\(\int (a+b x)^6 (A+B x) (d+e x)^4 \, dx\) [45]

Optimal result

Integrand size = 20, antiderivative size = 204 \[ \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx=\frac {(A b-a B) (b d-a e)^4 (a+b x)^7}{7 b^6}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e) (a+b x)^8}{8 b^6}+\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) (a+b x)^9}{9 b^6}+\frac {e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^{10}}{5 b^6}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)^{11}}{11 b^6}+\frac {B e^4 (a+b x)^{12}}{12 b^6} \] Output:

1/7*(A*b-B*a)*(-a*e+b*d)^4*(b*x+a)^7/b^6+1/8*(-a*e+b*d)^3*(4*A*b*e-5*B*a*e 
+B*b*d)*(b*x+a)^8/b^6+2/9*e*(-a*e+b*d)^2*(3*A*b*e-5*B*a*e+2*B*b*d)*(b*x+a) 
^9/b^6+1/5*e^2*(-a*e+b*d)*(2*A*b*e-5*B*a*e+3*B*b*d)*(b*x+a)^10/b^6+1/11*e^ 
3*(A*b*e-5*B*a*e+4*B*b*d)*(b*x+a)^11/b^6+1/12*B*e^4*(b*x+a)^12/b^6
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(762\) vs. \(2(204)=408\).

Time = 0.16 (sec) , antiderivative size = 762, normalized size of antiderivative = 3.74 \[ \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx=a^6 A d^4 x+\frac {1}{2} a^5 d^3 (6 A b d+a B d+4 a A e) x^2+\frac {1}{3} a^4 d^2 \left (2 a B d (3 b d+2 a e)+3 A \left (5 b^2 d^2+8 a b d e+2 a^2 e^2\right )\right ) x^3+\frac {1}{4} a^3 d \left (3 a B d \left (5 b^2 d^2+8 a b d e+2 a^2 e^2\right )+4 A \left (5 b^3 d^3+15 a b^2 d^2 e+9 a^2 b d e^2+a^3 e^3\right )\right ) x^4+\frac {1}{5} a^2 \left (4 a B d \left (5 b^3 d^3+15 a b^2 d^2 e+9 a^2 b d e^2+a^3 e^3\right )+A \left (15 b^4 d^4+80 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+24 a^3 b d e^3+a^4 e^4\right )\right ) x^5+\frac {1}{6} a \left (6 A b \left (b^4 d^4+10 a b^3 d^3 e+20 a^2 b^2 d^2 e^2+10 a^3 b d e^3+a^4 e^4\right )+a B \left (15 b^4 d^4+80 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+24 a^3 b d e^3+a^4 e^4\right )\right ) x^6+\frac {1}{7} b \left (6 a B \left (b^4 d^4+10 a b^3 d^3 e+20 a^2 b^2 d^2 e^2+10 a^3 b d e^3+a^4 e^4\right )+A b \left (b^4 d^4+24 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+80 a^3 b d e^3+15 a^4 e^4\right )\right ) x^7+\frac {1}{8} b^2 \left (15 a^4 B e^4+20 a^3 b e^3 (4 B d+A e)+30 a^2 b^2 d e^2 (3 B d+2 A e)+12 a b^3 d^2 e (2 B d+3 A e)+b^4 d^3 (B d+4 A e)\right ) x^8+\frac {1}{9} b^3 e \left (20 a^3 B e^3+15 a^2 b e^2 (4 B d+A e)+12 a b^2 d e (3 B d+2 A e)+2 b^3 d^2 (2 B d+3 A e)\right ) x^9+\frac {1}{10} b^4 e^2 \left (15 a^2 B e^2+6 a b e (4 B d+A e)+2 b^2 d (3 B d+2 A e)\right ) x^{10}+\frac {1}{11} b^5 e^3 (4 b B d+A b e+6 a B e) x^{11}+\frac {1}{12} b^6 B e^4 x^{12} \] Input:

Integrate[(a + b*x)^6*(A + B*x)*(d + e*x)^4,x]
 

Output:

a^6*A*d^4*x + (a^5*d^3*(6*A*b*d + a*B*d + 4*a*A*e)*x^2)/2 + (a^4*d^2*(2*a* 
B*d*(3*b*d + 2*a*e) + 3*A*(5*b^2*d^2 + 8*a*b*d*e + 2*a^2*e^2))*x^3)/3 + (a 
^3*d*(3*a*B*d*(5*b^2*d^2 + 8*a*b*d*e + 2*a^2*e^2) + 4*A*(5*b^3*d^3 + 15*a* 
b^2*d^2*e + 9*a^2*b*d*e^2 + a^3*e^3))*x^4)/4 + (a^2*(4*a*B*d*(5*b^3*d^3 + 
15*a*b^2*d^2*e + 9*a^2*b*d*e^2 + a^3*e^3) + A*(15*b^4*d^4 + 80*a*b^3*d^3*e 
 + 90*a^2*b^2*d^2*e^2 + 24*a^3*b*d*e^3 + a^4*e^4))*x^5)/5 + (a*(6*A*b*(b^4 
*d^4 + 10*a*b^3*d^3*e + 20*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + a^4*e^4) + a 
*B*(15*b^4*d^4 + 80*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 24*a^3*b*d*e^3 + a^ 
4*e^4))*x^6)/6 + (b*(6*a*B*(b^4*d^4 + 10*a*b^3*d^3*e + 20*a^2*b^2*d^2*e^2 
+ 10*a^3*b*d*e^3 + a^4*e^4) + A*b*(b^4*d^4 + 24*a*b^3*d^3*e + 90*a^2*b^2*d 
^2*e^2 + 80*a^3*b*d*e^3 + 15*a^4*e^4))*x^7)/7 + (b^2*(15*a^4*B*e^4 + 20*a^ 
3*b*e^3*(4*B*d + A*e) + 30*a^2*b^2*d*e^2*(3*B*d + 2*A*e) + 12*a*b^3*d^2*e* 
(2*B*d + 3*A*e) + b^4*d^3*(B*d + 4*A*e))*x^8)/8 + (b^3*e*(20*a^3*B*e^3 + 1 
5*a^2*b*e^2*(4*B*d + A*e) + 12*a*b^2*d*e*(3*B*d + 2*A*e) + 2*b^3*d^2*(2*B* 
d + 3*A*e))*x^9)/9 + (b^4*e^2*(15*a^2*B*e^2 + 6*a*b*e*(4*B*d + A*e) + 2*b^ 
2*d*(3*B*d + 2*A*e))*x^10)/10 + (b^5*e^3*(4*b*B*d + A*b*e + 6*a*B*e)*x^11) 
/11 + (b^6*B*e^4*x^12)/12
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 204, 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.

