\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{10}} \, dx\) [88]

Optimal result

Integrand size = 20, antiderivative size = 441 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{10}} \, dx=-\frac {b^9 (10 b B d-A b e-10 a B e) x}{e^{11}}+\frac {b^{10} B x^2}{2 e^{10}}+\frac {(b d-a e)^{10} (B d-A e)}{9 e^{12} (d+e x)^9}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{8 e^{12} (d+e x)^8}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{7 e^{12} (d+e x)^7}-\frac {5 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{2 e^{12} (d+e x)^6}+\frac {6 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)^5}-\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{2 e^{12} (d+e x)^4}+\frac {14 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{e^{12} (d+e x)^3}-\frac {15 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{e^{12} (d+e x)^2}+\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{e^{12} (d+e x)}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) \log (d+e x)}{e^{12}} \] Output:

-b^9*(-A*b*e-10*B*a*e+10*B*b*d)*x/e^11+1/2*b^10*B*x^2/e^10+1/9*(-a*e+b*d)^ 
10*(-A*e+B*d)/e^12/(e*x+d)^9-1/8*(-a*e+b*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)/e 
^12/(e*x+d)^8+5/7*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)^ 
7-5/2*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d)/e^12/(e*x+d)^6+6*b^3*(- 
a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)/e^12/(e*x+d)^5-21/2*b^4*(-a*e+b*d)^ 
5*(-6*A*b*e-5*B*a*e+11*B*b*d)/e^12/(e*x+d)^4+14*b^5*(-a*e+b*d)^4*(-5*A*b*e 
-6*B*a*e+11*B*b*d)/e^12/(e*x+d)^3-15*b^6*(-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11 
*B*b*d)/e^12/(e*x+d)^2+15*b^7*(-a*e+b*d)^2*(-3*A*b*e-8*B*a*e+11*B*b*d)/e^1 
2/(e*x+d)+5*b^8*(-a*e+b*d)*(-2*A*b*e-9*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. \(1460\) vs. \(2(441)=882\).

Time = 0.56 (sec) , antiderivative size = 1460, normalized size of antiderivative = 3.31 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{10}} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/504*(7*a^10*e^10*(8*A*e + B*(d + 9*e*x)) + 10*a^9*b*e^9*(7*A*e*(d + 9*e 
*x) + 2*B*(d^2 + 9*d*e*x + 36*e^2*x^2)) + 45*a^8*b^2*e^8*(2*A*e*(d^2 + 9*d 
*e*x + 36*e^2*x^2) + B*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3)) + 24 
*a^7*b^3*e^7*(5*A*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 4*B*(d 
^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4)) + 42*a^6*b^ 
4*e^6*(4*A*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^ 
4) + 5*B*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^ 
4 + 126*e^5*x^5)) + 252*a^5*b^5*e^5*(A*e*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 
 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5) + 2*B*(d^6 + 9*d^5*e*x + 
36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6 
*x^6)) + 210*a^4*b^6*e^4*(2*A*e*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3 
*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6) + 7*B*(d^7 + 9*d^ 
6*e*x + 36*d^5*e^2*x^2 + 84*d^4*e^3*x^3 + 126*d^3*e^4*x^4 + 126*d^2*e^5*x^ 
5 + 84*d*e^6*x^6 + 36*e^7*x^7)) + 840*a^3*b^7*e^3*(A*e*(d^7 + 9*d^6*e*x + 
36*d^5*e^2*x^2 + 84*d^4*e^3*x^3 + 126*d^3*e^4*x^4 + 126*d^2*e^5*x^5 + 84*d 
*e^6*x^6 + 36*e^7*x^7) + 8*B*(d^8 + 9*d^7*e*x + 36*d^6*e^2*x^2 + 84*d^5*e^ 
3*x^3 + 126*d^4*e^4*x^4 + 126*d^3*e^5*x^5 + 84*d^2*e^6*x^6 + 36*d*e^7*x^7 
+ 9*e^8*x^8)) - 9*a^2*b^8*e^2*(-280*A*e*(d^8 + 9*d^7*e*x + 36*d^6*e^2*x^2 
+ 84*d^5*e^3*x^3 + 126*d^4*e^4*x^4 + 126*d^3*e^5*x^5 + 84*d^2*e^6*x^6 + 36 
*d*e^7*x^7 + 9*e^8*x^8) + B*d*(7129*d^8 + 61641*d^7*e*x + 235224*d^6*e^...
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 441, 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.

