Integrand size = 35, antiderivative size = 130 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=-\frac {\left (c d^2-a e^2\right )^3 (d+e x)^{4+m}}{e^4 (4+m)}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{5+m}}{e^4 (5+m)}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{6+m}}{e^4 (6+m)}+\frac {c^3 d^3 (d+e x)^{7+m}}{e^4 (7+m)} \]
-(-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)
Time = 0.12 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.88 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {(d+e x)^{4+m} \left (-\frac {\left (c d^2-a e^2\right )^3}{4+m}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)}{5+m}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^2}{6+m}+\frac {c^3 d^3 (d+e x)^3}{7+m}\right )}{e^4} \]
((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
Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {1121, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3 \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{m+5}}{e^3}+\frac {\left (a e^2-c d^2\right )^3 (d+e x)^{m+3}}{e^3}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{m+4}}{e^3}+\frac {c^3 d^3 (d+e x)^{m+6}}{e^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{m+6}}{e^4 (m+6)}-\frac {\left (c d^2-a e^2\right )^3 (d+e x)^{m+4}}{e^4 (m+4)}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{m+5}}{e^4 (m+5)}+\frac {c^3 d^3 (d+e x)^{m+7}}{e^4 (m+7)}\) |
-(((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))
3.21.84.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(435\) vs. \(2(130)=260\).
Time = 2.96 (sec) , antiderivative size = 436, normalized size of antiderivative = 3.35
method | result | size |
gosper | \(\frac {\left (e x +d \right )^{4+m} \left (c^{3} d^{3} e^{3} m^{3} x^{3}+3 a \,c^{2} d^{2} e^{4} m^{3} x^{2}+15 c^{3} d^{3} e^{3} m^{2} x^{3}+3 a^{2} c d \,e^{5} m^{3} x +48 a \,c^{2} d^{2} e^{4} m^{2} x^{2}-3 c^{3} d^{4} e^{2} m^{2} x^{2}+74 c^{3} d^{3} e^{3} m \,x^{3}+a^{3} e^{6} m^{3}+51 a^{2} c d \,e^{5} m^{2} x -6 a \,c^{2} d^{3} e^{3} m^{2} x +249 a \,c^{2} d^{2} e^{4} m \,x^{2}-27 c^{3} d^{4} e^{2} m \,x^{2}+120 x^{3} c^{3} d^{3} e^{3}+18 a^{3} e^{6} m^{2}-3 a^{2} c \,d^{2} e^{4} m^{2}+282 a^{2} c d \,e^{5} m x -66 a \,c^{2} d^{3} e^{3} m x +420 x^{2} a \,c^{2} d^{2} e^{4}+6 c^{3} d^{5} e m x -60 x^{2} c^{3} d^{4} e^{2}+107 a^{3} e^{6} m -39 a^{2} c \,d^{2} e^{4} m +504 x \,a^{2} c d \,e^{5}+6 a \,c^{2} d^{4} e^{2} m -168 x a \,c^{2} d^{3} e^{3}+24 x \,c^{3} d^{5} e +210 e^{6} a^{3}-126 d^{2} e^{4} a^{2} c +42 d^{4} e^{2} c^{2} a -6 c^{3} d^{6}\right )}{e^{4} \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}\) | \(436\) |
norman | \(\text {Expression too large to display}\) | \(1150\) |
risch | \(\text {Expression too large to display}\) | \(1377\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1986\) |
1/e^4*(e*x+d)^(4+m)/(m^4+22*m^3+179*m^2+638*m+840)*(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)
Leaf count of result is larger than twice the leaf count of optimal. 1156 vs. \(2 (130) = 260\).
