\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx\) [90]

Optimal result

Integrand size = 20, antiderivative size = 321 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=-\frac {(B d-A e) (a+b x)^{11}}{11 e (b d-a e) (d+e x)^{11}}-\frac {B (b d-a e)^{10}}{10 e^{12} (d+e x)^{10}}+\frac {10 b B (b d-a e)^9}{9 e^{12} (d+e x)^9}-\frac {45 b^2 B (b d-a e)^8}{8 e^{12} (d+e x)^8}+\frac {120 b^3 B (b d-a e)^7}{7 e^{12} (d+e x)^7}-\frac {35 b^4 B (b d-a e)^6}{e^{12} (d+e x)^6}+\frac {252 b^5 B (b d-a e)^5}{5 e^{12} (d+e x)^5}-\frac {105 b^6 B (b d-a e)^4}{2 e^{12} (d+e x)^4}+\frac {40 b^7 B (b d-a e)^3}{e^{12} (d+e x)^3}-\frac {45 b^8 B (b d-a e)^2}{2 e^{12} (d+e x)^2}+\frac {10 b^9 B (b d-a e)}{e^{12} (d+e x)}+\frac {b^{10} B \log (d+e x)}{e^{12}} \] Output:

-1/11*(-A*e+B*d)*(b*x+a)^11/e/(-a*e+b*d)/(e*x+d)^11-1/10*B*(-a*e+b*d)^10/e 
^12/(e*x+d)^10+10/9*b*B*(-a*e+b*d)^9/e^12/(e*x+d)^9-45/8*b^2*B*(-a*e+b*d)^ 
8/e^12/(e*x+d)^8+120/7*b^3*B*(-a*e+b*d)^7/e^12/(e*x+d)^7-35*b^4*B*(-a*e+b* 
d)^6/e^12/(e*x+d)^6+252/5*b^5*B*(-a*e+b*d)^5/e^12/(e*x+d)^5-105/2*b^6*B*(- 
a*e+b*d)^4/e^12/(e*x+d)^4+40*b^7*B*(-a*e+b*d)^3/e^12/(e*x+d)^3-45/2*b^8*B* 
(-a*e+b*d)^2/e^12/(e*x+d)^2+10*b^9*B*(-a*e+b*d)/e^12/(e*x+d)+b^10*B*ln(e*x 
+d)/e^12
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1443\) vs. \(2(321)=642\).

Time = 1.22 (sec) , antiderivative size = 1443, normalized size of antiderivative = 4.50 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/27720*(252*a^10*e^10*(10*A*e + B*(d + 11*e*x)) + 280*a^9*b*e^9*(9*A*e*( 
d + 11*e*x) + 2*B*(d^2 + 11*d*e*x + 55*e^2*x^2)) + 315*a^8*b^2*e^8*(8*A*e* 
(d^2 + 11*d*e*x + 55*e^2*x^2) + 3*B*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165 
*e^3*x^3)) + 360*a^7*b^3*e^7*(7*A*e*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165 
*e^3*x^3) + 4*B*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e 
^4*x^4)) + 420*a^6*b^4*e^6*(6*A*e*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165 
*d*e^3*x^3 + 330*e^4*x^4) + 5*B*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d 
^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5)) + 504*a^5*b^5*e^5*(5*A*e*(d^5 + 
 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x 
^5) + 6*B*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e 
^4*x^4 + 462*d*e^5*x^5 + 462*e^6*x^6)) + 630*a^4*b^6*e^4*(4*A*e*(d^6 + 11* 
d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x 
^5 + 462*e^6*x^6) + 7*B*(d^7 + 11*d^6*e*x + 55*d^5*e^2*x^2 + 165*d^4*e^3*x 
^3 + 330*d^3*e^4*x^4 + 462*d^2*e^5*x^5 + 462*d*e^6*x^6 + 330*e^7*x^7)) + 8 
40*a^3*b^7*e^3*(3*A*e*(d^7 + 11*d^6*e*x + 55*d^5*e^2*x^2 + 165*d^4*e^3*x^3 
 + 330*d^3*e^4*x^4 + 462*d^2*e^5*x^5 + 462*d*e^6*x^6 + 330*e^7*x^7) + 8*B* 
(d^8 + 11*d^7*e*x + 55*d^6*e^2*x^2 + 165*d^5*e^3*x^3 + 330*d^4*e^4*x^4 + 4 
62*d^3*e^5*x^5 + 462*d^2*e^6*x^6 + 330*d*e^7*x^7 + 165*e^8*x^8)) + 1260*a^ 
2*b^8*e^2*(2*A*e*(d^8 + 11*d^7*e*x + 55*d^6*e^2*x^2 + 165*d^5*e^3*x^3 + 33 
0*d^4*e^4*x^4 + 462*d^3*e^5*x^5 + 462*d^2*e^6*x^6 + 330*d*e^7*x^7 + 165...
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {87, 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.