\(\displaystyle \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx\)

\(\Big \downarrow \) 86

\(\displaystyle \int \left (\frac {e^3 (a+b x)^{10} (-5 a B e+A b e+4 b B d)}{b^5}+\frac {2 e^2 (a+b x)^9 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^5}+\frac {2 e (a+b x)^8 (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^5}+\frac {(a+b x)^7 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{b^5}+\frac {(a+b x)^6 (A b-a B) (b d-a e)^4}{b^5}+\frac {B e^4 (a+b x)^{11}}{b^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^3 (a+b x)^{11} (-5 a B e+A b e+4 b B d)}{11 b^6}+\frac {e^2 (a+b x)^{10} (b d-a e) (-5 a B e+2 A b e+3 b B d)}{5 b^6}+\frac {2 e (a+b x)^9 (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{9 b^6}+\frac {(a+b x)^8 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{8 b^6}+\frac {(a+b x)^7 (A b-a B) (b d-a e)^4}{7 b^6}+\frac {B e^4 (a+b x)^{12}}{12 b^6}\)

Input:

Int[(a + b*x)^6*(A + B*x)*(d + e*x)^4,x]
 

Output:

((A*b - a*B)*(b*d - a*e)^4*(a + b*x)^7)/(7*b^6) + ((b*d - a*e)^3*(b*B*d + 
4*A*b*e - 5*a*B*e)*(a + b*x)^8)/(8*b^6) + (2*e*(b*d - a*e)^2*(2*b*B*d + 3* 
A*b*e - 5*a*B*e)*(a + b*x)^9)/(9*b^6) + (e^2*(b*d - a*e)*(3*b*B*d + 2*A*b* 
e - 5*a*B*e)*(a + b*x)^10)/(5*b^6) + (e^3*(4*b*B*d + A*b*e - 5*a*B*e)*(a + 
 b*x)^11)/(11*b^6) + (B*e^4*(a + b*x)^12)/(12*b^6)
 

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. \(820\) vs. \(2(192)=384\).

Time = 0.20 (sec) , antiderivative size = 821, normalized size of antiderivative = 4.02

