\(\int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx\) [79]

Optimal result

Integrand size = 20, antiderivative size = 348 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\frac {b (b d-a e)^9 (B d-A e) x}{e^{11}}-\frac {(b d-a e)^8 (B d-A e) (a+b x)^2}{2 e^{10}}+\frac {(b d-a e)^7 (B d-A e) (a+b x)^3}{3 e^9}-\frac {(b d-a e)^6 (B d-A e) (a+b x)^4}{4 e^8}+\frac {(b d-a e)^5 (B d-A e) (a+b x)^5}{5 e^7}-\frac {(b d-a e)^4 (B d-A e) (a+b x)^6}{6 e^6}+\frac {(b d-a e)^3 (B d-A e) (a+b x)^7}{7 e^5}-\frac {(b d-a e)^2 (B d-A e) (a+b x)^8}{8 e^4}+\frac {(b d-a e) (B d-A e) (a+b x)^9}{9 e^3}-\frac {(B d-A e) (a+b x)^{10}}{10 e^2}+\frac {B (a+b x)^{11}}{11 b e}-\frac {(b d-a e)^{10} (B d-A e) \log (d+e x)}{e^{12}} \] Output:

b*(-a*e+b*d)^9*(-A*e+B*d)*x/e^11-1/2*(-a*e+b*d)^8*(-A*e+B*d)*(b*x+a)^2/e^1 
0+1/3*(-a*e+b*d)^7*(-A*e+B*d)*(b*x+a)^3/e^9-1/4*(-a*e+b*d)^6*(-A*e+B*d)*(b 
*x+a)^4/e^8+1/5*(-a*e+b*d)^5*(-A*e+B*d)*(b*x+a)^5/e^7-1/6*(-a*e+b*d)^4*(-A 
*e+B*d)*(b*x+a)^6/e^6+1/7*(-a*e+b*d)^3*(-A*e+B*d)*(b*x+a)^7/e^5-1/8*(-a*e+ 
b*d)^2*(-A*e+B*d)*(b*x+a)^8/e^4+1/9*(-a*e+b*d)*(-A*e+B*d)*(b*x+a)^9/e^3-1/ 
10*(-A*e+B*d)*(b*x+a)^10/e^2+1/11*B*(b*x+a)^11/b/e-(-a*e+b*d)^10*(-A*e+B*d 
)*ln(e*x+d)/e^12
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1252\) vs. \(2(348)=696\).

Time = 0.77 (sec) , antiderivative size = 1252, normalized size of antiderivative = 3.60 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx =\text {Too large to display} \] Input:

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

Output:

(x*(27720*a^10*B*e^10 + 138600*a^9*b*e^9*(-2*B*d + 2*A*e + B*e*x) + 207900 
*a^8*b^2*e^8*(3*A*e*(-2*d + e*x) + B*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 2772 
00*a^7*b^3*e^7*(2*A*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + B*(-12*d^3 + 6*d^2*e 
*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 97020*a^6*b^4*e^6*(5*A*e*(-12*d^3 + 6*d^2 
*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + B*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 
- 15*d*e^3*x^3 + 12*e^4*x^4)) + 116424*a^5*b^5*e^5*(A*e*(60*d^4 - 30*d^3*e 
*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + B*(-60*d^5 + 30*d^4*e*x 
 - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5)) + 13860*a 
^4*b^6*e^4*(7*A*e*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 
- 12*d*e^4*x^4 + 10*e^5*x^5) + B*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 
- 105*d^3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e^6*x^6)) + 3960*a^ 
3*b^7*e^3*(2*A*e*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^ 
3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e^6*x^6) + B*(-840*d^7 + 420*d^6*e* 
x - 280*d^5*e^2*x^2 + 210*d^4*e^3*x^3 - 168*d^3*e^4*x^4 + 140*d^2*e^5*x^5 
- 120*d*e^6*x^6 + 105*e^7*x^7)) + 495*a^2*b^8*e^2*(3*A*e*(-840*d^7 + 420*d 
^6*e*x - 280*d^5*e^2*x^2 + 210*d^4*e^3*x^3 - 168*d^3*e^4*x^4 + 140*d^2*e^5 
*x^5 - 120*d*e^6*x^6 + 105*e^7*x^7) + B*(2520*d^8 - 1260*d^7*e*x + 840*d^6 
*e^2*x^2 - 630*d^5*e^3*x^3 + 504*d^4*e^4*x^4 - 420*d^3*e^5*x^5 + 360*d^2*e 
^6*x^6 - 315*d*e^7*x^7 + 280*e^8*x^8)) + 110*a*b^9*e*(A*e*(2520*d^8 - 1260 
*d^7*e*x + 840*d^6*e^2*x^2 - 630*d^5*e^3*x^3 + 504*d^4*e^4*x^4 - 420*d^...
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 348, 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.

