\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx\) [99]

Optimal result

Integrand size = 20, antiderivative size = 462 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx=\frac {(b d-a e)^{10} (B d-A e)}{20 e^{12} (d+e x)^{20}}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{19 e^{12} (d+e x)^{19}}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{18 e^{12} (d+e x)^{18}}-\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{17 e^{12} (d+e x)^{17}}+\frac {15 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{8 e^{12} (d+e x)^{16}}-\frac {14 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{5 e^{12} (d+e x)^{15}}+\frac {3 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{e^{12} (d+e x)^{14}}-\frac {30 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{13 e^{12} (d+e x)^{13}}+\frac {5 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{4 e^{12} (d+e x)^{12}}-\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e)}{11 e^{12} (d+e x)^{11}}+\frac {b^9 (11 b B d-A b e-10 a B e)}{10 e^{12} (d+e x)^{10}}-\frac {b^{10} B}{9 e^{12} (d+e x)^9} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1428\) vs. \(2(462)=924\).

Time = 0.52 (sec) , antiderivative size = 1428, normalized size of antiderivative = 3.09 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx =\text {Too large to display} \] Input:

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^21,x]
 

Output:

-1/16628040*(43758*a^10*e^10*(19*A*e + B*(d + 20*e*x)) + 48620*a^9*b*e^9*( 
9*A*e*(d + 20*e*x) + B*(d^2 + 20*d*e*x + 190*e^2*x^2)) + 12870*a^8*b^2*e^8 
*(17*A*e*(d^2 + 20*d*e*x + 190*e^2*x^2) + 3*B*(d^3 + 20*d^2*e*x + 190*d*e^ 
2*x^2 + 1140*e^3*x^3)) + 25740*a^7*b^3*e^7*(4*A*e*(d^3 + 20*d^2*e*x + 190* 
d*e^2*x^2 + 1140*e^3*x^3) + B*(d^4 + 20*d^3*e*x + 190*d^2*e^2*x^2 + 1140*d 
*e^3*x^3 + 4845*e^4*x^4)) + 15015*a^6*b^4*e^6*(3*A*e*(d^4 + 20*d^3*e*x + 1 
90*d^2*e^2*x^2 + 1140*d*e^3*x^3 + 4845*e^4*x^4) + B*(d^5 + 20*d^4*e*x + 19 
0*d^3*e^2*x^2 + 1140*d^2*e^3*x^3 + 4845*d*e^4*x^4 + 15504*e^5*x^5)) + 2574 
*a^5*b^5*e^5*(7*A*e*(d^5 + 20*d^4*e*x + 190*d^3*e^2*x^2 + 1140*d^2*e^3*x^3 
 + 4845*d*e^4*x^4 + 15504*e^5*x^5) + 3*B*(d^6 + 20*d^5*e*x + 190*d^4*e^2*x 
^2 + 1140*d^3*e^3*x^3 + 4845*d^2*e^4*x^4 + 15504*d*e^5*x^5 + 38760*e^6*x^6 
)) + 495*a^4*b^6*e^4*(13*A*e*(d^6 + 20*d^5*e*x + 190*d^4*e^2*x^2 + 1140*d^ 
3*e^3*x^3 + 4845*d^2*e^4*x^4 + 15504*d*e^5*x^5 + 38760*e^6*x^6) + 7*B*(d^7 
 + 20*d^6*e*x + 190*d^5*e^2*x^2 + 1140*d^4*e^3*x^3 + 4845*d^3*e^4*x^4 + 15 
504*d^2*e^5*x^5 + 38760*d*e^6*x^6 + 77520*e^7*x^7)) + 660*a^3*b^7*e^3*(3*A 
*e*(d^7 + 20*d^6*e*x + 190*d^5*e^2*x^2 + 1140*d^4*e^3*x^3 + 4845*d^3*e^4*x 
^4 + 15504*d^2*e^5*x^5 + 38760*d*e^6*x^6 + 77520*e^7*x^7) + 2*B*(d^8 + 20* 
d^7*e*x + 190*d^6*e^2*x^2 + 1140*d^5*e^3*x^3 + 4845*d^4*e^4*x^4 + 15504*d^ 
3*e^5*x^5 + 38760*d^2*e^6*x^6 + 77520*d*e^7*x^7 + 125970*e^8*x^8)) + 45*a^ 
2*b^8*e^2*(11*A*e*(d^8 + 20*d^7*e*x + 190*d^6*e^2*x^2 + 1140*d^5*e^3*x^...
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 462, 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.

