\(\int (a+b x)^3 (A+B x) (d+e x)^4 \, dx\) [26]

Optimal result

Integrand size = 20, antiderivative size = 163 \[ \int (a+b x)^3 (A+B x) (d+e x)^4 \, dx=\frac {(b d-a e)^3 (B d-A e) (d+e x)^5}{5 e^5}-\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^6}{6 e^5}+\frac {3 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^7}{7 e^5}-\frac {b^2 (4 b B d-A b e-3 a B e) (d+e x)^8}{8 e^5}+\frac {b^3 B (d+e x)^9}{9 e^5} \] Output:

1/5*(-a*e+b*d)^3*(-A*e+B*d)*(e*x+d)^5/e^5-1/6*(-a*e+b*d)^2*(-3*A*b*e-B*a*e 
+4*B*b*d)*(e*x+d)^6/e^5+3/7*b*(-a*e+b*d)*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^7/ 
e^5-1/8*b^2*(-A*b*e-3*B*a*e+4*B*b*d)*(e*x+d)^8/e^5+1/9*b^3*B*(e*x+d)^9/e^5
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(397\) vs. \(2(163)=326\).

Time = 0.09 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.44 \[ \int (a+b x)^3 (A+B x) (d+e x)^4 \, dx=a^3 A d^4 x+\frac {1}{2} a^2 d^3 (3 A b d+a B d+4 a A e) x^2+\frac {1}{3} a d^2 \left (a B d (3 b d+4 a e)+3 A \left (b^2 d^2+4 a b d e+2 a^2 e^2\right )\right ) x^3+\frac {1}{4} d \left (3 a B d \left (b^2 d^2+4 a b d e+2 a^2 e^2\right )+A \left (b^3 d^3+12 a b^2 d^2 e+18 a^2 b d e^2+4 a^3 e^3\right )\right ) x^4+\frac {1}{5} \left (a^3 e^3 (4 B d+A e)+6 a^2 b d e^2 (3 B d+2 A e)+6 a b^2 d^2 e (2 B d+3 A e)+b^3 d^3 (B d+4 A e)\right ) x^5+\frac {1}{6} e \left (a^3 B e^3+3 a^2 b e^2 (4 B d+A e)+6 a b^2 d e (3 B d+2 A e)+2 b^3 d^2 (2 B d+3 A e)\right ) x^6+\frac {1}{7} b e^2 \left (3 a^2 B e^2+3 a b e (4 B d+A e)+2 b^2 d (3 B d+2 A e)\right ) x^7+\frac {1}{8} b^2 e^3 (4 b B d+A b e+3 a B e) x^8+\frac {1}{9} b^3 B e^4 x^9 \] Input:

Integrate[(a + b*x)^3*(A + B*x)*(d + e*x)^4,x]
 

Output:

a^3*A*d^4*x + (a^2*d^3*(3*A*b*d + a*B*d + 4*a*A*e)*x^2)/2 + (a*d^2*(a*B*d* 
(3*b*d + 4*a*e) + 3*A*(b^2*d^2 + 4*a*b*d*e + 2*a^2*e^2))*x^3)/3 + (d*(3*a* 
B*d*(b^2*d^2 + 4*a*b*d*e + 2*a^2*e^2) + A*(b^3*d^3 + 12*a*b^2*d^2*e + 18*a 
^2*b*d*e^2 + 4*a^3*e^3))*x^4)/4 + ((a^3*e^3*(4*B*d + A*e) + 6*a^2*b*d*e^2* 
(3*B*d + 2*A*e) + 6*a*b^2*d^2*e*(2*B*d + 3*A*e) + b^3*d^3*(B*d + 4*A*e))*x 
^5)/5 + (e*(a^3*B*e^3 + 3*a^2*b*e^2*(4*B*d + A*e) + 6*a*b^2*d*e*(3*B*d + 2 
*A*e) + 2*b^3*d^2*(2*B*d + 3*A*e))*x^6)/6 + (b*e^2*(3*a^2*B*e^2 + 3*a*b*e* 
(4*B*d + A*e) + 2*b^2*d*(3*B*d + 2*A*e))*x^7)/7 + (b^2*e^3*(4*b*B*d + A*b* 
e + 3*a*B*e)*x^8)/8 + (b^3*B*e^4*x^9)/9
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {86, 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.

\(\displaystyle \int (a+b x)^3 (A+B x) (d+e x)^4 \, dx\)

\(\Big \downarrow \) 86

\(\displaystyle \int \left (\frac {b^2 (d+e x)^7 (3 a B e+A b e-4 b B d)}{e^4}-\frac {3 b (d+e x)^6 (b d-a e) (a B e+A b e-2 b B d)}{e^4}+\frac {(d+e x)^5 (a e-b d)^2 (a B e+3 A b e-4 b B d)}{e^4}+\frac {(d+e x)^4 (a e-b d)^3 (A e-B d)}{e^4}+\frac {b^3 B (d+e x)^8}{e^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^2 (d+e x)^8 (-3 a B e-A b e+4 b B d)}{8 e^5}+\frac {3 b (d+e x)^7 (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5}-\frac {(d+e x)^6 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5}+\frac {(d+e x)^5 (b d-a e)^3 (B d-A e)}{5 e^5}+\frac {b^3 B (d+e x)^9}{9 e^5}\)

Input:

Int[(a + b*x)^3*(A + B*x)*(d + e*x)^4,x]
 

Output:

((b*d - a*e)^3*(B*d - A*e)*(d + e*x)^5)/(5*e^5) - ((b*d - a*e)^2*(4*b*B*d 
- 3*A*b*e - a*B*e)*(d + e*x)^6)/(6*e^5) + (3*b*(b*d - a*e)*(2*b*B*d - A*b* 
e - a*B*e)*(d + e*x)^7)/(7*e^5) - (b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e* 
x)^8)/(8*e^5) + (b^3*B*(d + e*x)^9)/(9*e^5)
 

Defintions of rubi rules used

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(433\) vs. \(2(153)=306\).

