\(\int (a+b x)^3 (A+B x) (d+e x)^3 \, dx\) [27]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.87 \[ \int (a+b x)^3 (A+B x) (d+e x)^3 \, dx=a^3 A d^3 x+\frac {1}{2} a^2 d^2 (a B d+3 A (b d+a e)) x^2+a d \left (a B d (b d+a e)+A \left (b^2 d^2+3 a b d e+a^2 e^2\right )\right ) x^3+\frac {1}{4} \left (3 a B d \left (b^2 d^2+3 a b d e+a^2 e^2\right )+A \left (b^3 d^3+9 a b^2 d^2 e+9 a^2 b d e^2+a^3 e^3\right )\right ) x^4+\frac {1}{5} \left (a^3 B e^3+9 a b^2 d e (B d+A e)+3 a^2 b e^2 (3 B d+A e)+b^3 d^2 (B d+3 A e)\right ) x^5+\frac {1}{2} b e \left (a^2 B e^2+b^2 d (B d+A e)+a b e (3 B d+A e)\right ) x^6+\frac {1}{7} b^2 e^2 (3 b B d+A b e+3 a B e) x^7+\frac {1}{8} b^3 B e^3 x^8 \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.49 (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.

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

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((A*b - a*B)*(b*d - a*e)^3*(a + b*x)^4)/(4*b^5) + ((b*d - a*e)^2*(b*B*d + 
3*A*b*e - 4*a*B*e)*(a + b*x)^5)/(5*b^5) + (e*(b*d - a*e)*(b*B*d + A*b*e - 
2*a*B*e)*(a + b*x)^6)/(2*b^5) + (e^2*(3*b*B*d + A*b*e - 4*a*B*e)*(a + b*x) 
^7)/(7*b^5) + (B*e^3*(a + b*x)^8)/(8*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. \(338\) vs. \(2(149)=298\).

