\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx\) [89]

Optimal result

Integrand size = 20, antiderivative size = 446 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx=\frac {b^{10} B x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{10 e^{12} (d+e x)^{10}}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{9 e^{12} (d+e x)^9}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{8 e^{12} (d+e x)^8}-\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{7 e^{12} (d+e x)^7}+\frac {5 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)^6}-\frac {42 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{5 e^{12} (d+e x)^5}+\frac {21 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{2 e^{12} (d+e x)^4}-\frac {10 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{e^{12} (d+e x)^3}+\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{2 e^{12} (d+e x)^2}-\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e)}{e^{12} (d+e x)}-\frac {b^9 (11 b B d-A b e-10 a B e) \log (d+e x)}{e^{12}} \] Output:

b^10*B*x/e^11+1/10*(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)^10-1/9*(-a*e+b*d) 
^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/(e*x+d)^9+5/8*b*(-a*e+b*d)^8*(-9*A*b*e- 
2*B*a*e+11*B*b*d)/e^12/(e*x+d)^8-15/7*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+1 
1*B*b*d)/e^12/(e*x+d)^7+5*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)/e^1 
2/(e*x+d)^6-42/5*b^4*(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)/e^12/(e*x+d) 
^5+21/2*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)/e^12/(e*x+d)^4-10*b^6 
*(-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11*B*b*d)/e^12/(e*x+d)^3+15/2*b^7*(-a*e+b* 
d)^2*(-3*A*b*e-8*B*a*e+11*B*b*d)/e^12/(e*x+d)^2-5*b^8*(-a*e+b*d)*(-2*A*b*e 
-9*B*a*e+11*B*b*d)/e^12/(e*x+d)-b^9*(-A*b*e-10*B*a*e+11*B*b*d)*ln(e*x+d)/e 
^12
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1447\) vs. \(2(446)=892\).

Time = 0.54 (sec) , antiderivative size = 1447, normalized size of antiderivative = 3.24 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx =\text {Too large to display} \] Input:

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^11,x]
 

Output:

-1/2520*(28*a^10*e^10*(9*A*e + B*(d + 10*e*x)) + 70*a^9*b*e^9*(4*A*e*(d + 
10*e*x) + B*(d^2 + 10*d*e*x + 45*e^2*x^2)) + 45*a^8*b^2*e^8*(7*A*e*(d^2 + 
10*d*e*x + 45*e^2*x^2) + 3*B*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^ 
3)) + 120*a^7*b^3*e^7*(3*A*e*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^ 
3) + 2*B*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4) 
) + 420*a^6*b^4*e^6*(A*e*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^ 
3 + 210*e^4*x^4) + B*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 
+ 210*d*e^4*x^4 + 252*e^5*x^5)) + 252*a^5*b^5*e^5*(2*A*e*(d^5 + 10*d^4*e*x 
 + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5) + 3*B*( 
d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 25 
2*d*e^5*x^5 + 210*e^6*x^6)) + 210*a^4*b^6*e^4*(3*A*e*(d^6 + 10*d^5*e*x + 4 
5*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^ 
6*x^6) + 7*B*(d^7 + 10*d^6*e*x + 45*d^5*e^2*x^2 + 120*d^4*e^3*x^3 + 210*d^ 
3*e^4*x^4 + 252*d^2*e^5*x^5 + 210*d*e^6*x^6 + 120*e^7*x^7)) + 840*a^3*b^7* 
e^3*(A*e*(d^7 + 10*d^6*e*x + 45*d^5*e^2*x^2 + 120*d^4*e^3*x^3 + 210*d^3*e^ 
4*x^4 + 252*d^2*e^5*x^5 + 210*d*e^6*x^6 + 120*e^7*x^7) + 4*B*(d^8 + 10*d^7 
*e*x + 45*d^6*e^2*x^2 + 120*d^5*e^3*x^3 + 210*d^4*e^4*x^4 + 252*d^3*e^5*x^ 
5 + 210*d^2*e^6*x^6 + 120*d*e^7*x^7 + 45*e^8*x^8)) + 1260*a^2*b^8*e^2*(A*e 
*(d^8 + 10*d^7*e*x + 45*d^6*e^2*x^2 + 120*d^5*e^3*x^3 + 210*d^4*e^4*x^4 + 
252*d^3*e^5*x^5 + 210*d^2*e^6*x^6 + 120*d*e^7*x^7 + 45*e^8*x^8) + 9*B*(...
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 446, 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.