method	result	size
default	\(\frac {b^{6} B \,e^{4} x^{12}}{12}+\frac {\left (\left (b^{6} A +6 a \,b^{5} B \right ) e^{4}+4 b^{6} B d \,e^{3}\right ) x^{11}}{11}+\frac {\left (\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) e^{4}+4 \left (b^{6} A +6 a \,b^{5} B \right ) d \,e^{3}+6 b^{6} B \,d^{2} e^{2}\right ) x^{10}}{10}+\frac {\left (\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) e^{4}+4 \left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d \,e^{3}+6 \left (b^{6} A +6 a \,b^{5} B \right ) d^{2} e^{2}+4 b^{6} B \,d^{3} e \right ) x^{9}}{9}+\frac {\left (\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) e^{4}+4 \left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d \,e^{3}+6 \left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d^{2} e^{2}+4 \left (b^{6} A +6 a \,b^{5} B \right ) d^{3} e +b^{6} B \,d^{4}\right ) x^{8}}{8}+\frac {\left (\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) e^{4}+4 \left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d \,e^{3}+6 \left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d^{2} e^{2}+4 \left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d^{3} e +\left (b^{6} A +6 a \,b^{5} B \right ) d^{4}\right ) x^{7}}{7}+\frac {\left (\left (6 a^{5} b A +a^{6} B \right ) e^{4}+4 \left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d \,e^{3}+6 \left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d^{2} e^{2}+4 \left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d^{3} e +\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d^{4}\right ) x^{6}}{6}+\frac {\left (a^{6} A \,e^{4}+4 \left (6 a^{5} b A +a^{6} B \right ) d \,e^{3}+6 \left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d^{2} e^{2}+4 \left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d^{3} e +\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d^{4}\right ) x^{5}}{5}+\frac {\left (4 a^{6} A d \,e^{3}+6 \left (6 a^{5} b A +a^{6} B \right ) d^{2} e^{2}+4 \left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d^{3} e +\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d^{4}\right ) x^{4}}{4}+\frac {\left (6 a^{6} A \,d^{2} e^{2}+4 \left (6 a^{5} b A +a^{6} B \right ) d^{3} e +\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d^{4}\right ) x^{3}}{3}+\frac {\left (4 a^{6} A \,d^{3} e +\left (6 a^{5} b A +a^{6} B \right ) d^{4}\right ) x^{2}}{2}+a^{6} A \,d^{4} x\)	\(821\)
norman	\(\frac {b^{6} B \,e^{4} x^{12}}{12}+\left (\frac {1}{11} A \,b^{6} e^{4}+\frac {6}{11} B a \,b^{5} e^{4}+\frac {4}{11} b^{6} B d \,e^{3}\right ) x^{11}+\left (\frac {3}{5} A a \,b^{5} e^{4}+\frac {2}{5} A \,b^{6} d \,e^{3}+\frac {3}{2} B \,a^{2} b^{4} e^{4}+\frac {12}{5} B a \,b^{5} d \,e^{3}+\frac {3}{5} b^{6} B \,d^{2} e^{2}\right ) x^{10}+\left (\frac {5}{3} A \,a^{2} b^{4} e^{4}+\frac {8}{3} A a \,b^{5} d \,e^{3}+\frac {2}{3} A \,b^{6} d^{2} e^{2}+\frac {20}{9} B \,a^{3} b^{3} e^{4}+\frac {20}{3} B \,a^{2} b^{4} d \,e^{3}+4 B a \,b^{5} d^{2} e^{2}+\frac {4}{9} b^{6} B \,d^{3} e \right ) x^{9}+\left (\frac {5}{2} A \,a^{3} b^{3} e^{4}+\frac {15}{2} A \,a^{2} b^{4} d \,e^{3}+\frac {9}{2} A a \,b^{5} d^{2} e^{2}+\frac {1}{2} A \,b^{6} d^{3} e +\frac {15}{8} B \,a^{4} b^{2} e^{4}+10 B \,a^{3} b^{3} d \,e^{3}+\frac {45}{4} B \,a^{2} b^{4} d^{2} e^{2}+3 B a \,b^{5} d^{3} e +\frac {1}{8} b^{6} B \,d^{4}\right ) x^{8}+\left (\frac {15}{7} A \,a^{4} b^{2} e^{4}+\frac {80}{7} A \,a^{3} b^{3} d \,e^{3}+\frac {90}{7} A \,a^{2} b^{4} d^{2} e^{2}+\frac {24}{7} A a \,b^{5} d^{3} e +\frac {1}{7} A \,b^{6} d^{4}+\frac {6}{7} B \,a^{5} b \,e^{4}+\frac {60}{7} B \,a^{4} b^{2} d \,e^{3}+\frac {120}{7} B \,a^{3} b^{3} d^{2} e^{2}+\frac {60}{7} B \,a^{2} b^{4} d^{3} e +\frac {6}{7} B a \,b^{5} d^{4}\right ) x^{7}+\left (A \,a^{5} b \,e^{4}+10 A \,a^{4} b^{2} d \,e^{3}+20 A \,a^{3} b^{3} d^{2} e^{2}+10 A \,a^{2} b^{4} d^{3} e +A a \,b^{5} d^{4}+\frac {1}{6} B \,a^{6} e^{4}+4 B \,a^{5} b d \,e^{3}+15 B \,a^{4} b^{2} d^{2} e^{2}+\frac {40}{3} B \,a^{3} b^{3} d^{3} e +\frac {5}{2} B \,a^{2} b^{4} d^{4}\right ) x^{6}+\left (\frac {1}{5} a^{6} A \,e^{4}+\frac {24}{5} A \,a^{5} b d \,e^{3}+18 A \,a^{4} b^{2} d^{2} e^{2}+16 A \,a^{3} b^{3} d^{3} e +3 A \,a^{2} b^{4} d^{4}+\frac {4}{5} B \,a^{6} d \,e^{3}+\frac {36}{5} B \,a^{5} b \,d^{2} e^{2}+12 B \,a^{4} b^{2} d^{3} e +4 B \,a^{3} b^{3} d^{4}\right ) x^{5}+\left (a^{6} A d \,e^{3}+9 A \,a^{5} b \,d^{2} e^{2}+15 A \,a^{4} b^{2} d^{3} e +5 A \,a^{3} b^{3} d^{4}+\frac {3}{2} B \,a^{6} d^{2} e^{2}+6 B \,a^{5} b \,d^{3} e +\frac {15}{4} B \,a^{4} b^{2} d^{4}\right ) x^{4}+\left (2 a^{6} A \,d^{2} e^{2}+8 A \,a^{5} b \,d^{3} e +5 A \,a^{4} b^{2} d^{4}+\frac {4}{3} B \,a^{6} d^{3} e +2 B \,a^{5} b \,d^{4}\right ) x^{3}+\left (2 a^{6} A \,d^{3} e +3 A \,a^{5} b \,d^{4}+\frac {1}{2} B \,a^{6} d^{4}\right ) x^{2}+a^{6} A \,d^{4} x\)	\(862\)
gosper	\(\text {Expression too large to display}\)	\(1016\)
risch	\(\text {Expression too large to display}\)	\(1016\)
parallelrisch	\(\text {Expression too large to display}\)	\(1016\)
orering	\(\text {Expression too large to display}\)	\(1016\)

Input:

int((b*x+a)^6*(B*x+A)*(e*x+d)^4,x,method=_RETURNVERBOSE)
 

Output:

