\(\int (a+b x)^6 (A+B x) (d+e x)^3 \, dx\) [46]

Optimal result

Integrand size = 20, antiderivative size = 159 \[ \int (a+b x)^6 (A+B x) (d+e x)^3 \, dx=\frac {(A b-a B) (b d-a e)^3 (a+b x)^7}{7 b^5}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^8}{8 b^5}+\frac {e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^9}{3 b^5}+\frac {e^2 (3 b B d+A b e-4 a B e) (a+b x)^{10}}{10 b^5}+\frac {B e^3 (a+b x)^{11}}{11 b^5} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(586\) vs. \(2(159)=318\).

Time = 0.13 (sec) , antiderivative size = 586, normalized size of antiderivative = 3.69 \[ \int (a+b x)^6 (A+B x) (d+e x)^3 \, dx=a^6 A d^3 x+\frac {1}{2} a^5 d^2 (6 A b d+a B d+3 a A e) x^2+a^4 d \left (a B d (2 b d+a e)+A \left (5 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x^3+\frac {1}{4} a^3 \left (3 a B d \left (5 b^2 d^2+6 a b d e+a^2 e^2\right )+A \left (20 b^3 d^3+45 a b^2 d^2 e+18 a^2 b d e^2+a^3 e^3\right )\right ) x^4+\frac {1}{5} a^2 \left (a B \left (20 b^3 d^3+45 a b^2 d^2 e+18 a^2 b d e^2+a^3 e^3\right )+3 A b \left (5 b^3 d^3+20 a b^2 d^2 e+15 a^2 b d e^2+2 a^3 e^3\right )\right ) x^5+\frac {1}{2} a b \left (a B \left (5 b^3 d^3+20 a b^2 d^2 e+15 a^2 b d e^2+2 a^3 e^3\right )+A b \left (2 b^3 d^3+15 a b^2 d^2 e+20 a^2 b d e^2+5 a^3 e^3\right )\right ) x^6+\frac {1}{7} b^2 \left (3 a B \left (2 b^3 d^3+15 a b^2 d^2 e+20 a^2 b d e^2+5 a^3 e^3\right )+A b \left (b^3 d^3+18 a b^2 d^2 e+45 a^2 b d e^2+20 a^3 e^3\right )\right ) x^7+\frac {1}{8} b^3 \left (20 a^3 B e^3+18 a b^2 d e (B d+A e)+15 a^2 b e^2 (3 B d+A e)+b^3 d^2 (B d+3 A e)\right ) x^8+\frac {1}{3} b^4 e \left (5 a^2 B e^2+b^2 d (B d+A e)+2 a b e (3 B d+A e)\right ) x^9+\frac {1}{10} b^5 e^2 (3 b B d+A b e+6 a B e) x^{10}+\frac {1}{11} b^6 B e^3 x^{11} \] Input:

Integrate[(a + b*x)^6*(A + B*x)*(d + e*x)^3,x]
 

Output:

a^6*A*d^3*x + (a^5*d^2*(6*A*b*d + a*B*d + 3*a*A*e)*x^2)/2 + a^4*d*(a*B*d*( 
2*b*d + a*e) + A*(5*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*x^3 + (a^3*(3*a*B*d*(5 
*b^2*d^2 + 6*a*b*d*e + a^2*e^2) + A*(20*b^3*d^3 + 45*a*b^2*d^2*e + 18*a^2* 
b*d*e^2 + a^3*e^3))*x^4)/4 + (a^2*(a*B*(20*b^3*d^3 + 45*a*b^2*d^2*e + 18*a 
^2*b*d*e^2 + a^3*e^3) + 3*A*b*(5*b^3*d^3 + 20*a*b^2*d^2*e + 15*a^2*b*d*e^2 
 + 2*a^3*e^3))*x^5)/5 + (a*b*(a*B*(5*b^3*d^3 + 20*a*b^2*d^2*e + 15*a^2*b*d 
*e^2 + 2*a^3*e^3) + A*b*(2*b^3*d^3 + 15*a*b^2*d^2*e + 20*a^2*b*d*e^2 + 5*a 
^3*e^3))*x^6)/2 + (b^2*(3*a*B*(2*b^3*d^3 + 15*a*b^2*d^2*e + 20*a^2*b*d*e^2 
 + 5*a^3*e^3) + A*b*(b^3*d^3 + 18*a*b^2*d^2*e + 45*a^2*b*d*e^2 + 20*a^3*e^ 
3))*x^7)/7 + (b^3*(20*a^3*B*e^3 + 18*a*b^2*d*e*(B*d + A*e) + 15*a^2*b*e^2* 
(3*B*d + A*e) + b^3*d^2*(B*d + 3*A*e))*x^8)/8 + (b^4*e*(5*a^2*B*e^2 + b^2* 
d*(B*d + A*e) + 2*a*b*e*(3*B*d + A*e))*x^9)/3 + (b^5*e^2*(3*b*B*d + A*b*e 
+ 6*a*B*e)*x^10)/10 + (b^6*B*e^3*x^11)/11
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 159, 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.

\(\displaystyle \int (a+b x)^6 (A+B x) (d+e x)^3 \, dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*x)^6*(A + B*x)*(d + e*x)^3,x]
 

Output:

((A*b - a*B)*(b*d - a*e)^3*(a + b*x)^7)/(7*b^5) + ((b*d - a*e)^2*(b*B*d + 
3*A*b*e - 4*a*B*e)*(a + b*x)^8)/(8*b^5) + (e*(b*d - a*e)*(b*B*d + A*b*e - 
2*a*B*e)*(a + b*x)^9)/(3*b^5) + (e^2*(3*b*B*d + A*b*e - 4*a*B*e)*(a + b*x) 
^10)/(10*b^5) + (B*e^3*(a + b*x)^11)/(11*b^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. \(644\) vs. \(2(149)=298\).

