Integrand size = 20, antiderivative size = 348 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\frac {b (b d-a e)^9 (B d-A e) x}{e^{11}}-\frac {(b d-a e)^8 (B d-A e) (a+b x)^2}{2 e^{10}}+\frac {(b d-a e)^7 (B d-A e) (a+b x)^3}{3 e^9}-\frac {(b d-a e)^6 (B d-A e) (a+b x)^4}{4 e^8}+\frac {(b d-a e)^5 (B d-A e) (a+b x)^5}{5 e^7}-\frac {(b d-a e)^4 (B d-A e) (a+b x)^6}{6 e^6}+\frac {(b d-a e)^3 (B d-A e) (a+b x)^7}{7 e^5}-\frac {(b d-a e)^2 (B d-A e) (a+b x)^8}{8 e^4}+\frac {(b d-a e) (B d-A e) (a+b x)^9}{9 e^3}-\frac {(B d-A e) (a+b x)^{10}}{10 e^2}+\frac {B (a+b x)^{11}}{11 b e}-\frac {(b d-a e)^{10} (B d-A e) \log (d+e x)}{e^{12}} \]
b*(-a*e+b*d)^9*(-A*e+B*d)*x/e^11-1/2*(-a*e+b*d)^8*(-A*e+B*d)*(b*x+a)^2/e^1 0+1/3*(-a*e+b*d)^7*(-A*e+B*d)*(b*x+a)^3/e^9-1/4*(-a*e+b*d)^6*(-A*e+B*d)*(b *x+a)^4/e^8+1/5*(-a*e+b*d)^5*(-A*e+B*d)*(b*x+a)^5/e^7-1/6*(-a*e+b*d)^4*(-A *e+B*d)*(b*x+a)^6/e^6+1/7*(-a*e+b*d)^3*(-A*e+B*d)*(b*x+a)^7/e^5-1/8*(-a*e+ b*d)^2*(-A*e+B*d)*(b*x+a)^8/e^4+1/9*(-a*e+b*d)*(-A*e+B*d)*(b*x+a)^9/e^3-1/ 10*(-A*e+B*d)*(b*x+a)^10/e^2+1/11*B*(b*x+a)^11/b/e-(-a*e+b*d)^10*(-A*e+B*d )*ln(e*x+d)/e^12
Leaf count is larger than twice the leaf count of optimal. \(1252\) vs. \(2(348)=696\).
Time = 0.80 (sec) , antiderivative size = 1252, normalized size of antiderivative = 3.60 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\frac {x \left (27720 a^{10} B e^{10}+138600 a^9 b e^9 (-2 B d+2 A e+B e x)+207900 a^8 b^2 e^8 \left (3 A e (-2 d+e x)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+277200 a^7 b^3 e^7 \left (2 A e \left (6 d^2-3 d e x+2 e^2 x^2\right )+B \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+97020 a^6 b^4 e^6 \left (5 A e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+B \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+116424 a^5 b^5 e^5 \left (A e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+B \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+13860 a^4 b^6 e^4 \left (7 A e \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+B \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )\right )+3960 a^3 b^7 e^3 \left (2 A e \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )+B \left (-840 d^7+420 d^6 e x-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5-120 d e^6 x^6+105 e^7 x^7\right )\right )+495 a^2 b^8 e^2 \left (3 A e \left (-840 d^7+420 d^6 e x-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5-120 d e^6 x^6+105 e^7 x^7\right )+B \left (2520 d^8-1260 d^7 e x+840 d^6 e^2 x^2-630 d^5 