1/12*b^6*B*e^4*x^12+1/11*((A*b^6+6*B*a*b^5)*e^4+4*b^6*B*d*e^3)*x^11+1/10*( 
(6*A*a*b^5+15*B*a^2*b^4)*e^4+4*(A*b^6+6*B*a*b^5)*d*e^3+6*b^6*B*d^2*e^2)*x^ 
10+1/9*((15*A*a^2*b^4+20*B*a^3*b^3)*e^4+4*(6*A*a*b^5+15*B*a^2*b^4)*d*e^3+6 
*(A*b^6+6*B*a*b^5)*d^2*e^2+4*b^6*B*d^3*e)*x^9+1/8*((20*A*a^3*b^3+15*B*a^4* 
b^2)*e^4+4*(15*A*a^2*b^4+20*B*a^3*b^3)*d*e^3+6*(6*A*a*b^5+15*B*a^2*b^4)*d^ 
2*e^2+4*(A*b^6+6*B*a*b^5)*d^3*e+b^6*B*d^4)*x^8+1/7*((15*A*a^4*b^2+6*B*a^5* 
b)*e^4+4*(20*A*a^3*b^3+15*B*a^4*b^2)*d*e^3+6*(15*A*a^2*b^4+20*B*a^3*b^3)*d 
^2*e^2+4*(6*A*a*b^5+15*B*a^2*b^4)*d^3*e+(A*b^6+6*B*a*b^5)*d^4)*x^7+1/6*((6 
*A*a^5*b+B*a^6)*e^4+4*(15*A*a^4*b^2+6*B*a^5*b)*d*e^3+6*(20*A*a^3*b^3+15*B* 
a^4*b^2)*d^2*e^2+4*(15*A*a^2*b^4+20*B*a^3*b^3)*d^3*e+(6*A*a*b^5+15*B*a^2*b 
^4)*d^4)*x^6+1/5*(a^6*A*e^4+4*(6*A*a^5*b+B*a^6)*d*e^3+6*(15*A*a^4*b^2+6*B* 
a^5*b)*d^2*e^2+4*(20*A*a^3*b^3+15*B*a^4*b^2)*d^3*e+(15*A*a^2*b^4+20*B*a^3* 
b^3)*d^4)*x^5+1/4*(4*a^6*A*d*e^3+6*(6*A*a^5*b+B*a^6)*d^2*e^2+4*(15*A*a^4*b 
^2+6*B*a^5*b)*d^3*e+(20*A*a^3*b^3+15*B*a^4*b^2)*d^4)*x^4+1/3*(6*a^6*A*d^2* 
e^2+4*(6*A*a^5*b+B*a^6)*d^3*e+(15*A*a^4*b^2+6*B*a^5*b)*d^4)*x^3+1/2*(4*a^6 
*A*d^3*e+(6*A*a^5*b+B*a^6)*d^4)*x^2+a^6*A*d^4*x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 828 vs. \(2 (192) = 384\).

Time = 0.09 (sec) , antiderivative size = 828, normalized size of antiderivative = 4.06 \[ \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^4,x, algorithm="fricas")
 

Output:

1/12*B*b^6*e^4*x^12 + A*a^6*d^4*x + 1/11*(4*B*b^6*d*e^3 + (6*B*a*b^5 + A*b 
^6)*e^4)*x^11 + 1/10*(6*B*b^6*d^2*e^2 + 4*(6*B*a*b^5 + A*b^6)*d*e^3 + 3*(5 
*B*a^2*b^4 + 2*A*a*b^5)*e^4)*x^10 + 1/9*(4*B*b^6*d^3*e + 6*(6*B*a*b^5 + A* 
b^6)*d^2*e^2 + 12*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^3 + 5*(4*B*a^3*b^3 + 3*A*a 
^2*b^4)*e^4)*x^9 + 1/8*(B*b^6*d^4 + 4*(6*B*a*b^5 + A*b^6)*d^3*e + 18*(5*B* 
a^2*b^4 + 2*A*a*b^5)*d^2*e^2 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^3 + 5*(3 
*B*a^4*b^2 + 4*A*a^3*b^3)*e^4)*x^8 + 1/7*((6*B*a*b^5 + A*b^6)*d^4 + 12*(5* 
B*a^2*b^4 + 2*A*a*b^5)*d^3*e + 30*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^2 + 20 
*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^3 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^4)*x^7 
+ 1/6*(3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^ 
3*e + 30*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^2 + 12*(2*B*a^5*b + 5*A*a^4*b^2 
)*d*e^3 + (B*a^6 + 6*A*a^5*b)*e^4)*x^6 + 1/5*(A*a^6*e^4 + 5*(4*B*a^3*b^3 + 
 3*A*a^2*b^4)*d^4 + 20*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e + 18*(2*B*a^5*b + 
 5*A*a^4*b^2)*d^2*e^2 + 4*(B*a^6 + 6*A*a^5*b)*d*e^3)*x^5 + 1/4*(4*A*a^6*d* 
e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4 + 12*(2*B*a^5*b + 5*A*a^4*b^2)*d^3 
*e + 6*(B*a^6 + 6*A*a^5*b)*d^2*e^2)*x^4 + 1/3*(6*A*a^6*d^2*e^2 + 3*(2*B*a^ 
5*b + 5*A*a^4*b^2)*d^4 + 4*(B*a^6 + 6*A*a^5*b)*d^3*e)*x^3 + 1/2*(4*A*a^6*d 
^3*e + (B*a^6 + 6*A*a^5*b)*d^4)*x^2
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1035 vs. \(2 (204) = 408\).

