Integrand size = 26, antiderivative size = 170 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{12}} \, dx=-\frac {(b d-a e)^6}{11 e^7 (d+e x)^{11}}+\frac {3 b (b d-a e)^5}{5 e^7 (d+e x)^{10}}-\frac {5 b^2 (b d-a e)^4}{3 e^7 (d+e x)^9}+\frac {5 b^3 (b d-a e)^3}{2 e^7 (d+e x)^8}-\frac {15 b^4 (b d-a e)^2}{7 e^7 (d+e x)^7}+\frac {b^5 (b d-a e)}{e^7 (d+e x)^6}-\frac {b^6}{5 e^7 (d+e x)^5} \]
-1/11*(-a*e+b*d)^6/e^7/(e*x+d)^11+3/5*b*(-a*e+b*d)^5/e^7/(e*x+d)^10-5/3*b^ 2*(-a*e+b*d)^4/e^7/(e*x+d)^9+5/2*b^3*(-a*e+b*d)^3/e^7/(e*x+d)^8-15/7*b^4*( -a*e+b*d)^2/e^7/(e*x+d)^7+b^5*(-a*e+b*d)/e^7/(e*x+d)^6-1/5*b^6/e^7/(e*x+d) ^5
Time = 0.06 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{12}} \, dx=-\frac {210 a^6 e^6+126 a^5 b e^5 (d+11 e x)+70 a^4 b^2 e^4 \left (d^2+11 d e x+55 e^2 x^2\right )+35 a^3 b^3 e^3 \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+15 a^2 b^4 e^2 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 a b^5 e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+b^6 \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )}{2310 e^7 (d+e x)^{11}} \]
-1/2310*(210*a^6*e^6 + 126*a^5*b*e^5*(d + 11*e*x) + 70*a^4*b^2*e^4*(d^2 + 11*d*e*x + 55*e^2*x^2) + 35*a^3*b^3*e^3*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 15*a^2*b^4*e^2*(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*a*b^5*e*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 16 5*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5) + b^6*(d^6 + 11*d^5*e*x + 55* d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e^6* x^6))/(e^7*(d + e*x)^11)
Time = 0.36 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1098, 27, 53, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{12}} \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^6}{(d+e x)^{12}}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^6}{(d+e x)^{12}}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^7}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^8}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^9}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{10}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{11}}+\frac {(a e-b d)^6}{e^6 (d+e x)^{12}}+\frac {b^6}{e^6 (d+e x)^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^5 (b d-a e)}{e^7 (d+e x)^6}-\frac {15 b^4 (b d-a e)^2}{7 e^7 (d+e x)^7}+\frac {5 b^3 (b d-a e)^3}{2 e^7 (d+e x)^8}-\frac {5 b^2 (b d-a e)^4}{3 e^7 (d+e x)^9}+\frac {3 b (b d-a e)^5}{5 e^7 (d+e x)^{10}}-\frac {(b d-a e)^6}{11 e^7 (d+e x)^{11}}-\frac {b^6}{5 e^7 (d+e x)^5}\) |
-1/11*(b*d - a*e)^6/(e^7*(d + e*x)^11) + (3*b*(b*d - a*e)^5)/(5*e^7*(d + e *x)^10) - (5*b^2*(b*d - a*e)^4)/(3*e^7*(d + e*x)^9) + (5*b^3*(b*d - a*e)^3 )/(2*e^7*(d + e*x)^8) - (15*b^4*(b*d - a*e)^2)/(7*e^7*(d + e*x)^7) + (b^5* (b*d - a*e))/(e^7*(d + e*x)^6) - b^6/(5*e^7*(d + e*x)^5)
3.16.1.3.1 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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(334\) vs. \(2(158)=316\).
