\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx\) [100]

Optimal result

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1431\) vs. \(2(464)=928\).

Time = 0.54 (sec) , antiderivative size = 1431, normalized size of antiderivative = 3.08 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx =\text {Too large to display} \] Input:

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^22,x]
 

Output:

-1/38798760*(92378*a^10*e^10*(20*A*e + B*(d + 21*e*x)) + 48620*a^9*b*e^9*( 
19*A*e*(d + 21*e*x) + 2*B*(d^2 + 21*d*e*x + 210*e^2*x^2)) + 72930*a^8*b^2* 
e^8*(6*A*e*(d^2 + 21*d*e*x + 210*e^2*x^2) + B*(d^3 + 21*d^2*e*x + 210*d*e^ 
2*x^2 + 1330*e^3*x^3)) + 11440*a^7*b^3*e^7*(17*A*e*(d^3 + 21*d^2*e*x + 210 
*d*e^2*x^2 + 1330*e^3*x^3) + 4*B*(d^4 + 21*d^3*e*x + 210*d^2*e^2*x^2 + 133 
0*d*e^3*x^3 + 5985*e^4*x^4)) + 5005*a^6*b^4*e^6*(16*A*e*(d^4 + 21*d^3*e*x 
+ 210*d^2*e^2*x^2 + 1330*d*e^3*x^3 + 5985*e^4*x^4) + 5*B*(d^5 + 21*d^4*e*x 
 + 210*d^3*e^2*x^2 + 1330*d^2*e^3*x^3 + 5985*d*e^4*x^4 + 20349*e^5*x^5)) + 
 6006*a^5*b^5*e^5*(5*A*e*(d^5 + 21*d^4*e*x + 210*d^3*e^2*x^2 + 1330*d^2*e^ 
3*x^3 + 5985*d*e^4*x^4 + 20349*e^5*x^5) + 2*B*(d^6 + 21*d^5*e*x + 210*d^4* 
e^2*x^2 + 1330*d^3*e^3*x^3 + 5985*d^2*e^4*x^4 + 20349*d*e^5*x^5 + 54264*e^ 
6*x^6)) + 5005*a^4*b^6*e^4*(2*A*e*(d^6 + 21*d^5*e*x + 210*d^4*e^2*x^2 + 13 
30*d^3*e^3*x^3 + 5985*d^2*e^4*x^4 + 20349*d*e^5*x^5 + 54264*e^6*x^6) + B*( 
d^7 + 21*d^6*e*x + 210*d^5*e^2*x^2 + 1330*d^4*e^3*x^3 + 5985*d^3*e^4*x^4 + 
 20349*d^2*e^5*x^5 + 54264*d*e^6*x^6 + 116280*e^7*x^7)) + 220*a^3*b^7*e^3* 
(13*A*e*(d^7 + 21*d^6*e*x + 210*d^5*e^2*x^2 + 1330*d^4*e^3*x^3 + 5985*d^3* 
e^4*x^4 + 20349*d^2*e^5*x^5 + 54264*d*e^6*x^6 + 116280*e^7*x^7) + 8*B*(d^8 
 + 21*d^7*e*x + 210*d^6*e^2*x^2 + 1330*d^5*e^3*x^3 + 5985*d^4*e^4*x^4 + 20 
349*d^3*e^5*x^5 + 54264*d^2*e^6*x^6 + 116280*d*e^7*x^7 + 203490*e^8*x^8)) 
+ 165*a^2*b^8*e^2*(4*A*e*(d^8 + 21*d^7*e*x + 210*d^6*e^2*x^2 + 1330*d^5...
 

Rubi [A] (verified)

Time = 1.21 (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.