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 49

\(\displaystyle \frac {B \int \left (\frac {b^{10}}{e^{10} (d+e x)}-\frac {10 (b d-a e) b^9}{e^{10} (d+e x)^2}+\frac {45 (b d-a e)^2 b^8}{e^{10} (d+e x)^3}-\frac {120 (b d-a e)^3 b^7}{e^{10} (d+e x)^4}+\frac {210 (b d-a e)^4 b^6}{e^{10} (d+e x)^5}-\frac {252 (b d-a e)^5 b^5}{e^{10} (d+e x)^6}+\frac {210 (b d-a e)^6 b^4}{e^{10} (d+e x)^7}-\frac {120 (b d-a e)^7 b^3}{e^{10} (d+e x)^8}+\frac {45 (b d-a e)^8 b^2}{e^{10} (d+e x)^9}-\frac {10 (b d-a e)^9 b}{e^{10} (d+e x)^{10}}+\frac {(a e-b d)^{10}}{e^{10} (d+e x)^{11}}\right )dx}{e}-\frac {(a+b x)^{11} (B d-A e)}{11 e (d+e x)^{11} (b d-a e)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {B \left (\frac {10 b^9 (b d-a e)}{e^{11} (d+e x)}-\frac {45 b^8 (b d-a e)^2}{2 e^{11} (d+e x)^2}+\frac {40 b^7 (b d-a e)^3}{e^{11} (d+e x)^3}-\frac {105 b^6 (b d-a e)^4}{2 e^{11} (d+e x)^4}+\frac {252 b^5 (b d-a e)^5}{5 e^{11} (d+e x)^5}-\frac {35 b^4 (b d-a e)^6}{e^{11} (d+e x)^6}+\frac {120 b^3 (b d-a e)^7}{7 e^{11} (d+e x)^7}-\frac {45 b^2 (b d-a e)^8}{8 e^{11} (d+e x)^8}+\frac {10 b (b d-a e)^9}{9 e^{11} (d+e x)^9}-\frac {(b d-a e)^{10}}{10 e^{11} (d+e x)^{10}}+\frac {b^{10} \log (d+e x)}{e^{11}}\right )}{e}-\frac {(a+b x)^{11} (B d-A e)}{11 e (d+e x)^{11} (b d-a e)}\)

Input:

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

Output:

-1/11*((B*d - A*e)*(a + b*x)^11)/(e*(b*d - a*e)*(d + e*x)^11) + (B*(-1/10* 
(b*d - a*e)^10/(e^11*(d + e*x)^10) + (10*b*(b*d - a*e)^9)/(9*e^11*(d + e*x 
)^9) - (45*b^2*(b*d - a*e)^8)/(8*e^11*(d + e*x)^8) + (120*b^3*(b*d - a*e)^ 
7)/(7*e^11*(d + e*x)^7) - (35*b^4*(b*d - a*e)^6)/(e^11*(d + e*x)^6) + (252 
*b^5*(b*d - a*e)^5)/(5*e^11*(d + e*x)^5) - (105*b^6*(b*d - a*e)^4)/(2*e^11 
*(d + e*x)^4) + (40*b^7*(b*d - a*e)^3)/(e^11*(d + e*x)^3) - (45*b^8*(b*d - 
 a*e)^2)/(2*e^11*(d + e*x)^2) + (10*b^9*(b*d - a*e))/(e^11*(d + e*x)) + (b 
^10*Log[d + e*x])/e^11))/e
 

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 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

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. \(1902\) vs. \(2(305)=610\).