Time = 0.18 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.66

method	result	size
default	\(\frac {b^{3} B \,e^{4} x^{9}}{9}+\frac {\left (\left (b^{3} A +3 a \,b^{2} B \right ) e^{4}+4 b^{3} B d \,e^{3}\right ) x^{8}}{8}+\frac {\left (\left (3 a \,b^{2} A +3 a^{2} b B \right ) e^{4}+4 \left (b^{3} A +3 a \,b^{2} B \right ) d \,e^{3}+6 b^{3} B \,d^{2} e^{2}\right ) x^{7}}{7}+\frac {\left (\left (3 a^{2} b A +a^{3} B \right ) e^{4}+4 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d \,e^{3}+6 \left (b^{3} A +3 a \,b^{2} B \right ) d^{2} e^{2}+4 b^{3} B \,d^{3} e \right ) x^{6}}{6}+\frac {\left (a^{3} A \,e^{4}+4 \left (3 a^{2} b A +a^{3} B \right ) d \,e^{3}+6 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{2} e^{2}+4 \left (b^{3} A +3 a \,b^{2} B \right ) d^{3} e +b^{3} B \,d^{4}\right ) x^{5}}{5}+\frac {\left (4 a^{3} A d \,e^{3}+6 \left (3 a^{2} b A +a^{3} B \right ) d^{2} e^{2}+4 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{3} e +\left (b^{3} A +3 a \,b^{2} B \right ) d^{4}\right ) x^{4}}{4}+\frac {\left (6 a^{3} A \,d^{2} e^{2}+4 \left (3 a^{2} b A +a^{3} B \right ) d^{3} e +\left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{4}\right ) x^{3}}{3}+\frac {\left (4 a^{3} A \,d^{3} e +\left (3 a^{2} b A +a^{3} B \right ) d^{4}\right ) x^{2}}{2}+a^{3} A \,d^{4} x\)	\(434\)
norman	\(\frac {b^{3} B \,e^{4} x^{9}}{9}+\left (\frac {1}{8} A \,b^{3} e^{4}+\frac {3}{8} B a \,b^{2} e^{4}+\frac {1}{2} b^{3} B d \,e^{3}\right ) x^{8}+\left (\frac {3}{7} A a \,b^{2} e^{4}+\frac {4}{7} A \,b^{3} d \,e^{3}+\frac {3}{7} B \,a^{2} b \,e^{4}+\frac {12}{7} B a \,b^{2} d \,e^{3}+\frac {6}{7} b^{3} B \,d^{2} e^{2}\right ) x^{7}+\left (\frac {1}{2} A \,a^{2} b \,e^{4}+2 A a \,b^{2} d \,e^{3}+A \,b^{3} d^{2} e^{2}+\frac {1}{6} B \,a^{3} e^{4}+2 B \,a^{2} b d \,e^{3}+3 B a \,b^{2} d^{2} e^{2}+\frac {2}{3} b^{3} B \,d^{3} e \right ) x^{6}+\left (\frac {1}{5} a^{3} A \,e^{4}+\frac {12}{5} A \,a^{2} b d \,e^{3}+\frac {18}{5} A a \,b^{2} d^{2} e^{2}+\frac {4}{5} A \,b^{3} d^{3} e +\frac {4}{5} B \,a^{3} d \,e^{3}+\frac {18}{5} B \,a^{2} b \,d^{2} e^{2}+\frac {12}{5} B a \,b^{2} d^{3} e +\frac {1}{5} b^{3} B \,d^{4}\right ) x^{5}+\left (a^{3} A d \,e^{3}+\frac {9}{2} A \,a^{2} b \,d^{2} e^{2}+3 A a \,b^{2} d^{3} e +\frac {1}{4} A \,b^{3} d^{4}+\frac {3}{2} B \,a^{3} d^{2} e^{2}+3 B \,a^{2} b \,d^{3} e +\frac {3}{4} B a \,b^{2} d^{4}\right ) x^{4}+\left (2 a^{3} A \,d^{2} e^{2}+4 A \,a^{2} b \,d^{3} e +A a \,b^{2} d^{4}+\frac {4}{3} B \,a^{3} d^{3} e +B \,a^{2} b \,d^{4}\right ) x^{3}+\left (2 a^{3} A \,d^{3} e +\frac {3}{2} A \,a^{2} b \,d^{4}+\frac {1}{2} B \,a^{3} d^{4}\right ) x^{2}+a^{3} A \,d^{4} x\)	\(456\)
