\(\int (a+b x)^6 (A+B x) (d+e x)^7 \, dx\) [42]

Optimal result

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1224\) vs. \(2(292)=584\).

Time = 0.29 (sec) , antiderivative size = 1224, normalized size of antiderivative = 4.19 \[ \int (a+b x)^6 (A+B x) (d+e x)^7 \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*x)^6*(A + B*x)*(d + e*x)^7,x]
 

Output:

a^6*A*d^7*x + (a^5*d^6*(6*A*b*d + a*B*d + 7*a*A*e)*x^2)/2 + (a^4*d^5*(a*B* 
d*(6*b*d + 7*a*e) + 3*A*(5*b^2*d^2 + 14*a*b*d*e + 7*a^2*e^2))*x^3)/3 + (a^ 
3*d^4*(3*a*B*d*(5*b^2*d^2 + 14*a*b*d*e + 7*a^2*e^2) + A*(20*b^3*d^3 + 105* 
a*b^2*d^2*e + 126*a^2*b*d*e^2 + 35*a^3*e^3))*x^4)/4 + (a^2*d^3*(a*B*d*(20* 
b^3*d^3 + 105*a*b^2*d^2*e + 126*a^2*b*d*e^2 + 35*a^3*e^3) + 5*A*(3*b^4*d^4 
 + 28*a*b^3*d^3*e + 63*a^2*b^2*d^2*e^2 + 42*a^3*b*d*e^3 + 7*a^4*e^4))*x^5) 
/5 + (a*d^2*(5*a*B*d*(3*b^4*d^4 + 28*a*b^3*d^3*e + 63*a^2*b^2*d^2*e^2 + 42 
*a^3*b*d*e^3 + 7*a^4*e^4) + 3*A*(2*b^5*d^5 + 35*a*b^4*d^4*e + 140*a^2*b^3* 
d^3*e^2 + 175*a^3*b^2*d^2*e^3 + 70*a^4*b*d*e^4 + 7*a^5*e^5))*x^6)/6 + (d*( 
3*a*B*d*(2*b^5*d^5 + 35*a*b^4*d^4*e + 140*a^2*b^3*d^3*e^2 + 175*a^3*b^2*d^ 
2*e^3 + 70*a^4*b*d*e^4 + 7*a^5*e^5) + A*(b^6*d^6 + 42*a*b^5*d^5*e + 315*a^ 
2*b^4*d^4*e^2 + 700*a^3*b^3*d^3*e^3 + 525*a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^ 
5 + 7*a^6*e^6))*x^7)/7 + ((700*a^3*b^3*d^3*e^3*(B*d + A*e) + 42*a^5*b*d*e^ 
5*(3*B*d + A*e) + a^6*e^6*(7*B*d + A*e) + 42*a*b^5*d^5*e*(B*d + 3*A*e) + 1 
05*a^4*b^2*d^2*e^4*(5*B*d + 3*A*e) + 105*a^2*b^4*d^4*e^2*(3*B*d + 5*A*e) + 
 b^6*d^6*(B*d + 7*A*e))*x^8)/8 + (e*(a^6*B*e^6 + 525*a^2*b^4*d^3*e^2*(B*d 
+ A*e) + 105*a^4*b^2*d*e^4*(3*B*d + A*e) + 6*a^5*b*e^5*(7*B*d + A*e) + 7*b 
^6*d^5*(B*d + 3*A*e) + 140*a^3*b^3*d^2*e^3*(5*B*d + 3*A*e) + 42*a*b^5*d^4* 
e*(3*B*d + 5*A*e))*x^9)/9 + (b*e^2*(6*a^5*B*e^5 + 210*a*b^4*d^3*e*(B*d + A 
*e) + 140*a^3*b^2*d*e^3*(3*B*d + A*e) + 15*a^4*b*e^4*(7*B*d + A*e) + 10...
 

Rubi [A] (verified)

Time = 1.49 (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.

