Integrand size = 31, antiderivative size = 206 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx=\frac {(b d-a e)^4 (B d-A e)}{10 e^6 (d+e x)^{10}}-\frac {(b d-a e)^3 (5 b B d-4 A b e-a B e)}{9 e^6 (d+e x)^9}+\frac {b (b d-a e)^2 (5 b B d-3 A b e-2 a B e)}{4 e^6 (d+e x)^8}-\frac {2 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e)}{7 e^6 (d+e x)^7}+\frac {b^3 (5 b B d-A b e-4 a B e)}{6 e^6 (d+e x)^6}-\frac {b^4 B}{5 e^6 (d+e x)^5} \]
1/10*(-a*e+b*d)^4*(-A*e+B*d)/e^6/(e*x+d)^10-1/9*(-a*e+b*d)^3*(-4*A*b*e-B*a *e+5*B*b*d)/e^6/(e*x+d)^9+1/4*b*(-a*e+b*d)^2*(-3*A*b*e-2*B*a*e+5*B*b*d)/e^ 6/(e*x+d)^8-2/7*b^2*(-a*e+b*d)*(-2*A*b*e-3*B*a*e+5*B*b*d)/e^6/(e*x+d)^7+1/ 6*b^3*(-A*b*e-4*B*a*e+5*B*b*d)/e^6/(e*x+d)^6-1/5*b^4*B/e^6/(e*x+d)^5
Time = 0.09 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.55 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx=-\frac {14 a^4 e^4 (9 A e+B (d+10 e x))+14 a^3 b e^3 \left (4 A e (d+10 e x)+B \left (d^2+10 d e x+45 e^2 x^2\right )\right )+3 a^2 b^2 e^2 \left (7 A e \left (d^2+10 d e x+45 e^2 x^2\right )+3 B \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )\right )+2 a b^3 e \left (3 A e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+2 B \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )\right )+b^4 \left (A e \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+B \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )\right )}{1260 e^6 (d+e x)^{10}} \]
-1/1260*(14*a^4*e^4*(9*A*e + B*(d + 10*e*x)) + 14*a^3*b*e^3*(4*A*e*(d + 10 *e*x) + B*(d^2 + 10*d*e*x + 45*e^2*x^2)) + 3*a^2*b^2*e^2*(7*A*e*(d^2 + 10* d*e*x + 45*e^2*x^2) + 3*B*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3)) + 2*a*b^3*e*(3*A*e*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 2*B* (d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4)) + b^4*( A*e*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4) + B* (d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252 *e^5*x^5)))/(e^6*(d + e*x)^10)
Time = 0.44 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1184, 27, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2 (A+B x)}{(d+e x)^{11}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^4 (a+b x)^4 (A+B x)}{(d+e x)^{11}}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^4 (A+B x)}{(d+e x)^{11}}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^3 (4 a B e+A b e-5 b B d)}{e^5 (d+e x)^7}-\frac {2 b^2 (b d-a e) (3 a B e+2 A b e-5 b B d)}{e^5 (d+e x)^8}+\frac {2 b (b d-a e)^2 (2 a B e+3 A b e-5 b B d)}{e^5 (d+e x)^9}+\frac {(a e-b d)^3 (a B e+4 A b e-5 b B d)}{e^5 (d+e x)^{10}}+\frac {(a e-b d)^4 (A e-B d)}{e^5 (d+e x)^{11}}+\frac {b^4 B}{e^5 (d+e x)^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^3 (-4 a B e-A b e+5 b B d)}{6 e^6 (d+e x)^6}-\frac {2 b^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{7 e^6 (d+e x)^7}+\frac {b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{4 e^6 (d+e x)^8}-\frac {(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{9 e^6 (d+e x)^9}+\frac {(b d-a e)^4 (B d-A e)}{10 e^6 (d+e x)^{10}}-\frac {b^4 B}{5 e^6 (d+e x)^5}\) |
((b*d - a*e)^4*(B*d - A*e))/(10*e^6*(d + e*x)^10) - ((b*d - a*e)^3*(5*b*B* d - 4*A*b*e - a*B*e))/(9*e^6*(d + e*x)^9) + (b*(b*d - a*e)^2*(5*b*B*d - 3* A*b*e - 2*a*B*e))/(4*e^6*(d + e*x)^8) - (2*b^2*(b*d - a*e)*(5*b*B*d - 2*A* b*e - 3*a*B*e))/(7*e^6*(d + e*x)^7) + (b^3*(5*b*B*d - A*b*e - 4*a*B*e))/(6 *e^6*(d + e*x)^6) - (b^4*B)/(5*e^6*(d + e*x)^5)
3.17.91.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]))))
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(402\) vs. \(2(194)=388\).