gosper	\(\frac {12}{7} x^{7} B a \,b^{2} d \,e^{3}+\frac {2}{3} x^{6} b^{3} B \,d^{3} e +\frac {4}{5} x^{5} A \,b^{3} d^{3} e +\frac {4}{5} x^{5} B \,a^{3} d \,e^{3}+x^{4} a^{3} A d \,e^{3}+\frac {3}{7} x^{7} A a \,b^{2} e^{4}+\frac {4}{7} x^{7} A \,b^{3} d \,e^{3}+\frac {3}{7} x^{7} B \,a^{2} b \,e^{4}+\frac {6}{7} x^{7} b^{3} B \,d^{2} e^{2}+\frac {1}{2} x^{6} A \,a^{2} b \,e^{4}+x^{6} A \,b^{3} d^{2} e^{2}+2 x^{6} A a \,b^{2} d \,e^{3}+2 x^{6} B \,a^{2} b d \,e^{3}+3 x^{6} B a \,b^{2} d^{2} e^{2}+\frac {12}{5} x^{5} A \,a^{2} b d \,e^{3}+\frac {1}{2} x^{8} b^{3} B d \,e^{3}+\frac {3}{8} x^{8} B a \,b^{2} e^{4}+\frac {1}{6} x^{6} B \,a^{3} e^{4}+\frac {1}{5} x^{5} a^{3} A \,e^{4}+\frac {1}{5} x^{5} b^{3} B \,d^{4}+\frac {1}{9} b^{3} B \,e^{4} x^{9}+a^{3} A \,d^{4} x +\frac {1}{4} x^{4} A \,b^{3} d^{4}+\frac {1}{2} x^{2} B \,a^{3} d^{4}+\frac {3}{4} x^{4} B a \,b^{2} d^{4}+2 x^{3} a^{3} A \,d^{2} e^{2}+x^{3} A a \,b^{2} d^{4}+\frac {4}{3} x^{3} B \,a^{3} d^{3} e +x^{3} B \,a^{2} b \,d^{4}+\frac {3}{2} x^{2} A \,a^{2} b \,d^{4}+\frac {3}{2} x^{4} B \,a^{3} d^{2} e^{2}+2 x^{2} a^{3} A \,d^{3} e +\frac {1}{8} x^{8} A \,b^{3} e^{4}+4 x^{3} A \,a^{2} b \,d^{3} e +3 x^{4} B \,a^{2} b \,d^{3} e +\frac {18}{5} x^{5} A a \,b^{2} d^{2} e^{2}+\frac {18}{5} x^{5} B \,a^{2} b \,d^{2} e^{2}+\frac {12}{5} x^{5} B a \,b^{2} d^{3} e +\frac {9}{2} x^{4} A \,a^{2} b \,d^{2} e^{2}+3 x^{4} A a \,b^{2} d^{3} e\)	\(535\)
risch	\(\frac {12}{7} x^{7} B a \,b^{2} d \,e^{3}+\frac {2}{3} x^{6} b^{3} B \,d^{3} e +\frac {4}{5} x^{5} A \,b^{3} d^{3} e +\frac {4}{5} x^{5} B \,a^{3} d \,e^{3}+x^{4} a^{3} A d \,e^{3}+\frac {3}{7} x^{7} A a \,b^{2} e^{4}+\frac {4}{7} x^{7} A \,b^{3} d \,e^{3}+\frac {3}{7} x^{7} B \,a^{2} b \,e^{4}+\frac {6}{7} x^{7} b^{3} B \,d^{2} e^{2}+\frac {1}{2} x^{6} A \,a^{2} b \,e^{4}+x^{6} A \,b^{3} d^{2} e^{2}+2 x^{6} A a \,b^{2} d \,e^{3}+2 x^{6} B \,a^{2} b d \,e^{3}+3 x^{6} B a \,b^{2} d^{2} e^{2}+\frac {12}{5} x^{5} A \,a^{2} b d \,e^{3}+\frac {1}{2} x^{8} b^{3} B d \,e^{3}+\frac {3}{8} x^{8} B a \,b^{2} e^{4}+\frac {1}{6} x^{6} B \,a^{3} e^{4}+\frac {1}{5} x^{5} a^{3} A \,e^{4}+\frac {1}{5} x^{5} b^{3} B \,d^{4}+\frac {1}{9} b^{3} B \,e^{4} x^{9}+a^{3} A \,d^{4} x +\frac {1}{4} x^{4} A \,b^{3} d^{4}+\frac {1}{2} x^{2} B \,a^{3} d^{4}+\frac {3}{4} x^{4} B a \,b^{2} d^{4}+2 x^{3} a^{3} A \,d^{2} e^{2}+x^{3} A a \,b^{2} d^{4}+\frac {4}{3} x^{3} B \,a^{3} d^{3} e +x^{3} B \,a^{2} b \,d^{4}+\frac {3}{2} x^{2} A \,a^{2} b \,d^{4}+\frac {3}{2} x^{4} B \,a^{3} d^{2} e^{2}+2 x^{2} a^{3} A \,d^{3} e +\frac {1}{8} x^{8} A \,b^{3} e^{4}+4 x^{3} A \,a^{2} b \,d^{3} e +3 x^{4} B \,a^{2} b \,d^{3} e +\frac {18}{5} x^{5} A a \,b^{2} d^{2} e^{2}+\frac {18}{5} x^{5} B \,a^{2} b \,d^{2} e^{2}+\frac {12}{5} x^{5} B a \,b^{2} d^{3} e +\frac {9}{2} x^{4} A \,a^{2} b \,d^{2} e^{2}+3 x^{4} A a \,b^{2} d^{3} e\)	\(535\)