\(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^21,x]
 

Output:

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

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. \(1900\) vs. \(2(440)=880\).

Time = 0.37 (sec) , antiderivative size = 1901, normalized size of antiderivative = 4.11

method	result	size
risch	\(\text {Expression too large to display}\)	\(1901\)
default	\(\text {Expression too large to display}\)	\(1942\)
norman	\(\text {Expression too large to display}\)	\(2014\)
gosper	\(\text {Expression too large to display}\)	\(2233\)
orering	\(\text {Expression too large to display}\)	\(2233\)
parallelrisch	\(\text {Expression too large to display}\)	\(2242\)

Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^21,x,method=_RETURNVERBOSE)
 

Output:

(-1/16628040/e^12*(831402*A*a^10*e^11+437580*A*a^9*b*d*e^10+218790*A*a^8*b 
^2*d^2*e^9+102960*A*a^7*b^3*d^3*e^8+45045*A*a^6*b^4*d^4*e^7+18018*A*a^5*b^ 
5*d^5*e^6+6435*A*a^4*b^6*d^6*e^5+1980*A*a^3*b^7*d^7*e^4+495*A*a^2*b^8*d^8* 
e^3+90*A*a*b^9*d^9*e^2+9*A*b^10*d^10*e+43758*B*a^10*d*e^10+48620*B*a^9*b*d 
^2*e^9+38610*B*a^8*b^2*d^3*e^8+25740*B*a^7*b^3*d^4*e^7+15015*B*a^6*b^4*d^5 
*e^6+7722*B*a^5*b^5*d^6*e^5+3465*B*a^4*b^6*d^7*e^4+1320*B*a^3*b^7*d^8*e^3+ 
405*B*a^2*b^8*d^9*e^2+90*B*a*b^9*d^10*e+11*B*b^10*d^11)-1/831402/e^11*(437 
580*A*a^9*b*e^10+218790*A*a^8*b^2*d*e^9+102960*A*a^7*b^3*d^2*e^8+45045*A*a 
^6*b^4*d^3*e^7+18018*A*a^5*b^5*d^4*e^6+6435*A*a^4*b^6*d^5*e^5+1980*A*a^3*b 
^7*d^6*e^4+495*A*a^2*b^8*d^7*e^3+90*A*a*b^9*d^8*e^2+9*A*b^10*d^9*e+43758*B 
*a^10*e^10+48620*B*a^9*b*d*e^9+38610*B*a^8*b^2*d^2*e^8+25740*B*a^7*b^3*d^3 
*e^7+15015*B*a^6*b^4*d^4*e^6+7722*B*a^5*b^5*d^5*e^5+3465*B*a^4*b^6*d^6*e^4 
+1320*B*a^3*b^7*d^7*e^3+405*B*a^2*b^8*d^8*e^2+90*B*a*b^9*d^9*e+11*B*b^10*d 
^10)*x-1/87516*b/e^10*(218790*A*a^8*b*e^9+102960*A*a^7*b^2*d*e^8+45045*A*a 
^6*b^3*d^2*e^7+18018*A*a^5*b^4*d^3*e^6+6435*A*a^4*b^5*d^4*e^5+1980*A*a^3*b 
^6*d^5*e^4+495*A*a^2*b^7*d^6*e^3+90*A*a*b^8*d^7*e^2+9*A*b^9*d^8*e+48620*B* 
a^9*e^9+38610*B*a^8*b*d*e^8+25740*B*a^7*b^2*d^2*e^7+15015*B*a^6*b^3*d^3*e^ 
6+7722*B*a^5*b^4*d^4*e^5+3465*B*a^4*b^5*d^5*e^4+1320*B*a^3*b^6*d^6*e^3+405 
*B*a^2*b^7*d^7*e^2+90*B*a*b^8*d^8*e+11*B*b^9*d^9)*x^2-1/14586*b^2/e^9*(102 
960*A*a^7*b*e^8+45045*A*a^6*b^2*d*e^7+18018*A*a^5*b^3*d^2*e^6+6435*A*a^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2028 vs. \(2 (440) = 880\).