Time = 0.25 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.96
| method | result | size |
| risch | \(\frac {-\frac {B \,b^{4} x^{5}}{5 e}-\frac {b^{3} \left (A b e +4 B a e +B b d \right ) x^{4}}{6 e^{2}}-\frac {2 b^{2} \left (6 A a b \,e^{2}+A \,b^{2} d e +9 a^{2} B \,e^{2}+4 B a b d e +B \,b^{2} d^{2}\right ) x^{3}}{21 e^{3}}-\frac {b \left (21 A \,a^{2} b \,e^{3}+6 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +14 B \,e^{3} a^{3}+9 B \,a^{2} b d \,e^{2}+4 B a \,b^{2} d^{2} e +B \,b^{3} d^{3}\right ) x^{2}}{28 e^{4}}-\frac {\left (56 A \,a^{3} b \,e^{4}+21 A \,a^{2} b^{2} d \,e^{3}+6 A a \,b^{3} d^{2} e^{2}+A \,b^{4} d^{3} e +14 B \,a^{4} e^{4}+14 B \,a^{3} b d \,e^{3}+9 B \,a^{2} b^{2} d^{2} e^{2}+4 B a \,b^{3} d^{3} e +b^{4} B \,d^{4}\right ) x}{126 e^{5}}-\frac {126 A \,a^{4} e^{5}+56 A \,a^{3} b d \,e^{4}+21 A \,a^{2} b^{2} d^{2} e^{3}+6 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e +14 B \,a^{4} d \,e^{4}+14 B \,a^{3} b \,d^{2} e^{3}+9 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e +b^{4} B \,d^{5}}{1260 e^{6}}}{\left (e x +d \right )^{10}}\) | \(403\) |
| default | \(-\frac {b^{3} \left (A b e +4 B a e -5 B b d \right )}{6 e^{6} \left (e x +d \right )^{6}}-\frac {b^{4} B}{5 e^{6} \left (e x +d \right )^{5}}-\frac {b \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +2 B \,e^{3} a^{3}-9 B \,a^{2} b d \,e^{2}+12 B a \,b^{2} d^{2} e -5 B \,b^{3} d^{3}\right )}{4 e^{6} \left (e x +d \right )^{8}}-\frac {4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +B \,a^{4} e^{4}-8 B \,a^{3} b d \,e^{3}+18 B \,a^{2} b^{2} d^{2} e^{2}-16 B a \,b^{3} d^{3} e +5 b^{4} B \,d^{4}}{9 e^{6} \left (e x +d \right )^{9}}-\frac {A \,a^{4} e^{5}-4 A \,a^{3} b d \,e^{4}+6 A \,a^{2} b^{2} d^{2} e^{3}-4 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e -B \,a^{4} d \,e^{4}+4 B \,a^{3} b \,d^{2} e^{3}-6 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e -b^{4} B \,d^{5}}{10 e^{6} \left (e x +d \right )^{10}}-\frac {2 b^{2} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +3 a^{2} B \,e^{2}-8 B a b d e +5 B \,b^{2} d^{2}\right )}{7 e^{6} \left (e x +d \right )^{7}}\) | \(430\) |
