Integrand size = 31, antiderivative size = 206 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{10}} \, dx=\frac {(b d-a e)^4 (B d-A e)}{9 e^6 (d+e x)^9}-\frac {(b d-a e)^3 (5 b B d-4 A b e-a B e)}{8 e^6 (d+e x)^8}+\frac {2 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e)}{7 e^6 (d+e x)^7}-\frac {b^2 (b d-a e) (5 b B d-2 A b e-3 a B e)}{3 e^6 (d+e x)^6}+\frac {b^3 (5 b B d-A b e-4 a B e)}{5 e^6 (d+e x)^5}-\frac {b^4 B}{4 e^6 (d+e x)^4} \]
1/9*(-a*e+b*d)^4*(-A*e+B*d)/e^6/(e*x+d)^9-1/8*(-a*e+b*d)^3*(-4*A*b*e-B*a*e +5*B*b*d)/e^6/(e*x+d)^8+2/7*b*(-a*e+b*d)^2*(-3*A*b*e-2*B*a*e+5*B*b*d)/e^6/ (e*x+d)^7-1/3*b^2*(-a*e+b*d)*(-2*A*b*e-3*B*a*e+5*B*b*d)/e^6/(e*x+d)^6+1/5* b^3*(-A*b*e-4*B*a*e+5*B*b*d)/e^6/(e*x+d)^5-1/4*b^4*B/e^6/(e*x+d)^4
Time = 0.11 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.56 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{10}} \, dx=-\frac {35 a^4 e^4 (8 A e+B (d+9 e x))+20 a^3 b e^3 \left (7 A e (d+9 e x)+2 B \left (d^2+9 d e x+36 e^2 x^2\right )\right )+30 a^2 b^2 e^2 \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+4 a b^3 e \left (5 A e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+b^4 \left (4 A e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 B \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )}{2520 e^6 (d+e x)^9} \]
-1/2520*(35*a^4*e^4*(8*A*e + B*(d + 9*e*x)) + 20*a^3*b*e^3*(7*A*e*(d + 9*e *x) + 2*B*(d^2 + 9*d*e*x + 36*e^2*x^2)) + 30*a^2*b^2*e^2*(2*A*e*(d^2 + 9*d *e*x + 36*e^2*x^2) + B*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3)) + 4* a*b^3*e*(5*A*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 4*B*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4)) + b^4*(4*A*e*(d^ 4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 5*B*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5) ))/(e^6*(d + e*x)^9)
Time = 0.47 (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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2 (A+B x)}{(d+e x)^{10}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^4 (a+b x)^4 (A+B x)}{(d+e x)^{10}}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^4 (A+B x)}{(d+e x)^{10}}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^3 (4 a B e+A b e-5 b B d)}{e^5 (d+e x)^6}-\frac {2 b^2 (b d-a e) (3 a B e+2 A b e-5 b B d)}{e^5 (d+e x)^7}+\frac {2 b (b d-a e)^2 (2 a B e+3 A b e-5 b B d)}{e^5 (d+e x)^8}+\frac {(a e-b d)^3 (a B e+4 A b e-5 b B d)}{e^5 (d+e x)^9}+\frac {(a e-b d)^4 (A e-B d)}{e^5 (d+e x)^{10}}+\frac {b^4 B}{e^5 (d+e x)^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^3 (-4 a B e-A b e+5 b B d)}{5 e^6 (d+e x)^5}-\frac {b^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{3 e^6 (d+e x)^6}+\frac {2 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{7 e^6 (d+e x)^7}-\frac {(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{8 e^6 (d+e x)^8}+\frac {(b d-a e)^4 (B d-A e)}{9 e^6 (d+e x)^9}-\frac {b^4 B}{4 e^6 (d+e x)^4}\) |
((b*d - a*e)^4*(B*d - A*e))/(9*e^6*(d + e*x)^9) - ((b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e))/(8*e^6*(d + e*x)^8) + (2*b*(b*d - a*e)^2*(5*b*B*d - 3* A*b*e - 2*a*B*e))/(7*e^6*(d + e*x)^7) - (b^2*(b*d - a*e)*(5*b*B*d - 2*A*b* e - 3*a*B*e))/(3*e^6*(d + e*x)^6) + (b^3*(5*b*B*d - A*b*e - 4*a*B*e))/(5*e ^6*(d + e*x)^5) - (b^4*B)/(4*e^6*(d + e*x)^4)
3.17.90.3.1 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. \(412\) vs. \(2(194)=388\).