Time = 0.26 (sec) , antiderivative size = 1903, normalized size of antiderivative = 5.93

method	result	size
risch	\(\text {Expression too large to display}\)	\(1903\)
norman	\(\text {Expression too large to display}\)	\(1939\)
default	\(\text {Expression too large to display}\)	\(1940\)
parallelrisch	\(\text {Expression too large to display}\)	\(2458\)

Input:

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

Output:

(-b^9*(A*b*e+10*B*a*e-11*B*b*d)/e^2*x^10-5/2*b^8*(2*A*a*b*e^2+2*A*b^2*d*e+ 
9*B*a^2*e^2+20*B*a*b*d*e-33*B*b^2*d^2)/e^3*x^9-5/2*b^7*(6*A*a^2*b*e^3+6*A* 
a*b^2*d*e^2+6*A*b^3*d^2*e+16*B*a^3*e^3+27*B*a^2*b*d*e^2+60*B*a*b^2*d^2*e-1 
21*B*b^3*d^3)/e^4*x^8-5/2*b^6*(12*A*a^3*b*e^4+12*A*a^2*b^2*d*e^3+12*A*a*b^ 
3*d^2*e^2+12*A*b^4*d^3*e+21*B*a^4*e^4+32*B*a^3*b*d*e^3+54*B*a^2*b^2*d^2*e^ 
2+120*B*a*b^3*d^3*e-275*B*b^4*d^4)/e^5*x^7-7/10*b^5*(60*A*a^4*b*e^5+60*A*a 
^3*b^2*d*e^4+60*A*a^2*b^3*d^2*e^3+60*A*a*b^4*d^3*e^2+60*A*b^5*d^4*e+72*B*a 
^5*e^5+105*B*a^4*b*d*e^4+160*B*a^3*b^2*d^2*e^3+270*B*a^2*b^3*d^3*e^2+600*B 
*a*b^4*d^4*e-1507*B*b^5*d^5)/e^6*x^6-7/10*b^4*(60*A*a^5*b*e^6+60*A*a^4*b^2 
*d*e^5+60*A*a^3*b^3*d^2*e^4+60*A*a^2*b^4*d^3*e^3+60*A*a*b^5*d^4*e^2+60*A*b 
^6*d^5*e+50*B*a^6*e^6+72*B*a^5*b*d*e^5+105*B*a^4*b^2*d^2*e^4+160*B*a^3*b^3 
*d^3*e^3+270*B*a^2*b^4*d^4*e^2+600*B*a*b^5*d^5*e-1617*B*b^6*d^6)/e^7*x^5-1 
/14*b^3*(420*A*a^6*b*e^7+420*A*a^5*b^2*d*e^6+420*A*a^4*b^3*d^2*e^5+420*A*a 
^3*b^4*d^3*e^4+420*A*a^2*b^5*d^4*e^3+420*A*a*b^6*d^5*e^2+420*A*b^7*d^6*e+2 
40*B*a^7*e^7+350*B*a^6*b*d*e^6+504*B*a^5*b^2*d^2*e^5+735*B*a^4*b^3*d^3*e^4 
+1120*B*a^3*b^4*d^4*e^3+1890*B*a^2*b^5*d^5*e^2+4200*B*a*b^6*d^6*e-11979*B* 
b^7*d^7)/e^8*x^4-1/56*b^2*(840*A*a^7*b*e^8+840*A*a^6*b^2*d*e^7+840*A*a^5*b 
^3*d^2*e^6+840*A*a^4*b^4*d^3*e^5+840*A*a^3*b^5*d^4*e^4+840*A*a^2*b^6*d^5*e 
^3+840*A*a*b^7*d^6*e^2+840*A*b^8*d^7*e+315*B*a^8*e^8+480*B*a^7*b*d*e^7+700 
*B*a^6*b^2*d^2*e^6+1008*B*a^5*b^3*d^3*e^5+1470*B*a^4*b^4*d^4*e^4+2240*B...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2089 vs. \(2 (305) = 610\).

