\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^5} \, dx\) [83]

Optimal result

Integrand size = 20, antiderivative size = 444 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^5} \, dx=\frac {42 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{4 e^{12} (d+e x)^4}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{3 e^{12} (d+e x)^3}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{2 e^{12} (d+e x)^2}-\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{e^{12} (d+e x)}-\frac {21 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) (d+e x)^2}{e^{12}}+\frac {10 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) (d+e x)^3}{e^{12}}-\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^4}{4 e^{12}}+\frac {b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^5}{e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^6}{6 e^{12}}+\frac {b^{10} B (d+e x)^7}{7 e^{12}}-\frac {30 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e) \log (d+e x)}{e^{12}} \] Output:

42*b^4*(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)*x/e^11+1/4*(-a*e+b*d)^10*( 
-A*e+B*d)/e^12/(e*x+d)^4-1/3*(-a*e+b*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/ 
(e*x+d)^3+5/2*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)^2-15 
*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d)/e^12/(e*x+d)-21*b^5*(-a*e+b* 
d)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)*(e*x+d)^2/e^12+10*b^6*(-a*e+b*d)^3*(-4*A* 
b*e-7*B*a*e+11*B*b*d)*(e*x+d)^3/e^12-15/4*b^7*(-a*e+b*d)^2*(-3*A*b*e-8*B*a 
*e+11*B*b*d)*(e*x+d)^4/e^12+b^8*(-a*e+b*d)*(-2*A*b*e-9*B*a*e+11*B*b*d)*(e* 
x+d)^5/e^12-1/6*b^9*(-A*b*e-10*B*a*e+11*B*b*d)*(e*x+d)^6/e^12+1/7*b^10*B*( 
e*x+d)^7/e^12-30*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)*ln(e*x+d)/e^ 
12
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 686, normalized size of antiderivative = 1.55 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^5} \, dx=\frac {-84 b^4 e \left (-210 a^6 B e^6+140 a b^5 d^4 e (9 B d-5 A e)-42 b^6 d^5 (5 B d-3 A e)+600 a^3 b^3 d^2 e^3 (7 B d-3 A e)-1575 a^2 b^4 d^3 e^2 (2 B d-A e)-1050 a^4 b^2 d e^4 (3 B d-A e)-252 a^5 b e^5 (-5 B d+A e)\right ) x+42 b^5 e^2 \left (252 a^5 B e^5-14 b^5 d^4 (9 B d-5 A e)-225 a^2 b^3 d^2 e^2 (7 B d-3 A e)+350 a b^4 d^3 e (2 B d-A e)+600 a^3 b^2 d e^3 (3 B d-A e)+210 a^4 b e^4 (-5 B d+A e)\right ) x^2-140 b^6 e^3 \left (-42 a^4 B e^4+10 a b^3 d^2 e (7 B d-3 A e)-45 a^2 b^2 d e^2 (3 B d-A e)-24 a^3 b e^3 (-5 B d+A e)+7 b^4 d^3 (-2 B d+A e)\right ) x^3+105 b^7 e^4 \left (24 a^3 B e^3+10 a b^2 d e (3 B d-A e)+9 a^2 b e^2 (-5 B d+A e)+b^3 d^2 (-7 B d+3 A e)\right ) x^4-84 b^8 e^5 \left (-9 a^2 B e^2-2 a b e (-5 B d+A e)+b^2 d (-3 B d+A e)\right ) x^5+14 b^9 e^6 (-5 b B d+A b e+10 a B e) x^6+12 b^{10} B e^7 x^7+\frac {21 (b d-a e)^{10} (B d-A e)}{(d+e x)^4}-\frac {28 (b d-a e)^9 (11 b B d-10 A b e-a B e)}{(d+e x)^3}+\frac {210 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{(d+e x)^2}-\frac {1260 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{d+e x}-2520 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e) \log (d+e x)}{84 e^{12}} \] Input:

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^5,x]
 

Output:

