\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^3} \, dx\) [81]

Optimal result

Integrand size = 20, antiderivative size = 445 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^3} \, dx=\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{2 e^{12} (d+e x)^2}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{e^{12} (d+e x)}-\frac {15 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e) (d+e x)^2}{e^{12}}+\frac {14 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) (d+e x)^3}{e^{12}}-\frac {21 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) (d+e x)^4}{2 e^{12}}+\frac {6 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) (d+e x)^5}{e^{12}}-\frac {5 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^6}{2 e^{12}}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^7}{7 e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^8}{8 e^{12}}+\frac {b^{10} B (d+e x)^9}{9 e^{12}}-\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e) \log (d+e x)}{e^{12}} \] Output:

15*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d)*x/e^11+1/2*(-a*e+b*d)^10*( 
-A*e+B*d)/e^12/(e*x+d)^2-(-a*e+b*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/(e*x 
+d)-15*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)*(e*x+d)^2/e^12+14*b^4* 
(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)*(e*x+d)^3/e^12-21/2*b^5*(-a*e+b*d 
)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)*(e*x+d)^4/e^12+6*b^6*(-a*e+b*d)^3*(-4*A*b* 
e-7*B*a*e+11*B*b*d)*(e*x+d)^5/e^12-5/2*b^7*(-a*e+b*d)^2*(-3*A*b*e-8*B*a*e+ 
11*B*b*d)*(e*x+d)^6/e^12+5/7*b^8*(-a*e+b*d)*(-2*A*b*e-9*B*a*e+11*B*b*d)*(e 
*x+d)^7/e^12-1/8*b^9*(-A*b*e-10*B*a*e+11*B*b*d)*(e*x+d)^8/e^12+1/9*b^10*B* 
(e*x+d)^9/e^12-5*b*(-a*e+b*d)^8*(-9*A*b*e-2*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. \(1480\) vs. \(2(445)=890\).

Time = 0.42 (sec) , antiderivative size = 1480, normalized size of antiderivative = 3.33 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^3,x]
 

Output:

(-252*a^10*e^10*(A*e + B*(d + 2*e*x)) - 2520*a^9*b*e^9*(A*e*(d + 2*e*x) - 
B*d*(3*d + 4*e*x)) + 11340*a^8*b^2*e^8*(A*d*e*(3*d + 4*e*x) + B*(-5*d^3 - 
4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3)) + 30240*a^7*b^3*e^7*(A*e*(-5*d^3 - 4 
*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) + B*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^ 
2 - 4*d*e^3*x^3 + e^4*x^4)) + 17640*a^6*b^4*e^6*(3*A*e*(7*d^4 + 2*d^3*e*x 
- 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + B*(-27*d^5 + 6*d^4*e*x + 63*d^ 
3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5)) + 10584*a^5*b^5*e^5 
*(2*A*e*(-27*d^5 + 6*d^4*e*x + 63*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x 
^4 + 2*e^5*x^5) + 3*B*(22*d^6 - 16*d^5*e*x - 68*d^4*e^2*x^2 - 20*d^3*e^3*x 
^3 + 5*d^2*e^4*x^4 - 2*d*e^5*x^5 + e^6*x^6)) + 5292*a^4*b^6*e^4*(5*A*e*(22 
*d^6 - 16*d^5*e*x - 68*d^4*e^2*x^2 - 20*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 2*d* 
e^5*x^5 + e^6*x^6) + B*(-130*d^7 + 160*d^6*e*x + 500*d^5*e^2*x^2 + 140*d^4 
*e^3*x^3 - 35*d^3*e^4*x^4 + 14*d^2*e^5*x^5 - 7*d*e^6*x^6 + 4*e^7*x^7)) + 1 
008*a^3*b^7*e^3*(3*A*e*(-130*d^7 + 160*d^6*e*x + 500*d^5*e^2*x^2 + 140*d^4 
*e^3*x^3 - 35*d^3*e^4*x^4 + 14*d^2*e^5*x^5 - 7*d*e^6*x^6 + 4*e^7*x^7) + 2* 
B*(225*d^8 - 390*d^7*e*x - 1035*d^6*e^2*x^2 - 280*d^5*e^3*x^3 + 70*d^4*e^4 
*x^4 - 28*d^3*e^5*x^5 + 14*d^2*e^6*x^6 - 8*d*e^7*x^7 + 5*e^8*x^8)) + 108*a 
^2*b^8*e^2*(7*A*e*(225*d^8 - 390*d^7*e*x - 1035*d^6*e^2*x^2 - 280*d^5*e^3* 
x^3 + 70*d^4*e^4*x^4 - 28*d^3*e^5*x^5 + 14*d^2*e^6*x^6 - 8*d*e^7*x^7 + 5*e 
^8*x^8) - 3*B*(595*d^9 - 1330*d^8*e*x - 3185*d^7*e^2*x^2 - 840*d^6*e^3*...
 

