Integrand size = 20, antiderivative size = 321 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=-\frac {(B d-A e) (a+b x)^{11}}{11 e (b d-a e) (d+e x)^{11}}-\frac {B (b d-a e)^{10}}{10 e^{12} (d+e x)^{10}}+\frac {10 b B (b d-a e)^9}{9 e^{12} (d+e x)^9}-\frac {45 b^2 B (b d-a e)^8}{8 e^{12} (d+e x)^8}+\frac {120 b^3 B (b d-a e)^7}{7 e^{12} (d+e x)^7}-\frac {35 b^4 B (b d-a e)^6}{e^{12} (d+e x)^6}+\frac {252 b^5 B (b d-a e)^5}{5 e^{12} (d+e x)^5}-\frac {105 b^6 B (b d-a e)^4}{2 e^{12} (d+e x)^4}+\frac {40 b^7 B (b d-a e)^3}{e^{12} (d+e x)^3}-\frac {45 b^8 B (b d-a e)^2}{2 e^{12} (d+e x)^2}+\frac {10 b^9 B (b d-a e)}{e^{12} (d+e x)}+\frac {b^{10} B \log (d+e x)}{e^{12}} \]
-1/11*(-A*e+B*d)*(b*x+a)^11/e/(-a*e+b*d)/(e*x+d)^11-1/10*B*(-a*e+b*d)^10/e ^12/(e*x+d)^10+10/9*b*B*(-a*e+b*d)^9/e^12/(e*x+d)^9-45/8*b^2*B*(-a*e+b*d)^ 8/e^12/(e*x+d)^8+120/7*b^3*B*(-a*e+b*d)^7/e^12/(e*x+d)^7-35*b^4*B*(-a*e+b* d)^6/e^12/(e*x+d)^6+252/5*b^5*B*(-a*e+b*d)^5/e^12/(e*x+d)^5-105/2*b^6*B*(- a*e+b*d)^4/e^12/(e*x+d)^4+40*b^7*B*(-a*e+b*d)^3/e^12/(e*x+d)^3-45/2*b^8*B* (-a*e+b*d)^2/e^12/(e*x+d)^2+10*b^9*B*(-a*e+b*d)/e^12/(e*x+d)+b^10*B*ln(e*x +d)/e^12
Leaf count is larger than twice the leaf count of optimal. \(1443\) vs. \(2(321)=642\).
Time = 1.25 (sec) , antiderivative size = 1443, normalized size of antiderivative = 4.50 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=-\frac {252 a^{10} e^{10} (10 A e+B (d+11 e x))+280 a^9 b e^9 \left (9 A e (d+11 e x)+2 B \left (d^2+11 d e x+55 e^2 x^2\right )\right )+315 a^8 b^2 e^8 \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 )+360 a^7 b^3 e^7 \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 )+420 a^6 b^4 e^6 \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 )+504 a^5 b^5 e^5 \left (5 A 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 )+6 B \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 )\right )+630 a^4 b^6 e^4 \left (4 A e \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 )+7 B \left (d^7+11 d^6 e x+55 d^5 e^2 x^2+165 d^4 e^3 x^3+330 d^3 e^4 x^4+462 d^2 e^5 x^5+462 d e^6 x^6+330 e^7 x^7\right )\right )+840 a^3 b^7 e^3 \left (3 A e \left (d^7+11 d^6 e x+55 d^5 e^2 x^2+165 d^4 e^3 x^3+330 d^3 e^4 x^4+462 d^2 e^5 x^5+462 d e^6 x^6+330 e^7 x^7\right )+8 B \left (d^8+11 d^7 e x+55 d^6 e^2 x^2+165 d^5 e^3 x^3+330 d^4 e^4 x^4+462 d^3 e^5 x^5+462 d^2 e^6 x^6+330 d e^7 x^7+165 e^8 x^8\right )\right )+1260 a^2 b^8 e^2 \left (2 A e \left (d^8+11 