(-84*b^4*e*(-210*a^6*B*e^6 + 140*a*b^5*d^4*e*(9*B*d - 5*A*e) - 42*b^6*d^5* 
(5*B*d - 3*A*e) + 600*a^3*b^3*d^2*e^3*(7*B*d - 3*A*e) - 1575*a^2*b^4*d^3*e 
^2*(2*B*d - A*e) - 1050*a^4*b^2*d*e^4*(3*B*d - A*e) - 252*a^5*b*e^5*(-5*B* 
d + A*e))*x + 42*b^5*e^2*(252*a^5*B*e^5 - 14*b^5*d^4*(9*B*d - 5*A*e) - 225 
*a^2*b^3*d^2*e^2*(7*B*d - 3*A*e) + 350*a*b^4*d^3*e*(2*B*d - A*e) + 600*a^3 
*b^2*d*e^3*(3*B*d - A*e) + 210*a^4*b*e^4*(-5*B*d + A*e))*x^2 - 140*b^6*e^3 
*(-42*a^4*B*e^4 + 10*a*b^3*d^2*e*(7*B*d - 3*A*e) - 45*a^2*b^2*d*e^2*(3*B*d 
 - A*e) - 24*a^3*b*e^3*(-5*B*d + A*e) + 7*b^4*d^3*(-2*B*d + A*e))*x^3 + 10 
5*b^7*e^4*(24*a^3*B*e^3 + 10*a*b^2*d*e*(3*B*d - A*e) + 9*a^2*b*e^2*(-5*B*d 
 + A*e) + b^3*d^2*(-7*B*d + 3*A*e))*x^4 - 84*b^8*e^5*(-9*a^2*B*e^2 - 2*a*b 
*e*(-5*B*d + A*e) + b^2*d*(-3*B*d + A*e))*x^5 + 14*b^9*e^6*(-5*b*B*d + A*b 
*e + 10*a*B*e)*x^6 + 12*b^10*B*e^7*x^7 + (21*(b*d - a*e)^10*(B*d - A*e))/( 
d + e*x)^4 - (28*(b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(d + e*x)^3 
+ (210*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(d + e*x)^2 - (1260 
*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(d + e*x) - 2520*b^3*(b 
*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e)*Log[d + e*x])/(84*e^12)
 

Rubi [A] (verified)

Time = 1.44 (sec) , antiderivative size = 444, 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.

\(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^5} \, dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^9 (d+e x)^6 (-10 a B e-A b e+11 b B d)}{6 e^{12}}+\frac {b^8 (d+e x)^5 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{e^{12}}-\frac {15 b^7 (d+e x)^4 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{4 e^{12}}+\frac {10 b^6 (d+e x)^3 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12}}-\frac {21 b^5 (d+e x)^2 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12}}+\frac {42 b^4 x (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{11}}-\frac {30 b^3 (b d-a e)^6 \log (d+e x) (-4 a B e-7 A b e+11 b B d)}{e^{12}}-\frac {15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{e^{12} (d+e x)}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{2 e^{12} (d+e x)^2}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{3 e^{12} (d+e x)^3}+\frac {(b d-a e)^{10} (B d-A e)}{4 e^{12} (d+e x)^4}+\frac {b^{10} B (d+e x)^7}{7 e^{12}}\)

Input:

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^5,x]
 

Output:

(42*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e)*x)/e^11 + ((b*d - a*e 
)^10*(B*d - A*e))/(4*e^12*(d + e*x)^4) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b 
*e - a*B*e))/(3*e^12*(d + e*x)^3) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e 
 - 2*a*B*e))/(2*e^12*(d + e*x)^2) - (15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A* 
b*e - 3*a*B*e))/(e^12*(d + e*x)) - (21*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b 
*e - 6*a*B*e)*(d + e*x)^2)/e^12 + (10*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b* 
e - 7*a*B*e)*(d + e*x)^3)/e^12 - (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e 
 - 8*a*B*e)*(d + e*x)^4)/(4*e^12) + (b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 
 9*a*B*e)*(d + e*x)^5)/e^12 - (b^9*(11*b*B*d - A*b*e - 10*a*B*e)*(d + e*x) 
^6)/(6*e^12) + (b^10*B*(d + e*x)^7)/(7*e^12) - (30*b^3*(b*d - a*e)^6*(11*b 
*B*d - 7*A*b*e - 4*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. \(1909\) vs. \(2(432)=864\).