Time = 0.25 (sec) , antiderivative size = 2028, normalized size of antiderivative = 4.39 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^21,x, algorithm="fricas")
 

Output:

-1/16628040*(1847560*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 831402*A*a^10*e^1 
1 + 9*(10*B*a*b^9 + A*b^10)*d^10*e + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 
+ 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^7 
)*d^7*e^4 + 1287*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 3003*(5*B*a^6*b^4 + 
 6*A*a^5*b^5)*d^5*e^6 + 6435*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 12870*( 
3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 24310*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e 
^9 + 43758*(B*a^10 + 10*A*a^9*b)*d*e^10 + 184756*(11*B*b^10*d*e^10 + 9*(10 
*B*a*b^9 + A*b^10)*e^11)*x^10 + 167960*(11*B*b^10*d^2*e^9 + 9*(10*B*a*b^9 
+ A*b^10)*d*e^10 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 125970*(11*B*b 
^10*d^3*e^8 + 9*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 45*(9*B*a^2*b^8 + 2*A*a*b^ 
9)*d*e^10 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 77520*(11*B*b^10*d 
^4*e^7 + 9*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^ 
2*e^9 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 495*(7*B*a^4*b^6 + 4*A*a^ 
3*b^7)*e^11)*x^7 + 38760*(11*B*b^10*d^5*e^6 + 9*(10*B*a*b^9 + A*b^10)*d^4* 
e^7 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^ 
8)*d^2*e^9 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 1287*(6*B*a^5*b^5 + 
5*A*a^4*b^6)*e^11)*x^6 + 15504*(11*B*b^10*d^6*e^5 + 9*(10*B*a*b^9 + A*b^10 
)*d^5*e^6 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 165*(8*B*a^3*b^7 + 3*A* 
a^2*b^8)*d^3*e^8 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 1287*(6*B*a^5 
*b^5 + 5*A*a^4*b^6)*d*e^10 + 3003*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**21,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2028 vs. \(2 (440) = 880\).

Time = 0.15 (sec) , antiderivative size = 2028, normalized size of antiderivative = 4.39 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^21,x, algorithm="maxima")
 

Output:

-1/16628040*(1847560*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 831402*A*a^10*e^1 
1 + 9*(10*B*a*b^9 + A*b^10)*d^10*e + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 
+ 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^7 
)*d^7*e^4 + 1287*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 3003*(5*B*a^6*b^4 + 
 6*A*a^5*b^5)*d^5*e^6 + 6435*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 12870*( 
3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 24310*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e 
^9 + 43758*(B*a^10 + 10*A*a^9*b)*d*e^10 + 184756*(11*B*b^10*d*e^10 + 9*(10 
*B*a*b^9 + A*b^10)*e^11)*x^10 + 167960*(11*B*b^10*d^2*e^9 + 9*(10*B*a*b^9 
+ A*b^10)*d*e^10 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 125970*(11*B*b 
^10*d^3*e^8 + 9*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 45*(9*B*a^2*b^8 + 2*A*a*b^ 
9)*d*e^10 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 77520*(11*B*b^10*d 
^4*e^7 + 9*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^ 
2*e^9 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 495*(7*B*a^4*b^6 + 4*A*a^ 
3*b^7)*e^11)*x^7 + 38760*(11*B*b^10*d^5*e^6 + 9*(10*B*a*b^9 + A*b^10)*d^4* 
e^7 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^ 
8)*d^2*e^9 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 1287*(6*B*a^5*b^5 + 
5*A*a^4*b^6)*e^11)*x^6 + 15504*(11*B*b^10*d^6*e^5 + 9*(10*B*a*b^9 + A*b^10 
)*d^5*e^6 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 165*(8*B*a^3*b^7 + 3*A* 
a^2*b^8)*d^3*e^8 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 1287*(6*B*a^5 
*b^5 + 5*A*a^4*b^6)*d*e^10 + 3003*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2232 vs. \(2 (440) = 880\).

Time = 0.14 (sec) , antiderivative size = 2232, normalized size of antiderivative = 4.83 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^21,x, algorithm="giac")
 

Output:

-1/16628040*(1847560*B*b^10*e^11*x^11 + 2032316*B*b^10*d*e^10*x^10 + 16628 
040*B*a*b^9*e^11*x^10 + 1662804*A*b^10*e^11*x^10 + 1847560*B*b^10*d^2*e^9* 
x^9 + 15116400*B*a*b^9*d*e^10*x^9 + 1511640*A*b^10*d*e^10*x^9 + 68023800*B 
*a^2*b^8*e^11*x^9 + 15116400*A*a*b^9*e^11*x^9 + 1385670*B*b^10*d^3*e^8*x^8 
 + 11337300*B*a*b^9*d^2*e^9*x^8 + 1133730*A*b^10*d^2*e^9*x^8 + 51017850*B* 
a^2*b^8*d*e^10*x^8 + 11337300*A*a*b^9*d*e^10*x^8 + 166280400*B*a^3*b^7*e^1 
1*x^8 + 62355150*A*a^2*b^8*e^11*x^8 + 852720*B*b^10*d^4*e^7*x^7 + 6976800* 
B*a*b^9*d^3*e^8*x^7 + 697680*A*b^10*d^3*e^8*x^7 + 31395600*B*a^2*b^8*d^2*e 
^9*x^7 + 6976800*A*a*b^9*d^2*e^9*x^7 + 102326400*B*a^3*b^7*d*e^10*x^7 + 38 
372400*A*a^2*b^8*d*e^10*x^7 + 268606800*B*a^4*b^6*e^11*x^7 + 153489600*A*a 
^3*b^7*e^11*x^7 + 426360*B*b^10*d^5*e^6*x^6 + 3488400*B*a*b^9*d^4*e^7*x^6 
+ 348840*A*b^10*d^4*e^7*x^6 + 15697800*B*a^2*b^8*d^3*e^8*x^6 + 3488400*A*a 
*b^9*d^3*e^8*x^6 + 51163200*B*a^3*b^7*d^2*e^9*x^6 + 19186200*A*a^2*b^8*d^2 
*e^9*x^6 + 134303400*B*a^4*b^6*d*e^10*x^6 + 76744800*A*a^3*b^7*d*e^10*x^6 
+ 299304720*B*a^5*b^5*e^11*x^6 + 249420600*A*a^4*b^6*e^11*x^6 + 170544*B*b 
^10*d^6*e^5*x^5 + 1395360*B*a*b^9*d^5*e^6*x^5 + 139536*A*b^10*d^5*e^6*x^5 
+ 6279120*B*a^2*b^8*d^4*e^7*x^5 + 1395360*A*a*b^9*d^4*e^7*x^5 + 20465280*B 
*a^3*b^7*d^3*e^8*x^5 + 7674480*A*a^2*b^8*d^3*e^8*x^5 + 53721360*B*a^4*b^6* 
d^2*e^9*x^5 + 30697920*A*a^3*b^7*d^2*e^9*x^5 + 119721888*B*a^5*b^5*d*e^10* 
x^5 + 99768240*A*a^4*b^6*d*e^10*x^5 + 232792560*B*a^6*b^4*e^11*x^5 + 27...
 

Mupad [B] (verification not implemented)

Time = 1.46 (sec) , antiderivative size = 2110, normalized size of antiderivative = 4.57 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx=\text {Too large to display} \] Input:

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^21,x)
 

Output:

-((831402*A*a^10*e^11 + 11*B*b^10*d^11 + 9*A*b^10*d^10*e + 43758*B*a^10*d* 
e^10 + 90*A*a*b^9*d^9*e^2 + 48620*B*a^9*b*d^2*e^9 + 495*A*a^2*b^8*d^8*e^3 
+ 1980*A*a^3*b^7*d^7*e^4 + 6435*A*a^4*b^6*d^6*e^5 + 18018*A*a^5*b^5*d^5*e^ 
6 + 45045*A*a^6*b^4*d^4*e^7 + 102960*A*a^7*b^3*d^3*e^8 + 218790*A*a^8*b^2* 
d^2*e^9 + 405*B*a^2*b^8*d^9*e^2 + 1320*B*a^3*b^7*d^8*e^3 + 3465*B*a^4*b^6* 
d^7*e^4 + 7722*B*a^5*b^5*d^6*e^5 + 15015*B*a^6*b^4*d^5*e^6 + 25740*B*a^7*b 
^3*d^4*e^7 + 38610*B*a^8*b^2*d^3*e^8 + 437580*A*a^9*b*d*e^10 + 90*B*a*b^9* 
d^10*e)/(16628040*e^12) + (x*(43758*B*a^10*e^10 + 11*B*b^10*d^10 + 437580* 
A*a^9*b*e^10 + 9*A*b^10*d^9*e + 90*A*a*b^9*d^8*e^2 + 218790*A*a^8*b^2*d*e^ 
9 + 495*A*a^2*b^8*d^7*e^3 + 1980*A*a^3*b^7*d^6*e^4 + 6435*A*a^4*b^6*d^5*e^ 
5 + 18018*A*a^5*b^5*d^4*e^6 + 45045*A*a^6*b^4*d^3*e^7 + 102960*A*a^7*b^3*d 
^2*e^8 + 405*B*a^2*b^8*d^8*e^2 + 1320*B*a^3*b^7*d^7*e^3 + 3465*B*a^4*b^6*d 
^6*e^4 + 7722*B*a^5*b^5*d^5*e^5 + 15015*B*a^6*b^4*d^4*e^6 + 25740*B*a^7*b^ 
3*d^3*e^7 + 38610*B*a^8*b^2*d^2*e^8 + 90*B*a*b^9*d^9*e + 48620*B*a^9*b*d*e 
^9))/(831402*e^11) + (b^7*x^8*(1320*B*a^3*e^3 + 11*B*b^3*d^3 + 495*A*a^2*b 
*e^3 + 9*A*b^3*d^2*e + 90*A*a*b^2*d*e^2 + 90*B*a*b^2*d^2*e + 405*B*a^2*b*d 
*e^2))/(132*e^4) + (2*b^4*x^5*(15015*B*a^6*e^6 + 11*B*b^6*d^6 + 18018*A*a^ 
5*b*e^6 + 9*A*b^6*d^5*e + 90*A*a*b^5*d^4*e^2 + 6435*A*a^4*b^2*d*e^5 + 495* 
A*a^2*b^4*d^3*e^3 + 1980*A*a^3*b^3*d^2*e^4 + 405*B*a^2*b^4*d^4*e^2 + 1320* 
B*a^3*b^3*d^3*e^3 + 3465*B*a^4*b^2*d^2*e^4 + 90*B*a*b^5*d^5*e + 7722*B*...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1360, normalized size of antiderivative = 2.94 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^21,x)
 

