Integrand size = 20, antiderivative size = 447 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx=-\frac {42 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{5 e^{12} (d+e x)^5}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{4 e^{12} (d+e x)^4}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{3 e^{12} (d+e x)^3}-\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{2 e^{12} (d+e x)^2}+\frac {30 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)}+\frac {15 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) (d+e x)^2}{e^{12}}-\frac {5 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^3}{e^{12}}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^4}{4 e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^5}{5 e^{12}}+\frac {b^{10} B (d+e x)^6}{6 e^{12}}+\frac {42 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) \log (d+e x)}{e^{12}} \]
-42*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)*x/e^11+1/5*(-a*e+b*d)^10* (-A*e+B*d)/e^12/(e*x+d)^5-1/4*(-a*e+b*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12 /(e*x+d)^4+5/3*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)^3-1 5/2*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d)/e^12/(e*x+d)^2+30*b^3*(-a *e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)/e^12/(e*x+d)+15*b^6*(-a*e+b*d)^3*(-4 *A*b*e-7*B*a*e+11*B*b*d)*(e*x+d)^2/e^12-5*b^7*(-a*e+b*d)^2*(-3*A*b*e-8*B*a *e+11*B*b*d)*(e*x+d)^3/e^12+5/4*b^8*(-a*e+b*d)*(-2*A*b*e-9*B*a*e+11*B*b*d) *(e*x+d)^4/e^12-1/5*b^9*(-A*b*e-10*B*a*e+11*B*b*d)*(e*x+d)^5/e^12+1/6*b^10 *B*(e*x+d)^6/e^12+42*b^4*(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)*ln(e*x+d )/e^12
Time = 0.29 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx=\frac {60 b^5 e \left (252 a^5 B e^5+140 a b^4 d^3 e (9 B d-4 A e)-315 a^2 b^3 d^2 e^2 (8 B d-3 A e)+360 a^3 b^2 d e^3 (7 B d-2 A e)-126 b^5 d^4 (2 B d-A e)+210 a^4 b e^4 (-6 B d+A e)\right ) x-30 b^6 e^2 \left (-210 a^4 B e^4-14 b^4 d^3 (9 B d-4 A e)+70 a b^3 d^2 e (8 B d-3 A e)-135 a^2 b^2 d e^2 (7 B d-2 A e)-120 a^3 b e^3 (-6 B d+A e)\right ) x^2+20 b^7 e^3 \left (120 a^3 B e^3-7 b^3 d^2 (8 B d-3 A e)+30 a b^2 d e (7 B d-2 A e)+45 a^2 b e^2 (-6 B d+A e)\right ) x^3-15 b^8 e^4 \left (-45 a^2 B e^2-10 a b e (-6 B d+A e)+3 b^2 d (-7 B d+2 A e)\right ) x^4+12 b^9 e^5 (-6 b B d+A b e+10 a B e) x^5+10 b^{10} B e^6 x^6+\frac {12 (b d-a e)^{10} (B d-A e)}{(d+e x)^5}-\frac {15 (b d-a e)^9 (11 b B d-10 A b e-a B e)}{(d+e x)^4}+\frac {100 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{(d+e x)^3}-\frac {450 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{(d+e x)^2}+\frac {1800 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{d+e x}+2520 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) \log (d+e x)}{60 e^{12}} \]
(60*b^5*e*(252*a^5*B*e^5 + 140*a*b^4*d^3*e*(9*B*d - 4*A*e) - 315*a^2*b^3*d ^2*e^2*(8*B*d - 3*A*e) + 360*a^3*b^2*d*e^3*(7*B*d - 2*A*e) - 126*b^5*d^4*( 2*B*d - A*e) + 210*a^4*b*e^4*(-6*B*d + A*e))*x - 30*b^6*e^2*(-210*a^4*B*e^ 4 - 14*b^4*d^3*(9*B*d - 4*A*e) + 70*a*b^3*d^2*e*(8*B*d - 3*A*e) - 135*a^2* b^2*d*e^2*(7*B*d - 2*A*e) - 120*a^3*b*e^3*(-6*B*d + A*e))*x^2 + 20*b^7*e^3 *(120*a^3*B*e^3 - 7*b^3*d^2*(8*B*d - 3*A*e) + 30*a*b^2*d*e*(7*B*d - 2*A*e) + 45*a^2*b*e^2*(-6*B*d + A*e))*x^3 - 15*b^8*e^4*(-45*a^2*B*e^2 - 10*a*b*e *(-6*B*d + A*e) + 3*b^2*d*(-7*B*d + 2*A*e))*x^4 + 12*b^9*e^5*(-6*b*B*d + A *b*e + 10*a*B*e)*x^5 + 10*b^10*B*e^6*x^6 + (12*(b*d - a*e)^10*(B*d - A*e)) /(d + e*x)^5 - (15*(b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(d + e*x)^ 4 + (100*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(d + e*x)^3 - (45 0*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(d + e*x)^2 + (1800*b^ 3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(d + e*x) + 2520*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e)*Log[d + e*x])/(60*e^12)
Time = 1.34 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {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.
