Integrand size = 31, antiderivative size = 206 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{12}} \, dx=\frac {(b d-a e)^4 (B d-A e)}{11 e^6 (d+e x)^{11}}-\frac {(b d-a e)^3 (5 b B d-4 A b e-a B e)}{10 e^6 (d+e x)^{10}}+\frac {2 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e)}{9 e^6 (d+e x)^9}-\frac {b^2 (b d-a e) (5 b B d-2 A b e-3 a B e)}{4 e^6 (d+e x)^8}+\frac {b^3 (5 b B d-A b e-4 a B e)}{7 e^6 (d+e x)^7}-\frac {b^4 B}{6 e^6 (d+e x)^6} \]
1/11*(-a*e+b*d)^4*(-A*e+B*d)/e^6/(e*x+d)^11-1/10*(-a*e+b*d)^3*(-4*A*b*e-B* a*e+5*B*b*d)/e^6/(e*x+d)^10+2/9*b*(-a*e+b*d)^2*(-3*A*b*e-2*B*a*e+5*B*b*d)/ e^6/(e*x+d)^9-1/4*b^2*(-a*e+b*d)*(-2*A*b*e-3*B*a*e+5*B*b*d)/e^6/(e*x+d)^8+ 1/7*b^3*(-A*b*e-4*B*a*e+5*B*b*d)/e^6/(e*x+d)^7-1/6*b^4*B/e^6/(e*x+d)^6
Time = 0.09 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.57 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{12}} \, dx=-\frac {126 a^4 e^4 (10 A e+B (d+11 e x))+56 a^3 b e^3 \left (9 A e (d+11 e x)+2 B \left (d^2+11 d e x+55 e^2 x^2\right )\right )+21 a^2 b^2 e^2 \left (8 A e \left (d^2+11 d e x+55 e^2 x^2\right )+3 B \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )\right )+6 a b^3 e \left (7 A e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+4 B \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )\right )+b^4 \left (6 A e \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 B \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )\right )}{13860 e^6 (d+e x)^{11}} \]
-1/13860*(126*a^4*e^4*(10*A*e + B*(d + 11*e*x)) + 56*a^3*b*e^3*(9*A*e*(d + 11*e*x) + 2*B*(d^2 + 11*d*e*x + 55*e^2*x^2)) + 21*a^2*b^2*e^2*(8*A*e*(d^2 + 11*d*e*x + 55*e^2*x^2) + 3*B*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3 *x^3)) + 6*a*b^3*e*(7*A*e*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 4*B*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4)) + b^4*(6*A*e*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x ^4) + 5*B*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4 *x^4 + 462*e^5*x^5)))/(e^6*(d + e*x)^11)
Time = 0.42 (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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2 (A+B x)}{(d+e x)^{12}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^4 (a+b x)^4 (A+B x)}{(d+e x)^{12}}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^4 (A+B x)}{(d+e x)^{12}}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^3 (4 a B e+A b e-5 b B d)}{e^5 (d+e x)^8}-\frac {2 b^2 (b d-a