Rubi [A] (verified)

Time = 1.61 (sec) , antiderivative size = 445, 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.

\(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^3} \, dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^3,x]
 

Output:

(15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e)*x)/e^11 + ((b*d - a*e 
)^10*(B*d - A*e))/(2*e^12*(d + e*x)^2) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b 
*e - a*B*e))/(e^12*(d + e*x)) - (15*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e 
- 4*a*B*e)*(d + e*x)^2)/e^12 + (14*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 
 5*a*B*e)*(d + e*x)^3)/e^12 - (21*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 
6*a*B*e)*(d + e*x)^4)/(2*e^12) + (6*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e 
- 7*a*B*e)*(d + e*x)^5)/e^12 - (5*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 
8*a*B*e)*(d + e*x)^6)/(2*e^12) + (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 
9*a*B*e)*(d + e*x)^7)/(7*e^12) - (b^9*(11*b*B*d - A*b*e - 10*a*B*e)*(d + e 
*x)^8)/(8*e^12) + (b^10*B*(d + e*x)^9)/(9*e^12) - (5*b*(b*d - a*e)^8*(11*b 
*B*d - 9*A*b*e - 2*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. \(1901\) vs. \(2(433)=866\).

Time = 0.24 (sec) , antiderivative size = 1902, normalized size of antiderivative = 4.27

method	result	size
norman	\(\text {Expression too large to display}\)	\(1902\)
default	\(\text {Expression too large to display}\)	\(2117\)
risch	\(\text {Expression too large to display}\)	\(2236\)
parallelrisch	\(\text {Expression too large to display}\)	\(3127\)

Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^3,x,method=_RETURNVERBOSE)
 

Output:

(-1/2*(A*a^10*e^11+10*A*a^9*b*d*e^10-135*A*a^8*b^2*d^2*e^9+1080*A*a^7*b^3* 
d^3*e^8-3780*A*a^6*b^4*d^4*e^7+7560*A*a^5*b^5*d^5*e^6-9450*A*a^4*b^6*d^6*e 
^5+7560*A*a^3*b^7*d^7*e^4-3780*A*a^2*b^8*d^8*e^3+1080*A*a*b^9*d^9*e^2-135* 
A*b^10*d^10*e+B*a^10*d*e^10-30*B*a^9*b*d^2*e^9+405*B*a^8*b^2*d^3*e^8-2160* 
B*a^7*b^3*d^4*e^7+6300*B*a^6*b^4*d^5*e^6-11340*B*a^5*b^5*d^6*e^5+13230*B*a 
^4*b^6*d^7*e^4-10080*B*a^3*b^7*d^8*e^3+4860*B*a^2*b^8*d^9*e^2-1350*B*a*b^9 
*d^10*e+165*B*b^10*d^11)/e^12-(10*A*a^9*b*e^10-90*A*a^8*b^2*d*e^9+720*A*a^ 
7*b^3*d^2*e^8-2520*A*a^6*b^4*d^3*e^7+5040*A*a^5*b^5*d^4*e^6-6300*A*a^4*b^6 
*d^5*e^5+5040*A*a^3*b^7*d^6*e^4-2520*A*a^2*b^8*d^7*e^3+720*A*a*b^9*d^8*e^2 
-90*A*b^10*d^9*e+B*a^10*e^10-20*B*a^9*b*d*e^9+270*B*a^8*b^2*d^2*e^8-1440*B 
*a^7*b^3*d^3*e^7+4200*B*a^6*b^4*d^4*e^6-7560*B*a^5*b^5*d^5*e^5+8820*B*a^4* 
b^6*d^6*e^4-6720*B*a^3*b^7*d^7*e^3+3240*B*a^2*b^8*d^8*e^2-900*B*a*b^9*d^9* 
e+110*B*b^10*d^10)/e^11*x+5/3*b^2*(72*A*a^7*b*e^8-252*A*a^6*b^2*d*e^7+504* 
A*a^5*b^3*d^2*e^6-630*A*a^4*b^4*d^3*e^5+504*A*a^3*b^5*d^4*e^4-252*A*a^2*b^ 
6*d^5*e^3+72*A*a*b^7*d^6*e^2-9*A*b^8*d^7*e+27*B*a^8*e^8-144*B*a^7*b*d*e^7+ 
420*B*a^6*b^2*d^2*e^6-756*B*a^5*b^3*d^3*e^5+882*B*a^4*b^4*d^4*e^4-672*B*a^ 
3*b^5*d^5*e^3+324*B*a^2*b^6*d^6*e^2-90*B*a*b^7*d^7*e+11*B*b^8*d^8)/e^9*x^3 
+5/12*b^3*(252*A*a^6*b*e^7-504*A*a^5*b^2*d*e^6+630*A*a^4*b^3*d^2*e^5-504*A 
*a^3*b^4*d^3*e^4+252*A*a^2*b^5*d^4*e^3-72*A*a*b^6*d^5*e^2+9*A*b^7*d^6*e+14 
4*B*a^7*e^7-420*B*a^6*b*d*e^6+756*B*a^5*b^2*d^2*e^5-882*B*a^4*b^3*d^3*e...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2547 vs. \(2 (433) = 866\).

