\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx\) [80]

Optimal result

Integrand size = 20, antiderivative size = 445 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx=-\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{e^{12} (d+e x)}+\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) (d+e x)^2}{2 e^{12}}-\frac {10 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e) (d+e x)^3}{e^{12}}+\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) (d+e x)^4}{2 e^{12}}-\frac {42 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) (d+e x)^5}{5 e^{12}}+\frac {5 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) (d+e x)^6}{e^{12}}-\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^7}{7 e^{12}}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^8}{8 e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^9}{9 e^{12}}+\frac {b^{10} B (d+e x)^{10}}{10 e^{12}}+\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e) \log (d+e x)}{e^{12}} \] Output:

-5*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)*x/e^11+(-a*e+b*d)^10*(-A*e+B 
*d)/e^12/(e*x+d)+15/2*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d)*(e*x+d) 
^2/e^12-10*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)*(e*x+d)^3/e^12+21/ 
2*b^4*(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)*(e*x+d)^4/e^12-42/5*b^5*(-a 
*e+b*d)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)*(e*x+d)^5/e^12+5*b^6*(-a*e+b*d)^3*(- 
4*A*b*e-7*B*a*e+11*B*b*d)*(e*x+d)^6/e^12-15/7*b^7*(-a*e+b*d)^2*(-3*A*b*e-8 
*B*a*e+11*B*b*d)*(e*x+d)^7/e^12+5/8*b^8*(-a*e+b*d)*(-2*A*b*e-9*B*a*e+11*B* 
b*d)*(e*x+d)^8/e^12-1/9*b^9*(-A*b*e-10*B*a*e+11*B*b*d)*(e*x+d)^9/e^12+1/10 
*b^10*B*(e*x+d)^10/e^12+(-a*e+b*d)^9*(-10*A*b*e-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. \(1486\) vs. \(2(445)=890\).

Time = 0.42 (sec) , antiderivative size = 1486, normalized size of antiderivative = 3.34 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^2,x]
 

Output:

(-2520*a^10*e^10*(-(B*d) + A*e) + 25200*a^9*b*e^9*(A*d*e + B*(-d^2 + d*e*x 
 + e^2*x^2)) + 56700*a^8*b^2*e^8*(2*A*e*(-d^2 + d*e*x + e^2*x^2) + B*(2*d^ 
3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3)) + 50400*a^7*b^3*e^7*(3*A*e*(2*d^3 
- 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 2*B*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2 
*x^2 - 2*d*e^3*x^3 + e^4*x^4)) + 44100*a^6*b^4*e^6*(4*A*e*(-3*d^4 + 9*d^3* 
e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + B*(12*d^5 - 48*d^4*e*x - 30 
*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5)) + 10584*a^5*b^5* 
e^5*(5*A*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^ 
4*x^4 + 3*e^5*x^5) - 6*B*(10*d^6 - 50*d^5*e*x - 30*d^4*e^2*x^2 + 10*d^3*e^ 
3*x^3 - 5*d^2*e^4*x^4 + 3*d*e^5*x^5 - 2*e^6*x^6)) + 8820*a^4*b^6*e^4*(6*A* 
e*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 
- 3*d*e^5*x^5 + 2*e^6*x^6) + B*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 7 
0*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x^ 
7)) + 720*a^3*b^7*e^3*(7*A*e*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70* 
d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x^7) 
 - 4*B*(105*d^8 - 735*d^7*e*x - 420*d^6*e^2*x^2 + 140*d^5*e^3*x^3 - 70*d^4 
*e^4*x^4 + 42*d^3*e^5*x^5 - 28*d^2*e^6*x^6 + 20*d*e^7*x^7 - 15*e^8*x^8)) + 
 135*a^2*b^8*e^2*(8*A*e*(-105*d^8 + 735*d^7*e*x + 420*d^6*e^2*x^2 - 140*d^ 
5*e^3*x^3 + 70*d^4*e^4*x^4 - 42*d^3*e^5*x^5 + 28*d^2*e^6*x^6 - 20*d*e^7*x^ 
7 + 15*e^8*x^8) + 3*B*(280*d^9 - 2240*d^8*e*x - 1260*d^7*e^2*x^2 + 420*...
 

Rubi [A] (verified)

Time = 1.71 (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.