parallelrisch	\(\frac {12}{7} x^{7} B a \,b^{2} d \,e^{3}+\frac {2}{3} x^{6} b^{3} B \,d^{3} e +\frac {4}{5} x^{5} A \,b^{3} d^{3} e +\frac {4}{5} x^{5} B \,a^{3} d \,e^{3}+x^{4} a^{3} A d \,e^{3}+\frac {3}{7} x^{7} A a \,b^{2} e^{4}+\frac {4}{7} x^{7} A \,b^{3} d \,e^{3}+\frac {3}{7} x^{7} B \,a^{2} b \,e^{4}+\frac {6}{7} x^{7} b^{3} B \,d^{2} e^{2}+\frac {1}{2} x^{6} A \,a^{2} b \,e^{4}+x^{6} A \,b^{3} d^{2} e^{2}+2 x^{6} A a \,b^{2} d \,e^{3}+2 x^{6} B \,a^{2} b d \,e^{3}+3 x^{6} B a \,b^{2} d^{2} e^{2}+\frac {12}{5} x^{5} A \,a^{2} b d \,e^{3}+\frac {1}{2} x^{8} b^{3} B d \,e^{3}+\frac {3}{8} x^{8} B a \,b^{2} e^{4}+\frac {1}{6} x^{6} B \,a^{3} e^{4}+\frac {1}{5} x^{5} a^{3} A \,e^{4}+\frac {1}{5} x^{5} b^{3} B \,d^{4}+\frac {1}{9} b^{3} B \,e^{4} x^{9}+a^{3} A \,d^{4} x +\frac {1}{4} x^{4} A \,b^{3} d^{4}+\frac {1}{2} x^{2} B \,a^{3} d^{4}+\frac {3}{4} x^{4} B a \,b^{2} d^{4}+2 x^{3} a^{3} A \,d^{2} e^{2}+x^{3} A a \,b^{2} d^{4}+\frac {4}{3} x^{3} B \,a^{3} d^{3} e +x^{3} B \,a^{2} b \,d^{4}+\frac {3}{2} x^{2} A \,a^{2} b \,d^{4}+\frac {3}{2} x^{4} B \,a^{3} d^{2} e^{2}+2 x^{2} a^{3} A \,d^{3} e +\frac {1}{8} x^{8} A \,b^{3} e^{4}+4 x^{3} A \,a^{2} b \,d^{3} e +3 x^{4} B \,a^{2} b \,d^{3} e +\frac {18}{5} x^{5} A a \,b^{2} d^{2} e^{2}+\frac {18}{5} x^{5} B \,a^{2} b \,d^{2} e^{2}+\frac {12}{5} x^{5} B a \,b^{2} d^{3} e +\frac {9}{2} x^{4} A \,a^{2} b \,d^{2} e^{2}+3 x^{4} A a \,b^{2} d^{3} e\)	\(535\)
orering	\(\frac {x \left (280 b^{3} B \,e^{4} x^{8}+315 A \,b^{3} e^{4} x^{7}+945 B a \,b^{2} e^{4} x^{7}+1260 B \,b^{3} d \,e^{3} x^{7}+1080 A a \,b^{2} e^{4} x^{6}+1440 A \,b^{3} d \,e^{3} x^{6}+1080 B \,a^{2} b \,e^{4} x^{6}+4320 B a \,b^{2} d \,e^{3} x^{6}+2160 B \,b^{3} d^{2} e^{2} x^{6}+1260 A \,a^{2} b \,e^{4} x^{5}+5040 A a \,b^{2} d \,e^{3} x^{5}+2520 A \,b^{3} d^{2} e^{2} x^{5}+420 B \,a^{3} e^{4} x^{5}+5040 B \,a^{2} b d \,e^{3} x^{5}+7560 B a \,b^{2} d^{2} e^{2} x^{5}+1680 B \,b^{3} d^{3} e \,x^{5}+504 A \,a^{3} e^{4} x^{4}+6048 A \,a^{2} b d \,e^{3} x^{4}+9072 A a \,b^{2} d^{2} e^{2} x^{4}+2016 A \,b^{3} d^{3} e \,x^{4}+2016 B \,a^{3} d \,e^{3} x^{4}+9072 B \,a^{2} b \,d^{2} e^{2} x^{4}+6048 B a \,b^{2} d^{3} e \,x^{4}+504 B \,b^{3} d^{4} x^{4}+2520 A \,a^{3} d \,e^{3} x^{3}+11340 A \,a^{2} b \,d^{2} e^{2} x^{3}+7560 A a \,b^{2} d^{3} e \,x^{3}+630 A \,b^{3} d^{4} x^{3}+3780 B \,a^{3} d^{2} e^{2} x^{3}+7560 B \,a^{2} b \,d^{3} e \,x^{3}+1890 B a \,b^{2} d^{4} x^{3}+5040 A \,a^{3} d^{2} e^{2} x^{2}+10080 A \,a^{2} b \,d^{3} e \,x^{2}+2520 A a \,b^{2} d^{4} x^{2}+3360 B \,a^{3} d^{3} e \,x^{2}+2520 B \,a^{2} b \,d^{4} x^{2}+5040 A \,a^{3} d^{3} e x +3780 A \,a^{2} b \,d^{4} x +1260 B \,a^{3} d^{4} x +2520 a^{3} A \,d^{4}\right )}{2520}\)	\(536\)

