\(\int (a+b x)^3 (A+B x) (d+e x)^5 \, dx\) [25]

Optimal result

Integrand size = 20, antiderivative size = 163 \[ \int (a+b x)^3 (A+B x) (d+e x)^5 \, dx=\frac {(b d-a e)^3 (B d-A e) (d+e x)^6}{6 e^5}-\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^7}{7 e^5}+\frac {3 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^8}{8 e^5}-\frac {b^2 (4 b B d-A b e-3 a B e) (d+e x)^9}{9 e^5}+\frac {b^3 B (d+e x)^{10}}{10 e^5} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(471\) vs. \(2(163)=326\).

Time = 0.10 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.89 \[ \int (a+b x)^3 (A+B x) (d+e x)^5 \, dx=a^3 A d^5 x+\frac {1}{2} a^2 d^4 (3 A b d+a B d+5 a A e) x^2+\frac {1}{3} a d^3 \left (a B d (3 b d+5 a e)+A \left (3 b^2 d^2+15 a b d e+10 a^2 e^2\right )\right ) x^3+\frac {1}{4} d^2 \left (a B d \left (3 b^2 d^2+15 a b d e+10 a^2 e^2\right )+A \left (b^3 d^3+15 a b^2 d^2 e+30 a^2 b d e^2+10 a^3 e^3\right )\right ) x^4+\frac {1}{5} d \left (30 a^2 b d e^2 (B d+A e)+5 a^3 e^3 (2 B d+A e)+15 a b^2 d^2 e (B d+2 A e)+b^3 d^3 (B d+5 A e)\right ) x^5+\frac {1}{6} e \left (30 a b^2 d^2 e (B d+A e)+15 a^2 b d e^2 (2 B d+A e)+a^3 e^3 (5 B d+A e)+5 b^3 d^3 (B d+2 A e)\right ) x^6+\frac {1}{7} e^2 \left (a^3 B e^3+10 b^3 d^2 (B d+A e)+15 a b^2 d e (2 B d+A e)+3 a^2 b e^2 (5 B d+A e)\right ) x^7+\frac {1}{8} b e^3 \left (3 a^2 B e^2+5 b^2 d (2 B d+A e)+3 a b e (5 B d+A e)\right ) x^8+\frac {1}{9} b^2 e^4 (5 b B d+A b e+3 a B e) x^9+\frac {1}{10} b^3 B e^5 x^{10} \] Input:

Integrate[(a + b*x)^3*(A + B*x)*(d + e*x)^5,x]
 

Output:

a^3*A*d^5*x + (a^2*d^4*(3*A*b*d + a*B*d + 5*a*A*e)*x^2)/2 + (a*d^3*(a*B*d* 
(3*b*d + 5*a*e) + A*(3*b^2*d^2 + 15*a*b*d*e + 10*a^2*e^2))*x^3)/3 + (d^2*( 
a*B*d*(3*b^2*d^2 + 15*a*b*d*e + 10*a^2*e^2) + A*(b^3*d^3 + 15*a*b^2*d^2*e 
+ 30*a^2*b*d*e^2 + 10*a^3*e^3))*x^4)/4 + (d*(30*a^2*b*d*e^2*(B*d + A*e) + 
5*a^3*e^3*(2*B*d + A*e) + 15*a*b^2*d^2*e*(B*d + 2*A*e) + b^3*d^3*(B*d + 5* 
A*e))*x^5)/5 + (e*(30*a*b^2*d^2*e*(B*d + A*e) + 15*a^2*b*d*e^2*(2*B*d + A* 
e) + a^3*e^3*(5*B*d + A*e) + 5*b^3*d^3*(B*d + 2*A*e))*x^6)/6 + (e^2*(a^3*B 
*e^3 + 10*b^3*d^2*(B*d + A*e) + 15*a*b^2*d*e*(2*B*d + A*e) + 3*a^2*b*e^2*( 
5*B*d + A*e))*x^7)/7 + (b*e^3*(3*a^2*B*e^2 + 5*b^2*d*(2*B*d + A*e) + 3*a*b 
*e*(5*B*d + A*e))*x^8)/8 + (b^2*e^4*(5*b*B*d + A*b*e + 3*a*B*e)*x^9)/9 + ( 
b^3*B*e^5*x^10)/10
 

Rubi [A] (verified)

Time = 0.63 (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.

\(\displaystyle \int (a+b x)^3 (A+B x) (d+e x)^5 \, dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*x)^3*(A + B*x)*(d + e*x)^5,x]
 

Output:

((b*d - a*e)^3*(B*d - A*e)*(d + e*x)^6)/(6*e^5) - ((b*d - a*e)^2*(4*b*B*d 
- 3*A*b*e - a*B*e)*(d + e*x)^7)/(7*e^5) + (3*b*(b*d - a*e)*(2*b*B*d - A*b* 
e - a*B*e)*(d + e*x)^8)/(8*e^5) - (b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e* 
x)^9)/(9*e^5) + (b^3*B*(d + e*x)^10)/(10*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. \(528\) vs. \(2(153)=306\).