Time = 0.07 (sec) , antiderivative size = 1035, normalized size of antiderivative = 5.07 \[ \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)**6*(B*x+A)*(e*x+d)**4,x)
 

Output:

A*a**6*d**4*x + B*b**6*e**4*x**12/12 + x**11*(A*b**6*e**4/11 + 6*B*a*b**5* 
e**4/11 + 4*B*b**6*d*e**3/11) + x**10*(3*A*a*b**5*e**4/5 + 2*A*b**6*d*e**3 
/5 + 3*B*a**2*b**4*e**4/2 + 12*B*a*b**5*d*e**3/5 + 3*B*b**6*d**2*e**2/5) + 
 x**9*(5*A*a**2*b**4*e**4/3 + 8*A*a*b**5*d*e**3/3 + 2*A*b**6*d**2*e**2/3 + 
 20*B*a**3*b**3*e**4/9 + 20*B*a**2*b**4*d*e**3/3 + 4*B*a*b**5*d**2*e**2 + 
4*B*b**6*d**3*e/9) + x**8*(5*A*a**3*b**3*e**4/2 + 15*A*a**2*b**4*d*e**3/2 
+ 9*A*a*b**5*d**2*e**2/2 + A*b**6*d**3*e/2 + 15*B*a**4*b**2*e**4/8 + 10*B* 
a**3*b**3*d*e**3 + 45*B*a**2*b**4*d**2*e**2/4 + 3*B*a*b**5*d**3*e + B*b**6 
*d**4/8) + x**7*(15*A*a**4*b**2*e**4/7 + 80*A*a**3*b**3*d*e**3/7 + 90*A*a* 
*2*b**4*d**2*e**2/7 + 24*A*a*b**5*d**3*e/7 + A*b**6*d**4/7 + 6*B*a**5*b*e* 
*4/7 + 60*B*a**4*b**2*d*e**3/7 + 120*B*a**3*b**3*d**2*e**2/7 + 60*B*a**2*b 
**4*d**3*e/7 + 6*B*a*b**5*d**4/7) + x**6*(A*a**5*b*e**4 + 10*A*a**4*b**2*d 
*e**3 + 20*A*a**3*b**3*d**2*e**2 + 10*A*a**2*b**4*d**3*e + A*a*b**5*d**4 + 
 B*a**6*e**4/6 + 4*B*a**5*b*d*e**3 + 15*B*a**4*b**2*d**2*e**2 + 40*B*a**3* 
b**3*d**3*e/3 + 5*B*a**2*b**4*d**4/2) + x**5*(A*a**6*e**4/5 + 24*A*a**5*b* 
d*e**3/5 + 18*A*a**4*b**2*d**2*e**2 + 16*A*a**3*b**3*d**3*e + 3*A*a**2*b** 
4*d**4 + 4*B*a**6*d*e**3/5 + 36*B*a**5*b*d**2*e**2/5 + 12*B*a**4*b**2*d**3 
*e + 4*B*a**3*b**3*d**4) + x**4*(A*a**6*d*e**3 + 9*A*a**5*b*d**2*e**2 + 15 
*A*a**4*b**2*d**3*e + 5*A*a**3*b**3*d**4 + 3*B*a**6*d**2*e**2/2 + 6*B*a**5 
*b*d**3*e + 15*B*a**4*b**2*d**4/4) + x**3*(2*A*a**6*d**2*e**2 + 8*A*a**...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 828 vs. \(2 (192) = 384\).

Time = 0.04 (sec) , antiderivative size = 828, normalized size of antiderivative = 4.06 \[ \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^4,x, algorithm="maxima")
 

Output:

1/12*B*b^6*e^4*x^12 + A*a^6*d^4*x + 1/11*(4*B*b^6*d*e^3 + (6*B*a*b^5 + A*b 
^6)*e^4)*x^11 + 1/10*(6*B*b^6*d^2*e^2 + 4*(6*B*a*b^5 + A*b^6)*d*e^3 + 3*(5 
*B*a^2*b^4 + 2*A*a*b^5)*e^4)*x^10 + 1/9*(4*B*b^6*d^3*e + 6*(6*B*a*b^5 + A* 
b^6)*d^2*e^2 + 12*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^3 + 5*(4*B*a^3*b^3 + 3*A*a 
^2*b^4)*e^4)*x^9 + 1/8*(B*b^6*d^4 + 4*(6*B*a*b^5 + A*b^6)*d^3*e + 18*(5*B* 
a^2*b^4 + 2*A*a*b^5)*d^2*e^2 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^3 + 5*(3 
*B*a^4*b^2 + 4*A*a^3*b^3)*e^4)*x^8 + 1/7*((6*B*a*b^5 + A*b^6)*d^4 + 12*(5* 
B*a^2*b^4 + 2*A*a*b^5)*d^3*e + 30*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^2 + 20 
*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^3 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^4)*x^7 
+ 1/6*(3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^ 
3*e + 30*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^2 + 12*(2*B*a^5*b + 5*A*a^4*b^2 
)*d*e^3 + (B*a^6 + 6*A*a^5*b)*e^4)*x^6 + 1/5*(A*a^6*e^4 + 5*(4*B*a^3*b^3 + 
 3*A*a^2*b^4)*d^4 + 20*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e + 18*(2*B*a^5*b + 
 5*A*a^4*b^2)*d^2*e^2 + 4*(B*a^6 + 6*A*a^5*b)*d*e^3)*x^5 + 1/4*(4*A*a^6*d* 
e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4 + 12*(2*B*a^5*b + 5*A*a^4*b^2)*d^3 
*e + 6*(B*a^6 + 6*A*a^5*b)*d^2*e^2)*x^4 + 1/3*(6*A*a^6*d^2*e^2 + 3*(2*B*a^ 
5*b + 5*A*a^4*b^2)*d^4 + 4*(B*a^6 + 6*A*a^5*b)*d^3*e)*x^3 + 1/2*(4*A*a^6*d 
^3*e + (B*a^6 + 6*A*a^5*b)*d^4)*x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1015 vs. \(2 (192) = 384\).