\(\displaystyle \int (a+b x)^6 (A+B x) (d+e x)^7 \, dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*x)^6*(A + B*x)*(d + e*x)^7,x]
 

Output:

-1/8*((b*d - a*e)^6*(B*d - A*e)*(d + e*x)^8)/e^8 + ((b*d - a*e)^5*(7*b*B*d 
 - 6*A*b*e - a*B*e)*(d + e*x)^9)/(9*e^8) - (3*b*(b*d - a*e)^4*(7*b*B*d - 5 
*A*b*e - 2*a*B*e)*(d + e*x)^10)/(10*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)^11)/(11*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)^12)/(12*e^8) + (3*b^4*(b*d - a*e)*(7*b*B*d 
 - 2*A*b*e - 5*a*B*e)*(d + e*x)^13)/(13*e^8) - (b^5*(7*b*B*d - A*b*e - 6*a 
*B*e)*(d + e*x)^14)/(14*e^8) + (b^6*B*(d + e*x)^15)/(15*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. \(1348\) vs. \(2(276)=552\).

Time = 0.23 (sec) , antiderivative size = 1349, normalized size of antiderivative = 4.62

method	result	size
default	\(\text {Expression too large to display}\)	\(1349\)
norman	\(\text {Expression too large to display}\)	\(1448\)
orering	\(\text {Expression too large to display}\)	\(1712\)
gosper	\(\text {Expression too large to display}\)	\(1713\)
risch	\(\text {Expression too large to display}\)	\(1713\)
parallelrisch	\(\text {Expression too large to display}\)	\(1713\)

Input:

int((b*x+a)^6*(B*x+A)*(e*x+d)^7,x,method=_RETURNVERBOSE)
 

Output:

1/15*b^6*B*e^7*x^15+1/14*((A*b^6+6*B*a*b^5)*e^7+7*b^6*B*d*e^6)*x^14+1/13*( 
(6*A*a*b^5+15*B*a^2*b^4)*e^7+7*(A*b^6+6*B*a*b^5)*d*e^6+21*b^6*B*d^2*e^5)*x 
^13+1/12*((15*A*a^2*b^4+20*B*a^3*b^3)*e^7+7*(6*A*a*b^5+15*B*a^2*b^4)*d*e^6 
+21*(A*b^6+6*B*a*b^5)*d^2*e^5+35*b^6*B*d^3*e^4)*x^12+1/11*((20*A*a^3*b^3+1 
5*B*a^4*b^2)*e^7+7*(15*A*a^2*b^4+20*B*a^3*b^3)*d*e^6+21*(6*A*a*b^5+15*B*a^ 
2*b^4)*d^2*e^5+35*(A*b^6+6*B*a*b^5)*d^3*e^4+35*b^6*B*d^4*e^3)*x^11+1/10*(( 
15*A*a^4*b^2+6*B*a^5*b)*e^7+7*(20*A*a^3*b^3+15*B*a^4*b^2)*d*e^6+21*(15*A*a 
^2*b^4+20*B*a^3*b^3)*d^2*e^5+35*(6*A*a*b^5+15*B*a^2*b^4)*d^3*e^4+35*(A*b^6 
+6*B*a*b^5)*d^4*e^3+21*b^6*B*d^5*e^2)*x^10+1/9*((6*A*a^5*b+B*a^6)*e^7+7*(1 
5*A*a^4*b^2+6*B*a^5*b)*d*e^6+21*(20*A*a^3*b^3+15*B*a^4*b^2)*d^2*e^5+35*(15 
*A*a^2*b^4+20*B*a^3*b^3)*d^3*e^4+35*(6*A*a*b^5+15*B*a^2*b^4)*d^4*e^3+21*(A 
*b^6+6*B*a*b^5)*d^5*e^2+7*b^6*B*d^6*e)*x^9+1/8*(a^6*A*e^7+7*(6*A*a^5*b+B*a 
^6)*d*e^6+21*(15*A*a^4*b^2+6*B*a^5*b)*d^2*e^5+35*(20*A*a^3*b^3+15*B*a^4*b^ 
2)*d^3*e^4+35*(15*A*a^2*b^4+20*B*a^3*b^3)*d^4*e^3+21*(6*A*a*b^5+15*B*a^2*b 
^4)*d^5*e^2+7*(A*b^6+6*B*a*b^5)*d^6*e+b^6*B*d^7)*x^8+1/7*(7*a^6*A*d*e^6+21 
*(6*A*a^5*b+B*a^6)*d^2*e^5+35*(15*A*a^4*b^2+6*B*a^5*b)*d^3*e^4+35*(20*A*a^ 
3*b^3+15*B*a^4*b^2)*d^4*e^3+21*(15*A*a^2*b^4+20*B*a^3*b^3)*d^5*e^2+7*(6*A* 
a*b^5+15*B*a^2*b^4)*d^6*e+(A*b^6+6*B*a*b^5)*d^7)*x^7+1/6*(21*a^6*A*d^2*e^5 
+35*(6*A*a^5*b+B*a^6)*d^3*e^4+35*(15*A*a^4*b^2+6*B*a^5*b)*d^4*e^3+21*(20*A 
*a^3*b^3+15*B*a^4*b^2)*d^5*e^2+7*(15*A*a^2*b^4+20*B*a^3*b^3)*d^6*e+(6*A...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1356 vs. \(2 (276) = 552\).