Time = 0.18 (sec) , antiderivative size = 529, normalized size of antiderivative = 3.25

method	result	size
default	\(\frac {b^{3} B \,e^{5} x^{10}}{10}+\frac {\left (\left (b^{3} A +3 a \,b^{2} B \right ) e^{5}+5 b^{3} B d \,e^{4}\right ) x^{9}}{9}+\frac {\left (\left (3 a \,b^{2} A +3 a^{2} b B \right ) e^{5}+5 \left (b^{3} A +3 a \,b^{2} B \right ) d \,e^{4}+10 b^{3} B \,d^{2} e^{3}\right ) x^{8}}{8}+\frac {\left (\left (3 a^{2} b A +a^{3} B \right ) e^{5}+5 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d \,e^{4}+10 \left (b^{3} A +3 a \,b^{2} B \right ) d^{2} e^{3}+10 b^{3} B \,d^{3} e^{2}\right ) x^{7}}{7}+\frac {\left (a^{3} A \,e^{5}+5 \left (3 a^{2} b A +a^{3} B \right ) d \,e^{4}+10 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{2} e^{3}+10 \left (b^{3} A +3 a \,b^{2} B \right ) d^{3} e^{2}+5 b^{3} B \,d^{4} e \right ) x^{6}}{6}+\frac {\left (5 a^{3} A d \,e^{4}+10 \left (3 a^{2} b A +a^{3} B \right ) d^{2} e^{3}+10 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{3} e^{2}+5 \left (b^{3} A +3 a \,b^{2} B \right ) d^{4} e +b^{3} B \,d^{5}\right ) x^{5}}{5}+\frac {\left (10 a^{3} A \,d^{2} e^{3}+10 \left (3 a^{2} b A +a^{3} B \right ) d^{3} e^{2}+5 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{4} e +\left (b^{3} A +3 a \,b^{2} B \right ) d^{5}\right ) x^{4}}{4}+\frac {\left (10 a^{3} A \,d^{3} e^{2}+5 \left (3 a^{2} b A +a^{3} B \right ) d^{4} e +\left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{5}\right ) x^{3}}{3}+\frac {\left (5 a^{3} A \,d^{4} e +\left (3 a^{2} b A +a^{3} B \right ) d^{5}\right ) x^{2}}{2}+a^{3} A \,d^{5} x\)	\(529\)
norman	\(\frac {b^{3} B \,e^{5} x^{10}}{10}+\left (\frac {1}{9} A \,b^{3} e^{5}+\frac {1}{3} B a \,b^{2} e^{5}+\frac {5}{9} b^{3} B d \,e^{4}\right ) x^{9}+\left (\frac {3}{8} A a \,b^{2} e^{5}+\frac {5}{8} A \,b^{3} d \,e^{4}+\frac {3}{8} B \,a^{2} b \,e^{5}+\frac {15}{8} B a \,b^{2} d \,e^{4}+\frac {5}{4} b^{3} B \,d^{2} e^{3}\right ) x^{8}+\left (\frac {3}{7} A \,a^{2} b \,e^{5}+\frac {15}{7} A a \,b^{2} d \,e^{4}+\frac {10}{7} A \,b^{3} d^{2} e^{3}+\frac {1}{7} B \,a^{3} e^{5}+\frac {15}{7} B \,a^{2} b d \,e^{4}+\frac {30}{7} B a \,b^{2} d^{2} e^{3}+\frac {10}{7} b^{3} B \,d^{3} e^{2}\right ) x^{7}+\left (\frac {1}{6} a^{3} A \,e^{5}+\frac {5}{2} A \,a^{2} b d \,e^{4}+5 A a \,b^{2} d^{2} e^{3}+\frac {5}{3} A \,b^{3} d^{3} e^{2}+\frac {5}{6} B \,a^{3} d \,e^{4}+5 B \,a^{2} b \,d^{2} e^{3}+5 B a \,b^{2} d^{3} e^{2}+\frac {5}{6} b^{3} B \,d^{4} e \right ) x^{6}+\left (a^{3} A d \,e^{4}+6 A \,a^{2} b \,d^{2} e^{3}+6 A a \,b^{2} d^{3} e^{2}+A \,b^{3} d^{4} e +2 B \,a^{3} d^{2} e^{3}+6 B \,a^{2} b \,d^{3} e^{2}+3 B a \,b^{2} d^{4} e +\frac {1}{5} b^{3} B \,d^{5}\right ) x^{5}+\left (\frac {5}{2} a^{3} A \,d^{2} e^{3}+\frac {15}{2} A \,a^{2} b \,d^{3} e^{2}+\frac {15}{4} A a \,b^{2} d^{4} e +\frac {1}{4} A \,b^{3} d^{5}+\frac {5}{2} B \,a^{3} d^{3} e^{2}+\frac {15}{4} B \,a^{2} b \,d^{4} e +\frac {3}{4} B a \,b^{2} d^{5}\right ) x^{4}+\left (\frac {10}{3} a^{3} A \,d^{3} e^{2}+5 A \,a^{2} b \,d^{4} e +A a \,b^{2} d^{5}+\frac {5}{3} B \,a^{3} d^{4} e +B \,a^{2} b \,d^{5}\right ) x^{3}+\left (\frac {5}{2} a^{3} A \,d^{4} e +\frac {3}{2} A \,a^{2} b \,d^{5}+\frac {1}{2} B \,a^{3} d^{5}\right ) x^{2}+a^{3} A \,d^{5} x\)	\(561\)