| \(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^9 (d+e x)^4 (10 a B e+A b e-11 b B d)}{e^{11}}-\frac {5 b^8 (d+e x)^3 (b d-a e) (9 a B e+2 A b e-11 b B d)}{e^{11}}+\frac {15 b^7 (d+e x)^2 (b d-a e)^2 (8 a B e+3 A b e-11 b B d)}{e^{11}}-\frac {30 b^6 (d+e x) (b d-a e)^3 (7 a B e+4 A b e-11 b B d)}{e^{11}}+\frac {42 b^5 (b d-a e)^4 (6 a B e+5 A b e-11 b B d)}{e^{11}}-\frac {42 b^4 (b d-a e)^5 (5 a B e+6 A b e-11 b B d)}{e^{11} (d+e x)}+\frac {30 b^3 (b d-a e)^6 (4 a B e+7 A b e-11 b B d)}{e^{11} (d+e x)^2}-\frac {15 b^2 (b d-a e)^7 (3 a B e+8 A b e-11 b B d)}{e^{11} (d+e x)^3}+\frac {5 b (b d-a e)^8 (2 a B e+9 A b e-11 b B d)}{e^{11} (d+e x)^4}+\frac {(a e-b d)^9 (a B e+10 A b e-11 b B d)}{e^{11} (d+e x)^5}+\frac {(a e-b d)^{10} (A e-B d)}{e^{11} (d+e x)^6}+\frac {b^{10} B (d+e x)^5}{e^{11}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^9 (d+e x)^5 (-10 a B e-A b e+11 b B d)}{5 e^{12}}+\frac {5 b^8 (d+e x)^4 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{4 e^{12}}-\frac {5 b^7 (d+e x)^3 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{12}}+\frac {15 b^6 (d+e x)^2 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12}}-\frac {42 b^5 x (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{11}}+\frac {42 b^4 (b d-a e)^5 \log (d+e x) (-5 a B e-6 A b e+11 b B d)}{e^{12}}+\frac {30 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)}-\frac {15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{2 e^{12} (d+e x)^2}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{3 e^{12} (d+e x)^3}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{4 e^{12} (d+e x)^4}+\frac {(b d-a e)^{10} (B d-A e)}{5 e^{12} (d+e x)^5}+\frac {b^{10} B (d+e x)^6}{6 e^{12}}\) |
(-42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*x)/e^11 + ((b*d - a* e)^10*(B*d - A*e))/(5*e^12*(d + e*x)^5) - ((b*d - a*e)^9*(11*b*B*d - 10*A* b*e - a*B*e))/(4*e^12*(d + e*x)^4) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b* e - 2*a*B*e))/(3*e^12*(d + e*x)^3) - (15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A *b*e - 3*a*B*e))/(2*e^12*(d + e*x)^2) + (30*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(e^12*(d + e*x)) + (15*b^6*(b*d - a*e)^3*(11*b*B*d - 4 *A*b*e - 7*a*B*e)*(d + e*x)^2)/e^12 - (5*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A *b*e - 8*a*B*e)*(d + e*x)^3)/e^12 + (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^4)/(4*e^12) - (b^9*(11*b*B*d - A*b*e - 10*a*B*e)*(d + e*x)^5)/(5*e^12) + (b^10*B*(d + e*x)^6)/(6*e^12) + (42*b^4*(b*d - a*e)^5 *(11*b*B*d - 6*A*b*e - 5*a*B*e)*Log[d + e*x])/e^12
3.11.94.3.1 Defintions of rubi rules used
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]))))
Leaf count of result is larger than twice the leaf count of optimal. \(1915\) vs. \(2(433)=866\).
