\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{20}} \, dx\) [1108]

Optimal result

Integrand size = 20, antiderivative size = 460 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{20}} \, dx=\frac {(b d-a e)^{10} (B d-A e)}{19 e^{12} (d+e x)^{19}}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{18 e^{12} (d+e x)^{18}}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{17 e^{12} (d+e x)^{17}}-\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{16 e^{12} (d+e x)^{16}}+\frac {2 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)^{15}}-\frac {3 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{e^{12} (d+e x)^{14}}+\frac {42 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{13 e^{12} (d+e x)^{13}}-\frac {5 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{2 e^{12} (d+e x)^{12}}+\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{11 e^{12} (d+e x)^{11}}-\frac {b^8 (b d-a e) (11 b B d-2 A b e-9 a B e)}{2 e^{12} (d+e x)^{10}}+\frac {b^9 (11 b B d-A b e-10 a B e)}{9 e^{12} (d+e x)^9}-\frac {b^{10} B}{8 e^{12} (d+e x)^8} \]

[Out]

1/19*(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)^19-1/18*(-a*e+b*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/(e*x+d)^18+5/1
7*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)^17-15/16*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d
)/e^12/(e*x+d)^16+2*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)/e^12/(e*x+d)^15-3*b^4*(-a*e+b*d)^5*(-6*A*b*e-
5*B*a*e+11*B*b*d)/e^12/(e*x+d)^14+42/13*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)/e^12/(e*x+d)^13-5/2*b^6*(
-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11*B*b*d)/e^12/(e*x+d)^12+15/11*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)^11-1/2*b^8*(-a*e+b*d)*(-2*A*b*e-9*B*a*e+11*B*b*d)/e^12/(e*x+d)^10+1/9*b^9*(-A*b*e-10*B*a*e+11*B*b*d)
/e^12/(e*x+d)^9-1/8*b^10*B/e^12/(e*x+d)^8

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{20}} \, dx=\frac {b^9 (-10 a B e-A b e+11 b B d)}{9 e^{12} (d+e x)^9}-\frac {b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{2 e^{12} (d+e x)^{10}}+\frac {15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{11 e^{12} (d+e x)^{11}}-\frac {5 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{2 e^{12} (d+e x)^{12}}+\frac {42 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{13 e^{12} (d+e x)^{13}}-\frac {3 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)^{14}}+\frac {2 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^{15}}-\frac {15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{16 e^{12} (d+e x)^{16}}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{17 e^{12} (d+e x)^{17}}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{18 e^{12} (d+e x)^{18}}+\frac {(b d-a e)^{10} (B d-A e)}{19 e^{12} (d+e x)^{19}}-\frac {b^{10} B}{8 e^{12} (d+e x)^8} \]

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^20,x]

[Out]

((b*d - a*e)^10*(B*d - A*e))/(19*e^12*(d + e*x)^19) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(18*e^12*(
d + e*x)^18) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(17*e^12*(d + e*x)^17) - (15*b^2*(b*d - a*e)
^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(16*e^12*(d + e*x)^16) + (2*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e
))/(e^12*(d + e*x)^15) - (3*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(e^12*(d + e*x)^14) + (42*b^5*(b
*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(13*e^12*(d + e*x)^13) - (5*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e
- 7*a*B*e))/(2*e^12*(d + e*x)^12) + (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e))/(11*e^12*(d + e*x)^1
1) - (b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e))/(2*e^12*(d + e*x)^10) + (b^9*(11*b*B*d - A*b*e - 10*a*B*
e))/(9*e^12*(d + e*x)^9) - (b^10*B)/(8*e^12*(d + e*x)^8)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((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]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e)^{10} (-B d+A e)}{e^{11} (d+e x)^{20}}+\frac {(-b d+a e)^9 (-11 b B d+10 A b e+a B e)}{e^{11} (d+e x)^{19}}+\frac {5 b (b d-a e)^8 (-11 b B d+9 A b e+2 a B e)}{e^{11} (d+e x)^{18}}-\frac {15 b^2 (b d-a e)^7 (-11 b B d+8 A b e+3 a B e)}{e^{11} (d+e x)^{17}}+\frac {30 b^3 (b d-a e)^6 (-11 b B d+7 A b e+4 a B e)}{e^{11} (d+e x)^{16}}-\frac {42 b^4 (b d-a e)^5 (-11 b B d+6 A b e+5 a B e)}{e^{11} (d+e x)^{15}}+\frac {42 b^5 (b d-a e)^4 (-11 b B d+5 A b e+6 a B e)}{e^{11} (d+e x)^{14}}-\frac {30 b^6 (b d-a e)^3 (-11 b B d+4 A b e+7 a B e)}{e^{11} (d+e x)^{13}}+\frac {15 b^7 (b d-a e)^2 (-11 b B d+3 A b e+8 a B e)}{e^{11} (d+e x)^{12}}-\frac {5 b^8 (b d-a e) (-11 b B d+2 A b e+9 a B e)}{e^{11} (d+e x)^{11}}+\frac {b^9 (-11 b B d+A b e+10 a B e)}{e^{11} (d+e x)^{10}}+\frac {b^{10} B}{e^{11} (d+e x)^9}\right ) \, dx \\ & = \frac {(b d-a e)^{10} (B d-A e)}{19 e^{12} (d+e x)^{19}}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{18 e^{12} (d+e x)^{18}}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{17 e^{12} (d+e x)^{17}}-\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{16 e^{12} (d+e x)^{16}}+\frac {2 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)^{15}}-\frac {3 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{e^{12} (d+e x)^{14}}+\frac {42 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{13 e^{12} (d+e x)^{13}}-\frac {5 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{2 e^{12} (d+e x)^{12}}+\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{11 e^{12} (d+e x)^{11}}-\frac {b^8 (b d-a e) (11 b B d-2 A b e-9 a B e)}{2 e^{12} (d+e x)^{10}}+\frac {b^9 (11 b B d-A b e-10 a B e)}{9 e^{12} (d+e x)^9}-\frac {b^{10} B}{8 e^{12} (d+e x)^8} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1433\) vs. \(2(460)=920\).

