\(\int (a+b x)^{10} (A+B x) (d+e x)^{12} \, dx\) [66]

Optimal result

Integrand size = 20, antiderivative size = 464 \[ \int (a+b x)^{10} (A+B x) (d+e x)^{12} \, dx=-\frac {(b d-a e)^{10} (B d-A e) (d+e x)^{13}}{13 e^{12}}+\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e) (d+e x)^{14}}{14 e^{12}}-\frac {b (b d-a e)^8 (11 b B d-9 A b e-2 a B e) (d+e x)^{15}}{3 e^{12}}+\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) (d+e x)^{16}}{16 e^{12}}-\frac {30 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e) (d+e x)^{17}}{17 e^{12}}+\frac {7 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) (d+e x)^{18}}{3 e^{12}}-\frac {42 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) (d+e x)^{19}}{19 e^{12}}+\frac {3 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) (d+e x)^{20}}{2 e^{12}}-\frac {5 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^{21}}{7 e^{12}}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^{22}}{22 e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^{23}}{23 e^{12}}+\frac {b^{10} B (d+e x)^{24}}{24 e^{12}} \] Output:

-1/13*(-a*e+b*d)^10*(-A*e+B*d)*(e*x+d)^13/e^12+1/14*(-a*e+b*d)^9*(-10*A*b* 
e-B*a*e+11*B*b*d)*(e*x+d)^14/e^12-1/3*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11* 
B*b*d)*(e*x+d)^15/e^12+15/16*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d)* 
(e*x+d)^16/e^12-30/17*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)*(e*x+d) 
^17/e^12+7/3*b^4*(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)*(e*x+d)^18/e^12- 
42/19*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)*(e*x+d)^19/e^12+3/2*b^6 
*(-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11*B*b*d)*(e*x+d)^20/e^12-5/7*b^7*(-a*e+b* 
d)^2*(-3*A*b*e-8*B*a*e+11*B*b*d)*(e*x+d)^21/e^12+5/22*b^8*(-a*e+b*d)*(-2*A 
*b*e-9*B*a*e+11*B*b*d)*(e*x+d)^22/e^12-1/23*b^9*(-A*b*e-10*B*a*e+11*B*b*d) 
*(e*x+d)^23/e^12+1/24*b^10*B*(e*x+d)^24/e^12
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(3320\) vs. \(2(464)=928\).

Time = 0.88 (sec) , antiderivative size = 3320, normalized size of antiderivative = 7.16 \[ \int (a+b x)^{10} (A+B x) (d+e x)^{12} \, dx=\text {Result too large to show} \] Input:

Integrate[(a + b*x)^10*(A + B*x)*(d + e*x)^12,x]
 

Output:

a^10*A*d^12*x + (a^9*d^11*(a*B*d + 2*A*(5*b*d + 6*a*e))*x^2)/2 + (a^8*d^10 
*(2*a*B*d*(5*b*d + 6*a*e) + 3*A*(15*b^2*d^2 + 40*a*b*d*e + 22*a^2*e^2))*x^ 
3)/3 + (a^7*d^9*(3*a*B*d*(15*b^2*d^2 + 40*a*b*d*e + 22*a^2*e^2) + 20*A*(6* 
b^3*d^3 + 27*a*b^2*d^2*e + 33*a^2*b*d*e^2 + 11*a^3*e^3))*x^4)/4 + a^6*d^8* 
(4*a*B*d*(6*b^3*d^3 + 27*a*b^2*d^2*e + 33*a^2*b*d*e^2 + 11*a^3*e^3) + A*(4 
2*b^4*d^4 + 288*a*b^3*d^3*e + 594*a^2*b^2*d^2*e^2 + 440*a^3*b*d*e^3 + 99*a 
^4*e^4))*x^5 + (a^5*d^7*(5*a*B*d*(42*b^4*d^4 + 288*a*b^3*d^3*e + 594*a^2*b 
^2*d^2*e^2 + 440*a^3*b*d*e^3 + 99*a^4*e^4) + 18*A*(14*b^5*d^5 + 140*a*b^4* 
d^4*e + 440*a^2*b^3*d^3*e^2 + 550*a^3*b^2*d^2*e^3 + 275*a^4*b*d*e^4 + 44*a 
^5*e^5))*x^6)/6 + (3*a^4*d^6*(6*a*B*d*(14*b^5*d^5 + 140*a*b^4*d^4*e + 440* 
a^2*b^3*d^3*e^2 + 550*a^3*b^2*d^2*e^3 + 275*a^4*b*d*e^4 + 44*a^5*e^5) + A* 
(70*b^6*d^6 + 1008*a*b^5*d^5*e + 4620*a^2*b^4*d^4*e^2 + 8800*a^3*b^3*d^3*e 
^3 + 7425*a^4*b^2*d^2*e^4 + 2640*a^5*b*d*e^5 + 308*a^6*e^6))*x^7)/7 + (3*a 
^3*d^5*(a*B*d*(70*b^6*d^6 + 1008*a*b^5*d^5*e + 4620*a^2*b^4*d^4*e^2 + 8800 
*a^3*b^3*d^3*e^3 + 7425*a^4*b^2*d^2*e^4 + 2640*a^5*b*d*e^5 + 308*a^6*e^6) 
+ 8*A*(5*b^7*d^7 + 105*a*b^6*d^6*e + 693*a^2*b^5*d^5*e^2 + 1925*a^3*b^4*d^ 
4*e^3 + 2475*a^4*b^3*d^3*e^4 + 1485*a^5*b^2*d^2*e^5 + 385*a^6*b*d*e^6 + 33 
*a^7*e^7))*x^8)/8 + (a^2*d^4*(8*a*B*d*(5*b^7*d^7 + 105*a*b^6*d^6*e + 693*a 
^2*b^5*d^5*e^2 + 1925*a^3*b^4*d^4*e^3 + 2475*a^4*b^3*d^3*e^4 + 1485*a^5*b^ 
2*d^2*e^5 + 385*a^6*b*d*e^6 + 33*a^7*e^7) + 15*A*(b^8*d^8 + 32*a*b^7*d^...
 

Rubi [A] (verified)

Time = 5.35 (sec) , antiderivative size = 464, 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.

\(\displaystyle \int (a+b x)^{10} (A+B x) (d+e x)^{12} \, dx\)

\(\Big \downarrow \) 86

\(\displaystyle \int \left (\frac {b^9 (d+e x)^{22} (10 a B e+A b e-11 b B d)}{e^{11}}-\frac {5 b^8 (d+e x)^{21} (b d-a e) (9 a B e+2 A b e-11 b B d)}{e^{11}}+\frac {15 b^7 (d+e x)^{20} (b d-a e)^2 (8 a B e+3 A b e-11 b B d)}{e^{11}}-\frac {30 b^6 (d+e x)^{19} (b d-a e)^3 (7 a B e+4 A b e-11 b B d)}{e^{11}}+\frac {42 b^5 (d+e x)^{18} (b d-a e)^4 (6 a B e+5 A b e-11 b B d)}{e^{11}}-\frac {42 b^4 (d+e x)^{17} (b d-a e)^5 (5 a B e+6 A b e-11 b B d)}{e^{11}}+\frac {30 b^3 (d+e x)^{16} (b d-a e)^6 (4 a B e+7 A b e-11 b B d)}{e^{11}}-\frac {15 b^2 (d+e x)^{15} (b d-a e)^7 (3 a B e+8 A b e-11 b B d)}{e^{11}}+\frac {5 b (d+e x)^{14} (b d-a e)^8 (2 a B e+9 A b e-11 b B d)}{e^{11}}+\frac {(d+e x)^{13} (a e-b d)^9 (a B e+10 A b e-11 b B d)}{e^{11}}+\frac {(d+e x)^{12} (a e-b d)^{10} (A e-B d)}{e^{11}}+\frac {b^{10} B (d+e x)^{23}}{e^{11}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^9 (d+e x)^{23} (-10 a B e-A b e+11 b B d)}{23 e^{12}}+\frac {5 b^8 (d+e x)^{22} (b d-a e) (-9 a B e-2 A b e+11 b B d)}{22 e^{12}}-\frac {5 b^7 (d+e x)^{21} (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{7 e^{12}}+\frac {3 b^6 (d+e x)^{20} (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{2 e^{12}}-\frac {42 b^5 (d+e x)^{19} (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{19 e^{12}}+\frac {7 b^4 (d+e x)^{18} (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{3 e^{12}}-\frac {30 b^3 (d+e x)^{17} (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{17 e^{12}}+\frac {15 b^2 (d+e x)^{16} (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{16 e^{12}}-\frac {b (d+e x)^{15} (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{3 e^{12}}+\frac {(d+e x)^{14} (b d-a e)^9 (-a B e-10 A b e+11 b B d)}{14 e^{12}}-\frac {(d+e x)^{13} (b d-a e)^{10} (B d-A e)}{13 e^{12}}+\frac {b^{10} B (d+e x)^{24}}{24 e^{12}}\)

