\(\int (a+b x)^6 (A+B x) (d+e x)^8 \, dx\) [41]

Optimal result

Integrand size = 20, antiderivative size = 292 \[ \int (a+b x)^6 (A+B x) (d+e x)^8 \, dx=-\frac {(b d-a e)^6 (B d-A e) (d+e x)^9}{9 e^8}+\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e) (d+e x)^{10}}{10 e^8}-\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) (d+e x)^{11}}{11 e^8}+\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) (d+e x)^{12}}{12 e^8}-\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^{13}}{13 e^8}+\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^{14}}{14 e^8}-\frac {b^5 (7 b B d-A b e-6 a B e) (d+e x)^{15}}{15 e^8}+\frac {b^6 B (d+e x)^{16}}{16 e^8} \] Output:

-1/9*(-a*e+b*d)^6*(-A*e+B*d)*(e*x+d)^9/e^8+1/10*(-a*e+b*d)^5*(-6*A*b*e-B*a 
*e+7*B*b*d)*(e*x+d)^10/e^8-3/11*b*(-a*e+b*d)^4*(-5*A*b*e-2*B*a*e+7*B*b*d)* 
(e*x+d)^11/e^8+5/12*b^2*(-a*e+b*d)^3*(-4*A*b*e-3*B*a*e+7*B*b*d)*(e*x+d)^12 
/e^8-5/13*b^3*(-a*e+b*d)^2*(-3*A*b*e-4*B*a*e+7*B*b*d)*(e*x+d)^13/e^8+3/14* 
b^4*(-a*e+b*d)*(-2*A*b*e-5*B*a*e+7*B*b*d)*(e*x+d)^14/e^8-1/15*b^5*(-A*b*e- 
6*B*a*e+7*B*b*d)*(e*x+d)^15/e^8+1/16*b^6*B*(e*x+d)^16/e^8
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1385\) vs. \(2(292)=584\).

Time = 0.33 (sec) , antiderivative size = 1385, normalized size of antiderivative = 4.74 \[ \int (a+b x)^6 (A+B x) (d+e x)^8 \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*x)^6*(A + B*x)*(d + e*x)^8,x]
 

Output:

a^6*A*d^8*x + (a^5*d^7*(6*A*b*d + a*B*d + 8*a*A*e)*x^2)/2 + (a^4*d^6*(2*a* 
B*d*(3*b*d + 4*a*e) + A*(15*b^2*d^2 + 48*a*b*d*e + 28*a^2*e^2))*x^3)/3 + ( 
a^3*d^5*(a*B*d*(15*b^2*d^2 + 48*a*b*d*e + 28*a^2*e^2) + 4*A*(5*b^3*d^3 + 3 
0*a*b^2*d^2*e + 42*a^2*b*d*e^2 + 14*a^3*e^3))*x^4)/4 + (a^2*d^4*(4*a*B*d*( 
5*b^3*d^3 + 30*a*b^2*d^2*e + 42*a^2*b*d*e^2 + 14*a^3*e^3) + A*(15*b^4*d^4 
+ 160*a*b^3*d^3*e + 420*a^2*b^2*d^2*e^2 + 336*a^3*b*d*e^3 + 70*a^4*e^4))*x 
^5)/5 + (a*d^3*(a*B*d*(15*b^4*d^4 + 160*a*b^3*d^3*e + 420*a^2*b^2*d^2*e^2 
+ 336*a^3*b*d*e^3 + 70*a^4*e^4) + 2*A*(3*b^5*d^5 + 60*a*b^4*d^4*e + 280*a^ 
2*b^3*d^3*e^2 + 420*a^3*b^2*d^2*e^3 + 210*a^4*b*d*e^4 + 28*a^5*e^5))*x^6)/ 
6 + (d^2*(2*a*B*d*(3*b^5*d^5 + 60*a*b^4*d^4*e + 280*a^2*b^3*d^3*e^2 + 420* 
a^3*b^2*d^2*e^3 + 210*a^4*b*d*e^4 + 28*a^5*e^5) + A*(b^6*d^6 + 48*a*b^5*d^ 
5*e + 420*a^2*b^4*d^4*e^2 + 1120*a^3*b^3*d^3*e^3 + 1050*a^4*b^2*d^2*e^4 + 
336*a^5*b*d*e^5 + 28*a^6*e^6))*x^7)/7 + (d*(168*a^5*b*d*e^5*(2*B*d + A*e) 
+ 420*a^2*b^4*d^4*e^2*(B*d + 2*A*e) + 4*a^6*e^6*(7*B*d + 2*A*e) + 210*a^4* 
b^2*d^2*e^4*(5*B*d + 4*A*e) + 280*a^3*b^3*d^3*e^3*(4*B*d + 5*A*e) + 24*a*b 
^5*d^5*e*(2*B*d + 7*A*e) + b^6*d^6*(B*d + 8*A*e))*x^8)/8 + (e*(420*a^4*b^2 
*d^2*e^4*(2*B*d + A*e) + a^6*e^6*(8*B*d + A*e) + 168*a*b^5*d^5*e*(B*d + 2* 
A*e) + 24*a^5*b*d*e^5*(7*B*d + 2*A*e) + 280*a^3*b^3*d^3*e^3*(5*B*d + 4*A*e 
) + 210*a^2*b^4*d^4*e^2*(4*B*d + 5*A*e) + 4*b^6*d^6*(2*B*d + 7*A*e))*x^9)/ 
9 + (e^2*(a^6*B*e^6 + 560*a^3*b^3*d^2*e^3*(2*B*d + A*e) + 6*a^5*b*e^5*(...
 