Time = 2.30 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.97
| method | result | size |
| risch | \(\frac {-\frac {b^{6} x^{6}}{5 e}-\frac {b^{5} \left (5 a e +b d \right ) x^{5}}{5 e^{2}}-\frac {b^{4} \left (15 a^{2} e^{2}+5 a b d e +b^{2} d^{2}\right ) x^{4}}{7 e^{3}}-\frac {b^{3} \left (35 a^{3} e^{3}+15 a^{2} b d \,e^{2}+5 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{14 e^{4}}-\frac {b^{2} \left (70 e^{4} a^{4}+35 b \,e^{3} d \,a^{3}+15 b^{2} e^{2} d^{2} a^{2}+5 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{42 e^{5}}-\frac {b \left (126 a^{5} e^{5}+70 a^{4} b d \,e^{4}+35 a^{3} b^{2} d^{2} e^{3}+15 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{210 e^{6}}-\frac {210 a^{6} e^{6}+126 a^{5} b d \,e^{5}+70 a^{4} b^{2} d^{2} e^{4}+35 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}+5 a \,b^{5} d^{5} e +b^{6} d^{6}}{2310 e^{7}}}{\left (e x +d \right )^{11}}\) | \(335\) |
| default | \(-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{11 e^{7} \left (e x +d \right )^{11}}-\frac {b^{6}}{5 e^{7} \left (e x +d \right )^{5}}-\frac {5 b^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{2 e^{7} \left (e x +d \right )^{8}}-\frac {3 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{5 e^{7} \left (e x +d \right )^{10}}-\frac {15 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{7 e^{7} \left (e x +d \right )^{7}}-\frac {b^{5} \left (a e -b d \right )}{e^{7} \left (e x +d \right )^{6}}-\frac {5 b^{2} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{3 e^{7} \left (e x +d \right )^{9}}\) | \(357\) |
| norman | \(\frac {-\frac {b^{6} x^{6}}{5 e}-\frac {\left (5 e^{5} a \,b^{5}+d \,e^{4} b^{6}\right ) x^{5}}{5 e^{6}}-\frac {\left (15 a^{2} b^{4} e^{6}+5 a \,b^{5} d \,e^{5}+b^{6} d^{2} e^{4}\right ) x^{4}}{7 e^{7}}-\frac {\left (35 a^{3} b^{3} e^{7}+15 a^{2} b^{4} d \,e^{6}+5 a \,b^{5} d^{2} e^{5}+b^{6} d^{3} e^{4}\right ) x^{3}}{14 e^{8}}-\frac {\left (70 a^{4} b^{2} e^{8}+35 a^{3} b^{3} d \,e^{7}+15 a^{2} b^{4} d^{2} e^{6}+5 a \,b^{5} d^{3} e^{5}+b^{6} d^{4} e^{4}\right ) x^{2}}{42 e^{9}}-\frac {\left (126 a^{5} b \,e^{9}+70 a^{4} b^{2} d \,e^{8}+35 a^{3} b^{3} d^{2} e^{7}+15 a^{2} b^{4} d^{3} e^{6}+5 a \,b^{5} d^{4} e^{5}+b^{6} d^{5} e^{4}\right ) x}{210 e^{10}}-\frac {210 a^{6} e^{10}+126 a^{5} b d \,e^{9}+70 a^{4} b^{2} d^{2} e^{8}+35 a^{3} b^{3} d^{3} e^{7}+15 a^{2} b^{4} d^{4} e^{6}+5 a \,b^{5} d^{5} e^{5}+b^{6} d^{6} e^{4}}{2310 e^{11}}}{\left (e x +d \right )^{11}}\) | \(375\) |