Input:

int((b*x+a)^3*(B*x+A)*(e*x+d)^4,x,method=_RETURNVERBOSE)
 

Output:

1/9*b^3*B*e^4*x^9+1/8*((A*b^3+3*B*a*b^2)*e^4+4*b^3*B*d*e^3)*x^8+1/7*((3*A* 
a*b^2+3*B*a^2*b)*e^4+4*(A*b^3+3*B*a*b^2)*d*e^3+6*b^3*B*d^2*e^2)*x^7+1/6*(( 
3*A*a^2*b+B*a^3)*e^4+4*(3*A*a*b^2+3*B*a^2*b)*d*e^3+6*(A*b^3+3*B*a*b^2)*d^2 
*e^2+4*b^3*B*d^3*e)*x^6+1/5*(a^3*A*e^4+4*(3*A*a^2*b+B*a^3)*d*e^3+6*(3*A*a* 
b^2+3*B*a^2*b)*d^2*e^2+4*(A*b^3+3*B*a*b^2)*d^3*e+b^3*B*d^4)*x^5+1/4*(4*a^3 
*A*d*e^3+6*(3*A*a^2*b+B*a^3)*d^2*e^2+4*(3*A*a*b^2+3*B*a^2*b)*d^3*e+(A*b^3+ 
3*B*a*b^2)*d^4)*x^4+1/3*(6*a^3*A*d^2*e^2+4*(3*A*a^2*b+B*a^3)*d^3*e+(3*A*a* 
b^2+3*B*a^2*b)*d^4)*x^3+1/2*(4*a^3*A*d^3*e+(3*A*a^2*b+B*a^3)*d^4)*x^2+a^3* 
A*d^4*x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (153) = 306\).

Time = 0.08 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.61 \[ \int (a+b x)^3 (A+B x) (d+e x)^4 \, dx=\frac {1}{9} \, B b^{3} e^{4} x^{9} + A a^{3} d^{4} x + \frac {1}{8} \, {\left (4 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, B b^{3} d^{2} e^{2} + 4 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (4 \, B b^{3} d^{3} e + 6 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{4} + A a^{3} e^{4} + 4 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 18 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e + 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{3} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{3} d^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4}\right )} x^{2} \] Input:

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^4,x, algorithm="fricas")
 

Output:

1/9*B*b^3*e^4*x^9 + A*a^3*d^4*x + 1/8*(4*B*b^3*d*e^3 + (3*B*a*b^2 + A*b^3) 
*e^4)*x^8 + 1/7*(6*B*b^3*d^2*e^2 + 4*(3*B*a*b^2 + A*b^3)*d*e^3 + 3*(B*a^2* 
b + A*a*b^2)*e^4)*x^7 + 1/6*(4*B*b^3*d^3*e + 6*(3*B*a*b^2 + A*b^3)*d^2*e^2 
 + 12*(B*a^2*b + A*a*b^2)*d*e^3 + (B*a^3 + 3*A*a^2*b)*e^4)*x^6 + 1/5*(B*b^ 
3*d^4 + A*a^3*e^4 + 4*(3*B*a*b^2 + A*b^3)*d^3*e + 18*(B*a^2*b + A*a*b^2)*d 
^2*e^2 + 4*(B*a^3 + 3*A*a^2*b)*d*e^3)*x^5 + 1/4*(4*A*a^3*d*e^3 + (3*B*a*b^ 
2 + A*b^3)*d^4 + 12*(B*a^2*b + A*a*b^2)*d^3*e + 6*(B*a^3 + 3*A*a^2*b)*d^2* 
e^2)*x^4 + 1/3*(6*A*a^3*d^2*e^2 + 3*(B*a^2*b + A*a*b^2)*d^4 + 4*(B*a^3 + 3 
*A*a^2*b)*d^3*e)*x^3 + 1/2*(4*A*a^3*d^3*e + (B*a^3 + 3*A*a^2*b)*d^4)*x^2
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (155) = 310\).

Time = 0.05 (sec) , antiderivative size = 546, normalized size of antiderivative = 3.35 \[ \int (a+b x)^3 (A+B x) (d+e x)^4 \, dx=A a^{3} d^{4} x + \frac {B b^{3} e^{4} x^{9}}{9} + x^{8} \left (\frac {A b^{3} e^{4}}{8} + \frac {3 B a b^{2} e^{4}}{8} + \frac {B b^{3} d e^{3}}{2}\right ) + x^{7} \cdot \left (\frac {3 A a b^{2} e^{4}}{7} + \frac {4 A b^{3} d e^{3}}{7} + \frac {3 B a^{2} b e^{4}}{7} + \frac {12 B a b^{2} d e^{3}}{7} + \frac {6 B b^{3} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac {A a^{2} b e^{4}}{2} + 2 A a b^{2} d e^{3} + A b^{3} d^{2} e^{2} + \frac {B a^{3} e^{4}}{6} + 2 B a^{2} b d e^{3} + 3 B a b^{2} d^{2} e^{2} + \frac {2 B b^{3} d^{3} e}{3}\right ) + x^{5} \left (\frac {A a^{3} e^{4}}{5} + \frac {12 A a^{2} b d e^{3}}{5} + \frac {18 A a b^{2} d^{2} e^{2}}{5} + \frac {4 A b^{3} d^{3} e}{5} + \frac {4 B a^{3} d e^{3}}{5} + \frac {18 B a^{2} b d^{2} e^{2}}{5} + \frac {12 B a b^{2} d^{3} e}{5} + \frac {B b^{3} d^{4}}{5}\right ) + x^{4} \left (A a^{3} d e^{3} + \frac {9 A a^{2} b d^{2} e^{2}}{2} + 3 A a b^{2} d^{3} e + \frac {A b^{3} d^{4}}{4} + \frac {3 B a^{3} d^{2} e^{2}}{2} + 3 B a^{2} b d^{3} e + \frac {3 B a b^{2} d^{4}}{4}\right ) + x^{3} \cdot \left (2 A a^{3} d^{2} e^{2} + 4 A a^{2} b d^{3} e + A a b^{2} d^{4} + \frac {4 B a^{3} d^{3} e}{3} + B a^{2} b d^{4}\right ) + x^{2} \cdot \left (2 A a^{3} d^{3} e + \frac {3 A a^{2} b d^{4}}{2} + \frac {B a^{3} d^{4}}{2}\right ) \] Input:

integrate((b*x+a)**3*(B*x+A)*(e*x+d)**4,x)
 