Time = 0.22 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.00
| method | result | size |
| risch | \(\frac {-\frac {B \,b^{4} x^{5}}{4 e}-\frac {b^{3} \left (4 A b e +16 B a e +5 B b d \right ) x^{4}}{20 e^{2}}-\frac {b^{2} \left (20 A a b \,e^{2}+4 A \,b^{2} d e +30 a^{2} B \,e^{2}+16 B a b d e +5 B \,b^{2} d^{2}\right ) x^{3}}{30 e^{3}}-\frac {b \left (60 A \,a^{2} b \,e^{3}+20 A a \,b^{2} d \,e^{2}+4 A \,b^{3} d^{2} e +40 B \,e^{3} a^{3}+30 B \,a^{2} b d \,e^{2}+16 B a \,b^{2} d^{2} e +5 B \,b^{3} d^{3}\right ) x^{2}}{70 e^{4}}-\frac {\left (140 A \,a^{3} b \,e^{4}+60 A \,a^{2} b^{2} d \,e^{3}+20 A a \,b^{3} d^{2} e^{2}+4 A \,b^{4} d^{3} e +35 B \,a^{4} e^{4}+40 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}+16 B a \,b^{3} d^{3} e +5 b^{4} B \,d^{4}\right ) x}{280 e^{5}}-\frac {280 A \,a^{4} e^{5}+140 A \,a^{3} b d \,e^{4}+60 A \,a^{2} b^{2} d^{2} e^{3}+20 A a \,b^{3} d^{3} e^{2}+4 A \,b^{4} d^{4} e +35 B \,a^{4} d \,e^{4}+40 B \,a^{3} b \,d^{2} e^{3}+30 B \,a^{2} b^{2} d^{3} e^{2}+16 B a \,b^{3} d^{4} e +5 b^{4} B \,d^{5}}{2520 e^{6}}}{\left (e x +d \right )^{9}}\) | \(413\) |
| default | \(-\frac {b^{2} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +3 a^{2} B \,e^{2}-8 B a b d e +5 B \,b^{2} d^{2}\right )}{3 e^{6} \left (e x +d \right )^{6}}-\frac {b^{3} \left (A b e +4 B a e -5 B b d \right )}{5 e^{6} \left (e x +d \right )^{5}}-\frac {4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +B \,a^{4} e^{4}-8 B \,a^{3} b d \,e^{3}+18 B \,a^{2} b^{2} d^{2} e^{2}-16 B a \,b^{3} d^{3} e +5 b^{4} B \,d^{4}}{8 e^{6} \left (e x +d \right )^{8}}-\frac {A \,a^{4} e^{5}-4 A \,a^{3} b d \,e^{4}+6 A \,a^{2} b^{2} d^{2} e^{3}-4 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e -B \,a^{4} d \,e^{4}+4 B \,a^{3} b \,d^{2} e^{3}-6 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e -b^{4} B \,d^{5}}{9 e^{6} \left (e x +d \right )^{9}}-\frac {b^{4} B}{4 e^{6} \left (e x +d \right )^{4}}-\frac {2 b \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +2 B \,e^{3} a^{3}-9 B \,a^{2} b d \,e^{2}+12 B a \,b^{2} d^{2} e -5 B \,b^{3} d^{3}\right )}{7 e^{6} \left (e x +d \right )^{7}}\) | \(430\) |