Rubi [A] (verified)

Time = 1.71 (sec) , antiderivative size = 292, 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.

\(\displaystyle \int (a+b x)^6 (A+B x) (d+e x)^8 \, dx\)

\(\Big \downarrow \) 86

\(\displaystyle \int \left (\frac {b^5 (d+e x)^{14} (6 a B e+A b e-7 b B d)}{e^7}-\frac {3 b^4 (d+e x)^{13} (b d-a e) (5 a B e+2 A b e-7 b B d)}{e^7}+\frac {5 b^3 (d+e x)^{12} (b d-a e)^2 (4 a B e+3 A b e-7 b B d)}{e^7}-\frac {5 b^2 (d+e x)^{11} (b d-a e)^3 (3 a B e+4 A b e-7 b B d)}{e^7}+\frac {3 b (d+e x)^{10} (b d-a e)^4 (2 a B e+5 A b e-7 b B d)}{e^7}+\frac {(d+e x)^9 (a e-b d)^5 (a B e+6 A b e-7 b B d)}{e^7}+\frac {(d+e x)^8 (a e-b d)^6 (A e-B d)}{e^7}+\frac {b^6 B (d+e x)^{15}}{e^7}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^5 (d+e x)^{15} (-6 a B e-A b e+7 b B d)}{15 e^8}+\frac {3 b^4 (d+e x)^{14} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{14 e^8}-\frac {5 b^3 (d+e x)^{13} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{13 e^8}+\frac {5 b^2 (d+e x)^{12} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{12 e^8}-\frac {3 b (d+e x)^{11} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{11 e^8}+\frac {(d+e x)^{10} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{10 e^8}-\frac {(d+e x)^9 (b d-a e)^6 (B d-A e)}{9 e^8}+\frac {b^6 B (d+e x)^{16}}{16 e^8}\)

Input:

Int[(a + b*x)^6*(A + B*x)*(d + e*x)^8,x]
 

Output:

-1/9*((b*d - a*e)^6*(B*d - A*e)*(d + e*x)^9)/e^8 + ((b*d - a*e)^5*(7*b*B*d 
 - 6*A*b*e - a*B*e)*(d + e*x)^10)/(10*e^8) - (3*b*(b*d - a*e)^4*(7*b*B*d - 
 5*A*b*e - 2*a*B*e)*(d + e*x)^11)/(11*e^8) + (5*b^2*(b*d - a*e)^3*(7*b*B*d 
 - 4*A*b*e - 3*a*B*e)*(d + e*x)^12)/(12*e^8) - (5*b^3*(b*d - a*e)^2*(7*b*B 
*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^13)/(13*e^8) + (3*b^4*(b*d - a*e)*(7*b*B 
*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^14)/(14*e^8) - (b^5*(7*b*B*d - A*b*e - 6 
*a*B*e)*(d + e*x)^15)/(15*e^8) + (b^6*B*(d + e*x)^16)/(16*e^8)
 

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. \(1524\) vs. \(2(276)=552\).