Time = 0.52 (sec) , antiderivative size = 1433, normalized size of antiderivative = 3.12 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{20}} \, dx=-\frac {19448 a^{10} e^{10} (18 A e+B (d+19 e x))+11440 a^9 b e^9 \left (17 A e (d+19 e x)+2 B \left (d^2+19 d e x+171 e^2 x^2\right )\right )+6435 a^8 b^2 e^8 \left (16 A e \left (d^2+19 d e x+171 e^2 x^2\right )+3 B \left (d^3+19 d^2 e x+171 d e^2 x^2+969 e^3 x^3\right )\right )+3432 a^7 b^3 e^7 \left (15 A e \left (d^3+19 d^2 e x+171 d e^2 x^2+969 e^3 x^3\right )+4 B \left (d^4+19 d^3 e x+171 d^2 e^2 x^2+969 d e^3 x^3+3876 e^4 x^4\right )\right )+1716 a^6 b^4 e^6 \left (14 A e \left (d^4+19 d^3 e x+171 d^2 e^2 x^2+969 d e^3 x^3+3876 e^4 x^4\right )+5 B \left (d^5+19 d^4 e x+171 d^3 e^2 x^2+969 d^2 e^3 x^3+3876 d e^4 x^4+11628 e^5 x^5\right )\right )+792 a^5 b^5 e^5 \left (13 A e \left (d^5+19 d^4 e x+171 d^3 e^2 x^2+969 d^2 e^3 x^3+3876 d e^4 x^4+11628 e^5 x^5\right )+6 B \left (d^6+19 d^5 e x+171 d^4 e^2 x^2+969 d^3 e^3 x^3+3876 d^2 e^4 x^4+11628 d e^5 x^5+27132 e^6 x^6\right )\right )+330 a^4 b^6 e^4 \left (12 A e \left (d^6+19 d^5 e x+171 d^4 e^2 x^2+969 d^3 e^3 x^3+3876 d^2 e^4 x^4+11628 d e^5 x^5+27132 e^6 x^6\right )+7 B \left (d^7+19 d^6 e x+171 d^5 e^2 x^2+969 d^4 e^3 x^3+3876 d^3 e^4 x^4+11628 d^2 e^5 x^5+27132 d e^6 x^6+50388 e^7 x^7\right )\right )+120 a^3 b^7 e^3 \left (11 A e \left (d^7+19 d^6 e x+171 d^5 e^2 x^2+969 d^4 e^3 x^3+3876 d^3 e^4 x^4+11628 d^2 e^5 x^5+27132 d e^6 x^6+50388 e^7 x^7\right )+8 B \left (d^8+19 d^7 e x+171 d^6 e^2 x^2+969 d^5 e^3 x^3+3876 d^4 e^4 x^4+11628 d^3 e^5 x^5+27132 d^2 e^6 x^6+50388 d e^7 x^7+75582 e^8 x^8\right )\right )+36 a^2 b^8 e^2 \left (10 A e \left (d^8+19 d^7 e x+171 d^6 e^2 x^2+969 d^5 e^3 x^3+3876 d^4 e^4 x^4+11628 d^3 e^5 x^5+27132 d^2 e^6 x^6+50388 d e^7 x^7+75582 e^8 x^8\right )+9 B \left (d^9+19 d^8 e x+171 d^7 e^2 x^2+969 d^6 e^3 x^3+3876 d^5 e^4 x^4+11628 d^4 e^5 x^5+27132 d^3 e^6 x^6+50388 d^2 e^7 x^7+75582 d e^8 x^8+92378 e^9 x^9\right )\right )+8 a b^9 e \left (9 A e \left (d^9+19 d^8 e x+171 d^7 e^2 x^2+969 d^6 e^3 x^3+3876 d^5 e^4 x^4+11628 d^4 e^5 x^5+27132 d^3 e^6 x^6+50388 d^2 e^7 x^7+75582 d e^8 x^8+92378 e^9 x^9\right )+10 B \left (d^{10}+19 d^9 e x+171 d^8 e^2 x^2+969 d^7 e^3 x^3+3876 d^6 e^4 x^4+11628 d^5 e^5 x^5+27132 d^4 e^6 x^6+50388 d^3 e^7 x^7+75582 d^2 e^8 x^8+92378 d e^9 x^9+92378 e^{10} x^{10}\right )\right )+b^{10} \left (8 A e \left (d^{10}+19 d^9 e x+171 d^8 e^2 x^2+969 d^7 e^3 x^3+3876 d^6 e^4 x^4+11628 d^5 e^5 x^5+27132 d^4 e^6 x^6+50388 d^3 e^7 x^7+75582 d^2 e^8 x^8+92378 d e^9 x^9+92378 e^{10} x^{10}\right )+11 B \left (d^{11}+19 d^{10} e x+171 d^9 e^2 x^2+969 d^8 e^3 x^3+3876 d^7 e^4 x^4+11628 d^6 e^5 x^5+27132 d^5 e^6 x^6+50388 d^4 e^7 x^7+75582 d^3 e^8 x^8+92378 d^2 e^9 x^9+92378 d e^{10} x^{10}+75582 e^{11} x^{11}\right )\right )}{6651216 e^{12} (d+e x)^{19}} \]

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^20,x]

[Out]