gosper	\(\frac {15}{8} x^{8} B a \,b^{2} d \,e^{4}+\frac {10}{7} x^{7} A \,b^{3} d^{2} e^{3}+\frac {1}{10} b^{3} B \,e^{5} x^{10}+a^{3} A \,d^{5} x +\frac {1}{9} x^{9} A \,b^{3} e^{5}+\frac {1}{7} x^{7} B \,a^{3} e^{5}+\frac {1}{6} x^{6} a^{3} A \,e^{5}+\frac {15}{4} x^{4} A a \,b^{2} d^{4} e +\frac {15}{4} x^{4} B \,a^{2} b \,d^{4} e +6 x^{5} B \,a^{2} b \,d^{3} e^{2}+3 x^{5} B a \,b^{2} d^{4} e +\frac {15}{2} x^{4} A \,a^{2} b \,d^{3} e^{2}+6 x^{5} A a \,b^{2} d^{3} e^{2}+\frac {1}{5} x^{5} b^{3} B \,d^{5}+\frac {1}{4} x^{4} A \,b^{3} d^{5}+\frac {1}{2} x^{2} B \,a^{3} d^{5}+\frac {10}{7} x^{7} b^{3} B \,d^{3} e^{2}+\frac {5}{3} x^{6} A \,b^{3} d^{3} e^{2}+\frac {5}{6} x^{6} B \,a^{3} d \,e^{4}+\frac {5}{6} x^{6} b^{3} B \,d^{4} e +x^{5} a^{3} A d \,e^{4}+x^{5} A \,b^{3} d^{4} e +5 x^{6} A a \,b^{2} d^{2} e^{3}+6 x^{5} A \,a^{2} b \,d^{2} e^{3}+\frac {3}{4} x^{4} B a \,b^{2} d^{5}+\frac {10}{3} x^{3} a^{3} A \,d^{3} e^{2}+x^{3} A a \,b^{2} d^{5}+\frac {5}{3} x^{3} B \,a^{3} d^{4} e +x^{3} B \,a^{2} b \,d^{5}+\frac {5}{2} x^{2} a^{3} A \,d^{4} e +\frac {3}{2} x^{2} A \,a^{2} b \,d^{5}+\frac {5}{2} x^{6} A \,a^{2} b d \,e^{4}+\frac {15}{7} x^{7} A a \,b^{2} d \,e^{4}+\frac {15}{7} x^{7} B \,a^{2} b d \,e^{4}+\frac {30}{7} x^{7} B a \,b^{2} d^{2} e^{3}+5 x^{6} B \,a^{2} b \,d^{2} e^{3}+5 x^{6} B a \,b^{2} d^{3} e^{2}+\frac {5}{4} x^{8} b^{3} B \,d^{2} e^{3}+\frac {1}{3} x^{9} B a \,b^{2} e^{5}+\frac {5}{9} x^{9} b^{3} B d \,e^{4}+\frac {3}{8} x^{8} A a \,b^{2} e^{5}+\frac {5}{8} x^{8} A \,b^{3} d \,e^{4}+\frac {3}{8} x^{8} B \,a^{2} b \,e^{5}+2 x^{5} B \,a^{3} d^{2} e^{3}+\frac {5}{2} x^{4} a^{3} A \,d^{2} e^{3}+\frac {5}{2} x^{4} B \,a^{3} d^{3} e^{2}+\frac {3}{7} x^{7} A \,a^{2} b \,e^{5}+5 x^{3} A \,a^{2} b \,d^{4} e\)	\(659\)
risch	\(\frac {15}{8} x^{8} B a \,b^{2} d \,e^{4}+\frac {10}{7} x^{7} A \,b^{3} d^{2} e^{3}+\frac {1}{10} b^{3} B \,e^{5} x^{10}+a^{3} A \,d^{5} x +\frac {1}{9} x^{9} A \,b^{3} e^{5}+\frac {1}{7} x^{7} B \,a^{3} e^{5}+\frac {1}{6} x^{6} a^{3} A \,e^{5}+\frac {15}{4} x^{4} A a \,b^{2} d^{4} e +\frac {15}{4} x^{4} B \,a^{2} b \,d^{4} e +6 x^{5} B \,a^{2} b \,d^{3} e^{2}+3 x^{5} B a \,b^{2} d^{4} e +\frac {15}{2} x^{4} A \,a^{2} b \,d^{3} e^{2}+6 x^{5} A a \,b^{2} d^{3} e^{2}+\frac {1}{5} x^{5} b^{3} B \,d^{5}+\frac {1}{4} x^{4} A \,b^{3} d^{5}+\frac {1}{2} x^{2} B \,a^{3} d^{5}+\frac {10}{7} x^{7} b^{3} B \,d^{3} e^{2}+\frac {5}{3} x^{6} A \,b^{3} d^{3} e^{2}+\frac {5}{6} x^{6} B \,a^{3} d \,e^{4}+\frac {5}{6} x^{6} b^{3} B \,d^{4} e +x^{5} a^{3} A d \,e^{4}+x^{5} A \,b^{3} d^{4} e +5 x^{6} A a \,b^{2} d^{2} e^{3}+6 x^{5} A \,a^{2} b \,d^{2} e^{3}+\frac {3}{4} x^{4} B a \,b^{2} d^{5}+\frac {10}{3} x^{3} a^{3} A \,d^{3} e^{2}+x^{3} A a \,b^{2} d^{5}+\frac {5}{3} x^{3} B \,a^{3} d^{4} e +x^{3} B \,a^{2} b \,d^{5}+\frac {5}{2} x^{2} a^{3} A \,d^{4} e +\frac {3}{2} x^{2} A \,a^{2} b \,d^{5}+\frac {5}{2} x^{6} A \,a^{2} b d \,e^{4}+\frac {15}{7} x^{7} A a \,b^{2} d \,e^{4}+\frac {15}{7} x^{7} B \,a^{2} b d \,e^{4}+\frac {30}{7} x^{7} B a \,b^{2} d^{2} e^{3}+5 x^{6} B \,a^{2} b \,d^{2} e^{3}+5 x^{6} B a \,b^{2} d^{3} e^{2}+\frac {5}{4} x^{8} b^{3} B \,d^{2} e^{3}+\frac {1}{3} x^{9} B a \,b^{2} e^{5}+\frac {5}{9} x^{9} b^{3} B d \,e^{4}+\frac {3}{8} x^{8} A a \,b^{2} e^{5}+\frac {5}{8} x^{8} A \,b^{3} d \,e^{4}+\frac {3}{8} x^{8} B \,a^{2} b \,e^{5}+2 x^{5} B \,a^{3} d^{2} e^{3}+\frac {5}{2} x^{4} a^{3} A \,d^{2} e^{3}+\frac {5}{2} x^{4} B \,a^{3} d^{3} e^{2}+\frac {3}{7} x^{7} A \,a^{2} b \,e^{5}+5 x^{3} A \,a^{2} b \,d^{4} e\)	\(659\)