| norman | \(\frac {-\frac {B \,b^{4} x^{5}}{5 e}-\frac {\left (A \,b^{4} e^{5}+4 B a \,b^{3} e^{5}+B \,b^{4} d \,e^{4}\right ) x^{4}}{6 e^{6}}-\frac {2 \left (6 A a \,b^{3} e^{6}+A \,b^{4} d \,e^{5}+9 B \,a^{2} b^{2} e^{6}+4 B a \,b^{3} d \,e^{5}+b^{4} B \,d^{2} e^{4}\right ) x^{3}}{21 e^{7}}-\frac {\left (21 A \,a^{2} b^{2} e^{7}+6 A a \,b^{3} d \,e^{6}+A \,b^{4} d^{2} e^{5}+14 B \,a^{3} b \,e^{7}+9 B \,a^{2} b^{2} d \,e^{6}+4 B a \,b^{3} d^{2} e^{5}+b^{4} B \,d^{3} e^{4}\right ) x^{2}}{28 e^{8}}-\frac {\left (56 A \,a^{3} b \,e^{8}+21 A \,a^{2} b^{2} d \,e^{7}+6 A a \,b^{3} d^{2} e^{6}+A \,b^{4} d^{3} e^{5}+14 B \,a^{4} e^{8}+14 B \,a^{3} b d \,e^{7}+9 B \,a^{2} b^{2} d^{2} e^{6}+4 B a \,b^{3} d^{3} e^{5}+b^{4} B \,d^{4} e^{4}\right ) x}{126 e^{9}}-\frac {126 A \,a^{4} e^{9}+56 A \,a^{3} b d \,e^{8}+21 A \,a^{2} b^{2} d^{2} e^{7}+6 A a \,b^{3} d^{3} e^{6}+A \,b^{4} d^{4} e^{5}+14 B \,a^{4} d \,e^{8}+14 B \,a^{3} b \,d^{2} e^{7}+9 B \,a^{2} b^{2} d^{3} e^{6}+4 B a \,b^{3} d^{4} e^{5}+B \,b^{4} d^{5} e^{4}}{1260 e^{10}}}{\left (e x +d \right )^{10}}\) | \(450\) |
| gosper | \(-\frac {252 B \,x^{5} b^{4} e^{5}+210 A \,b^{4} e^{5} x^{4}+840 B \,x^{4} a \,b^{3} e^{5}+210 B \,x^{4} b^{4} d \,e^{4}+720 A \,x^{3} a \,b^{3} e^{5}+120 A \,x^{3} b^{4} d \,e^{4}+1080 B \,x^{3} a^{2} b^{2} e^{5}+480 B \,x^{3} a \,b^{3} d \,e^{4}+120 B \,x^{3} b^{4} d^{2} e^{3}+945 A \,x^{2} a^{2} b^{2} e^{5}+270 A \,x^{2} a \,b^{3} d \,e^{4}+45 A \,x^{2} b^{4} d^{2} e^{3}+630 B \,x^{2} a^{3} b \,e^{5}+405 B \,x^{2} a^{2} b^{2} d \,e^{4}+180 B \,x^{2} a \,b^{3} d^{2} e^{3}+45 B \,x^{2} b^{4} d^{3} e^{2}+560 A x \,a^{3} b \,e^{5}+210 A x \,a^{2} b^{2} d \,e^{4}+60 A x a \,b^{3} d^{2} e^{3}+10 A x \,b^{4} d^{3} e^{2}+140 B x \,a^{4} e^{5}+140 B x \,a^{3} b d \,e^{4}+90 B x \,a^{2} b^{2} d^{2} e^{3}+40 B x a \,b^{3} d^{3} e^{2}+10 B x \,b^{4} d^{4} e +126 A \,a^{4} e^{5}+56 A \,a^{3} b d \,e^{4}+21 A \,a^{2} b^{2} d^{2} e^{3}+6 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e +14 B \,a^{4} d \,e^{4}+14 B \,a^{3} b \,d^{2} e^{3}+9 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e +b^{4} B \,d^{5}}{1260 e^{6} \left (e x +d \right )^{10}}\) | \(467\) |