Time = 0.19 (sec) , antiderivative size = 645, normalized size of antiderivative = 4.06

method	result	size
default	\(\frac {b^{6} B \,e^{3} x^{11}}{11}+\frac {\left (\left (b^{6} A +6 a \,b^{5} B \right ) e^{3}+3 b^{6} B d \,e^{2}\right ) x^{10}}{10}+\frac {\left (\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) e^{3}+3 \left (b^{6} A +6 a \,b^{5} B \right ) d \,e^{2}+3 b^{6} B \,d^{2} e \right ) x^{9}}{9}+\frac {\left (\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) e^{3}+3 \left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d \,e^{2}+3 \left (b^{6} A +6 a \,b^{5} B \right ) d^{2} e +b^{6} B \,d^{3}\right ) x^{8}}{8}+\frac {\left (\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) e^{3}+3 \left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d \,e^{2}+3 \left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d^{2} e +\left (b^{6} A +6 a \,b^{5} B \right ) d^{3}\right ) x^{7}}{7}+\frac {\left (\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) e^{3}+3 \left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d \,e^{2}+3 \left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d^{2} e +\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d^{3}\right ) x^{6}}{6}+\frac {\left (\left (6 a^{5} b A +a^{6} B \right ) e^{3}+3 \left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d \,e^{2}+3 \left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d^{2} e +\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d^{3}\right ) x^{5}}{5}+\frac {\left (a^{6} A \,e^{3}+3 \left (6 a^{5} b A +a^{6} B \right ) d \,e^{2}+3 \left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d^{2} e +\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d^{3}\right ) x^{4}}{4}+\frac {\left (3 a^{6} A d \,e^{2}+3 \left (6 a^{5} b A +a^{6} B \right ) d^{2} e +\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d^{3}\right ) x^{3}}{3}+\frac {\left (3 a^{6} A \,d^{2} e +\left (6 a^{5} b A +a^{6} B \right ) d^{3}\right ) x^{2}}{2}+a^{6} A \,d^{3} x\)	\(645\)
norman	\(\frac {b^{6} B \,e^{3} x^{11}}{11}+\left (\frac {1}{10} A \,b^{6} e^{3}+\frac {3}{5} B a \,b^{5} e^{3}+\frac {3}{10} b^{6} B d \,e^{2}\right ) x^{10}+\left (\frac {2}{3} A a \,b^{5} e^{3}+\frac {1}{3} A \,b^{6} d \,e^{2}+\frac {5}{3} B \,a^{2} b^{4} e^{3}+2 B a \,b^{5} d \,e^{2}+\frac {1}{3} b^{6} B \,d^{2} e \right ) x^{9}+\left (\frac {15}{8} A \,a^{2} b^{4} e^{3}+\frac {9}{4} A a \,b^{5} d \,e^{2}+\frac {3}{8} A \,b^{6} d^{2} e +\frac {5}{2} B \,a^{3} b^{3} e^{3}+\frac {45}{8} B \,a^{2} b^{4} d \,e^{2}+\frac {9}{4} B a \,b^{5} d^{2} e +\frac {1}{8} b^{6} B \,d^{3}\right ) x^{8}+\left (\frac {20}{7} A \,a^{3} b^{3} e^{3}+\frac {45}{7} A \,a^{2} b^{4} d \,e^{2}+\frac {18}{7} A a \,b^{5} d^{2} e +\frac {1}{7} A \,b^{6} d^{3}+\frac {15}{7} B \,a^{4} b^{2} e^{3}+\frac {60}{7} B \,a^{3} b^{3} d \,e^{2}+\frac {45}{7} B \,a^{2} b^{4} d^{2} e +\frac {6}{7} B a \,b^{5} d^{3}\right ) x^{7}+\left (\frac {5}{2} A \,a^{4} b^{2} e^{3}+10 A \,a^{3} b^{3} d \,e^{2}+\frac {15}{2} A \,a^{2} b^{4} d^{2} e +A a \,b^{5} d^{3}+B \,a^{5} b \,e^{3}+\frac {15}{2} B \,a^{4} b^{2} d \,e^{2}+10 B \,a^{3} b^{3} d^{2} e +\frac {5}{2} B \,a^{2} b^{4} d^{3}\right ) x^{6}+\left (\frac {6}{5} A \,a^{5} b \,e^{3}+9 A \,a^{4} b^{2} d \,e^{2}+12 A \,a^{3} b^{3} d^{2} e +3 A \,a^{2} b^{4} d^{3}+\frac {1}{5} B \,a^{6} e^{3}+\frac {18}{5} B \,a^{5} b d \,e^{2}+9 B \,a^{4} b^{2} d^{2} e +4 B \,a^{3} b^{3} d^{3}\right ) x^{5}+\left (\frac {1}{4} a^{6} A \,e^{3}+\frac {9}{2} A \,a^{5} b d \,e^{2}+\frac {45}{4} A \,a^{4} b^{2} d^{2} e +5 A \,a^{3} b^{3} d^{3}+\frac {3}{4} B \,a^{6} d \,e^{2}+\frac {9}{2} B \,a^{5} b \,d^{2} e +\frac {15}{4} B \,a^{4} b^{2} d^{3}\right ) x^{4}+\left (a^{6} A d \,e^{2}+6 A \,a^{5} b \,d^{2} e +5 A \,a^{4} b^{2} d^{3}+B \,a^{6} d^{2} e +2 B \,a^{5} b \,d^{3}\right ) x^{3}+\left (\frac {3}{2} a^{6} A \,d^{2} e +3 A \,a^{5} b \,d^{3}+\frac {1}{2} B \,a^{6} d^{3}\right ) x^{2}+a^{6} A \,d^{3} x\)	\(666\)
gosper	\(\frac {5}{2} x^{8} B \,a^{3} b^{3} e^{3}+\frac {20}{7} x^{7} A \,a^{3} b^{3} e^{3}+\frac {15}{7} x^{7} B \,a^{4} b^{2} e^{3}+\frac {6}{7} x^{7} B a \,b^{5} d^{3}+\frac {5}{2} x^{6} A \,a^{4} b^{2} e^{3}+x^{6} A a \,b^{5} d^{3}+x^{6} B \,a^{5} b \,e^{3}+\frac {5}{2} x^{6} B \,a^{2} b^{4} d^{3}+\frac {1}{10} x^{10} A \,b^{6} e^{3}+\frac {1}{8} x^{8} b^{6} B \,d^{3}+\frac {1}{7} x^{7} A \,b^{6} d^{3}+\frac {1}{5} x^{5} B \,a^{6} e^{3}+\frac {1}{4} x^{4} a^{6} A \,e^{3}+\frac {1}{2} x^{2} B \,a^{6} d^{3}+\frac {6}{5} x^{5} A \,a^{5} b \,e^{3}+3 x^{5} A \,a^{2} b^{4} d^{3}+4 x^{5} B \,a^{3} b^{3} d^{3}+5 x^{4} A \,a^{3} b^{3} d^{3}+\frac {3}{4} x^{4} B \,a^{6} d \,e^{2}+\frac {15}{4} x^{4} B \,a^{4} b^{2} d^{3}+\frac {3}{2} x^{2} a^{6} A \,d^{2} e +3 x^{2} A \,a^{5} b \,d^{3}+\frac {3}{5} x^{10} B a \,b^{5} e^{3}+\frac {3}{10} x^{10} b^{6} B d \,e^{2}+\frac {2}{3} x^{9} A a \,b^{5} e^{3}+\frac {1}{3} x^{9} A \,b^{6} d \,e^{2}+\frac {5}{3} x^{9} B \,a^{2} b^{4} e^{3}+\frac {1}{3} x^{9} b^{6} B \,d^{2} e +\frac {15}{8} x^{8} A \,a^{2} b^{4} e^{3}+\frac {3}{8} x^{8} A \,b^{6} d^{2} e +\frac {18}{5} x^{5} B \,a^{5} b d \,e^{2}+9 x^{5} B \,a^{4} b^{2} d^{2} e +10 x^{6} B \,a^{3} b^{3} d^{2} e +9 x^{5} A \,a^{4} b^{2} d \,e^{2}+12 x^{5} A \,a^{3} b^{3} d^{2} e +A \,a^{6} d \,e^{2} x^{3}+5 A \,a^{4} b^{2} d^{3} x^{3}+B \,a^{6} d^{2} e \,x^{3}+2 B \,a^{5} b \,d^{3} x^{3}+2 x^{9} B a \,b^{5} d \,e^{2}+\frac {9}{2} x^{4} A \,a^{5} b d \,e^{2}+\frac {45}{4} x^{4} A \,a^{4} b^{2} d^{2} e +\frac {9}{2} x^{4} B \,a^{5} b \,d^{2} e +6 A \,a^{5} b \,d^{2} e \,x^{3}+\frac {9}{4} x^{8} A a \,b^{5} d \,e^{2}+\frac {45}{8} x^{8} B \,a^{2} b^{4} d \,e^{2}+\frac {15}{2} x^{6} B \,a^{4} b^{2} d \,e^{2}+\frac {9}{4} x^{8} B a \,b^{5} d^{2} e +\frac {45}{7} x^{7} A \,a^{2} b^{4} d \,e^{2}+\frac {18}{7} x^{7} A a \,b^{5} d^{2} e +\frac {60}{7} x^{7} B \,a^{3} b^{3} d \,e^{2}+\frac {45}{7} x^{7} B \,a^{2} b^{4} d^{2} e +10 x^{6} A \,a^{3} b^{3} d \,e^{2}+\frac {15}{2} x^{6} A \,a^{2} b^{4} d^{2} e +\frac {1}{11} b^{6} B \,e^{3} x^{11}+a^{6} A \,d^{3} x\)	\(783\)