Time = 0.07 (sec) , antiderivative size = 1356, normalized size of antiderivative = 4.64 \[ \int (a+b x)^6 (A+B x) (d+e x)^7 \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^7,x, algorithm="fricas")
 

Output:

1/15*B*b^6*e^7*x^15 + A*a^6*d^7*x + 1/14*(7*B*b^6*d*e^6 + (6*B*a*b^5 + A*b 
^6)*e^7)*x^14 + 1/13*(21*B*b^6*d^2*e^5 + 7*(6*B*a*b^5 + A*b^6)*d*e^6 + 3*( 
5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^13 + 1/12*(35*B*b^6*d^3*e^4 + 21*(6*B*a*b^ 
5 + A*b^6)*d^2*e^5 + 21*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 5*(4*B*a^3*b^3 + 
 3*A*a^2*b^4)*e^7)*x^12 + 1/11*(35*B*b^6*d^4*e^3 + 35*(6*B*a*b^5 + A*b^6)* 
d^3*e^4 + 63*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 35*(4*B*a^3*b^3 + 3*A*a^2 
*b^4)*d*e^6 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^11 + 1/10*(21*B*b^6*d^5 
*e^2 + 35*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 105*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3* 
e^4 + 105*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 35*(3*B*a^4*b^2 + 4*A*a^3* 
b^3)*d*e^6 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^10 + 1/9*(7*B*b^6*d^6*e + 
21*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 105*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 1 
75*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 105*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d 
^2*e^5 + 21*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + (B*a^6 + 6*A*a^5*b)*e^7)*x^9 
 + 1/8*(B*b^6*d^7 + A*a^6*e^7 + 7*(6*B*a*b^5 + A*b^6)*d^6*e + 63*(5*B*a^2* 
b^4 + 2*A*a*b^5)*d^5*e^2 + 175*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 175*( 
3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 63*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 
+ 7*(B*a^6 + 6*A*a^5*b)*d*e^6)*x^8 + 1/7*(7*A*a^6*d*e^6 + (6*B*a*b^5 + A*b 
^6)*d^7 + 21*(5*B*a^2*b^4 + 2*A*a*b^5)*d^6*e + 105*(4*B*a^3*b^3 + 3*A*a^2* 
b^4)*d^5*e^2 + 175*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4*e^3 + 105*(2*B*a^5*b + 
5*A*a^4*b^2)*d^3*e^4 + 21*(B*a^6 + 6*A*a^5*b)*d^2*e^5)*x^7 + 1/6*(21*A*...
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1756 vs. \(2 (296) = 592\).