Time = 0.13 (sec) , antiderivative size = 1015, normalized size of antiderivative = 4.98 \[ \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^4,x, algorithm="giac")
 

Output:

1/12*B*b^6*e^4*x^12 + 4/11*B*b^6*d*e^3*x^11 + 6/11*B*a*b^5*e^4*x^11 + 1/11 
*A*b^6*e^4*x^11 + 3/5*B*b^6*d^2*e^2*x^10 + 12/5*B*a*b^5*d*e^3*x^10 + 2/5*A 
*b^6*d*e^3*x^10 + 3/2*B*a^2*b^4*e^4*x^10 + 3/5*A*a*b^5*e^4*x^10 + 4/9*B*b^ 
6*d^3*e*x^9 + 4*B*a*b^5*d^2*e^2*x^9 + 2/3*A*b^6*d^2*e^2*x^9 + 20/3*B*a^2*b 
^4*d*e^3*x^9 + 8/3*A*a*b^5*d*e^3*x^9 + 20/9*B*a^3*b^3*e^4*x^9 + 5/3*A*a^2* 
b^4*e^4*x^9 + 1/8*B*b^6*d^4*x^8 + 3*B*a*b^5*d^3*e*x^8 + 1/2*A*b^6*d^3*e*x^ 
8 + 45/4*B*a^2*b^4*d^2*e^2*x^8 + 9/2*A*a*b^5*d^2*e^2*x^8 + 10*B*a^3*b^3*d* 
e^3*x^8 + 15/2*A*a^2*b^4*d*e^3*x^8 + 15/8*B*a^4*b^2*e^4*x^8 + 5/2*A*a^3*b^ 
3*e^4*x^8 + 6/7*B*a*b^5*d^4*x^7 + 1/7*A*b^6*d^4*x^7 + 60/7*B*a^2*b^4*d^3*e 
*x^7 + 24/7*A*a*b^5*d^3*e*x^7 + 120/7*B*a^3*b^3*d^2*e^2*x^7 + 90/7*A*a^2*b 
^4*d^2*e^2*x^7 + 60/7*B*a^4*b^2*d*e^3*x^7 + 80/7*A*a^3*b^3*d*e^3*x^7 + 6/7 
*B*a^5*b*e^4*x^7 + 15/7*A*a^4*b^2*e^4*x^7 + 5/2*B*a^2*b^4*d^4*x^6 + A*a*b^ 
5*d^4*x^6 + 40/3*B*a^3*b^3*d^3*e*x^6 + 10*A*a^2*b^4*d^3*e*x^6 + 15*B*a^4*b 
^2*d^2*e^2*x^6 + 20*A*a^3*b^3*d^2*e^2*x^6 + 4*B*a^5*b*d*e^3*x^6 + 10*A*a^4 
*b^2*d*e^3*x^6 + 1/6*B*a^6*e^4*x^6 + A*a^5*b*e^4*x^6 + 4*B*a^3*b^3*d^4*x^5 
 + 3*A*a^2*b^4*d^4*x^5 + 12*B*a^4*b^2*d^3*e*x^5 + 16*A*a^3*b^3*d^3*e*x^5 + 
 36/5*B*a^5*b*d^2*e^2*x^5 + 18*A*a^4*b^2*d^2*e^2*x^5 + 4/5*B*a^6*d*e^3*x^5 
 + 24/5*A*a^5*b*d*e^3*x^5 + 1/5*A*a^6*e^4*x^5 + 15/4*B*a^4*b^2*d^4*x^4 + 5 
*A*a^3*b^3*d^4*x^4 + 6*B*a^5*b*d^3*e*x^4 + 15*A*a^4*b^2*d^3*e*x^4 + 3/2*B* 
a^6*d^2*e^2*x^4 + 9*A*a^5*b*d^2*e^2*x^4 + A*a^6*d*e^3*x^4 + 2*B*a^5*b*d...
 

Mupad [B] (verification not implemented)