risch	\(\frac {5}{2} x^{8} B \,a^{3} b^{3} e^{3}+\frac {20}{7} x^{7} A \,a^{3} b^{3} e^{3}+\frac {15}{7} x^{7} B \,a^{4} b^{2} e^{3}+\frac {6}{7} x^{7} B a \,b^{5} d^{3}+\frac {5}{2} x^{6} A \,a^{4} b^{2} e^{3}+x^{6} A a \,b^{5} d^{3}+x^{6} B \,a^{5} b \,e^{3}+\frac {5}{2} x^{6} B \,a^{2} b^{4} d^{3}+\frac {1}{10} x^{10} A \,b^{6} e^{3}+\frac {1}{8} x^{8} b^{6} B \,d^{3}+\frac {1}{7} x^{7} A \,b^{6} d^{3}+\frac {1}{5} x^{5} B \,a^{6} e^{3}+\frac {1}{4} x^{4} a^{6} A \,e^{3}+\frac {1}{2} x^{2} B \,a^{6} d^{3}+\frac {6}{5} x^{5} A \,a^{5} b \,e^{3}+3 x^{5} A \,a^{2} b^{4} d^{3}+4 x^{5} B \,a^{3} b^{3} d^{3}+5 x^{4} A \,a^{3} b^{3} d^{3}+\frac {3}{4} x^{4} B \,a^{6} d \,e^{2}+\frac {15}{4} x^{4} B \,a^{4} b^{2} d^{3}+\frac {3}{2} x^{2} a^{6} A \,d^{2} e +3 x^{2} A \,a^{5} b \,d^{3}+\frac {3}{5} x^{10} B a \,b^{5} e^{3}+\frac {3}{10} x^{10} b^{6} B d \,e^{2}+\frac {2}{3} x^{9} A a \,b^{5} e^{3}+\frac {1}{3} x^{9} A \,b^{6} d \,e^{2}+\frac {5}{3} x^{9} B \,a^{2} b^{4} e^{3}+\frac {1}{3} x^{9} b^{6} B \,d^{2} e +\frac {15}{8} x^{8} A \,a^{2} b^{4} e^{3}+\frac {3}{8} x^{8} A \,b^{6} d^{2} e +\frac {18}{5} x^{5} B \,a^{5} b d \,e^{2}+9 x^{5} B \,a^{4} b^{2} d^{2} e +10 x^{6} B \,a^{3} b^{3} d^{2} e +9 x^{5} A \,a^{4} b^{2} d \,e^{2}+12 x^{5} A \,a^{3} b^{3} d^{2} e +A \,a^{6} d \,e^{2} x^{3}+5 A \,a^{4} b^{2} d^{3} x^{3}+B \,a^{6} d^{2} e \,x^{3}+2 B \,a^{5} b \,d^{3} x^{3}+2 x^{9} B a \,b^{5} d \,e^{2}+\frac {9}{2} x^{4} A \,a^{5} b d \,e^{2}+\frac {45}{4} x^{4} A \,a^{4} b^{2} d^{2} e +\frac {9}{2} x^{4} B \,a^{5} b \,d^{2} e +6 A \,a^{5} b \,d^{2} e \,x^{3}+\frac {9}{4} x^{8} A a \,b^{5} d \,e^{2}+\frac {45}{8} x^{8} B \,a^{2} b^{4} d \,e^{2}+\frac {15}{2} x^{6} B \,a^{4} b^{2} d \,e^{2}+\frac {9}{4} x^{8} B a \,b^{5} d^{2} e +\frac {45}{7} x^{7} A \,a^{2} b^{4} d \,e^{2}+\frac {18}{7} x^{7} A a \,b^{5} d^{2} e +\frac {60}{7} x^{7} B \,a^{3} b^{3} d \,e^{2}+\frac {45}{7} x^{7} B \,a^{2} b^{4} d^{2} e +10 x^{6} A \,a^{3} b^{3} d \,e^{2}+\frac {15}{2} x^{6} A \,a^{2} b^{4} d^{2} e +\frac {1}{11} b^{6} B \,e^{3} x^{11}+a^{6} A \,d^{3} x\)	\(783\)
parallelrisch	\(\frac {5}{2} x^{8} B \,a^{3} b^{3} e^{3}+\frac {20}{7} x^{7} A \,a^{3} b^{3} e^{3}+\frac {15}{7} x^{7} B \,a^{4} b^{2} e^{3}+\frac {6}{7} x^{7} B a \,b^{5} d^{3}+\frac {5}{2} x^{6} A \,a^{4} b^{2} e^{3}+x^{6} A a \,b^{5} d^{3}+x^{6} B \,a^{5} b \,e^{3}+\frac {5}{2} x^{6} B \,a^{2} b^{4} d^{3}+\frac {1}{10} x^{10} A \,b^{6} e^{3}+\frac {1}{8} x^{8} b^{6} B \,d^{3}+\frac {1}{7} x^{7} A \,b^{6} d^{3}+\frac {1}{5} x^{5} B \,a^{6} e^{3}+\frac {1}{4} x^{4} a^{6} A \,e^{3}+\frac {1}{2} x^{2} B \,a^{6} d^{3}+\frac {6}{5} x^{5} A \,a^{5} b \,e^{3}+3 x^{5} A \,a^{2} b^{4} d^{3}+4 x^{5} B \,a^{3} b^{3} d^{3}+5 x^{4} A \,a^{3} b^{3} d^{3}+\frac {3}{4} x^{4} B \,a^{6} d \,e^{2}+\frac {15}{4} x^{4} B \,a^{4} b^{2} d^{3}+\frac {3}{2} x^{2} a^{6} A \,d^{2} e +3 x^{2} A \,a^{5} b \,d^{3}+\frac {3}{5} x^{10} B a \,b^{5} e^{3}+\frac {3}{10} x^{10} b^{6} B d \,e^{2}+\frac {2}{3} x^{9} A a \,b^{5} e^{3}+\frac {1}{3} x^{9} A \,b^{6} d \,e^{2}+\frac {5}{3} x^{9} B \,a^{2} b^{4} e^{3}+\frac {1}{3} x^{9} b^{6} B \,d^{2} e +\frac {15}{8} x^{8} A \,a^{2} b^{4} e^{3}+\frac {3}{8} x^{8} A \,b^{6} d^{2} e +\frac {18}{5} x^{5} B \,a^{5} b d \,e^{2}+9 x^{5} B \,a^{4} b^{2} d^{2} e +10 x^{6} B \,a^{3} b^{3} d^{2} e +9 x^{5} A \,a^{4} b^{2} d \,e^{2}+12 x^{5} A \,a^{3} b^{3} d^{2} e +A \,a^{6} d \,e^{2} x^{3}+5 A \,a^{4} b^{2} d^{3} x^{3}+B \,a^{6} d^{2} e \,x^{3}+2 B \,a^{5} b \,d^{3} x^{3}+2 x^{9} B a \,b^{5} d \,e^{2}+\frac {9}{2} x^{4} A \,a^{5} b d \,e^{2}+\frac {45}{4} x^{4} A \,a^{4} b^{2} d^{2} e +\frac {9}{2} x^{4} B \,a^{5} b \,d^{2} e +6 A \,a^{5} b \,d^{2} e \,x^{3}+\frac {9}{4} x^{8} A a \,b^{5} d \,e^{2}+\frac {45}{8} x^{8} B \,a^{2} b^{4} d \,e^{2}+\frac {15}{2} x^{6} B \,a^{4} b^{2} d \,e^{2}+\frac {9}{4} x^{8} B a \,b^{5} d^{2} e +\frac {45}{7} x^{7} A \,a^{2} b^{4} d \,e^{2}+\frac {18}{7} x^{7} A a \,b^{5} d^{2} e +\frac {60}{7} x^{7} B \,a^{3} b^{3} d \,e^{2}+\frac {45}{7} x^{7} B \,a^{2} b^{4} d^{2} e +10 x^{6} A \,a^{3} b^{3} d \,e^{2}+\frac {15}{2} x^{6} A \,a^{2} b^{4} d^{2} e +\frac {1}{11} b^{6} B \,e^{3} x^{11}+a^{6} A \,d^{3} x\)	\(783\)
orering	\(\frac {x \left (840 b^{6} B \,e^{3} x^{10}+924 A \,b^{6} e^{3} x^{9}+5544 B a \,b^{5} e^{3} x^{9}+2772 B \,b^{6} d \,e^{2} x^{9}+6160 A a \,b^{5} e^{3} x^{8}+3080 A \,b^{6} d \,e^{2} x^{8}+15400 B \,a^{2} b^{4} e^{3} x^{8}+18480 B a \,b^{5} d \,e^{2} x^{8}+3080 B \,b^{6} d^{2} e \,x^{8}+17325 A \,a^{2} b^{4} e^{3} x^{7}+20790 A a \,b^{5} d \,e^{2} x^{7}+3465 A \,b^{6} d^{2} e \,x^{7}+23100 B \,a^{3} b^{3} e^{3} x^{7}+51975 B \,a^{2} b^{4} d \,e^{2} x^{7}+20790 B a \,b^{5} d^{2} e \,x^{7}+1155 B \,b^{6} d^{3} x^{7}+26400 A \,a^{3} b^{3} e^{3} x^{6}+59400 A \,a^{2} b^{4} d \,e^{2} x^{6}+23760 A a \,b^{5} d^{2} e \,x^{6}+1320 A \,b^{6} d^{3} x^{6}+19800 B \,a^{4} b^{2} e^{3} x^{6}+79200 B \,a^{3} b^{3} d \,e^{2} x^{6}+59400 B \,a^{2} b^{4} d^{2} e \,x^{6}+7920 B a \,b^{5} d^{3} x^{6}+23100 A \,a^{4} b^{2} e^{3} x^{5}+92400 A \,a^{3} b^{3} d \,e^{2} x^{5}+69300 A \,a^{2} b^{4} d^{2} e \,x^{5}+9240 A a \,b^{5} d^{3} x^{5}+9240 B \,a^{5} b \,e^{3} x^{5}+69300 B \,a^{4} b^{2} d \,e^{2} x^{5}+92400 B \,a^{3} b^{3} d^{2} e \,x^{5}+23100 B \,a^{2} b^{4} d^{3} x^{5}+11088 A \,a^{5} b \,e^{3} x^{4}+83160 A \,a^{4} b^{2} d \,e^{2} x^{4}+110880 A \,a^{3} b^{3} d^{2} e \,x^{4}+27720 A \,a^{2} b^{4} d^{3} x^{4}+1848 B \,a^{6} e^{3} x^{4}+33264 B \,a^{5} b d \,e^{2} x^{4}+83160 B \,a^{4} b^{2} d^{2} e \,x^{4}+36960 B \,a^{3} b^{3} d^{3} x^{4}+2310 A \,a^{6} e^{3} x^{3}+41580 A \,a^{5} b d \,e^{2} x^{3}+103950 A \,a^{4} b^{2} d^{2} e \,x^{3}+46200 A \,a^{3} b^{3} d^{3} x^{3}+6930 B \,a^{6} d \,e^{2} x^{3}+41580 B \,a^{5} b \,d^{2} e \,x^{3}+34650 B \,a^{4} b^{2} d^{3} x^{3}+9240 A \,a^{6} d \,e^{2} x^{2}+55440 A \,a^{5} b \,d^{2} e \,x^{2}+46200 A \,a^{4} b^{2} d^{3} x^{2}+9240 B \,a^{6} d^{2} e \,x^{2}+18480 B \,a^{5} b \,d^{3} x^{2}+13860 A \,a^{6} d^{2} e x +27720 A \,a^{5} b \,d^{3} x +4620 B \,a^{6} d^{3} x +9240 a^{6} A \,d^{3}\right )}{9240}\)	\(784\)