| parallelrisch | \(-\frac {252 b^{4} B \,x^{5} e^{9}+210 A \,b^{4} e^{9} x^{4}+840 B a \,b^{3} e^{9} x^{4}+210 B \,b^{4} d \,e^{8} x^{4}+720 A a \,b^{3} e^{9} x^{3}+120 A \,b^{4} d \,e^{8} x^{3}+1080 B \,a^{2} b^{2} e^{9} x^{3}+480 B a \,b^{3} d \,e^{8} x^{3}+120 B \,b^{4} d^{2} e^{7} x^{3}+945 A \,a^{2} b^{2} e^{9} x^{2}+270 A a \,b^{3} d \,e^{8} x^{2}+45 A \,b^{4} d^{2} e^{7} x^{2}+630 B \,a^{3} b \,e^{9} x^{2}+405 B \,a^{2} b^{2} d \,e^{8} x^{2}+180 B a \,b^{3} d^{2} e^{7} x^{2}+45 B \,b^{4} d^{3} e^{6} x^{2}+560 A \,a^{3} b \,e^{9} x +210 A \,a^{2} b^{2} d \,e^{8} x +60 A a \,b^{3} d^{2} e^{7} x +10 A \,b^{4} d^{3} e^{6} x +140 B \,a^{4} e^{9} x +140 B \,a^{3} b d \,e^{8} x +90 B \,a^{2} b^{2} d^{2} e^{7} x +40 B a \,b^{3} d^{3} e^{6} x +10 B \,b^{4} d^{4} e^{5} x +126 A \,a^{4} e^{9}+56 A \,a^{3} b d \,e^{8}+21 A \,a^{2} b^{2} d^{2} e^{7}+6 A a \,b^{3} d^{3} e^{6}+A \,b^{4} d^{4} e^{5}+14 B \,a^{4} d \,e^{8}+14 B \,a^{3} b \,d^{2} e^{7}+9 B \,a^{2} b^{2} d^{3} e^{6}+4 B a \,b^{3} d^{4} e^{5}+B \,b^{4} d^{5} e^{4}}{1260 e^{10} \left (e x +d \right )^{10}}\) | \(476\) |
(-1/5*B*b^4/e*x^5-1/6*b^3/e^2*(A*b*e+4*B*a*e+B*b*d)*x^4-2/21*b^2/e^3*(6*A* a*b*e^2+A*b^2*d*e+9*B*a^2*e^2+4*B*a*b*d*e+B*b^2*d^2)*x^3-1/28*b/e^4*(21*A* a^2*b*e^3+6*A*a*b^2*d*e^2+A*b^3*d^2*e+14*B*a^3*e^3+9*B*a^2*b*d*e^2+4*B*a*b ^2*d^2*e+B*b^3*d^3)*x^2-1/126/e^5*(56*A*a^3*b*e^4+21*A*a^2*b^2*d*e^3+6*A*a *b^3*d^2*e^2+A*b^4*d^3*e+14*B*a^4*e^4+14*B*a^3*b*d*e^3+9*B*a^2*b^2*d^2*e^2 +4*B*a*b^3*d^3*e+B*b^4*d^4)*x-1/1260/e^6*(126*A*a^4*e^5+56*A*a^3*b*d*e^4+2 1*A*a^2*b^2*d^2*e^3+6*A*a*b^3*d^3*e^2+A*b^4*d^4*e+14*B*a^4*d*e^4+14*B*a^3* b*d^2*e^3+9*B*a^2*b^2*d^3*e^2+4*B*a*b^3*d^4*e+B*b^4*d^5))/(e*x+d)^10
Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (194) = 388\).