| norman | \(\frac {-\frac {B \,b^{4} x^{5}}{4 e}-\frac {\left (4 A \,b^{4} e^{4}+16 B a \,b^{3} e^{4}+5 b^{4} B \,e^{3} d \right ) x^{4}}{20 e^{5}}-\frac {\left (20 A a \,b^{3} e^{5}+4 A \,b^{4} d \,e^{4}+30 B \,a^{2} b^{2} e^{5}+16 B a \,b^{3} d \,e^{4}+5 b^{4} B \,d^{2} e^{3}\right ) x^{3}}{30 e^{6}}-\frac {\left (60 A \,a^{2} b^{2} e^{6}+20 A a \,b^{3} d \,e^{5}+4 A \,b^{4} d^{2} e^{4}+40 B \,a^{3} b \,e^{6}+30 B \,a^{2} b^{2} d \,e^{5}+16 B a \,b^{3} d^{2} e^{4}+5 b^{4} B \,d^{3} e^{3}\right ) x^{2}}{70 e^{7}}-\frac {\left (140 A \,a^{3} b \,e^{7}+60 A \,a^{2} b^{2} d \,e^{6}+20 A a \,b^{3} d^{2} e^{5}+4 A \,b^{4} d^{3} e^{4}+35 B \,a^{4} e^{7}+40 B \,a^{3} b d \,e^{6}+30 B \,a^{2} b^{2} d^{2} e^{5}+16 B a \,b^{3} d^{3} e^{4}+5 b^{4} B \,d^{4} e^{3}\right ) x}{280 e^{8}}-\frac {280 A \,a^{4} e^{8}+140 A \,a^{3} b d \,e^{7}+60 A \,a^{2} b^{2} d^{2} e^{6}+20 A a \,b^{3} d^{3} e^{5}+4 A \,b^{4} d^{4} e^{4}+35 B \,a^{4} d \,e^{7}+40 B \,a^{3} b \,d^{2} e^{6}+30 B \,a^{2} b^{2} d^{3} e^{5}+16 B a \,b^{3} d^{4} e^{4}+5 B \,b^{4} d^{5} e^{3}}{2520 e^{9}}}{\left (e x +d \right )^{9}}\) | \(460\) |
| gosper | \(-\frac {630 B \,x^{5} b^{4} e^{5}+504 A \,b^{4} e^{5} x^{4}+2016 B \,x^{4} a \,b^{3} e^{5}+630 B \,x^{4} b^{4} d \,e^{4}+1680 A \,x^{3} a \,b^{3} e^{5}+336 A \,x^{3} b^{4} d \,e^{4}+2520 B \,x^{3} a^{2} b^{2} e^{5}+1344 B \,x^{3} a \,b^{3} d \,e^{4}+420 B \,x^{3} b^{4} d^{2} e^{3}+2160 A \,x^{2} a^{2} b^{2} e^{5}+720 A \,x^{2} a \,b^{3} d \,e^{4}+144 A \,x^{2} b^{4} d^{2} e^{3}+1440 B \,x^{2} a^{3} b \,e^{5}+1080 B \,x^{2} a^{2} b^{2} d \,e^{4}+576 B \,x^{2} a \,b^{3} d^{2} e^{3}+180 B \,x^{2} b^{4} d^{3} e^{2}+1260 A x \,a^{3} b \,e^{5}+540 A x \,a^{2} b^{2} d \,e^{4}+180 A x a \,b^{3} d^{2} e^{3}+36 A x \,b^{4} d^{3} e^{2}+315 B x \,a^{4} e^{5}+360 B x \,a^{3} b d \,e^{4}+270 B x \,a^{2} b^{2} d^{2} e^{3}+144 B x a \,b^{3} d^{3} e^{2}+45 B x \,b^{4} d^{4} e +280 A \,a^{4} e^{5}+140 A \,a^{3} b d \,e^{4}+60 A \,a^{2} b^{2} d^{2} e^{3}+20 A a \,b^{3} d^{3} e^{2}+4 A \,b^{4} d^{4} e +35 B \,a^{4} d \,e^{4}+40 B \,a^{3} b \,d^{2} e^{3}+30 B \,a^{2} b^{2} d^{3} e^{2}+16 B a \,b^{3} d^{4} e +5 b^{4} B \,d^{5}}{2520 e^{6} \left (e x +d \right )^{9}}\) | \(469\) |