Time = 0.40 (sec) , antiderivative size = 1156, normalized size of antiderivative = 8.89 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\text {Too large to display} \]
(a^3*d^4*e^6*m^3 - 6*c^3*d^10 + 42*a*c^2*d^8*e^2 - 126*a^2*c*d^6*e^4 + 210 *a^3*d^4*e^6 + (c^3*d^3*e^7*m^3 + 15*c^3*d^3*e^7*m^2 + 74*c^3*d^3*e^7*m + 120*c^3*d^3*e^7)*x^7 + (420*c^3*d^4*e^6 + 420*a*c^2*d^2*e^8 + (4*c^3*d^4*e ^6 + 3*a*c^2*d^2*e^8)*m^3 + 3*(19*c^3*d^4*e^6 + 16*a*c^2*d^2*e^8)*m^2 + (2 69*c^3*d^4*e^6 + 249*a*c^2*d^2*e^8)*m)*x^6 + 3*(168*c^3*d^5*e^5 + 504*a*c^ 2*d^3*e^7 + 168*a^2*c*d*e^9 + (2*c^3*d^5*e^5 + 4*a*c^2*d^3*e^7 + a^2*c*d*e ^9)*m^3 + (26*c^3*d^5*e^5 + 62*a*c^2*d^3*e^7 + 17*a^2*c*d*e^9)*m^2 + 2*(57 *c^3*d^5*e^5 + 155*a*c^2*d^3*e^7 + 47*a^2*c*d*e^9)*m)*x^5 + (210*c^3*d^6*e ^4 + 1890*a*c^2*d^4*e^6 + 1890*a^2*c*d^2*e^8 + 210*a^3*e^10 + (4*c^3*d^6*e ^4 + 18*a*c^2*d^4*e^6 + 12*a^2*c*d^2*e^8 + a^3*e^10)*m^3 + 3*(14*c^3*d^6*e ^4 + 88*a*c^2*d^4*e^6 + 67*a^2*c*d^2*e^8 + 6*a^3*e^10)*m^2 + (158*c^3*d^6* e^4 + 1236*a*c^2*d^4*e^6 + 1089*a^2*c*d^2*e^8 + 107*a^3*e^10)*m)*x^4 + (84 0*a*c^2*d^5*e^5 + 2520*a^2*c*d^3*e^7 + 840*a^3*d*e^9 + (c^3*d^7*e^3 + 12*a *c^2*d^5*e^5 + 18*a^2*c*d^3*e^7 + 4*a^3*d*e^9)*m^3 + 3*(c^3*d^7*e^3 + 52*a *c^2*d^5*e^5 + 98*a^2*c*d^3*e^7 + 24*a^3*d*e^9)*m^2 + 2*(c^3*d^7*e^3 + 312 *a*c^2*d^5*e^5 + 768*a^2*c*d^3*e^7 + 214*a^3*d*e^9)*m)*x^3 - 3*(a^2*c*d^6* e^4 - 6*a^3*d^4*e^6)*m^2 + 3*(420*a^2*c*d^4*e^6 + 420*a^3*d^2*e^8 + (a*c^2 *d^6*e^4 + 4*a^2*c*d^4*e^6 + 2*a^3*d^2*e^8)*m^3 - (c^3*d^8*e^2 - 8*a*c^2*d ^6*e^4 - 62*a^2*c*d^4*e^6 - 36*a^3*d^2*e^8)*m^2 - (c^3*d^8*e^2 - 7*a*c^2*d ^6*e^4 - 298*a^2*c*d^4*e^6 - 214*a^3*d^2*e^8)*m)*x^2 + (6*a*c^2*d^8*e^2...
Leaf count of result is larger than twice the leaf count of optimal. 7164 vs. \(2 (114) = 228\).
Time = 2.64 (sec) , antiderivative size = 7164, normalized size of antiderivative = 55.11 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\text {Too large to display} \]
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) + 1 8*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*...
Leaf count of result is larger than twice the leaf count of optimal. 1819 vs. \(2 (130) = 260\).
Time = 0.27 (sec) , antiderivative size = 1819, normalized size of antiderivative = 13.99 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\text {Too large to display} \]
3*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a^2*c*d^4/(m^2 + 3*m + 2) + 3*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a^3*d^2*e^2/(m^2 + 3*m + 2) + (e*x + d)^(m + 1)*a^3*d^3*e^2/(m + 1) + 9*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*a^2*c*d^3/(m^3 + 6 *m^2 + 11*m + 6) + 3*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^ 2*e*m*x + 2*d^3)*(e*x + d)^m*a*c^2*d^5/((m^3 + 6*m^2 + 11*m + 6)*e^2) + 3* ((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*a^3*d*e^2/(m^3 + 6*m^2 + 11*m + 6) + 9*((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*a^2*c*d^2/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) + ((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^6/((m^4 + 10*m^ 3 + 35*m^2 + 50*m + 24)*e^4) + 9*((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*a*c^2*d^4/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^2) + ((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*a^3*e^2/(m^4 + 10*m^3 + 35*m^ 2 + 50*m + 24) + 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*a^2*c*d/(m^5 +...