Time = 0.24 (sec) , antiderivative size = 1910, normalized size of antiderivative = 4.30

method	result	size
norman	\(\text {Expression too large to display}\)	\(1910\)
default	\(\text {Expression too large to display}\)	\(2034\)
risch	\(\text {Expression too large to display}\)	\(2127\)
parallelrisch	\(\text {Expression too large to display}\)	\(3635\)

Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^5,x,method=_RETURNVERBOSE)
 

Output:

(b^5*(105*A*a^4*b*e^5-140*A*a^3*b^2*d*e^4+105*A*a^2*b^3*d^2*e^3-42*A*a*b^4 
*d^3*e^2+7*A*b^5*d^4*e+126*B*a^5*e^5-245*B*a^4*b*d*e^4+280*B*a^3*b^2*d^2*e 
^3-189*B*a^2*b^3*d^3*e^2+70*B*a*b^4*d^4*e-11*B*b^5*d^5)/e^6*x^6-1/12*(3*A* 
a^10*e^11+10*A*a^9*b*d*e^10+45*A*a^8*b^2*d^2*e^9+360*A*a^7*b^3*d^3*e^8-525 
0*A*a^6*b^4*d^4*e^7+31500*A*a^5*b^5*d^5*e^6-78750*A*a^4*b^6*d^6*e^5+105000 
*A*a^3*b^7*d^7*e^4-78750*A*a^2*b^8*d^8*e^3+31500*A*a*b^9*d^9*e^2-5250*A*b^ 
10*d^10*e+B*a^10*d*e^10+10*B*a^9*b*d^2*e^9+135*B*a^8*b^2*d^3*e^8-3000*B*a^ 
7*b^3*d^4*e^7+26250*B*a^6*b^4*d^5*e^6-94500*B*a^5*b^5*d^6*e^5+183750*B*a^4 
*b^6*d^7*e^4-210000*B*a^3*b^7*d^8*e^3+141750*B*a^2*b^8*d^9*e^2-52500*B*a*b 
^9*d^10*e+8250*B*b^10*d^11)/e^12-(120*A*a^7*b^3*e^8-840*A*a^6*b^4*d*e^7+50 
40*A*a^5*b^5*d^2*e^6-12600*A*a^4*b^6*d^3*e^5+16800*A*a^3*b^7*d^4*e^4-12600 
*A*a^2*b^8*d^5*e^3+5040*A*a*b^9*d^6*e^2-840*A*b^10*d^7*e+45*B*a^8*b^2*e^8- 
480*B*a^7*b^3*d*e^7+4200*B*a^6*b^4*d^2*e^6-15120*B*a^5*b^5*d^3*e^5+29400*B 
*a^4*b^6*d^4*e^4-33600*B*a^3*b^7*d^5*e^3+22680*B*a^2*b^8*d^6*e^2-8400*B*a* 
b^9*d^7*e+1320*B*b^10*d^8)/e^9*x^3-1/2*(45*A*a^8*b^2*e^9+360*A*a^7*b^3*d*e 
^8-3780*A*a^6*b^4*d^2*e^7+22680*A*a^5*b^5*d^3*e^6-56700*A*a^4*b^6*d^4*e^5+ 
75600*A*a^3*b^7*d^5*e^4-56700*A*a^2*b^8*d^6*e^3+22680*A*a*b^9*d^7*e^2-3780 
*A*b^10*d^8*e+10*B*a^9*b*e^9+135*B*a^8*b^2*d*e^8-2160*B*a^7*b^3*d^2*e^7+18 
900*B*a^6*b^4*d^3*e^6-68040*B*a^5*b^5*d^4*e^5+132300*B*a^4*b^6*d^5*e^4-151 
200*B*a^3*b^7*d^6*e^3+102060*B*a^2*b^8*d^7*e^2-37800*B*a*b^9*d^8*e+5940...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2808 vs. \(2 (432) = 864\).

