\(\int (a+b x)^{11} (c+d x)^{10} \, dx\) [94]

Optimal result

Integrand size = 15, antiderivative size = 279 \[ \int (a+b x)^{11} (c+d x)^{10} \, dx=\frac {(b c-a d)^{10} (a+b x)^{12}}{12 b^{11}}+\frac {10 d (b c-a d)^9 (a+b x)^{13}}{13 b^{11}}+\frac {45 d^2 (b c-a d)^8 (a+b x)^{14}}{14 b^{11}}+\frac {8 d^3 (b c-a d)^7 (a+b x)^{15}}{b^{11}}+\frac {105 d^4 (b c-a d)^6 (a+b x)^{16}}{8 b^{11}}+\frac {252 d^5 (b c-a d)^5 (a+b x)^{17}}{17 b^{11}}+\frac {35 d^6 (b c-a d)^4 (a+b x)^{18}}{3 b^{11}}+\frac {120 d^7 (b c-a d)^3 (a+b x)^{19}}{19 b^{11}}+\frac {9 d^8 (b c-a d)^2 (a+b x)^{20}}{4 b^{11}}+\frac {10 d^9 (b c-a d) (a+b x)^{21}}{21 b^{11}}+\frac {d^{10} (a+b x)^{22}}{22 b^{11}} \] Output:

1/12*(-a*d+b*c)^10*(b*x+a)^12/b^11+10/13*d*(-a*d+b*c)^9*(b*x+a)^13/b^11+45 
/14*d^2*(-a*d+b*c)^8*(b*x+a)^14/b^11+8*d^3*(-a*d+b*c)^7*(b*x+a)^15/b^11+10 
5/8*d^4*(-a*d+b*c)^6*(b*x+a)^16/b^11+252/17*d^5*(-a*d+b*c)^5*(b*x+a)^17/b^ 
11+35/3*d^6*(-a*d+b*c)^4*(b*x+a)^18/b^11+120/19*d^7*(-a*d+b*c)^3*(b*x+a)^1 
9/b^11+9/4*d^8*(-a*d+b*c)^2*(b*x+a)^20/b^11+10/21*d^9*(-a*d+b*c)*(b*x+a)^2 
1/b^11+1/22*d^10*(b*x+a)^22/b^11
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1702\) vs. \(2(279)=558\).

Time = 0.11 (sec) , antiderivative size = 1702, normalized size of antiderivative = 6.10 \[ \int (a+b x)^{11} (c+d x)^{10} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*x)^11*(c + d*x)^10,x]
 

Output:

a^11*c^10*x + (a^10*c^9*(11*b*c + 10*a*d)*x^2)/2 + (5*a^9*c^8*(11*b^2*c^2 
+ 22*a*b*c*d + 9*a^2*d^2)*x^3)/3 + (5*a^8*c^7*(33*b^3*c^3 + 110*a*b^2*c^2* 
d + 99*a^2*b*c*d^2 + 24*a^3*d^3)*x^4)/4 + 3*a^7*c^6*(22*b^4*c^4 + 110*a*b^ 
3*c^3*d + 165*a^2*b^2*c^2*d^2 + 88*a^3*b*c*d^3 + 14*a^4*d^4)*x^5 + (a^6*c^ 
5*(154*b^5*c^5 + 1100*a*b^4*c^4*d + 2475*a^2*b^3*c^3*d^2 + 2200*a^3*b^2*c^ 
2*d^3 + 770*a^4*b*c*d^4 + 84*a^5*d^5)*x^6)/2 + (6*a^5*c^4*(77*b^6*c^6 + 77 
0*a*b^5*c^5*d + 2475*a^2*b^4*c^4*d^2 + 3300*a^3*b^3*c^3*d^3 + 1925*a^4*b^2 
*c^2*d^4 + 462*a^5*b*c*d^5 + 35*a^6*d^6)*x^7)/7 + (15*a^4*c^3*(11*b^7*c^7 
+ 154*a*b^6*c^6*d + 693*a^2*b^5*c^5*d^2 + 1320*a^3*b^4*c^4*d^3 + 1155*a^4* 
b^3*c^3*d^4 + 462*a^5*b^2*c^2*d^5 + 77*a^6*b*c*d^6 + 4*a^7*d^7)*x^8)/4 + ( 
5*a^3*c^2*(11*b^8*c^8 + 220*a*b^7*c^7*d + 1386*a^2*b^6*c^6*d^2 + 3696*a^3* 
b^5*c^5*d^3 + 4620*a^4*b^4*c^4*d^4 + 2772*a^5*b^3*c^3*d^5 + 770*a^6*b^2*c^ 
2*d^6 + 88*a^7*b*c*d^7 + 3*a^8*d^8)*x^9)/3 + (a^2*c*(11*b^9*c^9 + 330*a*b^ 
8*c^8*d + 2970*a^2*b^7*c^7*d^2 + 11088*a^3*b^6*c^6*d^3 + 19404*a^4*b^5*c^5 
*d^4 + 16632*a^5*b^4*c^4*d^5 + 6930*a^6*b^3*c^3*d^6 + 1320*a^7*b^2*c^2*d^7 
 + 99*a^8*b*c*d^8 + 2*a^9*d^9)*x^10)/2 + (a*(11*b^10*c^10 + 550*a*b^9*c^9* 
d + 7425*a^2*b^8*c^8*d^2 + 39600*a^3*b^7*c^7*d^3 + 97020*a^4*b^6*c^6*d^4 + 
 116424*a^5*b^5*c^5*d^5 + 69300*a^6*b^4*c^4*d^6 + 19800*a^7*b^3*c^3*d^7 + 
2475*a^8*b^2*c^2*d^8 + 110*a^9*b*c*d^9 + a^10*d^10)*x^11)/11 + (b*(b^10*c^ 
10 + 110*a*b^9*c^9*d + 2475*a^2*b^8*c^8*d^2 + 19800*a^3*b^7*c^7*d^3 + 6...
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {49, 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.