Time = 0.17 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.13

method	result	size
default	\(\frac {b^{3} B \,e^{3} x^{8}}{8}+\frac {\left (\left (b^{3} A +3 a \,b^{2} B \right ) e^{3}+3 b^{3} B d \,e^{2}\right ) x^{7}}{7}+\frac {\left (\left (3 a \,b^{2} A +3 a^{2} b B \right ) e^{3}+3 \left (b^{3} A +3 a \,b^{2} B \right ) d \,e^{2}+3 b^{3} B \,d^{2} e \right ) x^{6}}{6}+\frac {\left (\left (3 a^{2} b A +a^{3} B \right ) e^{3}+3 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d \,e^{2}+3 \left (b^{3} A +3 a \,b^{2} B \right ) d^{2} e +b^{3} B \,d^{3}\right ) x^{5}}{5}+\frac {\left (a^{3} A \,e^{3}+3 \left (3 a^{2} b A +a^{3} B \right ) d \,e^{2}+3 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{2} e +\left (b^{3} A +3 a \,b^{2} B \right ) d^{3}\right ) x^{4}}{4}+\frac {\left (3 a^{3} A d \,e^{2}+3 \left (3 a^{2} b A +a^{3} B \right ) d^{2} e +\left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{3}\right ) x^{3}}{3}+\frac {\left (3 a^{3} A \,d^{2} e +\left (3 a^{2} b A +a^{3} B \right ) d^{3}\right ) x^{2}}{2}+a^{3} A \,d^{3} x\)	\(339\)
norman	\(\frac {b^{3} B \,e^{3} x^{8}}{8}+\left (\frac {1}{7} A \,b^{3} e^{3}+\frac {3}{7} B a \,b^{2} e^{3}+\frac {3}{7} b^{3} B d \,e^{2}\right ) x^{7}+\left (\frac {1}{2} A a \,b^{2} e^{3}+\frac {1}{2} A \,b^{3} d \,e^{2}+\frac {1}{2} B \,a^{2} b \,e^{3}+\frac {3}{2} B a \,b^{2} d \,e^{2}+\frac {1}{2} b^{3} B \,d^{2} e \right ) x^{6}+\left (\frac {3}{5} A \,a^{2} b \,e^{3}+\frac {9}{5} A a \,b^{2} d \,e^{2}+\frac {3}{5} A \,b^{3} d^{2} e +\frac {1}{5} B \,a^{3} e^{3}+\frac {9}{5} B \,a^{2} b d \,e^{2}+\frac {9}{5} B a \,b^{2} d^{2} e +\frac {1}{5} b^{3} B \,d^{3}\right ) x^{5}+\left (\frac {1}{4} a^{3} A \,e^{3}+\frac {9}{4} A \,a^{2} b d \,e^{2}+\frac {9}{4} A a \,b^{2} d^{2} e +\frac {1}{4} A \,b^{3} d^{3}+\frac {3}{4} B \,a^{3} d \,e^{2}+\frac {9}{4} B \,a^{2} b \,d^{2} e +\frac {3}{4} B a \,b^{2} d^{3}\right ) x^{4}+\left (a^{3} A d \,e^{2}+3 A \,a^{2} b \,d^{2} e +A a \,b^{2} d^{3}+B \,a^{3} d^{2} e +B \,a^{2} b \,d^{3}\right ) x^{3}+\left (\frac {3}{2} a^{3} A \,d^{2} e +\frac {3}{2} A \,a^{2} b \,d^{3}+\frac {1}{2} B \,a^{3} d^{3}\right ) x^{2}+a^{3} A \,d^{3} x\)	\(351\)