Output:

A*a**3*d**4*x + B*b**3*e**4*x**9/9 + x**8*(A*b**3*e**4/8 + 3*B*a*b**2*e**4 
/8 + B*b**3*d*e**3/2) + x**7*(3*A*a*b**2*e**4/7 + 4*A*b**3*d*e**3/7 + 3*B* 
a**2*b*e**4/7 + 12*B*a*b**2*d*e**3/7 + 6*B*b**3*d**2*e**2/7) + x**6*(A*a** 
2*b*e**4/2 + 2*A*a*b**2*d*e**3 + A*b**3*d**2*e**2 + B*a**3*e**4/6 + 2*B*a* 
*2*b*d*e**3 + 3*B*a*b**2*d**2*e**2 + 2*B*b**3*d**3*e/3) + x**5*(A*a**3*e** 
4/5 + 12*A*a**2*b*d*e**3/5 + 18*A*a*b**2*d**2*e**2/5 + 4*A*b**3*d**3*e/5 + 
 4*B*a**3*d*e**3/5 + 18*B*a**2*b*d**2*e**2/5 + 12*B*a*b**2*d**3*e/5 + B*b* 
*3*d**4/5) + x**4*(A*a**3*d*e**3 + 9*A*a**2*b*d**2*e**2/2 + 3*A*a*b**2*d** 
3*e + A*b**3*d**4/4 + 3*B*a**3*d**2*e**2/2 + 3*B*a**2*b*d**3*e + 3*B*a*b** 
2*d**4/4) + x**3*(2*A*a**3*d**2*e**2 + 4*A*a**2*b*d**3*e + A*a*b**2*d**4 + 
 4*B*a**3*d**3*e/3 + B*a**2*b*d**4) + x**2*(2*A*a**3*d**3*e + 3*A*a**2*b*d 
**4/2 + B*a**3*d**4/2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (153) = 306\).

Time = 0.04 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.61 \[ \int (a+b x)^3 (A+B x) (d+e x)^4 \, dx=\frac {1}{9} \, B b^{3} e^{4} x^{9} + A a^{3} d^{4} x + \frac {1}{8} \, {\left (4 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, B b^{3} d^{2} e^{2} + 4 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (4 \, B b^{3} d^{3} e + 6 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{4} + A a^{3} e^{4} + 4 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 18 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e + 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{3} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{3} d^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4}\right )} x^{2} \] Input:

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^4,x, algorithm="maxima")
 

Output:

1/9*B*b^3*e^4*x^9 + A*a^3*d^4*x + 1/8*(4*B*b^3*d*e^3 + (3*B*a*b^2 + A*b^3) 
*e^4)*x^8 + 1/7*(6*B*b^3*d^2*e^2 + 4*(3*B*a*b^2 + A*b^3)*d*e^3 + 3*(B*a^2* 
b + A*a*b^2)*e^4)*x^7 + 1/6*(4*B*b^3*d^3*e + 6*(3*B*a*b^2 + A*b^3)*d^2*e^2 
 + 12*(B*a^2*b + A*a*b^2)*d*e^3 + (B*a^3 + 3*A*a^2*b)*e^4)*x^6 + 1/5*(B*b^ 
3*d^4 + A*a^3*e^4 + 4*(3*B*a*b^2 + A*b^3)*d^3*e + 18*(B*a^2*b + A*a*b^2)*d 
^2*e^2 + 4*(B*a^3 + 3*A*a^2*b)*d*e^3)*x^5 + 1/4*(4*A*a^3*d*e^3 + (3*B*a*b^ 
2 + A*b^3)*d^4 + 12*(B*a^2*b + A*a*b^2)*d^3*e + 6*(B*a^3 + 3*A*a^2*b)*d^2* 
e^2)*x^4 + 1/3*(6*A*a^3*d^2*e^2 + 3*(B*a^2*b + A*a*b^2)*d^4 + 4*(B*a^3 + 3 
*A*a^2*b)*d^3*e)*x^3 + 1/2*(4*A*a^3*d^3*e + (B*a^3 + 3*A*a^2*b)*d^4)*x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (153) = 306\).