\(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx\)

\(\Big \downarrow \) 86

\(\displaystyle \int \left (\frac {b^9 (10 a B e+A b e-11 b B d)}{e^{11} (d+e x)}-\frac {5 b^8 (b d-a e) (9 a B e+2 A b e-11 b B d)}{e^{11} (d+e x)^2}+\frac {15 b^7 (b d-a e)^2 (8 a B e+3 A b e-11 b B d)}{e^{11} (d+e x)^3}-\frac {30 b^6 (b d-a e)^3 (7 a B e+4 A b e-11 b B d)}{e^{11} (d+e x)^4}+\frac {42 b^5 (b d-a e)^4 (6 a B e+5 A b e-11 b B d)}{e^{11} (d+e x)^5}-\frac {42 b^4 (b d-a e)^5 (5 a B e+6 A b e-11 b B d)}{e^{11} (d+e x)^6}+\frac {30 b^3 (b d-a e)^6 (4 a B e+7 A b e-11 b B d)}{e^{11} (d+e x)^7}-\frac {15 b^2 (b d-a e)^7 (3 a B e+8 A b e-11 b B d)}{e^{11} (d+e x)^8}+\frac {5 b (b d-a e)^8 (2 a B e+9 A b e-11 b B d)}{e^{11} (d+e x)^9}+\frac {(a e-b d)^9 (a B e+10 A b e-11 b B d)}{e^{11} (d+e x)^{10}}+\frac {(a e-b d)^{10} (A e-B d)}{e^{11} (d+e x)^{11}}+\frac {b^{10} B}{e^{11}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^9 \log (d+e x) (-10 a B e-A b e+11 b B d)}{e^{12}}-\frac {5 b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{e^{12} (d+e x)}+\frac {15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{2 e^{12} (d+e x)^2}-\frac {10 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12} (d+e x)^3}+\frac {21 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{2 e^{12} (d+e x)^4}-\frac {42 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{5 e^{12} (d+e x)^5}+\frac {5 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^6}-\frac {15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{7 e^{12} (d+e x)^7}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{8 e^{12} (d+e x)^8}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{9 e^{12} (d+e x)^9}+\frac {(b d-a e)^{10} (B d-A e)}{10 e^{12} (d+e x)^{10}}+\frac {b^{10} B x}{e^{11}}\)

Input:

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^11,x]
 

Output:

(b^10*B*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(10*e^12*(d + e*x)^10) - (( 
b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(9*e^12*(d + e*x)^9) + (5*b*(b 
*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(8*e^12*(d + e*x)^8) - (15*b^2 
*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(7*e^12*(d + e*x)^7) + (5*b 
^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(e^12*(d + e*x)^6) - (42* 
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)^5) + ( 
21*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(2*e^12*(d + e*x)^4) 
- (10*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e))/(e^12*(d + e*x)^3) 
 + (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e))/(2*e^12*(d + e*x) 
^2) - (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e))/(e^12*(d + e*x)) 
- (b^9*(11*b*B*d - A*b*e - 10*a*B*e)*Log[d + e*x])/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. \(1921\) vs. \(2(432)=864\).

Time = 0.26 (sec) , antiderivative size = 1922, normalized size of antiderivative = 4.31

method	result	size
risch	\(\text {Expression too large to display}\)	\(1922\)
default	\(\text {Expression too large to display}\)	\(1933\)
norman	\(\text {Expression too large to display}\)	\(1934\)
parallelrisch	\(\text {Expression too large to display}\)	\(2863\)

Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^11,x,method=_RETURNVERBOSE)
 

Output:

b^10*B*x/e^11+((-10*A*a*b^9*e^10+10*A*b^10*d*e^9-45*B*a^2*b^8*e^10+100*B*a 
*b^9*d*e^9-55*B*b^10*d^2*e^8)*x^9-15/2*b^7*e^7*(3*A*a^2*b*e^3+6*A*a*b^2*d* 
e^2-9*A*b^3*d^2*e+8*B*a^3*e^3+27*B*a^2*b*d*e^2-90*B*a*b^2*d^2*e+55*B*b^3*d 
^3)*x^8-10*b^6*e^6*(4*A*a^3*b*e^4+6*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2-22* 
A*b^4*d^3*e+7*B*a^4*e^4+16*B*a^3*b*d*e^3+54*B*a^2*b^2*d^2*e^2-220*B*a*b^3* 
d^3*e+143*B*b^4*d^4)*x^7-7/2*b^5*e^5*(15*A*a^4*b*e^5+20*A*a^3*b^2*d*e^4+30 
*A*a^2*b^3*d^2*e^3+60*A*a*b^4*d^3*e^2-125*A*b^5*d^4*e+18*B*a^5*e^5+35*B*a^ 
4*b*d*e^4+80*B*a^3*b^2*d^2*e^3+270*B*a^2*b^3*d^3*e^2-1250*B*a*b^4*d^4*e+84 
7*B*b^5*d^5)*x^6-21/5*b^4*e^4*(12*A*a^5*b*e^6+15*A*a^4*b^2*d*e^5+20*A*a^3* 
b^3*d^2*e^4+30*A*a^2*b^4*d^3*e^3+60*A*a*b^5*d^4*e^2-137*A*b^6*d^5*e+10*B*a 
^6*e^6+18*B*a^5*b*d*e^5+35*B*a^4*b^2*d^2*e^4+80*B*a^3*b^3*d^3*e^3+270*B*a^ 
2*b^4*d^4*e^2-1370*B*a*b^5*d^5*e+957*B*b^6*d^6)*x^5-1/2*e^3*b^3*(70*A*a^6* 
b*e^7+84*A*a^5*b^2*d*e^6+105*A*a^4*b^3*d^2*e^5+140*A*a^3*b^4*d^3*e^4+210*A 
*a^2*b^5*d^4*e^3+420*A*a*b^6*d^5*e^2-1029*A*b^7*d^6*e+40*B*a^7*e^7+70*B*a^ 
6*b*d*e^6+126*B*a^5*b^2*d^2*e^5+245*B*a^4*b^3*d^3*e^4+560*B*a^3*b^4*d^4*e^ 
3+1890*B*a^2*b^5*d^5*e^2-10290*B*a*b^6*d^6*e+7359*B*b^7*d^7)*x^4-1/7*b^2*e 
^2*(120*A*a^7*b*e^8+140*A*a^6*b^2*d*e^7+168*A*a^5*b^3*d^2*e^6+210*A*a^4*b^ 
4*d^3*e^5+280*A*a^3*b^5*d^4*e^4+420*A*a^2*b^6*d^5*e^3+840*A*a*b^7*d^6*e^2- 
2178*A*b^8*d^7*e+45*B*a^8*e^8+80*B*a^7*b*d*e^7+140*B*a^6*b^2*d^2*e^6+252*B 
*a^5*b^3*d^3*e^5+490*B*a^4*b^4*d^4*e^4+1120*B*a^3*b^5*d^5*e^3+3780*B*a^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2309 vs. \(2 (432) = 864\).

Time = 0.24 (sec) , antiderivative size = 2309, normalized size of antiderivative = 5.18 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^11,x, algorithm="fricas")
 

Output:

1/2520*(2520*B*b^10*e^11*x^11 + 25200*B*b^10*d*e^10*x^10 - 55991*B*b^10*d^ 
11 - 252*A*a^10*e^11 + 7381*(10*B*a*b^9 + A*b^10)*d^10*e - 1260*(9*B*a^2*b 
^8 + 2*A*a*b^9)*d^9*e^2 - 420*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 - 210*(7 
*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^ 
5 - 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 60*(4*B*a^7*b^3 + 7*A*a^6*b^4 
)*d^4*e^7 - 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 35*(2*B*a^9*b + 9*A*a 
^8*b^2)*d^2*e^9 - 28*(B*a^10 + 10*A*a^9*b)*d*e^10 - 12600*(2*B*b^10*d^2*e^ 
9 - 2*(10*B*a*b^9 + A*b^10)*d*e^10 + (9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 
 18900*(39*B*b^10*d^3*e^8 - 9*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 3*(9*B*a^2*b 
^8 + 2*A*a*b^9)*d*e^10 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 - 25200*(12 
2*B*b^10*d^4*e^7 - 22*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 6*(9*B*a^2*b^8 + 2*A 
*a*b^9)*d^2*e^9 + 2*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + (7*B*a^4*b^6 + 4* 
A*a^3*b^7)*e^11)*x^7 - 8820*(775*B*b^10*d^5*e^6 - 125*(10*B*a*b^9 + A*b^10 
)*d^4*e^7 + 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 10*(8*B*a^3*b^7 + 3*A*a 
^2*b^8)*d^2*e^9 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 3*(6*B*a^5*b^5 + 
5*A*a^4*b^6)*e^11)*x^6 - 10584*(907*B*b^10*d^6*e^5 - 137*(10*B*a*b^9 + A*b 
^10)*d^5*e^6 + 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 10*(8*B*a^3*b^7 + 3* 
A*a^2*b^8)*d^3*e^8 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 3*(6*B*a^5*b^ 
5 + 5*A*a^4*b^6)*d*e^10 + 2*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 - 1260*( 
7119*B*b^10*d^7*e^4 - 1029*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 210*(9*B*a^2...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**11,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1914 vs. \(2 (432) = 864\).