-1/6651216*(19448*a^10*e^10*(18*A*e + B*(d + 19*e*x)) + 11440*a^9*b*e^9*(17*A*e*(d + 19*e*x) + 2*B*(d^2 + 19*d
*e*x + 171*e^2*x^2)) + 6435*a^8*b^2*e^8*(16*A*e*(d^2 + 19*d*e*x + 171*e^2*x^2) + 3*B*(d^3 + 19*d^2*e*x + 171*d
*e^2*x^2 + 969*e^3*x^3)) + 3432*a^7*b^3*e^7*(15*A*e*(d^3 + 19*d^2*e*x + 171*d*e^2*x^2 + 969*e^3*x^3) + 4*B*(d^
4 + 19*d^3*e*x + 171*d^2*e^2*x^2 + 969*d*e^3*x^3 + 3876*e^4*x^4)) + 1716*a^6*b^4*e^6*(14*A*e*(d^4 + 19*d^3*e*x
+ 171*d^2*e^2*x^2 + 969*d*e^3*x^3 + 3876*e^4*x^4) + 5*B*(d^5 + 19*d^4*e*x + 171*d^3*e^2*x^2 + 969*d^2*e^3*x^3
+ 3876*d*e^4*x^4 + 11628*e^5*x^5)) + 792*a^5*b^5*e^5*(13*A*e*(d^5 + 19*d^4*e*x + 171*d^3*e^2*x^2 + 969*d^2*e^
3*x^3 + 3876*d*e^4*x^4 + 11628*e^5*x^5) + 6*B*(d^6 + 19*d^5*e*x + 171*d^4*e^2*x^2 + 969*d^3*e^3*x^3 + 3876*d^2
*e^4*x^4 + 11628*d*e^5*x^5 + 27132*e^6*x^6)) + 330*a^4*b^6*e^4*(12*A*e*(d^6 + 19*d^5*e*x + 171*d^4*e^2*x^2 + 9
69*d^3*e^3*x^3 + 3876*d^2*e^4*x^4 + 11628*d*e^5*x^5 + 27132*e^6*x^6) + 7*B*(d^7 + 19*d^6*e*x + 171*d^5*e^2*x^2
+ 969*d^4*e^3*x^3 + 3876*d^3*e^4*x^4 + 11628*d^2*e^5*x^5 + 27132*d*e^6*x^6 + 50388*e^7*x^7)) + 120*a^3*b^7*e^
3*(11*A*e*(d^7 + 19*d^6*e*x + 171*d^5*e^2*x^2 + 969*d^4*e^3*x^3 + 3876*d^3*e^4*x^4 + 11628*d^2*e^5*x^5 + 27132
*d*e^6*x^6 + 50388*e^7*x^7) + 8*B*(d^8 + 19*d^7*e*x + 171*d^6*e^2*x^2 + 969*d^5*e^3*x^3 + 3876*d^4*e^4*x^4 + 1
1628*d^3*e^5*x^5 + 27132*d^2*e^6*x^6 + 50388*d*e^7*x^7 + 75582*e^8*x^8)) + 36*a^2*b^8*e^2*(10*A*e*(d^8 + 19*d^
7*e*x + 171*d^6*e^2*x^2 + 969*d^5*e^3*x^3 + 3876*d^4*e^4*x^4 + 11628*d^3*e^5*x^5 + 27132*d^2*e^6*x^6 + 50388*d
*e^7*x^7 + 75582*e^8*x^8) + 9*B*(d^9 + 19*d^8*e*x + 171*d^7*e^2*x^2 + 969*d^6*e^3*x^3 + 3876*d^5*e^4*x^4 + 116
28*d^4*e^5*x^5 + 27132*d^3*e^6*x^6 + 50388*d^2*e^7*x^7 + 75582*d*e^8*x^8 + 92378*e^9*x^9)) + 8*a*b^9*e*(9*A*e*
(d^9 + 19*d^8*e*x + 171*d^7*e^2*x^2 + 969*d^6*e^3*x^3 + 3876*d^5*e^4*x^4 + 11628*d^4*e^5*x^5 + 27132*d^3*e^6*x
^6 + 50388*d^2*e^7*x^7 + 75582*d*e^8*x^8 + 92378*e^9*x^9) + 10*B*(d^10 + 19*d^9*e*x + 171*d^8*e^2*x^2 + 969*d^
7*e^3*x^3 + 3876*d^6*e^4*x^4 + 11628*d^5*e^5*x^5 + 27132*d^4*e^6*x^6 + 50388*d^3*e^7*x^7 + 75582*d^2*e^8*x^8 +
92378*d*e^9*x^9 + 92378*e^10*x^10)) + b^10*(8*A*e*(d^10 + 19*d^9*e*x + 171*d^8*e^2*x^2 + 969*d^7*e^3*x^3 + 38
76*d^6*e^4*x^4 + 11628*d^5*e^5*x^5 + 27132*d^4*e^6*x^6 + 50388*d^3*e^7*x^7 + 75582*d^2*e^8*x^8 + 92378*d*e^9*x
^9 + 92378*e^10*x^10) + 11*B*(d^11 + 19*d^10*e*x + 171*d^9*e^2*x^2 + 969*d^8*e^3*x^3 + 3876*d^7*e^4*x^4 + 1162
8*d^6*e^5*x^5 + 27132*d^5*e^6*x^6 + 50388*d^4*e^7*x^7 + 75582*d^3*e^8*x^8 + 92378*d^2*e^9*x^9 + 92378*d*e^10*x
^10 + 75582*e^11*x^11)))/(e^12*(d + e*x)^19)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1900\) vs. \(2(440)=880\).

Time = 2.14 (sec) , antiderivative size = 1901, normalized size of antiderivative = 4.13

method	result	size
risch	\(\text {Expression too large to display}\)	\(1901\)
default	\(\text {Expression too large to display}\)	\(1942\)
norman	\(\text {Expression too large to display}\)	\(2014\)
gosper	\(\text {Expression too large to display}\)	\(2233\)
parallelrisch	\(\text {Expression too large to display}\)	\(2242\)

[In]

int((b*x+a)^10*(B*x+A)/(e*x+d)^20,x,method=_RETURNVERBOSE)

[Out]