e^3 x^3+504 d^4 e^4 x^4-420 d^3 e^5 x^5+360 d^2 e^6 x^6-315 d e^7 x^7+280 e^8 x^8\right )\right )+110 a b^9 e \left (A e \left (2520 d^8-1260 d^7 e x+840 d^6 e^2 x^2-630 d^5 e^3 x^3+504 d^4 e^4 x^4-420 d^3 e^5 x^5+360 d^2 e^6 x^6-315 d e^7 x^7+280 e^8 x^8\right )+B \left (-2520 d^9+1260 d^8 e x-840 d^7 e^2 x^2+630 d^6 e^3 x^3-504 d^5 e^4 x^4+420 d^4 e^5 x^5-360 d^3 e^6 x^6+315 d^2 e^7 x^7-280 d e^8 x^8+252 e^9 x^9\right )\right )+b^{10} \left (11 A e \left (-2520 d^9+1260 d^8 e x-840 d^7 e^2 x^2+630 d^6 e^3 x^3-504 d^5 e^4 x^4+420 d^4 e^5 x^5-360 d^3 e^6 x^6+315 d^2 e^7 x^7-280 d e^8 x^8+252 e^9 x^9\right )+B \left (27720 d^{10}-13860 d^9 e x+9240 d^8 e^2 x^2-6930 d^7 e^3 x^3+5544 d^6 e^4 x^4-4620 d^5 e^5 x^5+3960 d^4 e^6 x^6-3465 d^3 e^7 x^7+3080 d^2 e^8 x^8-2772 d e^9 x^9+2520 e^{10} x^{10}\right )\right )\right )}{27720 e^{11}}+\frac {(b d-a e)^{10} (-B d+A e) \log (d+e x)}{e^{12}} \]
(x*(27720*a^10*B*e^10 + 138600*a^9*b*e^9*(-2*B*d + 2*A*e + B*e*x) + 207900 *a^8*b^2*e^8*(3*A*e*(-2*d + e*x) + B*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 2772 00*a^7*b^3*e^7*(2*A*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + B*(-12*d^3 + 6*d^2*e *x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 97020*a^6*b^4*e^6*(5*A*e*(-12*d^3 + 6*d^2 *e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + B*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) + 116424*a^5*b^5*e^5*(A*e*(60*d^4 - 30*d^3*e *x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + B*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5)) + 13860*a ^4*b^6*e^4*(7*A*e*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + B*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e^6*x^6)) + 3960*a^ 3*b^7*e^3*(2*A*e*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^ 3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e^6*x^6) + B*(-840*d^7 + 420*d^6*e* x - 280*d^5*e^2*x^2 + 210*d^4*e^3*x^3 - 168*d^3*e^4*x^4 + 140*d^2*e^5*x^5 - 120*d*e^6*x^6 + 105*e^7*x^7)) + 495*a^2*b^8*e^2*(3*A*e*(-840*d^7 + 420*d ^6*e*x - 280*d^5*e^2*x^2 + 210*d^4*e^3*x^3 - 168*d^3*e^4*x^4 + 140*d^2*e^5 *x^5 - 120*d*e^6*x^6 + 105*e^7*x^7) + B*(2520*d^8 - 1260*d^7*e*x + 840*d^6 *e^2*x^2 - 630*d^5*e^3*x^3 + 504*d^4*e^4*x^4 - 420*d^3*e^5*x^5 + 360*d^2*e ^6*x^6 - 315*d*e^7*x^7 + 280*e^8*x^8)) + 110*a*b^9*e*(A*e*(2520*d^8 - 1260 *d^7*e*x + 840*d^6*e^2*x^2 - 630*d^5*e^3*x^3 + 504*d^4*e^4*x^4 - 420*d^...
Time = 0.63 (sec) , antiderivative size = 348, 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.