Time = 0.45 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.43 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx=-\frac {252 \, B b^{4} e^{5} x^{5} + B b^{4} d^{5} + 126 \, A a^{4} e^{5} + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 7 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 14 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 210 \, {\left (B b^{4} d e^{4} + {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 120 \, {\left (B b^{4} d^{2} e^{3} + {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 45 \, {\left (B b^{4} d^{3} e^{2} + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 7 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 10 \, {\left (B b^{4} d^{4} e + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 7 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 14 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{1260 \, {\left (e^{16} x^{10} + 10 \, d e^{15} x^{9} + 45 \, d^{2} e^{14} x^{8} + 120 \, d^{3} e^{13} x^{7} + 210 \, d^{4} e^{12} x^{6} + 252 \, d^{5} e^{11} x^{5} + 210 \, d^{6} e^{10} x^{4} + 120 \, d^{7} e^{9} x^{3} + 45 \, d^{8} e^{8} x^{2} + 10 \, d^{9} e^{7} x + d^{10} e^{6}\right )}} \]
-1/1260*(252*B*b^4*e^5*x^5 + B*b^4*d^5 + 126*A*a^4*e^5 + (4*B*a*b^3 + A*b^ 4)*d^4*e + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 7*(2*B*a^3*b + 3*A*a^2*b^ 2)*d^2*e^3 + 14*(B*a^4 + 4*A*a^3*b)*d*e^4 + 210*(B*b^4*d*e^4 + (4*B*a*b^3 + A*b^4)*e^5)*x^4 + 120*(B*b^4*d^2*e^3 + (4*B*a*b^3 + A*b^4)*d*e^4 + 3*(3* B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 45*(B*b^4*d^3*e^2 + (4*B*a*b^3 + A*b^4)* d^2*e^3 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 7*(2*B*a^3*b + 3*A*a^2*b^2)* e^5)*x^2 + 10*(B*b^4*d^4*e + (4*B*a*b^3 + A*b^4)*d^3*e^2 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 7*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 14*(B*a^4 + 4*A *a^3*b)*e^5)*x)/(e^16*x^10 + 10*d*e^15*x^9 + 45*d^2*e^14*x^8 + 120*d^3*e^1 3*x^7 + 210*d^4*e^12*x^6 + 252*d^5*e^11*x^5 + 210*d^6*e^10*x^4 + 120*d^7*e ^9*x^3 + 45*d^8*e^8*x^2 + 10*d^9*e^7*x + d^10*e^6)
Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (194) = 388\).
Time = 0.22 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.43 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx=-\frac {252 \, B b^{4} e^{5} x^{5} + B b^{4} d^{5} + 126 \, A a^{4} e^{5} + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 7 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 14 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 210 \, {\left (B b^{4} d e^{4} + {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 120 \, {\left (B b^{4} d^{2} e^{3} + {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 45 \, {\left (B b^{4} d^{3} e^{2} + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 7 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 10 \, {\left (B b^{4} d^{4} e + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 7 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 14 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{1260 \, {\left (e^{16} x^{10} + 10 \, d e^{15} x^{9} + 45 \, d^{2} e^{14} x^{8} + 120 \, d^{3} e^{13} x^{7} + 210 \, d^{4} e^{12} x^{6} + 252 \, d^{5} e^{11} x^{5} + 210 \, d^{6} e^{10} x^{4} + 120 \, d^{7} e^{9} x^{3} + 45 \, d^{8} e^{8} x^{2} + 10 \, d^{9} e^{7} x + d^{10} e^{6}\right )}} \]
-1/1260*(252*B*b^4*e^5*x^5 + B*b^4*d^5 + 126*A*a^4*e^5 + (4*B*a*b^3 + A*b^ 4)*d^4*e + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 7*(2*B*a^3*b + 3*A*a^2*b^ 2)*d^2*e^3 + 14*(B*a^4 + 4*A*a^3*b)*d*e^4 + 210*(B*b^4*d*e^4 + (4*B*a*b^3 + A*b^4)*e^5)*x^4 + 120*(B*b^4*d^2*e^3 + (4*B*a*b^3 + A*b^4)*d*e^4 + 3*(3* B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 45*(B*b^4*d^3*e^2 + (4*B*a*b^3 + A*b^4)* d^2*e^3 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 7*(2*B*a^3*b + 3*A*a^2*b^2)* e^5)*x^2 + 10*(B*b^4*d^4*e + (4*B*a*b^3 + A*b^4)*d^3*e^2 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 7*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 14*(B*a^4 + 4*A *a^3*b)*e^5)*x)/(e^16*x^10 + 10*d*e^15*x^9 + 45*d^2*e^14*x^8 + 120*d^3*e^1 3*x^7 + 210*d^4*e^12*x^6 + 252*d^5*e^11*x^5 + 210*d^6*e^10*x^4 + 120*d^7*e ^9*x^3 + 45*d^8*e^8*x^2 + 10*d^9*e^7*x + d^10*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (194) = 388\).