Time = 0.23 (sec) , antiderivative size = 1525, normalized size of antiderivative = 5.22

method	result	size
default	\(\text {Expression too large to display}\)	\(1525\)
norman	\(\text {Expression too large to display}\)	\(1642\)
gosper	\(\text {Expression too large to display}\)	\(1944\)
risch	\(\text {Expression too large to display}\)	\(1944\)
parallelrisch	\(\text {Expression too large to display}\)	\(1944\)
orering	\(\text {Expression too large to display}\)	\(1944\)

Input:

int((b*x+a)^6*(B*x+A)*(e*x+d)^8,x,method=_RETURNVERBOSE)
 

Output:

1/16*b^6*B*e^8*x^16+1/15*((A*b^6+6*B*a*b^5)*e^8+8*b^6*B*d*e^7)*x^15+1/14*( 
(6*A*a*b^5+15*B*a^2*b^4)*e^8+8*(A*b^6+6*B*a*b^5)*d*e^7+28*b^6*B*d^2*e^6)*x 
^14+1/13*((15*A*a^2*b^4+20*B*a^3*b^3)*e^8+8*(6*A*a*b^5+15*B*a^2*b^4)*d*e^7 
+28*(A*b^6+6*B*a*b^5)*d^2*e^6+56*b^6*B*d^3*e^5)*x^13+1/12*((20*A*a^3*b^3+1 
5*B*a^4*b^2)*e^8+8*(15*A*a^2*b^4+20*B*a^3*b^3)*d*e^7+28*(6*A*a*b^5+15*B*a^ 
2*b^4)*d^2*e^6+56*(A*b^6+6*B*a*b^5)*d^3*e^5+70*b^6*B*d^4*e^4)*x^12+1/11*(( 
15*A*a^4*b^2+6*B*a^5*b)*e^8+8*(20*A*a^3*b^3+15*B*a^4*b^2)*d*e^7+28*(15*A*a 
^2*b^4+20*B*a^3*b^3)*d^2*e^6+56*(6*A*a*b^5+15*B*a^2*b^4)*d^3*e^5+70*(A*b^6 
+6*B*a*b^5)*d^4*e^4+56*b^6*B*d^5*e^3)*x^11+1/10*((6*A*a^5*b+B*a^6)*e^8+8*( 
15*A*a^4*b^2+6*B*a^5*b)*d*e^7+28*(20*A*a^3*b^3+15*B*a^4*b^2)*d^2*e^6+56*(1 
5*A*a^2*b^4+20*B*a^3*b^3)*d^3*e^5+70*(6*A*a*b^5+15*B*a^2*b^4)*d^4*e^4+56*( 
A*b^6+6*B*a*b^5)*d^5*e^3+28*b^6*B*d^6*e^2)*x^10+1/9*(a^6*A*e^8+8*(6*A*a^5* 
b+B*a^6)*d*e^7+28*(15*A*a^4*b^2+6*B*a^5*b)*d^2*e^6+56*(20*A*a^3*b^3+15*B*a 
^4*b^2)*d^3*e^5+70*(15*A*a^2*b^4+20*B*a^3*b^3)*d^4*e^4+56*(6*A*a*b^5+15*B* 
a^2*b^4)*d^5*e^3+28*(A*b^6+6*B*a*b^5)*d^6*e^2+8*b^6*B*d^7*e)*x^9+1/8*(8*a^ 
6*A*d*e^7+28*(6*A*a^5*b+B*a^6)*d^2*e^6+56*(15*A*a^4*b^2+6*B*a^5*b)*d^3*e^5 
+70*(20*A*a^3*b^3+15*B*a^4*b^2)*d^4*e^4+56*(15*A*a^2*b^4+20*B*a^3*b^3)*d^5 
*e^3+28*(6*A*a*b^5+15*B*a^2*b^4)*d^6*e^2+8*(A*b^6+6*B*a*b^5)*d^7*e+b^6*B*d 
^8)*x^8+1/7*(28*a^6*A*d^2*e^6+56*(6*A*a^5*b+B*a^6)*d^3*e^5+70*(15*A*a^4*b^ 
2+6*B*a^5*b)*d^4*e^4+56*(20*A*a^3*b^3+15*B*a^4*b^2)*d^5*e^3+28*(15*A*a^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1532 vs. \(2 (276) = 552\).