e) (3 a B e+2 A b e-5 b B d)}{e^5 (d+e x)^9}+\frac {2 b (b d-a e)^2 (2 a B e+3 A b e-5 b B d)}{e^5 (d+e x)^{10}}+\frac {(a e-b d)^3 (a B e+4 A b e-5 b B d)}{e^5 (d+e x)^{11}}+\frac {(a e-b d)^4 (A e-B d)}{e^5 (d+e x)^{12}}+\frac {b^4 B}{e^5 (d+e x)^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^3 (-4 a B e-A b e+5 b B d)}{7 e^6 (d+e x)^7}-\frac {b^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{4 e^6 (d+e x)^8}+\frac {2 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{9 e^6 (d+e x)^9}-\frac {(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{10 e^6 (d+e x)^{10}}+\frac {(b d-a e)^4 (B d-A e)}{11 e^6 (d+e x)^{11}}-\frac {b^4 B}{6 e^6 (d+e x)^6}\) |
((b*d - a*e)^4*(B*d - A*e))/(11*e^6*(d + e*x)^11) - ((b*d - a*e)^3*(5*b*B* d - 4*A*b*e - a*B*e))/(10*e^6*(d + e*x)^10) + (2*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e))/(9*e^6*(d + e*x)^9) - (b^2*(b*d - a*e)*(5*b*B*d - 2* A*b*e - 3*a*B*e))/(4*e^6*(d + e*x)^8) + (b^3*(5*b*B*d - A*b*e - 4*a*B*e))/ (7*e^6*(d + e*x)^7) - (b^4*B)/(6*e^6*(d + e*x)^6)
3.17.92.3.1 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 = 0.25 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.00
| method | result | size |
| risch | \(\frac {-\frac {B \,b^{4} x^{5}}{6 e}-\frac {b^{3} \left (6 A b e +24 B a e +5 B b d \right ) x^{4}}{42 e^{2}}-\frac {b^{2} \left (42 A a b \,e^{2}+6 A \,b^{2} d e +63 a^{2} B \,e^{2}+24 B a b d e +5 B \,b^{2} d^{2}\right ) x^{3}}{84 e^{3}}-\frac {b \left (168 A \,a^{2} b \,e^{3}+42 A a \,b^{2} d \,e^{2}+6 A \,b^{3} d^{2} e +112 B \,e^{3} a^{3}+63 B \,a^{2} b d \,e^{2}+24 B a \,b^{2} d^{2} e +5 B \,b^{3} d^{3}\right ) x^{2}}{252 e^{4}}-\frac {\left (504 A \,a^{3} b \,e^{4}+168 A \,a^{2} b^{2} d \,e^{3}+42 A a \,b^{3} d^{2} e^{2}+6 A \,b^{4} d^{3} e +126 B \,a^{4} e^{4}+112 B \,a^{3} b d \,e^{3}+63 B \,a^{2} b^{2} d^{2} e^{2}+24 B a \,b^{3} d^{3} e +5 b^{4} B \,d^{4}\right ) x}{1260 e^{5}}-\frac {1260 A \,a^{4} e^{5}+504 A \,a^{3} b d \,e^{4}+168 A \,a^{2} b^{2} d^{2} e^{3}+42 A a \,b^{3} d^{3} e^{2}+6 A \,b^{4} d^{4} e +126 B \,a^{4} d \,e^{4}+112 B \,a^{3} b \,d^{2} e^{3}+63 B \,a^{2} b^{2} d^{3} e^{2}+24 B a \,b^{3} d^{4} e +5 b^{4} B \,d^{5}}{13860 e^{6}}}{\left (e x +d \right )^{11}}\) | \(413\) |