\(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^2,x]
 

Output:

(-5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e)*x)/e^11 + ((b*d - a*e)^ 
10*(B*d - A*e))/(e^12*(d + e*x)) + (15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b 
*e - 3*a*B*e)*(d + e*x)^2)/(2*e^12) - (10*b^3*(b*d - a*e)^6*(11*b*B*d - 7* 
A*b*e - 4*a*B*e)*(d + e*x)^3)/e^12 + (21*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A 
*b*e - 5*a*B*e)*(d + e*x)^4)/(2*e^12) - (42*b^5*(b*d - a*e)^4*(11*b*B*d - 
5*A*b*e - 6*a*B*e)*(d + e*x)^5)/(5*e^12) + (5*b^6*(b*d - a*e)^3*(11*b*B*d 
- 4*A*b*e - 7*a*B*e)*(d + e*x)^6)/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)^7)/(7*e^12) + (5*b^8*(b*d - a*e)*(11*b*B*d - 
 2*A*b*e - 9*a*B*e)*(d + e*x)^8)/(8*e^12) - (b^9*(11*b*B*d - A*b*e - 10*a* 
B*e)*(d + e*x)^9)/(9*e^12) + (b^10*B*(d + e*x)^10)/(10*e^12) + ((b*d - a*e 
)^9*(11*b*B*d - 10*A*b*e - 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. \(1906\) vs. \(2(431)=862\).

Time = 0.24 (sec) , antiderivative size = 1907, normalized size of antiderivative = 4.29

method	result	size
norman	\(\text {Expression too large to display}\)	\(1907\)
default	\(\text {Expression too large to display}\)	\(2167\)
risch	\(\text {Expression too large to display}\)	\(2447\)
parallelrisch	\(\text {Expression too large to display}\)	\(2772\)

Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^2,x,method=_RETURNVERBOSE)
 

Output:

((A*a^10*e^11-10*A*a^9*b*d*e^10+90*A*a^8*b^2*d^2*e^9-360*A*a^7*b^3*d^3*e^8 
+840*A*a^6*b^4*d^4*e^7-1260*A*a^5*b^5*d^5*e^6+1260*A*a^4*b^6*d^6*e^5-840*A 
*a^3*b^7*d^7*e^4+360*A*a^2*b^8*d^8*e^3-90*A*a*b^9*d^9*e^2+10*A*b^10*d^10*e 
-B*a^10*d*e^10+20*B*a^9*b*d^2*e^9-135*B*a^8*b^2*d^3*e^8+480*B*a^7*b^3*d^4* 
e^7-1050*B*a^6*b^4*d^5*e^6+1512*B*a^5*b^5*d^6*e^5-1470*B*a^4*b^6*d^7*e^4+9 
60*B*a^3*b^7*d^8*e^3-405*B*a^2*b^8*d^9*e^2+100*B*a*b^9*d^10*e-11*B*b^10*d^ 
11)/e^11/d*x+1/2*b*(90*A*a^8*b*e^9-360*A*a^7*b^2*d*e^8+840*A*a^6*b^3*d^2*e 
^7-1260*A*a^5*b^4*d^3*e^6+1260*A*a^4*b^5*d^4*e^5-840*A*a^3*b^6*d^5*e^4+360 
*A*a^2*b^7*d^6*e^3-90*A*a*b^8*d^7*e^2+10*A*b^9*d^8*e+20*B*a^9*e^9-135*B*a^ 
8*b*d*e^8+480*B*a^7*b^2*d^2*e^7-1050*B*a^6*b^3*d^3*e^6+1512*B*a^5*b^4*d^4* 
e^5-1470*B*a^4*b^5*d^5*e^4+960*B*a^3*b^6*d^6*e^3-405*B*a^2*b^7*d^7*e^2+100 
*B*a*b^8*d^8*e-11*B*b^9*d^9)/e^10*x^2+1/6*b^2*(360*A*a^7*b*e^8-840*A*a^6*b 
^2*d*e^7+1260*A*a^5*b^3*d^2*e^6-1260*A*a^4*b^4*d^3*e^5+840*A*a^3*b^5*d^4*e 
^4-360*A*a^2*b^6*d^5*e^3+90*A*a*b^7*d^6*e^2-10*A*b^8*d^7*e+135*B*a^8*e^8-4 
80*B*a^7*b*d*e^7+1050*B*a^6*b^2*d^2*e^6-1512*B*a^5*b^3*d^3*e^5+1470*B*a^4* 
b^4*d^4*e^4-960*B*a^3*b^5*d^5*e^3+405*B*a^2*b^6*d^6*e^2-100*B*a*b^7*d^7*e+ 
11*B*b^8*d^8)/e^9*x^3+1/12*b^3*(840*A*a^6*b*e^7-1260*A*a^5*b^2*d*e^6+1260* 
A*a^4*b^3*d^2*e^5-840*A*a^3*b^4*d^3*e^4+360*A*a^2*b^5*d^4*e^3-90*A*a*b^6*d 
^5*e^2+10*A*b^7*d^6*e+480*B*a^7*e^7-1050*B*a^6*b*d*e^6+1512*B*a^5*b^2*d^2* 
e^5-1470*B*a^4*b^3*d^3*e^4+960*B*a^3*b^4*d^4*e^3-405*B*a^2*b^5*d^5*e^2+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2329 vs. \(2 (431) = 862\).