Time = 0.15 (sec) , antiderivative size = 2547, normalized size of antiderivative = 5.72 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^3} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^3,x, algorithm="fricas")
 

Output:

1/504*(56*B*b^10*e^11*x^11 - 5292*B*b^10*d^11 - 252*A*a^10*e^11 + 4788*(10 
*B*a*b^9 + A*b^10)*d^10*e - 21420*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 5670 
0*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 - 98280*(7*B*a^4*b^6 + 4*A*a^3*b^7)* 
d^7*e^4 + 116424*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 - 95256*(5*B*a^6*b^4 
+ 6*A*a^5*b^5)*d^5*e^6 + 52920*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 - 18900 
*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 3780*(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 - 7*(11*B*b^10*d*e^10 - 9*(10*B*a*b 
^9 + A*b^10)*e^11)*x^10 + 10*(11*B*b^10*d^2*e^9 - 9*(10*B*a*b^9 + A*b^10)* 
d*e^10 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 15*(11*B*b^10*d^3*e^8 - 
9*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 84 
*(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 - 9*(10*B*a 
*b^9 + A*b^10)*d^3*e^8 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 84*(8*B*a^ 
3*b^7 + 3*A*a^2*b^8)*d*e^10 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 - 
42*(11*B*b^10*d^5*e^6 - 9*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 36*(9*B*a^2*b^8 
+ 2*A*a*b^9)*d^3*e^8 - 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 126*(7*B*a 
^4*b^6 + 4*A*a^3*b^7)*d*e^10 - 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 
 84*(11*B*b^10*d^6*e^5 - 9*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 36*(9*B*a^2*b^8 
 + 2*A*a*b^9)*d^4*e^7 - 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 126*(7*B* 
a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 
84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 - 210*(11*B*b^10*d^7*e^4 - 9*(...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2004 vs. \(2 (462) = 924\).

Time = 22.45 (sec) , antiderivative size = 2004, normalized size of antiderivative = 4.50 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^3} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**3,x)
 

Output:

B*b**10*x**9/(9*e**3) + 5*b*(a*e - b*d)**8*(9*A*b*e + 2*B*a*e - 11*B*b*d)* 
log(d + e*x)/e**12 + x**8*(A*b**10/(8*e**3) + 5*B*a*b**9/(4*e**3) - 3*B*b* 
*10*d/(8*e**4)) + x**7*(10*A*a*b**9/(7*e**3) - 3*A*b**10*d/(7*e**4) + 45*B 
*a**2*b**8/(7*e**3) - 30*B*a*b**9*d/(7*e**4) + 6*B*b**10*d**2/(7*e**5)) + 
x**6*(15*A*a**2*b**8/(2*e**3) - 5*A*a*b**9*d/e**4 + A*b**10*d**2/e**5 + 20 
*B*a**3*b**7/e**3 - 45*B*a**2*b**8*d/(2*e**4) + 10*B*a*b**9*d**2/e**5 - 5* 
B*b**10*d**3/(3*e**6)) + x**5*(24*A*a**3*b**7/e**3 - 27*A*a**2*b**8*d/e**4 
 + 12*A*a*b**9*d**2/e**5 - 2*A*b**10*d**3/e**6 + 42*B*a**4*b**6/e**3 - 72* 
B*a**3*b**7*d/e**4 + 54*B*a**2*b**8*d**2/e**5 - 20*B*a*b**9*d**3/e**6 + 3* 
B*b**10*d**4/e**7) + x**4*(105*A*a**4*b**6/(2*e**3) - 90*A*a**3*b**7*d/e** 
4 + 135*A*a**2*b**8*d**2/(2*e**5) - 25*A*a*b**9*d**3/e**6 + 15*A*b**10*d** 
4/(4*e**7) + 63*B*a**5*b**5/e**3 - 315*B*a**4*b**6*d/(2*e**4) + 180*B*a**3 
*b**7*d**2/e**5 - 225*B*a**2*b**8*d**3/(2*e**6) + 75*B*a*b**9*d**4/(2*e**7 
) - 21*B*b**10*d**5/(4*e**8)) + x**3*(84*A*a**5*b**5/e**3 - 210*A*a**4*b** 
6*d/e**4 + 240*A*a**3*b**7*d**2/e**5 - 150*A*a**2*b**8*d**3/e**6 + 50*A*a* 
b**9*d**4/e**7 - 7*A*b**10*d**5/e**8 + 70*B*a**6*b**4/e**3 - 252*B*a**5*b* 
*5*d/e**4 + 420*B*a**4*b**6*d**2/e**5 - 400*B*a**3*b**7*d**3/e**6 + 225*B* 
a**2*b**8*d**4/e**7 - 70*B*a*b**9*d**5/e**8 + 28*B*b**10*d**6/(3*e**9)) + 
x**2*(105*A*a**6*b**4/e**3 - 378*A*a**5*b**5*d/e**4 + 630*A*a**4*b**6*d**2 
/e**5 - 600*A*a**3*b**7*d**3/e**6 + 675*A*a**2*b**8*d**4/(2*e**7) - 105...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1826 vs. \(2 (433) = 866\).

Time = 0.08 (sec) , antiderivative size = 1826, normalized size of antiderivative = 4.10 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^3} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^3,x, algorithm="maxima")
 

Output:

-1/2*(21*B*b^10*d^11 + A*a^10*e^11 - 19*(10*B*a*b^9 + A*b^10)*d^10*e + 85* 
(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 225*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^ 
3 + 390*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 462*(6*B*a^5*b^5 + 5*A*a^4*b 
^6)*d^6*e^5 + 378*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 210*(4*B*a^7*b^3 + 
 7*A*a^6*b^4)*d^4*e^7 + 75*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 15*(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 + 2*(11*B*b^10* 
d^10*e - 10*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d 
^8*e^3 - 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 210*(7*B*a^4*b^6 + 4*A* 
a^3*b^7)*d^6*e^5 - 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 210*(5*B*a^6* 
b^4 + 6*A*a^5*b^5)*d^4*e^7 - 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 45* 
(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 
+ (B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^14*x^2 + 2*d*e^13*x + d^2*e^12) + 1/50 
4*(56*B*b^10*e^8*x^9 - 63*(3*B*b^10*d*e^7 - (10*B*a*b^9 + A*b^10)*e^8)*x^8 
 + 72*(6*B*b^10*d^2*e^6 - 3*(10*B*a*b^9 + A*b^10)*d*e^7 + 5*(9*B*a^2*b^8 + 
 2*A*a*b^9)*e^8)*x^7 - 84*(10*B*b^10*d^3*e^5 - 6*(10*B*a*b^9 + A*b^10)*d^2 
*e^6 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^7 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8) 
*e^8)*x^6 + 504*(3*B*b^10*d^4*e^4 - 2*(10*B*a*b^9 + A*b^10)*d^3*e^5 + 6*(9 
*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^6 - 9*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^7 + 6* 
(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^8)*x^5 - 126*(21*B*b^10*d^5*e^3 - 15*(10*B*a 
*b^9 + A*b^10)*d^4*e^4 + 50*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^5 - 90*(8*B...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2134 vs. \(2 (433) = 866\).

