^2 + 2*c^2*d^4*m
^2 + 126*a*c*d^2*e^2 + 46*a*c*d^2*e^2*m + 4*a*c*d^2*e^2*m^2))/(638*m + 179*m^2 + 22*m^3 + m^4 + 840) + (c^2*d^
2*e^2*x^6*(d + e*x)^m*(9*m + m^2 + 20)*(21*a*e^2 + 21*c*d^2 + 3*a*e^2*m + 4*c*d^2*m))/(638*m + 179*m^2 + 22*m^
3 + m^4 + 840)

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