Output:

( - 75582*a**11*e**11 - 43758*a**10*b*d*e**10 - 875160*a**10*b*e**11*x - 2 
4310*a**9*b**2*d**2*e**9 - 486200*a**9*b**2*d*e**10*x - 4618900*a**9*b**2* 
e**11*x**2 - 12870*a**8*b**3*d**3*e**8 - 257400*a**8*b**3*d**2*e**9*x - 24 
45300*a**8*b**3*d*e**10*x**2 - 14671800*a**8*b**3*e**11*x**3 - 6435*a**7*b 
**4*d**4*e**7 - 128700*a**7*b**4*d**3*e**8*x - 1222650*a**7*b**4*d**2*e**9 
*x**2 - 7335900*a**7*b**4*d*e**10*x**3 - 31177575*a**7*b**4*e**11*x**4 - 3 
003*a**6*b**5*d**5*e**6 - 60060*a**6*b**5*d**4*e**7*x - 570570*a**6*b**5*d 
**3*e**8*x**2 - 3423420*a**6*b**5*d**2*e**9*x**3 - 14549535*a**6*b**5*d*e* 
*10*x**4 - 46558512*a**6*b**5*e**11*x**5 - 1287*a**5*b**6*d**6*e**5 - 2574 
0*a**5*b**6*d**5*e**6*x - 244530*a**5*b**6*d**4*e**7*x**2 - 1467180*a**5*b 
**6*d**3*e**8*x**3 - 6235515*a**5*b**6*d**2*e**9*x**4 - 19953648*a**5*b**6 
*d*e**10*x**5 - 49884120*a**5*b**6*e**11*x**6 - 495*a**4*b**7*d**7*e**4 - 
9900*a**4*b**7*d**6*e**5*x - 94050*a**4*b**7*d**5*e**6*x**2 - 564300*a**4* 
b**7*d**4*e**7*x**3 - 2398275*a**4*b**7*d**3*e**8*x**4 - 7674480*a**4*b**7 
*d**2*e**9*x**5 - 19186200*a**4*b**7*d*e**10*x**6 - 38372400*a**4*b**7*e** 
11*x**7 - 165*a**3*b**8*d**8*e**3 - 3300*a**3*b**8*d**7*e**4*x - 31350*a** 
3*b**8*d**6*e**5*x**2 - 188100*a**3*b**8*d**5*e**6*x**3 - 799425*a**3*b**8 
*d**4*e**7*x**4 - 2558160*a**3*b**8*d**3*e**8*x**5 - 6395400*a**3*b**8*d** 
2*e**9*x**6 - 12790800*a**3*b**8*d*e**10*x**7 - 20785050*a**3*b**8*e**11*x 
**8 - 45*a**2*b**9*d**9*e**2 - 900*a**2*b**9*d**8*e**3*x - 8550*a**2*b*...