Time = 0.11 (sec) , antiderivative size = 1756, normalized size of antiderivative = 6.01 \[ \int (a+b x)^6 (A+B x) (d+e x)^7 \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)**6*(B*x+A)*(e*x+d)**7,x)
 

Output:

A*a**6*d**7*x + B*b**6*e**7*x**15/15 + x**14*(A*b**6*e**7/14 + 3*B*a*b**5* 
e**7/7 + B*b**6*d*e**6/2) + x**13*(6*A*a*b**5*e**7/13 + 7*A*b**6*d*e**6/13 
 + 15*B*a**2*b**4*e**7/13 + 42*B*a*b**5*d*e**6/13 + 21*B*b**6*d**2*e**5/13 
) + x**12*(5*A*a**2*b**4*e**7/4 + 7*A*a*b**5*d*e**6/2 + 7*A*b**6*d**2*e**5 
/4 + 5*B*a**3*b**3*e**7/3 + 35*B*a**2*b**4*d*e**6/4 + 21*B*a*b**5*d**2*e** 
5/2 + 35*B*b**6*d**3*e**4/12) + x**11*(20*A*a**3*b**3*e**7/11 + 105*A*a**2 
*b**4*d*e**6/11 + 126*A*a*b**5*d**2*e**5/11 + 35*A*b**6*d**3*e**4/11 + 15* 
B*a**4*b**2*e**7/11 + 140*B*a**3*b**3*d*e**6/11 + 315*B*a**2*b**4*d**2*e** 
5/11 + 210*B*a*b**5*d**3*e**4/11 + 35*B*b**6*d**4*e**3/11) + x**10*(3*A*a* 
*4*b**2*e**7/2 + 14*A*a**3*b**3*d*e**6 + 63*A*a**2*b**4*d**2*e**5/2 + 21*A 
*a*b**5*d**3*e**4 + 7*A*b**6*d**4*e**3/2 + 3*B*a**5*b*e**7/5 + 21*B*a**4*b 
**2*d*e**6/2 + 42*B*a**3*b**3*d**2*e**5 + 105*B*a**2*b**4*d**3*e**4/2 + 21 
*B*a*b**5*d**4*e**3 + 21*B*b**6*d**5*e**2/10) + x**9*(2*A*a**5*b*e**7/3 + 
35*A*a**4*b**2*d*e**6/3 + 140*A*a**3*b**3*d**2*e**5/3 + 175*A*a**2*b**4*d* 
*3*e**4/3 + 70*A*a*b**5*d**4*e**3/3 + 7*A*b**6*d**5*e**2/3 + B*a**6*e**7/9 
 + 14*B*a**5*b*d*e**6/3 + 35*B*a**4*b**2*d**2*e**5 + 700*B*a**3*b**3*d**3* 
e**4/9 + 175*B*a**2*b**4*d**4*e**3/3 + 14*B*a*b**5*d**5*e**2 + 7*B*b**6*d* 
*6*e/9) + x**8*(A*a**6*e**7/8 + 21*A*a**5*b*d*e**6/4 + 315*A*a**4*b**2*d** 
2*e**5/8 + 175*A*a**3*b**3*d**3*e**4/2 + 525*A*a**2*b**4*d**4*e**3/8 + 63* 
A*a*b**5*d**5*e**2/4 + 7*A*b**6*d**6*e/8 + 7*B*a**6*d*e**6/8 + 63*B*a**...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1356 vs. \(2 (276) = 552\).

Time = 0.04 (sec) , antiderivative size = 1356, normalized size of antiderivative = 4.64 \[ \int (a+b x)^6 (A+B x) (d+e x)^7 \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^7,x, algorithm="maxima")
 

Output:

1/15*B*b^6*e^7*x^15 + A*a^6*d^7*x + 1/14*(7*B*b^6*d*e^6 + (6*B*a*b^5 + A*b 
^6)*e^7)*x^14 + 1/13*(21*B*b^6*d^2*e^5 + 7*(6*B*a*b^5 + A*b^6)*d*e^6 + 3*( 
5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^13 + 1/12*(35*B*b^6*d^3*e^4 + 21*(6*B*a*b^ 
5 + A*b^6)*d^2*e^5 + 21*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 5*(4*B*a^3*b^3 + 
 3*A*a^2*b^4)*e^7)*x^12 + 1/11*(35*B*b^6*d^4*e^3 + 35*(6*B*a*b^5 + A*b^6)* 
d^3*e^4 + 63*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 35*(4*B*a^3*b^3 + 3*A*a^2 
*b^4)*d*e^6 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^11 + 1/10*(21*B*b^6*d^5 
*e^2 + 35*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 105*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3* 
e^4 + 105*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 35*(3*B*a^4*b^2 + 4*A*a^3* 
b^3)*d*e^6 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^10 + 1/9*(7*B*b^6*d^6*e + 
21*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 105*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 1 
75*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 105*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d 
^2*e^5 + 21*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + (B*a^6 + 6*A*a^5*b)*e^7)*x^9 
 + 1/8*(B*b^6*d^7 + A*a^6*e^7 + 7*(6*B*a*b^5 + A*b^6)*d^6*e + 63*(5*B*a^2* 
b^4 + 2*A*a*b^5)*d^5*e^2 + 175*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 175*( 
3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 63*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 
+ 7*(B*a^6 + 6*A*a^5*b)*d*e^6)*x^8 + 1/7*(7*A*a^6*d*e^6 + (6*B*a*b^5 + A*b 
^6)*d^7 + 21*(5*B*a^2*b^4 + 2*A*a*b^5)*d^6*e + 105*(4*B*a^3*b^3 + 3*A*a^2* 
b^4)*d^5*e^2 + 175*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4*e^3 + 105*(2*B*a^5*b + 
5*A*a^4*b^2)*d^3*e^4 + 21*(B*a^6 + 6*A*a^5*b)*d^2*e^5)*x^7 + 1/6*(21*A*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1712 vs. \(2 (276) = 552\).

Time = 0.13 (sec) , antiderivative size = 1712, normalized size of antiderivative = 5.86 \[ \int (a+b x)^6 (A+B x) (d+e x)^7 \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^7,x, algorithm="giac")
 

Output:

1/15*B*b^6*e^7*x^15 + 1/2*B*b^6*d*e^6*x^14 + 3/7*B*a*b^5*e^7*x^14 + 1/14*A 
*b^6*e^7*x^14 + 21/13*B*b^6*d^2*e^5*x^13 + 42/13*B*a*b^5*d*e^6*x^13 + 7/13 
*A*b^6*d*e^6*x^13 + 15/13*B*a^2*b^4*e^7*x^13 + 6/13*A*a*b^5*e^7*x^13 + 35/ 
12*B*b^6*d^3*e^4*x^12 + 21/2*B*a*b^5*d^2*e^5*x^12 + 7/4*A*b^6*d^2*e^5*x^12 
 + 35/4*B*a^2*b^4*d*e^6*x^12 + 7/2*A*a*b^5*d*e^6*x^12 + 5/3*B*a^3*b^3*e^7* 
x^12 + 5/4*A*a^2*b^4*e^7*x^12 + 35/11*B*b^6*d^4*e^3*x^11 + 210/11*B*a*b^5* 
d^3*e^4*x^11 + 35/11*A*b^6*d^3*e^4*x^11 + 315/11*B*a^2*b^4*d^2*e^5*x^11 + 
126/11*A*a*b^5*d^2*e^5*x^11 + 140/11*B*a^3*b^3*d*e^6*x^11 + 105/11*A*a^2*b 
^4*d*e^6*x^11 + 15/11*B*a^4*b^2*e^7*x^11 + 20/11*A*a^3*b^3*e^7*x^11 + 21/1 
0*B*b^6*d^5*e^2*x^10 + 21*B*a*b^5*d^4*e^3*x^10 + 7/2*A*b^6*d^4*e^3*x^10 + 
105/2*B*a^2*b^4*d^3*e^4*x^10 + 21*A*a*b^5*d^3*e^4*x^10 + 42*B*a^3*b^3*d^2* 
e^5*x^10 + 63/2*A*a^2*b^4*d^2*e^5*x^10 + 21/2*B*a^4*b^2*d*e^6*x^10 + 14*A* 
a^3*b^3*d*e^6*x^10 + 3/5*B*a^5*b*e^7*x^10 + 3/2*A*a^4*b^2*e^7*x^10 + 7/9*B 
*b^6*d^6*e*x^9 + 14*B*a*b^5*d^5*e^2*x^9 + 7/3*A*b^6*d^5*e^2*x^9 + 175/3*B* 
a^2*b^4*d^4*e^3*x^9 + 70/3*A*a*b^5*d^4*e^3*x^9 + 700/9*B*a^3*b^3*d^3*e^4*x 
^9 + 175/3*A*a^2*b^4*d^3*e^4*x^9 + 35*B*a^4*b^2*d^2*e^5*x^9 + 140/3*A*a^3* 
b^3*d^2*e^5*x^9 + 14/3*B*a^5*b*d*e^6*x^9 + 35/3*A*a^4*b^2*d*e^6*x^9 + 1/9* 
B*a^6*e^7*x^9 + 2/3*A*a^5*b*e^7*x^9 + 1/8*B*b^6*d^7*x^8 + 21/4*B*a*b^5*d^6 
*e*x^8 + 7/8*A*b^6*d^6*e*x^8 + 315/8*B*a^2*b^4*d^5*e^2*x^8 + 63/4*A*a*b^5* 
d^5*e^2*x^8 + 175/2*B*a^3*b^3*d^4*e^3*x^8 + 525/8*A*a^2*b^4*d^4*e^3*x^8...
 