parallelrisch	\(\frac {15}{8} x^{8} B a \,b^{2} d \,e^{4}+\frac {10}{7} x^{7} A \,b^{3} d^{2} e^{3}+\frac {1}{10} b^{3} B \,e^{5} x^{10}+a^{3} A \,d^{5} x +\frac {1}{9} x^{9} A \,b^{3} e^{5}+\frac {1}{7} x^{7} B \,a^{3} e^{5}+\frac {1}{6} x^{6} a^{3} A \,e^{5}+\frac {15}{4} x^{4} A a \,b^{2} d^{4} e +\frac {15}{4} x^{4} B \,a^{2} b \,d^{4} e +6 x^{5} B \,a^{2} b \,d^{3} e^{2}+3 x^{5} B a \,b^{2} d^{4} e +\frac {15}{2} x^{4} A \,a^{2} b \,d^{3} e^{2}+6 x^{5} A a \,b^{2} d^{3} e^{2}+\frac {1}{5} x^{5} b^{3} B \,d^{5}+\frac {1}{4} x^{4} A \,b^{3} d^{5}+\frac {1}{2} x^{2} B \,a^{3} d^{5}+\frac {10}{7} x^{7} b^{3} B \,d^{3} e^{2}+\frac {5}{3} x^{6} A \,b^{3} d^{3} e^{2}+\frac {5}{6} x^{6} B \,a^{3} d \,e^{4}+\frac {5}{6} x^{6} b^{3} B \,d^{4} e +x^{5} a^{3} A d \,e^{4}+x^{5} A \,b^{3} d^{4} e +5 x^{6} A a \,b^{2} d^{2} e^{3}+6 x^{5} A \,a^{2} b \,d^{2} e^{3}+\frac {3}{4} x^{4} B a \,b^{2} d^{5}+\frac {10}{3} x^{3} a^{3} A \,d^{3} e^{2}+x^{3} A a \,b^{2} d^{5}+\frac {5}{3} x^{3} B \,a^{3} d^{4} e +x^{3} B \,a^{2} b \,d^{5}+\frac {5}{2} x^{2} a^{3} A \,d^{4} e +\frac {3}{2} x^{2} A \,a^{2} b \,d^{5}+\frac {5}{2} x^{6} A \,a^{2} b d \,e^{4}+\frac {15}{7} x^{7} A a \,b^{2} d \,e^{4}+\frac {15}{7} x^{7} B \,a^{2} b d \,e^{4}+\frac {30}{7} x^{7} B a \,b^{2} d^{2} e^{3}+5 x^{6} B \,a^{2} b \,d^{2} e^{3}+5 x^{6} B a \,b^{2} d^{3} e^{2}+\frac {5}{4} x^{8} b^{3} B \,d^{2} e^{3}+\frac {1}{3} x^{9} B a \,b^{2} e^{5}+\frac {5}{9} x^{9} b^{3} B d \,e^{4}+\frac {3}{8} x^{8} A a \,b^{2} e^{5}+\frac {5}{8} x^{8} A \,b^{3} d \,e^{4}+\frac {3}{8} x^{8} B \,a^{2} b \,e^{5}+2 x^{5} B \,a^{3} d^{2} e^{3}+\frac {5}{2} x^{4} a^{3} A \,d^{2} e^{3}+\frac {5}{2} x^{4} B \,a^{3} d^{3} e^{2}+\frac {3}{7} x^{7} A \,a^{2} b \,e^{5}+5 x^{3} A \,a^{2} b \,d^{4} e\)	\(659\)
orering	\(\frac {x \left (252 b^{3} B \,e^{5} x^{9}+280 A \,b^{3} e^{5} x^{8}+840 B a \,b^{2} e^{5} x^{8}+1400 B \,b^{3} d \,e^{4} x^{8}+945 A a \,b^{2} e^{5} x^{7}+1575 A \,b^{3} d \,e^{4} x^{7}+945 B \,a^{2} b \,e^{5} x^{7}+4725 B a \,b^{2} d \,e^{4} x^{7}+3150 B \,b^{3} d^{2} e^{3} x^{7}+1080 A \,a^{2} b \,e^{5} x^{6}+5400 A a \,b^{2} d \,e^{4} x^{6}+3600 A \,b^{3} d^{2} e^{3} x^{6}+360 B \,a^{3} e^{5} x^{6}+5400 B \,a^{2} b d \,e^{4} x^{6}+10800 B a \,b^{2} d^{2} e^{3} x^{6}+3600 B \,b^{3} d^{3} e^{2} x^{6}+420 A \,a^{3} e^{5} x^{5}+6300 A \,a^{2} b d \,e^{4} x^{5}+12600 A a \,b^{2} d^{2} e^{3} x^{5}+4200 A \,b^{3} d^{3} e^{2} x^{5}+2100 B \,a^{3} d \,e^{4} x^{5}+12600 B \,a^{2} b \,d^{2} e^{3} x^{5}+12600 B a \,b^{2} d^{3} e^{2} x^{5}+2100 B \,b^{3} d^{4} e \,x^{5}+2520 A \,a^{3} d \,e^{4} x^{4}+15120 A \,a^{2} b \,d^{2} e^{3} x^{4}+15120 A a \,b^{2} d^{3} e^{2} x^{4}+2520 A \,b^{3} d^{4} e \,x^{4}+5040 B \,a^{3} d^{2} e^{3} x^{4}+15120 B \,a^{2} b \,d^{3} e^{2} x^{4}+7560 B a \,b^{2} d^{4} e \,x^{4}+504 B \,b^{3} d^{5} x^{4}+6300 A \,a^{3} d^{2} e^{3} x^{3}+18900 A \,a^{2} b \,d^{3} e^{2} x^{3}+9450 A a \,b^{2} d^{4} e \,x^{3}+630 A \,b^{3} d^{5} x^{3}+6300 B \,a^{3} d^{3} e^{2} x^{3}+9450 B \,a^{2} b \,d^{4} e \,x^{3}+1890 B a \,b^{2} d^{5} x^{3}+8400 A \,a^{3} d^{3} e^{2} x^{2}+12600 A \,a^{2} b \,d^{4} e \,x^{2}+2520 A a \,b^{2} d^{5} x^{2}+4200 B \,a^{3} d^{4} e \,x^{2}+2520 B \,a^{2} b \,d^{5} x^{2}+6300 A \,a^{3} d^{4} e x +3780 A \,a^{2} b \,d^{5} x +1260 B \,a^{3} d^{5} x +2520 a^{3} A \,d^{5}\right )}{2520}\)	\(660\)