Time = 0.08 (sec) , antiderivative size = 1532, normalized size of antiderivative = 5.25 \[ \int (a+b x)^6 (A+B x) (d+e x)^8 \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^8,x, algorithm="fricas")
 

Output:

1/16*B*b^6*e^8*x^16 + A*a^6*d^8*x + 1/15*(8*B*b^6*d*e^7 + (6*B*a*b^5 + A*b 
^6)*e^8)*x^15 + 1/14*(28*B*b^6*d^2*e^6 + 8*(6*B*a*b^5 + A*b^6)*d*e^7 + 3*( 
5*B*a^2*b^4 + 2*A*a*b^5)*e^8)*x^14 + 1/13*(56*B*b^6*d^3*e^5 + 28*(6*B*a*b^ 
5 + A*b^6)*d^2*e^6 + 24*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^7 + 5*(4*B*a^3*b^3 + 
 3*A*a^2*b^4)*e^8)*x^13 + 1/12*(70*B*b^6*d^4*e^4 + 56*(6*B*a*b^5 + A*b^6)* 
d^3*e^5 + 84*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^6 + 40*(4*B*a^3*b^3 + 3*A*a^2 
*b^4)*d*e^7 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^8)*x^12 + 1/11*(56*B*b^6*d^5 
*e^3 + 70*(6*B*a*b^5 + A*b^6)*d^4*e^4 + 168*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3* 
e^5 + 140*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^6 + 40*(3*B*a^4*b^2 + 4*A*a^3* 
b^3)*d*e^7 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^8)*x^11 + 1/10*(28*B*b^6*d^6*e^ 
2 + 56*(6*B*a*b^5 + A*b^6)*d^5*e^3 + 210*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^4 
 + 280*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^5 + 140*(3*B*a^4*b^2 + 4*A*a^3*b^ 
3)*d^2*e^6 + 24*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^7 + (B*a^6 + 6*A*a^5*b)*e^8) 
*x^10 + 1/9*(8*B*b^6*d^7*e + A*a^6*e^8 + 28*(6*B*a*b^5 + A*b^6)*d^6*e^2 + 
168*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^3 + 350*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^ 
4*e^4 + 280*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^5 + 84*(2*B*a^5*b + 5*A*a^4* 
b^2)*d^2*e^6 + 8*(B*a^6 + 6*A*a^5*b)*d*e^7)*x^9 + 1/8*(B*b^6*d^8 + 8*A*a^6 
*d*e^7 + 8*(6*B*a*b^5 + A*b^6)*d^7*e + 84*(5*B*a^2*b^4 + 2*A*a*b^5)*d^6*e^ 
2 + 280*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^5*e^3 + 350*(3*B*a^4*b^2 + 4*A*a^3*b 
^3)*d^4*e^4 + 168*(2*B*a^5*b + 5*A*a^4*b^2)*d^3*e^5 + 28*(B*a^6 + 6*A*a...
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1969 vs. \(2 (296) = 592\).