| default | \(-\frac {b^{4} B}{6 e^{6} \left (e x +d \right )^{6}}-\frac {b^{2} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +3 a^{2} B \,e^{2}-8 B a b d e +5 B \,b^{2} d^{2}\right )}{4 e^{6} \left (e x +d \right )^{8}}-\frac {A \,a^{4} e^{5}-4 A \,a^{3} b d \,e^{4}+6 A \,a^{2} b^{2} d^{2} e^{3}-4 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e -B \,a^{4} d \,e^{4}+4 B \,a^{3} b \,d^{2} e^{3}-6 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e -b^{4} B \,d^{5}}{11 e^{6} \left (e x +d \right )^{11}}-\frac {2 b \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +2 B \,e^{3} a^{3}-9 B \,a^{2} b d \,e^{2}+12 B a \,b^{2} d^{2} e -5 B \,b^{3} d^{3}\right )}{9 e^{6} \left (e x +d \right )^{9}}-\frac {4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +B \,a^{4} e^{4}-8 B \,a^{3} b d \,e^{3}+18 B \,a^{2} b^{2} d^{2} e^{2}-16 B a \,b^{3} d^{3} e +5 b^{4} B \,d^{4}}{10 e^{6} \left (e x +d \right )^{10}}-\frac {b^{3} \left (A b e +4 B a e -5 B b d \right )}{7 e^{6} \left (e x +d \right )^{7}}\) | \(430\) |
| norman | \(\frac {-\frac {B \,b^{4} x^{5}}{6 e}-\frac {\left (6 A \,b^{4} e^{6}+24 B a \,b^{3} e^{6}+5 b^{4} B d \,e^{5}\right ) x^{4}}{42 e^{7}}-\frac {\left (42 A a \,b^{3} e^{7}+6 A \,b^{4} d \,e^{6}+63 B \,a^{2} b^{2} e^{7}+24 B a \,b^{3} d \,e^{6}+5 b^{4} B \,d^{2} e^{5}\right ) x^{3}}{84 e^{8}}-\frac {\left (168 A \,a^{2} b^{2} e^{8}+42 A a \,b^{3} d \,e^{7}+6 A \,b^{4} d^{2} e^{6}+112 B \,a^{3} b \,e^{8}+63 B \,a^{2} b^{2} d \,e^{7}+24 B a \,b^{3} d^{2} e^{6}+5 B \,b^{4} d^{3} e^{5}\right ) x^{2}}{252 e^{9}}-\frac {\left (504 A \,a^{3} b \,e^{9}+168 A \,a^{2} b^{2} d \,e^{8}+42 A a \,b^{3} d^{2} e^{7}+6 A \,b^{4} d^{3} e^{6}+126 B \,a^{4} e^{9}+112 B \,a^{3} b d \,e^{8}+63 B \,a^{2} b^{2} d^{2} e^{7}+24 B a \,b^{3} d^{3} e^{6}+5 B \,b^{4} d^{4} e^{5}\right ) x}{1260 e^{10}}-\frac {1260 A \,a^{4} e^{10}+504 A \,a^{3} b d \,e^{9}+168 A \,a^{2} b^{2} d^{2} e^{8}+42 A a \,b^{3} d^{3} e^{7}+6 A \,b^{4} d^{4} e^{6}+126 B \,a^{4} d \,e^{9}+112 B \,a^{3} b \,d^{2} e^{8}+63 B \,a^{2} b^{2} d^{3} e^{7}+24 B a \,b^{3} d^{4} e^{6}+5 B \,b^{4} d^{5} e^{5}}{13860 e^{11}}}{\left (e x +d \right )^{11}}\) | \(460\) |
| gosper | \(-\frac {2310 B \,x^{5} b^{4} e^{5}+1980 A \,b^{4} e^{5} x^{4}+7920 B \,x^{4} a \,b^{3} e^{5}+1650 B \,x^{4} b^{4} d \,e^{4}+6930 A \,x^{3} a \,b^{3} e^{5}+990 A \,x^{3} b^{4} d \,e^{4}+10395 B \,x^{3} a^{2} b^{2} e^{5}+3960 B \,x^{3} a \,b^{3} d \,e^{4}+825 B \,x^{3} b^{4} d^{2} e^{3}+9240 A \,x^{2} a^{2} b^{2} e^{5}+2310 A \,x^{2} a \,b^{3} d \,e^{4}+330 A \,x^{2} b^{4} d^{2} e^{3}+6160 B \,x^{2} a^{3} b \,e^{5}+3465 B \,x^{2} a^{2} b^{2} d \,e^{4}+1320 B \,x^{2} a \,b^{3} d^{2} e^{3}+275 