Time = 0.12 (sec) , antiderivative size = 2329, normalized size of antiderivative = 5.23 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^2,x, algorithm="fricas")
 

Output:

1/2520*(252*B*b^10*e^11*x^11 + 2520*B*b^10*d^11 - 2520*A*a^10*e^11 - 2520* 
(10*B*a*b^9 + A*b^10)*d^10*e + 12600*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 3 
7800*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 75600*(7*B*a^4*b^6 + 4*A*a^3*b^ 
7)*d^7*e^4 - 105840*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 105840*(5*B*a^6* 
b^4 + 6*A*a^5*b^5)*d^5*e^6 - 75600*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 3 
7800*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 12600*(2*B*a^9*b + 9*A*a^8*b^2) 
*d^2*e^9 + 2520*(B*a^10 + 10*A*a^9*b)*d*e^10 - 28*(11*B*b^10*d*e^10 - 10*( 
10*B*a*b^9 + A*b^10)*e^11)*x^10 + 35*(11*B*b^10*d^2*e^9 - 10*(10*B*a*b^9 + 
 A*b^10)*d*e^10 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 45*(11*B*b^10*d 
^3*e^8 - 10*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d 
*e^10 - 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 60*(11*B*b^10*d^4*e^7 
- 10*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 
- 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7) 
*e^11)*x^7 - 84*(11*B*b^10*d^5*e^6 - 10*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 45 
*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e 
^9 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 - 252*(6*B*a^5*b^5 + 5*A*a^4*b 
^6)*e^11)*x^6 + 126*(11*B*b^10*d^6*e^5 - 10*(10*B*a*b^9 + A*b^10)*d^5*e^6 
+ 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d 
^3*e^8 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 252*(6*B*a^5*b^5 + 5*A* 
a^4*b^6)*d*e^10 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 - 210*(11*B...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1974 vs. \(2 (464) = 928\).

Time = 5.01 (sec) , antiderivative size = 1974, normalized size of antiderivative = 4.44 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**2,x)
 

Output:

B*b**10*x**10/(10*e**2) + x**9*(A*b**10/(9*e**2) + 10*B*a*b**9/(9*e**2) - 
2*B*b**10*d/(9*e**3)) + x**8*(5*A*a*b**9/(4*e**2) - A*b**10*d/(4*e**3) + 4 
5*B*a**2*b**8/(8*e**2) - 5*B*a*b**9*d/(2*e**3) + 3*B*b**10*d**2/(8*e**4)) 
+ x**7*(45*A*a**2*b**8/(7*e**2) - 20*A*a*b**9*d/(7*e**3) + 3*A*b**10*d**2/ 
(7*e**4) + 120*B*a**3*b**7/(7*e**2) - 90*B*a**2*b**8*d/(7*e**3) + 30*B*a*b 
**9*d**2/(7*e**4) - 4*B*b**10*d**3/(7*e**5)) + x**6*(20*A*a**3*b**7/e**2 - 
 15*A*a**2*b**8*d/e**3 + 5*A*a*b**9*d**2/e**4 - 2*A*b**10*d**3/(3*e**5) + 
35*B*a**4*b**6/e**2 - 40*B*a**3*b**7*d/e**3 + 45*B*a**2*b**8*d**2/(2*e**4) 
 - 20*B*a*b**9*d**3/(3*e**5) + 5*B*b**10*d**4/(6*e**6)) + x**5*(42*A*a**4* 
b**6/e**2 - 48*A*a**3*b**7*d/e**3 + 27*A*a**2*b**8*d**2/e**4 - 8*A*a*b**9* 
d**3/e**5 + A*b**10*d**4/e**6 + 252*B*a**5*b**5/(5*e**2) - 84*B*a**4*b**6* 
d/e**3 + 72*B*a**3*b**7*d**2/e**4 - 36*B*a**2*b**8*d**3/e**5 + 10*B*a*b**9 
*d**4/e**6 - 6*B*b**10*d**5/(5*e**7)) + x**4*(63*A*a**5*b**5/e**2 - 105*A* 
a**4*b**6*d/e**3 + 90*A*a**3*b**7*d**2/e**4 - 45*A*a**2*b**8*d**3/e**5 + 2 
5*A*a*b**9*d**4/(2*e**6) - 3*A*b**10*d**5/(2*e**7) + 105*B*a**6*b**4/(2*e* 
*2) - 126*B*a**5*b**5*d/e**3 + 315*B*a**4*b**6*d**2/(2*e**4) - 120*B*a**3* 
b**7*d**3/e**5 + 225*B*a**2*b**8*d**4/(4*e**6) - 15*B*a*b**9*d**5/e**7 + 7 
*B*b**10*d**6/(4*e**8)) + x**3*(70*A*a**6*b**4/e**2 - 168*A*a**5*b**5*d/e* 
*3 + 210*A*a**4*b**6*d**2/e**4 - 160*A*a**3*b**7*d**3/e**5 + 75*A*a**2*b** 
8*d**4/e**6 - 20*A*a*b**9*d**5/e**7 + 7*A*b**10*d**6/(3*e**8) + 40*B*a*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1817 vs. \(2 (431) = 862\).

Time = 0.05 (sec) , antiderivative size = 1817, normalized size of antiderivative = 4.08 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^2,x, algorithm="maxima")
 

Output:

(B*b^10*d^11 - A*a^10*e^11 - (10*B*a*b^9 + A*b^10)*d^10*e + 5*(9*B*a^2*b^8 
 + 2*A*a*b^9)*d^9*e^2 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 30*(7*B*a 
^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 4 
2*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 30*(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 - 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)/(e^13*x + d*e^12) + 1/2520*(252* 
B*b^10*e^9*x^10 - 280*(2*B*b^10*d*e^8 - (10*B*a*b^9 + A*b^10)*e^9)*x^9 + 3 
15*(3*B*b^10*d^2*e^7 - 2*(10*B*a*b^9 + A*b^10)*d*e^8 + 5*(9*B*a^2*b^8 + 2* 
A*a*b^9)*e^9)*x^8 - 360*(4*B*b^10*d^3*e^6 - 3*(10*B*a*b^9 + A*b^10)*d^2*e^ 
7 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^8 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^ 
9)*x^7 + 420*(5*B*b^10*d^4*e^5 - 4*(10*B*a*b^9 + A*b^10)*d^3*e^6 + 15*(9*B 
*a^2*b^8 + 2*A*a*b^9)*d^2*e^7 - 30*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^8 + 30* 
(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^9)*x^6 - 504*(6*B*b^10*d^5*e^4 - 5*(10*B*a*b 
^9 + A*b^10)*d^4*e^5 + 20*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^6 - 45*(8*B*a^3* 
b^7 + 3*A*a^2*b^8)*d^2*e^7 + 60*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^8 - 42*(6* 
B*a^5*b^5 + 5*A*a^4*b^6)*e^9)*x^5 + 630*(7*B*b^10*d^6*e^3 - 6*(10*B*a*b^9 
+ A*b^10)*d^5*e^4 + 25*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^5 - 60*(8*B*a^3*b^7 
 + 3*A*a^2*b^8)*d^3*e^6 + 90*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^7 - 84*(6*B 
*a^5*b^5 + 5*A*a^4*b^6)*d*e^8 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^9)*x^4 - 
840*(8*B*b^10*d^7*e^2 - 7*(10*B*a*b^9 + A*b^10)*d^6*e^3 + 30*(9*B*a^2*b...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2207 vs. \(2 (431) = 862\).