Input:

Int[(a + b*x)^10*(A + B*x)*(d + e*x)^12,x]
 

Output:

-1/13*((b*d - a*e)^10*(B*d - A*e)*(d + e*x)^13)/e^12 + ((b*d - a*e)^9*(11* 
b*B*d - 10*A*b*e - a*B*e)*(d + e*x)^14)/(14*e^12) - (b*(b*d - a*e)^8*(11*b 
*B*d - 9*A*b*e - 2*a*B*e)*(d + e*x)^15)/(3*e^12) + (15*b^2*(b*d - a*e)^7*( 
11*b*B*d - 8*A*b*e - 3*a*B*e)*(d + e*x)^16)/(16*e^12) - (30*b^3*(b*d - a*e 
)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e)*(d + e*x)^17)/(17*e^12) + (7*b^4*(b*d - 
 a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e)*(d + e*x)^18)/(3*e^12) - (42*b^5*(b 
*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*(d + e*x)^19)/(19*e^12) + (3*b^ 
6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e)*(d + e*x)^20)/(2*e^12) - (5 
*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*(d + e*x)^21)/(7*e^12) + 
 (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^22)/(22*e^12) 
 - (b^9*(11*b*B*d - A*b*e - 10*a*B*e)*(d + e*x)^23)/(23*e^12) + (b^10*B*(d 
 + e*x)^24)/(24*e^12)
 

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. \(3608\) vs. \(2(440)=880\).

Time = 0.33 (sec) , antiderivative size = 3609, normalized size of antiderivative = 7.78

method	result	size
default	\(\text {Expression too large to display}\)	\(3609\)
norman	\(\text {Expression too large to display}\)	\(3922\)
gosper	\(\text {Expression too large to display}\)	\(4664\)
risch	\(\text {Expression too large to display}\)	\(4664\)
parallelrisch	\(\text {Expression too large to display}\)	\(4664\)
orering	\(\text {Expression too large to display}\)	\(4664\)

Input:

int((b*x+a)^10*(B*x+A)*(e*x+d)^12,x,method=_RETURNVERBOSE)
 

Output:

1/24*b^10*B*e^12*x^24+1/23*((A*b^10+10*B*a*b^9)*e^12+12*b^10*B*d*e^11)*x^2 
3+1/22*((10*A*a*b^9+45*B*a^2*b^8)*e^12+12*(A*b^10+10*B*a*b^9)*d*e^11+66*b^ 
10*B*d^2*e^10)*x^22+1/21*((45*A*a^2*b^8+120*B*a^3*b^7)*e^12+12*(10*A*a*b^9 
+45*B*a^2*b^8)*d*e^11+66*(A*b^10+10*B*a*b^9)*d^2*e^10+220*b^10*B*d^3*e^9)* 
x^21+1/20*((120*A*a^3*b^7+210*B*a^4*b^6)*e^12+12*(45*A*a^2*b^8+120*B*a^3*b 
^7)*d*e^11+66*(10*A*a*b^9+45*B*a^2*b^8)*d^2*e^10+220*(A*b^10+10*B*a*b^9)*d 
^3*e^9+495*b^10*B*d^4*e^8)*x^20+1/19*((210*A*a^4*b^6+252*B*a^5*b^5)*e^12+1 
2*(120*A*a^3*b^7+210*B*a^4*b^6)*d*e^11+66*(45*A*a^2*b^8+120*B*a^3*b^7)*d^2 
*e^10+220*(10*A*a*b^9+45*B*a^2*b^8)*d^3*e^9+495*(A*b^10+10*B*a*b^9)*d^4*e^ 
8+792*b^10*B*d^5*e^7)*x^19+1/18*((252*A*a^5*b^5+210*B*a^6*b^4)*e^12+12*(21 
0*A*a^4*b^6+252*B*a^5*b^5)*d*e^11+66*(120*A*a^3*b^7+210*B*a^4*b^6)*d^2*e^1 
0+220*(45*A*a^2*b^8+120*B*a^3*b^7)*d^3*e^9+495*(10*A*a*b^9+45*B*a^2*b^8)*d 
^4*e^8+792*(A*b^10+10*B*a*b^9)*d^5*e^7+924*b^10*B*d^6*e^6)*x^18+1/17*((210 
*A*a^6*b^4+120*B*a^7*b^3)*e^12+12*(252*A*a^5*b^5+210*B*a^6*b^4)*d*e^11+66* 
(210*A*a^4*b^6+252*B*a^5*b^5)*d^2*e^10+220*(120*A*a^3*b^7+210*B*a^4*b^6)*d 
^3*e^9+495*(45*A*a^2*b^8+120*B*a^3*b^7)*d^4*e^8+792*(10*A*a*b^9+45*B*a^2*b 
^8)*d^5*e^7+924*(A*b^10+10*B*a*b^9)*d^6*e^6+792*b^10*B*d^7*e^5)*x^17+1/16* 
((120*A*a^7*b^3+45*B*a^8*b^2)*e^12+12*(210*A*a^6*b^4+120*B*a^7*b^3)*d*e^11 
+66*(252*A*a^5*b^5+210*B*a^6*b^4)*d^2*e^10+220*(210*A*a^4*b^6+252*B*a^5*b^ 
5)*d^3*e^9+495*(120*A*a^3*b^7+210*B*a^4*b^6)*d^4*e^8+792*(45*A*a^2*b^8+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3621 vs. \(2 (440) = 880\).