(-1/6651216/e^12*(350064*A*a^10*e^11+194480*A*a^9*b*d*e^10+102960*A*a^8*b^2*d^2*e^9+51480*A*a^7*b^3*d^3*e^8+24
024*A*a^6*b^4*d^4*e^7+10296*A*a^5*b^5*d^5*e^6+3960*A*a^4*b^6*d^6*e^5+1320*A*a^3*b^7*d^7*e^4+360*A*a^2*b^8*d^8*
e^3+72*A*a*b^9*d^9*e^2+8*A*b^10*d^10*e+19448*B*a^10*d*e^10+22880*B*a^9*b*d^2*e^9+19305*B*a^8*b^2*d^3*e^8+13728
*B*a^7*b^3*d^4*e^7+8580*B*a^6*b^4*d^5*e^6+4752*B*a^5*b^5*d^6*e^5+2310*B*a^4*b^6*d^7*e^4+960*B*a^3*b^7*d^8*e^3+
324*B*a^2*b^8*d^9*e^2+80*B*a*b^9*d^10*e+11*B*b^10*d^11)-1/350064/e^11*(194480*A*a^9*b*e^10+102960*A*a^8*b^2*d*
e^9+51480*A*a^7*b^3*d^2*e^8+24024*A*a^6*b^4*d^3*e^7+10296*A*a^5*b^5*d^4*e^6+3960*A*a^4*b^6*d^5*e^5+1320*A*a^3*
b^7*d^6*e^4+360*A*a^2*b^8*d^7*e^3+72*A*a*b^9*d^8*e^2+8*A*b^10*d^9*e+19448*B*a^10*e^10+22880*B*a^9*b*d*e^9+1930
5*B*a^8*b^2*d^2*e^8+13728*B*a^7*b^3*d^3*e^7+8580*B*a^6*b^4*d^4*e^6+4752*B*a^5*b^5*d^5*e^5+2310*B*a^4*b^6*d^6*e
^4+960*B*a^3*b^7*d^7*e^3+324*B*a^2*b^8*d^8*e^2+80*B*a*b^9*d^9*e+11*B*b^10*d^10)*x-1/38896*b/e^10*(102960*A*a^8
*b*e^9+51480*A*a^7*b^2*d*e^8+24024*A*a^6*b^3*d^2*e^7+10296*A*a^5*b^4*d^3*e^6+3960*A*a^4*b^5*d^4*e^5+1320*A*a^3
*b^6*d^5*e^4+360*A*a^2*b^7*d^6*e^3+72*A*a*b^8*d^7*e^2+8*A*b^9*d^8*e+22880*B*a^9*e^9+19305*B*a^8*b*d*e^8+13728*
B*a^7*b^2*d^2*e^7+8580*B*a^6*b^3*d^3*e^6+4752*B*a^5*b^4*d^4*e^5+2310*B*a^4*b^5*d^5*e^4+960*B*a^3*b^6*d^6*e^3+3
24*B*a^2*b^7*d^7*e^2+80*B*a*b^8*d^8*e+11*B*b^9*d^9)*x^2-1/6864*b^2/e^9*(51480*A*a^7*b*e^8+24024*A*a^6*b^2*d*e^
7+10296*A*a^5*b^3*d^2*e^6+3960*A*a^4*b^4*d^3*e^5+1320*A*a^3*b^5*d^4*e^4+360*A*a^2*b^6*d^5*e^3+72*A*a*b^7*d^6*e
^2+8*A*b^8*d^7*e+19305*B*a^8*e^8+13728*B*a^7*b*d*e^7+8580*B*a^6*b^2*d^2*e^6+4752*B*a^5*b^3*d^3*e^5+2310*B*a^4*
b^4*d^4*e^4+960*B*a^3*b^5*d^5*e^3+324*B*a^2*b^6*d^6*e^2+80*B*a*b^7*d^7*e+11*B*b^8*d^8)*x^3-1/1716*b^3/e^8*(240
24*A*a^6*b*e^7+10296*A*a^5*b^2*d*e^6+3960*A*a^4*b^3*d^2*e^5+1320*A*a^3*b^4*d^3*e^4+360*A*a^2*b^5*d^4*e^3+72*A*
a*b^6*d^5*e^2+8*A*b^7*d^6*e+13728*B*a^7*e^7+8580*B*a^6*b*d*e^6+4752*B*a^5*b^2*d^2*e^5+2310*B*a^4*b^3*d^3*e^4+9
60*B*a^3*b^4*d^4*e^3+324*B*a^2*b^5*d^5*e^2+80*B*a*b^6*d^6*e+11*B*b^7*d^7)*x^4-1/572*b^4/e^7*(10296*A*a^5*b*e^6
+3960*A*a^4*b^2*d*e^5+1320*A*a^3*b^3*d^2*e^4+360*A*a^2*b^4*d^3*e^3+72*A*a*b^5*d^4*e^2+8*A*b^6*d^5*e+8580*B*a^6
*e^6+4752*B*a^5*b*d*e^5+2310*B*a^4*b^2*d^2*e^4+960*B*a^3*b^3*d^3*e^3+324*B*a^2*b^4*d^4*e^2+80*B*a*b^5*d^5*e+11
*B*b^6*d^6)*x^5-7/1716*b^5/e^6*(3960*A*a^4*b*e^5+1320*A*a^3*b^2*d*e^4+360*A*a^2*b^3*d^2*e^3+72*A*a*b^4*d^3*e^2
+8*A*b^5*d^4*e+4752*B*a^5*e^5+2310*B*a^4*b*d*e^4+960*B*a^3*b^2*d^2*e^3+324*B*a^2*b^3*d^3*e^2+80*B*a*b^4*d^4*e+
11*B*b^5*d^5)*x^6-1/132*b^6/e^5*(1320*A*a^3*b*e^4+360*A*a^2*b^2*d*e^3+72*A*a*b^3*d^2*e^2+8*A*b^4*d^3*e+2310*B*
a^4*e^4+960*B*a^3*b*d*e^3+324*B*a^2*b^2*d^2*e^2+80*B*a*b^3*d^3*e+11*B*b^4*d^4)*x^7-1/88*b^7/e^4*(360*A*a^2*b*e
^3+72*A*a*b^2*d*e^2+8*A*b^3*d^2*e+960*B*a^3*e^3+324*B*a^2*b*d*e^2+80*B*a*b^2*d^2*e+11*B*b^3*d^3)*x^8-1/72*b^8/
e^3*(72*A*a*b*e^2+8*A*b^2*d*e+324*B*a^2*e^2+80*B*a*b*d*e+11*B*b^2*d^2)*x^9-1/72*b^9/e^2*(8*A*b*e+80*B*a*e+11*B
*b*d)*x^10-1/8*b^10*B/e*x^11)/(e*x+d)^19

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2017 vs. \(2 (440) = 880\).