\(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^22,x]
 

Output:

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

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 = 0.38 (sec) , antiderivative size = 1901, normalized size of antiderivative = 4.10

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)^22,x,method=_RETURNVERBOSE)
 

Output:

(-1/38798760/e^12*(1847560*A*a^10*e^11+923780*A*a^9*b*d*e^10+437580*A*a^8* 
b^2*d^2*e^9+194480*A*a^7*b^3*d^3*e^8+80080*A*a^6*b^4*d^4*e^7+30030*A*a^5*b 
^5*d^5*e^6+10010*A*a^4*b^6*d^6*e^5+2860*A*a^3*b^7*d^7*e^4+660*A*a^2*b^8*d^ 
8*e^3+110*A*a*b^9*d^9*e^2+10*A*b^10*d^10*e+92378*B*a^10*d*e^10+97240*B*a^9 
*b*d^2*e^9+72930*B*a^8*b^2*d^3*e^8+45760*B*a^7*b^3*d^4*e^7+25025*B*a^6*b^4 
*d^5*e^6+12012*B*a^5*b^5*d^6*e^5+5005*B*a^4*b^6*d^7*e^4+1760*B*a^3*b^7*d^8 
*e^3+495*B*a^2*b^8*d^9*e^2+100*B*a*b^9*d^10*e+11*B*b^10*d^11)-1/1847560/e^ 
11*(923780*A*a^9*b*e^10+437580*A*a^8*b^2*d*e^9+194480*A*a^7*b^3*d^2*e^8+80 
080*A*a^6*b^4*d^3*e^7+30030*A*a^5*b^5*d^4*e^6+10010*A*a^4*b^6*d^5*e^5+2860 
*A*a^3*b^7*d^6*e^4+660*A*a^2*b^8*d^7*e^3+110*A*a*b^9*d^8*e^2+10*A*b^10*d^9 
*e+92378*B*a^10*e^10+97240*B*a^9*b*d*e^9+72930*B*a^8*b^2*d^2*e^8+45760*B*a 
^7*b^3*d^3*e^7+25025*B*a^6*b^4*d^4*e^6+12012*B*a^5*b^5*d^5*e^5+5005*B*a^4* 
b^6*d^6*e^4+1760*B*a^3*b^7*d^7*e^3+495*B*a^2*b^8*d^8*e^2+100*B*a*b^9*d^9*e 
+11*B*b^10*d^10)*x-1/184756*b/e^10*(437580*A*a^8*b*e^9+194480*A*a^7*b^2*d* 
e^8+80080*A*a^6*b^3*d^2*e^7+30030*A*a^5*b^4*d^3*e^6+10010*A*a^4*b^5*d^4*e^ 
5+2860*A*a^3*b^6*d^5*e^4+660*A*a^2*b^7*d^6*e^3+110*A*a*b^8*d^7*e^2+10*A*b^ 
9*d^8*e+97240*B*a^9*e^9+72930*B*a^8*b*d*e^8+45760*B*a^7*b^2*d^2*e^7+25025* 
B*a^6*b^3*d^3*e^6+12012*B*a^5*b^4*d^4*e^5+5005*B*a^4*b^5*d^5*e^4+1760*B*a^ 
3*b^6*d^6*e^3+495*B*a^2*b^7*d^7*e^2+100*B*a*b^8*d^8*e+11*B*b^9*d^9)*x^2-1/ 
29172*b^2/e^9*(194480*A*a^7*b*e^8+80080*A*a^6*b^2*d*e^7+30030*A*a^5*b^3...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2039 vs. \(2 (440) = 880\).

Time = 0.26 (sec) , antiderivative size = 2039, normalized size of antiderivative = 4.39 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^22,x, algorithm="fricas")
 

Output:

-1/38798760*(3879876*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 1847560*A*a^10*e^ 
11 + 10*(10*B*a*b^9 + A*b^10)*d^10*e + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^ 
2 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 715*(7*B*a^4*b^6 + 4*A*a^3*b 
^7)*d^7*e^4 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 5005*(5*B*a^6*b^4 
 + 6*A*a^5*b^5)*d^5*e^6 + 11440*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 2431 
0*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 48620*(2*B*a^9*b + 9*A*a^8*b^2)*d^ 
2*e^9 + 92378*(B*a^10 + 10*A*a^9*b)*d*e^10 + 352716*(11*B*b^10*d*e^10 + 10 
*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 293930*(11*B*b^10*d^2*e^9 + 10*(10*B*a 
*b^9 + A*b^10)*d*e^10 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 203490*(1 
1*B*b^10*d^3*e^8 + 10*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 55*(9*B*a^2*b^8 + 2* 
A*a*b^9)*d*e^10 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 116280*(11*B 
*b^10*d^4*e^7 + 10*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 55*(9*B*a^2*b^8 + 2*A*a 
*b^9)*d^2*e^9 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 715*(7*B*a^4*b^6 
+ 4*A*a^3*b^7)*e^11)*x^7 + 54264*(11*B*b^10*d^5*e^6 + 10*(10*B*a*b^9 + A*b 
^10)*d^4*e^7 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 220*(8*B*a^3*b^7 + 3 
*A*a^2*b^8)*d^2*e^9 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 2002*(6*B*a 
^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 20349*(11*B*b^10*d^6*e^5 + 10*(10*B*a*b^ 
9 + A*b^10)*d^5*e^6 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 220*(8*B*a^3* 
b^7 + 3*A*a^2*b^8)*d^3*e^8 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 200 
2*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**22,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2039 vs. \(2 (440) = 880\).