gosper	\(\frac {3}{2} x^{6} B a \,b^{2} d \,e^{2}+\frac {9}{5} x^{5} A a \,b^{2} d \,e^{2}+\frac {9}{5} x^{5} B \,a^{2} b d \,e^{2}+\frac {9}{5} x^{5} B a \,b^{2} d^{2} e +\frac {9}{4} x^{4} A \,a^{2} b d \,e^{2}+\frac {9}{4} x^{4} A a \,b^{2} d^{2} e +\frac {3}{7} x^{7} B a \,b^{2} e^{3}+\frac {3}{7} x^{7} b^{3} B d \,e^{2}+\frac {1}{2} x^{6} A a \,b^{2} e^{3}+\frac {1}{2} x^{6} A \,b^{3} d \,e^{2}+\frac {1}{2} x^{6} B \,a^{2} b \,e^{3}+\frac {1}{2} x^{6} b^{3} B \,d^{2} e +\frac {3}{5} x^{5} A \,a^{2} b \,e^{3}+\frac {3}{5} x^{5} A \,b^{3} d^{2} e +\frac {3}{4} x^{4} B \,a^{3} d \,e^{2}+\frac {3}{4} x^{4} B a \,b^{2} d^{3}+\frac {3}{2} x^{2} a^{3} A \,d^{2} e +\frac {3}{2} x^{2} A \,a^{2} b \,d^{3}+A \,a^{3} d \,e^{2} x^{3}+A a \,b^{2} d^{3} x^{3}+B \,a^{3} d^{2} e \,x^{3}+B \,a^{2} b \,d^{3} x^{3}+\frac {9}{4} x^{4} B \,a^{2} b \,d^{2} e +\frac {1}{8} b^{3} B \,e^{3} x^{8}+a^{3} A \,d^{3} x +\frac {1}{7} x^{7} A \,b^{3} e^{3}+\frac {1}{5} x^{5} B \,a^{3} e^{3}+\frac {1}{5} x^{5} b^{3} B \,d^{3}+\frac {1}{4} x^{4} a^{3} A \,e^{3}+\frac {1}{4} x^{4} A \,b^{3} d^{3}+\frac {1}{2} x^{2} B \,a^{3} d^{3}+3 A \,a^{2} b \,d^{2} e \,x^{3}\)	\(411\)
risch	\(\frac {3}{2} x^{6} B a \,b^{2} d \,e^{2}+\frac {9}{5} x^{5} A a \,b^{2} d \,e^{2}+\frac {9}{5} x^{5} B \,a^{2} b d \,e^{2}+\frac {9}{5} x^{5} B a \,b^{2} d^{2} e +\frac {9}{4} x^{4} A \,a^{2} b d \,e^{2}+\frac {9}{4} x^{4} A a \,b^{2} d^{2} e +\frac {3}{7} x^{7} B a \,b^{2} e^{3}+\frac {3}{7} x^{7} b^{3} B d \,e^{2}+\frac {1}{2} x^{6} A a \,b^{2} e^{3}+\frac {1}{2} x^{6} A \,b^{3} d \,e^{2}+\frac {1}{2} x^{6} B \,a^{2} b \,e^{3}+\frac {1}{2} x^{6} b^{3} B \,d^{2} e +\frac {3}{5} x^{5} A \,a^{2} b \,e^{3}+\frac {3}{5} x^{5} A \,b^{3} d^{2} e +\frac {3}{4} x^{4} B \,a^{3} d \,e^{2}+\frac {3}{4} x^{4} B a \,b^{2} d^{3}+\frac {3}{2} x^{2} a^{3} A \,d^{2} e +\frac {3}{2} x^{2} A \,a^{2} b \,d^{3}+A \,a^{3} d \,e^{2} x^{3}+A a \,b^{2} d^{3} x^{3}+B \,a^{3} d^{2} e \,x^{3}+B \,a^{2} b \,d^{3} x^{3}+\frac {9}{4} x^{4} B \,a^{2} b \,d^{2} e +\frac {1}{8} b^{3} B \,e^{3} x^{8}+a^{3} A \,d^{3} x +\frac {1}{7} x^{7} A \,b^{3} e^{3}+\frac {1}{5} x^{5} B \,a^{3} e^{3}+\frac {1}{5} x^{5} b^{3} B \,d^{3}+\frac {1}{4} x^{4} a^{3} A \,e^{3}+\frac {1}{4} x^{4} A \,b^{3} d^{3}+\frac {1}{2} x^{2} B \,a^{3} d^{3}+3 A \,a^{2} b \,d^{2} e \,x^{3}\)	\(411\)