Time = 0.36 (sec) , antiderivative size = 2017, normalized size of antiderivative = 4.38 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{20}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^20,x, algorithm="fricas")

[Out]

-1/6651216*(831402*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 350064*A*a^10*e^11 + 8*(10*B*a*b^9 + A*b^10)*d^10*e + 3
6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7
)*d^7*e^4 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 3432*(4*B*a^7
*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 6435*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 11440*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e
^9 + 19448*(B*a^10 + 10*A*a^9*b)*d*e^10 + 92378*(11*B*b^10*d*e^10 + 8*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 92378
*(11*B*b^10*d^2*e^9 + 8*(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 + 75582*(11*B*b^
10*d^3*e^8 + 8*(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 + 120*(8*B*a^3*b^7 + 3*A*a^
2*b^8)*e^11)*x^8 + 50388*(11*B*b^10*d^4*e^7 + 8*(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 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 27132*(11*B*b^10
*d^5*e^6 + 8*(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 + 120*(8*B*a^3*b^7 + 3*A*a^2
*b^8)*d^2*e^9 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 11628*(11
*B*b^10*d^6*e^5 + 8*(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 + 120*(8*B*a^3*b^7 +
3*A*a^2*b^8)*d^3*e^8 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 1716
*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 3876*(11*B*b^10*d^7*e^4 + 8*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 36*(9*B*a
^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^
8 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 3432*(4*B*a^7*b^3 + 7*
A*a^6*b^4)*e^11)*x^4 + 969*(11*B*b^10*d^8*e^3 + 8*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)
*d^6*e^5 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 792*(6*B*a^5*b^
5 + 5*A*a^4*b^6)*d^3*e^8 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 3432*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10
+ 6435*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 171*(11*B*b^10*d^9*e^2 + 8*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 36*(
9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d
^5*e^6 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 3432*(4*B*a^7*b^
3 + 7*A*a^6*b^4)*d^2*e^9 + 6435*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 11440*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2
+ 19*(11*B*b^10*d^10*e + 8*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 36*(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 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e
^6 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 3432*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 6435*(3*B*a^8*b^2 +
8*A*a^7*b^3)*d^2*e^9 + 11440*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 19448*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^31*x^
19 + 19*d*e^30*x^18 + 171*d^2*e^29*x^17 + 969*d^3*e^28*x^16 + 3876*d^4*e^27*x^15 + 11628*d^5*e^26*x^14 + 27132
*d^6*e^25*x^13 + 50388*d^7*e^24*x^12 + 75582*d^8*e^23*x^11 + 92378*d^9*e^22*x^10 + 92378*d^10*e^21*x^9 + 75582
*d^11*e^20*x^8 + 50388*d^12*e^19*x^7 + 27132*d^13*e^18*x^6 + 11628*d^14*e^17*x^5 + 3876*d^15*e^16*x^4 + 969*d^
16*e^15*x^3 + 171*d^17*e^14*x^2 + 19*d^18*e^13*x + d^19*e^12)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{20}} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**20,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2017 vs. \(2 (440) = 880\).

Time = 0.32 (sec) , antiderivative size = 2017, normalized size of antiderivative = 4.38 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{20}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^20,x, algorithm="maxima")

[Out]

-1/6651216*(831402*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 350064*A*a^10*e^11 + 8*(10*B*a*b^9 + A*b^10)*d^10*e + 3
6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7
)*d^7*e^4 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 3432*(4*B*a^7
*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 6435*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 11440*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e
^9 + 19448*(B*a^10 + 10*A*a^9*b)*d*e^10 + 92378*(11*B*b^10*d*e^10 + 8*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 92378
*(11*B*b^10*d^2*e^9 + 8*(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 + 75582*(11*B*b^
10*d^3*e^8 + 8*(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 + 120*(8*B*a^3*b^7 + 3*A*a^
2*b^8)*e^11)*x^8 + 50388*(11*B*b^10*d^4*e^7 + 8*(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 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 27132*(11*B*b^10
*d^5*e^6 + 8*(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 + 120*(8*B*a^3*b^7 + 3*A*a^2
*b^8)*d^2*e^9 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 11628*(11
*B*b^10*d^6*e^5 + 8*(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 + 120*(8*B*a^3*b^7 +
3*A*a^2*b^8)*d^3*e^8 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 1716
*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 3876*(11*B*b^10*d^7*e^4 + 8*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 36*(9*B*a
^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^
8 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 3432*(4*B*a^7*b^3 + 7*
A*a^6*b^4)*e^11)*x^4 + 969*(11*B*b^10*d^8*e^3 + 8*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)
*d^6*e^5 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 792*(6*B*a^5*b^
5 + 5*A*a^4*b^6)*d^3*e^8 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 3432*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10
+ 6435*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 171*(11*B*b^10*d^9*e^2 + 8*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 36*(
9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d
^5*e^6 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 3432*(4*B*a^7*b^
3 + 7*A*a^6*b^4)*d^2*e^9 + 6435*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 11440*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2
+ 19*(11*B*b^10*d^10*e + 8*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 36*(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 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e
^6 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 3432*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 6435*(3*B*a^8*b^2 +
8*A*a^7*b^3)*d^2*e^9 + 11440*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 19448*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^31*x^
19 + 19*d*e^30*x^18 + 171*d^2*e^29*x^17 + 969*d^3*e^28*x^16 + 3876*d^4*e^27*x^15 + 11628*d^5*e^26*x^14 + 27132
*d^6*e^25*x^13 + 50388*d^7*e^24*x^12 + 75582*d^8*e^23*x^11 + 92378*d^9*e^22*x^10 + 92378*d^10*e^21*x^9 + 75582
*d^11*e^20*x^8 + 50388*d^12*e^19*x^7 + 27132*d^13*e^18*x^6 + 11628*d^14*e^17*x^5 + 3876*d^15*e^16*x^4 + 969*d^
16*e^15*x^3 + 171*d^17*e^14*x^2 + 19*d^18*e^13*x + d^19*e^12)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2232 vs. \(2 (440) = 880\).