Time = 0.12 (sec) , antiderivative size = 534, normalized size of antiderivative = 3.28 \[ \int (a+b x)^3 (A+B x) (d+e x)^4 \, dx=\frac {1}{9} \, B b^{3} e^{4} x^{9} + \frac {1}{2} \, B b^{3} d e^{3} x^{8} + \frac {3}{8} \, B a b^{2} e^{4} x^{8} + \frac {1}{8} \, A b^{3} e^{4} x^{8} + \frac {6}{7} \, B b^{3} d^{2} e^{2} x^{7} + \frac {12}{7} \, B a b^{2} d e^{3} x^{7} + \frac {4}{7} \, A b^{3} d e^{3} x^{7} + \frac {3}{7} \, B a^{2} b e^{4} x^{7} + \frac {3}{7} \, A a b^{2} e^{4} x^{7} + \frac {2}{3} \, B b^{3} d^{3} e x^{6} + 3 \, B a b^{2} d^{2} e^{2} x^{6} + A b^{3} d^{2} e^{2} x^{6} + 2 \, B a^{2} b d e^{3} x^{6} + 2 \, A a b^{2} d e^{3} x^{6} + \frac {1}{6} \, B a^{3} e^{4} x^{6} + \frac {1}{2} \, A a^{2} b e^{4} x^{6} + \frac {1}{5} \, B b^{3} d^{4} x^{5} + \frac {12}{5} \, B a b^{2} d^{3} e x^{5} + \frac {4}{5} \, A b^{3} d^{3} e x^{5} + \frac {18}{5} \, B a^{2} b d^{2} e^{2} x^{5} + \frac {18}{5} \, A a b^{2} d^{2} e^{2} x^{5} + \frac {4}{5} \, B a^{3} d e^{3} x^{5} + \frac {12}{5} \, A a^{2} b d e^{3} x^{5} + \frac {1}{5} \, A a^{3} e^{4} x^{5} + \frac {3}{4} \, B a b^{2} d^{4} x^{4} + \frac {1}{4} \, A b^{3} d^{4} x^{4} + 3 \, B a^{2} b d^{3} e x^{4} + 3 \, A a b^{2} d^{3} e x^{4} + \frac {3}{2} \, B a^{3} d^{2} e^{2} x^{4} + \frac {9}{2} \, A a^{2} b d^{2} e^{2} x^{4} + A a^{3} d e^{3} x^{4} + B a^{2} b d^{4} x^{3} + A a b^{2} d^{4} x^{3} + \frac {4}{3} \, B a^{3} d^{3} e x^{3} + 4 \, A a^{2} b d^{3} e x^{3} + 2 \, A a^{3} d^{2} e^{2} x^{3} + \frac {1}{2} \, B a^{3} d^{4} x^{2} + \frac {3}{2} \, A a^{2} b d^{4} x^{2} + 2 \, A a^{3} d^{3} e x^{2} + A a^{3} d^{4} x \] Input:

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^4,x, algorithm="giac")
 

Output:

1/9*B*b^3*e^4*x^9 + 1/2*B*b^3*d*e^3*x^8 + 3/8*B*a*b^2*e^4*x^8 + 1/8*A*b^3* 
e^4*x^8 + 6/7*B*b^3*d^2*e^2*x^7 + 12/7*B*a*b^2*d*e^3*x^7 + 4/7*A*b^3*d*e^3 
*x^7 + 3/7*B*a^2*b*e^4*x^7 + 3/7*A*a*b^2*e^4*x^7 + 2/3*B*b^3*d^3*e*x^6 + 3 
*B*a*b^2*d^2*e^2*x^6 + A*b^3*d^2*e^2*x^6 + 2*B*a^2*b*d*e^3*x^6 + 2*A*a*b^2 
*d*e^3*x^6 + 1/6*B*a^3*e^4*x^6 + 1/2*A*a^2*b*e^4*x^6 + 1/5*B*b^3*d^4*x^5 + 
 12/5*B*a*b^2*d^3*e*x^5 + 4/5*A*b^3*d^3*e*x^5 + 18/5*B*a^2*b*d^2*e^2*x^5 + 
 18/5*A*a*b^2*d^2*e^2*x^5 + 4/5*B*a^3*d*e^3*x^5 + 12/5*A*a^2*b*d*e^3*x^5 + 
 1/5*A*a^3*e^4*x^5 + 3/4*B*a*b^2*d^4*x^4 + 1/4*A*b^3*d^4*x^4 + 3*B*a^2*b*d 
^3*e*x^4 + 3*A*a*b^2*d^3*e*x^4 + 3/2*B*a^3*d^2*e^2*x^4 + 9/2*A*a^2*b*d^2*e 
^2*x^4 + A*a^3*d*e^3*x^4 + B*a^2*b*d^4*x^3 + A*a*b^2*d^4*x^3 + 4/3*B*a^3*d 
^3*e*x^3 + 4*A*a^2*b*d^3*e*x^3 + 2*A*a^3*d^2*e^2*x^3 + 1/2*B*a^3*d^4*x^2 + 
 3/2*A*a^2*b*d^4*x^2 + 2*A*a^3*d^3*e*x^2 + A*a^3*d^4*x
 

Mupad [B] (verification not implemented)

Time = 0.96 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.69 \[ \int (a+b x)^3 (A+B x) (d+e x)^4 \, dx=x^3\,\left (\frac {4\,B\,a^3\,d^3\,e}{3}+2\,A\,a^3\,d^2\,e^2+B\,a^2\,b\,d^4+4\,A\,a^2\,b\,d^3\,e+A\,a\,b^2\,d^4\right )+x^7\,\left (\frac {3\,B\,a^2\,b\,e^4}{7}+\frac {12\,B\,a\,b^2\,d\,e^3}{7}+\frac {3\,A\,a\,b^2\,e^4}{7}+\frac {6\,B\,b^3\,d^2\,e^2}{7}+\frac {4\,A\,b^3\,d\,e^3}{7}\right )+x^5\,\left (\frac {4\,B\,a^3\,d\,e^3}{5}+\frac {A\,a^3\,e^4}{5}+\frac {18\,B\,a^2\,b\,d^2\,e^2}{5}+\frac {12\,A\,a^2\,b\,d\,e^3}{5}+\frac {12\,B\,a\,b^2\,d^3\,e}{5}+\frac {18\,A\,a\,b^2\,d^2\,e^2}{5}+\frac {B\,b^3\,d^4}{5}+\frac {4\,A\,b^3\,d^3\,e}{5}\right )+x^4\,\left (\frac {3\,B\,a^3\,d^2\,e^2}{2}+A\,a^3\,d\,e^3+3\,B\,a^2\,b\,d^3\,e+\frac {9\,A\,a^2\,b\,d^2\,e^2}{2}+\frac {3\,B\,a\,b^2\,d^4}{4}+3\,A\,a\,b^2\,d^3\,e+\frac {A\,b^3\,d^4}{4}\right )+x^6\,\left (\frac {B\,a^3\,e^4}{6}+2\,B\,a^2\,b\,d\,e^3+\frac {A\,a^2\,b\,e^4}{2}+3\,B\,a\,b^2\,d^2\,e^2+2\,A\,a\,b^2\,d\,e^3+\frac {2\,B\,b^3\,d^3\,e}{3}+A\,b^3\,d^2\,e^2\right )+\frac {a^2\,d^3\,x^2\,\left (4\,A\,a\,e+3\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^2\,e^3\,x^8\,\left (A\,b\,e+3\,B\,a\,e+4\,B\,b\,d\right )}{8}+A\,a^3\,d^4\,x+\frac {B\,b^3\,e^4\,x^9}{9} \] Input:

int((A + B*x)*(a + b*x)^3*(d + e*x)^4,x)
 