Time = 0.12 (sec) , antiderivative size = 1969, normalized size of antiderivative = 6.74 \[ \int (a+b x)^6 (A+B x) (d+e x)^8 \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)**6*(B*x+A)*(e*x+d)**8,x)
 

Output:

A*a**6*d**8*x + B*b**6*e**8*x**16/16 + x**15*(A*b**6*e**8/15 + 2*B*a*b**5* 
e**8/5 + 8*B*b**6*d*e**7/15) + x**14*(3*A*a*b**5*e**8/7 + 4*A*b**6*d*e**7/ 
7 + 15*B*a**2*b**4*e**8/14 + 24*B*a*b**5*d*e**7/7 + 2*B*b**6*d**2*e**6) + 
x**13*(15*A*a**2*b**4*e**8/13 + 48*A*a*b**5*d*e**7/13 + 28*A*b**6*d**2*e** 
6/13 + 20*B*a**3*b**3*e**8/13 + 120*B*a**2*b**4*d*e**7/13 + 168*B*a*b**5*d 
**2*e**6/13 + 56*B*b**6*d**3*e**5/13) + x**12*(5*A*a**3*b**3*e**8/3 + 10*A 
*a**2*b**4*d*e**7 + 14*A*a*b**5*d**2*e**6 + 14*A*b**6*d**3*e**5/3 + 5*B*a* 
*4*b**2*e**8/4 + 40*B*a**3*b**3*d*e**7/3 + 35*B*a**2*b**4*d**2*e**6 + 28*B 
*a*b**5*d**3*e**5 + 35*B*b**6*d**4*e**4/6) + x**11*(15*A*a**4*b**2*e**8/11 
 + 160*A*a**3*b**3*d*e**7/11 + 420*A*a**2*b**4*d**2*e**6/11 + 336*A*a*b**5 
*d**3*e**5/11 + 70*A*b**6*d**4*e**4/11 + 6*B*a**5*b*e**8/11 + 120*B*a**4*b 
**2*d*e**7/11 + 560*B*a**3*b**3*d**2*e**6/11 + 840*B*a**2*b**4*d**3*e**5/1 
1 + 420*B*a*b**5*d**4*e**4/11 + 56*B*b**6*d**5*e**3/11) + x**10*(3*A*a**5* 
b*e**8/5 + 12*A*a**4*b**2*d*e**7 + 56*A*a**3*b**3*d**2*e**6 + 84*A*a**2*b* 
*4*d**3*e**5 + 42*A*a*b**5*d**4*e**4 + 28*A*b**6*d**5*e**3/5 + B*a**6*e**8 
/10 + 24*B*a**5*b*d*e**7/5 + 42*B*a**4*b**2*d**2*e**6 + 112*B*a**3*b**3*d* 
*3*e**5 + 105*B*a**2*b**4*d**4*e**4 + 168*B*a*b**5*d**5*e**3/5 + 14*B*b**6 
*d**6*e**2/5) + x**9*(A*a**6*e**8/9 + 16*A*a**5*b*d*e**7/3 + 140*A*a**4*b* 
*2*d**2*e**6/3 + 1120*A*a**3*b**3*d**3*e**5/9 + 350*A*a**2*b**4*d**4*e**4/ 
3 + 112*A*a*b**5*d**5*e**3/3 + 28*A*b**6*d**6*e**2/9 + 8*B*a**6*d*e**7/...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1532 vs. \(2 (276) = 552\).

Time = 0.05 (sec) , antiderivative size = 1532, normalized size of antiderivative = 5.25 \[ \int (a+b x)^6 (A+B x) (d+e x)^8 \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^8,x, algorithm="maxima")
 

Output:

1/16*B*b^6*e^8*x^16 + A*a^6*d^8*x + 1/15*(8*B*b^6*d*e^7 + (6*B*a*b^5 + A*b 
^6)*e^8)*x^15 + 1/14*(28*B*b^6*d^2*e^6 + 8*(6*B*a*b^5 + A*b^6)*d*e^7 + 3*( 
5*B*a^2*b^4 + 2*A*a*b^5)*e^8)*x^14 + 1/13*(56*B*b^6*d^3*e^5 + 28*(6*B*a*b^ 
5 + A*b^6)*d^2*e^6 + 24*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^7 + 5*(4*B*a^3*b^3 + 
 3*A*a^2*b^4)*e^8)*x^13 + 1/12*(70*B*b^6*d^4*e^4 + 56*(6*B*a*b^5 + A*b^6)* 
d^3*e^5 + 84*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^6 + 40*(4*B*a^3*b^3 + 3*A*a^2 
*b^4)*d*e^7 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^8)*x^12 + 1/11*(56*B*b^6*d^5 
*e^3 + 70*(6*B*a*b^5 + A*b^6)*d^4*e^4 + 168*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3* 
e^5 + 140*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^6 + 40*(3*B*a^4*b^2 + 4*A*a^3* 
b^3)*d*e^7 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^8)*x^11 + 1/10*(28*B*b^6*d^6*e^ 
2 + 56*(6*B*a*b^5 + A*b^6)*d^5*e^3 + 210*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^4 
 + 280*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^5 + 140*(3*B*a^4*b^2 + 4*A*a^3*b^ 
3)*d^2*e^6 + 24*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^7 + (B*a^6 + 6*A*a^5*b)*e^8) 
*x^10 + 1/9*(8*B*b^6*d^7*e + A*a^6*e^8 + 28*(6*B*a*b^5 + A*b^6)*d^6*e^2 + 
168*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^3 + 350*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^ 
4*e^4 + 280*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^5 + 84*(2*B*a^5*b + 5*A*a^4* 
b^2)*d^2*e^6 + 8*(B*a^6 + 6*A*a^5*b)*d*e^7)*x^9 + 1/8*(B*b^6*d^8 + 8*A*a^6 
*d*e^7 + 8*(6*B*a*b^5 + A*b^6)*d^7*e + 84*(5*B*a^2*b^4 + 2*A*a*b^5)*d^6*e^ 
2 + 280*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^5*e^3 + 350*(3*B*a^4*b^2 + 4*A*a^3*b 
^3)*d^4*e^4 + 168*(2*B*a^5*b + 5*A*a^4*b^2)*d^3*e^5 + 28*(B*a^6 + 6*A*a...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1943 vs. \(2 (276) = 552\).

Time = 0.13 (sec) , antiderivative size = 1943, normalized size of antiderivative = 6.65 \[ \int (a+b x)^6 (A+B x) (d+e x)^8 \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^8,x, algorithm="giac")
 

Output:

1/16*B*b^6*e^8*x^16 + 8/15*B*b^6*d*e^7*x^15 + 2/5*B*a*b^5*e^8*x^15 + 1/15* 
A*b^6*e^8*x^15 + 2*B*b^6*d^2*e^6*x^14 + 24/7*B*a*b^5*d*e^7*x^14 + 4/7*A*b^ 
6*d*e^7*x^14 + 15/14*B*a^2*b^4*e^8*x^14 + 3/7*A*a*b^5*e^8*x^14 + 56/13*B*b 
^6*d^3*e^5*x^13 + 168/13*B*a*b^5*d^2*e^6*x^13 + 28/13*A*b^6*d^2*e^6*x^13 + 
 120/13*B*a^2*b^4*d*e^7*x^13 + 48/13*A*a*b^5*d*e^7*x^13 + 20/13*B*a^3*b^3* 
e^8*x^13 + 15/13*A*a^2*b^4*e^8*x^13 + 35/6*B*b^6*d^4*e^4*x^12 + 28*B*a*b^5 
*d^3*e^5*x^12 + 14/3*A*b^6*d^3*e^5*x^12 + 35*B*a^2*b^4*d^2*e^6*x^12 + 14*A 
*a*b^5*d^2*e^6*x^12 + 40/3*B*a^3*b^3*d*e^7*x^12 + 10*A*a^2*b^4*d*e^7*x^12 
+ 5/4*B*a^4*b^2*e^8*x^12 + 5/3*A*a^3*b^3*e^8*x^12 + 56/11*B*b^6*d^5*e^3*x^ 
11 + 420/11*B*a*b^5*d^4*e^4*x^11 + 70/11*A*b^6*d^4*e^4*x^11 + 840/11*B*a^2 
*b^4*d^3*e^5*x^11 + 336/11*A*a*b^5*d^3*e^5*x^11 + 560/11*B*a^3*b^3*d^2*e^6 
*x^11 + 420/11*A*a^2*b^4*d^2*e^6*x^11 + 120/11*B*a^4*b^2*d*e^7*x^11 + 160/ 
11*A*a^3*b^3*d*e^7*x^11 + 6/11*B*a^5*b*e^8*x^11 + 15/11*A*a^4*b^2*e^8*x^11 
 + 14/5*B*b^6*d^6*e^2*x^10 + 168/5*B*a*b^5*d^5*e^3*x^10 + 28/5*A*b^6*d^5*e 
^3*x^10 + 105*B*a^2*b^4*d^4*e^4*x^10 + 42*A*a*b^5*d^4*e^4*x^10 + 112*B*a^3 
*b^3*d^3*e^5*x^10 + 84*A*a^2*b^4*d^3*e^5*x^10 + 42*B*a^4*b^2*d^2*e^6*x^10 
+ 56*A*a^3*b^3*d^2*e^6*x^10 + 24/5*B*a^5*b*d*e^7*x^10 + 12*A*a^4*b^2*d*e^7 
*x^10 + 1/10*B*a^6*e^8*x^10 + 3/5*A*a^5*b*e^8*x^10 + 8/9*B*b^6*d^7*e*x^9 + 
 56/3*B*a*b^5*d^6*e^2*x^9 + 28/9*A*b^6*d^6*e^2*x^9 + 280/3*B*a^2*b^4*d^5*e 
^3*x^9 + 112/3*A*a*b^5*d^5*e^3*x^9 + 1400/9*B*a^3*b^3*d^4*e^4*x^9 + 350...
 