B \,x^{2} b^{4} d^{3} e^{2}+5544 A x \,a^{3} b \,e^{5}+1848 A x \,a^{2} b^{2} d \,e^{4}+462 A x a \,b^{3} d^{2} e^{3}+66 A x \,b^{4} d^{3} e^{2}+1386 B x \,a^{4} e^{5}+1232 B x \,a^{3} b d \,e^{4}+693 B x \,a^{2} b^{2} d^{2} e^{3}+264 B x a \,b^{3} d^{3} e^{2}+55 B x \,b^{4} d^{4} e +1260 A \,a^{4} e^{5}+504 A \,a^{3} b d \,e^{4}+168 A \,a^{2} b^{2} d^{2} e^{3}+42 A a \,b^{3} d^{3} e^{2}+6 A \,b^{4} d^{4} e +126 B \,a^{4} d \,e^{4}+112 B \,a^{3} b \,d^{2} e^{3}+63 B \,a^{2} b^{2} d^{3} e^{2}+24 B a \,b^{3} d^{4} e +5 b^{4} B \,d^{5}}{13860 e^{6} \left (e x +d \right )^{11}}\) | \(469\) |
| parallelrisch | \(-\frac {2310 b^{4} B \,x^{5} e^{10}+1980 A \,b^{4} e^{10} x^{4}+7920 B a \,b^{3} e^{10} x^{4}+1650 B \,b^{4} d \,e^{9} x^{4}+6930 A a \,b^{3} e^{10} x^{3}+990 A \,b^{4} d \,e^{9} x^{3}+10395 B \,a^{2} b^{2} e^{10} x^{3}+3960 B a \,b^{3} d \,e^{9} x^{3}+825 B \,b^{4} d^{2} e^{8} x^{3}+9240 A \,a^{2} b^{2} e^{10} x^{2}+2310 A a \,b^{3} d \,e^{9} x^{2}+330 A \,b^{4} d^{2} e^{8} x^{2}+6160 B \,a^{3} b \,e^{10} x^{2}+3465 B \,a^{2} b^{2} d \,e^{9} x^{2}+1320 B a \,b^{3} d^{2} e^{8} x^{2}+275 B \,b^{4} d^{3} e^{7} x^{2}+5544 A \,a^{3} b \,e^{10} x +1848 A \,a^{2} b^{2} d \,e^{9} x +462 A a \,b^{3} d^{2} e^{8} x +66 A \,b^{4} d^{3} e^{7} x +1386 B \,a^{4} e^{10} x +1232 B \,a^{3} b d \,e^{9} x +693 B \,a^{2} b^{2} d^{2} e^{8} x +264 B a \,b^{3} d^{3} e^{7} x +55 B \,b^{4} d^{4} e^{6} x +1260 A \,a^{4} e^{10}+504 A \,a^{3} b d \,e^{9}+168 A \,a^{2} b^{2} d^{2} e^{8}+42 A a \,b^{3} d^{3} e^{7}+6 A \,b^{4} d^{4} e^{6}+126 B \,a^{4} d \,e^{9}+112 B \,a^{3} b \,d^{2} e^{8}+63 B \,a^{2} b^{2} d^{3} e^{7}+24 B a \,b^{3} d^{4} e^{6}+5 B \,b^{4} d^{5} e^{5}}{13860 e^{11} \left (e x +d \right )^{11}}\) | \(478\) |
(-1/6*B*b^4/e*x^5-1/42*b^3/e^2*(6*A*b*e+24*B*a*e+5*B*b*d)*x^4-1/84*b^2/e^3 *(42*A*a*b*e^2+6*A*b^2*d*e+63*B*a^2*e^2+24*B*a*b*d*e+5*B*b^2*d^2)*x^3-1/25 2*b/e^4*(168*A*a^2*b*e^3+42*A*a*b^2*d*e^2+6*A*b^3*d^2*e+112*B*a^3*e^3+63*B *a^2*b*d*e^2+24*B*a*b^2*d^2*e+5*B*b^3*d^3)*x^2-1/1260/e^5*(504*A*a^3*b*e^4 +168*A*a^2*b^2*d*e^3+42*A*a*b^3*d^2*e^2+6*A*b^4*d^3*e+126*B*a^4*e^4+112*B* a^3*b*d*e^3+63*B*a^2*b^2*d^2*e^2+24*B*a*b^3*d^3*e+5*B*b^4*d^4)*x-1/13860/e ^6*(1260*A*a^4*e^5+504*A*a^3*b*d*e^4+168*A*a^2*b^2*d^2*e^3+42*A*a*b^3*d^3* e^2+6*A*b^4*d^4*e+126*B*a^4*d*e^4+112*B*a^3*b*d^2*e^3+63*B*a^2*b^2*d^3*e^2 +24*B*a*b^3*d^4*e+5*B*b^4*d^5))/(e*x+d)^11
Leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (194) = 388\).