Output:

x^3*(A*a*b^2*d^4 + B*a^2*b*d^4 + (4*B*a^3*d^3*e)/3 + 2*A*a^3*d^2*e^2 + 4*A 
*a^2*b*d^3*e) + x^7*((3*A*a*b^2*e^4)/7 + (3*B*a^2*b*e^4)/7 + (4*A*b^3*d*e^ 
3)/7 + (6*B*b^3*d^2*e^2)/7 + (12*B*a*b^2*d*e^3)/7) + x^5*((A*a^3*e^4)/5 + 
(B*b^3*d^4)/5 + (4*A*b^3*d^3*e)/5 + (4*B*a^3*d*e^3)/5 + (18*A*a*b^2*d^2*e^ 
2)/5 + (18*B*a^2*b*d^2*e^2)/5 + (12*A*a^2*b*d*e^3)/5 + (12*B*a*b^2*d^3*e)/ 
5) + x^4*((A*b^3*d^4)/4 + (3*B*a*b^2*d^4)/4 + A*a^3*d*e^3 + (3*B*a^3*d^2*e 
^2)/2 + (9*A*a^2*b*d^2*e^2)/2 + 3*A*a*b^2*d^3*e + 3*B*a^2*b*d^3*e) + x^6*( 
(B*a^3*e^4)/6 + (A*a^2*b*e^4)/2 + (2*B*b^3*d^3*e)/3 + A*b^3*d^2*e^2 + 3*B* 
a*b^2*d^2*e^2 + 2*A*a*b^2*d*e^3 + 2*B*a^2*b*d*e^3) + (a^2*d^3*x^2*(4*A*a*e 
 + 3*A*b*d + B*a*d))/2 + (b^2*e^3*x^8*(A*b*e + 3*B*a*e + 4*B*b*d))/8 + A*a 
^3*d^4*x + (B*b^3*e^4*x^9)/9
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.98 \[ \int (a+b x)^3 (A+B x) (d+e x)^4 \, dx=\frac {x \left (70 b^{4} e^{4} x^{8}+315 a \,b^{3} e^{4} x^{7}+315 b^{4} d \,e^{3} x^{7}+540 a^{2} b^{2} e^{4} x^{6}+1440 a \,b^{3} d \,e^{3} x^{6}+540 b^{4} d^{2} e^{2} x^{6}+420 a^{3} b \,e^{4} x^{5}+2520 a^{2} b^{2} d \,e^{3} x^{5}+2520 a \,b^{3} d^{2} e^{2} x^{5}+420 b^{4} d^{3} e \,x^{5}+126 a^{4} e^{4} x^{4}+2016 a^{3} b d \,e^{3} x^{4}+4536 a^{2} b^{2} d^{2} e^{2} x^{4}+2016 a \,b^{3} d^{3} e \,x^{4}+126 b^{4} d^{4} x^{4}+630 a^{4} d \,e^{3} x^{3}+3780 a^{3} b \,d^{2} e^{2} x^{3}+3780 a^{2} b^{2} d^{3} e \,x^{3}+630 a \,b^{3} d^{4} x^{3}+1260 a^{4} d^{2} e^{2} x^{2}+3360 a^{3} b \,d^{3} e \,x^{2}+1260 a^{2} b^{2} d^{4} x^{2}+1260 a^{4} d^{3} e x +1260 a^{3} b \,d^{4} x +630 a^{4} d^{4}\right )}{630} \] Input:

int((b*x+a)^3*(B*x+A)*(e*x+d)^4,x)
 

Output:

(x*(630*a**4*d**4 + 1260*a**4*d**3*e*x + 1260*a**4*d**2*e**2*x**2 + 630*a* 
*4*d*e**3*x**3 + 126*a**4*e**4*x**4 + 1260*a**3*b*d**4*x + 3360*a**3*b*d** 
3*e*x**2 + 3780*a**3*b*d**2*e**2*x**3 + 2016*a**3*b*d*e**3*x**4 + 420*a**3 
*b*e**4*x**5 + 1260*a**2*b**2*d**4*x**2 + 3780*a**2*b**2*d**3*e*x**3 + 453 
6*a**2*b**2*d**2*e**2*x**4 + 2520*a**2*b**2*d*e**3*x**5 + 540*a**2*b**2*e* 
*4*x**6 + 630*a*b**3*d**4*x**3 + 2016*a*b**3*d**3*e*x**4 + 2520*a*b**3*d** 
2*e**2*x**5 + 1440*a*b**3*d*e**3*x**6 + 315*a*b**3*e**4*x**7 + 126*b**4*d* 
*4*x**4 + 420*b**4*d**3*e*x**5 + 540*b**4*d**2*e**2*x**6 + 315*b**4*d*e**3 
*x**7 + 70*b**4*e**4*x**8))/630