Time = 0.14 (sec) , antiderivative size = 2808, normalized size of antiderivative = 6.32 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^5} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^5,x, algorithm="fricas")
 

Output:

1/84*(12*B*b^10*e^11*x^11 - 11837*B*b^10*d^11 - 21*A*a^10*e^11 + 8449*(10* 
B*a*b^9 + A*b^10)*d^10*e - 28875*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 55965 
*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 - 66990*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d 
^7*e^4 + 50274*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 - 22638*(5*B*a^6*b^4 + 
6*A*a^5*b^5)*d^5*e^6 + 5250*(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 - 35*(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 - 2*(11*B*b^10*d*e^10 - 7*(10*B*a*b^9 + A*b^ 
10)*e^11)*x^10 + 4*(11*B*b^10*d^2*e^9 - 7*(10*B*a*b^9 + A*b^10)*d*e^10 + 2 
1*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 9*(11*B*b^10*d^3*e^8 - 7*(10*B*a*b 
^9 + A*b^10)*d^2*e^9 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 35*(8*B*a^3*b 
^7 + 3*A*a^2*b^8)*e^11)*x^8 + 24*(11*B*b^10*d^4*e^7 - 7*(10*B*a*b^9 + A*b^ 
10)*d^3*e^8 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 35*(8*B*a^3*b^7 + 3*A 
*a^2*b^8)*d*e^10 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 - 84*(11*B*b^1 
0*d^5*e^6 - 7*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 21*(9*B*a^2*b^8 + 2*A*a*b^9) 
*d^3*e^8 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 35*(7*B*a^4*b^6 + 4*A* 
a^3*b^7)*d*e^10 - 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 504*(11*B*b^1 
0*d^6*e^5 - 7*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 21*(9*B*a^2*b^8 + 2*A*a*b^9) 
*d^4*e^7 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 35*(7*B*a^4*b^6 + 4*A* 
a^3*b^7)*d^2*e^9 - 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 7*(5*B*a^6*b^4 
+ 6*A*a^5*b^5)*e^11)*x^5 + 7*(6559*B*b^10*d^7*e^4 - 4043*(10*B*a*b^9 + ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^5} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**5,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1846 vs. \(2 (432) = 864\).

Time = 0.12 (sec) , antiderivative size = 1846, normalized size of antiderivative = 4.16 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^5} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^5,x, algorithm="maxima")
 

Output:

-1/12*(1691*B*b^10*d^11 + 3*A*a^10*e^11 - 1207*(10*B*a*b^9 + A*b^10)*d^10* 
e + 4125*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 7995*(8*B*a^3*b^7 + 3*A*a^2*b 
^8)*d^8*e^3 + 9570*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 7182*(6*B*a^5*b^5 
 + 5*A*a^4*b^6)*d^6*e^5 + 3234*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 750*( 
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 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + (B*a^10 + 10*A*a^9*b)*d*e^10 + 1 
80*(11*B*b^10*d^8*e^3 - 8*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 28*(9*B*a^2*b^8 
+ 2*A*a*b^9)*d^6*e^5 - 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 70*(7*B*a^ 
4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 28 
*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^1 
0 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 30*(187*B*b^10*d^9*e^2 - 135*( 
10*B*a*b^9 + A*b^10)*d^8*e^3 + 468*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 924 
*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 1134*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^ 
5*e^6 - 882*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 420*(5*B*a^6*b^4 + 6*A*a 
^5*b^5)*d^3*e^8 - 108*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 9*(3*B*a^8*b^2 
 + 8*A*a^7*b^3)*d*e^10 + (2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 4*(1331*B*b 
^10*d^10*e - 955*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 3285*(9*B*a^2*b^8 + 2*A*a 
*b^9)*d^8*e^3 - 6420*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 7770*(7*B*a^4*b 
^6 + 4*A*a^3*b^7)*d^6*e^5 - 5922*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 273 
0*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 660*(4*B*a^7*b^3 + 7*A*a^6*b^4)...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2571 vs. \(2 (432) = 864\).