d^7 e x+55 d^6 e^2 x^2+165 d^5 e^3 x^3+330 d^4 e^4 x^4+462 d^3 e^5 x^5+462 d^2 e^6 x^6+330 d e^7 x^7+165 e^8 x^8\right )+9 B \left (d^9+11 d^8 e x+55 d^7 e^2 x^2+165 d^6 e^3 x^3+330 d^5 e^4 x^4+462 d^4 e^5 x^5+462 d^3 e^6 x^6+330 d^2 e^7 x^7+165 d e^8 x^8+55 e^9 x^9\right )\right )+2520 a b^9 e \left (A e \left (d^9+11 d^8 e x+55 d^7 e^2 x^2+165 d^6 e^3 x^3+330 d^5 e^4 x^4+462 d^4 e^5 x^5+462 d^3 e^6 x^6+330 d^2 e^7 x^7+165 d e^8 x^8+55 e^9 x^9\right )+10 B \left (d^{10}+11 d^9 e x+55 d^8 e^2 x^2+165 d^7 e^3 x^3+330 d^6 e^4 x^4+462 d^5 e^5 x^5+462 d^4 e^6 x^6+330 d^3 e^7 x^7+165 d^2 e^8 x^8+55 d e^9 x^9+11 e^{10} x^{10}\right )\right )+b^{10} \left (2520 A e \left (d^{10}+11 d^9 e x+55 d^8 e^2 x^2+165 d^7 e^3 x^3+330 d^6 e^4 x^4+462 d^5 e^5 x^5+462 d^4 e^6 x^6+330 d^3 e^7 x^7+165 d^2 e^8 x^8+55 d e^9 x^9+11 e^{10} x^{10}\right )-B d \left (83711 d^{10}+893101 d^9 e x+4313045 d^8 e^2 x^2+12430935 d^7 e^3 x^3+23718420 d^6 e^4 x^4+31376268 d^5 e^5 x^5+29241828 d^4 e^6 x^6+19057500 d^3 e^7 x^7+8385300 d^2 e^8 x^8+2286900 d e^9 x^9+304920 e^{10} x^{10}\right )\right )-27720 b^{10} B (d+e x)^{11} \log (d+e x)}{27720 e^{12} (d+e x)^{11}} \]
-1/27720*(252*a^10*e^10*(10*A*e + B*(d + 11*e*x)) + 280*a^9*b*e^9*(9*A*e*( d + 11*e*x) + 2*B*(d^2 + 11*d*e*x + 55*e^2*x^2)) + 315*a^8*b^2*e^8*(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)) + 360*a^7*b^3*e^7*(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)) + 420*a^6*b^4*e^6*(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)) + 504*a^5*b^5*e^5*(5*A*e*(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) + 6*B*(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)) + 630*a^4*b^6*e^4*(4*A*e*(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) + 7*B*(d^7 + 11*d^6*e*x + 55*d^5*e^2*x^2 + 165*d^4*e^3*x ^3 + 330*d^3*e^4*x^4 + 462*d^2*e^5*x^5 + 462*d*e^6*x^6 + 330*e^7*x^7)) + 8 40*a^3*b^7*e^3*(3*A*e*(d^7 + 11*d^6*e*x + 55*d^5*e^2*x^2 + 165*d^4*e^3*x^3 + 330*d^3*e^4*x^4 + 462*d^2*e^5*x^5 + 462*d*e^6*x^6 + 330*e^7*x^7) + 8*B* (d^8 + 11*d^7*e*x + 55*d^6*e^2*x^2 + 165*d^5*e^3*x^3 + 330*d^4*e^4*x^4 + 4 62*d^3*e^5*x^5 + 462*d^2*e^6*x^6 + 330*d*e^7*x^7 + 165*e^8*x^8)) + 1260*a^ 2*b^8*e^2*(2*A*e*(d^8 + 11*d^7*e*x + 55*d^6*e^2*x^2 + 165*d^5*e^3*x^3 + 33 0*d^4*e^4*x^4 + 462*d^3*e^5*x^5 + 462*d^2*e^6*x^6 + 330*d*e^7*x^7 + 165...
Time = 0.70 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {87, 49, 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.