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

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(b*(b*d - a*e)^9*(B*d - A*e)*x)/e^11 - ((b*d - a*e)^8*(B*d - A*e)*(a + b*x 
)^2)/(2*e^10) + ((b*d - a*e)^7*(B*d - A*e)*(a + b*x)^3)/(3*e^9) - ((b*d - 
a*e)^6*(B*d - A*e)*(a + b*x)^4)/(4*e^8) + ((b*d - a*e)^5*(B*d - A*e)*(a + 
b*x)^5)/(5*e^7) - ((b*d - a*e)^4*(B*d - A*e)*(a + b*x)^6)/(6*e^6) + ((b*d 
- a*e)^3*(B*d - A*e)*(a + b*x)^7)/(7*e^5) - ((b*d - a*e)^2*(B*d - A*e)*(a 
+ b*x)^8)/(8*e^4) + ((b*d - a*e)*(B*d - A*e)*(a + b*x)^9)/(9*e^3) - ((B*d 
- A*e)*(a + b*x)^10)/(10*e^2) + (B*(a + b*x)^11)/(11*b*e) - ((b*d - a*e)^1 
0*(B*d - A*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. \(1883\) vs. \(2(328)=656\).

Time = 0.24 (sec) , antiderivative size = 1884, normalized size of antiderivative = 5.41

method	result	size
norman	\(\text {Expression too large to display}\)	\(1884\)
default	\(\text {Expression too large to display}\)	\(2225\)
risch	\(\text {Expression too large to display}\)	\(2357\)
parallelrisch	\(\text {Expression too large to display}\)	\(2358\)

Input:

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

Output:

(10*A*a^9*b*e^10-45*A*a^8*b^2*d*e^9+120*A*a^7*b^3*d^2*e^8-210*A*a^6*b^4*d^ 
3*e^7+252*A*a^5*b^5*d^4*e^6-210*A*a^4*b^6*d^5*e^5+120*A*a^3*b^7*d^6*e^4-45 
*A*a^2*b^8*d^7*e^3+10*A*a*b^9*d^8*e^2-A*b^10*d^9*e+B*a^10*e^10-10*B*a^9*b* 
d*e^9+45*B*a^8*b^2*d^2*e^8-120*B*a^7*b^3*d^3*e^7+210*B*a^6*b^4*d^4*e^6-252 
*B*a^5*b^5*d^5*e^5+210*B*a^4*b^6*d^6*e^4-120*B*a^3*b^7*d^7*e^3+45*B*a^2*b^ 
8*d^8*e^2-10*B*a*b^9*d^9*e+B*b^10*d^10)/e^11*x+1/2*b/e^10*(45*A*a^8*b*e^9- 
120*A*a^7*b^2*d*e^8+210*A*a^6*b^3*d^2*e^7-252*A*a^5*b^4*d^3*e^6+210*A*a^4* 
b^5*d^4*e^5-120*A*a^3*b^6*d^5*e^4+45*A*a^2*b^7*d^6*e^3-10*A*a*b^8*d^7*e^2+ 
A*b^9*d^8*e+10*B*a^9*e^9-45*B*a^8*b*d*e^8+120*B*a^7*b^2*d^2*e^7-210*B*a^6* 
b^3*d^3*e^6+252*B*a^5*b^4*d^4*e^5-210*B*a^4*b^5*d^5*e^4+120*B*a^3*b^6*d^6* 
e^3-45*B*a^2*b^7*d^7*e^2+10*B*a*b^8*d^8*e-B*b^9*d^9)*x^2+1/3*b^2/e^9*(120* 
A*a^7*b*e^8-210*A*a^6*b^2*d*e^7+252*A*a^5*b^3*d^2*e^6-210*A*a^4*b^4*d^3*e^ 
5+120*A*a^3*b^5*d^4*e^4-45*A*a^2*b^6*d^5*e^3+10*A*a*b^7*d^6*e^2-A*b^8*d^7* 
e+45*B*a^8*e^8-120*B*a^7*b*d*e^7+210*B*a^6*b^2*d^2*e^6-252*B*a^5*b^3*d^3*e 
^5+210*B*a^4*b^4*d^4*e^4-120*B*a^3*b^5*d^5*e^3+45*B*a^2*b^6*d^6*e^2-10*B*a 
*b^7*d^7*e+B*b^8*d^8)*x^3+1/4*b^3/e^8*(210*A*a^6*b*e^7-252*A*a^5*b^2*d*e^6 
+210*A*a^4*b^3*d^2*e^5-120*A*a^3*b^4*d^3*e^4+45*A*a^2*b^5*d^4*e^3-10*A*a*b 
^6*d^5*e^2+A*b^7*d^6*e+120*B*a^7*e^7-210*B*a^6*b*d*e^6+252*B*a^5*b^2*d^2*e 
^5-210*B*a^4*b^3*d^3*e^4+120*B*a^3*b^4*d^4*e^3-45*B*a^2*b^5*d^5*e^2+10*B*a 
*b^6*d^6*e-B*b^7*d^7)*x^4+1/5*b^4/e^7*(252*A*a^5*b*e^6-210*A*a^4*b^2*d*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1805 vs. \(2 (328) = 656\).

Time = 0.08 (sec) , antiderivative size = 1805, normalized size of antiderivative = 5.19 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\text {Too large to display} \] Input:

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

Output:

1/27720*(2520*B*b^10*e^11*x^11 - 2772*(B*b^10*d*e^10 - (10*B*a*b^9 + A*b^1 
0)*e^11)*x^10 + 3080*(B*b^10*d^2*e^9 - (10*B*a*b^9 + A*b^10)*d*e^10 + 5*(9 
*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 3465*(B*b^10*d^3*e^8 - (10*B*a*b^9 + A 
*b^10)*d^2*e^9 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 15*(8*B*a^3*b^7 + 3* 
A*a^2*b^8)*e^11)*x^8 + 3960*(B*b^10*d^4*e^7 - (10*B*a*b^9 + A*b^10)*d^3*e^ 
8 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d 
*e^10 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 - 4620*(B*b^10*d^5*e^6 - 
(10*B*a*b^9 + A*b^10)*d^4*e^7 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 15*( 
8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 
 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 5544*(B*b^10*d^6*e^5 - (10*B 
*a*b^9 + A*b^10)*d^5*e^6 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 15*(8*B*a 
^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 4 
2*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11 
)*x^5 - 6930*(B*b^10*d^7*e^4 - (10*B*a*b^9 + A*b^10)*d^6*e^5 + 5*(9*B*a^2* 
b^8 + 2*A*a*b^9)*d^5*e^6 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 30*(7* 
B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 
+ 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e 
^11)*x^4 + 9240*(B*b^10*d^8*e^3 - (10*B*a*b^9 + A*b^10)*d^7*e^4 + 5*(9*B*a 
^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 30* 
(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1912 vs. \(2 (298) = 596\).