Time = 0.13 (sec) , antiderivative size = 3621, normalized size of antiderivative = 7.80 \[ \int (a+b x)^{10} (A+B x) (d+e x)^{12} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)*(e*x+d)^12,x, algorithm="fricas")
 

Output:

Too large to include
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4655 vs. \(2 (478) = 956\).

Time = 0.27 (sec) , antiderivative size = 4655, normalized size of antiderivative = 10.03 \[ \int (a+b x)^{10} (A+B x) (d+e x)^{12} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)**10*(B*x+A)*(e*x+d)**12,x)
 

Output:

A*a**10*d**12*x + B*b**10*e**12*x**24/24 + x**23*(A*b**10*e**12/23 + 10*B* 
a*b**9*e**12/23 + 12*B*b**10*d*e**11/23) + x**22*(5*A*a*b**9*e**12/11 + 6* 
A*b**10*d*e**11/11 + 45*B*a**2*b**8*e**12/22 + 60*B*a*b**9*d*e**11/11 + 3* 
B*b**10*d**2*e**10) + x**21*(15*A*a**2*b**8*e**12/7 + 40*A*a*b**9*d*e**11/ 
7 + 22*A*b**10*d**2*e**10/7 + 40*B*a**3*b**7*e**12/7 + 180*B*a**2*b**8*d*e 
**11/7 + 220*B*a*b**9*d**2*e**10/7 + 220*B*b**10*d**3*e**9/21) + x**20*(6* 
A*a**3*b**7*e**12 + 27*A*a**2*b**8*d*e**11 + 33*A*a*b**9*d**2*e**10 + 11*A 
*b**10*d**3*e**9 + 21*B*a**4*b**6*e**12/2 + 72*B*a**3*b**7*d*e**11 + 297*B 
*a**2*b**8*d**2*e**10/2 + 110*B*a*b**9*d**3*e**9 + 99*B*b**10*d**4*e**8/4) 
 + x**19*(210*A*a**4*b**6*e**12/19 + 1440*A*a**3*b**7*d*e**11/19 + 2970*A* 
a**2*b**8*d**2*e**10/19 + 2200*A*a*b**9*d**3*e**9/19 + 495*A*b**10*d**4*e* 
*8/19 + 252*B*a**5*b**5*e**12/19 + 2520*B*a**4*b**6*d*e**11/19 + 7920*B*a* 
*3*b**7*d**2*e**10/19 + 9900*B*a**2*b**8*d**3*e**9/19 + 4950*B*a*b**9*d**4 
*e**8/19 + 792*B*b**10*d**5*e**7/19) + x**18*(14*A*a**5*b**5*e**12 + 140*A 
*a**4*b**6*d*e**11 + 440*A*a**3*b**7*d**2*e**10 + 550*A*a**2*b**8*d**3*e** 
9 + 275*A*a*b**9*d**4*e**8 + 44*A*b**10*d**5*e**7 + 35*B*a**6*b**4*e**12/3 
 + 168*B*a**5*b**5*d*e**11 + 770*B*a**4*b**6*d**2*e**10 + 4400*B*a**3*b**7 
*d**3*e**9/3 + 2475*B*a**2*b**8*d**4*e**8/2 + 440*B*a*b**9*d**5*e**7 + 154 
*B*b**10*d**6*e**6/3) + x**17*(210*A*a**6*b**4*e**12/17 + 3024*A*a**5*b**5 
*d*e**11/17 + 13860*A*a**4*b**6*d**2*e**10/17 + 26400*A*a**3*b**7*d**3*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3621 vs. \(2 (440) = 880\).