| \(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {B \int \frac {(a+b x)^{10}}{(d+e x)^{11}}dx}{e}-\frac {(a+b x)^{11} (B d-A e)}{11 e (d+e x)^{11} (b d-a e)}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {B \int \left (\frac {b^{10}}{e^{10} (d+e x)}-\frac {10 (b d-a e) b^9}{e^{10} (d+e x)^2}+\frac {45 (b d-a e)^2 b^8}{e^{10} (d+e x)^3}-\frac {120 (b d-a e)^3 b^7}{e^{10} (d+e x)^4}+\frac {210 (b d-a e)^4 b^6}{e^{10} (d+e x)^5}-\frac {252 (b d-a e)^5 b^5}{e^{10} (d+e x)^6}+\frac {210 (b d-a e)^6 b^4}{e^{10} (d+e x)^7}-\frac {120 (b d-a e)^7 b^3}{e^{10} (d+e x)^8}+\frac {45 (b d-a e)^8 b^2}{e^{10} (d+e x)^9}-\frac {10 (b d-a e)^9 b}{e^{10} (d+e x)^{10}}+\frac {(a e-b d)^{10}}{e^{10} (d+e x)^{11}}\right )dx}{e}-\frac {(a+b x)^{11} (B d-A e)}{11 e (d+e x)^{11} (b d-a e)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {B \left (\frac {10 b^9 (b d-a e)}{e^{11} (d+e x)}-\frac {45 b^8 (b d-a e)^2}{2 e^{11} (d+e x)^2}+\frac {40 b^7 (b d-a e)^3}{e^{11} (d+e x)^3}-\frac {105 b^6 (b d-a e)^4}{2 e^{11} (d+e x)^4}+\frac {252 b^5 (b d-a e)^5}{5 e^{11} (d+e x)^5}-\frac {35 b^4 (b d-a e)^6}{e^{11} (d+e x)^6}+\frac {120 b^3 (b d-a e)^7}{7 e^{11} (d+e x)^7}-\frac {45 b^2 (b d-a e)^8}{8 e^{11} (d+e x)^8}+\frac {10 b (b d-a e)^9}{9 e^{11} (d+e x)^9}-\frac {(b d-a e)^{10}}{10 e^{11} (d+e x)^{10}}+\frac {b^{10} \log (d+e x)}{e^{11}}\right )}{e}-\frac {(a+b x)^{11} (B d-A e)}{11 e (d+e x)^{11} (b d-a e)}\) |
-1/11*((B*d - A*e)*(a + b*x)^11)/(e*(b*d - a*e)*(d + e*x)^11) + (B*(-1/10* (b*d - a*e)^10/(e^11*(d + e*x)^10) + (10*b*(b*d - a*e)^9)/(9*e^11*(d + e*x )^9) - (45*b^2*(b*d - a*e)^8)/(8*e^11*(d + e*x)^8) + (120*b^3*(b*d - a*e)^ 7)/(7*e^11*(d + e*x)^7) - (35*b^4*(b*d - a*e)^6)/(e^11*(d + e*x)^6) + (252 *b^5*(b*d - a*e)^5)/(5*e^11*(d + e*x)^5) - (105*b^6*(b*d - a*e)^4)/(2*e^11 *(d + e*x)^4) + (40*b^7*(b*d - a*e)^3)/(e^11*(d + e*x)^3) - (45*b^8*(b*d - a*e)^2)/(2*e^11*(d + e*x)^2) + (10*b^9*(b*d - a*e))/(e^11*(d + e*x)) + (b ^10*Log[d + e*x])/e^11))/e
3.11.100.3.1 Defintions of rubi rules used
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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Leaf count of result is larger than twice the leaf count of optimal. \(1902\) vs. \(2(305)=610\).
Time = 2.13 (sec) , antiderivative size = 1903, normalized size of antiderivative = 5.93
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1903\) |
| norman | \(\text {Expression too large to display}\) | \(1939\) |
| default | \(\text {Expression too large to display}\) | \(1940\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2458\) |