Time = 2.19 (sec) , antiderivative size = 1912, normalized size of antiderivative = 5.49 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\text {Too large to display} \] Input:

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

Output:

B*b**10*x**11/(11*e) + x**10*(A*b**10/(10*e) + B*a*b**9/e - B*b**10*d/(10* 
e**2)) + x**9*(10*A*a*b**9/(9*e) - A*b**10*d/(9*e**2) + 5*B*a**2*b**8/e - 
10*B*a*b**9*d/(9*e**2) + B*b**10*d**2/(9*e**3)) + x**8*(45*A*a**2*b**8/(8* 
e) - 5*A*a*b**9*d/(4*e**2) + A*b**10*d**2/(8*e**3) + 15*B*a**3*b**7/e - 45 
*B*a**2*b**8*d/(8*e**2) + 5*B*a*b**9*d**2/(4*e**3) - B*b**10*d**3/(8*e**4) 
) + x**7*(120*A*a**3*b**7/(7*e) - 45*A*a**2*b**8*d/(7*e**2) + 10*A*a*b**9* 
d**2/(7*e**3) - A*b**10*d**3/(7*e**4) + 30*B*a**4*b**6/e - 120*B*a**3*b**7 
*d/(7*e**2) + 45*B*a**2*b**8*d**2/(7*e**3) - 10*B*a*b**9*d**3/(7*e**4) + B 
*b**10*d**4/(7*e**5)) + x**6*(35*A*a**4*b**6/e - 20*A*a**3*b**7*d/e**2 + 1 
5*A*a**2*b**8*d**2/(2*e**3) - 5*A*a*b**9*d**3/(3*e**4) + A*b**10*d**4/(6*e 
**5) + 42*B*a**5*b**5/e - 35*B*a**4*b**6*d/e**2 + 20*B*a**3*b**7*d**2/e**3 
 - 15*B*a**2*b**8*d**3/(2*e**4) + 5*B*a*b**9*d**4/(3*e**5) - B*b**10*d**5/ 
(6*e**6)) + x**5*(252*A*a**5*b**5/(5*e) - 42*A*a**4*b**6*d/e**2 + 24*A*a** 
3*b**7*d**2/e**3 - 9*A*a**2*b**8*d**3/e**4 + 2*A*a*b**9*d**4/e**5 - A*b**1 
0*d**5/(5*e**6) + 42*B*a**6*b**4/e - 252*B*a**5*b**5*d/(5*e**2) + 42*B*a** 
4*b**6*d**2/e**3 - 24*B*a**3*b**7*d**3/e**4 + 9*B*a**2*b**8*d**4/e**5 - 2* 
B*a*b**9*d**5/e**6 + B*b**10*d**6/(5*e**7)) + x**4*(105*A*a**6*b**4/(2*e) 
- 63*A*a**5*b**5*d/e**2 + 105*A*a**4*b**6*d**2/(2*e**3) - 30*A*a**3*b**7*d 
**3/e**4 + 45*A*a**2*b**8*d**4/(4*e**5) - 5*A*a*b**9*d**5/(2*e**6) + A*b** 
10*d**6/(4*e**7) + 30*B*a**7*b**3/e - 105*B*a**6*b**4*d/(2*e**2) + 63*B...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1804 vs. \(2 (328) = 656\).