Time = 0.14 (sec) , antiderivative size = 2571, normalized size of antiderivative = 5.79 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^5} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^5,x, algorithm="giac")
 

Output:

1/84*(12*B*b^10 - 14*(11*B*b^10*d*e - 10*B*a*b^9*e^2 - A*b^10*e^2)/((e*x + 
 d)*e) + 84*(11*B*b^10*d^2*e^2 - 20*B*a*b^9*d*e^3 - 2*A*b^10*d*e^3 + 9*B*a 
^2*b^8*e^4 + 2*A*a*b^9*e^4)/((e*x + d)^2*e^2) - 315*(11*B*b^10*d^3*e^3 - 3 
0*B*a*b^9*d^2*e^4 - 3*A*b^10*d^2*e^4 + 27*B*a^2*b^8*d*e^5 + 6*A*a*b^9*d*e^ 
5 - 8*B*a^3*b^7*e^6 - 3*A*a^2*b^8*e^6)/((e*x + d)^3*e^3) + 840*(11*B*b^10* 
d^4*e^4 - 40*B*a*b^9*d^3*e^5 - 4*A*b^10*d^3*e^5 + 54*B*a^2*b^8*d^2*e^6 + 1 
2*A*a*b^9*d^2*e^6 - 32*B*a^3*b^7*d*e^7 - 12*A*a^2*b^8*d*e^7 + 7*B*a^4*b^6* 
e^8 + 4*A*a^3*b^7*e^8)/((e*x + d)^4*e^4) - 1764*(11*B*b^10*d^5*e^5 - 50*B* 
a*b^9*d^4*e^6 - 5*A*b^10*d^4*e^6 + 90*B*a^2*b^8*d^3*e^7 + 20*A*a*b^9*d^3*e 
^7 - 80*B*a^3*b^7*d^2*e^8 - 30*A*a^2*b^8*d^2*e^8 + 35*B*a^4*b^6*d*e^9 + 20 
*A*a^3*b^7*d*e^9 - 6*B*a^5*b^5*e^10 - 5*A*a^4*b^6*e^10)/((e*x + d)^5*e^5) 
+ 3528*(11*B*b^10*d^6*e^6 - 60*B*a*b^9*d^5*e^7 - 6*A*b^10*d^5*e^7 + 135*B* 
a^2*b^8*d^4*e^8 + 30*A*a*b^9*d^4*e^8 - 160*B*a^3*b^7*d^3*e^9 - 60*A*a^2*b^ 
8*d^3*e^9 + 105*B*a^4*b^6*d^2*e^10 + 60*A*a^3*b^7*d^2*e^10 - 36*B*a^5*b^5* 
d*e^11 - 30*A*a^4*b^6*d*e^11 + 5*B*a^6*b^4*e^12 + 6*A*a^5*b^5*e^12)/((e*x 
+ d)^6*e^6))*(e*x + d)^7/e^12 + 30*(11*B*b^10*d^7 - 70*B*a*b^9*d^6*e - 7*A 
*b^10*d^6*e + 189*B*a^2*b^8*d^5*e^2 + 42*A*a*b^9*d^5*e^2 - 280*B*a^3*b^7*d 
^4*e^3 - 105*A*a^2*b^8*d^4*e^3 + 245*B*a^4*b^6*d^3*e^4 + 140*A*a^3*b^7*d^3 
*e^4 - 126*B*a^5*b^5*d^2*e^5 - 105*A*a^4*b^6*d^2*e^5 + 35*B*a^6*b^4*d*e^6 
+ 42*A*a^5*b^5*d*e^6 - 4*B*a^7*b^3*e^7 - 7*A*a^6*b^4*e^7)*log(abs(e*x +...
 

Mupad [B] (verification not implemented)

Time = 1.16 (sec) , antiderivative size = 3655, normalized size of antiderivative = 8.23 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^5} \, dx=\text {Too large to display} \] Input:

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^5,x)
 