Time = 0.29 (sec) , antiderivative size = 2232, normalized size of antiderivative = 4.85 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{20}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^20,x, algorithm="giac")

[Out]

-1/6651216*(831402*B*b^10*e^11*x^11 + 1016158*B*b^10*d*e^10*x^10 + 7390240*B*a*b^9*e^11*x^10 + 739024*A*b^10*e
^11*x^10 + 1016158*B*b^10*d^2*e^9*x^9 + 7390240*B*a*b^9*d*e^10*x^9 + 739024*A*b^10*d*e^10*x^9 + 29930472*B*a^2
*b^8*e^11*x^9 + 6651216*A*a*b^9*e^11*x^9 + 831402*B*b^10*d^3*e^8*x^8 + 6046560*B*a*b^9*d^2*e^9*x^8 + 604656*A*
b^10*d^2*e^9*x^8 + 24488568*B*a^2*b^8*d*e^10*x^8 + 5441904*A*a*b^9*d*e^10*x^8 + 72558720*B*a^3*b^7*e^11*x^8 +
27209520*A*a^2*b^8*e^11*x^8 + 554268*B*b^10*d^4*e^7*x^7 + 4031040*B*a*b^9*d^3*e^8*x^7 + 403104*A*b^10*d^3*e^8*
x^7 + 16325712*B*a^2*b^8*d^2*e^9*x^7 + 3627936*A*a*b^9*d^2*e^9*x^7 + 48372480*B*a^3*b^7*d*e^10*x^7 + 18139680*
A*a^2*b^8*d*e^10*x^7 + 116396280*B*a^4*b^6*e^11*x^7 + 66512160*A*a^3*b^7*e^11*x^7 + 298452*B*b^10*d^5*e^6*x^6
+ 2170560*B*a*b^9*d^4*e^7*x^6 + 217056*A*b^10*d^4*e^7*x^6 + 8790768*B*a^2*b^8*d^3*e^8*x^6 + 1953504*A*a*b^9*d^
3*e^8*x^6 + 26046720*B*a^3*b^7*d^2*e^9*x^6 + 9767520*A*a^2*b^8*d^2*e^9*x^6 + 62674920*B*a^4*b^6*d*e^10*x^6 + 3
5814240*A*a^3*b^7*d*e^10*x^6 + 128931264*B*a^5*b^5*e^11*x^6 + 107442720*A*a^4*b^6*e^11*x^6 + 127908*B*b^10*d^6
*e^5*x^5 + 930240*B*a*b^9*d^5*e^6*x^5 + 93024*A*b^10*d^5*e^6*x^5 + 3767472*B*a^2*b^8*d^4*e^7*x^5 + 837216*A*a*
b^9*d^4*e^7*x^5 + 11162880*B*a^3*b^7*d^3*e^8*x^5 + 4186080*A*a^2*b^8*d^3*e^8*x^5 + 26860680*B*a^4*b^6*d^2*e^9*
x^5 + 15348960*A*a^3*b^7*d^2*e^9*x^5 + 55256256*B*a^5*b^5*d*e^10*x^5 + 46046880*A*a^4*b^6*d*e^10*x^5 + 9976824
0*B*a^6*b^4*e^11*x^5 + 119721888*A*a^5*b^5*e^11*x^5 + 42636*B*b^10*d^7*e^4*x^4 + 310080*B*a*b^9*d^6*e^5*x^4 +
31008*A*b^10*d^6*e^5*x^4 + 1255824*B*a^2*b^8*d^5*e^6*x^4 + 279072*A*a*b^9*d^5*e^6*x^4 + 3720960*B*a^3*b^7*d^4*
e^7*x^4 + 1395360*A*a^2*b^8*d^4*e^7*x^4 + 8953560*B*a^4*b^6*d^3*e^8*x^4 + 5116320*A*a^3*b^7*d^3*e^8*x^4 + 1841
8752*B*a^5*b^5*d^2*e^9*x^4 + 15348960*A*a^4*b^6*d^2*e^9*x^4 + 33256080*B*a^6*b^4*d*e^10*x^4 + 39907296*A*a^5*b
^5*d*e^10*x^4 + 53209728*B*a^7*b^3*e^11*x^4 + 93117024*A*a^6*b^4*e^11*x^4 + 10659*B*b^10*d^8*e^3*x^3 + 77520*B