Time = 0.41 (sec) , antiderivative size = 522, normalized size of antiderivative = 2.53 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{12}} \, dx=-\frac {2310 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 1260 \, A a^{4} e^{5} + 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 21 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 56 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 126 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 330 \, {\left (5 \, B b^{4} d e^{4} + 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 165 \, {\left (5 \, B b^{4} d^{2} e^{3} + 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 21 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 55 \, {\left (5 \, B b^{4} d^{3} e^{2} + 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 21 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 56 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 11 \, {\left (5 \, B b^{4} d^{4} e + 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 21 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 56 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 126 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{13860 \, {\left (e^{17} x^{11} + 11 \, d e^{16} x^{10} + 55 \, d^{2} e^{15} x^{9} + 165 \, d^{3} e^{14} x^{8} + 330 \, d^{4} e^{13} x^{7} + 462 \, d^{5} e^{12} x^{6} + 462 \, d^{6} e^{11} x^{5} + 330 \, d^{7} e^{10} x^{4} + 165 \, d^{8} e^{9} x^{3} + 55 \, d^{9} e^{8} x^{2} + 11 \, d^{10} e^{7} x + d^{11} e^{6}\right )}} \]
-1/13860*(2310*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 1260*A*a^4*e^5 + 6*(4*B*a*b^3 + A*b^4)*d^4*e + 21*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 56*(2*B*a^3*b + 3 *A*a^2*b^2)*d^2*e^3 + 126*(B*a^4 + 4*A*a^3*b)*d*e^4 + 330*(5*B*b^4*d*e^4 + 6*(4*B*a*b^3 + A*b^4)*e^5)*x^4 + 165*(5*B*b^4*d^2*e^3 + 6*(4*B*a*b^3 + A* b^4)*d*e^4 + 21*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 55*(5*B*b^4*d^3*e^2 + 6*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 21*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 56*( 2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 11*(5*B*b^4*d^4*e + 6*(4*B*a*b^3 + A*b ^4)*d^3*e^2 + 21*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 56*(2*B*a^3*b + 3*A*a ^2*b^2)*d*e^4 + 126*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^17*x^11 + 11*d*e^16*x^1 0 + 55*d^2*e^15*x^9 + 165*d^3*e^14*x^8 + 330*d^4*e^13*x^7 + 462*d^5*e^12*x ^6 + 462*d^6*e^11*x^5 + 330*d^7*e^10*x^4 + 165*d^8*e^9*x^3 + 55*d^9*e^8*x^ 2 + 11*d^10*e^7*x + d^11*e^6)
Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{12}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (194) = 388\).
Time = 0.21 (sec) , antiderivative size = 522, normalized size of antiderivative = 2.53 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{12}} \, dx=-\frac {2310 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 1260 \, A a^{4} e^{5} + 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 21 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 56 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 126 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 330 \, {\left (5 \, B b^{4} d e^{4} + 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 165 \, {\left (5 \, B b^{4} d^{2} e^{3} + 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 21 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 55 \, {\left (5 \, B b^{4} d^{3} e^{2} + 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 21 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 56 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 11 \, {\left (5 \, B b^{4} d^{4} e + 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 21 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 56 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 126 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{13860 \, {\left (e^{17} x^{11} + 11 \, d e^{16} x^{10} + 55 \, d^{2} e^{15} x^{9} + 165 \, d^{3} e^{14} x^{8} + 330 \, d^{4} e^{13} x^{7} + 462 \, d^{5} e^{12} x^{6} + 462 \, d^{6} e^{11} x^{5} + 330 \, d^{7} e^{10} x^{4} + 165 \, d^{8} e^{9} x^{3} + 55 \, d^{9} e^{8} x^{2} + 11 \, d^{10} e^{7} x + d^{11} e^{6}\right )}} \]
-1/13860*(2310*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 1260*A*a^4*e^5 + 6*(4*B*a*b^3 + A*b^4)*d^4*e + 21*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 56*(2*B*a^3*b + 3 *A*a^2*b^2)*d^2*e^3 + 126*(B*a^4 + 4*A*a^3*b)*d*e^4 + 330*(5*B*b^4*d*e^4 + 6*(4*B*a*b^3 + A*b^4)*e^5)*x^4 + 165*(5*B*b^4*d^2*e^3 + 6*(4*B*a*b^3 + A* b^4)*d*e^4 + 21*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 55*(5*B*b^4*d^3*e^2 + 6*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 21*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 56*( 2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 11*(5*B*b^4*d^4*e + 6*(4*B*a*b^3 + A*b ^4)*d^3*e^2 + 21*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 56*(2*B*a^3*b + 3*A*a ^2*b^2)*d*e^4 + 126*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^17*x^11 + 11*d*e^16*x^1 0 + 55*d^2*e^15*x^9 + 165*d^3*e^14*x^8 + 330*d^4*e^13*x^7 + 462*d^5*e^12*x ^6 + 462*d^6*e^11*x^5 + 330*d^7*e^10*x^4 + 165*d^8*e^9*x^3 + 55*d^9*e^8*x^ 2 + 11*d^10*e^7*x + d^11*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (194) = 388\).