(-b^9*(A*b*e+10*B*a*e-11*B*b*d)/e^2*x^10-5/2*b^8*(2*A*a*b*e^2+2*A*b^2*d*e+ 9*B*a^2*e^2+20*B*a*b*d*e-33*B*b^2*d^2)/e^3*x^9-5/2*b^7*(6*A*a^2*b*e^3+6*A* a*b^2*d*e^2+6*A*b^3*d^2*e+16*B*a^3*e^3+27*B*a^2*b*d*e^2+60*B*a*b^2*d^2*e-1 21*B*b^3*d^3)/e^4*x^8-5/2*b^6*(12*A*a^3*b*e^4+12*A*a^2*b^2*d*e^3+12*A*a*b^ 3*d^2*e^2+12*A*b^4*d^3*e+21*B*a^4*e^4+32*B*a^3*b*d*e^3+54*B*a^2*b^2*d^2*e^ 2+120*B*a*b^3*d^3*e-275*B*b^4*d^4)/e^5*x^7-7/10*b^5*(60*A*a^4*b*e^5+60*A*a ^3*b^2*d*e^4+60*A*a^2*b^3*d^2*e^3+60*A*a*b^4*d^3*e^2+60*A*b^5*d^4*e+72*B*a ^5*e^5+105*B*a^4*b*d*e^4+160*B*a^3*b^2*d^2*e^3+270*B*a^2*b^3*d^3*e^2+600*B *a*b^4*d^4*e-1507*B*b^5*d^5)/e^6*x^6-7/10*b^4*(60*A*a^5*b*e^6+60*A*a^4*b^2 *d*e^5+60*A*a^3*b^3*d^2*e^4+60*A*a^2*b^4*d^3*e^3+60*A*a*b^5*d^4*e^2+60*A*b ^6*d^5*e+50*B*a^6*e^6+72*B*a^5*b*d*e^5+105*B*a^4*b^2*d^2*e^4+160*B*a^3*b^3 *d^3*e^3+270*B*a^2*b^4*d^4*e^2+600*B*a*b^5*d^5*e-1617*B*b^6*d^6)/e^7*x^5-1 /14*b^3*(420*A*a^6*b*e^7+420*A*a^5*b^2*d*e^6+420*A*a^4*b^3*d^2*e^5+420*A*a ^3*b^4*d^3*e^4+420*A*a^2*b^5*d^4*e^3+420*A*a*b^6*d^5*e^2+420*A*b^7*d^6*e+2 40*B*a^7*e^7+350*B*a^6*b*d*e^6+504*B*a^5*b^2*d^2*e^5+735*B*a^4*b^3*d^3*e^4 +1120*B*a^3*b^4*d^4*e^3+1890*B*a^2*b^5*d^5*e^2+4200*B*a*b^6*d^6*e-11979*B* b^7*d^7)/e^8*x^4-1/56*b^2*(840*A*a^7*b*e^8+840*A*a^6*b^2*d*e^7+840*A*a^5*b ^3*d^2*e^6+840*A*a^4*b^4*d^3*e^5+840*A*a^3*b^5*d^4*e^4+840*A*a^2*b^6*d^5*e ^3+840*A*a*b^7*d^6*e^2+840*A*b^8*d^7*e+315*B*a^8*e^8+480*B*a^7*b*d*e^7+700 *B*a^6*b^2*d^2*e^6+1008*B*a^5*b^3*d^3*e^5+1470*B*a^4*b^4*d^4*e^4+2240*B...
Leaf count of result is larger than twice the leaf count of optimal. 2089 vs. \(2 (305) = 610\).
Time = 0.31 (sec) , antiderivative size = 2089, normalized size of antiderivative = 6.51 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=\text {Too large to display} \]
1/27720*(83711*B*b^10*d^11 - 2520*A*a^10*e^11 - 2520*(10*B*a*b^9 + A*b^10) *d^10*e - 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 840*(8*B*a^3*b^7 + 3*A* a^2*b^8)*d^8*e^3 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 504*(6*B*a^5* b^5 + 5*A*a^4*b^6)*d^6*e^5 - 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 360 *(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3 *e^8 - 280*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 252*(B*a^10 + 10*A*a^9*b)*d *e^10 + 27720*(11*B*b^10*d*e^10 - (10*B*a*b^9 + A*b^10)*e^11)*x^10 + 69300 *(33*B*b^10*d^2*e^9 - 2*(10*B*a*b^9 + A*b^10)*d*e^10 - (9*B*a^2*b^8 + 2*A* a*b^9)*e^11)*x^9 + 69300*(121*B*b^10*d^3*e^8 - 6*(10*B*a*b^9 + A*b^10)*d^2 *e^9 - 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 2*(8*B*a^3*b^7 + 3*A*a^2*b^8)* e^11)*x^8 + 69300*(275*B*b^10*d^4*e^7 - 12*(10*B*a*b^9 + A*b^10)*d^3*e^8 - 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^1 0 - 3*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 19404*(1507*B*b^10*d^5*e^6 - 60*(10*B*a*b^9 + A*b^10)*d^4*e^7 - 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 - 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d *e^10 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 19404*(1617*B*b^10*d^6* e^5 - 60*(10*B*a*b^9 + A*b^10)*d^5*e^6 - 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4* e^7 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 - 15*(7*B*a^4*b^6 + 4*A*a^3*b ^7)*d^2*e^9 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 - 10*(5*B*a^6*b^4 + 6* A*a^5*b^5)*e^11)*x^5 + 1980*(11979*B*b^10*d^7*e^4 - 420*(10*B*a*b^9 + A...
Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1932 vs. \(2 (305) = 610\).