Time = 0.17 (sec) , antiderivative size = 2089, normalized size of antiderivative = 6.51 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/27720*(83711*B*b^10*d^11 - 2520*A*a^10*e^11 - 2520*(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 - 840*(8*B*a^3*b^7 + 3*A* 
a^2*b^8)*d^8*e^3 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 504*(6*B*a^5* 
b^5 + 5*A*a^4*b^6)*d^6*e^5 - 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 360 
*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3 
*e^8 - 280*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 252*(B*a^10 + 10*A*a^9*b)*d 
*e^10 + 27720*(11*B*b^10*d*e^10 - (10*B*a*b^9 + A*b^10)*e^11)*x^10 + 69300 
*(33*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 + 69300*(121*B*b^10*d^3*e^8 - 6*(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 - 2*(8*B*a^3*b^7 + 3*A*a^2*b^8)* 
e^11)*x^8 + 69300*(275*B*b^10*d^4*e^7 - 12*(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 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^1 
0 - 3*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 19404*(1507*B*b^10*d^5*e^6 - 
 60*(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 - 
 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 - 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d 
*e^10 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 19404*(1617*B*b^10*d^6* 
e^5 - 60*(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 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 - 15*(7*B*a^4*b^6 + 4*A*a^3*b 
^7)*d^2*e^9 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 - 10*(5*B*a^6*b^4 + 6* 
A*a^5*b^5)*e^11)*x^5 + 1980*(11979*B*b^10*d^7*e^4 - 420*(10*B*a*b^9 + A...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1932 vs. \(2 (305) = 610\).

Time = 0.12 (sec) , antiderivative size = 1932, normalized size of antiderivative = 6.02 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/27720*(83711*B*b^10*d^11 - 2520*A*a^10*e^11 - 2520*(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 - 840*(8*B*a^3*b^7 + 3*A* 
a^2*b^8)*d^8*e^3 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 504*(6*B*a^5* 
b^5 + 5*A*a^4*b^6)*d^6*e^5 - 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 360 
*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3 
*e^8 - 280*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 252*(B*a^10 + 10*A*a^9*b)*d 
*e^10 + 27720*(11*B*b^10*d*e^10 - (10*B*a*b^9 + A*b^10)*e^11)*x^10 + 69300 
*(33*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 + 69300*(121*B*b^10*d^3*e^8 - 6*(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 - 2*(8*B*a^3*b^7 + 3*A*a^2*b^8)* 
e^11)*x^8 + 69300*(275*B*b^10*d^4*e^7 - 12*(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 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^1 
0 - 3*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 19404*(1507*B*b^10*d^5*e^6 - 
 60*(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 - 
 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 - 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d 
*e^10 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 19404*(1617*B*b^10*d^6* 
e^5 - 60*(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 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 - 15*(7*B*a^4*b^6 + 4*A*a^3*b 
^7)*d^2*e^9 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 - 10*(5*B*a^6*b^4 + 6* 
A*a^5*b^5)*e^11)*x^5 + 1980*(11979*B*b^10*d^7*e^4 - 420*(10*B*a*b^9 + A...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1974 vs. \(2 (305) = 610\).

Time = 0.13 (sec) , antiderivative size = 1974, normalized size of antiderivative = 6.15 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=\text {Too large to display} \] Input:

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

Output:

B*b^10*log(abs(e*x + d))/e^12 + 1/27720*(27720*(11*B*b^10*d*e^9 - 10*B*a*b 
^9*e^10 - A*b^10*e^10)*x^10 + 69300*(33*B*b^10*d^2*e^8 - 20*B*a*b^9*d*e^9 
- 2*A*b^10*d*e^9 - 9*B*a^2*b^8*e^10 - 2*A*a*b^9*e^10)*x^9 + 69300*(121*B*b 
^10*d^3*e^7 - 60*B*a*b^9*d^2*e^8 - 6*A*b^10*d^2*e^8 - 27*B*a^2*b^8*d*e^9 - 
 6*A*a*b^9*d*e^9 - 16*B*a^3*b^7*e^10 - 6*A*a^2*b^8*e^10)*x^8 + 69300*(275* 
B*b^10*d^4*e^6 - 120*B*a*b^9*d^3*e^7 - 12*A*b^10*d^3*e^7 - 54*B*a^2*b^8*d^ 
2*e^8 - 12*A*a*b^9*d^2*e^8 - 32*B*a^3*b^7*d*e^9 - 12*A*a^2*b^8*d*e^9 - 21* 
B*a^4*b^6*e^10 - 12*A*a^3*b^7*e^10)*x^7 + 19404*(1507*B*b^10*d^5*e^5 - 600 
*B*a*b^9*d^4*e^6 - 60*A*b^10*d^4*e^6 - 270*B*a^2*b^8*d^3*e^7 - 60*A*a*b^9* 
d^3*e^7 - 160*B*a^3*b^7*d^2*e^8 - 60*A*a^2*b^8*d^2*e^8 - 105*B*a^4*b^6*d*e 
^9 - 60*A*a^3*b^7*d*e^9 - 72*B*a^5*b^5*e^10 - 60*A*a^4*b^6*e^10)*x^6 + 194 
04*(1617*B*b^10*d^6*e^4 - 600*B*a*b^9*d^5*e^5 - 60*A*b^10*d^5*e^5 - 270*B* 
a^2*b^8*d^4*e^6 - 60*A*a*b^9*d^4*e^6 - 160*B*a^3*b^7*d^3*e^7 - 60*A*a^2*b^ 
8*d^3*e^7 - 105*B*a^4*b^6*d^2*e^8 - 60*A*a^3*b^7*d^2*e^8 - 72*B*a^5*b^5*d* 
e^9 - 60*A*a^4*b^6*d*e^9 - 50*B*a^6*b^4*e^10 - 60*A*a^5*b^5*e^10)*x^5 + 19 
80*(11979*B*b^10*d^7*e^3 - 4200*B*a*b^9*d^6*e^4 - 420*A*b^10*d^6*e^4 - 189 
0*B*a^2*b^8*d^5*e^5 - 420*A*a*b^9*d^5*e^5 - 1120*B*a^3*b^7*d^4*e^6 - 420*A 
*a^2*b^8*d^4*e^6 - 735*B*a^4*b^6*d^3*e^7 - 420*A*a^3*b^7*d^3*e^7 - 504*B*a 
^5*b^5*d^2*e^8 - 420*A*a^4*b^6*d^2*e^8 - 350*B*a^6*b^4*d*e^9 - 420*A*a^5*b 
^5*d*e^9 - 240*B*a^7*b^3*e^10 - 420*A*a^6*b^4*e^10)*x^4 + 495*(25113*B*...
 

Mupad [B] (verification not implemented)

Time = 1.29 (sec) , antiderivative size = 2446, normalized size of antiderivative = 7.62 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=\text {Too large to display} \] Input:

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

Output:

-((A*a^10*e^11)/11 - (83711*B*b^10*d^11)/27720 + (A*b^10*d^10*e)/11 + (B*a 
^10*d*e^10)/110 - B*b^10*d^11*log(d + e*x) + (B*a^10*e^11*x)/10 + A*b^10*e 
^11*x^10 - (81191*B*b^10*d^10*e*x)/2520 + (A*a*b^9*d^9*e^2)/11 + (2*B*a^9* 
b*d^2*e^9)/99 + 5*A*a*b^9*e^11*x^9 + (10*B*a^9*b*e^11*x^2)/9 + 10*B*a*b^9* 
e^11*x^10 + A*b^10*d^9*e^2*x + 5*A*b^10*d*e^10*x^9 - 11*B*b^10*d*e^10*x^10 
 - B*b^10*e^11*x^11*log(d + e*x) + (A*a^2*b^8*d^8*e^3)/11 + (A*a^3*b^7*d^7 
*e^4)/11 + (A*a^4*b^6*d^6*e^5)/11 + (A*a^5*b^5*d^5*e^6)/11 + (A*a^6*b^4*d^ 
4*e^7)/11 + (A*a^7*b^3*d^3*e^8)/11 + (A*a^8*b^2*d^2*e^9)/11 + (9*B*a^2*b^8 
*d^9*e^2)/22 + (8*B*a^3*b^7*d^8*e^3)/33 + (7*B*a^4*b^6*d^7*e^4)/44 + (6*B* 
a^5*b^5*d^6*e^5)/55 + (5*B*a^6*b^4*d^5*e^6)/66 + (4*B*a^7*b^3*d^4*e^7)/77 
+ (3*B*a^8*b^2*d^3*e^8)/88 + 5*A*a^8*b^2*e^11*x^2 + 15*A*a^7*b^3*e^11*x^3 
+ 30*A*a^6*b^4*e^11*x^4 + 42*A*a^5*b^5*e^11*x^5 + 42*A*a^4*b^6*e^11*x^6 + 
30*A*a^3*b^7*e^11*x^7 + 15*A*a^2*b^8*e^11*x^8 + (45*B*a^8*b^2*e^11*x^3)/8 
+ (120*B*a^7*b^3*e^11*x^4)/7 + 35*B*a^6*b^4*e^11*x^5 + (252*B*a^5*b^5*e^11 
*x^6)/5 + (105*B*a^4*b^6*e^11*x^7)/2 + 40*B*a^3*b^7*e^11*x^8 + (45*B*a^2*b 
^8*e^11*x^9)/2 + 5*A*b^10*d^8*e^3*x^2 + 15*A*b^10*d^7*e^4*x^3 + 30*A*b^10* 
d^6*e^5*x^4 + 42*A*b^10*d^5*e^6*x^5 + 42*A*b^10*d^4*e^7*x^6 + 30*A*b^10*d^ 
3*e^8*x^7 + 15*A*b^10*d^2*e^9*x^8 - (78419*B*b^10*d^9*e^2*x^2)/504 - (2511 
3*B*b^10*d^8*e^3*x^3)/56 - (11979*B*b^10*d^7*e^4*x^4)/14 - (11319*B*b^10*d 
^6*e^5*x^5)/10 - (10549*B*b^10*d^5*e^6*x^6)/10 - (1375*B*b^10*d^4*e^7*x...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1368, normalized size of antiderivative = 4.26 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx =\text {Too large to display} \] Input:

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

Output:

(27720*log(d + e*x)*b**11*d**12 + 304920*log(d + e*x)*b**11*d**11*e*x + 15 
24600*log(d + e*x)*b**11*d**10*e**2*x**2 + 4573800*log(d + e*x)*b**11*d**9 
*e**3*x**3 + 9147600*log(d + e*x)*b**11*d**8*e**4*x**4 + 12806640*log(d + 
e*x)*b**11*d**7*e**5*x**5 + 12806640*log(d + e*x)*b**11*d**6*e**6*x**6 + 9 
147600*log(d + e*x)*b**11*d**5*e**7*x**7 + 4573800*log(d + e*x)*b**11*d**4 
*e**8*x**8 + 1524600*log(d + e*x)*b**11*d**3*e**9*x**9 + 304920*log(d + e* 
x)*b**11*d**2*e**10*x**10 + 27720*log(d + e*x)*b**11*d*e**11*x**11 - 2520* 
a**11*d*e**11 - 2772*a**10*b*d**2*e**10 - 30492*a**10*b*d*e**11*x - 3080*a 
**9*b**2*d**3*e**9 - 33880*a**9*b**2*d**2*e**10*x - 169400*a**9*b**2*d*e** 
11*x**2 - 3465*a**8*b**3*d**4*e**8 - 38115*a**8*b**3*d**3*e**9*x - 190575* 
a**8*b**3*d**2*e**10*x**2 - 571725*a**8*b**3*d*e**11*x**3 - 3960*a**7*b**4 
*d**5*e**7 - 43560*a**7*b**4*d**4*e**8*x - 217800*a**7*b**4*d**3*e**9*x**2 
 - 653400*a**7*b**4*d**2*e**10*x**3 - 1306800*a**7*b**4*d*e**11*x**4 - 462 
0*a**6*b**5*d**6*e**6 - 50820*a**6*b**5*d**5*e**7*x - 254100*a**6*b**5*d** 
4*e**8*x**2 - 762300*a**6*b**5*d**3*e**9*x**3 - 1524600*a**6*b**5*d**2*e** 
10*x**4 - 2134440*a**6*b**5*d*e**11*x**5 - 5544*a**5*b**6*d**7*e**5 - 6098 
4*a**5*b**6*d**6*e**6*x - 304920*a**5*b**6*d**5*e**7*x**2 - 914760*a**5*b* 
*6*d**4*e**8*x**3 - 1829520*a**5*b**6*d**3*e**9*x**4 - 2561328*a**5*b**6*d 
**2*e**10*x**5 - 2561328*a**5*b**6*d*e**11*x**6 - 6930*a**4*b**7*d**8*e**4 
 - 76230*a**4*b**7*d**7*e**5*x - 381150*a**4*b**7*d**6*e**6*x**2 - 1143...