Time = 0.15 (sec) , antiderivative size = 2207, normalized size of antiderivative = 4.96 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^2,x, algorithm="giac")
 

Output:

1/2520*(252*B*b^10 - 280*(11*B*b^10*d*e - 10*B*a*b^9*e^2 - A*b^10*e^2)/((e 
*x + d)*e) + 1575*(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) - 5400*(11*B*b^10*d^3* 
e^3 - 30*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) + 12600*(1 
1*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 + 12*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) - 21168*(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) + 26460*(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) - 25200*(11*B*b^10*d^7*e^7 - 70*B*a*b^9*d^6*e^8 - 7 
*A*b^10*d^6*e^8 + 189*B*a^2*b^8*d^5*e^9 + 42*A*a*b^9*d^5*e^9 - 280*B*a^3*b 
^7*d^4*e^10 - 105*A*a^2*b^8*d^4*e^10 + 245*B*a^4*b^6*d^3*e^11 + 140*A*a^3* 
b^7*d^3*e^11 - 126*B*a^5*b^5*d^2*e^12 - 105*A*a^4*b^6*d^2*e^12 + 35*B*a^6* 
b^4*d*e^13 + 42*A*a^5*b^5*d*e^13 - 4*B*a^7*b^3*e^14 - 7*A*a^6*b^4*e^14)...
 

Mupad [B] (verification not implemented)

Time = 1.12 (sec) , antiderivative size = 7792, normalized size of antiderivative = 17.51 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx=\text {Too large to display} \] Input:

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^2,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 1.44 (sec) , antiderivative size = 1474, normalized size of antiderivative = 3.31 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^2,x)
 

Output:

(27720*log(d + e*x)*a**10*b*d**2*e**10 + 27720*log(d + e*x)*a**10*b*d*e**1 
1*x - 277200*log(d + e*x)*a**9*b**2*d**3*e**9 - 277200*log(d + e*x)*a**9*b 
**2*d**2*e**10*x + 1247400*log(d + e*x)*a**8*b**3*d**4*e**8 + 1247400*log( 
d + e*x)*a**8*b**3*d**3*e**9*x - 3326400*log(d + e*x)*a**7*b**4*d**5*e**7 
- 3326400*log(d + e*x)*a**7*b**4*d**4*e**8*x + 5821200*log(d + e*x)*a**6*b 
**5*d**6*e**6 + 5821200*log(d + e*x)*a**6*b**5*d**5*e**7*x - 6985440*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 + 
 5821200*log(d + e*x)*a**4*b**7*d**8*e**4 + 5821200*log(d + e*x)*a**4*b**7 
*d**7*e**5*x - 3326400*log(d + e*x)*a**3*b**8*d**9*e**3 - 3326400*log(d + 
e*x)*a**3*b**8*d**8*e**4*x + 1247400*log(d + e*x)*a**2*b**9*d**10*e**2 + 1 
247400*log(d + e*x)*a**2*b**9*d**9*e**3*x - 277200*log(d + e*x)*a*b**10*d* 
*11*e - 277200*log(d + e*x)*a*b**10*d**10*e**2*x + 27720*log(d + e*x)*b**1 
1*d**12 + 27720*log(d + e*x)*b**11*d**11*e*x + 2520*a**11*e**12*x - 27720* 
a**10*b*d*e**11*x + 277200*a**9*b**2*d**2*e**10*x + 138600*a**9*b**2*d*e** 
11*x**2 - 1247400*a**8*b**3*d**3*e**9*x - 623700*a**8*b**3*d**2*e**10*x**2 
 + 207900*a**8*b**3*d*e**11*x**3 + 3326400*a**7*b**4*d**4*e**8*x + 1663200 
*a**7*b**4*d**3*e**9*x**2 - 554400*a**7*b**4*d**2*e**10*x**3 + 277200*a**7 
*b**4*d*e**11*x**4 - 5821200*a**6*b**5*d**5*e**7*x - 2910600*a**6*b**5*d** 
4*e**8*x**2 + 970200*a**6*b**5*d**3*e**9*x**3 - 485100*a**6*b**5*d**2*e**1 
0*x**4 + 291060*a**6*b**5*d*e**11*x**5 + 6985440*a**5*b**6*d**6*e**6*x ...