Time = 2.10 (sec) , antiderivative size = 1916, normalized size of antiderivative = 4.29
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(1916\) |
| default | \(\text {Expression too large to display}\) | \(2001\) |
| risch | \(\text {Expression too large to display}\) | \(2079\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(3743\) |
(b^6*(60*A*a^3*b*e^4-60*A*a^2*b^2*d*e^3+30*A*a*b^3*d^2*e^2-6*A*b^4*d^3*e+1 05*B*a^4*e^4-160*B*a^3*b*d*e^3+135*B*a^2*b^2*d^2*e^2-60*B*a*b^3*d^3*e+11*B *b^4*d^4)/e^5*x^7-1/60*(12*A*a^10*e^11+30*A*a^9*b*d*e^10+90*A*a^8*b^2*d^2* e^9+360*A*a^7*b^3*d^3*e^8+2520*A*a^6*b^4*d^4*e^7-34524*A*a^5*b^5*d^5*e^6+1 72620*A*a^4*b^6*d^6*e^5-345240*A*a^3*b^7*d^7*e^4+345240*A*a^2*b^8*d^8*e^3- 172620*A*a*b^9*d^9*e^2+34524*A*b^10*d^10*e+3*B*a^10*d*e^10+20*B*a^9*b*d^2* e^9+135*B*a^8*b^2*d^3*e^8+1440*B*a^7*b^3*d^4*e^7-28770*B*a^6*b^4*d^5*e^6+2 07144*B*a^5*b^5*d^6*e^5-604170*B*a^4*b^6*d^7*e^4+920640*B*a^3*b^7*d^8*e^3- 776790*B*a^2*b^8*d^9*e^2+345240*B*a*b^9*d^10*e-63294*B*b^10*d^11)/e^12-5*( 42*A*a^6*b^4*e^7-252*A*a^5*b^5*d*e^6+1260*A*a^4*b^6*d^2*e^5-2520*A*a^3*b^7 *d^3*e^4+2520*A*a^2*b^8*d^4*e^3-1260*A*a*b^9*d^5*e^2+252*A*b^10*d^6*e+24*B *a^7*b^3*e^7-210*B*a^6*b^4*d*e^6+1512*B*a^5*b^5*d^2*e^5-4410*B*a^4*b^6*d^3 *e^4+6720*B*a^3*b^7*d^4*e^3-5670*B*a^2*b^8*d^5*e^2+2520*B*a*b^9*d^6*e-462* B*b^10*d^7)/e^8*x^4-5/2*(24*A*a^7*b^3*e^8+168*A*a^6*b^4*d*e^7-1512*A*a^5*b ^5*d^2*e^6+7560*A*a^4*b^6*d^3*e^5-15120*A*a^3*b^7*d^4*e^4+15120*A*a^2*b^8* d^5*e^3-7560*A*a*b^9*d^6*e^2+1512*A*b^10*d^7*e+9*B*a^8*b^2*e^8+96*B*a^7*b^ 3*d*e^7-1260*B*a^6*b^4*d^2*e^6+9072*B*a^5*b^5*d^3*e^5-26460*B*a^4*b^6*d^4* e^4+40320*B*a^3*b^7*d^5*e^3-34020*B*a^2*b^8*d^6*e^2+15120*B*a*b^9*d^7*e-27 72*B*b^10*d^8)/e^9*x^3-5/6*(18*A*a^8*b^2*e^9+72*A*a^7*b^3*d*e^8+504*A*a^6* b^4*d^2*e^7-5544*A*a^5*b^5*d^3*e^6+27720*A*a^4*b^6*d^4*e^5-55440*A*a^3*...
Leaf count of result is larger than twice the leaf count of optimal. 2851 vs. \(2 (433) = 866\).
Time = 0.30 (sec) , antiderivative size = 2851, normalized size of antiderivative = 6.38 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx=\text {Too large to display} \]
1/60*(10*B*b^10*e^11*x^11 + 15797*B*b^10*d^11 - 12*A*a^10*e^11 - 9762*(10* B*a*b^9 + A*b^10)*d^10*e + 28185*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 44580 *(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 41310*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d ^7*e^4 - 21924*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 5754*(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 - 45*(3*B*a^ 8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 3*(B *a^10 + 10*A*a^9*b)*d*e^10 - 2*(11*B*b^10*d*e^10 - 6*(10*B*a*b^9 + A*b^10) *e^11)*x^10 + 5*(11*B*b^10*d^2*e^9 - 6*(10*B*a*b^9 + A*b^10)*d*e^10 + 15*( 9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 15*(11*B*b^10*d^3*e^8 - 6*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 60*(11*B*b^10*d^4*e^7 - 6*(10*B*a*b^9 + A*b^10 )*d^3*e^8 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 20*(8*B*a^3*b^7 + 3*A*a ^2*b^8)*d*e^10 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 - 420*(11*B*b^10 *d^5*e^6 - 6*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 15*(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 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 - (47497*B*b^10*d ^6*e^5 - 24762*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 58125*(9*B*a^2*b^8 + 2*A*a* b^9)*d^4*e^7 - 70500*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 45000*(7*B*a^4* b^6 + 4*A*a^3*b^7)*d^2*e^9 - 12600*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10)*x^5 - 5*(19777*B*b^10*d^7*e^4 - 9642*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 20325...
Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1861 vs. \(2 (433) = 866\).
Time = 0.31 (sec) , antiderivative size = 1861, normalized size of antiderivative = 4.16 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx=\text {Too large to display} \]
1/60*(15797*B*b^10*d^11 - 12*A*a^10*e^11 - 9762*(10*B*a*b^9 + A*b^10)*d^10 *e + 28185*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 44580*(8*B*a^3*b^7 + 3*A*a^ 2*b^8)*d^8*e^3 + 41310*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 21924*(6*B*a^ 5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 5754*(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 - 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d ^3*e^8 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 3*(B*a^10 + 10*A*a^9*b)*d* e^10 + 1800*(11*B*b^10*d^7*e^4 - 7*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 21*(9*B *a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 3 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2 *e^9 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - (4*B*a^7*b^3 + 7*A*a^6*b^4)* e^11)*x^4 + 450*(165*B*b^10*d^8*e^3 - 104*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 308*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 504*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^ 5*e^6 + 490*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 280*(6*B*a^5*b^5 + 5*A*a ^4*b^6)*d^3*e^8 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 - (3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 50*(2101*B *b^10*d^9*e^2 - 1314*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 3852*(9*B*a^2*b^8 + 2 *A*a*b^9)*d^7*e^4 - 6216*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 5922*(7*B*a ^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 3276*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 72*(4*B*a^7*b^3 + 7*A*a^6*b^4)* d^2*e^9 - 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 - 2*(2*B*a^9*b + 9*A*a^8...
Leaf count of result is larger than twice the leaf count of optimal. 2032 vs. \(2 (433) = 866\).
Time = 0.30 (sec) , antiderivative size = 2032, normalized size of antiderivative = 4.55 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx=\text {Too large to display} \]
42*(11*B*b^10*d^6 - 60*B*a*b^9*d^5*e - 6*A*b^10*d^5*e + 135*B*a^2*b^8*d^4* e^2 + 30*A*a*b^9*d^4*e^2 - 160*B*a^3*b^7*d^3*e^3 - 60*A*a^2*b^8*d^3*e^3 + 105*B*a^4*b^6*d^2*e^4 + 60*A*a^3*b^7*d^2*e^4 - 36*B*a^5*b^5*d*e^5 - 30*A*a ^4*b^6*d*e^5 + 5*B*a^6*b^4*e^6 + 6*A*a^5*b^5*e^6)*log(abs(e*x + d))/e^12 + 1/60*(15797*B*b^10*d^11 - 97620*B*a*b^9*d^10*e - 9762*A*b^10*d^10*e + 253 665*B*a^2*b^8*d^9*e^2 + 56370*A*a*b^9*d^9*e^2 - 356640*B*a^3*b^7*d^8*e^3 - 133740*A*a^2*b^8*d^8*e^3 + 289170*B*a^4*b^6*d^7*e^4 + 165240*A*a^3*b^7*d^ 7*e^4 - 131544*B*a^5*b^5*d^6*e^5 - 109620*A*a^4*b^6*d^6*e^5 + 28770*B*a^6* b^4*d^5*e^6 + 34524*A*a^5*b^5*d^5*e^6 - 1440*B*a^7*b^3*d^4*e^7 - 2520*A*a^ 6*b^4*d^4*e^7 - 135*B*a^8*b^2*d^3*e^8 - 360*A*a^7*b^3*d^3*e^8 - 20*B*a^9*b *d^2*e^9 - 90*A*a^8*b^2*d^2*e^9 - 3*B*a^10*d*e^10 - 30*A*a^9*b*d*e^10 - 12 *A*a^10*e^11 + 1800*(11*B*b^10*d^7*e^4 - 70*B*a*b^9*d^6*e^5 - 7*A*b^10*d^6 *e^5 + 189*B*a^2*b^8*d^5*e^6 + 42*A*a*b^9*d^5*e^6 - 280*B*a^3*b^7*d^4*e^7 - 105*A*a^2*b^8*d^4*e^7 + 245*B*a^4*b^6*d^3*e^8 + 140*A*a^3*b^7*d^3*e^8 - 126*B*a^5*b^5*d^2*e^9 - 105*A*a^4*b^6*d^2*e^9 + 35*B*a^6*b^4*d*e^10 + 42*A *a^5*b^5*d*e^10 - 4*B*a^7*b^3*e^11 - 7*A*a^6*b^4*e^11)*x^4 + 450*(165*B*b^ 10*d^8*e^3 - 1040*B*a*b^9*d^7*e^4 - 104*A*b^10*d^7*e^4 + 2772*B*a^2*b^8*d^ 6*e^5 + 616*A*a*b^9*d^6*e^5 - 4032*B*a^3*b^7*d^5*e^6 - 1512*A*a^2*b^8*d^5* e^6 + 3430*B*a^4*b^6*d^4*e^7 + 1960*A*a^3*b^7*d^4*e^7 - 1680*B*a^5*b^5*d^3 *e^8 - 1400*A*a^4*b^6*d^3*e^8 + 420*B*a^6*b^4*d^2*e^9 + 504*A*a^5*b^5*d...