\(\displaystyle \int (a+b x)^{11} (c+d x)^{10} \, dx\)

\(\Big \downarrow \) 49

\(\displaystyle \int \left (\frac {10 d^9 (a+b x)^{20} (b c-a d)}{b^{10}}+\frac {45 d^8 (a+b x)^{19} (b c-a d)^2}{b^{10}}+\frac {120 d^7 (a+b x)^{18} (b c-a d)^3}{b^{10}}+\frac {210 d^6 (a+b x)^{17} (b c-a d)^4}{b^{10}}+\frac {252 d^5 (a+b x)^{16} (b c-a d)^5}{b^{10}}+\frac {210 d^4 (a+b x)^{15} (b c-a d)^6}{b^{10}}+\frac {120 d^3 (a+b x)^{14} (b c-a d)^7}{b^{10}}+\frac {45 d^2 (a+b x)^{13} (b c-a d)^8}{b^{10}}+\frac {10 d (a+b x)^{12} (b c-a d)^9}{b^{10}}+\frac {(a+b x)^{11} (b c-a d)^{10}}{b^{10}}+\frac {d^{10} (a+b x)^{21}}{b^{10}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {10 d^9 (a+b x)^{21} (b c-a d)}{21 b^{11}}+\frac {9 d^8 (a+b x)^{20} (b c-a d)^2}{4 b^{11}}+\frac {120 d^7 (a+b x)^{19} (b c-a d)^3}{19 b^{11}}+\frac {35 d^6 (a+b x)^{18} (b c-a d)^4}{3 b^{11}}+\frac {252 d^5 (a+b x)^{17} (b c-a d)^5}{17 b^{11}}+\frac {105 d^4 (a+b x)^{16} (b c-a d)^6}{8 b^{11}}+\frac {8 d^3 (a+b x)^{15} (b c-a d)^7}{b^{11}}+\frac {45 d^2 (a+b x)^{14} (b c-a d)^8}{14 b^{11}}+\frac {10 d (a+b x)^{13} (b c-a d)^9}{13 b^{11}}+\frac {(a+b x)^{12} (b c-a d)^{10}}{12 b^{11}}+\frac {d^{10} (a+b x)^{22}}{22 b^{11}}\)

Input:

Int[(a + b*x)^11*(c + d*x)^10,x]
 

Output:

((b*c - a*d)^10*(a + b*x)^12)/(12*b^11) + (10*d*(b*c - a*d)^9*(a + b*x)^13 
)/(13*b^11) + (45*d^2*(b*c - a*d)^8*(a + b*x)^14)/(14*b^11) + (8*d^3*(b*c 
- a*d)^7*(a + b*x)^15)/b^11 + (105*d^4*(b*c - a*d)^6*(a + b*x)^16)/(8*b^11 
) + (252*d^5*(b*c - a*d)^5*(a + b*x)^17)/(17*b^11) + (35*d^6*(b*c - a*d)^4 
*(a + b*x)^18)/(3*b^11) + (120*d^7*(b*c - a*d)^3*(a + b*x)^19)/(19*b^11) + 
 (9*d^8*(b*c - a*d)^2*(a + b*x)^20)/(4*b^11) + (10*d^9*(b*c - a*d)*(a + b* 
x)^21)/(21*b^11) + (d^10*(a + b*x)^22)/(22*b^11)
 

Defintions of rubi rules used

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

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. \(1720\) vs. \(2(259)=518\).

Time = 0.14 (sec) , antiderivative size = 1721, normalized size of antiderivative = 6.17

method	result	size
norman	\(\text {Expression too large to display}\)	\(1721\)
default	\(\text {Expression too large to display}\)	\(1741\)
gosper	\(\text {Expression too large to display}\)	\(2011\)
risch	\(\text {Expression too large to display}\)	\(2011\)
parallelrisch	\(\text {Expression too large to display}\)	\(2011\)
orering	\(\text {Expression too large to display}\)	\(2012\)

Input:

int((b*x+a)^11*(d*x+c)^10,x,method=_RETURNVERBOSE)
 

Output:

a^11*c^10*x+(5*a^11*c^9*d+11/2*a^10*b*c^10)*x^2+(15*a^11*c^8*d^2+110/3*a^1 
0*b*c^9*d+55/3*b^2*a^9*c^10)*x^3+(30*a^11*c^7*d^3+495/4*a^10*b*c^8*d^2+275 
/2*b^2*a^9*c^9*d+165/4*b^3*a^8*c^10)*x^4+(42*a^11*c^6*d^4+264*a^10*b*c^7*d 
^3+495*a^9*b^2*c^8*d^2+330*a^8*b^3*c^9*d+66*a^7*b^4*c^10)*x^5+(42*a^11*c^5 
*d^5+385*a^10*b*c^6*d^4+1100*b^2*a^9*c^7*d^3+2475/2*b^3*a^8*c^8*d^2+550*a^ 
7*b^4*c^9*d+77*a^6*b^5*c^10)*x^6+(30*a^11*c^4*d^6+396*a^10*b*c^5*d^5+1650* 
b^2*a^9*c^6*d^4+19800/7*b^3*a^8*c^7*d^3+14850/7*a^7*b^4*c^8*d^2+660*a^6*b^ 
5*c^9*d+66*a^5*b^6*c^10)*x^7+(15*a^11*c^3*d^7+1155/4*a^10*b*c^4*d^6+3465/2 
*b^2*a^9*c^5*d^5+17325/4*b^3*a^8*c^6*d^4+4950*a^7*b^4*c^7*d^3+10395/4*a^6* 
b^5*c^8*d^2+1155/2*a^5*b^6*c^9*d+165/4*a^4*b^7*c^10)*x^8+(5*a^11*c^2*d^8+4 
40/3*a^10*b*c^3*d^7+3850/3*b^2*a^9*c^4*d^6+4620*b^3*a^8*c^5*d^5+7700*a^7*b 
^4*c^6*d^4+6160*a^6*b^5*c^7*d^3+2310*a^5*b^6*c^8*d^2+1100/3*a^4*b^7*c^9*d+ 
55/3*a^3*b^8*c^10)*x^9+(a^11*c*d^9+99/2*a^10*b*c^2*d^8+660*b^2*a^9*c^3*d^7 
+3465*b^3*a^8*c^4*d^6+8316*a^7*b^4*c^5*d^5+9702*a^6*b^5*c^6*d^4+5544*a^5*b 
^6*c^7*d^3+1485*a^4*b^7*c^8*d^2+165*a^3*b^8*c^9*d+11/2*a^2*b^9*c^10)*x^10+ 
(1/11*a^11*d^10+10*a^10*b*c*d^9+225*b^2*a^9*c^2*d^8+1800*b^3*a^8*c^3*d^7+6 
300*a^7*b^4*c^4*d^6+10584*a^6*b^5*c^5*d^5+8820*a^5*b^6*c^6*d^4+3600*a^4*b^ 
7*c^7*d^3+675*a^3*b^8*c^8*d^2+50*a^2*b^9*c^9*d+a*b^10*c^10)*x^11+(11/12*a^ 
10*b*d^10+275/6*b^2*a^9*c*d^9+2475/4*b^3*a^8*c^2*d^8+3300*a^7*b^4*c^3*d^7+ 
8085*a^6*b^5*c^4*d^6+9702*a^5*b^6*c^5*d^5+5775*a^4*b^7*c^6*d^4+1650*a^3...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1740 vs. \(2 (259) = 518\).