| gosper | \(-\frac {462 x^{6} b^{6} e^{6}+2310 x^{5} a \,b^{5} e^{6}+462 x^{5} b^{6} d \,e^{5}+4950 x^{4} a^{2} b^{4} e^{6}+1650 x^{4} a \,b^{5} d \,e^{5}+330 x^{4} b^{6} d^{2} e^{4}+5775 x^{3} a^{3} b^{3} e^{6}+2475 x^{3} a^{2} b^{4} d \,e^{5}+825 x^{3} a \,b^{5} d^{2} e^{4}+165 x^{3} b^{6} d^{3} e^{3}+3850 x^{2} a^{4} b^{2} e^{6}+1925 x^{2} a^{3} b^{3} d \,e^{5}+825 x^{2} a^{2} b^{4} d^{2} e^{4}+275 x^{2} a \,b^{5} d^{3} e^{3}+55 x^{2} b^{6} d^{4} e^{2}+1386 x \,a^{5} b \,e^{6}+770 x \,a^{4} b^{2} d \,e^{5}+385 x \,a^{3} b^{3} d^{2} e^{4}+165 x \,a^{2} b^{4} d^{3} e^{3}+55 x a \,b^{5} d^{4} e^{2}+11 x \,b^{6} d^{5} e +210 a^{6} e^{6}+126 a^{5} b d \,e^{5}+70 a^{4} b^{2} d^{2} e^{4}+35 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}+5 a \,b^{5} d^{5} e +b^{6} d^{6}}{2310 e^{7} \left (e x +d \right )^{11}}\) | \(376\) |
| parallelrisch | \(\frac {-462 b^{6} x^{6} e^{10}-2310 a \,b^{5} e^{10} x^{5}-462 b^{6} d \,e^{9} x^{5}-4950 a^{2} b^{4} e^{10} x^{4}-1650 a \,b^{5} d \,e^{9} x^{4}-330 b^{6} d^{2} e^{8} x^{4}-5775 a^{3} b^{3} e^{10} x^{3}-2475 a^{2} b^{4} d \,e^{9} x^{3}-825 a \,b^{5} d^{2} e^{8} x^{3}-165 b^{6} d^{3} e^{7} x^{3}-3850 a^{4} b^{2} e^{10} x^{2}-1925 a^{3} b^{3} d \,e^{9} x^{2}-825 a^{2} b^{4} d^{2} e^{8} x^{2}-275 a \,b^{5} d^{3} e^{7} x^{2}-55 b^{6} d^{4} e^{6} x^{2}-1386 a^{5} b \,e^{10} x -770 a^{4} b^{2} d \,e^{9} x -385 a^{3} b^{3} d^{2} e^{8} x -165 a^{2} b^{4} d^{3} e^{7} x -55 a \,b^{5} d^{4} e^{6} x -11 b^{6} d^{5} e^{5} x -210 a^{6} e^{10}-126 a^{5} b d \,e^{9}-70 a^{4} b^{2} d^{2} e^{8}-35 a^{3} b^{3} d^{3} e^{7}-15 a^{2} b^{4} d^{4} e^{6}-5 a \,b^{5} d^{5} e^{5}-b^{6} d^{6} e^{4}}{2310 e^{11} \left (e x +d \right )^{11}}\) | \(384\) |
(-1/5*b^6/e*x^6-1/5*b^5/e^2*(5*a*e+b*d)*x^5-1/7*b^4/e^3*(15*a^2*e^2+5*a*b* d*e+b^2*d^2)*x^4-1/14*b^3/e^4*(35*a^3*e^3+15*a^2*b*d*e^2+5*a*b^2*d^2*e+b^3 *d^3)*x^3-1/42*b^2/e^5*(70*a^4*e^4+35*a^3*b*d*e^3+15*a^2*b^2*d^2*e^2+5*a*b ^3*d^3*e+b^4*d^4)*x^2-1/210*b/e^6*(126*a^5*e^5+70*a^4*b*d*e^4+35*a^3*b^2*d ^2*e^3+15*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e+b^5*d^5)*x-1/2310/e^7*(210*a^6*e^6 +126*a^5*b*d*e^5+70*a^4*b^2*d^2*e^4+35*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2+ 5*a*b^5*d^5*e+b^6*d^6))/(e*x+d)^11
Leaf count of result is larger than twice the leaf count of optimal. 463 vs. \(2 (158) = 316\).