Time = 0.07 (sec) , antiderivative size = 3621, normalized size of antiderivative = 7.80 \[ \int (a+b x)^{10} (A+B x) (d+e x)^{12} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)*(e*x+d)^12,x, algorithm="maxima")
 

Output:

1/24*B*b^10*e^12*x^24 + A*a^10*d^12*x + 1/23*(12*B*b^10*d*e^11 + (10*B*a*b 
^9 + A*b^10)*e^12)*x^23 + 1/22*(66*B*b^10*d^2*e^10 + 12*(10*B*a*b^9 + A*b^ 
10)*d*e^11 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^12)*x^22 + 1/21*(220*B*b^10*d^3 
*e^9 + 66*(10*B*a*b^9 + A*b^10)*d^2*e^10 + 60*(9*B*a^2*b^8 + 2*A*a*b^9)*d* 
e^11 + 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^12)*x^21 + 1/4*(99*B*b^10*d^4*e^8 
+ 44*(10*B*a*b^9 + A*b^10)*d^3*e^9 + 66*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^10 
 + 36*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^11 + 6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e 
^12)*x^20 + 1/19*(792*B*b^10*d^5*e^7 + 495*(10*B*a*b^9 + A*b^10)*d^4*e^8 + 
 1100*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^9 + 990*(8*B*a^3*b^7 + 3*A*a^2*b^8)* 
d^2*e^10 + 360*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^11 + 42*(6*B*a^5*b^5 + 5*A* 
a^4*b^6)*e^12)*x^19 + 1/6*(308*B*b^10*d^6*e^6 + 264*(10*B*a*b^9 + A*b^10)* 
d^5*e^7 + 825*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^8 + 1100*(8*B*a^3*b^7 + 3*A* 
a^2*b^8)*d^3*e^9 + 660*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^10 + 168*(6*B*a^5 
*b^5 + 5*A*a^4*b^6)*d*e^11 + 14*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^12)*x^18 + 3 
/17*(264*B*b^10*d^7*e^5 + 308*(10*B*a*b^9 + A*b^10)*d^6*e^6 + 1320*(9*B*a^ 
2*b^8 + 2*A*a*b^9)*d^5*e^7 + 2475*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^8 + 22 
00*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^9 + 924*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d 
^2*e^10 + 168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^11 + 10*(4*B*a^7*b^3 + 7*A*a 
^6*b^4)*e^12)*x^17 + 3/16*(165*B*b^10*d^8*e^4 + 264*(10*B*a*b^9 + A*b^10)* 
d^7*e^5 + 1540*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^6 + 3960*(8*B*a^3*b^7 + ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4663 vs. \(2 (440) = 880\).

Time = 0.13 (sec) , antiderivative size = 4663, normalized size of antiderivative = 10.05 \[ \int (a+b x)^{10} (A+B x) (d+e x)^{12} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)*(e*x+d)^12,x, algorithm="giac")
 

Output:

1/24*B*b^10*e^12*x^24 + 12/23*B*b^10*d*e^11*x^23 + 10/23*B*a*b^9*e^12*x^23 
 + 1/23*A*b^10*e^12*x^23 + 3*B*b^10*d^2*e^10*x^22 + 60/11*B*a*b^9*d*e^11*x 
^22 + 6/11*A*b^10*d*e^11*x^22 + 45/22*B*a^2*b^8*e^12*x^22 + 5/11*A*a*b^9*e 
^12*x^22 + 220/21*B*b^10*d^3*e^9*x^21 + 220/7*B*a*b^9*d^2*e^10*x^21 + 22/7 
*A*b^10*d^2*e^10*x^21 + 180/7*B*a^2*b^8*d*e^11*x^21 + 40/7*A*a*b^9*d*e^11* 
x^21 + 40/7*B*a^3*b^7*e^12*x^21 + 15/7*A*a^2*b^8*e^12*x^21 + 99/4*B*b^10*d 
^4*e^8*x^20 + 110*B*a*b^9*d^3*e^9*x^20 + 11*A*b^10*d^3*e^9*x^20 + 297/2*B* 
a^2*b^8*d^2*e^10*x^20 + 33*A*a*b^9*d^2*e^10*x^20 + 72*B*a^3*b^7*d*e^11*x^2 
0 + 27*A*a^2*b^8*d*e^11*x^20 + 21/2*B*a^4*b^6*e^12*x^20 + 6*A*a^3*b^7*e^12 
*x^20 + 792/19*B*b^10*d^5*e^7*x^19 + 4950/19*B*a*b^9*d^4*e^8*x^19 + 495/19 
*A*b^10*d^4*e^8*x^19 + 9900/19*B*a^2*b^8*d^3*e^9*x^19 + 2200/19*A*a*b^9*d^ 
3*e^9*x^19 + 7920/19*B*a^3*b^7*d^2*e^10*x^19 + 2970/19*A*a^2*b^8*d^2*e^10* 
x^19 + 2520/19*B*a^4*b^6*d*e^11*x^19 + 1440/19*A*a^3*b^7*d*e^11*x^19 + 252 
/19*B*a^5*b^5*e^12*x^19 + 210/19*A*a^4*b^6*e^12*x^19 + 154/3*B*b^10*d^6*e^ 
6*x^18 + 440*B*a*b^9*d^5*e^7*x^18 + 44*A*b^10*d^5*e^7*x^18 + 2475/2*B*a^2* 
b^8*d^4*e^8*x^18 + 275*A*a*b^9*d^4*e^8*x^18 + 4400/3*B*a^3*b^7*d^3*e^9*x^1 
8 + 550*A*a^2*b^8*d^3*e^9*x^18 + 770*B*a^4*b^6*d^2*e^10*x^18 + 440*A*a^3*b 
^7*d^2*e^10*x^18 + 168*B*a^5*b^5*d*e^11*x^18 + 140*A*a^4*b^6*d*e^11*x^18 + 
 35/3*B*a^6*b^4*e^12*x^18 + 14*A*a^5*b^5*e^12*x^18 + 792/17*B*b^10*d^7*e^5 
*x^17 + 9240/17*B*a*b^9*d^6*e^6*x^17 + 924/17*A*b^10*d^6*e^6*x^17 + 356...
 

Mupad [B] (verification not implemented)

Time = 1.82 (sec) , antiderivative size = 3891, normalized size of antiderivative = 8.39 \[ \int (a+b x)^{10} (A+B x) (d+e x)^{12} \, dx=\text {Too large to display} \] Input:

int((A + B*x)*(a + b*x)^10*(d + e*x)^12,x)
 

Output:

x^5*(42*A*a^6*b^4*d^12 + 24*B*a^7*b^3*d^12 + 99*A*a^10*d^8*e^4 + 44*B*a^10 
*d^9*e^3 + 288*A*a^7*b^3*d^11*e + 440*A*a^9*b*d^9*e^3 + 108*B*a^8*b^2*d^11 
*e + 132*B*a^9*b*d^10*e^2 + 594*A*a^8*b^2*d^10*e^2) + x^8*(15*A*a^3*b^7*d^ 
12 + (105*B*a^4*b^6*d^12)/4 + 99*A*a^10*d^5*e^7 + (231*B*a^10*d^6*e^6)/2 + 
 315*A*a^4*b^6*d^11*e + 1155*A*a^9*b*d^6*e^6 + 378*B*a^5*b^5*d^11*e + 990* 
B*a^9*b*d^7*e^5 + 2079*A*a^5*b^5*d^10*e^2 + 5775*A*a^6*b^4*d^9*e^3 + 7425* 
A*a^7*b^3*d^8*e^4 + 4455*A*a^8*b^2*d^7*e^5 + (3465*B*a^6*b^4*d^10*e^2)/2 + 
 3300*B*a^7*b^3*d^9*e^3 + (22275*B*a^8*b^2*d^8*e^4)/8) + x^20*(6*A*a^3*b^7 
*e^12 + (21*B*a^4*b^6*e^12)/2 + 11*A*b^10*d^3*e^9 + (99*B*b^10*d^4*e^8)/4 
+ 33*A*a*b^9*d^2*e^10 + 27*A*a^2*b^8*d*e^11 + 110*B*a*b^9*d^3*e^9 + 72*B*a 
^3*b^7*d*e^11 + (297*B*a^2*b^8*d^2*e^10)/2) + x^17*((210*A*a^6*b^4*e^12)/1 
7 + (120*B*a^7*b^3*e^12)/17 + (924*A*b^10*d^6*e^6)/17 + (792*B*b^10*d^7*e^ 
5)/17 + (7920*A*a*b^9*d^5*e^7)/17 + (3024*A*a^5*b^5*d*e^11)/17 + (9240*B*a 
*b^9*d^6*e^6)/17 + (2520*B*a^6*b^4*d*e^11)/17 + (22275*A*a^2*b^8*d^4*e^8)/ 
17 + (26400*A*a^3*b^7*d^3*e^9)/17 + (13860*A*a^4*b^6*d^2*e^10)/17 + (35640 
*B*a^2*b^8*d^5*e^7)/17 + (59400*B*a^3*b^7*d^4*e^8)/17 + (46200*B*a^4*b^6*d 
^3*e^9)/17 + (16632*B*a^5*b^5*d^2*e^10)/17) + x^10*(A*a*b^9*d^12 + (9*B*a^ 
2*b^8*d^12)/2 + 22*A*a^10*d^3*e^9 + (99*B*a^10*d^4*e^8)/2 + 54*A*a^2*b^8*d 
^11*e + 495*A*a^9*b*d^4*e^8 + 144*B*a^3*b^7*d^11*e + 792*B*a^9*b*d^5*e^7 + 
 792*A*a^3*b^7*d^10*e^2 + 4620*A*a^4*b^6*d^9*e^3 + 12474*A*a^5*b^5*d^8*...
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 2399, normalized size of antiderivative = 5.17 \[ \int (a+b x)^{10} (A+B x) (d+e x)^{12} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^10*(B*x+A)*(e*x+d)^12,x)
 

Output:

(x*(32449872*a**11*d**12 + 194699232*a**11*d**11*e*x + 713897184*a**11*d** 
10*e**2*x**2 + 1784742960*a**11*d**9*e**3*x**3 + 3212537328*a**11*d**8*e** 
4*x**4 + 4283383104*a**11*d**7*e**5*x**5 + 4283383104*a**11*d**6*e**6*x**6 
 + 3212537328*a**11*d**5*e**7*x**7 + 1784742960*a**11*d**4*e**8*x**8 + 713 
897184*a**11*d**3*e**9*x**9 + 194699232*a**11*d**2*e**10*x**10 + 32449872* 
a**11*d*e**11*x**11 + 2496144*a**11*e**12*x**12 + 178474296*a**10*b*d**12* 
x + 1427794368*a**10*b*d**11*e*x**2 + 5889651768*a**10*b*d**10*e**2*x**3 + 
 15705738048*a**10*b*d**9*e**3*x**4 + 29448258840*a**10*b*d**8*e**4*x**5 + 
 40386183552*a**10*b*d**7*e**5*x**6 + 41227562376*a**10*b*d**6*e**6*x**7 + 
 31411476096*a**10*b*d**5*e**7*x**8 + 17668955304*a**10*b*d**4*e**8*x**9 + 
 7138971840*a**10*b*d**3*e**9*x**10 + 1963217256*a**10*b*d**2*e**10*x**11 
+ 329491008*a**10*b*d*e**11*x**12 + 25496328*a**10*b*e**12*x**13 + 5949143 
20*a**9*b**2*d**12*x**2 + 5354228880*a**9*b**2*d**11*e*x**3 + 23558607072* 
a**9*b**2*d**10*e**2*x**4 + 65440575200*a**9*b**2*d**9*e**3*x**5 + 1262068 
23600*a**9*b**2*d**8*e**4*x**6 + 176689553040*a**9*b**2*d**7*e**5*x**7 + 1 
83233610560*a**9*b**2*d**6*e**6*x**8 + 141351642432*a**9*b**2*d**5*e**7*x* 
*9 + 80313433200*a**9*b**2*d**4*e**8*x**10 + 32720287600*a**9*b**2*d**3*e* 
*9*x**11 + 9061002720*a**9*b**2*d**2*e**10*x**12 + 1529779680*a**9*b**2*d* 
e**11*x**13 + 118982864*a**9*b**2*e**12*x**14 + 1338557220*a**8*b**3*d**12 
*x**3 + 12850149312*a**8*b**3*d**11*e*x**4 + 58896517680*a**8*b**3*d**1...