Time = 0.13 (sec) , antiderivative size = 1914, normalized size of antiderivative = 4.29 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^11,x, algorithm="maxima")
 

Output:

B*b^10*x/e^11 - 1/2520*(55991*B*b^10*d^11 + 252*A*a^10*e^11 - 7381*(10*B*a 
*b^9 + A*b^10)*d^10*e + 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 420*(8*B* 
a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 
 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)* 
d^5*e^6 + 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 45*(3*B*a^8*b^2 + 8*A*a 
^7*b^3)*d^3*e^8 + 35*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 28*(B*a^10 + 10*A 
*a^9*b)*d*e^10 + 12600*(11*B*b^10*d^2*e^9 - 2*(10*B*a*b^9 + A*b^10)*d*e^10 
 + (9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 18900*(55*B*b^10*d^3*e^8 - 9*(10* 
B*a*b^9 + A*b^10)*d^2*e^9 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + (8*B*a^3* 
b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 25200*(143*B*b^10*d^4*e^7 - 22*(10*B*a*b^9 
+ A*b^10)*d^3*e^8 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 2*(8*B*a^3*b^7 + 
 3*A*a^2*b^8)*d*e^10 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 8820*(847*B 
*b^10*d^5*e^6 - 125*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 30*(9*B*a^2*b^8 + 2*A* 
a*b^9)*d^3*e^8 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 5*(7*B*a^4*b^6 + 
 4*A*a^3*b^7)*d*e^10 + 3*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 10584*(95 
7*B*b^10*d^6*e^5 - 137*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 30*(9*B*a^2*b^8 + 2 
*A*a*b^9)*d^4*e^7 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 5*(7*B*a^4*b^ 
6 + 4*A*a^3*b^7)*d^2*e^9 + 3*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 2*(5*B*a 
^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 1260*(7359*B*b^10*d^7*e^4 - 1029*(10*B*a 
*b^9 + A*b^10)*d^6*e^5 + 210*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 70*(8*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1970 vs. \(2 (432) = 864\).

Time = 0.13 (sec) , antiderivative size = 1970, normalized size of antiderivative = 4.42 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^11,x, algorithm="giac")
 

Output:

B*b^10*x/e^11 - (11*B*b^10*d - 10*B*a*b^9*e - A*b^10*e)*log(abs(e*x + d))/ 
e^12 - 1/2520*(55991*B*b^10*d^11 - 73810*B*a*b^9*d^10*e - 7381*A*b^10*d^10 
*e + 11340*B*a^2*b^8*d^9*e^2 + 2520*A*a*b^9*d^9*e^2 + 3360*B*a^3*b^7*d^8*e 
^3 + 1260*A*a^2*b^8*d^8*e^3 + 1470*B*a^4*b^6*d^7*e^4 + 840*A*a^3*b^7*d^7*e 
^4 + 756*B*a^5*b^5*d^6*e^5 + 630*A*a^4*b^6*d^6*e^5 + 420*B*a^6*b^4*d^5*e^6 
 + 504*A*a^5*b^5*d^5*e^6 + 240*B*a^7*b^3*d^4*e^7 + 420*A*a^6*b^4*d^4*e^7 + 
 135*B*a^8*b^2*d^3*e^8 + 360*A*a^7*b^3*d^3*e^8 + 70*B*a^9*b*d^2*e^9 + 315* 
A*a^8*b^2*d^2*e^9 + 28*B*a^10*d*e^10 + 280*A*a^9*b*d*e^10 + 252*A*a^10*e^1 
1 + 12600*(11*B*b^10*d^2*e^9 - 20*B*a*b^9*d*e^10 - 2*A*b^10*d*e^10 + 9*B*a 
^2*b^8*e^11 + 2*A*a*b^9*e^11)*x^9 + 18900*(55*B*b^10*d^3*e^8 - 90*B*a*b^9* 
d^2*e^9 - 9*A*b^10*d^2*e^9 + 27*B*a^2*b^8*d*e^10 + 6*A*a*b^9*d*e^10 + 8*B* 
a^3*b^7*e^11 + 3*A*a^2*b^8*e^11)*x^8 + 25200*(143*B*b^10*d^4*e^7 - 220*B*a 
*b^9*d^3*e^8 - 22*A*b^10*d^3*e^8 + 54*B*a^2*b^8*d^2*e^9 + 12*A*a*b^9*d^2*e 
^9 + 16*B*a^3*b^7*d*e^10 + 6*A*a^2*b^8*d*e^10 + 7*B*a^4*b^6*e^11 + 4*A*a^3 
*b^7*e^11)*x^7 + 8820*(847*B*b^10*d^5*e^6 - 1250*B*a*b^9*d^4*e^7 - 125*A*b 
^10*d^4*e^7 + 270*B*a^2*b^8*d^3*e^8 + 60*A*a*b^9*d^3*e^8 + 80*B*a^3*b^7*d^ 
2*e^9 + 30*A*a^2*b^8*d^2*e^9 + 35*B*a^4*b^6*d*e^10 + 20*A*a^3*b^7*d*e^10 + 
 18*B*a^5*b^5*e^11 + 15*A*a^4*b^6*e^11)*x^6 + 10584*(957*B*b^10*d^6*e^5 - 
1370*B*a*b^9*d^5*e^6 - 137*A*b^10*d^5*e^6 + 270*B*a^2*b^8*d^4*e^7 + 60*A*a 
*b^9*d^4*e^7 + 80*B*a^3*b^7*d^3*e^8 + 30*A*a^2*b^8*d^3*e^8 + 35*B*a^4*b...
 

Mupad [B] (verification not implemented)

Time = 1.36 (sec) , antiderivative size = 2874, normalized size of antiderivative = 6.44 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx=\text {Too large to display} \] Input:

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^11,x)
 

Output:

-((A*a^10*e^11)/10 + (55991*B*b^10*d^11)/2520 - (7381*A*b^10*d^10*e)/2520 
+ (B*a^10*d*e^10)/90 + 11*B*b^10*d^11*log(d + e*x) + (B*a^10*e^11*x)/9 - B 
*b^10*e^11*x^11 + (53219*B*b^10*d^10*e*x)/252 + A*a*b^9*d^9*e^2 + (B*a^9*b 
*d^2*e^9)/36 + 10*A*a*b^9*e^11*x^9 + (5*B*a^9*b*e^11*x^2)/4 - (7129*A*b^10 
*d^9*e^2*x)/252 - 10*A*b^10*d*e^10*x^9 - 10*B*b^10*d*e^10*x^10 - A*b^10*e^ 
11*x^10*log(d + e*x) + (A*a^2*b^8*d^8*e^3)/2 + (A*a^3*b^7*d^7*e^4)/3 + (A* 
a^4*b^6*d^6*e^5)/4 + (A*a^5*b^5*d^5*e^6)/5 + (A*a^6*b^4*d^4*e^7)/6 + (A*a^ 
7*b^3*d^3*e^8)/7 + (A*a^8*b^2*d^2*e^9)/8 + (9*B*a^2*b^8*d^9*e^2)/2 + (4*B* 
a^3*b^7*d^8*e^3)/3 + (7*B*a^4*b^6*d^7*e^4)/12 + (3*B*a^5*b^5*d^6*e^5)/10 + 
 (B*a^6*b^4*d^5*e^6)/6 + (2*B*a^7*b^3*d^4*e^7)/21 + (3*B*a^8*b^2*d^3*e^8)/ 
56 + (45*A*a^8*b^2*e^11*x^2)/8 + (120*A*a^7*b^3*e^11*x^3)/7 + 35*A*a^6*b^4 
*e^11*x^4 + (252*A*a^5*b^5*e^11*x^5)/5 + (105*A*a^4*b^6*e^11*x^6)/2 + 40*A 
*a^3*b^7*e^11*x^7 + (45*A*a^2*b^8*e^11*x^8)/2 + (45*B*a^8*b^2*e^11*x^3)/7 
+ 20*B*a^7*b^3*e^11*x^4 + 42*B*a^6*b^4*e^11*x^5 + 63*B*a^5*b^5*e^11*x^6 + 
70*B*a^4*b^6*e^11*x^7 + 60*B*a^3*b^7*e^11*x^8 + 45*B*a^2*b^8*e^11*x^9 - (6 
849*A*b^10*d^8*e^3*x^2)/56 - (2178*A*b^10*d^7*e^4*x^3)/7 - (1029*A*b^10*d^ 
6*e^5*x^4)/2 - (2877*A*b^10*d^5*e^6*x^5)/5 - (875*A*b^10*d^4*e^7*x^6)/2 - 
220*A*b^10*d^3*e^8*x^7 - (135*A*b^10*d^2*e^9*x^8)/2 + (50139*B*b^10*d^9*e^ 
2*x^2)/56 + (15558*B*b^10*d^8*e^3*x^3)/7 + (7119*B*b^10*d^7*e^4*x^4)/2 + ( 
19047*B*b^10*d^6*e^5*x^5)/5 + (5425*B*b^10*d^5*e^6*x^6)/2 + 1220*B*b^10...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1541, normalized size of antiderivative = 3.46 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^11,x)
                                                                                    