Time = 0.05 (sec) , antiderivative size = 1804, normalized size of antiderivative = 5.18 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\text {Too large to display} \] Input:

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

Output:

1/27720*(2520*B*b^10*e^10*x^11 - 2772*(B*b^10*d*e^9 - (10*B*a*b^9 + A*b^10 
)*e^10)*x^10 + 3080*(B*b^10*d^2*e^8 - (10*B*a*b^9 + A*b^10)*d*e^9 + 5*(9*B 
*a^2*b^8 + 2*A*a*b^9)*e^10)*x^9 - 3465*(B*b^10*d^3*e^7 - (10*B*a*b^9 + A*b 
^10)*d^2*e^8 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^9 - 15*(8*B*a^3*b^7 + 3*A*a 
^2*b^8)*e^10)*x^8 + 3960*(B*b^10*d^4*e^6 - (10*B*a*b^9 + A*b^10)*d^3*e^7 + 
 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^8 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^ 
9 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^10)*x^7 - 4620*(B*b^10*d^5*e^5 - (10* 
B*a*b^9 + A*b^10)*d^4*e^6 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^7 - 15*(8*B* 
a^3*b^7 + 3*A*a^2*b^8)*d^2*e^8 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^9 - 42 
*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^10)*x^6 + 5544*(B*b^10*d^6*e^4 - (10*B*a*b^ 
9 + A*b^10)*d^5*e^5 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^6 - 15*(8*B*a^3*b^ 
7 + 3*A*a^2*b^8)*d^3*e^7 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^8 - 42*(6* 
B*a^5*b^5 + 5*A*a^4*b^6)*d*e^9 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^10)*x^5 
- 6930*(B*b^10*d^7*e^3 - (10*B*a*b^9 + A*b^10)*d^6*e^4 + 5*(9*B*a^2*b^8 + 
2*A*a*b^9)*d^5*e^5 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^6 + 30*(7*B*a^4* 
b^6 + 4*A*a^3*b^7)*d^3*e^7 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^8 + 42*( 
5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^9 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^10)*x^ 
4 + 9240*(B*b^10*d^8*e^2 - (10*B*a*b^9 + A*b^10)*d^7*e^3 + 5*(9*B*a^2*b^8 
+ 2*A*a*b^9)*d^6*e^4 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^5 + 30*(7*B*a^ 
4*b^6 + 4*A*a^3*b^7)*d^4*e^6 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^7 +...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2230 vs. \(2 (328) = 656\).

Time = 0.12 (sec) , antiderivative size = 2230, normalized size of antiderivative = 6.41 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\text {Too large to display} \] Input:

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

Output:

1/27720*(2520*B*b^10*e^10*x^11 - 2772*B*b^10*d*e^9*x^10 + 27720*B*a*b^9*e^ 
10*x^10 + 2772*A*b^10*e^10*x^10 + 3080*B*b^10*d^2*e^8*x^9 - 30800*B*a*b^9* 
d*e^9*x^9 - 3080*A*b^10*d*e^9*x^9 + 138600*B*a^2*b^8*e^10*x^9 + 30800*A*a* 
b^9*e^10*x^9 - 3465*B*b^10*d^3*e^7*x^8 + 34650*B*a*b^9*d^2*e^8*x^8 + 3465* 
A*b^10*d^2*e^8*x^8 - 155925*B*a^2*b^8*d*e^9*x^8 - 34650*A*a*b^9*d*e^9*x^8 
+ 415800*B*a^3*b^7*e^10*x^8 + 155925*A*a^2*b^8*e^10*x^8 + 3960*B*b^10*d^4* 
e^6*x^7 - 39600*B*a*b^9*d^3*e^7*x^7 - 3960*A*b^10*d^3*e^7*x^7 + 178200*B*a 
^2*b^8*d^2*e^8*x^7 + 39600*A*a*b^9*d^2*e^8*x^7 - 475200*B*a^3*b^7*d*e^9*x^ 
7 - 178200*A*a^2*b^8*d*e^9*x^7 + 831600*B*a^4*b^6*e^10*x^7 + 475200*A*a^3* 
b^7*e^10*x^7 - 4620*B*b^10*d^5*e^5*x^6 + 46200*B*a*b^9*d^4*e^6*x^6 + 4620* 
A*b^10*d^4*e^6*x^6 - 207900*B*a^2*b^8*d^3*e^7*x^6 - 46200*A*a*b^9*d^3*e^7* 
x^6 + 554400*B*a^3*b^7*d^2*e^8*x^6 + 207900*A*a^2*b^8*d^2*e^8*x^6 - 970200 
*B*a^4*b^6*d*e^9*x^6 - 554400*A*a^3*b^7*d*e^9*x^6 + 1164240*B*a^5*b^5*e^10 
*x^6 + 970200*A*a^4*b^6*e^10*x^6 + 5544*B*b^10*d^6*e^4*x^5 - 55440*B*a*b^9 
*d^5*e^5*x^5 - 5544*A*b^10*d^5*e^5*x^5 + 249480*B*a^2*b^8*d^4*e^6*x^5 + 55 
440*A*a*b^9*d^4*e^6*x^5 - 665280*B*a^3*b^7*d^3*e^7*x^5 - 249480*A*a^2*b^8* 
d^3*e^7*x^5 + 1164240*B*a^4*b^6*d^2*e^8*x^5 + 665280*A*a^3*b^7*d^2*e^8*x^5 
 - 1397088*B*a^5*b^5*d*e^9*x^5 - 1164240*A*a^4*b^6*d*e^9*x^5 + 1164240*B*a 
^6*b^4*e^10*x^5 + 1397088*A*a^5*b^5*e^10*x^5 - 6930*B*b^10*d^7*e^3*x^4 + 6 
9300*B*a*b^9*d^6*e^4*x^4 + 6930*A*b^10*d^6*e^4*x^4 - 311850*B*a^2*b^8*d...
 