| \(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {(a e-b d)^{10} (A e-B d)}{e^{11} (d+e x)}-\frac {b (b d-a e)^9 (A e-B d)}{e^{11}}+\frac {b (a+b x) (b d-a e)^8 (A e-B d)}{e^{10}}-\frac {b (a+b x)^2 (b d-a e)^7 (A e-B d)}{e^9}+\frac {b (a+b x)^3 (b d-a e)^6 (A e-B d)}{e^8}-\frac {b (a+b x)^4 (b d-a e)^5 (A e-B d)}{e^7}+\frac {b (a+b x)^5 (b d-a e)^4 (A e-B d)}{e^6}-\frac {b (a+b x)^6 (b d-a e)^3 (A e-B d)}{e^5}+\frac {b (a+b x)^7 (b d-a e)^2 (A e-B d)}{e^4}-\frac {b (a+b x)^8 (b d-a e) (A e-B d)}{e^3}+\frac {b (a+b x)^9 (A e-B d)}{e^2}+\frac {B (a+b x)^{10}}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(b d-a e)^{10} (B d-A e) \log (d+e x)}{e^{12}}+\frac {b x (b d-a e)^9 (B d-A e)}{e^{11}}-\frac {(a+b x)^2 (b d-a e)^8 (B d-A e)}{2 e^{10}}+\frac {(a+b x)^3 (b d-a e)^7 (B d-A e)}{3 e^9}-\frac {(a+b x)^4 (b d-a e)^6 (B d-A e)}{4 e^8}+\frac {(a+b x)^5 (b d-a e)^5 (B d-A e)}{5 e^7}-\frac {(a+b x)^6 (b d-a e)^4 (B d-A e)}{6 e^6}+\frac {(a+b x)^7 (b d-a e)^3 (B d-A e)}{7 e^5}-\frac {(a+b x)^8 (b d-a e)^2 (B d-A e)}{8 e^4}+\frac {(a+b x)^9 (b d-a e) (B d-A e)}{9 e^3}-\frac {(a+b x)^{10} (B d-A e)}{10 e^2}+\frac {B (a+b x)^{11}}{11 b e}\) |
(b*(b*d - a*e)^9*(B*d - A*e)*x)/e^11 - ((b*d - a*e)^8*(B*d - A*e)*(a + b*x )^2)/(2*e^10) + ((b*d - a*e)^7*(B*d - A*e)*(a + b*x)^3)/(3*e^9) - ((b*d - a*e)^6*(B*d - A*e)*(a + b*x)^4)/(4*e^8) + ((b*d - a*e)^5*(B*d - A*e)*(a + b*x)^5)/(5*e^7) - ((b*d - a*e)^4*(B*d - A*e)*(a + b*x)^6)/(6*e^6) + ((b*d - a*e)^3*(B*d - A*e)*(a + b*x)^7)/(7*e^5) - ((b*d - a*e)^2*(B*d - A*e)*(a + b*x)^8)/(8*e^4) + ((b*d - a*e)*(B*d - A*e)*(a + b*x)^9)/(9*e^3) - ((B*d - A*e)*(a + b*x)^10)/(10*e^2) + (B*(a + b*x)^11)/(11*b*e) - ((b*d - a*e)^1 0*(B*d - A*e)*Log[d + e*x])/e^12
3.11.89.3.1 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. \(1883\) vs. \(2(328)=656\).
Time = 0.72 (sec) , antiderivative size = 1884, normalized size of antiderivative = 5.41
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(1884\) |
| default | \(\text {Expression too large to display}\) | \(2225\) |
| risch | \(\text {Expression too large to display}\) | \(2357\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2358\) |
(10*A*a^9*b*e^10-45*A*a^8*b^2*d*e^9+120*A*a^7*b^3*d^2*e^8-210*A*a^6*b^4*d^ 3*e^7+252*A*a^5*b^5*d^4*e^6-210*A*a^4*b^6*d^5*e^5+120*A*a^3*b^7*d^6*e^4-45 *A*a^2*b^8*d^7*e^3+10*A*a*b^9*d^8*e^2-A*b^10*d^9*e+B*a^10*e^10-10*B*a^9*b* d*e^9+45*B*a^8*b^2*d^2*e^8-120*B*a^7*b^3*d^3*e^7+210*B*a^6*b^4*d^4*e^6-252 *B*a^5*b^5*d^5*e^5+210*B*a^4*b^6*d^6*e^4-120*B*a^3*b^7*d^7*e^3+45*B*a^2*b^ 8*d^8*e^2-10*B*a*b^9*d^9*e+B*b^10*d^10)/e^11*x+1/2*b/e^10*(45*A*a^8*b*e^9- 