parallelrisch	\(\frac {3}{2} x^{6} B a \,b^{2} d \,e^{2}+\frac {9}{5} x^{5} A a \,b^{2} d \,e^{2}+\frac {9}{5} x^{5} B \,a^{2} b d \,e^{2}+\frac {9}{5} x^{5} B a \,b^{2} d^{2} e +\frac {9}{4} x^{4} A \,a^{2} b d \,e^{2}+\frac {9}{4} x^{4} A a \,b^{2} d^{2} e +\frac {3}{7} x^{7} B a \,b^{2} e^{3}+\frac {3}{7} x^{7} b^{3} B d \,e^{2}+\frac {1}{2} x^{6} A a \,b^{2} e^{3}+\frac {1}{2} x^{6} A \,b^{3} d \,e^{2}+\frac {1}{2} x^{6} B \,a^{2} b \,e^{3}+\frac {1}{2} x^{6} b^{3} B \,d^{2} e +\frac {3}{5} x^{5} A \,a^{2} b \,e^{3}+\frac {3}{5} x^{5} A \,b^{3} d^{2} e +\frac {3}{4} x^{4} B \,a^{3} d \,e^{2}+\frac {3}{4} x^{4} B a \,b^{2} d^{3}+\frac {3}{2} x^{2} a^{3} A \,d^{2} e +\frac {3}{2} x^{2} A \,a^{2} b \,d^{3}+A \,a^{3} d \,e^{2} x^{3}+A a \,b^{2} d^{3} x^{3}+B \,a^{3} d^{2} e \,x^{3}+B \,a^{2} b \,d^{3} x^{3}+\frac {9}{4} x^{4} B \,a^{2} b \,d^{2} e +\frac {1}{8} b^{3} B \,e^{3} x^{8}+a^{3} A \,d^{3} x +\frac {1}{7} x^{7} A \,b^{3} e^{3}+\frac {1}{5} x^{5} B \,a^{3} e^{3}+\frac {1}{5} x^{5} b^{3} B \,d^{3}+\frac {1}{4} x^{4} a^{3} A \,e^{3}+\frac {1}{4} x^{4} A \,b^{3} d^{3}+\frac {1}{2} x^{2} B \,a^{3} d^{3}+3 A \,a^{2} b \,d^{2} e \,x^{3}\)	\(411\)
orering	\(\frac {x \left (35 b^{3} B \,e^{3} x^{7}+40 A \,b^{3} e^{3} x^{6}+120 B a \,b^{2} e^{3} x^{6}+120 B \,b^{3} d \,e^{2} x^{6}+140 A a \,b^{2} e^{3} x^{5}+140 A \,b^{3} d \,e^{2} x^{5}+140 B \,a^{2} b \,e^{3} x^{5}+420 B a \,b^{2} d \,e^{2} x^{5}+140 B \,b^{3} d^{2} e \,x^{5}+168 A \,a^{2} b \,e^{3} x^{4}+504 A a \,b^{2} d \,e^{2} x^{4}+168 A \,b^{3} d^{2} e \,x^{4}+56 B \,a^{3} e^{3} x^{4}+504 B \,a^{2} b d \,e^{2} x^{4}+504 B a \,b^{2} d^{2} e \,x^{4}+56 B \,b^{3} d^{3} x^{4}+70 A \,a^{3} e^{3} x^{3}+630 A \,a^{2} b d \,e^{2} x^{3}+630 A a \,b^{2} d^{2} e \,x^{3}+70 A \,b^{3} d^{3} x^{3}+210 B \,a^{3} d \,e^{2} x^{3}+630 B \,a^{2} b \,d^{2} e \,x^{3}+210 B a \,b^{2} d^{3} x^{3}+280 A \,a^{3} d \,e^{2} x^{2}+840 A \,a^{2} b \,d^{2} e \,x^{2}+280 A a \,b^{2} d^{3} x^{2}+280 B \,a^{3} d^{2} e \,x^{2}+280 B \,a^{2} b \,d^{3} x^{2}+420 A \,a^{3} d^{2} e x +420 A \,a^{2} b \,d^{3} x +140 B \,a^{3} d^{3} x +280 a^{3} A \,d^{3}\right )}{280}\)	\(412\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (149) = 298\).