Mupad [B] (verification not implemented)

Time = 1.02 (sec) , antiderivative size = 1795, normalized size of antiderivative = 5.16 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\text {Too large to display} \] Input:

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

Output:

x^4*((d*((d*((d*((d*((d*((d*((A*b^10 + 10*B*a*b^9)/e - (B*b^10*d)/e^2))/e 
- (5*a*b^8*(2*A*b + 9*B*a))/e))/e + (15*a^2*b^7*(3*A*b + 8*B*a))/e))/e - ( 
30*a^3*b^6*(4*A*b + 7*B*a))/e))/e + (42*a^4*b^5*(5*A*b + 6*B*a))/e))/e - ( 
42*a^5*b^4*(6*A*b + 5*B*a))/e))/(4*e) + (15*a^6*b^3*(7*A*b + 4*B*a))/(2*e) 
) - x^3*((d*((d*((d*((d*((d*((d*((d*((A*b^10 + 10*B*a*b^9)/e - (B*b^10*d)/ 
e^2))/e - (5*a*b^8*(2*A*b + 9*B*a))/e))/e + (15*a^2*b^7*(3*A*b + 8*B*a))/e 
))/e - (30*a^3*b^6*(4*A*b + 7*B*a))/e))/e + (42*a^4*b^5*(5*A*b + 6*B*a))/e 
))/e - (42*a^5*b^4*(6*A*b + 5*B*a))/e))/e + (30*a^6*b^3*(7*A*b + 4*B*a))/e 
))/(3*e) - (5*a^7*b^2*(8*A*b + 3*B*a))/e) - x^5*((d*((d*((d*((d*((d*((A*b^ 
10 + 10*B*a*b^9)/e - (B*b^10*d)/e^2))/e - (5*a*b^8*(2*A*b + 9*B*a))/e))/e 
+ (15*a^2*b^7*(3*A*b + 8*B*a))/e))/e - (30*a^3*b^6*(4*A*b + 7*B*a))/e))/e 
+ (42*a^4*b^5*(5*A*b + 6*B*a))/e))/(5*e) - (42*a^5*b^4*(6*A*b + 5*B*a))/(5 
*e)) + x^6*((d*((d*((d*((d*((A*b^10 + 10*B*a*b^9)/e - (B*b^10*d)/e^2))/e - 
 (5*a*b^8*(2*A*b + 9*B*a))/e))/e + (15*a^2*b^7*(3*A*b + 8*B*a))/e))/e - (3 
0*a^3*b^6*(4*A*b + 7*B*a))/e))/(6*e) + (7*a^4*b^5*(5*A*b + 6*B*a))/e) - x^ 
7*((d*((d*((d*((A*b^10 + 10*B*a*b^9)/e - (B*b^10*d)/e^2))/e - (5*a*b^8*(2* 
A*b + 9*B*a))/e))/e + (15*a^2*b^7*(3*A*b + 8*B*a))/e))/(7*e) - (30*a^3*b^6 
*(4*A*b + 7*B*a))/(7*e)) + x^8*((d*((d*((A*b^10 + 10*B*a*b^9)/e - (B*b^10* 
d)/e^2))/e - (5*a*b^8*(2*A*b + 9*B*a))/e))/(8*e) + (15*a^2*b^7*(3*A*b + 8* 
B*a))/(8*e)) - x^9*((d*((A*b^10 + 10*B*a*b^9)/e - (B*b^10*d)/e^2))/(9*e...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1216, normalized size of antiderivative = 3.49 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx =\text {Too large to display} \] Input:

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

Output:

(27720*log(d + e*x)*a**11*e**11 - 304920*log(d + e*x)*a**10*b*d*e**10 + 15 
24600*log(d + e*x)*a**9*b**2*d**2*e**9 - 4573800*log(d + e*x)*a**8*b**3*d* 
*3*e**8 + 9147600*log(d + e*x)*a**7*b**4*d**4*e**7 - 12806640*log(d + e*x) 
*a**6*b**5*d**5*e**6 + 12806640*log(d + e*x)*a**5*b**6*d**6*e**5 - 9147600 
*log(d + e*x)*a**4*b**7*d**7*e**4 + 4573800*log(d + e*x)*a**3*b**8*d**8*e* 
*3 - 1524600*log(d + e*x)*a**2*b**9*d**9*e**2 + 304920*log(d + e*x)*a*b**1 
0*d**10*e - 27720*log(d + e*x)*b**11*d**11 + 304920*a**10*b*e**11*x - 1524 
600*a**9*b**2*d*e**10*x + 762300*a**9*b**2*e**11*x**2 + 4573800*a**8*b**3* 
d**2*e**9*x - 2286900*a**8*b**3*d*e**10*x**2 + 1524600*a**8*b**3*e**11*x** 
3 - 9147600*a**7*b**4*d**3*e**8*x + 4573800*a**7*b**4*d**2*e**9*x**2 - 304 
9200*a**7*b**4*d*e**10*x**3 + 2286900*a**7*b**4*e**11*x**4 + 12806640*a**6 
*b**5*d**4*e**7*x - 6403320*a**6*b**5*d**3*e**8*x**2 + 4268880*a**6*b**5*d 
**2*e**9*x**3 - 3201660*a**6*b**5*d*e**10*x**4 + 2561328*a**6*b**5*e**11*x 
**5 - 12806640*a**5*b**6*d**5*e**6*x + 6403320*a**5*b**6*d**4*e**7*x**2 - 
4268880*a**5*b**6*d**3*e**8*x**3 + 3201660*a**5*b**6*d**2*e**9*x**4 - 2561 
328*a**5*b**6*d*e**10*x**5 + 2134440*a**5*b**6*e**11*x**6 + 9147600*a**4*b 
**7*d**6*e**5*x - 4573800*a**4*b**7*d**5*e**6*x**2 + 3049200*a**4*b**7*d** 
4*e**7*x**3 - 2286900*a**4*b**7*d**3*e**8*x**4 + 1829520*a**4*b**7*d**2*e* 
*9*x**5 - 1524600*a**4*b**7*d*e**10*x**6 + 1306800*a**4*b**7*e**11*x**7 - 
4573800*a**3*b**8*d**7*e**4*x + 2286900*a**3*b**8*d**6*e**5*x**2 - 1524...