Input:

int((b*x+a)^6*(B*x+A)*(e*x+d)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 643 vs. \(2 (149) = 298\).

Time = 0.08 (sec) , antiderivative size = 643, normalized size of antiderivative = 4.04 \[ \int (a+b x)^6 (A+B x) (d+e x)^3 \, dx=\frac {1}{11} \, B b^{6} e^{3} x^{11} + A a^{6} d^{3} x + \frac {1}{10} \, {\left (3 \, B b^{6} d e^{2} + {\left (6 \, B a b^{5} + A b^{6}\right )} e^{3}\right )} x^{10} + \frac {1}{3} \, {\left (B b^{6} d^{2} e + {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{2} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (B b^{6} d^{3} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e + 9 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{2} + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left ({\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} + 9 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e + 15 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{2} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left ({\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{2} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e + 9 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{2} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (A a^{6} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} + 9 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e + 3 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{2}\right )} x^{4} + {\left (A a^{6} d e^{2} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{3} + {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{6} d^{2} e + {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{3}\right )} x^{2} \] Input:

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^3,x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (155) = 310\).

Time = 0.06 (sec) , antiderivative size = 802, normalized size of antiderivative = 5.04 \[ \int (a+b x)^6 (A+B x) (d+e x)^3 \, dx=A a^{6} d^{3} x + \frac {B b^{6} e^{3} x^{11}}{11} + x^{10} \left (\frac {A b^{6} e^{3}}{10} + \frac {3 B a b^{5} e^{3}}{5} + \frac {3 B b^{6} d e^{2}}{10}\right ) + x^{9} \cdot \left (\frac {2 A a b^{5} e^{3}}{3} + \frac {A b^{6} d e^{2}}{3} + \frac {5 B a^{2} b^{4} e^{3}}{3} + 2 B a b^{5} d e^{2} + \frac {B b^{6} d^{2} e}{3}\right ) + x^{8} \cdot \left (\frac {15 A a^{2} b^{4} e^{3}}{8} + \frac {9 A a b^{5} d e^{2}}{4} + \frac {3 A b^{6} d^{2} e}{8} + \frac {5 B a^{3} b^{3} e^{3}}{2} + \frac {45 B a^{2} b^{4} d e^{2}}{8} + \frac {9 B a b^{5} d^{2} e}{4} + \frac {B b^{6} d^{3}}{8}\right ) + x^{7} \cdot \left (\frac {20 A a^{3} b^{3} e^{3}}{7} + \frac {45 A a^{2} b^{4} d e^{2}}{7} + \frac {18 A a b^{5} d^{2} e}{7} + \frac {A b^{6} d^{3}}{7} + \frac {15 B a^{4} b^{2} e^{3}}{7} + \frac {60 B a^{3} b^{3} d e^{2}}{7} + \frac {45 B a^{2} b^{4} d^{2} e}{7} + \frac {6 B a b^{5} d^{3}}{7}\right ) + x^{6} \cdot \left (\frac {5 A a^{4} b^{2} e^{3}}{2} + 10 A a^{3} b^{3} d e^{2} + \frac {15 A a^{2} b^{4} d^{2} e}{2} + A a b^{5} d^{3} + B a^{5} b e^{3} + \frac {15 B a^{4} b^{2} d e^{2}}{2} + 10 B a^{3} b^{3} d^{2} e + \frac {5 B a^{2} b^{4} d^{3}}{2}\right ) + x^{5} \cdot \left (\frac {6 A a^{5} b e^{3}}{5} + 9 A a^{4} b^{2} d e^{2} + 12 A a^{3} b^{3} d^{2} e + 3 A a^{2} b^{4} d^{3} + \frac {B a^{6} e^{3}}{5} + \frac {18 B a^{5} b d e^{2}}{5} + 9 B a^{4} b^{2} d^{2} e + 4 B a^{3} b^{3} d^{3}\right ) + x^{4} \left (\frac {A a^{6} e^{3}}{4} + \frac {9 A a^{5} b d e^{2}}{2} + \frac {45 A a^{4} b^{2} d^{2} e}{4} + 5 A a^{3} b^{3} d^{3} + \frac {3 B a^{6} d e^{2}}{4} + \frac {9 B a^{5} b d^{2} e}{2} + \frac {15 B a^{4} b^{2} d^{3}}{4}\right ) + x^{3} \left (A a^{6} d e^{2} + 6 A a^{5} b d^{2} e + 5 A a^{4} b^{2} d^{3} + B a^{6} d^{2} e + 2 B a^{5} b d^{3}\right ) + x^{2} \cdot \left (\frac {3 A a^{6} d^{2} e}{2} + 3 A a^{5} b d^{3} + \frac {B a^{6} d^{3}}{2}\right ) \] Input:

integrate((b*x+a)**6*(B*x+A)*(e*x+d)**3,x)
 

Output:

A*a**6*d**3*x + B*b**6*e**3*x**11/11 + x**10*(A*b**6*e**3/10 + 3*B*a*b**5* 
e**3/5 + 3*B*b**6*d*e**2/10) + x**9*(2*A*a*b**5*e**3/3 + A*b**6*d*e**2/3 + 
 5*B*a**2*b**4*e**3/3 + 2*B*a*b**5*d*e**2 + B*b**6*d**2*e/3) + x**8*(15*A* 
a**2*b**4*e**3/8 + 9*A*a*b**5*d*e**2/4 + 3*A*b**6*d**2*e/8 + 5*B*a**3*b**3 
*e**3/2 + 45*B*a**2*b**4*d*e**2/8 + 9*B*a*b**5*d**2*e/4 + B*b**6*d**3/8) + 
 x**7*(20*A*a**3*b**3*e**3/7 + 45*A*a**2*b**4*d*e**2/7 + 18*A*a*b**5*d**2* 
e/7 + A*b**6*d**3/7 + 15*B*a**4*b**2*e**3/7 + 60*B*a**3*b**3*d*e**2/7 + 45 
*B*a**2*b**4*d**2*e/7 + 6*B*a*b**5*d**3/7) + x**6*(5*A*a**4*b**2*e**3/2 + 
10*A*a**3*b**3*d*e**2 + 15*A*a**2*b**4*d**2*e/2 + A*a*b**5*d**3 + B*a**5*b 
*e**3 + 15*B*a**4*b**2*d*e**2/2 + 10*B*a**3*b**3*d**2*e + 5*B*a**2*b**4*d* 
*3/2) + x**5*(6*A*a**5*b*e**3/5 + 9*A*a**4*b**2*d*e**2 + 12*A*a**3*b**3*d* 
*2*e + 3*A*a**2*b**4*d**3 + B*a**6*e**3/5 + 18*B*a**5*b*d*e**2/5 + 9*B*a** 
4*b**2*d**2*e + 4*B*a**3*b**3*d**3) + x**4*(A*a**6*e**3/4 + 9*A*a**5*b*d*e 
**2/2 + 45*A*a**4*b**2*d**2*e/4 + 5*A*a**3*b**3*d**3 + 3*B*a**6*d*e**2/4 + 
 9*B*a**5*b*d**2*e/2 + 15*B*a**4*b**2*d**3/4) + x**3*(A*a**6*d*e**2 + 6*A* 
a**5*b*d**2*e + 5*A*a**4*b**2*d**3 + B*a**6*d**2*e + 2*B*a**5*b*d**3) + x* 
*2*(3*A*a**6*d**2*e/2 + 3*A*a**5*b*d**3 + B*a**6*d**3/2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 643 vs. \(2 (149) = 298\).