Time = 0.45 (sec) , antiderivative size = 2681, normalized size of antiderivative = 6.00 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx=\text {Too large to display} \]
x^5*((A*b^10 + 10*B*a*b^9)/(5*e^6) - (6*B*b^10*d)/(5*e^7)) - x^4*((3*d*((A *b^10 + 10*B*a*b^9)/e^6 - (6*B*b^10*d)/e^7))/(2*e) - (5*a*b^8*(2*A*b + 9*B *a))/(4*e^6) + (15*B*b^10*d^2)/(4*e^8)) - (x^4*(210*A*a^6*b^4*e^10 + 120*B *a^7*b^3*e^10 + 210*A*b^10*d^6*e^4 - 330*B*b^10*d^7*e^3 - 1260*A*a*b^9*d^5 *e^5 - 1260*A*a^5*b^5*d*e^9 + 2100*B*a*b^9*d^6*e^4 - 1050*B*a^6*b^4*d*e^9 + 3150*A*a^2*b^8*d^4*e^6 - 4200*A*a^3*b^7*d^3*e^7 + 3150*A*a^4*b^6*d^2*e^8 - 5670*B*a^2*b^8*d^5*e^5 + 8400*B*a^3*b^7*d^4*e^6 - 7350*B*a^4*b^6*d^3*e^ 7 + 3780*B*a^5*b^5*d^2*e^8) + x^3*(60*A*a^7*b^3*e^10 + (45*B*a^8*b^2*e^10) /2 + 780*A*b^10*d^7*e^3 - (2475*B*b^10*d^8*e^2)/2 - 4620*A*a*b^9*d^6*e^4 + 420*A*a^6*b^4*d*e^9 + 7800*B*a*b^9*d^7*e^3 + 240*B*a^7*b^3*d*e^9 + 11340* A*a^2*b^8*d^5*e^5 - 14700*A*a^3*b^7*d^4*e^6 + 10500*A*a^4*b^6*d^3*e^7 - 37 80*A*a^5*b^5*d^2*e^8 - 20790*B*a^2*b^8*d^6*e^4 + 30240*B*a^3*b^7*d^5*e^5 - 25725*B*a^4*b^6*d^4*e^6 + 12600*B*a^5*b^5*d^3*e^7 - 3150*B*a^6*b^4*d^2*e^ 8) + (12*A*a^10*e^11 - 15797*B*b^10*d^11 + 9762*A*b^10*d^10*e + 3*B*a^10*d *e^10 - 56370*A*a*b^9*d^9*e^2 + 20*B*a^9*b*d^2*e^9 + 133740*A*a^2*b^8*d^8* e^3 - 165240*A*a^3*b^7*d^7*e^4 + 109620*A*a^4*b^6*d^6*e^5 - 34524*A*a^5*b^ 5*d^5*e^6 + 2520*A*a^6*b^4*d^4*e^7 + 360*A*a^7*b^3*d^3*e^8 + 90*A*a^8*b^2* d^2*e^9 - 253665*B*a^2*b^8*d^9*e^2 + 356640*B*a^3*b^7*d^8*e^3 - 289170*B*a ^4*b^6*d^7*e^4 + 131544*B*a^5*b^5*d^6*e^5 - 28770*B*a^6*b^4*d^5*e^6 + 1440 *B*a^7*b^3*d^4*e^7 + 135*B*a^8*b^2*d^3*e^8 + 30*A*a^9*b*d*e^10 + 97620*...