| parallelrisch | \(-\frac {630 b^{4} B \,x^{5} e^{8}+504 A \,b^{4} e^{8} x^{4}+2016 B a \,b^{3} e^{8} x^{4}+630 B \,b^{4} d \,e^{7} x^{4}+1680 A a \,b^{3} e^{8} x^{3}+336 A \,b^{4} d \,e^{7} x^{3}+2520 B \,a^{2} b^{2} e^{8} x^{3}+1344 B a \,b^{3} d \,e^{7} x^{3}+420 B \,b^{4} d^{2} e^{6} x^{3}+2160 A \,a^{2} b^{2} e^{8} x^{2}+720 A a \,b^{3} d \,e^{7} x^{2}+144 A \,b^{4} d^{2} e^{6} x^{2}+1440 B \,a^{3} b \,e^{8} x^{2}+1080 B \,a^{2} b^{2} d \,e^{7} x^{2}+576 B a \,b^{3} d^{2} e^{6} x^{2}+180 B \,b^{4} d^{3} e^{5} x^{2}+1260 A \,a^{3} b \,e^{8} x +540 A \,a^{2} b^{2} d \,e^{7} x +180 A a \,b^{3} d^{2} e^{6} x +36 A \,b^{4} d^{3} e^{5} x +315 B \,a^{4} e^{8} x +360 B \,a^{3} b d \,e^{7} x +270 B \,a^{2} b^{2} d^{2} e^{6} x +144 B a \,b^{3} d^{3} e^{5} x +45 B \,b^{4} d^{4} e^{4} x +280 A \,a^{4} e^{8}+140 A \,a^{3} b d \,e^{7}+60 A \,a^{2} b^{2} d^{2} e^{6}+20 A a \,b^{3} d^{3} e^{5}+4 A \,b^{4} d^{4} e^{4}+35 B \,a^{4} d \,e^{7}+40 B \,a^{3} b \,d^{2} e^{6}+30 B \,a^{2} b^{2} d^{3} e^{5}+16 B a \,b^{3} d^{4} e^{4}+5 B \,b^{4} d^{5} e^{3}}{2520 e^{9} \left (e x +d \right )^{9}}\) | \(478\) |
(-1/4*B*b^4/e*x^5-1/20*b^3/e^2*(4*A*b*e+16*B*a*e+5*B*b*d)*x^4-1/30*b^2/e^3 *(20*A*a*b*e^2+4*A*b^2*d*e+30*B*a^2*e^2+16*B*a*b*d*e+5*B*b^2*d^2)*x^3-1/70 *b/e^4*(60*A*a^2*b*e^3+20*A*a*b^2*d*e^2+4*A*b^3*d^2*e+40*B*a^3*e^3+30*B*a^ 2*b*d*e^2+16*B*a*b^2*d^2*e+5*B*b^3*d^3)*x^2-1/280/e^5*(140*A*a^3*b*e^4+60* A*a^2*b^2*d*e^3+20*A*a*b^3*d^2*e^2+4*A*b^4*d^3*e+35*B*a^4*e^4+40*B*a^3*b*d *e^3+30*B*a^2*b^2*d^2*e^2+16*B*a*b^3*d^3*e+5*B*b^4*d^4)*x-1/2520/e^6*(280* A*a^4*e^5+140*A*a^3*b*d*e^4+60*A*a^2*b^2*d^2*e^3+20*A*a*b^3*d^3*e^2+4*A*b^ 4*d^4*e+35*B*a^4*d*e^4+40*B*a^3*b*d^2*e^3+30*B*a^2*b^2*d^3*e^2+16*B*a*b^3* d^4*e+5*B*b^4*d^5))/(e*x+d)^9
Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (194) = 388\).