Time = 0.28 (sec) , antiderivative size = 1932, normalized size of antiderivative = 6.02 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=\text {Too large to display} \]
1/27720*(83711*B*b^10*d^11 - 2520*A*a^10*e^11 - 2520*(10*B*a*b^9 + A*b^10) *d^10*e - 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 840*(8*B*a^3*b^7 + 3*A* a^2*b^8)*d^8*e^3 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 504*(6*B*a^5* b^5 + 5*A*a^4*b^6)*d^6*e^5 - 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 360 *(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3 *e^8 - 280*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 252*(B*a^10 + 10*A*a^9*b)*d *e^10 + 27720*(11*B*b^10*d*e^10 - (10*B*a*b^9 + A*b^10)*e^11)*x^10 + 69300 *(33*B*b^10*d^2*e^9 - 2*(10*B*a*b^9 + A*b^10)*d*e^10 - (9*B*a^2*b^8 + 2*A* a*b^9)*e^11)*x^9 + 69300*(121*B*b^10*d^3*e^8 - 6*(10*B*a*b^9 + A*b^10)*d^2 *e^9 - 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 2*(8*B*a^3*b^7 + 3*A*a^2*b^8)* e^11)*x^8 + 69300*(275*B*b^10*d^4*e^7 - 12*(10*B*a*b^9 + A*b^10)*d^3*e^8 - 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^1 0 - 3*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 19404*(1507*B*b^10*d^5*e^6 - 60*(10*B*a*b^9 + A*b^10)*d^4*e^7 - 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 - 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d *e^10 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 19404*(1617*B*b^10*d^6* e^5 - 60*(10*B*a*b^9 + A*b^10)*d^5*e^6 - 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4* e^7 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 - 15*(7*B*a^4*b^6 + 4*A*a^3*b ^7)*d^2*e^9 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 - 10*(5*B*a^6*b^4 + 6* A*a^5*b^5)*e^11)*x^5 + 1980*(11979*B*b^10*d^7*e^4 - 420*(10*B*a*b^9 + A...
Leaf count of result is larger than twice the leaf count of optimal. 1974 vs. \(2 (305) = 610\).
Time = 0.30 (sec) , antiderivative size = 1974, normalized size of antiderivative = 6.15 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=\text {Too large to display} \]
B*b^10*log(abs(e*x + d))/e^12 + 1/27720*(27720*(11*B*b^10*d*e^9 - 10*B*a*b ^9*e^10 - A*b^10*e^10)*x^10 + 69300*(33*B*b^10*d^2*e^8 - 20*B*a*b^9*d*e^9 - 2*A*b^10*d*e^9 - 9*B*a^2*b^8*e^10 - 2*A*a*b^9*e^10)*x^9 + 69300*(121*B*b ^10*d^3*e^7 - 60*B*a*b^9*d^2*e^8 - 6*A*b^10*d^2*e^8 - 27*B*a^2*b^8*d*e^9 - 6*A*a*b^9*d*e^9 - 16*B*a^3*b^7*e^10 - 6*A*a^2*b^8*e^10)*x^8 + 69300*(275* B*b^10*d^4*e^6 - 120*B*a*b^9*d^3*e^7 - 12*A*b^10*d^3*e^7 - 54*B*a^2*b^8*d^ 2*e^8 - 12*A*a*b^9*d^2*e^8 - 32*B*a^3*b^7*d*e^9 - 12*A*a^2*b^8*d*e^9 - 21* B*a^4*b^6*e^10 - 12*A*a^3*b^7*e^10)*x^7 + 19404*(1507*B*b^10*d^5*e^5 - 600 *B*a*b^9*d^4*e^6 - 60*A*b^10*d^4*e^6 - 270*B*a^2*b^8*d^3*e^7 - 60*A*a*b^9* d^3*e^7 - 160*B*a^3*b^7*d^2*e^8 - 60*A*a^2*b^8*d^2*e^8 - 105*B*a^4*b^6*d*e ^9 - 60*A*a^3*b^7*d*e^9 - 72*B*a^5*b^5*e^10 - 60*A*a^4*b^6*e^10)*x^6 + 194 04*(1617*B*b^10*d^6*e^4 - 600*B*a*b^9*d^5*e^5 - 60*A*b^10*d^5*e^5 - 270*B* a^2*b^8*d^4*e^6 - 60*A*a*b^9*d^4*e^6 - 160*B*a^3*b^7*d^3*e^7 - 60*A*a^2*b^ 8*d^3*e^7 - 105*B*a^4*b^6*d^2*e^8 - 60*A*a^3*b^7*d^2*e^8 - 72*B*a^5*b^5*d* e^9 - 60*A*a^4*b^6*d*e^9 - 50*B*a^6*b^4*e^10 - 60*A*a^5*b^5*e^10)*x^5 + 19 80*(11979*B*b^10*d^7*e^3 - 4200*B*a*b^9*d^6*e^4 - 420*A*b^10*d^6*e^4 - 189 0*B*a^2*b^8*d^5*e^5 - 420*A*a*b^9*d^5*e^5 - 1120*B*a^3*b^7*d^4*e^6 - 420*A *a^2*b^8*d^4*e^6 - 735*B*a^4*b^6*d^3*e^7 - 420*A*a^3*b^7*d^3*e^7 - 504*B*a ^5*b^5*d^2*e^8 - 420*A*a^4*b^6*d^2*e^8 - 350*B*a^6*b^4*d*e^9 - 420*A*a^5*b ^5*d*e^9 - 240*B*a^7*b^3*e^10 - 420*A*a^6*b^4*e^10)*x^4 + 495*(25113*B*...