Output:

x*((10*d^3*((10*d^2*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e^2 - 
(5*d*((5*d*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e - (5*a*b^8*(2 
*A*b + 9*B*a))/e^5 + (10*B*b^10*d^2)/e^7))/e - (15*a^2*b^7*(3*A*b + 8*B*a) 
)/e^5 + (10*B*b^10*d^3)/e^8))/e^3 - (d^5*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B 
*b^10*d)/e^6))/e^5 - (10*d^2*((5*d*((10*d^2*((A*b^10 + 10*B*a*b^9)/e^5 - ( 
5*B*b^10*d)/e^6))/e^2 - (5*d*((5*d*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10* 
d)/e^6))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^5 + (10*B*b^10*d^2)/e^7))/e - (15 
*a^2*b^7*(3*A*b + 8*B*a))/e^5 + (10*B*b^10*d^3)/e^8))/e - (10*d^3*((A*b^10 
 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e^3 + (10*d^2*((5*d*((A*b^10 + 10* 
B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^5 + (10* 
B*b^10*d^2)/e^7))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^5 - (5*B*b^10*d^4)/ 
e^9))/e^2 + (5*d*((5*d*((5*d*((10*d^2*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^ 
10*d)/e^6))/e^2 - (5*d*((5*d*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6 
))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^5 + (10*B*b^10*d^2)/e^7))/e - (15*a^2*b 
^7*(3*A*b + 8*B*a))/e^5 + (10*B*b^10*d^3)/e^8))/e - (10*d^3*((A*b^10 + 10* 
B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e^3 + (10*d^2*((5*d*((A*b^10 + 10*B*a*b^ 
9)/e^5 - (5*B*b^10*d)/e^6))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^5 + (10*B*b^10 
*d^2)/e^7))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^5 - (5*B*b^10*d^4)/e^9))/ 
e - (10*d^2*((10*d^2*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e^2 - 
 (5*d*((5*d*((A*b^10 + 10*B*a*b^9)/e^5 - (5*B*b^10*d)/e^6))/e - (5*a*b^...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1889, normalized size of antiderivative = 4.25 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^5,x)
                                                                                    
                                                                                    
 

Output:

(27720*log(d + e*x)*a**7*b**4*d**5*e**7 + 110880*log(d + e*x)*a**7*b**4*d* 
*4*e**8*x + 166320*log(d + e*x)*a**7*b**4*d**3*e**9*x**2 + 110880*log(d + 
e*x)*a**7*b**4*d**2*e**10*x**3 + 27720*log(d + e*x)*a**7*b**4*d*e**11*x**4 
 - 194040*log(d + e*x)*a**6*b**5*d**6*e**6 - 776160*log(d + e*x)*a**6*b**5 
*d**5*e**7*x - 1164240*log(d + e*x)*a**6*b**5*d**4*e**8*x**2 - 776160*log( 
d + e*x)*a**6*b**5*d**3*e**9*x**3 - 194040*log(d + e*x)*a**6*b**5*d**2*e** 
10*x**4 + 582120*log(d + e*x)*a**5*b**6*d**7*e**5 + 2328480*log(d + e*x)*a 
**5*b**6*d**6*e**6*x + 3492720*log(d + e*x)*a**5*b**6*d**5*e**7*x**2 + 232 
8480*log(d + e*x)*a**5*b**6*d**4*e**8*x**3 + 582120*log(d + e*x)*a**5*b**6 
*d**3*e**9*x**4 - 970200*log(d + e*x)*a**4*b**7*d**8*e**4 - 3880800*log(d 
+ e*x)*a**4*b**7*d**7*e**5*x - 5821200*log(d + e*x)*a**4*b**7*d**6*e**6*x* 
*2 - 3880800*log(d + e*x)*a**4*b**7*d**5*e**7*x**3 - 970200*log(d + e*x)*a 
**4*b**7*d**4*e**8*x**4 + 970200*log(d + e*x)*a**3*b**8*d**9*e**3 + 388080 
0*log(d + e*x)*a**3*b**8*d**8*e**4*x + 5821200*log(d + e*x)*a**3*b**8*d**7 
*e**5*x**2 + 3880800*log(d + e*x)*a**3*b**8*d**6*e**6*x**3 + 970200*log(d 
+ e*x)*a**3*b**8*d**5*e**7*x**4 - 582120*log(d + e*x)*a**2*b**9*d**10*e**2 
 - 2328480*log(d + e*x)*a**2*b**9*d**9*e**3*x - 3492720*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 - 5821 
20*log(d + e*x)*a**2*b**9*d**6*e**6*x**4 + 194040*log(d + e*x)*a*b**10*d** 
11*e + 776160*log(d + e*x)*a*b**10*d**10*e**2*x + 1164240*log(d + e*x)*...