*a*b^9*d^7*e^4*x^3 + 7752*A*b^10*d^7*e^4*x^3 + 313956*B*a^2*b^8*d^6*e^5*x^3 + 69768*A*a*b^9*d^6*e^5*x^3 + 9302
40*B*a^3*b^7*d^5*e^6*x^3 + 348840*A*a^2*b^8*d^5*e^6*x^3 + 2238390*B*a^4*b^6*d^4*e^7*x^3 + 1279080*A*a^3*b^7*d^
4*e^7*x^3 + 4604688*B*a^5*b^5*d^3*e^8*x^3 + 3837240*A*a^4*b^6*d^3*e^8*x^3 + 8314020*B*a^6*b^4*d^2*e^9*x^3 + 99
76824*A*a^5*b^5*d^2*e^9*x^3 + 13302432*B*a^7*b^3*d*e^10*x^3 + 23279256*A*a^6*b^4*d*e^10*x^3 + 18706545*B*a^8*b
^2*e^11*x^3 + 49884120*A*a^7*b^3*e^11*x^3 + 1881*B*b^10*d^9*e^2*x^2 + 13680*B*a*b^9*d^8*e^3*x^2 + 1368*A*b^10*
d^8*e^3*x^2 + 55404*B*a^2*b^8*d^7*e^4*x^2 + 12312*A*a*b^9*d^7*e^4*x^2 + 164160*B*a^3*b^7*d^6*e^5*x^2 + 61560*A
*a^2*b^8*d^6*e^5*x^2 + 395010*B*a^4*b^6*d^5*e^6*x^2 + 225720*A*a^3*b^7*d^5*e^6*x^2 + 812592*B*a^5*b^5*d^4*e^7*
x^2 + 677160*A*a^4*b^6*d^4*e^7*x^2 + 1467180*B*a^6*b^4*d^3*e^8*x^2 + 1760616*A*a^5*b^5*d^3*e^8*x^2 + 2347488*B
*a^7*b^3*d^2*e^9*x^2 + 4108104*A*a^6*b^4*d^2*e^9*x^2 + 3301155*B*a^8*b^2*d*e^10*x^2 + 8803080*A*a^7*b^3*d*e^10
*x^2 + 3912480*B*a^9*b*e^11*x^2 + 17606160*A*a^8*b^2*e^11*x^2 + 209*B*b^10*d^10*e*x + 1520*B*a*b^9*d^9*e^2*x +
152*A*b^10*d^9*e^2*x + 6156*B*a^2*b^8*d^8*e^3*x + 1368*A*a*b^9*d^8*e^3*x + 18240*B*a^3*b^7*d^7*e^4*x + 6840*A
*a^2*b^8*d^7*e^4*x + 43890*B*a^4*b^6*d^6*e^5*x + 25080*A*a^3*b^7*d^6*e^5*x + 90288*B*a^5*b^5*d^5*e^6*x + 75240
*A*a^4*b^6*d^5*e^6*x + 163020*B*a^6*b^4*d^4*e^7*x + 195624*A*a^5*b^5*d^4*e^7*x + 260832*B*a^7*b^3*d^3*e^8*x +
456456*A*a^6*b^4*d^3*e^8*x + 366795*B*a^8*b^2*d^2*e^9*x + 978120*A*a^7*b^3*d^2*e^9*x + 434720*B*a^9*b*d*e^10*x
+ 1956240*A*a^8*b^2*d*e^10*x + 369512*B*a^10*e^11*x + 3695120*A*a^9*b*e^11*x + 11*B*b^10*d^11 + 80*B*a*b^9*d^
10*e + 8*A*b^10*d^10*e + 324*B*a^2*b^8*d^9*e^2 + 72*A*a*b^9*d^9*e^2 + 960*B*a^3*b^7*d^8*e^3 + 360*A*a^2*b^8*d^
8*e^3 + 2310*B*a^4*b^6*d^7*e^4 + 1320*A*a^3*b^7*d^7*e^4 + 4752*B*a^5*b^5*d^6*e^5 + 3960*A*a^4*b^6*d^6*e^5 + 85
80*B*a^6*b^4*d^5*e^6 + 10296*A*a^5*b^5*d^5*e^6 + 13728*B*a^7*b^3*d^4*e^7 + 24024*A*a^6*b^4*d^4*e^7 + 19305*B*a
^8*b^2*d^3*e^8 + 51480*A*a^7*b^3*d^3*e^8 + 22880*B*a^9*b*d^2*e^9 + 102960*A*a^8*b^2*d^2*e^9 + 19448*B*a^10*d*e
^10 + 194480*A*a^9*b*d*e^10 + 350064*A*a^10*e^11)/((e*x + d)^19*e^12)