Time = 0.14 (sec) , antiderivative size = 2134, normalized size of antiderivative = 4.80 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^3} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^3,x, algorithm="giac")
 

Output:

-5*(11*B*b^10*d^9 - 90*B*a*b^9*d^8*e - 9*A*b^10*d^8*e + 324*B*a^2*b^8*d^7* 
e^2 + 72*A*a*b^9*d^7*e^2 - 672*B*a^3*b^7*d^6*e^3 - 252*A*a^2*b^8*d^6*e^3 + 
 882*B*a^4*b^6*d^5*e^4 + 504*A*a^3*b^7*d^5*e^4 - 756*B*a^5*b^5*d^4*e^5 - 6 
30*A*a^4*b^6*d^4*e^5 + 420*B*a^6*b^4*d^3*e^6 + 504*A*a^5*b^5*d^3*e^6 - 144 
*B*a^7*b^3*d^2*e^7 - 252*A*a^6*b^4*d^2*e^7 + 27*B*a^8*b^2*d*e^8 + 72*A*a^7 
*b^3*d*e^8 - 2*B*a^9*b*e^9 - 9*A*a^8*b^2*e^9)*log(abs(e*x + d))/e^12 - 1/2 
*(21*B*b^10*d^11 - 190*B*a*b^9*d^10*e - 19*A*b^10*d^10*e + 765*B*a^2*b^8*d 
^9*e^2 + 170*A*a*b^9*d^9*e^2 - 1800*B*a^3*b^7*d^8*e^3 - 675*A*a^2*b^8*d^8* 
e^3 + 2730*B*a^4*b^6*d^7*e^4 + 1560*A*a^3*b^7*d^7*e^4 - 2772*B*a^5*b^5*d^6 
*e^5 - 2310*A*a^4*b^6*d^6*e^5 + 1890*B*a^6*b^4*d^5*e^6 + 2268*A*a^5*b^5*d^ 
5*e^6 - 840*B*a^7*b^3*d^4*e^7 - 1470*A*a^6*b^4*d^4*e^7 + 225*B*a^8*b^2*d^3 
*e^8 + 600*A*a^7*b^3*d^3*e^8 - 30*B*a^9*b*d^2*e^9 - 135*A*a^8*b^2*d^2*e^9 
+ B*a^10*d*e^10 + 10*A*a^9*b*d*e^10 + A*a^10*e^11 + 2*(11*B*b^10*d^10*e - 
100*B*a*b^9*d^9*e^2 - 10*A*b^10*d^9*e^2 + 405*B*a^2*b^8*d^8*e^3 + 90*A*a*b 
^9*d^8*e^3 - 960*B*a^3*b^7*d^7*e^4 - 360*A*a^2*b^8*d^7*e^4 + 1470*B*a^4*b^ 
6*d^6*e^5 + 840*A*a^3*b^7*d^6*e^5 - 1512*B*a^5*b^5*d^5*e^6 - 1260*A*a^4*b^ 
6*d^5*e^6 + 1050*B*a^6*b^4*d^4*e^7 + 1260*A*a^5*b^5*d^4*e^7 - 480*B*a^7*b^ 
3*d^3*e^8 - 840*A*a^6*b^4*d^3*e^8 + 135*B*a^8*b^2*d^2*e^9 + 360*A*a^7*b^3* 
d^2*e^9 - 20*B*a^9*b*d*e^10 - 90*A*a^8*b^2*d*e^10 + B*a^10*e^11 + 10*A*a^9 
*b*e^11)*x)/((e*x + d)^2*e^12) + 1/504*(56*B*b^10*e^24*x^9 - 189*B*b^10...
 

Mupad [B] (verification not implemented)

Time = 1.21 (sec) , antiderivative size = 8104, normalized size of antiderivative = 18.21 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^3} \, dx=\text {Too large to display} \] Input:

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^3,x)
 

Output:

x^5*((3*d*((3*d^2*((A*b^10 + 10*B*a*b^9)/e^3 - (3*B*b^10*d)/e^4))/e^2 - (3 
*d*((3*d*((A*b^10 + 10*B*a*b^9)/e^3 - (3*B*b^10*d)/e^4))/e - (5*a*b^8*(2*A 
*b + 9*B*a))/e^3 + (3*B*b^10*d^2)/e^5))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e 
^3 + (B*b^10*d^3)/e^6))/(5*e) - (d^3*((A*b^10 + 10*B*a*b^9)/e^3 - (3*B*b^1 
0*d)/e^4))/(5*e^3) + (3*d^2*((3*d*((A*b^10 + 10*B*a*b^9)/e^3 - (3*B*b^10*d 
)/e^4))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^3 + (3*B*b^10*d^2)/e^5))/(5*e^2) + 
 (6*a^3*b^6*(4*A*b + 7*B*a))/e^3) + x*((3*d*((3*d*((d^3*((3*d^2*((A*b^10 + 
 10*B*a*b^9)/e^3 - (3*B*b^10*d)/e^4))/e^2 - (3*d*((3*d*((A*b^10 + 10*B*a*b 
^9)/e^3 - (3*B*b^10*d)/e^4))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^3 + (3*B*b^10 
*d^2)/e^5))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^3 + (B*b^10*d^3)/e^6))/e^3 
- (3*d*((3*d^2*((3*d^2*((A*b^10 + 10*B*a*b^9)/e^3 - (3*B*b^10*d)/e^4))/e^2 
 - (3*d*((3*d*((A*b^10 + 10*B*a*b^9)/e^3 - (3*B*b^10*d)/e^4))/e - (5*a*b^8 
*(2*A*b + 9*B*a))/e^3 + (3*B*b^10*d^2)/e^5))/e - (15*a^2*b^7*(3*A*b + 8*B* 
a))/e^3 + (B*b^10*d^3)/e^6))/e^2 - (3*d*((3*d*((3*d^2*((A*b^10 + 10*B*a*b^ 
9)/e^3 - (3*B*b^10*d)/e^4))/e^2 - (3*d*((3*d*((A*b^10 + 10*B*a*b^9)/e^3 - 
(3*B*b^10*d)/e^4))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^3 + (3*B*b^10*d^2)/e^5) 
)/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^3 + (B*b^10*d^3)/e^6))/e - (d^3*((A*b 
^10 + 10*B*a*b^9)/e^3 - (3*B*b^10*d)/e^4))/e^3 + (3*d^2*((3*d*((A*b^10 + 1 
0*B*a*b^9)/e^3 - (3*B*b^10*d)/e^4))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^3 + (3 
*B*b^10*d^2)/e^5))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^3))/e + (d^3*((...
 

Reduce [B] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 1661, normalized size of antiderivative = 3.73 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^3,x)
 

Output:

(27720*log(d + e*x)*a**9*b**2*d**3*e**9 + 55440*log(d + e*x)*a**9*b**2*d** 
2*e**10*x + 27720*log(d + e*x)*a**9*b**2*d*e**11*x**2 - 249480*log(d + e*x 
)*a**8*b**3*d**4*e**8 - 498960*log(d + e*x)*a**8*b**3*d**3*e**9*x - 249480 
*log(d + e*x)*a**8*b**3*d**2*e**10*x**2 + 997920*log(d + e*x)*a**7*b**4*d* 
*5*e**7 + 1995840*log(d + e*x)*a**7*b**4*d**4*e**8*x + 997920*log(d + e*x) 
*a**7*b**4*d**3*e**9*x**2 - 2328480*log(d + e*x)*a**6*b**5*d**6*e**6 - 465 
6960*log(d + e*x)*a**6*b**5*d**5*e**7*x - 2328480*log(d + e*x)*a**6*b**5*d 
**4*e**8*x**2 + 3492720*log(d + e*x)*a**5*b**6*d**7*e**5 + 6985440*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 - 3492720*log(d + e*x)*a**4*b**7*d**8*e**4 - 6985440*log(d + e*x)*a**4*b 
**7*d**7*e**5*x - 3492720*log(d + e*x)*a**4*b**7*d**6*e**6*x**2 + 2328480* 
log(d + e*x)*a**3*b**8*d**9*e**3 + 4656960*log(d + e*x)*a**3*b**8*d**8*e** 
4*x + 2328480*log(d + e*x)*a**3*b**8*d**7*e**5*x**2 - 997920*log(d + e*x)* 
a**2*b**9*d**10*e**2 - 1995840*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 + 249480*log(d + e*x)*a*b**10*d**11 
*e + 498960*log(d + e*x)*a*b**10*d**10*e**2*x + 249480*log(d + e*x)*a*b**1 
0*d**9*e**3*x**2 - 27720*log(d + e*x)*b**11*d**12 - 55440*log(d + e*x)*b** 
11*d**11*e*x - 27720*log(d + e*x)*b**11*d**10*e**2*x**2 - 252*a**11*d*e**1 
1 + 2772*a**10*b*e**12*x**2 + 13860*a**9*b**2*d**3*e**9 - 27720*a**9*b**2* 
d*e**11*x**2 - 124740*a**8*b**3*d**4*e**8 + 249480*a**8*b**3*d**2*e**10...