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 1431, normalized size of antiderivative = 4.90 \[ \int (a+b x)^6 (A+B x) (d+e x)^7 \, dx=\text {Too large to display} \] Input:

int((A + B*x)*(a + b*x)^6*(d + e*x)^7,x)
 

Output:

x^6*(A*a*b^5*d^7 + (5*B*a^2*b^4*d^7)/2 + (7*A*a^6*d^2*e^5)/2 + (35*B*a^6*d 
^3*e^4)/6 + (35*A*a^2*b^4*d^6*e)/2 + 35*A*a^5*b*d^3*e^4 + (70*B*a^3*b^3*d^ 
6*e)/3 + 35*B*a^5*b*d^4*e^3 + 70*A*a^3*b^3*d^5*e^2 + (175*A*a^4*b^2*d^4*e^ 
3)/2 + (105*B*a^4*b^2*d^5*e^2)/2) + x^10*((3*B*a^5*b*e^7)/5 + (3*A*a^4*b^2 
*e^7)/2 + (7*A*b^6*d^4*e^3)/2 + (21*B*b^6*d^5*e^2)/10 + 21*A*a*b^5*d^3*e^4 
 + 14*A*a^3*b^3*d*e^6 + 21*B*a*b^5*d^4*e^3 + (21*B*a^4*b^2*d*e^6)/2 + (63* 
A*a^2*b^4*d^2*e^5)/2 + (105*B*a^2*b^4*d^3*e^4)/2 + 42*B*a^3*b^3*d^2*e^5) + 
 x^5*(3*A*a^2*b^4*d^7 + 4*B*a^3*b^3*d^7 + 7*A*a^6*d^3*e^4 + 7*B*a^6*d^4*e^ 
3 + 28*A*a^3*b^3*d^6*e + 42*A*a^5*b*d^4*e^3 + 21*B*a^4*b^2*d^6*e + (126*B* 
a^5*b*d^5*e^2)/5 + 63*A*a^4*b^2*d^5*e^2) + x^11*((20*A*a^3*b^3*e^7)/11 + ( 
15*B*a^4*b^2*e^7)/11 + (35*A*b^6*d^3*e^4)/11 + (35*B*b^6*d^4*e^3)/11 + (12 
6*A*a*b^5*d^2*e^5)/11 + (105*A*a^2*b^4*d*e^6)/11 + (210*B*a*b^5*d^3*e^4)/1 
1 + (140*B*a^3*b^3*d*e^6)/11 + (315*B*a^2*b^4*d^2*e^5)/11) + x^3*(2*B*a^5* 
b*d^7 + (7*B*a^6*d^6*e)/3 + 5*A*a^4*b^2*d^7 + 7*A*a^6*d^5*e^2 + 14*A*a^5*b 
*d^6*e) + x^8*((A*a^6*e^7)/8 + (B*b^6*d^7)/8 + (7*A*b^6*d^6*e)/8 + (7*B*a^ 
6*d*e^6)/8 + (63*A*a*b^5*d^5*e^2)/4 + (63*B*a^5*b*d^2*e^5)/4 + (525*A*a^2* 
b^4*d^4*e^3)/8 + (175*A*a^3*b^3*d^3*e^4)/2 + (315*A*a^4*b^2*d^2*e^5)/8 + ( 
315*B*a^2*b^4*d^5*e^2)/8 + (175*B*a^3*b^3*d^4*e^3)/2 + (525*B*a^4*b^2*d^3* 
e^4)/8 + (21*A*a^5*b*d*e^6)/4 + (21*B*a*b^5*d^6*e)/4) + x^13*((6*A*a*b^5*e 
^7)/13 + (7*A*b^6*d*e^6)/13 + (15*B*a^2*b^4*e^7)/13 + (21*B*b^6*d^2*e^5...
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 925, normalized size of antiderivative = 3.17 \[ \int (a+b x)^6 (A+B x) (d+e x)^7 \, dx =\text {Too large to display} \] Input:

int((b*x+a)^6*(B*x+A)*(e*x+d)^7,x)
 

Output:

(x*(51480*a**7*d**7 + 180180*a**7*d**6*e*x + 360360*a**7*d**5*e**2*x**2 + 
450450*a**7*d**4*e**3*x**3 + 360360*a**7*d**3*e**4*x**4 + 180180*a**7*d**2 
*e**5*x**5 + 51480*a**7*d*e**6*x**6 + 6435*a**7*e**7*x**7 + 180180*a**6*b* 
d**7*x + 840840*a**6*b*d**6*e*x**2 + 1891890*a**6*b*d**5*e**2*x**3 + 25225 
20*a**6*b*d**4*e**3*x**4 + 2102100*a**6*b*d**3*e**4*x**5 + 1081080*a**6*b* 
d**2*e**5*x**6 + 315315*a**6*b*d*e**6*x**7 + 40040*a**6*b*e**7*x**8 + 3603 
60*a**5*b**2*d**7*x**2 + 1891890*a**5*b**2*d**6*e*x**3 + 4540536*a**5*b**2 
*d**5*e**2*x**4 + 6306300*a**5*b**2*d**4*e**3*x**5 + 5405400*a**5*b**2*d** 
3*e**4*x**6 + 2837835*a**5*b**2*d**2*e**5*x**7 + 840840*a**5*b**2*d*e**6*x 
**8 + 108108*a**5*b**2*e**7*x**9 + 450450*a**4*b**3*d**7*x**3 + 2522520*a* 
*4*b**3*d**6*e*x**4 + 6306300*a**4*b**3*d**5*e**2*x**5 + 9009000*a**4*b**3 
*d**4*e**3*x**6 + 7882875*a**4*b**3*d**3*e**4*x**7 + 4204200*a**4*b**3*d** 
2*e**5*x**8 + 1261260*a**4*b**3*d*e**6*x**9 + 163800*a**4*b**3*e**7*x**10 
+ 360360*a**3*b**4*d**7*x**4 + 2102100*a**3*b**4*d**6*e*x**5 + 5405400*a** 
3*b**4*d**5*e**2*x**6 + 7882875*a**3*b**4*d**4*e**3*x**7 + 7007000*a**3*b* 
*4*d**3*e**4*x**8 + 3783780*a**3*b**4*d**2*e**5*x**9 + 1146600*a**3*b**4*d 
*e**6*x**10 + 150150*a**3*b**4*e**7*x**11 + 180180*a**2*b**5*d**7*x**5 + 1 
081080*a**2*b**5*d**6*e*x**6 + 2837835*a**2*b**5*d**5*e**2*x**7 + 4204200* 
a**2*b**5*d**4*e**3*x**8 + 3783780*a**2*b**5*d**3*e**4*x**9 + 2063880*a**2 
*b**5*d**2*e**5*x**10 + 630630*a**2*b**5*d*e**6*x**11 + 83160*a**2*b**5...