Time = 0.30 (sec) , antiderivative size = 463, normalized size of antiderivative = 2.72 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{12}} \, dx=-\frac {462 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 5 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} + 35 \, a^{3} b^{3} d^{3} e^{3} + 70 \, a^{4} b^{2} d^{2} e^{4} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6} + 462 \, {\left (b^{6} d e^{5} + 5 \, a b^{5} e^{6}\right )} x^{5} + 330 \, {\left (b^{6} d^{2} e^{4} + 5 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 165 \, {\left (b^{6} d^{3} e^{3} + 5 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} + 35 \, a^{3} b^{3} e^{6}\right )} x^{3} + 55 \, {\left (b^{6} d^{4} e^{2} + 5 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 35 \, a^{3} b^{3} d e^{5} + 70 \, a^{4} b^{2} e^{6}\right )} x^{2} + 11 \, {\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 126 \, a^{5} b e^{6}\right )} x}{2310 \, {\left (e^{18} x^{11} + 11 \, d e^{17} x^{10} + 55 \, d^{2} e^{16} x^{9} + 165 \, d^{3} e^{15} x^{8} + 330 \, d^{4} e^{14} x^{7} + 462 \, d^{5} e^{13} x^{6} + 462 \, d^{6} e^{12} x^{5} + 330 \, d^{7} e^{11} x^{4} + 165 \, d^{8} e^{10} x^{3} + 55 \, d^{9} e^{9} x^{2} + 11 \, d^{10} e^{8} x + d^{11} e^{7}\right )}} \]
-1/2310*(462*b^6*e^6*x^6 + b^6*d^6 + 5*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 + 35*a^3*b^3*d^3*e^3 + 70*a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^5 + 210*a^6*e^6 + 462*(b^6*d*e^5 + 5*a*b^5*e^6)*x^5 + 330*(b^6*d^2*e^4 + 5*a*b^5*d*e^5 + 15* a^2*b^4*e^6)*x^4 + 165*(b^6*d^3*e^3 + 5*a*b^5*d^2*e^4 + 15*a^2*b^4*d*e^5 + 35*a^3*b^3*e^6)*x^3 + 55*(b^6*d^4*e^2 + 5*a*b^5*d^3*e^3 + 15*a^2*b^4*d^2* e^4 + 35*a^3*b^3*d*e^5 + 70*a^4*b^2*e^6)*x^2 + 11*(b^6*d^5*e + 5*a*b^5*d^4 *e^2 + 15*a^2*b^4*d^3*e^3 + 35*a^3*b^3*d^2*e^4 + 70*a^4*b^2*d*e^5 + 126*a^ 5*b*e^6)*x)/(e^18*x^11 + 11*d*e^17*x^10 + 55*d^2*e^16*x^9 + 165*d^3*e^15*x ^8 + 330*d^4*e^14*x^7 + 462*d^5*e^13*x^6 + 462*d^6*e^12*x^5 + 330*d^7*e^11 *x^4 + 165*d^8*e^10*x^3 + 55*d^9*e^9*x^2 + 11*d^10*e^8*x + d^11*e^7)
Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{12}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 463 vs. \(2 (158) = 316\).