                                                                                    
 

Output:

(27720*log(d + e*x)*a*b**10*d**11*e + 277200*log(d + e*x)*a*b**10*d**10*e* 
*2*x + 1247400*log(d + e*x)*a*b**10*d**9*e**3*x**2 + 3326400*log(d + e*x)* 
a*b**10*d**8*e**4*x**3 + 5821200*log(d + e*x)*a*b**10*d**7*e**5*x**4 + 698 
5440*log(d + e*x)*a*b**10*d**6*e**6*x**5 + 5821200*log(d + e*x)*a*b**10*d* 
*5*e**7*x**6 + 3326400*log(d + e*x)*a*b**10*d**4*e**8*x**7 + 1247400*log(d 
 + e*x)*a*b**10*d**3*e**9*x**8 + 277200*log(d + e*x)*a*b**10*d**2*e**10*x* 
*9 + 27720*log(d + e*x)*a*b**10*d*e**11*x**10 - 27720*log(d + e*x)*b**11*d 
**12 - 277200*log(d + e*x)*b**11*d**11*e*x - 1247400*log(d + e*x)*b**11*d* 
*10*e**2*x**2 - 3326400*log(d + e*x)*b**11*d**9*e**3*x**3 - 5821200*log(d 
+ e*x)*b**11*d**8*e**4*x**4 - 6985440*log(d + e*x)*b**11*d**7*e**5*x**5 - 
5821200*log(d + e*x)*b**11*d**6*e**6*x**6 - 3326400*log(d + e*x)*b**11*d** 
5*e**7*x**7 - 1247400*log(d + e*x)*b**11*d**4*e**8*x**8 - 277200*log(d + e 
*x)*b**11*d**3*e**9*x**9 - 27720*log(d + e*x)*b**11*d**2*e**10*x**10 - 252 
*a**11*d*e**11 - 308*a**10*b*d**2*e**10 - 3080*a**10*b*d*e**11*x - 385*a** 
9*b**2*d**3*e**9 - 3850*a**9*b**2*d**2*e**10*x - 17325*a**9*b**2*d*e**11*x 
**2 - 495*a**8*b**3*d**4*e**8 - 4950*a**8*b**3*d**3*e**9*x - 22275*a**8*b* 
*3*d**2*e**10*x**2 - 59400*a**8*b**3*d*e**11*x**3 - 660*a**7*b**4*d**5*e** 
7 - 6600*a**7*b**4*d**4*e**8*x - 29700*a**7*b**4*d**3*e**9*x**2 - 79200*a* 
*7*b**4*d**2*e**10*x**3 - 138600*a**7*b**4*d*e**11*x**4 - 924*a**6*b**5*d* 
*6*e**6 - 9240*a**6*b**5*d**5*e**7*x - 41580*a**6*b**5*d**4*e**8*x**2 -...