Input:

int((b*x+a)^3*(B*x+A)*(e*x+d)^5,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (153) = 306\).

Time = 0.06 (sec) , antiderivative size = 518, normalized size of antiderivative = 3.18 \[ \int (a+b x)^3 (A+B x) (d+e x)^5 \, dx=\frac {1}{10} \, B b^{3} e^{5} x^{10} + A a^{3} d^{5} x + \frac {1}{9} \, {\left (5 \, B b^{3} d e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{5}\right )} x^{9} + \frac {1}{8} \, {\left (10 \, B b^{3} d^{2} e^{3} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{4} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, B b^{3} d^{3} e^{2} + 10 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{3} + 15 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{4} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (5 \, B b^{3} d^{4} e + A a^{3} e^{5} + 10 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{2} + 30 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{3} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{5} + 5 \, A a^{3} d e^{4} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e + 30 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{2} + 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (10 \, A a^{3} d^{2} e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} + 15 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} e + 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, A a^{3} d^{3} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{5} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e\right )} x^{3} + \frac {1}{2} \, {\left (5 \, A a^{3} d^{4} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )} x^{2} \] Input:

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^5,x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 678 vs. \(2 (155) = 310\).

Time = 0.07 (sec) , antiderivative size = 678, normalized size of antiderivative = 4.16 \[ \int (a+b x)^3 (A+B x) (d+e x)^5 \, dx=A a^{3} d^{5} x + \frac {B b^{3} e^{5} x^{10}}{10} + x^{9} \left (\frac {A b^{3} e^{5}}{9} + \frac {B a b^{2} e^{5}}{3} + \frac {5 B b^{3} d e^{4}}{9}\right ) + x^{8} \cdot \left (\frac {3 A a b^{2} e^{5}}{8} + \frac {5 A b^{3} d e^{4}}{8} + \frac {3 B a^{2} b e^{5}}{8} + \frac {15 B a b^{2} d e^{4}}{8} + \frac {5 B b^{3} d^{2} e^{3}}{4}\right ) + x^{7} \cdot \left (\frac {3 A a^{2} b e^{5}}{7} + \frac {15 A a b^{2} d e^{4}}{7} + \frac {10 A b^{3} d^{2} e^{3}}{7} + \frac {B a^{3} e^{5}}{7} + \frac {15 B a^{2} b d e^{4}}{7} + \frac {30 B a b^{2} d^{2} e^{3}}{7} + \frac {10 B b^{3} d^{3} e^{2}}{7}\right ) + x^{6} \left (\frac {A a^{3} e^{5}}{6} + \frac {5 A a^{2} b d e^{4}}{2} + 5 A a b^{2} d^{2} e^{3} + \frac {5 A b^{3} d^{3} e^{2}}{3} + \frac {5 B a^{3} d e^{4}}{6} + 5 B a^{2} b d^{2} e^{3} + 5 B a b^{2} d^{3} e^{2} + \frac {5 B b^{3} d^{4} e}{6}\right ) + x^{5} \left (A a^{3} d e^{4} + 6 A a^{2} b d^{2} e^{3} + 6 A a b^{2} d^{3} e^{2} + A b^{3} d^{4} e + 2 B a^{3} d^{2} e^{3} + 6 B a^{2} b d^{3} e^{2} + 3 B a b^{2} d^{4} e + \frac {B b^{3} d^{5}}{5}\right ) + x^{4} \cdot \left (\frac {5 A a^{3} d^{2} e^{3}}{2} + \frac {15 A a^{2} b d^{3} e^{2}}{2} + \frac {15 A a b^{2} d^{4} e}{4} + \frac {A b^{3} d^{5}}{4} + \frac {5 B a^{3} d^{3} e^{2}}{2} + \frac {15 B a^{2} b d^{4} e}{4} + \frac {3 B a b^{2} d^{5}}{4}\right ) + x^{3} \cdot \left (\frac {10 A a^{3} d^{3} e^{2}}{3} + 5 A a^{2} b d^{4} e + A a b^{2} d^{5} + \frac {5 B a^{3} d^{4} e}{3} + B a^{2} b d^{5}\right ) + x^{2} \cdot \left (\frac {5 A a^{3} d^{4} e}{2} + \frac {3 A a^{2} b d^{5}}{2} + \frac {B a^{3} d^{5}}{2}\right ) \] Input:

integrate((b*x+a)**3*(B*x+A)*(e*x+d)**5,x)
 

Output:

A*a**3*d**5*x + B*b**3*e**5*x**10/10 + x**9*(A*b**3*e**5/9 + B*a*b**2*e**5 
/3 + 5*B*b**3*d*e**4/9) + x**8*(3*A*a*b**2*e**5/8 + 5*A*b**3*d*e**4/8 + 3* 
B*a**2*b*e**5/8 + 15*B*a*b**2*d*e**4/8 + 5*B*b**3*d**2*e**3/4) + x**7*(3*A 
*a**2*b*e**5/7 + 15*A*a*b**2*d*e**4/7 + 10*A*b**3*d**2*e**3/7 + B*a**3*e** 
5/7 + 15*B*a**2*b*d*e**4/7 + 30*B*a*b**2*d**2*e**3/7 + 10*B*b**3*d**3*e**2 
/7) + x**6*(A*a**3*e**5/6 + 5*A*a**2*b*d*e**4/2 + 5*A*a*b**2*d**2*e**3 + 5 
*A*b**3*d**3*e**2/3 + 5*B*a**3*d*e**4/6 + 5*B*a**2*b*d**2*e**3 + 5*B*a*b** 
2*d**3*e**2 + 5*B*b**3*d**4*e/6) + x**5*(A*a**3*d*e**4 + 6*A*a**2*b*d**2*e 
**3 + 6*A*a*b**2*d**3*e**2 + A*b**3*d**4*e + 2*B*a**3*d**2*e**3 + 6*B*a**2 
*b*d**3*e**2 + 3*B*a*b**2*d**4*e + B*b**3*d**5/5) + x**4*(5*A*a**3*d**2*e* 
*3/2 + 15*A*a**2*b*d**3*e**2/2 + 15*A*a*b**2*d**4*e/4 + A*b**3*d**5/4 + 5* 
B*a**3*d**3*e**2/2 + 15*B*a**2*b*d**4*e/4 + 3*B*a*b**2*d**5/4) + x**3*(10* 
A*a**3*d**3*e**2/3 + 5*A*a**2*b*d**4*e + A*a*b**2*d**5 + 5*B*a**3*d**4*e/3 
 + B*a**2*b*d**5) + x**2*(5*A*a**3*d**4*e/2 + 3*A*a**2*b*d**5/2 + B*a**3*d 
**5/2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (153) = 306\).

Time = 0.04 (sec) , antiderivative size = 518, normalized size of antiderivative = 3.18 \[ \int (a+b x)^3 (A+B x) (d+e x)^5 \, dx=\frac {1}{10} \, B b^{3} e^{5} x^{10} + A a^{3} d^{5} x + \frac {1}{9} \, {\left (5 \, B b^{3} d e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{5}\right )} x^{9} + \frac {1}{8} \, {\left (10 \, B b^{3} d^{2} e^{3} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{4} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, B b^{3} d^{3} e^{2} + 10 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{3} + 15 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{4} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (5 \, B b^{3} d^{4} e + A a^{3} e^{5} + 10 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{2} + 30 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{3} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{5} + 5 \, A a^{3} d e^{4} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e + 30 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{2} + 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (10 \, A a^{3} d^{2} e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} + 15 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} e + 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, A a^{3} d^{3} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{5} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e\right )} x^{3} + \frac {1}{2} \, {\left (5 \, A a^{3} d^{4} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )} x^{2} \] Input:

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^5,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 658 vs. \(2 (153) = 306\).

Time = 0.12 (sec) , antiderivative size = 658, normalized size of antiderivative = 4.04 \[ \int (a+b x)^3 (A+B x) (d+e x)^5 \, dx=\frac {1}{10} \, B b^{3} e^{5} x^{10} + \frac {5}{9} \, B b^{3} d e^{4} x^{9} + \frac {1}{3} \, B a b^{2} e^{5} x^{9} + \frac {1}{9} \, A b^{3} e^{5} x^{9} + \frac {5}{4} \, B b^{3} d^{2} e^{3} x^{8} + \frac {15}{8} \, B a b^{2} d e^{4} x^{8} + \frac {5}{8} \, A b^{3} d e^{4} x^{8} + \frac {3}{8} \, B a^{2} b e^{5} x^{8} + \frac {3}{8} \, A a b^{2} e^{5} x^{8} + \frac {10}{7} \, B b^{3} d^{3} e^{2} x^{7} + \frac {30}{7} \, B a b^{2} d^{2} e^{3} x^{7} + \frac {10}{7} \, A b^{3} d^{2} e^{3} x^{7} + \frac {15}{7} \, B a^{2} b d e^{4} x^{7} + \frac {15}{7} \, A a b^{2} d e^{4} x^{7} + \frac {1}{7} \, B a^{3} e^{5} x^{7} + \frac {3}{7} \, A a^{2} b e^{5} x^{7} + \frac {5}{6} \, B b^{3} d^{4} e x^{6} + 5 \, B a b^{2} d^{3} e^{2} x^{6} + \frac {5}{3} \, A b^{3} d^{3} e^{2} x^{6} + 5 \, B a^{2} b d^{2} e^{3} x^{6} + 5 \, A a b^{2} d^{2} e^{3} x^{6} + \frac {5}{6} \, B a^{3} d e^{4} x^{6} + \frac {5}{2} \, A a^{2} b d e^{4} x^{6} + \frac {1}{6} \, A a^{3} e^{5} x^{6} + \frac {1}{5} \, B b^{3} d^{5} x^{5} + 3 \, B a b^{2} d^{4} e x^{5} + A b^{3} d^{4} e x^{5} + 6 \, B a^{2} b d^{3} e^{2} x^{5} + 6 \, A a b^{2} d^{3} e^{2} x^{5} + 2 \, B a^{3} d^{2} e^{3} x^{5} + 6 \, A a^{2} b d^{2} e^{3} x^{5} + A a^{3} d e^{4} x^{5} + \frac {3}{4} \, B a b^{2} d^{5} x^{4} + \frac {1}{4} \, A b^{3} d^{5} x^{4} + \frac {15}{4} \, B a^{2} b d^{4} e x^{4} + \frac {15}{4} \, A a b^{2} d^{4} e x^{4} + \frac {5}{2} \, B a^{3} d^{3} e^{2} x^{4} + \frac {15}{2} \, A a^{2} b d^{3} e^{2} x^{4} + \frac {5}{2} \, A a^{3} d^{2} e^{3} x^{4} + B a^{2} b d^{5} x^{3} + A a b^{2} d^{5} x^{3} + \frac {5}{3} \, B a^{3} d^{4} e x^{3} + 5 \, A a^{2} b d^{4} e x^{3} + \frac {10}{3} \, A a^{3} d^{3} e^{2} x^{3} + \frac {1}{2} \, B a^{3} d^{5} x^{2} + \frac {3}{2} \, A a^{2} b d^{5} x^{2} + \frac {5}{2} \, A a^{3} d^{4} e x^{2} + A a^{3} d^{5} x \] Input:

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^5,x, algorithm="giac")
 

Output:

1/10*B*b^3*e^5*x^10 + 5/9*B*b^3*d*e^4*x^9 + 1/3*B*a*b^2*e^5*x^9 + 1/9*A*b^ 
3*e^5*x^9 + 5/4*B*b^3*d^2*e^3*x^8 + 15/8*B*a*b^2*d*e^4*x^8 + 5/8*A*b^3*d*e 
^4*x^8 + 3/8*B*a^2*b*e^5*x^8 + 3/8*A*a*b^2*e^5*x^8 + 10/7*B*b^3*d^3*e^2*x^ 
7 + 30/7*B*a*b^2*d^2*e^3*x^7 + 10/7*A*b^3*d^2*e^3*x^7 + 15/7*B*a^2*b*d*e^4 
*x^7 + 15/7*A*a*b^2*d*e^4*x^7 + 1/7*B*a^3*e^5*x^7 + 3/7*A*a^2*b*e^5*x^7 + 
5/6*B*b^3*d^4*e*x^6 + 5*B*a*b^2*d^3*e^2*x^6 + 5/3*A*b^3*d^3*e^2*x^6 + 5*B* 
a^2*b*d^2*e^3*x^6 + 5*A*a*b^2*d^2*e^3*x^6 + 5/6*B*a^3*d*e^4*x^6 + 5/2*A*a^ 
2*b*d*e^4*x^6 + 1/6*A*a^3*e^5*x^6 + 1/5*B*b^3*d^5*x^5 + 3*B*a*b^2*d^4*e*x^ 
5 + A*b^3*d^4*e*x^5 + 6*B*a^2*b*d^3*e^2*x^5 + 6*A*a*b^2*d^3*e^2*x^5 + 2*B* 
a^3*d^2*e^3*x^5 + 6*A*a^2*b*d^2*e^3*x^5 + A*a^3*d*e^4*x^5 + 3/4*B*a*b^2*d^ 
5*x^4 + 1/4*A*b^3*d^5*x^4 + 15/4*B*a^2*b*d^4*e*x^4 + 15/4*A*a*b^2*d^4*e*x^ 
4 + 5/2*B*a^3*d^3*e^2*x^4 + 15/2*A*a^2*b*d^3*e^2*x^4 + 5/2*A*a^3*d^2*e^3*x 
^4 + B*a^2*b*d^5*x^3 + A*a*b^2*d^5*x^3 + 5/3*B*a^3*d^4*e*x^3 + 5*A*a^2*b*d 
^4*e*x^3 + 10/3*A*a^3*d^3*e^2*x^3 + 1/2*B*a^3*d^5*x^2 + 3/2*A*a^2*b*d^5*x^ 
2 + 5/2*A*a^3*d^4*e*x^2 + A*a^3*d^5*x
 

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 544, normalized size of antiderivative = 3.34 \[ \int (a+b x)^3 (A+B x) (d+e x)^5 \, dx=x^3\,\left (\frac {5\,B\,a^3\,d^4\,e}{3}+\frac {10\,A\,a^3\,d^3\,e^2}{3}+B\,a^2\,b\,d^5+5\,A\,a^2\,b\,d^4\,e+A\,a\,b^2\,d^5\right )+x^8\,\left (\frac {3\,B\,a^2\,b\,e^5}{8}+\frac {15\,B\,a\,b^2\,d\,e^4}{8}+\frac {3\,A\,a\,b^2\,e^5}{8}+\frac {5\,B\,b^3\,d^2\,e^3}{4}+\frac {5\,A\,b^3\,d\,e^4}{8}\right )+x^4\,\left (\frac {5\,B\,a^3\,d^3\,e^2}{2}+\frac {5\,A\,a^3\,d^2\,e^3}{2}+\frac {15\,B\,a^2\,b\,d^4\,e}{4}+\frac {15\,A\,a^2\,b\,d^3\,e^2}{2}+\frac {3\,B\,a\,b^2\,d^5}{4}+\frac {15\,A\,a\,b^2\,d^4\,e}{4}+\frac {A\,b^3\,d^5}{4}\right )+x^7\,\left (\frac {B\,a^3\,e^5}{7}+\frac {15\,B\,a^2\,b\,d\,e^4}{7}+\frac {3\,A\,a^2\,b\,e^5}{7}+\frac {30\,B\,a\,b^2\,d^2\,e^3}{7}+\frac {15\,A\,a\,b^2\,d\,e^4}{7}+\frac {10\,B\,b^3\,d^3\,e^2}{7}+\frac {10\,A\,b^3\,d^2\,e^3}{7}\right )+x^5\,\left (2\,B\,a^3\,d^2\,e^3+A\,a^3\,d\,e^4+6\,B\,a^2\,b\,d^3\,e^2+6\,A\,a^2\,b\,d^2\,e^3+3\,B\,a\,b^2\,d^4\,e+6\,A\,a\,b^2\,d^3\,e^2+\frac {B\,b^3\,d^5}{5}+A\,b^3\,d^4\,e\right )+x^6\,\left (\frac {5\,B\,a^3\,d\,e^4}{6}+\frac {A\,a^3\,e^5}{6}+5\,B\,a^2\,b\,d^2\,e^3+\frac {5\,A\,a^2\,b\,d\,e^4}{2}+5\,B\,a\,b^2\,d^3\,e^2+5\,A\,a\,b^2\,d^2\,e^3+\frac {5\,B\,b^3\,d^4\,e}{6}+\frac {5\,A\,b^3\,d^3\,e^2}{3}\right )+\frac {a^2\,d^4\,x^2\,\left (5\,A\,a\,e+3\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^2\,e^4\,x^9\,\left (A\,b\,e+3\,B\,a\,e+5\,B\,b\,d\right )}{9}+A\,a^3\,d^5\,x+\frac {B\,b^3\,e^5\,x^{10}}{10} \] Input:

int((A + B*x)*(a + b*x)^3*(d + e*x)^5,x)
 