Time = 0.05 (sec) , antiderivative size = 643, normalized size of antiderivative = 4.04 \[ \int (a+b x)^6 (A+B x) (d+e x)^3 \, dx=\frac {1}{11} \, B b^{6} e^{3} x^{11} + A a^{6} d^{3} x + \frac {1}{10} \, {\left (3 \, B b^{6} d e^{2} + {\left (6 \, B a b^{5} + A b^{6}\right )} e^{3}\right )} x^{10} + \frac {1}{3} \, {\left (B b^{6} d^{2} e + {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{2} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (B b^{6} d^{3} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e + 9 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{2} + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left ({\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} + 9 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e + 15 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{2} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left ({\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{2} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e + 9 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{2} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (A a^{6} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} + 9 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e + 3 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{2}\right )} x^{4} + {\left (A a^{6} d e^{2} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{3} + {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{6} d^{2} e + {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{3}\right )} x^{2} \] Input:

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^3,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 782 vs. \(2 (149) = 298\).

Time = 0.12 (sec) , antiderivative size = 782, normalized size of antiderivative = 4.92 \[ \int (a+b x)^6 (A+B x) (d+e x)^3 \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^3,x, algorithm="giac")
 

Output:

1/11*B*b^6*e^3*x^11 + 3/10*B*b^6*d*e^2*x^10 + 3/5*B*a*b^5*e^3*x^10 + 1/10* 
A*b^6*e^3*x^10 + 1/3*B*b^6*d^2*e*x^9 + 2*B*a*b^5*d*e^2*x^9 + 1/3*A*b^6*d*e 
^2*x^9 + 5/3*B*a^2*b^4*e^3*x^9 + 2/3*A*a*b^5*e^3*x^9 + 1/8*B*b^6*d^3*x^8 + 
 9/4*B*a*b^5*d^2*e*x^8 + 3/8*A*b^6*d^2*e*x^8 + 45/8*B*a^2*b^4*d*e^2*x^8 + 
9/4*A*a*b^5*d*e^2*x^8 + 5/2*B*a^3*b^3*e^3*x^8 + 15/8*A*a^2*b^4*e^3*x^8 + 6 
/7*B*a*b^5*d^3*x^7 + 1/7*A*b^6*d^3*x^7 + 45/7*B*a^2*b^4*d^2*e*x^7 + 18/7*A 
*a*b^5*d^2*e*x^7 + 60/7*B*a^3*b^3*d*e^2*x^7 + 45/7*A*a^2*b^4*d*e^2*x^7 + 1 
5/7*B*a^4*b^2*e^3*x^7 + 20/7*A*a^3*b^3*e^3*x^7 + 5/2*B*a^2*b^4*d^3*x^6 + A 
*a*b^5*d^3*x^6 + 10*B*a^3*b^3*d^2*e*x^6 + 15/2*A*a^2*b^4*d^2*e*x^6 + 15/2* 
B*a^4*b^2*d*e^2*x^6 + 10*A*a^3*b^3*d*e^2*x^6 + B*a^5*b*e^3*x^6 + 5/2*A*a^4 
*b^2*e^3*x^6 + 4*B*a^3*b^3*d^3*x^5 + 3*A*a^2*b^4*d^3*x^5 + 9*B*a^4*b^2*d^2 
*e*x^5 + 12*A*a^3*b^3*d^2*e*x^5 + 18/5*B*a^5*b*d*e^2*x^5 + 9*A*a^4*b^2*d*e 
^2*x^5 + 1/5*B*a^6*e^3*x^5 + 6/5*A*a^5*b*e^3*x^5 + 15/4*B*a^4*b^2*d^3*x^4 
+ 5*A*a^3*b^3*d^3*x^4 + 9/2*B*a^5*b*d^2*e*x^4 + 45/4*A*a^4*b^2*d^2*e*x^4 + 
 3/4*B*a^6*d*e^2*x^4 + 9/2*A*a^5*b*d*e^2*x^4 + 1/4*A*a^6*e^3*x^4 + 2*B*a^5 
*b*d^3*x^3 + 5*A*a^4*b^2*d^3*x^3 + B*a^6*d^2*e*x^3 + 6*A*a^5*b*d^2*e*x^3 + 
 A*a^6*d*e^2*x^3 + 1/2*B*a^6*d^3*x^2 + 3*A*a^5*b*d^3*x^2 + 3/2*A*a^6*d^2*e 
*x^2 + A*a^6*d^3*x
 

Mupad [B] (verification not implemented)