Time = 0.07 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.04 \[ \int (a+b x)^3 (A+B x) (d+e x)^3 \, dx=\frac {1}{8} \, B b^{3} e^{3} x^{8} + A a^{3} d^{3} x + \frac {1}{7} \, {\left (3 \, B b^{3} d e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (B b^{3} d^{2} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{2} + {\left (B a^{2} b + A a b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (A a^{3} e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{2}\right )} x^{4} + {\left (A a^{3} d e^{2} + {\left (B a^{2} b + A a b^{2}\right )} d^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{3} d^{2} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3}\right )} x^{2} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (155) = 310\).

Time = 0.05 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.65 \[ \int (a+b x)^3 (A+B x) (d+e x)^3 \, dx=A a^{3} d^{3} x + \frac {B b^{3} e^{3} x^{8}}{8} + x^{7} \left (\frac {A b^{3} e^{3}}{7} + \frac {3 B a b^{2} e^{3}}{7} + \frac {3 B b^{3} d e^{2}}{7}\right ) + x^{6} \left (\frac {A a b^{2} e^{3}}{2} + \frac {A b^{3} d e^{2}}{2} + \frac {B a^{2} b e^{3}}{2} + \frac {3 B a b^{2} d e^{2}}{2} + \frac {B b^{3} d^{2} e}{2}\right ) + x^{5} \cdot \left (\frac {3 A a^{2} b e^{3}}{5} + \frac {9 A a b^{2} d e^{2}}{5} + \frac {3 A b^{3} d^{2} e}{5} + \frac {B a^{3} e^{3}}{5} + \frac {9 B a^{2} b d e^{2}}{5} + \frac {9 B a b^{2} d^{2} e}{5} + \frac {B b^{3} d^{3}}{5}\right ) + x^{4} \left (\frac {A a^{3} e^{3}}{4} + \frac {9 A a^{2} b d e^{2}}{4} + \frac {9 A a b^{2} d^{2} e}{4} + \frac {A b^{3} d^{3}}{4} + \frac {3 B a^{3} d e^{2}}{4} + \frac {9 B a^{2} b d^{2} e}{4} + \frac {3 B a b^{2} d^{3}}{4}\right ) + x^{3} \left (A a^{3} d e^{2} + 3 A a^{2} b d^{2} e + A a b^{2} d^{3} + B a^{3} d^{2} e + B a^{2} b d^{3}\right ) + x^{2} \cdot \left (\frac {3 A a^{3} d^{2} e}{2} + \frac {3 A a^{2} b d^{3}}{2} + \frac {B a^{3} d^{3}}{2}\right ) \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (149) = 298\).