Time = 1.01 (sec) , antiderivative size = 845, normalized size of antiderivative = 4.14 \[ \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx=x^4\,\left (\frac {3\,B\,a^6\,d^2\,e^2}{2}+A\,a^6\,d\,e^3+6\,B\,a^5\,b\,d^3\,e+9\,A\,a^5\,b\,d^2\,e^2+\frac {15\,B\,a^4\,b^2\,d^4}{4}+15\,A\,a^4\,b^2\,d^3\,e+5\,A\,a^3\,b^3\,d^4\right )+x^9\,\left (\frac {20\,B\,a^3\,b^3\,e^4}{9}+\frac {20\,B\,a^2\,b^4\,d\,e^3}{3}+\frac {5\,A\,a^2\,b^4\,e^4}{3}+4\,B\,a\,b^5\,d^2\,e^2+\frac {8\,A\,a\,b^5\,d\,e^3}{3}+\frac {4\,B\,b^6\,d^3\,e}{9}+\frac {2\,A\,b^6\,d^2\,e^2}{3}\right )+x^3\,\left (\frac {4\,B\,a^6\,d^3\,e}{3}+2\,A\,a^6\,d^2\,e^2+2\,B\,a^5\,b\,d^4+8\,A\,a^5\,b\,d^3\,e+5\,A\,a^4\,b^2\,d^4\right )+x^{10}\,\left (\frac {3\,B\,a^2\,b^4\,e^4}{2}+\frac {12\,B\,a\,b^5\,d\,e^3}{5}+\frac {3\,A\,a\,b^5\,e^4}{5}+\frac {3\,B\,b^6\,d^2\,e^2}{5}+\frac {2\,A\,b^6\,d\,e^3}{5}\right )+x^5\,\left (\frac {4\,B\,a^6\,d\,e^3}{5}+\frac {A\,a^6\,e^4}{5}+\frac {36\,B\,a^5\,b\,d^2\,e^2}{5}+\frac {24\,A\,a^5\,b\,d\,e^3}{5}+12\,B\,a^4\,b^2\,d^3\,e+18\,A\,a^4\,b^2\,d^2\,e^2+4\,B\,a^3\,b^3\,d^4+16\,A\,a^3\,b^3\,d^3\,e+3\,A\,a^2\,b^4\,d^4\right )+x^8\,\left (\frac {15\,B\,a^4\,b^2\,e^4}{8}+10\,B\,a^3\,b^3\,d\,e^3+\frac {5\,A\,a^3\,b^3\,e^4}{2}+\frac {45\,B\,a^2\,b^4\,d^2\,e^2}{4}+\frac {15\,A\,a^2\,b^4\,d\,e^3}{2}+3\,B\,a\,b^5\,d^3\,e+\frac {9\,A\,a\,b^5\,d^2\,e^2}{2}+\frac {B\,b^6\,d^4}{8}+\frac {A\,b^6\,d^3\,e}{2}\right )+x^6\,\left (\frac {B\,a^6\,e^4}{6}+4\,B\,a^5\,b\,d\,e^3+A\,a^5\,b\,e^4+15\,B\,a^4\,b^2\,d^2\,e^2+10\,A\,a^4\,b^2\,d\,e^3+\frac {40\,B\,a^3\,b^3\,d^3\,e}{3}+20\,A\,a^3\,b^3\,d^2\,e^2+\frac {5\,B\,a^2\,b^4\,d^4}{2}+10\,A\,a^2\,b^4\,d^3\,e+A\,a\,b^5\,d^4\right )+x^7\,\left (\frac {6\,B\,a^5\,b\,e^4}{7}+\frac {60\,B\,a^4\,b^2\,d\,e^3}{7}+\frac {15\,A\,a^4\,b^2\,e^4}{7}+\frac {120\,B\,a^3\,b^3\,d^2\,e^2}{7}+\frac {80\,A\,a^3\,b^3\,d\,e^3}{7}+\frac {60\,B\,a^2\,b^4\,d^3\,e}{7}+\frac {90\,A\,a^2\,b^4\,d^2\,e^2}{7}+\frac {6\,B\,a\,b^5\,d^4}{7}+\frac {24\,A\,a\,b^5\,d^3\,e}{7}+\frac {A\,b^6\,d^4}{7}\right )+\frac {a^5\,d^3\,x^2\,\left (4\,A\,a\,e+6\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^5\,e^3\,x^{11}\,\left (A\,b\,e+6\,B\,a\,e+4\,B\,b\,d\right )}{11}+A\,a^6\,d^4\,x+\frac {B\,b^6\,e^4\,x^{12}}{12} \] Input:

int((A + B*x)*(a + b*x)^6*(d + e*x)^4,x)
 

Output:

x^4*(A*a^6*d*e^3 + 5*A*a^3*b^3*d^4 + (15*B*a^4*b^2*d^4)/4 + (3*B*a^6*d^2*e 
^2)/2 + 15*A*a^4*b^2*d^3*e + 9*A*a^5*b*d^2*e^2 + 6*B*a^5*b*d^3*e) + x^9*(( 
4*B*b^6*d^3*e)/9 + (5*A*a^2*b^4*e^4)/3 + (20*B*a^3*b^3*e^4)/9 + (2*A*b^6*d 
^2*e^2)/3 + 4*B*a*b^5*d^2*e^2 + (20*B*a^2*b^4*d*e^3)/3 + (8*A*a*b^5*d*e^3) 
/3) + x^3*(2*B*a^5*b*d^4 + (4*B*a^6*d^3*e)/3 + 5*A*a^4*b^2*d^4 + 2*A*a^6*d 
^2*e^2 + 8*A*a^5*b*d^3*e) + x^10*((3*A*a*b^5*e^4)/5 + (2*A*b^6*d*e^3)/5 + 
(3*B*a^2*b^4*e^4)/2 + (3*B*b^6*d^2*e^2)/5 + (12*B*a*b^5*d*e^3)/5) + x^5*(( 
A*a^6*e^4)/5 + (4*B*a^6*d*e^3)/5 + 3*A*a^2*b^4*d^4 + 4*B*a^3*b^3*d^4 + 16* 
A*a^3*b^3*d^3*e + 12*B*a^4*b^2*d^3*e + (36*B*a^5*b*d^2*e^2)/5 + 18*A*a^4*b 
^2*d^2*e^2 + (24*A*a^5*b*d*e^3)/5) + x^8*((B*b^6*d^4)/8 + (A*b^6*d^3*e)/2 
+ (5*A*a^3*b^3*e^4)/2 + (15*B*a^4*b^2*e^4)/8 + (9*A*a*b^5*d^2*e^2)/2 + (15 
*A*a^2*b^4*d*e^3)/2 + 10*B*a^3*b^3*d*e^3 + (45*B*a^2*b^4*d^2*e^2)/4 + 3*B* 
a*b^5*d^3*e) + x^6*((B*a^6*e^4)/6 + A*a*b^5*d^4 + A*a^5*b*e^4 + (5*B*a^2*b 
^4*d^4)/2 + 10*A*a^2*b^4*d^3*e + 10*A*a^4*b^2*d*e^3 + (40*B*a^3*b^3*d^3*e) 
/3 + 20*A*a^3*b^3*d^2*e^2 + 15*B*a^4*b^2*d^2*e^2 + 4*B*a^5*b*d*e^3) + x^7* 
((A*b^6*d^4)/7 + (6*B*a*b^5*d^4)/7 + (6*B*a^5*b*e^4)/7 + (15*A*a^4*b^2*e^4 
)/7 + (80*A*a^3*b^3*d*e^3)/7 + (60*B*a^2*b^4*d^3*e)/7 + (60*B*a^4*b^2*d*e^ 
3)/7 + (90*A*a^2*b^4*d^2*e^2)/7 + (120*B*a^3*b^3*d^2*e^2)/7 + (24*A*a*b^5* 
d^3*e)/7) + (a^5*d^3*x^2*(4*A*a*e + 6*A*b*d + B*a*d))/2 + (b^5*e^3*x^11*(A 
*b*e + 6*B*a*e + 4*B*b*d))/11 + A*a^6*d^4*x + (B*b^6*e^4*x^12)/12
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.68 \[ \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx=\frac {x \left (330 b^{7} e^{4} x^{11}+2520 a \,b^{6} e^{4} x^{10}+1440 b^{7} d \,e^{3} x^{10}+8316 a^{2} b^{5} e^{4} x^{9}+11088 a \,b^{6} d \,e^{3} x^{9}+2376 b^{7} d^{2} e^{2} x^{9}+15400 a^{3} b^{4} e^{4} x^{8}+36960 a^{2} b^{5} d \,e^{3} x^{8}+18480 a \,b^{6} d^{2} e^{2} x^{8}+1760 b^{7} d^{3} e \,x^{8}+17325 a^{4} b^{3} e^{4} x^{7}+69300 a^{3} b^{4} d \,e^{3} x^{7}+62370 a^{2} b^{5} d^{2} e^{2} x^{7}+13860 a \,b^{6} d^{3} e \,x^{7}+495 b^{7} d^{4} x^{7}+11880 a^{5} b^{2} e^{4} x^{6}+79200 a^{4} b^{3} d \,e^{3} x^{6}+118800 a^{3} b^{4} d^{2} e^{2} x^{6}+47520 a^{2} b^{5} d^{3} e \,x^{6}+3960 a \,b^{6} d^{4} x^{6}+4620 a^{6} b \,e^{4} x^{5}+55440 a^{5} b^{2} d \,e^{3} x^{5}+138600 a^{4} b^{3} d^{2} e^{2} x^{5}+92400 a^{3} b^{4} d^{3} e \,x^{5}+13860 a^{2} b^{5} d^{4} x^{5}+792 a^{7} e^{4} x^{4}+22176 a^{6} b d \,e^{3} x^{4}+99792 a^{5} b^{2} d^{2} e^{2} x^{4}+110880 a^{4} b^{3} d^{3} e \,x^{4}+27720 a^{3} b^{4} d^{4} x^{4}+3960 a^{7} d \,e^{3} x^{3}+41580 a^{6} b \,d^{2} e^{2} x^{3}+83160 a^{5} b^{2} d^{3} e \,x^{3}+34650 a^{4} b^{3} d^{4} x^{3}+7920 a^{7} d^{2} e^{2} x^{2}+36960 a^{6} b \,d^{3} e \,x^{2}+27720 a^{5} b^{2} d^{4} x^{2}+7920 a^{7} d^{3} e x +13860 a^{6} b \,d^{4} x +3960 a^{7} d^{4}\right )}{3960} \] Input:

int((b*x+a)^6*(B*x+A)*(e*x+d)^4,x)
 

Output:

(x*(3960*a**7*d**4 + 7920*a**7*d**3*e*x + 7920*a**7*d**2*e**2*x**2 + 3960* 
a**7*d*e**3*x**3 + 792*a**7*e**4*x**4 + 13860*a**6*b*d**4*x + 36960*a**6*b 
*d**3*e*x**2 + 41580*a**6*b*d**2*e**2*x**3 + 22176*a**6*b*d*e**3*x**4 + 46 
20*a**6*b*e**4*x**5 + 27720*a**5*b**2*d**4*x**2 + 83160*a**5*b**2*d**3*e*x 
**3 + 99792*a**5*b**2*d**2*e**2*x**4 + 55440*a**5*b**2*d*e**3*x**5 + 11880 
*a**5*b**2*e**4*x**6 + 34650*a**4*b**3*d**4*x**3 + 110880*a**4*b**3*d**3*e 
*x**4 + 138600*a**4*b**3*d**2*e**2*x**5 + 79200*a**4*b**3*d*e**3*x**6 + 17 
325*a**4*b**3*e**4*x**7 + 27720*a**3*b**4*d**4*x**4 + 92400*a**3*b**4*d**3 
*e*x**5 + 118800*a**3*b**4*d**2*e**2*x**6 + 69300*a**3*b**4*d*e**3*x**7 + 
15400*a**3*b**4*e**4*x**8 + 13860*a**2*b**5*d**4*x**5 + 47520*a**2*b**5*d* 
*3*e*x**6 + 62370*a**2*b**5*d**2*e**2*x**7 + 36960*a**2*b**5*d*e**3*x**8 + 
 8316*a**2*b**5*e**4*x**9 + 3960*a*b**6*d**4*x**6 + 13860*a*b**6*d**3*e*x* 
*7 + 18480*a*b**6*d**2*e**2*x**8 + 11088*a*b**6*d*e**3*x**9 + 2520*a*b**6* 
e**4*x**10 + 495*b**7*d**4*x**7 + 1760*b**7*d**3*e*x**8 + 2376*b**7*d**2*e 
**2*x**9 + 1440*b**7*d*e**3*x**10 + 330*b**7*e**4*x**11))/3960