Time = 0.08 (sec) , antiderivative size = 1740, normalized size of antiderivative = 6.24 \[ \int (a+b x)^{11} (c+d x)^{10} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^11*(d*x+c)^10,x, algorithm="fricas")
 

Output:

1/22*b^11*d^10*x^22 + a^11*c^10*x + 1/21*(10*b^11*c*d^9 + 11*a*b^10*d^10)* 
x^21 + 1/4*(9*b^11*c^2*d^8 + 22*a*b^10*c*d^9 + 11*a^2*b^9*d^10)*x^20 + 5/1 
9*(24*b^11*c^3*d^7 + 99*a*b^10*c^2*d^8 + 110*a^2*b^9*c*d^9 + 33*a^3*b^8*d^ 
10)*x^19 + 5/6*(14*b^11*c^4*d^6 + 88*a*b^10*c^3*d^7 + 165*a^2*b^9*c^2*d^8 
+ 110*a^3*b^8*c*d^9 + 22*a^4*b^7*d^10)*x^18 + 3/17*(84*b^11*c^5*d^5 + 770* 
a*b^10*c^4*d^6 + 2200*a^2*b^9*c^3*d^7 + 2475*a^3*b^8*c^2*d^8 + 1100*a^4*b^ 
7*c*d^9 + 154*a^5*b^6*d^10)*x^17 + 3/8*(35*b^11*c^6*d^4 + 462*a*b^10*c^5*d 
^5 + 1925*a^2*b^9*c^4*d^6 + 3300*a^3*b^8*c^3*d^7 + 2475*a^4*b^7*c^2*d^8 + 
770*a^5*b^6*c*d^9 + 77*a^6*b^5*d^10)*x^16 + 2*(4*b^11*c^7*d^3 + 77*a*b^10* 
c^6*d^4 + 462*a^2*b^9*c^5*d^5 + 1155*a^3*b^8*c^4*d^6 + 1320*a^4*b^7*c^3*d^ 
7 + 693*a^5*b^6*c^2*d^8 + 154*a^6*b^5*c*d^9 + 11*a^7*b^4*d^10)*x^15 + 15/1 
4*(3*b^11*c^8*d^2 + 88*a*b^10*c^7*d^3 + 770*a^2*b^9*c^6*d^4 + 2772*a^3*b^8 
*c^5*d^5 + 4620*a^4*b^7*c^4*d^6 + 3696*a^5*b^6*c^3*d^7 + 1386*a^6*b^5*c^2* 
d^8 + 220*a^7*b^4*c*d^9 + 11*a^8*b^3*d^10)*x^14 + 5/13*(2*b^11*c^9*d + 99* 
a*b^10*c^8*d^2 + 1320*a^2*b^9*c^7*d^3 + 6930*a^3*b^8*c^6*d^4 + 16632*a^4*b 
^7*c^5*d^5 + 19404*a^5*b^6*c^4*d^6 + 11088*a^6*b^5*c^3*d^7 + 2970*a^7*b^4* 
c^2*d^8 + 330*a^8*b^3*c*d^9 + 11*a^9*b^2*d^10)*x^13 + 1/12*(b^11*c^10 + 11 
0*a*b^10*c^9*d + 2475*a^2*b^9*c^8*d^2 + 19800*a^3*b^8*c^7*d^3 + 69300*a^4* 
b^7*c^6*d^4 + 116424*a^5*b^6*c^5*d^5 + 97020*a^6*b^5*c^4*d^6 + 39600*a^7*b 
^4*c^3*d^7 + 7425*a^8*b^3*c^2*d^8 + 550*a^9*b^2*c*d^9 + 11*a^10*b*d^10)...
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1965 vs. \(2 (258) = 516\).