Time = 0.45 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.43 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{10}} \, dx=-\frac {630 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 280 \, A a^{4} e^{5} + 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 10 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 20 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 35 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 126 \, {\left (5 \, B b^{4} d e^{4} + 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 84 \, {\left (5 \, B b^{4} d^{2} e^{3} + 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 10 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 36 \, {\left (5 \, B b^{4} d^{3} e^{2} + 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 10 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 20 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 9 \, {\left (5 \, B b^{4} d^{4} e + 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 10 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 20 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 35 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{2520 \, {\left (e^{15} x^{9} + 9 \, d e^{14} x^{8} + 36 \, d^{2} e^{13} x^{7} + 84 \, d^{3} e^{12} x^{6} + 126 \, d^{4} e^{11} x^{5} + 126 \, d^{5} e^{10} x^{4} + 84 \, d^{6} e^{9} x^{3} + 36 \, d^{7} e^{8} x^{2} + 9 \, d^{8} e^{7} x + d^{9} e^{6}\right )}} \]
-1/2520*(630*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 280*A*a^4*e^5 + 4*(4*B*a*b^3 + A*b^4)*d^4*e + 10*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 20*(2*B*a^3*b + 3*A* a^2*b^2)*d^2*e^3 + 35*(B*a^4 + 4*A*a^3*b)*d*e^4 + 126*(5*B*b^4*d*e^4 + 4*( 4*B*a*b^3 + A*b^4)*e^5)*x^4 + 84*(5*B*b^4*d^2*e^3 + 4*(4*B*a*b^3 + A*b^4)* d*e^4 + 10*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 36*(5*B*b^4*d^3*e^2 + 4*(4 *B*a*b^3 + A*b^4)*d^2*e^3 + 10*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 20*(2*B*a ^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 9*(5*B*b^4*d^4*e + 4*(4*B*a*b^3 + A*b^4)*d^ 3*e^2 + 10*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 20*(2*B*a^3*b + 3*A*a^2*b^2 )*d*e^4 + 35*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^15*x^9 + 9*d*e^14*x^8 + 36*d^2 *e^13*x^7 + 84*d^3*e^12*x^6 + 126*d^4*e^11*x^5 + 126*d^5*e^10*x^4 + 84*d^6 *e^9*x^3 + 36*d^7*e^8*x^2 + 9*d^8*e^7*x + d^9*e^6)
Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{10}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (194) = 388\).
Time = 0.21 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.43 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{10}} \, dx=-\frac {630 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 280 \, A a^{4} e^{5} + 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 10 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 20 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 35 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 126 \, {\left (5 \, B b^{4} d e^{4} + 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 84 \, {\left (5 \, B b^{4} d^{2} e^{3} + 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 10 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 36 \, {\left (5 \, B b^{4} d^{3} e^{2} + 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 10 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 20 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 9 \, {\left (5 \, B b^{4} d^{4} e + 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 10 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 20 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 35 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{2520 \, {\left (e^{15} x^{9} + 9 \, d e^{14} x^{8} + 36 \, d^{2} e^{13} x^{7} + 84 \, d^{3} e^{12} x^{6} + 126 \, d^{4} e^{11} x^{5} + 126 \, d^{5} e^{10} x^{4} + 84 \, d^{6} e^{9} x^{3} + 36 \, d^{7} e^{8} x^{2} + 9 \, d^{8} e^{7} x + d^{9} e^{6}\right )}} \]
-1/2520*(630*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 280*A*a^4*e^5 + 4*(4*B*a*b^3 + A*b^4)*d^4*e + 10*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 20*(2*B*a^3*b + 3*A* a^2*b^2)*d^2*e^3 + 35*(B*a^4 + 4*A*a^3*b)*d*e^4 + 126*(5*B*b^4*d*e^4 + 4*( 4*B*a*b^3 + A*b^4)*e^5)*x^4 + 84*(5*B*b^4*d^2*e^3 + 4*(4*B*a*b^3 + A*b^4)* d*e^4 + 10*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 36*(5*B*b^4*d^3*e^2 + 4*(4 *B*a*b^3 + A*b^4)*d^2*e^3 + 10*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 20*(2*B*a ^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 9*(5*B*b^4*d^4*e + 4*(4*B*a*b^3 + A*b^4)*d^ 3*e^2 + 10*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 20*(2*B*a^3*b + 3*A*a^2*b^2 )*d*e^4 + 35*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^15*x^9 + 9*d*e^14*x^8 + 36*d^2 *e^13*x^7 + 84*d^3*e^12*x^6 + 126*d^4*e^11*x^5 + 126*d^5*e^10*x^4 + 84*d^6 *e^9*x^3 + 36*d^7*e^8*x^2 + 9*d^8*e^7*x + d^9*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (194) = 388\).