Mupad [B] (verification not implemented)

Time = 1.20 (sec) , antiderivative size = 1625, normalized size of antiderivative = 5.57 \[ \int (a+b x)^6 (A+B x) (d+e x)^8 \, dx=\text {Too large to display} \] Input:

int((A + B*x)*(a + b*x)^6*(d + e*x)^8,x)
 

Output:

x^6*(A*a*b^5*d^8 + (5*B*a^2*b^4*d^8)/2 + (28*A*a^6*d^3*e^5)/3 + (35*B*a^6* 
d^4*e^4)/3 + 20*A*a^2*b^4*d^7*e + 70*A*a^5*b*d^4*e^4 + (80*B*a^3*b^3*d^7*e 
)/3 + 56*B*a^5*b*d^5*e^3 + (280*A*a^3*b^3*d^6*e^2)/3 + 140*A*a^4*b^2*d^5*e 
^3 + 70*B*a^4*b^2*d^6*e^2) + x^11*((6*B*a^5*b*e^8)/11 + (15*A*a^4*b^2*e^8) 
/11 + (70*A*b^6*d^4*e^4)/11 + (56*B*b^6*d^5*e^3)/11 + (336*A*a*b^5*d^3*e^5 
)/11 + (160*A*a^3*b^3*d*e^7)/11 + (420*B*a*b^5*d^4*e^4)/11 + (120*B*a^4*b^ 
2*d*e^7)/11 + (420*A*a^2*b^4*d^2*e^6)/11 + (840*B*a^2*b^4*d^3*e^5)/11 + (5 
60*B*a^3*b^3*d^2*e^6)/11) + x^5*(3*A*a^2*b^4*d^8 + 4*B*a^3*b^3*d^8 + 14*A* 
a^6*d^4*e^4 + (56*B*a^6*d^5*e^3)/5 + 32*A*a^3*b^3*d^7*e + (336*A*a^5*b*d^5 
*e^3)/5 + 24*B*a^4*b^2*d^7*e + (168*B*a^5*b*d^6*e^2)/5 + 84*A*a^4*b^2*d^6* 
e^2) + x^12*((5*A*a^3*b^3*e^8)/3 + (5*B*a^4*b^2*e^8)/4 + (14*A*b^6*d^3*e^5 
)/3 + (35*B*b^6*d^4*e^4)/6 + 14*A*a*b^5*d^2*e^6 + 10*A*a^2*b^4*d*e^7 + 28* 
B*a*b^5*d^3*e^5 + (40*B*a^3*b^3*d*e^7)/3 + 35*B*a^2*b^4*d^2*e^6) + x^7*((A 
*b^6*d^8)/7 + (6*B*a*b^5*d^8)/7 + 4*A*a^6*d^2*e^6 + 8*B*a^6*d^3*e^5 + 48*A 
*a^5*b*d^3*e^5 + (120*B*a^2*b^4*d^7*e)/7 + 60*B*a^5*b*d^4*e^4 + 60*A*a^2*b 
^4*d^6*e^2 + 160*A*a^3*b^3*d^5*e^3 + 150*A*a^4*b^2*d^4*e^4 + 80*B*a^3*b^3* 
d^6*e^2 + 120*B*a^4*b^2*d^5*e^3 + (48*A*a*b^5*d^7*e)/7) + x^10*((B*a^6*e^8 
)/10 + (3*A*a^5*b*e^8)/5 + (28*A*b^6*d^5*e^3)/5 + (14*B*b^6*d^6*e^2)/5 + 4 
2*A*a*b^5*d^4*e^4 + 12*A*a^4*b^2*d*e^7 + (168*B*a*b^5*d^5*e^3)/5 + 84*A*a^ 
2*b^4*d^3*e^5 + 56*A*a^3*b^3*d^2*e^6 + 105*B*a^2*b^4*d^4*e^4 + 112*B*a^...
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 1051, normalized size of antiderivative = 3.60 \[ \int (a+b x)^6 (A+B x) (d+e x)^8 \, dx =\text {Too large to display} \] Input:

int((b*x+a)^6*(B*x+A)*(e*x+d)^8,x)
 

Output:

(x*(102960*a**7*d**8 + 411840*a**7*d**7*e*x + 960960*a**7*d**6*e**2*x**2 + 
 1441440*a**7*d**5*e**3*x**3 + 1441440*a**7*d**4*e**4*x**4 + 960960*a**7*d 
**3*e**5*x**5 + 411840*a**7*d**2*e**6*x**6 + 102960*a**7*d*e**7*x**7 + 114 
40*a**7*e**8*x**8 + 360360*a**6*b*d**8*x + 1921920*a**6*b*d**7*e*x**2 + 50 
45040*a**6*b*d**6*e**2*x**3 + 8072064*a**6*b*d**5*e**3*x**4 + 8408400*a**6 
*b*d**4*e**4*x**5 + 5765760*a**6*b*d**3*e**5*x**6 + 2522520*a**6*b*d**2*e* 
*6*x**7 + 640640*a**6*b*d*e**7*x**8 + 72072*a**6*b*e**8*x**9 + 720720*a**5 
*b**2*d**8*x**2 + 4324320*a**5*b**2*d**7*e*x**3 + 12108096*a**5*b**2*d**6* 
e**2*x**4 + 20180160*a**5*b**2*d**5*e**3*x**5 + 21621600*a**5*b**2*d**4*e* 
*4*x**6 + 15135120*a**5*b**2*d**3*e**5*x**7 + 6726720*a**5*b**2*d**2*e**6* 
x**8 + 1729728*a**5*b**2*d*e**7*x**9 + 196560*a**5*b**2*e**8*x**10 + 90090 
0*a**4*b**3*d**8*x**3 + 5765760*a**4*b**3*d**7*e*x**4 + 16816800*a**4*b**3 
*d**6*e**2*x**5 + 28828800*a**4*b**3*d**5*e**3*x**6 + 31531500*a**4*b**3*d 
**4*e**4*x**7 + 22422400*a**4*b**3*d**3*e**5*x**8 + 10090080*a**4*b**3*d** 
2*e**6*x**9 + 2620800*a**4*b**3*d*e**7*x**10 + 300300*a**4*b**3*e**8*x**11 
 + 720720*a**3*b**4*d**8*x**4 + 4804800*a**3*b**4*d**7*e*x**5 + 14414400*a 
**3*b**4*d**6*e**2*x**6 + 25225200*a**3*b**4*d**5*e**3*x**7 + 28028000*a** 
3*b**4*d**4*e**4*x**8 + 20180160*a**3*b**4*d**3*e**5*x**9 + 9172800*a**3*b 
**4*d**2*e**6*x**10 + 2402400*a**3*b**4*d*e**7*x**11 + 277200*a**3*b**4*e* 
*8*x**12 + 360360*a**2*b**5*d**8*x**5 + 2471040*a**2*b**5*d**7*e*x**6 +...