Time = 0.16 (sec) , antiderivative size = 2039, normalized size of antiderivative = 4.39 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^22,x, algorithm="maxima")
 

Output:

-1/38798760*(3879876*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 1847560*A*a^10*e^ 
11 + 10*(10*B*a*b^9 + A*b^10)*d^10*e + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^ 
2 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 715*(7*B*a^4*b^6 + 4*A*a^3*b 
^7)*d^7*e^4 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 5005*(5*B*a^6*b^4 
 + 6*A*a^5*b^5)*d^5*e^6 + 11440*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 2431 
0*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 48620*(2*B*a^9*b + 9*A*a^8*b^2)*d^ 
2*e^9 + 92378*(B*a^10 + 10*A*a^9*b)*d*e^10 + 352716*(11*B*b^10*d*e^10 + 10 
*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 293930*(11*B*b^10*d^2*e^9 + 10*(10*B*a 
*b^9 + A*b^10)*d*e^10 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 203490*(1 
1*B*b^10*d^3*e^8 + 10*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 55*(9*B*a^2*b^8 + 2* 
A*a*b^9)*d*e^10 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 116280*(11*B 
*b^10*d^4*e^7 + 10*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 55*(9*B*a^2*b^8 + 2*A*a 
*b^9)*d^2*e^9 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 715*(7*B*a^4*b^6 
+ 4*A*a^3*b^7)*e^11)*x^7 + 54264*(11*B*b^10*d^5*e^6 + 10*(10*B*a*b^9 + A*b 
^10)*d^4*e^7 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 220*(8*B*a^3*b^7 + 3 
*A*a^2*b^8)*d^2*e^9 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 2002*(6*B*a 
^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 20349*(11*B*b^10*d^6*e^5 + 10*(10*B*a*b^ 
9 + A*b^10)*d^5*e^6 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 220*(8*B*a^3* 
b^7 + 3*A*a^2*b^8)*d^3*e^8 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 200 
2*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^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 = 0.13 (sec) , antiderivative size = 2232, normalized size of antiderivative = 4.81 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^22,x, algorithm="giac")
 

Output:

-1/38798760*(3879876*B*b^10*e^11*x^11 + 3879876*B*b^10*d*e^10*x^10 + 35271 
600*B*a*b^9*e^11*x^10 + 3527160*A*b^10*e^11*x^10 + 3233230*B*b^10*d^2*e^9* 
x^9 + 29393000*B*a*b^9*d*e^10*x^9 + 2939300*A*b^10*d*e^10*x^9 + 145495350* 
B*a^2*b^8*e^11*x^9 + 32332300*A*a*b^9*e^11*x^9 + 2238390*B*b^10*d^3*e^8*x^ 
8 + 20349000*B*a*b^9*d^2*e^9*x^8 + 2034900*A*b^10*d^2*e^9*x^8 + 100727550* 
B*a^2*b^8*d*e^10*x^8 + 22383900*A*a*b^9*d*e^10*x^8 + 358142400*B*a^3*b^7*e 
^11*x^8 + 134303400*A*a^2*b^8*e^11*x^8 + 1279080*B*b^10*d^4*e^7*x^7 + 1162 
8000*B*a*b^9*d^3*e^8*x^7 + 1162800*A*b^10*d^3*e^8*x^7 + 57558600*B*a^2*b^8 
*d^2*e^9*x^7 + 12790800*A*a*b^9*d^2*e^9*x^7 + 204652800*B*a^3*b^7*d*e^10*x 
^7 + 76744800*A*a^2*b^8*d*e^10*x^7 + 581981400*B*a^4*b^6*e^11*x^7 + 332560 
800*A*a^3*b^7*e^11*x^7 + 596904*B*b^10*d^5*e^6*x^6 + 5426400*B*a*b^9*d^4*e 
^7*x^6 + 542640*A*b^10*d^4*e^7*x^6 + 26860680*B*a^2*b^8*d^3*e^8*x^6 + 5969 
040*A*a*b^9*d^3*e^8*x^6 + 95504640*B*a^3*b^7*d^2*e^9*x^6 + 35814240*A*a^2* 
b^8*d^2*e^9*x^6 + 271591320*B*a^4*b^6*d*e^10*x^6 + 155195040*A*a^3*b^7*d*e 
^10*x^6 + 651819168*B*a^5*b^5*e^11*x^6 + 543182640*A*a^4*b^6*e^11*x^6 + 22 
3839*B*b^10*d^6*e^5*x^5 + 2034900*B*a*b^9*d^5*e^6*x^5 + 203490*A*b^10*d^5* 
e^6*x^5 + 10072755*B*a^2*b^8*d^4*e^7*x^5 + 2238390*A*a*b^9*d^4*e^7*x^5 + 3 
5814240*B*a^3*b^7*d^3*e^8*x^5 + 13430340*A*a^2*b^8*d^3*e^8*x^5 + 101846745 
*B*a^4*b^6*d^2*e^9*x^5 + 58198140*A*a^3*b^7*d^2*e^9*x^5 + 244432188*B*a^5* 
b^5*d*e^10*x^5 + 203693490*A*a^4*b^6*d*e^10*x^5 + 509233725*B*a^6*b^4*e...
 

Mupad [B] (verification not implemented)

Time = 1.63 (sec) , antiderivative size = 2121, normalized size of antiderivative = 4.57 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx=\text {Too large to display} \] Input:

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^22,x)
 

Output:

-((1847560*A*a^10*e^11 + 11*B*b^10*d^11 + 10*A*b^10*d^10*e + 92378*B*a^10* 
d*e^10 + 110*A*a*b^9*d^9*e^2 + 97240*B*a^9*b*d^2*e^9 + 660*A*a^2*b^8*d^8*e 
^3 + 2860*A*a^3*b^7*d^7*e^4 + 10010*A*a^4*b^6*d^6*e^5 + 30030*A*a^5*b^5*d^ 
5*e^6 + 80080*A*a^6*b^4*d^4*e^7 + 194480*A*a^7*b^3*d^3*e^8 + 437580*A*a^8* 
b^2*d^2*e^9 + 495*B*a^2*b^8*d^9*e^2 + 1760*B*a^3*b^7*d^8*e^3 + 5005*B*a^4* 
b^6*d^7*e^4 + 12012*B*a^5*b^5*d^6*e^5 + 25025*B*a^6*b^4*d^5*e^6 + 45760*B* 
a^7*b^3*d^4*e^7 + 72930*B*a^8*b^2*d^3*e^8 + 923780*A*a^9*b*d*e^10 + 100*B* 
a*b^9*d^10*e)/(38798760*e^12) + (x*(92378*B*a^10*e^10 + 11*B*b^10*d^10 + 9 
23780*A*a^9*b*e^10 + 10*A*b^10*d^9*e + 110*A*a*b^9*d^8*e^2 + 437580*A*a^8* 
b^2*d*e^9 + 660*A*a^2*b^8*d^7*e^3 + 2860*A*a^3*b^7*d^6*e^4 + 10010*A*a^4*b 
^6*d^5*e^5 + 30030*A*a^5*b^5*d^4*e^6 + 80080*A*a^6*b^4*d^3*e^7 + 194480*A* 
a^7*b^3*d^2*e^8 + 495*B*a^2*b^8*d^8*e^2 + 1760*B*a^3*b^7*d^7*e^3 + 5005*B* 
a^4*b^6*d^6*e^4 + 12012*B*a^5*b^5*d^5*e^5 + 25025*B*a^6*b^4*d^4*e^6 + 4576 
0*B*a^7*b^3*d^3*e^7 + 72930*B*a^8*b^2*d^2*e^8 + 100*B*a*b^9*d^9*e + 97240* 
B*a^9*b*d*e^9))/(1847560*e^11) + (3*b^7*x^8*(1760*B*a^3*e^3 + 11*B*b^3*d^3 
 + 660*A*a^2*b*e^3 + 10*A*b^3*d^2*e + 110*A*a*b^2*d*e^2 + 100*B*a*b^2*d^2* 
e + 495*B*a^2*b*d*e^2))/(572*e^4) + (3*b^4*x^5*(25025*B*a^6*e^6 + 11*B*b^6 
*d^6 + 30030*A*a^5*b*e^6 + 10*A*b^6*d^5*e + 110*A*a*b^5*d^4*e^2 + 10010*A* 
a^4*b^2*d*e^5 + 660*A*a^2*b^4*d^3*e^3 + 2860*A*a^3*b^3*d^2*e^4 + 495*B*a^2 
*b^4*d^4*e^2 + 1760*B*a^3*b^3*d^3*e^3 + 5005*B*a^4*b^2*d^2*e^4 + 100*B*...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1371, normalized size of antiderivative = 2.95 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^22,x)
 