Time = 0.28 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.27 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{12}} \, dx=-\frac {2310 \, B b^{4} e^{5} x^{5} + 1650 \, B b^{4} d e^{4} x^{4} + 7920 \, B a b^{3} e^{5} x^{4} + 1980 \, A b^{4} e^{5} x^{4} + 825 \, B b^{4} d^{2} e^{3} x^{3} + 3960 \, B a b^{3} d e^{4} x^{3} + 990 \, A b^{4} d e^{4} x^{3} + 10395 \, B a^{2} b^{2} e^{5} x^{3} + 6930 \, A a b^{3} e^{5} x^{3} + 275 \, B b^{4} d^{3} e^{2} x^{2} + 1320 \, B a b^{3} d^{2} e^{3} x^{2} + 330 \, A b^{4} d^{2} e^{3} x^{2} + 3465 \, B a^{2} b^{2} d e^{4} x^{2} + 2310 \, A a b^{3} d e^{4} x^{2} + 6160 \, B a^{3} b e^{5} x^{2} + 9240 \, A a^{2} b^{2} e^{5} x^{2} + 55 \, B b^{4} d^{4} e x + 264 \, B a b^{3} d^{3} e^{2} x + 66 \, A b^{4} d^{3} e^{2} x + 693 \, B a^{2} b^{2} d^{2} e^{3} x + 462 \, A a b^{3} d^{2} e^{3} x + 1232 \, B a^{3} b d e^{4} x + 1848 \, A a^{2} b^{2} d e^{4} x + 1386 \, B a^{4} e^{5} x + 5544 \, A a^{3} b e^{5} x + 5 \, B b^{4} d^{5} + 24 \, B a b^{3} d^{4} e + 6 \, A b^{4} d^{4} e + 63 \, B a^{2} b^{2} d^{3} e^{2} + 42 \, A a b^{3} d^{3} e^{2} + 112 \, B a^{3} b d^{2} e^{3} + 168 \, A a^{2} b^{2} d^{2} e^{3} + 126 \, B a^{4} d e^{4} + 504 \, A a^{3} b d e^{4} + 1260 \, A a^{4} e^{5}}{13860 \, {\left (e x + d\right )}^{11} e^{6}} \]
-1/13860*(2310*B*b^4*e^5*x^5 + 1650*B*b^4*d*e^4*x^4 + 7920*B*a*b^3*e^5*x^4 + 1980*A*b^4*e^5*x^4 + 825*B*b^4*d^2*e^3*x^3 + 3960*B*a*b^3*d*e^4*x^3 + 9 90*A*b^4*d*e^4*x^3 + 10395*B*a^2*b^2*e^5*x^3 + 6930*A*a*b^3*e^5*x^3 + 275* B*b^4*d^3*e^2*x^2 + 1320*B*a*b^3*d^2*e^3*x^2 + 330*A*b^4*d^2*e^3*x^2 + 346 5*B*a^2*b^2*d*e^4*x^2 + 2310*A*a*b^3*d*e^4*x^2 + 6160*B*a^3*b*e^5*x^2 + 92 40*A*a^2*b^2*e^5*x^2 + 55*B*b^4*d^4*e*x + 264*B*a*b^3*d^3*e^2*x + 66*A*b^4 *d^3*e^2*x + 693*B*a^2*b^2*d^2*e^3*x + 462*A*a*b^3*d^2*e^3*x + 1232*B*a^3* b*d*e^4*x + 1848*A*a^2*b^2*d*e^4*x + 1386*B*a^4*e^5*x + 5544*A*a^3*b*e^5*x + 5*B*b^4*d^5 + 24*B*a*b^3*d^4*e + 6*A*b^4*d^4*e + 63*B*a^2*b^2*d^3*e^2 + 42*A*a*b^3*d^3*e^2 + 112*B*a^3*b*d^2*e^3 + 168*A*a^2*b^2*d^2*e^3 + 126*B* a^4*d*e^4 + 504*A*a^3*b*d*e^4 + 1260*A*a^4*e^5)/((e*x + d)^11*e^6)