Time = 0.27 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.26 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx=-\frac {252 \, B b^{4} e^{5} x^{5} + 210 \, B b^{4} d e^{4} x^{4} + 840 \, B a b^{3} e^{5} x^{4} + 210 \, A b^{4} e^{5} x^{4} + 120 \, B b^{4} d^{2} e^{3} x^{3} + 480 \, B a b^{3} d e^{4} x^{3} + 120 \, A b^{4} d e^{4} x^{3} + 1080 \, B a^{2} b^{2} e^{5} x^{3} + 720 \, A a b^{3} e^{5} x^{3} + 45 \, B b^{4} d^{3} e^{2} x^{2} + 180 \, B a b^{3} d^{2} e^{3} x^{2} + 45 \, A b^{4} d^{2} e^{3} x^{2} + 405 \, B a^{2} b^{2} d e^{4} x^{2} + 270 \, A a b^{3} d e^{4} x^{2} + 630 \, B a^{3} b e^{5} x^{2} + 945 \, A a^{2} b^{2} e^{5} x^{2} + 10 \, B b^{4} d^{4} e x + 40 \, B a b^{3} d^{3} e^{2} x + 10 \, A b^{4} d^{3} e^{2} x + 90 \, B a^{2} b^{2} d^{2} e^{3} x + 60 \, A a b^{3} d^{2} e^{3} x + 140 \, B a^{3} b d e^{4} x + 210 \, A a^{2} b^{2} d e^{4} x + 140 \, B a^{4} e^{5} x + 560 \, A a^{3} b e^{5} x + B b^{4} d^{5} + 4 \, B a b^{3} d^{4} e + A b^{4} d^{4} e + 9 \, B a^{2} b^{2} d^{3} e^{2} + 6 \, A a b^{3} d^{3} e^{2} + 14 \, B a^{3} b d^{2} e^{3} + 21 \, A a^{2} b^{2} d^{2} e^{3} + 14 \, B a^{4} d e^{4} + 56 \, A a^{3} b d e^{4} + 126 \, A a^{4} e^{5}}{1260 \, {\left (e x + d\right )}^{10} e^{6}} \]
-1/1260*(252*B*b^4*e^5*x^5 + 210*B*b^4*d*e^4*x^4 + 840*B*a*b^3*e^5*x^4 + 2 10*A*b^4*e^5*x^4 + 120*B*b^4*d^2*e^3*x^3 + 480*B*a*b^3*d*e^4*x^3 + 120*A*b ^4*d*e^4*x^3 + 1080*B*a^2*b^2*e^5*x^3 + 720*A*a*b^3*e^5*x^3 + 45*B*b^4*d^3 *e^2*x^2 + 180*B*a*b^3*d^2*e^3*x^2 + 45*A*b^4*d^2*e^3*x^2 + 405*B*a^2*b^2* d*e^4*x^2 + 270*A*a*b^3*d*e^4*x^2 + 630*B*a^3*b*e^5*x^2 + 945*A*a^2*b^2*e^ 5*x^2 + 10*B*b^4*d^4*e*x + 40*B*a*b^3*d^3*e^2*x + 10*A*b^4*d^3*e^2*x + 90* B*a^2*b^2*d^2*e^3*x + 60*A*a*b^3*d^2*e^3*x + 140*B*a^3*b*d*e^4*x + 210*A*a ^2*b^2*d*e^4*x + 140*B*a^4*e^5*x + 560*A*a^3*b*e^5*x + B*b^4*d^5 + 4*B*a*b ^3*d^4*e + A*b^4*d^4*e + 9*B*a^2*b^2*d^3*e^2 + 6*A*a*b^3*d^3*e^2 + 14*B*a^ 3*b*d^2*e^3 + 21*A*a^2*b^2*d^2*e^3 + 14*B*a^4*d*e^4 + 56*A*a^3*b*d*e^4 + 1 26*A*a^4*e^5)/((e*x + d)^10*e^6)