Time = 0.12 (sec) , antiderivative size = 1965, normalized size of antiderivative = 7.04 \[ \int (a+b x)^{11} (c+d x)^{10} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)**11*(d*x+c)**10,x)
 

Output:

a**11*c**10*x + b**11*d**10*x**22/22 + x**21*(11*a*b**10*d**10/21 + 10*b** 
11*c*d**9/21) + x**20*(11*a**2*b**9*d**10/4 + 11*a*b**10*c*d**9/2 + 9*b**1 
1*c**2*d**8/4) + x**19*(165*a**3*b**8*d**10/19 + 550*a**2*b**9*c*d**9/19 + 
 495*a*b**10*c**2*d**8/19 + 120*b**11*c**3*d**7/19) + x**18*(55*a**4*b**7* 
d**10/3 + 275*a**3*b**8*c*d**9/3 + 275*a**2*b**9*c**2*d**8/2 + 220*a*b**10 
*c**3*d**7/3 + 35*b**11*c**4*d**6/3) + x**17*(462*a**5*b**6*d**10/17 + 330 
0*a**4*b**7*c*d**9/17 + 7425*a**3*b**8*c**2*d**8/17 + 6600*a**2*b**9*c**3* 
d**7/17 + 2310*a*b**10*c**4*d**6/17 + 252*b**11*c**5*d**5/17) + x**16*(231 
*a**6*b**5*d**10/8 + 1155*a**5*b**6*c*d**9/4 + 7425*a**4*b**7*c**2*d**8/8 
+ 2475*a**3*b**8*c**3*d**7/2 + 5775*a**2*b**9*c**4*d**6/8 + 693*a*b**10*c* 
*5*d**5/4 + 105*b**11*c**6*d**4/8) + x**15*(22*a**7*b**4*d**10 + 308*a**6* 
b**5*c*d**9 + 1386*a**5*b**6*c**2*d**8 + 2640*a**4*b**7*c**3*d**7 + 2310*a 
**3*b**8*c**4*d**6 + 924*a**2*b**9*c**5*d**5 + 154*a*b**10*c**6*d**4 + 8*b 
**11*c**7*d**3) + x**14*(165*a**8*b**3*d**10/14 + 1650*a**7*b**4*c*d**9/7 
+ 1485*a**6*b**5*c**2*d**8 + 3960*a**5*b**6*c**3*d**7 + 4950*a**4*b**7*c** 
4*d**6 + 2970*a**3*b**8*c**5*d**5 + 825*a**2*b**9*c**6*d**4 + 660*a*b**10* 
c**7*d**3/7 + 45*b**11*c**8*d**2/14) + x**13*(55*a**9*b**2*d**10/13 + 1650 
*a**8*b**3*c*d**9/13 + 14850*a**7*b**4*c**2*d**8/13 + 55440*a**6*b**5*c**3 
*d**7/13 + 97020*a**5*b**6*c**4*d**6/13 + 83160*a**4*b**7*c**5*d**5/13 + 3 
4650*a**3*b**8*c**6*d**4/13 + 6600*a**2*b**9*c**7*d**3/13 + 495*a*b**10...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1740 vs. \(2 (259) = 518\).

Time = 0.05 (sec) , antiderivative size = 1740, normalized size of antiderivative = 6.24 \[ \int (a+b x)^{11} (c+d x)^{10} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^11*(d*x+c)^10,x, algorithm="maxima")
 

Output:

1/22*b^11*d^10*x^22 + a^11*c^10*x + 1/21*(10*b^11*c*d^9 + 11*a*b^10*d^10)* 
x^21 + 1/4*(9*b^11*c^2*d^8 + 22*a*b^10*c*d^9 + 11*a^2*b^9*d^10)*x^20 + 5/1 
9*(24*b^11*c^3*d^7 + 99*a*b^10*c^2*d^8 + 110*a^2*b^9*c*d^9 + 33*a^3*b^8*d^ 
10)*x^19 + 5/6*(14*b^11*c^4*d^6 + 88*a*b^10*c^3*d^7 + 165*a^2*b^9*c^2*d^8 
+ 110*a^3*b^8*c*d^9 + 22*a^4*b^7*d^10)*x^18 + 3/17*(84*b^11*c^5*d^5 + 770* 
a*b^10*c^4*d^6 + 2200*a^2*b^9*c^3*d^7 + 2475*a^3*b^8*c^2*d^8 + 1100*a^4*b^ 
7*c*d^9 + 154*a^5*b^6*d^10)*x^17 + 3/8*(35*b^11*c^6*d^4 + 462*a*b^10*c^5*d 
^5 + 1925*a^2*b^9*c^4*d^6 + 3300*a^3*b^8*c^3*d^7 + 2475*a^4*b^7*c^2*d^8 + 
770*a^5*b^6*c*d^9 + 77*a^6*b^5*d^10)*x^16 + 2*(4*b^11*c^7*d^3 + 77*a*b^10* 
c^6*d^4 + 462*a^2*b^9*c^5*d^5 + 1155*a^3*b^8*c^4*d^6 + 1320*a^4*b^7*c^3*d^ 
7 + 693*a^5*b^6*c^2*d^8 + 154*a^6*b^5*c*d^9 + 11*a^7*b^4*d^10)*x^15 + 15/1 
4*(3*b^11*c^8*d^2 + 88*a*b^10*c^7*d^3 + 770*a^2*b^9*c^6*d^4 + 2772*a^3*b^8 
*c^5*d^5 + 4620*a^4*b^7*c^4*d^6 + 3696*a^5*b^6*c^3*d^7 + 1386*a^6*b^5*c^2* 
d^8 + 220*a^7*b^4*c*d^9 + 11*a^8*b^3*d^10)*x^14 + 5/13*(2*b^11*c^9*d + 99* 
a*b^10*c^8*d^2 + 1320*a^2*b^9*c^7*d^3 + 6930*a^3*b^8*c^6*d^4 + 16632*a^4*b 
^7*c^5*d^5 + 19404*a^5*b^6*c^4*d^6 + 11088*a^6*b^5*c^3*d^7 + 2970*a^7*b^4* 
c^2*d^8 + 330*a^8*b^3*c*d^9 + 11*a^9*b^2*d^10)*x^13 + 1/12*(b^11*c^10 + 11 
0*a*b^10*c^9*d + 2475*a^2*b^9*c^8*d^2 + 19800*a^3*b^8*c^7*d^3 + 69300*a^4* 
b^7*c^6*d^4 + 116424*a^5*b^6*c^5*d^5 + 97020*a^6*b^5*c^4*d^6 + 39600*a^7*b 
^4*c^3*d^7 + 7425*a^8*b^3*c^2*d^8 + 550*a^9*b^2*c*d^9 + 11*a^10*b*d^10)...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2010 vs. \(2 (259) = 518\).

Time = 0.13 (sec) , antiderivative size = 2010, normalized size of antiderivative = 7.20 \[ \int (a+b x)^{11} (c+d x)^{10} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^11*(d*x+c)^10,x, algorithm="giac")
 

Output:

1/22*b^11*d^10*x^22 + 10/21*b^11*c*d^9*x^21 + 11/21*a*b^10*d^10*x^21 + 9/4 
*b^11*c^2*d^8*x^20 + 11/2*a*b^10*c*d^9*x^20 + 11/4*a^2*b^9*d^10*x^20 + 120 
/19*b^11*c^3*d^7*x^19 + 495/19*a*b^10*c^2*d^8*x^19 + 550/19*a^2*b^9*c*d^9* 
x^19 + 165/19*a^3*b^8*d^10*x^19 + 35/3*b^11*c^4*d^6*x^18 + 220/3*a*b^10*c^ 
3*d^7*x^18 + 275/2*a^2*b^9*c^2*d^8*x^18 + 275/3*a^3*b^8*c*d^9*x^18 + 55/3* 
a^4*b^7*d^10*x^18 + 252/17*b^11*c^5*d^5*x^17 + 2310/17*a*b^10*c^4*d^6*x^17 
 + 6600/17*a^2*b^9*c^3*d^7*x^17 + 7425/17*a^3*b^8*c^2*d^8*x^17 + 3300/17*a 
^4*b^7*c*d^9*x^17 + 462/17*a^5*b^6*d^10*x^17 + 105/8*b^11*c^6*d^4*x^16 + 6 
93/4*a*b^10*c^5*d^5*x^16 + 5775/8*a^2*b^9*c^4*d^6*x^16 + 2475/2*a^3*b^8*c^ 
3*d^7*x^16 + 7425/8*a^4*b^7*c^2*d^8*x^16 + 1155/4*a^5*b^6*c*d^9*x^16 + 231 
/8*a^6*b^5*d^10*x^16 + 8*b^11*c^7*d^3*x^15 + 154*a*b^10*c^6*d^4*x^15 + 924 
*a^2*b^9*c^5*d^5*x^15 + 2310*a^3*b^8*c^4*d^6*x^15 + 2640*a^4*b^7*c^3*d^7*x 
^15 + 1386*a^5*b^6*c^2*d^8*x^15 + 308*a^6*b^5*c*d^9*x^15 + 22*a^7*b^4*d^10 
*x^15 + 45/14*b^11*c^8*d^2*x^14 + 660/7*a*b^10*c^7*d^3*x^14 + 825*a^2*b^9* 
c^6*d^4*x^14 + 2970*a^3*b^8*c^5*d^5*x^14 + 4950*a^4*b^7*c^4*d^6*x^14 + 396 
0*a^5*b^6*c^3*d^7*x^14 + 1485*a^6*b^5*c^2*d^8*x^14 + 1650/7*a^7*b^4*c*d^9* 
x^14 + 165/14*a^8*b^3*d^10*x^14 + 10/13*b^11*c^9*d*x^13 + 495/13*a*b^10*c^ 
8*d^2*x^13 + 6600/13*a^2*b^9*c^7*d^3*x^13 + 34650/13*a^3*b^8*c^6*d^4*x^13 
+ 83160/13*a^4*b^7*c^5*d^5*x^13 + 97020/13*a^5*b^6*c^4*d^6*x^13 + 55440/13 
*a^6*b^5*c^3*d^7*x^13 + 14850/13*a^7*b^4*c^2*d^8*x^13 + 1650/13*a^8*b^3...
 

Mupad [B] (verification not implemented)

Time = 0.63 (sec) , antiderivative size = 1702, normalized size of antiderivative = 6.10 \[ \int (a+b x)^{11} (c+d x)^{10} \, dx=\text {Too large to display} \] Input:

int((a + b*x)^11*(c + d*x)^10,x)
 

Output:

x^7*(66*a^5*b^6*c^10 + 30*a^11*c^4*d^6 + 660*a^6*b^5*c^9*d + 396*a^10*b*c^ 
5*d^5 + (14850*a^7*b^4*c^8*d^2)/7 + (19800*a^8*b^3*c^7*d^3)/7 + 1650*a^9*b 
^2*c^6*d^4) + x^16*((231*a^6*b^5*d^10)/8 + (105*b^11*c^6*d^4)/8 + (693*a*b 
^10*c^5*d^5)/4 + (1155*a^5*b^6*c*d^9)/4 + (5775*a^2*b^9*c^4*d^6)/8 + (2475 
*a^3*b^8*c^3*d^7)/2 + (7425*a^4*b^7*c^2*d^8)/8) + x^11*((a^11*d^10)/11 + a 
*b^10*c^10 + 50*a^2*b^9*c^9*d + 675*a^3*b^8*c^8*d^2 + 3600*a^4*b^7*c^7*d^3 
 + 8820*a^5*b^6*c^6*d^4 + 10584*a^6*b^5*c^5*d^5 + 6300*a^7*b^4*c^4*d^6 + 1 
800*a^8*b^3*c^3*d^7 + 225*a^9*b^2*c^2*d^8 + 10*a^10*b*c*d^9) + x^12*((b^11 
*c^10)/12 + (11*a^10*b*d^10)/12 + (275*a^9*b^2*c*d^9)/6 + (825*a^2*b^9*c^8 
*d^2)/4 + 1650*a^3*b^8*c^7*d^3 + 5775*a^4*b^7*c^6*d^4 + 9702*a^5*b^6*c^5*d 
^5 + 8085*a^6*b^5*c^4*d^6 + 3300*a^7*b^4*c^3*d^7 + (2475*a^8*b^3*c^2*d^8)/ 
4 + (55*a*b^10*c^9*d)/6) + x^5*(66*a^7*b^4*c^10 + 42*a^11*c^6*d^4 + 330*a^ 
8*b^3*c^9*d + 264*a^10*b*c^7*d^3 + 495*a^9*b^2*c^8*d^2) + x^18*((55*a^4*b^ 
7*d^10)/3 + (35*b^11*c^4*d^6)/3 + (220*a*b^10*c^3*d^7)/3 + (275*a^3*b^8*c* 
d^9)/3 + (275*a^2*b^9*c^2*d^8)/2) + x^8*((165*a^4*b^7*c^10)/4 + 15*a^11*c^ 
3*d^7 + (1155*a^5*b^6*c^9*d)/2 + (1155*a^10*b*c^4*d^6)/4 + (10395*a^6*b^5* 
c^8*d^2)/4 + 4950*a^7*b^4*c^7*d^3 + (17325*a^8*b^3*c^6*d^4)/4 + (3465*a^9* 
b^2*c^5*d^5)/2) + x^15*(22*a^7*b^4*d^10 + 8*b^11*c^7*d^3 + 154*a*b^10*c^6* 
d^4 + 308*a^6*b^5*c*d^9 + 924*a^2*b^9*c^5*d^5 + 2310*a^3*b^8*c^4*d^6 + 264 
0*a^4*b^7*c^3*d^7 + 1386*a^5*b^6*c^2*d^8) + x^6*(77*a^6*b^5*c^10 + 42*a...
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 2011, normalized size of antiderivative = 7.21 \[ \int (a+b x)^{11} (c+d x)^{10} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^11*(d*x+c)^10,x)
 