Leaf count of result is larger than twice the leaf count of optimal. 1999 vs. \(2 (130) = 260\).
Time = 0.30 (sec) , antiderivative size = 1999, normalized size of antiderivative = 15.38 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\text {Too large to display} \]
((e*x + d)^m*c^3*d^3*e^7*m^3*x^7 + 4*(e*x + d)^m*c^3*d^4*e^6*m^3*x^6 + 3*( e*x + d)^m*a*c^2*d^2*e^8*m^3*x^6 + 15*(e*x + d)^m*c^3*d^3*e^7*m^2*x^7 + 6* (e*x + d)^m*c^3*d^5*e^5*m^3*x^5 + 12*(e*x + d)^m*a*c^2*d^3*e^7*m^3*x^5 + 3 *(e*x + d)^m*a^2*c*d*e^9*m^3*x^5 + 57*(e*x + d)^m*c^3*d^4*e^6*m^2*x^6 + 48 *(e*x + d)^m*a*c^2*d^2*e^8*m^2*x^6 + 74*(e*x + d)^m*c^3*d^3*e^7*m*x^7 + 4* (e*x + d)^m*c^3*d^6*e^4*m^3*x^4 + 18*(e*x + d)^m*a*c^2*d^4*e^6*m^3*x^4 + 1 2*(e*x + d)^m*a^2*c*d^2*e^8*m^3*x^4 + (e*x + d)^m*a^3*e^10*m^3*x^4 + 78*(e *x + d)^m*c^3*d^5*e^5*m^2*x^5 + 186*(e*x + d)^m*a*c^2*d^3*e^7*m^2*x^5 + 51 *(e*x + d)^m*a^2*c*d*e^9*m^2*x^5 + 269*(e*x + d)^m*c^3*d^4*e^6*m*x^6 + 249 *(e*x + d)^m*a*c^2*d^2*e^8*m*x^6 + 120*(e*x + d)^m*c^3*d^3*e^7*x^7 + (e*x + d)^m*c^3*d^7*e^3*m^3*x^3 + 12*(e*x + d)^m*a*c^2*d^5*e^5*m^3*x^3 + 18*(e* x + d)^m*a^2*c*d^3*e^7*m^3*x^3 + 4*(e*x + d)^m*a^3*d*e^9*m^3*x^3 + 42*(e*x + d)^m*c^3*d^6*e^4*m^2*x^4 + 264*(e*x + d)^m*a*c^2*d^4*e^6*m^2*x^4 + 201* (e*x + d)^m*a^2*c*d^2*e^8*m^2*x^4 + 18*(e*x + d)^m*a^3*e^10*m^2*x^4 + 342* (e*x + d)^m*c^3*d^5*e^5*m*x^5 + 930*(e*x + d)^m*a*c^2*d^3*e^7*m*x^5 + 282* (e*x + d)^m*a^2*c*d*e^9*m*x^5 + 420*(e*x + d)^m*c^3*d^4*e^6*x^6 + 420*(e*x + d)^m*a*c^2*d^2*e^8*x^6 + 3*(e*x + d)^m*a*c^2*d^6*e^4*m^3*x^2 + 12*(e*x + d)^m*a^2*c*d^4*e^6*m^3*x^2 + 6*(e*x + d)^m*a^3*d^2*e^8*m^3*x^2 + 3*(e*x + d)^m*c^3*d^7*e^3*m^2*x^3 + 156*(e*x + d)^m*a*c^2*d^5*e^5*m^2*x^3 + 294*( e*x + d)^m*a^2*c*d^3*e^7*m^2*x^3 + 72*(e*x + d)^m*a^3*d*e^9*m^2*x^3 + 1...
Time = 10.61 (sec) , antiderivative size = 1202, normalized size of antiderivative = 9.25 \[ \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=\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} \]
(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*(63 8*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*(84 0*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 +...