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

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^9 x (-10 a B e-A b e+10 b B d)}{e^{11}}+\frac {5 b^8 (b d-a e) \log (d+e x) (-9 a B e-2 A b e+11 b B d)}{e^{12}}+\frac {15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{12} (d+e x)}-\frac {15 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12} (d+e x)^2}+\frac {14 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)^3}-\frac {21 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{2 e^{12} (d+e x)^4}+\frac {6 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^5}-\frac {5 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{2 e^{12} (d+e x)^6}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{7 e^{12} (d+e x)^7}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{8 e^{12} (d+e x)^8}+\frac {(b d-a e)^{10} (B d-A e)}{9 e^{12} (d+e x)^9}+\frac {b^{10} B x^2}{2 e^{10}}\)

Input:

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

Output:

-((b^9*(10*b*B*d - A*b*e - 10*a*B*e)*x)/e^11) + (b^10*B*x^2)/(2*e^10) + (( 
b*d - a*e)^10*(B*d - A*e))/(9*e^12*(d + e*x)^9) - ((b*d - a*e)^9*(11*b*B*d 
 - 10*A*b*e - a*B*e))/(8*e^12*(d + e*x)^8) + (5*b*(b*d - a*e)^8*(11*b*B*d 
- 9*A*b*e - 2*a*B*e))/(7*e^12*(d + e*x)^7) - (5*b^2*(b*d - a*e)^7*(11*b*B* 
d - 8*A*b*e - 3*a*B*e))/(2*e^12*(d + e*x)^6) + (6*b^3*(b*d - a*e)^6*(11*b* 
B*d - 7*A*b*e - 4*a*B*e))/(e^12*(d + e*x)^5) - (21*b^4*(b*d - a*e)^5*(11*b 
*B*d - 6*A*b*e - 5*a*B*e))/(2*e^12*(d + e*x)^4) + (14*b^5*(b*d - a*e)^4*(1 
1*b*B*d - 5*A*b*e - 6*a*B*e))/(e^12*(d + e*x)^3) - (15*b^6*(b*d - a*e)^3*( 
11*b*B*d - 4*A*b*e - 7*a*B*e))/(e^12*(d + e*x)^2) + (15*b^7*(b*d - a*e)^2* 
(11*b*B*d - 3*A*b*e - 8*a*B*e))/(e^12*(d + e*x)) + (5*b^8*(b*d - a*e)*(11* 
b*B*d - 2*A*b*e - 9*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. \(1927\) vs. \(2(429)=858\).

Time = 0.24 (sec) , antiderivative size = 1928, normalized size of antiderivative = 4.37

method	result	size
default	\(\text {Expression too large to display}\)	\(1928\)
norman	\(\text {Expression too large to display}\)	\(1933\)
risch	\(\text {Expression too large to display}\)	\(1942\)
parallelrisch	\(\text {Expression too large to display}\)	\(3215\)

Input:

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

Output:

b^9/e^11*(1/2*B*b*e*x^2+A*b*e*x+10*B*a*e*x-10*B*b*d*x)-5/2*b^2/e^12*(8*A*a 
^7*b*e^8-56*A*a^6*b^2*d*e^7+168*A*a^5*b^3*d^2*e^6-280*A*a^4*b^4*d^3*e^5+28 
0*A*a^3*b^5*d^4*e^4-168*A*a^2*b^6*d^5*e^3+56*A*a*b^7*d^6*e^2-8*A*b^8*d^7*e 
+3*B*a^8*e^8-32*B*a^7*b*d*e^7+140*B*a^6*b^2*d^2*e^6-336*B*a^5*b^3*d^3*e^5+ 
490*B*a^4*b^4*d^4*e^4-448*B*a^3*b^5*d^5*e^3+252*B*a^2*b^6*d^6*e^2-80*B*a*b 
^7*d^7*e+11*B*b^8*d^8)/(e*x+d)^6-5/7*b/e^12*(9*A*a^8*b*e^9-72*A*a^7*b^2*d* 
e^8+252*A*a^6*b^3*d^2*e^7-504*A*a^5*b^4*d^3*e^6+630*A*a^4*b^5*d^4*e^5-504* 
A*a^3*b^6*d^5*e^4+252*A*a^2*b^7*d^6*e^3-72*A*a*b^8*d^7*e^2+9*A*b^9*d^8*e+2 
*B*a^9*e^9-27*B*a^8*b*d*e^8+144*B*a^7*b^2*d^2*e^7-420*B*a^6*b^3*d^3*e^6+75 
6*B*a^5*b^4*d^4*e^5-882*B*a^4*b^5*d^5*e^4+672*B*a^3*b^6*d^6*e^3-324*B*a^2* 
b^7*d^7*e^2+90*B*a*b^8*d^8*e-11*B*b^9*d^9)/(e*x+d)^7-1/8/e^12*(10*A*a^9*b* 
e^10-90*A*a^8*b^2*d*e^9+360*A*a^7*b^3*d^2*e^8-840*A*a^6*b^4*d^3*e^7+1260*A 
*a^5*b^5*d^4*e^6-1260*A*a^4*b^6*d^5*e^5+840*A*a^3*b^7*d^6*e^4-360*A*a^2*b^ 
8*d^7*e^3+90*A*a*b^9*d^8*e^2-10*A*b^10*d^9*e+B*a^10*e^10-20*B*a^9*b*d*e^9+ 
135*B*a^8*b^2*d^2*e^8-480*B*a^7*b^3*d^3*e^7+1050*B*a^6*b^4*d^4*e^6-1512*B* 
a^5*b^5*d^5*e^5+1470*B*a^4*b^6*d^6*e^4-960*B*a^3*b^7*d^7*e^3+405*B*a^2*b^8 
*d^8*e^2-100*B*a*b^9*d^9*e+11*B*b^10*d^10)/(e*x+d)^8-15*b^7/e^12*(3*A*a^2* 
b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+8*B*a^3*e^3-27*B*a^2*b*d*e^2+30*B*a*b^ 
2*d^2*e-11*B*b^3*d^3)/(e*x+d)-15*b^6/e^12*(4*A*a^3*b*e^4-12*A*a^2*b^2*d*e^ 
3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e+7*B*a^4*e^4-32*B*a^3*b*d*e^3+54*B*a^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2501 vs. \(2 (429) = 858\).

Time = 0.23 (sec) , antiderivative size = 2501, normalized size of antiderivative = 5.67 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{10}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/504*(252*B*b^10*e^11*x^11 + 42131*B*b^10*d^11 - 56*A*a^10*e^11 - 9722*(1 
0*B*a*b^9 + A*b^10)*d^10*e + 7129*(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 - 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7* 
e^4 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 - 42*(5*B*a^6*b^4 + 6*A*a^5*b 
^5)*d^5*e^6 - 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 - 15*(3*B*a^8*b^2 + 8 
*A*a^7*b^3)*d^3*e^8 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 7*(B*a^10 + 1 
0*A*a^9*b)*d*e^10 - 252*(11*B*b^10*d*e^10 - 2*(10*B*a*b^9 + A*b^10)*e^11)* 
x^10 - 4536*(8*B*b^10*d^2*e^9 - (10*B*a*b^9 + A*b^10)*d*e^10)*x^9 - 1512*( 
51*B*b^10*d^3*e^8 + 3*(10*B*a*b^9 + A*b^10)*d^2*e^9 - 15*(9*B*a^2*b^8 + 2* 
A*a*b^9)*d*e^10 + 5*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 1512*(126*B*b^ 
10*d^4*e^7 - 72*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 90*(9*B*a^2*b^8 + 2*A*a*b^ 
9)*d^2*e^9 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 - 5*(7*B*a^4*b^6 + 4*A* 
a^3*b^7)*e^11)*x^7 + 3528*(346*B*b^10*d^5*e^6 - 112*(10*B*a*b^9 + A*b^10)* 
d^4*e^7 + 110*(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 - 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 - 2*(6*B*a^5*b^5 + 5 
*A*a^4*b^6)*e^11)*x^6 + 5292*(511*B*b^10*d^6*e^5 - 142*(10*B*a*b^9 + A*b^1 
0)*d^5*e^6 + 125*(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 - 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 2*(6*B*a^5*b^5 
 + 5*A*a^4*b^6)*d*e^10 - (5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 756*(4501 
*B*b^10*d^7*e^4 - 1162*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 959*(9*B*a^2*b^8...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1904 vs. \(2 (429) = 858\).