Time = 0.27 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.27 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{10}} \, dx=-\frac {630 \, B b^{4} e^{5} x^{5} + 630 \, B b^{4} d e^{4} x^{4} + 2016 \, B a b^{3} e^{5} x^{4} + 504 \, A b^{4} e^{5} x^{4} + 420 \, B b^{4} d^{2} e^{3} x^{3} + 1344 \, B a b^{3} d e^{4} x^{3} + 336 \, A b^{4} d e^{4} x^{3} + 2520 \, B a^{2} b^{2} e^{5} x^{3} + 1680 \, A a b^{3} e^{5} x^{3} + 180 \, B b^{4} d^{3} e^{2} x^{2} + 576 \, B a b^{3} d^{2} e^{3} x^{2} + 144 \, A b^{4} d^{2} e^{3} x^{2} + 1080 \, B a^{2} b^{2} d e^{4} x^{2} + 720 \, A a b^{3} d e^{4} x^{2} + 1440 \, B a^{3} b e^{5} x^{2} + 2160 \, A a^{2} b^{2} e^{5} x^{2} + 45 \, B b^{4} d^{4} e x + 144 \, B a b^{3} d^{3} e^{2} x + 36 \, A b^{4} d^{3} e^{2} x + 270 \, B a^{2} b^{2} d^{2} e^{3} x + 180 \, A a b^{3} d^{2} e^{3} x + 360 \, B a^{3} b d e^{4} x + 540 \, A a^{2} b^{2} d e^{4} x + 315 \, B a^{4} e^{5} x + 1260 \, A a^{3} b e^{5} x + 5 \, B b^{4} d^{5} + 16 \, B a b^{3} d^{4} e + 4 \, A b^{4} d^{4} e + 30 \, B a^{2} b^{2} d^{3} e^{2} + 20 \, A a b^{3} d^{3} e^{2} + 40 \, B a^{3} b d^{2} e^{3} + 60 \, A a^{2} b^{2} d^{2} e^{3} + 35 \, B a^{4} d e^{4} + 140 \, A a^{3} b d e^{4} + 280 \, A a^{4} e^{5}}{2520 \, {\left (e x + d\right )}^{9} e^{6}} \]
-1/2520*(630*B*b^4*e^5*x^5 + 630*B*b^4*d*e^4*x^4 + 2016*B*a*b^3*e^5*x^4 + 504*A*b^4*e^5*x^4 + 420*B*b^4*d^2*e^3*x^3 + 1344*B*a*b^3*d*e^4*x^3 + 336*A *b^4*d*e^4*x^3 + 2520*B*a^2*b^2*e^5*x^3 + 1680*A*a*b^3*e^5*x^3 + 180*B*b^4 *d^3*e^2*x^2 + 576*B*a*b^3*d^2*e^3*x^2 + 144*A*b^4*d^2*e^3*x^2 + 1080*B*a^ 2*b^2*d*e^4*x^2 + 720*A*a*b^3*d*e^4*x^2 + 1440*B*a^3*b*e^5*x^2 + 2160*A*a^ 2*b^2*e^5*x^2 + 45*B*b^4*d^4*e*x + 144*B*a*b^3*d^3*e^2*x + 36*A*b^4*d^3*e^ 2*x + 270*B*a^2*b^2*d^2*e^3*x + 180*A*a*b^3*d^2*e^3*x + 360*B*a^3*b*d*e^4* x + 540*A*a^2*b^2*d*e^4*x + 315*B*a^4*e^5*x + 1260*A*a^3*b*e^5*x + 5*B*b^4 *d^5 + 16*B*a*b^3*d^4*e + 4*A*b^4*d^4*e + 30*B*a^2*b^2*d^3*e^2 + 20*A*a*b^ 3*d^3*e^2 + 40*B*a^3*b*d^2*e^3 + 60*A*a^2*b^2*d^2*e^3 + 35*B*a^4*d*e^4 + 1 40*A*a^3*b*d*e^4 + 280*A*a^4*e^5)/((e*x + d)^9*e^6)