Time = 2.05 (sec) , antiderivative size = 2446, normalized size of antiderivative = 7.62 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=\text {Too large to display} \]
-((A*a^10*e^11)/11 - (83711*B*b^10*d^11)/27720 + (A*b^10*d^10*e)/11 + (B*a ^10*d*e^10)/110 - B*b^10*d^11*log(d + e*x) + (B*a^10*e^11*x)/10 + A*b^10*e ^11*x^10 - (81191*B*b^10*d^10*e*x)/2520 + (A*a*b^9*d^9*e^2)/11 + (2*B*a^9* b*d^2*e^9)/99 + 5*A*a*b^9*e^11*x^9 + (10*B*a^9*b*e^11*x^2)/9 + 10*B*a*b^9* e^11*x^10 + A*b^10*d^9*e^2*x + 5*A*b^10*d*e^10*x^9 - 11*B*b^10*d*e^10*x^10 - B*b^10*e^11*x^11*log(d + e*x) + (A*a^2*b^8*d^8*e^3)/11 + (A*a^3*b^7*d^7 *e^4)/11 + (A*a^4*b^6*d^6*e^5)/11 + (A*a^5*b^5*d^5*e^6)/11 + (A*a^6*b^4*d^ 4*e^7)/11 + (A*a^7*b^3*d^3*e^8)/11 + (A*a^8*b^2*d^2*e^9)/11 + (9*B*a^2*b^8 *d^9*e^2)/22 + (8*B*a^3*b^7*d^8*e^3)/33 + (7*B*a^4*b^6*d^7*e^4)/44 + (6*B* a^5*b^5*d^6*e^5)/55 + (5*B*a^6*b^4*d^5*e^6)/66 + (4*B*a^7*b^3*d^4*e^7)/77 + (3*B*a^8*b^2*d^3*e^8)/88 + 5*A*a^8*b^2*e^11*x^2 + 15*A*a^7*b^3*e^11*x^3 + 30*A*a^6*b^4*e^11*x^4 + 42*A*a^5*b^5*e^11*x^5 + 42*A*a^4*b^6*e^11*x^6 + 30*A*a^3*b^7*e^11*x^7 + 15*A*a^2*b^8*e^11*x^8 + (45*B*a^8*b^2*e^11*x^3)/8 + (120*B*a^7*b^3*e^11*x^4)/7 + 35*B*a^6*b^4*e^11*x^5 + (252*B*a^5*b^5*e^11 *x^6)/5 + (105*B*a^4*b^6*e^11*x^7)/2 + 40*B*a^3*b^7*e^11*x^8 + (45*B*a^2*b ^8*e^11*x^9)/2 + 5*A*b^10*d^8*e^3*x^2 + 15*A*b^10*d^7*e^4*x^3 + 30*A*b^10* d^6*e^5*x^4 + 42*A*b^10*d^5*e^6*x^5 + 42*A*b^10*d^4*e^7*x^6 + 30*A*b^10*d^ 3*e^8*x^7 + 15*A*b^10*d^2*e^9*x^8 - (78419*B*b^10*d^9*e^2*x^2)/504 - (2511 3*B*b^10*d^8*e^3*x^3)/56 - (11979*B*b^10*d^7*e^4*x^4)/14 - (11319*B*b^10*d ^6*e^5*x^5)/10 - (10549*B*b^10*d^5*e^6*x^6)/10 - (1375*B*b^10*d^4*e^7*x...