Time = 1.00 (sec) , antiderivative size = 649, normalized size of antiderivative = 4.08 \[ \int (a+b x)^6 (A+B x) (d+e x)^3 \, dx=x^3\,\left (B\,a^6\,d^2\,e+A\,a^6\,d\,e^2+2\,B\,a^5\,b\,d^3+6\,A\,a^5\,b\,d^2\,e+5\,A\,a^4\,b^2\,d^3\right )+x^9\,\left (\frac {5\,B\,a^2\,b^4\,e^3}{3}+2\,B\,a\,b^5\,d\,e^2+\frac {2\,A\,a\,b^5\,e^3}{3}+\frac {B\,b^6\,d^2\,e}{3}+\frac {A\,b^6\,d\,e^2}{3}\right )+x^4\,\left (\frac {3\,B\,a^6\,d\,e^2}{4}+\frac {A\,a^6\,e^3}{4}+\frac {9\,B\,a^5\,b\,d^2\,e}{2}+\frac {9\,A\,a^5\,b\,d\,e^2}{2}+\frac {15\,B\,a^4\,b^2\,d^3}{4}+\frac {45\,A\,a^4\,b^2\,d^2\,e}{4}+5\,A\,a^3\,b^3\,d^3\right )+x^8\,\left (\frac {5\,B\,a^3\,b^3\,e^3}{2}+\frac {45\,B\,a^2\,b^4\,d\,e^2}{8}+\frac {15\,A\,a^2\,b^4\,e^3}{8}+\frac {9\,B\,a\,b^5\,d^2\,e}{4}+\frac {9\,A\,a\,b^5\,d\,e^2}{4}+\frac {B\,b^6\,d^3}{8}+\frac {3\,A\,b^6\,d^2\,e}{8}\right )+x^6\,\left (B\,a^5\,b\,e^3+\frac {15\,B\,a^4\,b^2\,d\,e^2}{2}+\frac {5\,A\,a^4\,b^2\,e^3}{2}+10\,B\,a^3\,b^3\,d^2\,e+10\,A\,a^3\,b^3\,d\,e^2+\frac {5\,B\,a^2\,b^4\,d^3}{2}+\frac {15\,A\,a^2\,b^4\,d^2\,e}{2}+A\,a\,b^5\,d^3\right )+x^5\,\left (\frac {B\,a^6\,e^3}{5}+\frac {18\,B\,a^5\,b\,d\,e^2}{5}+\frac {6\,A\,a^5\,b\,e^3}{5}+9\,B\,a^4\,b^2\,d^2\,e+9\,A\,a^4\,b^2\,d\,e^2+4\,B\,a^3\,b^3\,d^3+12\,A\,a^3\,b^3\,d^2\,e+3\,A\,a^2\,b^4\,d^3\right )+x^7\,\left (\frac {15\,B\,a^4\,b^2\,e^3}{7}+\frac {60\,B\,a^3\,b^3\,d\,e^2}{7}+\frac {20\,A\,a^3\,b^3\,e^3}{7}+\frac {45\,B\,a^2\,b^4\,d^2\,e}{7}+\frac {45\,A\,a^2\,b^4\,d\,e^2}{7}+\frac {6\,B\,a\,b^5\,d^3}{7}+\frac {18\,A\,a\,b^5\,d^2\,e}{7}+\frac {A\,b^6\,d^3}{7}\right )+\frac {a^5\,d^2\,x^2\,\left (3\,A\,a\,e+6\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^5\,e^2\,x^{10}\,\left (A\,b\,e+6\,B\,a\,e+3\,B\,b\,d\right )}{10}+A\,a^6\,d^3\,x+\frac {B\,b^6\,e^3\,x^{11}}{11} \] Input:

int((A + B*x)*(a + b*x)^6*(d + e*x)^3,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.65 \[ \int (a+b x)^6 (A+B x) (d+e x)^3 \, dx=\frac {x \left (120 b^{7} e^{3} x^{10}+924 a \,b^{6} e^{3} x^{9}+396 b^{7} d \,e^{2} x^{9}+3080 a^{2} b^{5} e^{3} x^{8}+3080 a \,b^{6} d \,e^{2} x^{8}+440 b^{7} d^{2} e \,x^{8}+5775 a^{3} b^{4} e^{3} x^{7}+10395 a^{2} b^{5} d \,e^{2} x^{7}+3465 a \,b^{6} d^{2} e \,x^{7}+165 b^{7} d^{3} x^{7}+6600 a^{4} b^{3} e^{3} x^{6}+19800 a^{3} b^{4} d \,e^{2} x^{6}+11880 a^{2} b^{5} d^{2} e \,x^{6}+1320 a \,b^{6} d^{3} x^{6}+4620 a^{5} b^{2} e^{3} x^{5}+23100 a^{4} b^{3} d \,e^{2} x^{5}+23100 a^{3} b^{4} d^{2} e \,x^{5}+4620 a^{2} b^{5} d^{3} x^{5}+1848 a^{6} b \,e^{3} x^{4}+16632 a^{5} b^{2} d \,e^{2} x^{4}+27720 a^{4} b^{3} d^{2} e \,x^{4}+9240 a^{3} b^{4} d^{3} x^{4}+330 a^{7} e^{3} x^{3}+6930 a^{6} b d \,e^{2} x^{3}+20790 a^{5} b^{2} d^{2} e \,x^{3}+11550 a^{4} b^{3} d^{3} x^{3}+1320 a^{7} d \,e^{2} x^{2}+9240 a^{6} b \,d^{2} e \,x^{2}+9240 a^{5} b^{2} d^{3} x^{2}+1980 a^{7} d^{2} e x +4620 a^{6} b \,d^{3} x +1320 a^{7} d^{3}\right )}{1320} \] Input:

int((b*x+a)^6*(B*x+A)*(e*x+d)^3,x)
 

Output:

(x*(1320*a**7*d**3 + 1980*a**7*d**2*e*x + 1320*a**7*d*e**2*x**2 + 330*a**7 
*e**3*x**3 + 4620*a**6*b*d**3*x + 9240*a**6*b*d**2*e*x**2 + 6930*a**6*b*d* 
e**2*x**3 + 1848*a**6*b*e**3*x**4 + 9240*a**5*b**2*d**3*x**2 + 20790*a**5* 
b**2*d**2*e*x**3 + 16632*a**5*b**2*d*e**2*x**4 + 4620*a**5*b**2*e**3*x**5 
+ 11550*a**4*b**3*d**3*x**3 + 27720*a**4*b**3*d**2*e*x**4 + 23100*a**4*b** 
3*d*e**2*x**5 + 6600*a**4*b**3*e**3*x**6 + 9240*a**3*b**4*d**3*x**4 + 2310 
0*a**3*b**4*d**2*e*x**5 + 19800*a**3*b**4*d*e**2*x**6 + 5775*a**3*b**4*e** 
3*x**7 + 4620*a**2*b**5*d**3*x**5 + 11880*a**2*b**5*d**2*e*x**6 + 10395*a* 
*2*b**5*d*e**2*x**7 + 3080*a**2*b**5*e**3*x**8 + 1320*a*b**6*d**3*x**6 + 3 
465*a*b**6*d**2*e*x**7 + 3080*a*b**6*d*e**2*x**8 + 924*a*b**6*e**3*x**9 + 
165*b**7*d**3*x**7 + 440*b**7*d**2*e*x**8 + 396*b**7*d*e**2*x**9 + 120*b** 
7*e**3*x**10))/1320