Time = 10.72 (sec) , antiderivative size = 523, normalized size of antiderivative = 2.54 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{12}} \, dx=-\frac {\frac {126\,B\,a^4\,d\,e^4+1260\,A\,a^4\,e^5+112\,B\,a^3\,b\,d^2\,e^3+504\,A\,a^3\,b\,d\,e^4+63\,B\,a^2\,b^2\,d^3\,e^2+168\,A\,a^2\,b^2\,d^2\,e^3+24\,B\,a\,b^3\,d^4\,e+42\,A\,a\,b^3\,d^3\,e^2+5\,B\,b^4\,d^5+6\,A\,b^4\,d^4\,e}{13860\,e^6}+\frac {x\,\left (126\,B\,a^4\,e^4+112\,B\,a^3\,b\,d\,e^3+504\,A\,a^3\,b\,e^4+63\,B\,a^2\,b^2\,d^2\,e^2+168\,A\,a^2\,b^2\,d\,e^3+24\,B\,a\,b^3\,d^3\,e+42\,A\,a\,b^3\,d^2\,e^2+5\,B\,b^4\,d^4+6\,A\,b^4\,d^3\,e\right )}{1260\,e^5}+\frac {b^3\,x^4\,\left (6\,A\,b\,e+24\,B\,a\,e+5\,B\,b\,d\right )}{42\,e^2}+\frac {b\,x^2\,\left (112\,B\,a^3\,e^3+63\,B\,a^2\,b\,d\,e^2+168\,A\,a^2\,b\,e^3+24\,B\,a\,b^2\,d^2\,e+42\,A\,a\,b^2\,d\,e^2+5\,B\,b^3\,d^3+6\,A\,b^3\,d^2\,e\right )}{252\,e^4}+\frac {b^2\,x^3\,\left (63\,B\,a^2\,e^2+24\,B\,a\,b\,d\,e+42\,A\,a\,b\,e^2+5\,B\,b^2\,d^2+6\,A\,b^2\,d\,e\right )}{84\,e^3}+\frac {B\,b^4\,x^5}{6\,e}}{d^{11}+11\,d^{10}\,e\,x+55\,d^9\,e^2\,x^2+165\,d^8\,e^3\,x^3+330\,d^7\,e^4\,x^4+462\,d^6\,e^5\,x^5+462\,d^5\,e^6\,x^6+330\,d^4\,e^7\,x^7+165\,d^3\,e^8\,x^8+55\,d^2\,e^9\,x^9+11\,d\,e^{10}\,x^{10}+e^{11}\,x^{11}} \]
-((1260*A*a^4*e^5 + 5*B*b^4*d^5 + 6*A*b^4*d^4*e + 126*B*a^4*d*e^4 + 42*A*a *b^3*d^3*e^2 + 112*B*a^3*b*d^2*e^3 + 168*A*a^2*b^2*d^2*e^3 + 63*B*a^2*b^2* d^3*e^2 + 504*A*a^3*b*d*e^4 + 24*B*a*b^3*d^4*e)/(13860*e^6) + (x*(126*B*a^ 4*e^4 + 5*B*b^4*d^4 + 504*A*a^3*b*e^4 + 6*A*b^4*d^3*e + 42*A*a*b^3*d^2*e^2 + 168*A*a^2*b^2*d*e^3 + 63*B*a^2*b^2*d^2*e^2 + 24*B*a*b^3*d^3*e + 112*B*a ^3*b*d*e^3))/(1260*e^5) + (b^3*x^4*(6*A*b*e + 24*B*a*e + 5*B*b*d))/(42*e^2 ) + (b*x^2*(112*B*a^3*e^3 + 5*B*b^3*d^3 + 168*A*a^2*b*e^3 + 6*A*b^3*d^2*e + 42*A*a*b^2*d*e^2 + 24*B*a*b^2*d^2*e + 63*B*a^2*b*d*e^2))/(252*e^4) + (b^ 2*x^3*(63*B*a^2*e^2 + 5*B*b^2*d^2 + 42*A*a*b*e^2 + 6*A*b^2*d*e + 24*B*a*b* d*e))/(84*e^3) + (B*b^4*x^5)/(6*e))/(d^11 + e^11*x^11 + 11*d*e^10*x^10 + 5 5*d^9*e^2*x^2 + 165*d^8*e^3*x^3 + 330*d^7*e^4*x^4 + 462*d^6*e^5*x^5 + 462* d^5*e^6*x^6 + 330*d^4*e^7*x^7 + 165*d^3*e^8*x^8 + 55*d^2*e^9*x^9 + 11*d^10 *e*x)