120*A*a^7*b^2*d*e^8+210*A*a^6*b^3*d^2*e^7-252*A*a^5*b^4*d^3*e^6+210*A*a^4* b^5*d^4*e^5-120*A*a^3*b^6*d^5*e^4+45*A*a^2*b^7*d^6*e^3-10*A*a*b^8*d^7*e^2+ A*b^9*d^8*e+10*B*a^9*e^9-45*B*a^8*b*d*e^8+120*B*a^7*b^2*d^2*e^7-210*B*a^6* b^3*d^3*e^6+252*B*a^5*b^4*d^4*e^5-210*B*a^4*b^5*d^5*e^4+120*B*a^3*b^6*d^6* e^3-45*B*a^2*b^7*d^7*e^2+10*B*a*b^8*d^8*e-B*b^9*d^9)*x^2+1/3*b^2/e^9*(120* A*a^7*b*e^8-210*A*a^6*b^2*d*e^7+252*A*a^5*b^3*d^2*e^6-210*A*a^4*b^4*d^3*e^ 5+120*A*a^3*b^5*d^4*e^4-45*A*a^2*b^6*d^5*e^3+10*A*a*b^7*d^6*e^2-A*b^8*d^7* e+45*B*a^8*e^8-120*B*a^7*b*d*e^7+210*B*a^6*b^2*d^2*e^6-252*B*a^5*b^3*d^3*e ^5+210*B*a^4*b^4*d^4*e^4-120*B*a^3*b^5*d^5*e^3+45*B*a^2*b^6*d^6*e^2-10*B*a *b^7*d^7*e+B*b^8*d^8)*x^3+1/4*b^3/e^8*(210*A*a^6*b*e^7-252*A*a^5*b^2*d*e^6 +210*A*a^4*b^3*d^2*e^5-120*A*a^3*b^4*d^3*e^4+45*A*a^2*b^5*d^4*e^3-10*A*a*b ^6*d^5*e^2+A*b^7*d^6*e+120*B*a^7*e^7-210*B*a^6*b*d*e^6+252*B*a^5*b^2*d^2*e ^5-210*B*a^4*b^3*d^3*e^4+120*B*a^3*b^4*d^4*e^3-45*B*a^2*b^5*d^5*e^2+10*B*a *b^6*d^6*e-B*b^7*d^7)*x^4+1/5*b^4/e^7*(252*A*a^5*b*e^6-210*A*a^4*b^2*d*...
Leaf count of result is larger than twice the leaf count of optimal. 1805 vs. \(2 (328) = 656\).
Time = 0.23 (sec) , antiderivative size = 1805, normalized size of antiderivative = 5.19 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\text {Too large to display} \]
1/27720*(2520*B*b^10*e^11*x^11 - 2772*(B*b^10*d*e^10 - (10*B*a*b^9 + A*b^1 0)*e^11)*x^10 + 3080*(B*b^10*d^2*e^9 - (10*B*a*b^9 + A*b^10)*d*e^10 + 5*(9 *B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 3465*(B*b^10*d^3*e^8 - (10*B*a*b^9 + A *b^10)*d^2*e^9 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 15*(8*B*a^3*b^7 + 3* A*a^2*b^8)*e^11)*x^8 + 3960*(B*b^10*d^4*e^7 - (10*B*a*b^9 + A*b^10)*d^3*e^ 8 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d *e^10 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 - 4620*(B*b^10*d^5*e^6 - (10*B*a*b^9 + A*b^10)*d^4*e^7 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 15*( 8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 5544*(B*b^10*d^6*e^5 - (10*B *a*b^9 + A*b^10)*d^5*e^6 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 15*(8*B*a ^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 4 2*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11 )*x^5 - 6930*(B*b^10*d^7*e^4 - (10*B*a*b^9 + A*b^10)*d^6*e^5 + 5*(9*B*a^2* b^8 + 2*A*a*b^9)*d^5*e^6 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 30*(7* B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e ^11)*x^4 + 9240*(B*b^10*d^8*e^3 - (10*B*a*b^9 + A*b^10)*d^7*e^4 + 5*(9*B*a ^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 30* (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^...
Leaf count of result is larger than twice the leaf count of optimal. 1912 vs. \(2 (298) = 596\).