Output:

x^3*(A*a*b^2*d^5 + B*a^2*b*d^5 + (5*B*a^3*d^4*e)/3 + (10*A*a^3*d^3*e^2)/3 
+ 5*A*a^2*b*d^4*e) + x^8*((3*A*a*b^2*e^5)/8 + (3*B*a^2*b*e^5)/8 + (5*A*b^3 
*d*e^4)/8 + (5*B*b^3*d^2*e^3)/4 + (15*B*a*b^2*d*e^4)/8) + x^4*((A*b^3*d^5) 
/4 + (3*B*a*b^2*d^5)/4 + (5*A*a^3*d^2*e^3)/2 + (5*B*a^3*d^3*e^2)/2 + (15*A 
*a^2*b*d^3*e^2)/2 + (15*A*a*b^2*d^4*e)/4 + (15*B*a^2*b*d^4*e)/4) + x^7*((B 
*a^3*e^5)/7 + (3*A*a^2*b*e^5)/7 + (10*A*b^3*d^2*e^3)/7 + (10*B*b^3*d^3*e^2 
)/7 + (30*B*a*b^2*d^2*e^3)/7 + (15*A*a*b^2*d*e^4)/7 + (15*B*a^2*b*d*e^4)/7 
) + x^5*((B*b^3*d^5)/5 + A*a^3*d*e^4 + A*b^3*d^4*e + 2*B*a^3*d^2*e^3 + 6*A 
*a*b^2*d^3*e^2 + 6*A*a^2*b*d^2*e^3 + 6*B*a^2*b*d^3*e^2 + 3*B*a*b^2*d^4*e) 
+ x^6*((A*a^3*e^5)/6 + (5*B*a^3*d*e^4)/6 + (5*B*b^3*d^4*e)/6 + (5*A*b^3*d^ 
3*e^2)/3 + 5*A*a*b^2*d^2*e^3 + 5*B*a*b^2*d^3*e^2 + 5*B*a^2*b*d^2*e^3 + (5* 
A*a^2*b*d*e^4)/2) + (a^2*d^4*x^2*(5*A*a*e + 3*A*b*d + B*a*d))/2 + (b^2*e^4 
*x^9*(A*b*e + 3*B*a*e + 5*B*b*d))/9 + A*a^3*d^5*x + (B*b^3*e^5*x^10)/10
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.44 \[ \int (a+b x)^3 (A+B x) (d+e x)^5 \, dx=\frac {x \left (126 b^{4} e^{5} x^{9}+560 a \,b^{3} e^{5} x^{8}+700 b^{4} d \,e^{4} x^{8}+945 a^{2} b^{2} e^{5} x^{7}+3150 a \,b^{3} d \,e^{4} x^{7}+1575 b^{4} d^{2} e^{3} x^{7}+720 a^{3} b \,e^{5} x^{6}+5400 a^{2} b^{2} d \,e^{4} x^{6}+7200 a \,b^{3} d^{2} e^{3} x^{6}+1800 b^{4} d^{3} e^{2} x^{6}+210 a^{4} e^{5} x^{5}+4200 a^{3} b d \,e^{4} x^{5}+12600 a^{2} b^{2} d^{2} e^{3} x^{5}+8400 a \,b^{3} d^{3} e^{2} x^{5}+1050 b^{4} d^{4} e \,x^{5}+1260 a^{4} d \,e^{4} x^{4}+10080 a^{3} b \,d^{2} e^{3} x^{4}+15120 a^{2} b^{2} d^{3} e^{2} x^{4}+5040 a \,b^{3} d^{4} e \,x^{4}+252 b^{4} d^{5} x^{4}+3150 a^{4} d^{2} e^{3} x^{3}+12600 a^{3} b \,d^{3} e^{2} x^{3}+9450 a^{2} b^{2} d^{4} e \,x^{3}+1260 a \,b^{3} d^{5} x^{3}+4200 a^{4} d^{3} e^{2} x^{2}+8400 a^{3} b \,d^{4} e \,x^{2}+2520 a^{2} b^{2} d^{5} x^{2}+3150 a^{4} d^{4} e x +2520 a^{3} b \,d^{5} x +1260 a^{4} d^{5}\right )}{1260} \] Input:

int((b*x+a)^3*(B*x+A)*(e*x+d)^5,x)
 

Output:

(x*(1260*a**4*d**5 + 3150*a**4*d**4*e*x + 4200*a**4*d**3*e**2*x**2 + 3150* 
a**4*d**2*e**3*x**3 + 1260*a**4*d*e**4*x**4 + 210*a**4*e**5*x**5 + 2520*a* 
*3*b*d**5*x + 8400*a**3*b*d**4*e*x**2 + 12600*a**3*b*d**3*e**2*x**3 + 1008 
0*a**3*b*d**2*e**3*x**4 + 4200*a**3*b*d*e**4*x**5 + 720*a**3*b*e**5*x**6 + 
 2520*a**2*b**2*d**5*x**2 + 9450*a**2*b**2*d**4*e*x**3 + 15120*a**2*b**2*d 
**3*e**2*x**4 + 12600*a**2*b**2*d**2*e**3*x**5 + 5400*a**2*b**2*d*e**4*x** 
6 + 945*a**2*b**2*e**5*x**7 + 1260*a*b**3*d**5*x**3 + 5040*a*b**3*d**4*e*x 
**4 + 8400*a*b**3*d**3*e**2*x**5 + 7200*a*b**3*d**2*e**3*x**6 + 3150*a*b** 
3*d*e**4*x**7 + 560*a*b**3*e**5*x**8 + 252*b**4*d**5*x**4 + 1050*b**4*d**4 
*e*x**5 + 1800*b**4*d**3*e**2*x**6 + 1575*b**4*d**2*e**3*x**7 + 700*b**4*d 
*e**4*x**8 + 126*b**4*e**5*x**9))/1260