Time = 0.17 (sec) , antiderivative size = 1904, normalized size of antiderivative = 4.32 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{10}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/504*(42131*B*b^10*d^11 - 56*A*a^10*e^11 - 9722*(10*B*a*b^9 + A*b^10)*d^1 
0*e + 7129*(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 - 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 84*(6*B*a^5*b^5 + 
 5*A*a^4*b^6)*d^6*e^5 - 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 24*(4*B*a 
^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 - 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 1 
0*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 7*(B*a^10 + 10*A*a^9*b)*d*e^10 + 756 
0*(11*B*b^10*d^3*e^8 - 3*(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 + 7560*(77*B*b^1 
0*d^4*e^7 - 20*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 18*(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^10 - (7*B*a^4*b^6 + 4*A*a^3* 
b^7)*e^11)*x^7 + 3528*(517*B*b^10*d^5*e^6 - 130*(10*B*a*b^9 + A*b^10)*d^4* 
e^7 + 110*(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 - 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 - 2*(6*B*a^5*b^5 + 5*A*a 
^4*b^6)*e^11)*x^6 + 5292*(627*B*b^10*d^6*e^5 - 154*(10*B*a*b^9 + A*b^10)*d 
^5*e^6 + 125*(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 - 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 2*(6*B*a^5*b^5 + 5 
*A*a^4*b^6)*d*e^10 - (5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 756*(5049*B*b 
^10*d^7*e^4 - 1218*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 959*(9*B*a^2*b^8 + 2*A* 
a*b^9)*d^5*e^6 - 140*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 - 35*(7*B*a^4*b^6 
 + 4*A*a^3*b^7)*d^3*e^8 - 14*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 - 7*(5...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1975 vs. \(2 (429) = 858\).

Time = 0.13 (sec) , antiderivative size = 1975, normalized size of antiderivative = 4.48 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{10}} \, dx=\text {Too large to display} \] Input:

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

Output:

5*(11*B*b^10*d^2 - 20*B*a*b^9*d*e - 2*A*b^10*d*e + 9*B*a^2*b^8*e^2 + 2*A*a 
*b^9*e^2)*log(abs(e*x + d))/e^12 + 1/2*(B*b^10*e^10*x^2 - 20*B*b^10*d*e^9* 
x + 20*B*a*b^9*e^10*x + 2*A*b^10*e^10*x)/e^20 + 1/504*(42131*B*b^10*d^11 - 
 97220*B*a*b^9*d^10*e - 9722*A*b^10*d^10*e + 64161*B*a^2*b^8*d^9*e^2 + 142 
58*A*a*b^9*d^9*e^2 - 6720*B*a^3*b^7*d^8*e^3 - 2520*A*a^2*b^8*d^8*e^3 - 147 
0*B*a^4*b^6*d^7*e^4 - 840*A*a^3*b^7*d^7*e^4 - 504*B*a^5*b^5*d^6*e^5 - 420* 
A*a^4*b^6*d^6*e^5 - 210*B*a^6*b^4*d^5*e^6 - 252*A*a^5*b^5*d^5*e^6 - 96*B*a 
^7*b^3*d^4*e^7 - 168*A*a^6*b^4*d^4*e^7 - 45*B*a^8*b^2*d^3*e^8 - 120*A*a^7* 
b^3*d^3*e^8 - 20*B*a^9*b*d^2*e^9 - 90*A*a^8*b^2*d^2*e^9 - 7*B*a^10*d*e^10 
- 70*A*a^9*b*d*e^10 - 56*A*a^10*e^11 + 7560*(11*B*b^10*d^3*e^8 - 30*B*a*b^ 
9*d^2*e^9 - 3*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 + 7560*(77*B*b^10*d^4*e^7 - 200*B*a 
*b^9*d^3*e^8 - 20*A*b^10*d^3*e^8 + 162*B*a^2*b^8*d^2*e^9 + 36*A*a*b^9*d^2* 
e^9 - 32*B*a^3*b^7*d*e^10 - 12*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 + 3528*(517*B*b^10*d^5*e^6 - 1300*B*a*b^9*d^4*e^7 - 130*A 
*b^10*d^4*e^7 + 990*B*a^2*b^8*d^3*e^8 + 220*A*a*b^9*d^3*e^8 - 160*B*a^3*b^ 
7*d^2*e^9 - 60*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 - 12*B*a^5*b^5*e^11 - 10*A*a^4*b^6*e^11)*x^6 + 5292*(627*B*b^10*d^6*e^5 
 - 1540*B*a*b^9*d^5*e^6 - 154*A*b^10*d^5*e^6 + 1125*B*a^2*b^8*d^4*e^7 + 25 
0*A*a*b^9*d^4*e^7 - 160*B*a^3*b^7*d^3*e^8 - 60*A*a^2*b^8*d^3*e^8 - 35*B...
 

Mupad [B] (verification not implemented)

Time = 1.25 (sec) , antiderivative size = 2048, normalized size of antiderivative = 4.64 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{10}} \, dx=\text {Too large to display} \] Input:

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

Output:

x*((A*b^10 + 10*B*a*b^9)/e^10 - (10*B*b^10*d)/e^11) - (x^7*(60*A*a^3*b^7*e 
^10 + 105*B*a^4*b^6*e^10 + 300*A*b^10*d^3*e^7 - 1155*B*b^10*d^4*e^6 - 540* 
A*a*b^9*d^2*e^8 + 180*A*a^2*b^8*d*e^9 + 3000*B*a*b^9*d^3*e^7 + 480*B*a^3*b 
^7*d*e^9 - 2430*B*a^2*b^8*d^2*e^8) + x^4*(42*A*a^6*b^4*e^10 + 24*B*a^7*b^3 
*e^10 + 1827*A*b^10*d^6*e^4 - (15147*B*b^10*d^7*e^3)/2 - 2877*A*a*b^9*d^5* 
e^5 + 63*A*a^5*b^5*d*e^9 + 18270*B*a*b^9*d^6*e^4 + (105*B*a^6*b^4*d*e^9)/2 
 + 630*A*a^2*b^8*d^4*e^6 + 210*A*a^3*b^7*d^3*e^7 + 105*A*a^4*b^6*d^2*e^8 - 
 (25893*B*a^2*b^8*d^5*e^5)/2 + 1680*B*a^3*b^7*d^4*e^6 + (735*B*a^4*b^6*d^3 
*e^7)/2 + 126*B*a^5*b^5*d^2*e^8) + x^6*(70*A*a^4*b^6*e^10 + 84*B*a^5*b^5*e 
^10 + 910*A*b^10*d^4*e^6 - 3619*B*b^10*d^5*e^5 - 1540*A*a*b^9*d^3*e^7 + 14 
0*A*a^3*b^7*d*e^9 + 9100*B*a*b^9*d^4*e^6 + 245*B*a^4*b^6*d*e^9 + 420*A*a^2 
*b^8*d^2*e^8 - 6930*B*a^2*b^8*d^3*e^7 + 1120*B*a^3*b^7*d^2*e^8) + x^3*(20* 
A*a^7*b^3*e^10 + (15*B*a^8*b^2*e^10)/2 + 1338*A*b^10*d^7*e^3 - (11253*B*b^ 
10*d^8*e^2)/2 - 2058*A*a*b^9*d^6*e^4 + 28*A*a^6*b^4*d*e^9 + 13380*B*a*b^9* 
d^7*e^3 + 16*B*a^7*b^3*d*e^9 + 420*A*a^2*b^8*d^5*e^5 + 140*A*a^3*b^7*d^4*e 
^6 + 70*A*a^4*b^6*d^3*e^7 + 42*A*a^5*b^5*d^2*e^8 - 9261*B*a^2*b^8*d^6*e^4 
+ 1120*B*a^3*b^7*d^5*e^5 + 245*B*a^4*b^6*d^4*e^6 + 84*B*a^5*b^5*d^3*e^7 + 
35*B*a^6*b^4*d^2*e^8) + (56*A*a^10*e^11 - 42131*B*b^10*d^11 + 9722*A*b^10* 
d^10*e + 7*B*a^10*d*e^10 - 14258*A*a*b^9*d^9*e^2 + 20*B*a^9*b*d^2*e^9 + 25 
20*A*a^2*b^8*d^8*e^3 + 840*A*a^3*b^7*d^7*e^4 + 420*A*a^4*b^6*d^6*e^5 + ...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1714, normalized size of antiderivative = 3.89 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{10}} \, dx =\text {Too large to display} \] Input:

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

Output:

(27720*log(d + e*x)*a**2*b**9*d**10*e**2 + 249480*log(d + e*x)*a**2*b**9*d 
**9*e**3*x + 997920*log(d + e*x)*a**2*b**9*d**8*e**4*x**2 + 2328480*log(d 
+ e*x)*a**2*b**9*d**7*e**5*x**3 + 3492720*log(d + e*x)*a**2*b**9*d**6*e**6 
*x**4 + 3492720*log(d + e*x)*a**2*b**9*d**5*e**7*x**5 + 2328480*log(d + e* 
x)*a**2*b**9*d**4*e**8*x**6 + 997920*log(d + e*x)*a**2*b**9*d**3*e**9*x**7 
 + 249480*log(d + e*x)*a**2*b**9*d**2*e**10*x**8 + 27720*log(d + e*x)*a**2 
*b**9*d*e**11*x**9 - 55440*log(d + e*x)*a*b**10*d**11*e - 498960*log(d + e 
*x)*a*b**10*d**10*e**2*x - 1995840*log(d + e*x)*a*b**10*d**9*e**3*x**2 - 4 
656960*log(d + e*x)*a*b**10*d**8*e**4*x**3 - 6985440*log(d + e*x)*a*b**10* 
d**7*e**5*x**4 - 6985440*log(d + e*x)*a*b**10*d**6*e**6*x**5 - 4656960*log 
(d + e*x)*a*b**10*d**5*e**7*x**6 - 1995840*log(d + e*x)*a*b**10*d**4*e**8* 
x**7 - 498960*log(d + e*x)*a*b**10*d**3*e**9*x**8 - 55440*log(d + e*x)*a*b 
**10*d**2*e**10*x**9 + 27720*log(d + e*x)*b**11*d**12 + 249480*log(d + e*x 
)*b**11*d**11*e*x + 997920*log(d + e*x)*b**11*d**10*e**2*x**2 + 2328480*lo 
g(d + e*x)*b**11*d**9*e**3*x**3 + 3492720*log(d + e*x)*b**11*d**8*e**4*x** 
4 + 3492720*log(d + e*x)*b**11*d**7*e**5*x**5 + 2328480*log(d + e*x)*b**11 
*d**6*e**6*x**6 + 997920*log(d + e*x)*b**11*d**5*e**7*x**7 + 249480*log(d 
+ e*x)*b**11*d**4*e**8*x**8 + 27720*log(d + e*x)*b**11*d**3*e**9*x**9 - 56 
*a**11*d*e**11 - 77*a**10*b*d**2*e**10 - 693*a**10*b*d*e**11*x - 110*a**9* 
b**2*d**3*e**9 - 990*a**9*b**2*d**2*e**10*x - 3960*a**9*b**2*d*e**11*x*...