Output:

(x*(7759752*a**11*c**10 + 38798760*a**11*c**9*d*x + 116396280*a**11*c**8*d 
**2*x**2 + 232792560*a**11*c**7*d**3*x**3 + 325909584*a**11*c**6*d**4*x**4 
 + 325909584*a**11*c**5*d**5*x**5 + 232792560*a**11*c**4*d**6*x**6 + 11639 
6280*a**11*c**3*d**7*x**7 + 38798760*a**11*c**2*d**8*x**8 + 7759752*a**11* 
c*d**9*x**9 + 705432*a**11*d**10*x**10 + 42678636*a**10*b*c**10*x + 284524 
240*a**10*b*c**9*d*x**2 + 960269310*a**10*b*c**8*d**2*x**3 + 2048574528*a* 
*10*b*c**7*d**3*x**4 + 2987504520*a**10*b*c**6*d**4*x**5 + 3072861792*a**1 
0*b*c**5*d**5*x**6 + 2240628390*a**10*b*c**4*d**6*x**7 + 1138096960*a**10* 
b*c**3*d**7*x**8 + 384107724*a**10*b*c**2*d**8*x**9 + 77597520*a**10*b*c*d 
**9*x**10 + 7113106*a**10*b*d**10*x**11 + 142262120*a**9*b**2*c**10*x**2 + 
 1066965900*a**9*b**2*c**9*d*x**3 + 3841077240*a**9*b**2*c**8*d**2*x**4 + 
8535727200*a**9*b**2*c**7*d**3*x**5 + 12803590800*a**9*b**2*c**6*d**4*x**6 
 + 13443770340*a**9*b**2*c**5*d**5*x**7 + 9958348400*a**9*b**2*c**4*d**6*x 
**8 + 5121436320*a**9*b**2*c**3*d**7*x**9 + 1745944200*a**9*b**2*c**2*d**8 
*x**10 + 355655300*a**9*b**2*c*d**9*x**11 + 32829720*a**9*b**2*d**10*x**12 
 + 320089770*a**8*b**3*c**10*x**3 + 2560718160*a**8*b**3*c**9*d*x**4 + 960 
2693100*a**8*b**3*c**8*d**2*x**5 + 21949012800*a**8*b**3*c**7*d**3*x**6 + 
33609425850*a**8*b**3*c**6*d**4*x**7 + 35850054240*a**8*b**3*c**5*d**5*x** 
8 + 26887540680*a**8*b**3*c**4*d**6*x**9 + 13967553600*a**8*b**3*c**3*d**7 
*x**10 + 4801346550*a**8*b**3*c**2*d**8*x**11 + 984891600*a**8*b**3*c*d...