Output:

( - 167960*a**11*e**11 - 92378*a**10*b*d*e**10 - 1939938*a**10*b*e**11*x - 
 48620*a**9*b**2*d**2*e**9 - 1021020*a**9*b**2*d*e**10*x - 10210200*a**9*b 
**2*e**11*x**2 - 24310*a**8*b**3*d**3*e**8 - 510510*a**8*b**3*d**2*e**9*x 
- 5105100*a**8*b**3*d*e**10*x**2 - 32332300*a**8*b**3*e**11*x**3 - 11440*a 
**7*b**4*d**4*e**7 - 240240*a**7*b**4*d**3*e**8*x - 2402400*a**7*b**4*d**2 
*e**9*x**2 - 15215200*a**7*b**4*d*e**10*x**3 - 68468400*a**7*b**4*e**11*x* 
*4 - 5005*a**6*b**5*d**5*e**6 - 105105*a**6*b**5*d**4*e**7*x - 1051050*a** 
6*b**5*d**3*e**8*x**2 - 6656650*a**6*b**5*d**2*e**9*x**3 - 29954925*a**6*b 
**5*d*e**10*x**4 - 101846745*a**6*b**5*e**11*x**5 - 2002*a**5*b**6*d**6*e* 
*5 - 42042*a**5*b**6*d**5*e**6*x - 420420*a**5*b**6*d**4*e**7*x**2 - 26626 
60*a**5*b**6*d**3*e**8*x**3 - 11981970*a**5*b**6*d**2*e**9*x**4 - 40738698 
*a**5*b**6*d*e**10*x**5 - 108636528*a**5*b**6*e**11*x**6 - 715*a**4*b**7*d 
**7*e**4 - 15015*a**4*b**7*d**6*e**5*x - 150150*a**4*b**7*d**5*e**6*x**2 - 
 950950*a**4*b**7*d**4*e**7*x**3 - 4279275*a**4*b**7*d**3*e**8*x**4 - 1454 
9535*a**4*b**7*d**2*e**9*x**5 - 38798760*a**4*b**7*d*e**10*x**6 - 83140200 
*a**4*b**7*e**11*x**7 - 220*a**3*b**8*d**8*e**3 - 4620*a**3*b**8*d**7*e**4 
*x - 46200*a**3*b**8*d**6*e**5*x**2 - 292600*a**3*b**8*d**5*e**6*x**3 - 13 
16700*a**3*b**8*d**4*e**7*x**4 - 4476780*a**3*b**8*d**3*e**8*x**5 - 119380 
80*a**3*b**8*d**2*e**9*x**6 - 25581600*a**3*b**8*d*e**10*x**7 - 44767800*a 
**3*b**8*e**11*x**8 - 55*a**2*b**9*d**9*e**2 - 1155*a**2*b**9*d**8*e**3...