Time = 0.22 (sec) , antiderivative size = 463, normalized size of antiderivative = 2.72 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{12}} \, dx=-\frac {462 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 5 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} + 35 \, a^{3} b^{3} d^{3} e^{3} + 70 \, a^{4} b^{2} d^{2} e^{4} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6} + 462 \, {\left (b^{6} d e^{5} + 5 \, a b^{5} e^{6}\right )} x^{5} + 330 \, {\left (b^{6} d^{2} e^{4} + 5 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 165 \, {\left (b^{6} d^{3} e^{3} + 5 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} + 35 \, a^{3} b^{3} e^{6}\right )} x^{3} + 55 \, {\left (b^{6} d^{4} e^{2} + 5 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 35 \, a^{3} b^{3} d e^{5} + 70 \, a^{4} b^{2} e^{6}\right )} x^{2} + 11 \, {\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 126 \, a^{5} b e^{6}\right )} x}{2310 \, {\left (e^{18} x^{11} + 11 \, d e^{17} x^{10} + 55 \, d^{2} e^{16} x^{9} + 165 \, d^{3} e^{15} x^{8} + 330 \, d^{4} e^{14} x^{7} + 462 \, d^{5} e^{13} x^{6} + 462 \, d^{6} e^{12} x^{5} + 330 \, d^{7} e^{11} x^{4} + 165 \, d^{8} e^{10} x^{3} + 55 \, d^{9} e^{9} x^{2} + 11 \, d^{10} e^{8} x + d^{11} e^{7}\right )}} \]
-1/2310*(462*b^6*e^6*x^6 + b^6*d^6 + 5*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 + 35*a^3*b^3*d^3*e^3 + 70*a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^5 + 210*a^6*e^6 + 462*(b^6*d*e^5 + 5*a*b^5*e^6)*x^5 + 330*(b^6*d^2*e^4 + 5*a*b^5*d*e^5 + 15* a^2*b^4*e^6)*x^4 + 165*(b^6*d^3*e^3 + 5*a*b^5*d^2*e^4 + 15*a^2*b^4*d*e^5 + 35*a^3*b^3*e^6)*x^3 + 55*(b^6*d^4*e^2 + 5*a*b^5*d^3*e^3 + 15*a^2*b^4*d^2* e^4 + 35*a^3*b^3*d*e^5 + 70*a^4*b^2*e^6)*x^2 + 11*(b^6*d^5*e + 5*a*b^5*d^4 *e^2 + 15*a^2*b^4*d^3*e^3 + 35*a^3*b^3*d^2*e^4 + 70*a^4*b^2*d*e^5 + 126*a^ 5*b*e^6)*x)/(e^18*x^11 + 11*d*e^17*x^10 + 55*d^2*e^16*x^9 + 165*d^3*e^15*x ^8 + 330*d^4*e^14*x^7 + 462*d^5*e^13*x^6 + 462*d^6*e^12*x^5 + 330*d^7*e^11 *x^4 + 165*d^8*e^10*x^3 + 55*d^9*e^9*x^2 + 11*d^10*e^8*x + d^11*e^7)
Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (158) = 316\).
Time = 0.26 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.21 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{12}} \, dx=-\frac {462 \, b^{6} e^{6} x^{6} + 462 \, b^{6} d e^{5} x^{5} + 2310 \, a b^{5} e^{6} x^{5} + 330 \, b^{6} d^{2} e^{4} x^{4} + 1650 \, a b^{5} d e^{5} x^{4} + 4950 \, a^{2} b^{4} e^{6} x^{4} + 165 \, b^{6} d^{3} e^{3} x^{3} + 825 \, a b^{5} d^{2} e^{4} x^{3} + 2475 \, a^{2} b^{4} d e^{5} x^{3} + 5775 \, a^{3} b^{3} e^{6} x^{3} + 55 \, b^{6} d^{4} e^{2} x^{2} + 275 \, a b^{5} d^{3} e^{3} x^{2} + 825 \, a^{2} b^{4} d^{2} e^{4} x^{2} + 1925 \, a^{3} b^{3} d e^{5} x^{2} + 3850 \, a^{4} b^{2} e^{6} x^{2} + 11 \, b^{6} d^{5} e x + 55 \, a b^{5} d^{4} e^{2} x + 165 \, a^{2} b^{4} d^{3} e^{3} x + 385 \, a^{3} b^{3} d^{2} e^{4} x + 770 \, a^{4} b^{2} d e^{5} x + 1386 \, a^{5} b e^{6} x + b^{6} d^{6} + 5 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} + 35 \, a^{3} b^{3} d^{3} e^{3} + 70 \, a^{4} b^{2} d^{2} e^{4} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6}}{2310 \, {\left (e x + d\right )}^{11} e^{7}} \]