Time = 0.30 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.44 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx=-\frac {\frac {14\,B\,a^4\,d\,e^4+126\,A\,a^4\,e^5+14\,B\,a^3\,b\,d^2\,e^3+56\,A\,a^3\,b\,d\,e^4+9\,B\,a^2\,b^2\,d^3\,e^2+21\,A\,a^2\,b^2\,d^2\,e^3+4\,B\,a\,b^3\,d^4\,e+6\,A\,a\,b^3\,d^3\,e^2+B\,b^4\,d^5+A\,b^4\,d^4\,e}{1260\,e^6}+\frac {x\,\left (14\,B\,a^4\,e^4+14\,B\,a^3\,b\,d\,e^3+56\,A\,a^3\,b\,e^4+9\,B\,a^2\,b^2\,d^2\,e^2+21\,A\,a^2\,b^2\,d\,e^3+4\,B\,a\,b^3\,d^3\,e+6\,A\,a\,b^3\,d^2\,e^2+B\,b^4\,d^4+A\,b^4\,d^3\,e\right )}{126\,e^5}+\frac {b^3\,x^4\,\left (A\,b\,e+4\,B\,a\,e+B\,b\,d\right )}{6\,e^2}+\frac {b\,x^2\,\left (14\,B\,a^3\,e^3+9\,B\,a^2\,b\,d\,e^2+21\,A\,a^2\,b\,e^3+4\,B\,a\,b^2\,d^2\,e+6\,A\,a\,b^2\,d\,e^2+B\,b^3\,d^3+A\,b^3\,d^2\,e\right )}{28\,e^4}+\frac {2\,b^2\,x^3\,\left (9\,B\,a^2\,e^2+4\,B\,a\,b\,d\,e+6\,A\,a\,b\,e^2+B\,b^2\,d^2+A\,b^2\,d\,e\right )}{21\,e^3}+\frac {B\,b^4\,x^5}{5\,e}}{d^{10}+10\,d^9\,e\,x+45\,d^8\,e^2\,x^2+120\,d^7\,e^3\,x^3+210\,d^6\,e^4\,x^4+252\,d^5\,e^5\,x^5+210\,d^4\,e^6\,x^6+120\,d^3\,e^7\,x^7+45\,d^2\,e^8\,x^8+10\,d\,e^9\,x^9+e^{10}\,x^{10}} \]
-((126*A*a^4*e^5 + B*b^4*d^5 + A*b^4*d^4*e + 14*B*a^4*d*e^4 + 6*A*a*b^3*d^ 3*e^2 + 14*B*a^3*b*d^2*e^3 + 21*A*a^2*b^2*d^2*e^3 + 9*B*a^2*b^2*d^3*e^2 + 56*A*a^3*b*d*e^4 + 4*B*a*b^3*d^4*e)/(1260*e^6) + (x*(14*B*a^4*e^4 + B*b^4* d^4 + 56*A*a^3*b*e^4 + A*b^4*d^3*e + 6*A*a*b^3*d^2*e^2 + 21*A*a^2*b^2*d*e^ 3 + 9*B*a^2*b^2*d^2*e^2 + 4*B*a*b^3*d^3*e + 14*B*a^3*b*d*e^3))/(126*e^5) + (b^3*x^4*(A*b*e + 4*B*a*e + B*b*d))/(6*e^2) + (b*x^2*(14*B*a^3*e^3 + B*b^ 3*d^3 + 21*A*a^2*b*e^3 + A*b^3*d^2*e + 6*A*a*b^2*d*e^2 + 4*B*a*b^2*d^2*e + 9*B*a^2*b*d*e^2))/(28*e^4) + (2*b^2*x^3*(9*B*a^2*e^2 + B*b^2*d^2 + 6*A*a* b*e^2 + A*b^2*d*e + 4*B*a*b*d*e))/(21*e^3) + (B*b^4*x^5)/(5*e))/(d^10 + e^ 10*x^10 + 10*d*e^9*x^9 + 45*d^8*e^2*x^2 + 120*d^7*e^3*x^3 + 210*d^6*e^4*x^ 4 + 252*d^5*e^5*x^5 + 210*d^4*e^6*x^6 + 120*d^3*e^7*x^7 + 45*d^2*e^8*x^8 + 10*d^9*e*x)