Time = 0.03 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.04 \[ \int (a+b x)^3 (A+B x) (d+e x)^3 \, dx=\frac {1}{8} \, B b^{3} e^{3} x^{8} + A a^{3} d^{3} x + \frac {1}{7} \, {\left (3 \, B b^{3} d e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (B b^{3} d^{2} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{2} + {\left (B a^{2} b + A a b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (A a^{3} e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{2}\right )} x^{4} + {\left (A a^{3} d e^{2} + {\left (B a^{2} b + A a b^{2}\right )} d^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{3} d^{2} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3}\right )} x^{2} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (149) = 298\).

Time = 0.12 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.58 \[ \int (a+b x)^3 (A+B x) (d+e x)^3 \, dx=\frac {1}{8} \, B b^{3} e^{3} x^{8} + \frac {3}{7} \, B b^{3} d e^{2} x^{7} + \frac {3}{7} \, B a b^{2} e^{3} x^{7} + \frac {1}{7} \, A b^{3} e^{3} x^{7} + \frac {1}{2} \, B b^{3} d^{2} e x^{6} + \frac {3}{2} \, B a b^{2} d e^{2} x^{6} + \frac {1}{2} \, A b^{3} d e^{2} x^{6} + \frac {1}{2} \, B a^{2} b e^{3} x^{6} + \frac {1}{2} \, A a b^{2} e^{3} x^{6} + \frac {1}{5} \, B b^{3} d^{3} x^{5} + \frac {9}{5} \, B a b^{2} d^{2} e x^{5} + \frac {3}{5} \, A b^{3} d^{2} e x^{5} + \frac {9}{5} \, B a^{2} b d e^{2} x^{5} + \frac {9}{5} \, A a b^{2} d e^{2} x^{5} + \frac {1}{5} \, B a^{3} e^{3} x^{5} + \frac {3}{5} \, A a^{2} b e^{3} x^{5} + \frac {3}{4} \, B a b^{2} d^{3} x^{4} + \frac {1}{4} \, A b^{3} d^{3} x^{4} + \frac {9}{4} \, B a^{2} b d^{2} e x^{4} + \frac {9}{4} \, A a b^{2} d^{2} e x^{4} + \frac {3}{4} \, B a^{3} d e^{2} x^{4} + \frac {9}{4} \, A a^{2} b d e^{2} x^{4} + \frac {1}{4} \, A a^{3} e^{3} x^{4} + B a^{2} b d^{3} x^{3} + A a b^{2} d^{3} x^{3} + B a^{3} d^{2} e x^{3} + 3 \, A a^{2} b d^{2} e x^{3} + A a^{3} d e^{2} x^{3} + \frac {1}{2} \, B a^{3} d^{3} x^{2} + \frac {3}{2} \, A a^{2} b d^{3} x^{2} + \frac {3}{2} \, A a^{3} d^{2} e x^{2} + A a^{3} d^{3} x \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.10 \[ \int (a+b x)^3 (A+B x) (d+e x)^3 \, dx=x^3\,\left (B\,a^3\,d^2\,e+A\,a^3\,d\,e^2+B\,a^2\,b\,d^3+3\,A\,a^2\,b\,d^2\,e+A\,a\,b^2\,d^3\right )+x^6\,\left (\frac {B\,a^2\,b\,e^3}{2}+\frac {3\,B\,a\,b^2\,d\,e^2}{2}+\frac {A\,a\,b^2\,e^3}{2}+\frac {B\,b^3\,d^2\,e}{2}+\frac {A\,b^3\,d\,e^2}{2}\right )+x^4\,\left (\frac {3\,B\,a^3\,d\,e^2}{4}+\frac {A\,a^3\,e^3}{4}+\frac {9\,B\,a^2\,b\,d^2\,e}{4}+\frac {9\,A\,a^2\,b\,d\,e^2}{4}+\frac {3\,B\,a\,b^2\,d^3}{4}+\frac {9\,A\,a\,b^2\,d^2\,e}{4}+\frac {A\,b^3\,d^3}{4}\right )+x^5\,\left (\frac {B\,a^3\,e^3}{5}+\frac {9\,B\,a^2\,b\,d\,e^2}{5}+\frac {3\,A\,a^2\,b\,e^3}{5}+\frac {9\,B\,a\,b^2\,d^2\,e}{5}+\frac {9\,A\,a\,b^2\,d\,e^2}{5}+\frac {B\,b^3\,d^3}{5}+\frac {3\,A\,b^3\,d^2\,e}{5}\right )+\frac {a^2\,d^2\,x^2\,\left (3\,A\,a\,e+3\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^2\,e^2\,x^7\,\left (A\,b\,e+3\,B\,a\,e+3\,B\,b\,d\right )}{7}+A\,a^3\,d^3\,x+\frac {B\,b^3\,e^3\,x^8}{8} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.55 \[ \int (a+b x)^3 (A+B x) (d+e x)^3 \, dx=\frac {x \left (35 b^{4} e^{3} x^{7}+160 a \,b^{3} e^{3} x^{6}+120 b^{4} d \,e^{2} x^{6}+280 a^{2} b^{2} e^{3} x^{5}+560 a \,b^{3} d \,e^{2} x^{5}+140 b^{4} d^{2} e \,x^{5}+224 a^{3} b \,e^{3} x^{4}+1008 a^{2} b^{2} d \,e^{2} x^{4}+672 a \,b^{3} d^{2} e \,x^{4}+56 b^{4} d^{3} x^{4}+70 a^{4} e^{3} x^{3}+840 a^{3} b d \,e^{2} x^{3}+1260 a^{2} b^{2} d^{2} e \,x^{3}+280 a \,b^{3} d^{3} x^{3}+280 a^{4} d \,e^{2} x^{2}+1120 a^{3} b \,d^{2} e \,x^{2}+560 a^{2} b^{2} d^{3} x^{2}+420 a^{4} d^{2} e x +560 a^{3} b \,d^{3} x +280 a^{4} d^{3}\right )}{280} \] Input:

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

Output:

(x*(280*a**4*d**3 + 420*a**4*d**2*e*x + 280*a**4*d*e**2*x**2 + 70*a**4*e** 
3*x**3 + 560*a**3*b*d**3*x + 1120*a**3*b*d**2*e*x**2 + 840*a**3*b*d*e**2*x 
**3 + 224*a**3*b*e**3*x**4 + 560*a**2*b**2*d**3*x**2 + 1260*a**2*b**2*d**2 
*e*x**3 + 1008*a**2*b**2*d*e**2*x**4 + 280*a**2*b**2*e**3*x**5 + 280*a*b** 
3*d**3*x**3 + 672*a*b**3*d**2*e*x**4 + 560*a*b**3*d*e**2*x**5 + 160*a*b**3 
*e**3*x**6 + 56*b**4*d**3*x**4 + 140*b**4*d**2*e*x**5 + 120*b**4*d*e**2*x* 
*6 + 35*b**4*e**3*x**7))/280