-1/2310*(462*b^6*e^6*x^6 + 462*b^6*d*e^5*x^5 + 2310*a*b^5*e^6*x^5 + 330*b^ 6*d^2*e^4*x^4 + 1650*a*b^5*d*e^5*x^4 + 4950*a^2*b^4*e^6*x^4 + 165*b^6*d^3* e^3*x^3 + 825*a*b^5*d^2*e^4*x^3 + 2475*a^2*b^4*d*e^5*x^3 + 5775*a^3*b^3*e^ 6*x^3 + 55*b^6*d^4*e^2*x^2 + 275*a*b^5*d^3*e^3*x^2 + 825*a^2*b^4*d^2*e^4*x ^2 + 1925*a^3*b^3*d*e^5*x^2 + 3850*a^4*b^2*e^6*x^2 + 11*b^6*d^5*e*x + 55*a *b^5*d^4*e^2*x + 165*a^2*b^4*d^3*e^3*x + 385*a^3*b^3*d^2*e^4*x + 770*a^4*b ^2*d*e^5*x + 1386*a^5*b*e^6*x + b^6*d^6 + 5*a*b^5*d^5*e + 15*a^2*b^4*d^4*e ^2 + 35*a^3*b^3*d^3*e^3 + 70*a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^5 + 210*a^6*e ^6)/((e*x + d)^11*e^7)
Time = 0.19 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.62 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{12}} \, dx=-\frac {\frac {210\,a^6\,e^6+126\,a^5\,b\,d\,e^5+70\,a^4\,b^2\,d^2\,e^4+35\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2+5\,a\,b^5\,d^5\,e+b^6\,d^6}{2310\,e^7}+\frac {b^6\,x^6}{5\,e}+\frac {b^3\,x^3\,\left (35\,a^3\,e^3+15\,a^2\,b\,d\,e^2+5\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{14\,e^4}+\frac {b\,x\,\left (126\,a^5\,e^5+70\,a^4\,b\,d\,e^4+35\,a^3\,b^2\,d^2\,e^3+15\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{210\,e^6}+\frac {b^5\,x^5\,\left (5\,a\,e+b\,d\right )}{5\,e^2}+\frac {b^2\,x^2\,\left (70\,a^4\,e^4+35\,a^3\,b\,d\,e^3+15\,a^2\,b^2\,d^2\,e^2+5\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{42\,e^5}+\frac {b^4\,x^4\,\left (15\,a^2\,e^2+5\,a\,b\,d\,e+b^2\,d^2\right )}{7\,e^3}}{d^{11}+11\,d^{10}\,e\,x+55\,d^9\,e^2\,x^2+165\,d^8\,e^3\,x^3+330\,d^7\,e^4\,x^4+462\,d^6\,e^5\,x^5+462\,d^5\,e^6\,x^6+330\,d^4\,e^7\,x^7+165\,d^3\,e^8\,x^8+55\,d^2\,e^9\,x^9+11\,d\,e^{10}\,x^{10}+e^{11}\,x^{11}} \]
-((210*a^6*e^6 + b^6*d^6 + 15*a^2*b^4*d^4*e^2 + 35*a^3*b^3*d^3*e^3 + 70*a^ 4*b^2*d^2*e^4 + 5*a*b^5*d^5*e + 126*a^5*b*d*e^5)/(2310*e^7) + (b^6*x^6)/(5 *e) + (b^3*x^3*(35*a^3*e^3 + b^3*d^3 + 5*a*b^2*d^2*e + 15*a^2*b*d*e^2))/(1 4*e^4) + (b*x*(126*a^5*e^5 + b^5*d^5 + 15*a^2*b^3*d^3*e^2 + 35*a^3*b^2*d^2 *e^3 + 5*a*b^4*d^4*e + 70*a^4*b*d*e^4))/(210*e^6) + (b^5*x^5*(5*a*e + b*d) )/(5*e^2) + (b^2*x^2*(70*a^4*e^4 + b^4*d^4 + 15*a^2*b^2*d^2*e^2 + 5*a*b^3* d^3*e + 35*a^3*b*d*e^3))/(42*e^5) + (b^4*x^4*(15*a^2*e^2 + b^2*d^2 + 5*a*b *d*e))/(7*e^3))/(d^11 + e^11*x^11 + 11*d*e^10*x^10 + 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^10*e*x)