Mupad [B] (verification not implemented)

Time = 30.39 (sec) , antiderivative size = 2099, normalized size of antiderivative = 4.56 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{20}} \, dx=\text {Too large to display} \]

[In]

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^20,x)

[Out]

-((350064*A*a^10*e^11 + 11*B*b^10*d^11 + 8*A*b^10*d^10*e + 19448*B*a^10*d*e^10 + 72*A*a*b^9*d^9*e^2 + 22880*B*
a^9*b*d^2*e^9 + 360*A*a^2*b^8*d^8*e^3 + 1320*A*a^3*b^7*d^7*e^4 + 3960*A*a^4*b^6*d^6*e^5 + 10296*A*a^5*b^5*d^5*
e^6 + 24024*A*a^6*b^4*d^4*e^7 + 51480*A*a^7*b^3*d^3*e^8 + 102960*A*a^8*b^2*d^2*e^9 + 324*B*a^2*b^8*d^9*e^2 + 9
60*B*a^3*b^7*d^8*e^3 + 2310*B*a^4*b^6*d^7*e^4 + 4752*B*a^5*b^5*d^6*e^5 + 8580*B*a^6*b^4*d^5*e^6 + 13728*B*a^7*
b^3*d^4*e^7 + 19305*B*a^8*b^2*d^3*e^8 + 194480*A*a^9*b*d*e^10 + 80*B*a*b^9*d^10*e)/(6651216*e^12) + (x*(19448*
B*a^10*e^10 + 11*B*b^10*d^10 + 194480*A*a^9*b*e^10 + 8*A*b^10*d^9*e + 72*A*a*b^9*d^8*e^2 + 102960*A*a^8*b^2*d*
e^9 + 360*A*a^2*b^8*d^7*e^3 + 1320*A*a^3*b^7*d^6*e^4 + 3960*A*a^4*b^6*d^5*e^5 + 10296*A*a^5*b^5*d^4*e^6 + 2402
4*A*a^6*b^4*d^3*e^7 + 51480*A*a^7*b^3*d^2*e^8 + 324*B*a^2*b^8*d^8*e^2 + 960*B*a^3*b^7*d^7*e^3 + 2310*B*a^4*b^6
*d^6*e^4 + 4752*B*a^5*b^5*d^5*e^5 + 8580*B*a^6*b^4*d^4*e^6 + 13728*B*a^7*b^3*d^3*e^7 + 19305*B*a^8*b^2*d^2*e^8
+ 80*B*a*b^9*d^9*e + 22880*B*a^9*b*d*e^9))/(350064*e^11) + (b^7*x^8*(960*B*a^3*e^3 + 11*B*b^3*d^3 + 360*A*a^2
*b*e^3 + 8*A*b^3*d^2*e + 72*A*a*b^2*d*e^2 + 80*B*a*b^2*d^2*e + 324*B*a^2*b*d*e^2))/(88*e^4) + (b^4*x^5*(8580*B
*a^6*e^6 + 11*B*b^6*d^6 + 10296*A*a^5*b*e^6 + 8*A*b^6*d^5*e + 72*A*a*b^5*d^4*e^2 + 3960*A*a^4*b^2*d*e^5 + 360*
A*a^2*b^4*d^3*e^3 + 1320*A*a^3*b^3*d^2*e^4 + 324*B*a^2*b^4*d^4*e^2 + 960*B*a^3*b^3*d^3*e^3 + 2310*B*a^4*b^2*d^
2*e^4 + 80*B*a*b^5*d^5*e + 4752*B*a^5*b*d*e^5))/(572*e^7) + (b^9*x^10*(8*A*b*e + 80*B*a*e + 11*B*b*d))/(72*e^2
) + (b^6*x^7*(2310*B*a^4*e^4 + 11*B*b^4*d^4 + 1320*A*a^3*b*e^4 + 8*A*b^4*d^3*e + 72*A*a*b^3*d^2*e^2 + 360*A*a^
2*b^2*d*e^3 + 324*B*a^2*b^2*d^2*e^2 + 80*B*a*b^3*d^3*e + 960*B*a^3*b*d*e^3))/(132*e^5) + (b^3*x^4*(13728*B*a^7
*e^7 + 11*B*b^7*d^7 + 24024*A*a^6*b*e^7 + 8*A*b^7*d^6*e + 72*A*a*b^6*d^5*e^2 + 10296*A*a^5*b^2*d*e^6 + 360*A*a
^2*b^5*d^4*e^3 + 1320*A*a^3*b^4*d^3*e^4 + 3960*A*a^4*b^3*d^2*e^5 + 324*B*a^2*b^5*d^5*e^2 + 960*B*a^3*b^4*d^4*e
^3 + 2310*B*a^4*b^3*d^3*e^4 + 4752*B*a^5*b^2*d^2*e^5 + 80*B*a*b^6*d^6*e + 8580*B*a^6*b*d*e^6))/(1716*e^8) + (b
*x^2*(22880*B*a^9*e^9 + 11*B*b^9*d^9 + 102960*A*a^8*b*e^9 + 8*A*b^9*d^8*e + 72*A*a*b^8*d^7*e^2 + 51480*A*a^7*b
^2*d*e^8 + 360*A*a^2*b^7*d^6*e^3 + 1320*A*a^3*b^6*d^5*e^4 + 3960*A*a^4*b^5*d^4*e^5 + 10296*A*a^5*b^4*d^3*e^6 +
24024*A*a^6*b^3*d^2*e^7 + 324*B*a^2*b^7*d^7*e^2 + 960*B*a^3*b^6*d^6*e^3 + 2310*B*a^4*b^5*d^5*e^4 + 4752*B*a^5
*b^4*d^4*e^5 + 8580*B*a^6*b^3*d^3*e^6 + 13728*B*a^7*b^2*d^2*e^7 + 80*B*a*b^8*d^8*e + 19305*B*a^8*b*d*e^8))/(38
896*e^10) + (b^8*x^9*(324*B*a^2*e^2 + 11*B*b^2*d^2 + 72*A*a*b*e^2 + 8*A*b^2*d*e + 80*B*a*b*d*e))/(72*e^3) + (7
*b^5*x^6*(4752*B*a^5*e^5 + 11*B*b^5*d^5 + 3960*A*a^4*b*e^5 + 8*A*b^5*d^4*e + 72*A*a*b^4*d^3*e^2 + 1320*A*a^3*b
^2*d*e^4 + 360*A*a^2*b^3*d^2*e^3 + 324*B*a^2*b^3*d^3*e^2 + 960*B*a^3*b^2*d^2*e^3 + 80*B*a*b^4*d^4*e + 2310*B*a
^4*b*d*e^4))/(1716*e^6) + (b^2*x^3*(19305*B*a^8*e^8 + 11*B*b^8*d^8 + 51480*A*a^7*b*e^8 + 8*A*b^8*d^7*e + 72*A*
a*b^7*d^6*e^2 + 24024*A*a^6*b^2*d*e^7 + 360*A*a^2*b^6*d^5*e^3 + 1320*A*a^3*b^5*d^4*e^4 + 3960*A*a^4*b^4*d^3*e^
5 + 10296*A*a^5*b^3*d^2*e^6 + 324*B*a^2*b^6*d^6*e^2 + 960*B*a^3*b^5*d^5*e^3 + 2310*B*a^4*b^4*d^4*e^4 + 4752*B*
a^5*b^3*d^3*e^5 + 8580*B*a^6*b^2*d^2*e^6 + 80*B*a*b^7*d^7*e + 13728*B*a^7*b*d*e^7))/(6864*e^9) + (B*b^10*x^11)
/(8*e))/(d^19 + e^19*x^19 + 19*d*e^18*x^18 + 171*d^17*e^2*x^2 + 969*d^16*e^3*x^3 + 3876*d^15*e^4*x^4 + 11628*d
^14*e^5*x^5 + 27132*d^13*e^6*x^6 + 50388*d^12*e^7*x^7 + 75582*d^11*e^8*x^8 + 92378*d^10*e^9*x^9 + 92378*d^9*e^
10*x^10 + 75582*d^8*e^11*x^11 + 50388*d^7*e^12*x^12 + 27132*d^6*e^13*x^13 + 11628*d^5*e^14*x^14 + 3876*d^4*e^1
5*x^15 + 969*d^3*e^16*x^16 + 171*d^2*e^17*x^17 + 19*d^18*e*x)