Time = 10.88 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.43 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{10}} \, dx=-\frac {\frac {35\,B\,a^4\,d\,e^4+280\,A\,a^4\,e^5+40\,B\,a^3\,b\,d^2\,e^3+140\,A\,a^3\,b\,d\,e^4+30\,B\,a^2\,b^2\,d^3\,e^2+60\,A\,a^2\,b^2\,d^2\,e^3+16\,B\,a\,b^3\,d^4\,e+20\,A\,a\,b^3\,d^3\,e^2+5\,B\,b^4\,d^5+4\,A\,b^4\,d^4\,e}{2520\,e^6}+\frac {x\,\left (35\,B\,a^4\,e^4+40\,B\,a^3\,b\,d\,e^3+140\,A\,a^3\,b\,e^4+30\,B\,a^2\,b^2\,d^2\,e^2+60\,A\,a^2\,b^2\,d\,e^3+16\,B\,a\,b^3\,d^3\,e+20\,A\,a\,b^3\,d^2\,e^2+5\,B\,b^4\,d^4+4\,A\,b^4\,d^3\,e\right )}{280\,e^5}+\frac {b^3\,x^4\,\left (4\,A\,b\,e+16\,B\,a\,e+5\,B\,b\,d\right )}{20\,e^2}+\frac {b\,x^2\,\left (40\,B\,a^3\,e^3+30\,B\,a^2\,b\,d\,e^2+60\,A\,a^2\,b\,e^3+16\,B\,a\,b^2\,d^2\,e+20\,A\,a\,b^2\,d\,e^2+5\,B\,b^3\,d^3+4\,A\,b^3\,d^2\,e\right )}{70\,e^4}+\frac {b^2\,x^3\,\left (30\,B\,a^2\,e^2+16\,B\,a\,b\,d\,e+20\,A\,a\,b\,e^2+5\,B\,b^2\,d^2+4\,A\,b^2\,d\,e\right )}{30\,e^3}+\frac {B\,b^4\,x^5}{4\,e}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \]
-((280*A*a^4*e^5 + 5*B*b^4*d^5 + 4*A*b^4*d^4*e + 35*B*a^4*d*e^4 + 20*A*a*b ^3*d^3*e^2 + 40*B*a^3*b*d^2*e^3 + 60*A*a^2*b^2*d^2*e^3 + 30*B*a^2*b^2*d^3* e^2 + 140*A*a^3*b*d*e^4 + 16*B*a*b^3*d^4*e)/(2520*e^6) + (x*(35*B*a^4*e^4 + 5*B*b^4*d^4 + 140*A*a^3*b*e^4 + 4*A*b^4*d^3*e + 20*A*a*b^3*d^2*e^2 + 60* A*a^2*b^2*d*e^3 + 30*B*a^2*b^2*d^2*e^2 + 16*B*a*b^3*d^3*e + 40*B*a^3*b*d*e ^3))/(280*e^5) + (b^3*x^4*(4*A*b*e + 16*B*a*e + 5*B*b*d))/(20*e^2) + (b*x^ 2*(40*B*a^3*e^3 + 5*B*b^3*d^3 + 60*A*a^2*b*e^3 + 4*A*b^3*d^2*e + 20*A*a*b^ 2*d*e^2 + 16*B*a*b^2*d^2*e + 30*B*a^2*b*d*e^2))/(70*e^4) + (b^2*x^3*(30*B* a^2*e^2 + 5*B*b^2*d^2 + 20*A*a*b*e^2 + 4*A*b^2*d*e + 16*B*a*b*d*e))/(30*e^ 3) + (B*b^4*x^5)/(4*e))/(d^9 + e^9*x^9 + 9*d*e^8*x^8 + 36*d^7*e^2*x^2 + 84 *d^6*e^3*x^3 + 126*d^5*e^4*x^4 + 126*d^4*e^5*x^5 + 84*d^3*e^6*x^6 + 36*d^2 *e^7*x^7 + 9*d^8*e*x)