Time = 1.96 (sec) , antiderivative size = 1912, normalized size of antiderivative = 5.49 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\text {Too large to display} \]
B*b**10*x**11/(11*e) + x**10*(A*b**10/(10*e) + B*a*b**9/e - B*b**10*d/(10* e**2)) + x**9*(10*A*a*b**9/(9*e) - A*b**10*d/(9*e**2) + 5*B*a**2*b**8/e - 10*B*a*b**9*d/(9*e**2) + B*b**10*d**2/(9*e**3)) + x**8*(45*A*a**2*b**8/(8* e) - 5*A*a*b**9*d/(4*e**2) + A*b**10*d**2/(8*e**3) + 15*B*a**3*b**7/e - 45 *B*a**2*b**8*d/(8*e**2) + 5*B*a*b**9*d**2/(4*e**3) - B*b**10*d**3/(8*e**4) ) + x**7*(120*A*a**3*b**7/(7*e) - 45*A*a**2*b**8*d/(7*e**2) + 10*A*a*b**9* d**2/(7*e**3) - A*b**10*d**3/(7*e**4) + 30*B*a**4*b**6/e - 120*B*a**3*b**7 *d/(7*e**2) + 45*B*a**2*b**8*d**2/(7*e**3) - 10*B*a*b**9*d**3/(7*e**4) + B *b**10*d**4/(7*e**5)) + x**6*(35*A*a**4*b**6/e - 20*A*a**3*b**7*d/e**2 + 1 5*A*a**2*b**8*d**2/(2*e**3) - 5*A*a*b**9*d**3/(3*e**4) + A*b**10*d**4/(6*e **5) + 42*B*a**5*b**5/e - 35*B*a**4*b**6*d/e**2 + 20*B*a**3*b**7*d**2/e**3 - 15*B*a**2*b**8*d**3/(2*e**4) + 5*B*a*b**9*d**4/(3*e**5) - B*b**10*d**5/ (6*e**6)) + x**5*(252*A*a**5*b**5/(5*e) - 42*A*a**4*b**6*d/e**2 + 24*A*a** 3*b**7*d**2/e**3 - 9*A*a**2*b**8*d**3/e**4 + 2*A*a*b**9*d**4/e**5 - A*b**1 0*d**5/(5*e**6) + 42*B*a**6*b**4/e - 252*B*a**5*b**5*d/(5*e**2) + 42*B*a** 4*b**6*d**2/e**3 - 24*B*a**3*b**7*d**3/e**4 + 9*B*a**2*b**8*d**4/e**5 - 2* B*a*b**9*d**5/e**6 + B*b**10*d**6/(5*e**7)) + x**4*(105*A*a**6*b**4/(2*e) - 63*A*a**5*b**5*d/e**2 + 105*A*a**4*b**6*d**2/(2*e**3) - 30*A*a**3*b**7*d **3/e**4 + 45*A*a**2*b**8*d**4/(4*e**5) - 5*A*a*b**9*d**5/(2*e**6) + A*b** 10*d**6/(4*e**7) + 30*B*a**7*b**3/e - 105*B*a**6*b**4*d/(2*e**2) + 63*B...
Leaf count of result is larger than twice the leaf count of optimal. 1804 vs. \(2 (328) = 656\).
Time = 0.22 (sec) , antiderivative size = 1804, normalized size of antiderivative = 5.18 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\text {Too large to display} \]
1/27720*(2520*B*b^10*e^10*x^11 - 2772*(B*b^10*d*e^9 - (10*B*a*b^9 + A*b^10 )*e^10)*x^10 + 3080*(B*b^10*d^2*e^8 - (10*B*a*b^9 + A*b^10)*d*e^9 + 5*(9*B *a^2*b^8 + 2*A*a*b^9)*e^10)*x^9 - 3465*(B*b^10*d^3*e^7 - (10*B*a*b^9 + A*b ^10)*d^2*e^8 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^9 - 15*(8*B*a^3*b^7 + 3*A*a ^2*b^8)*e^10)*x^8 + 3960*(B*b^10*d^4*e^6 - (10*B*a*b^9 + A*b^10)*d^3*e^7 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^8 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^ 9 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^10)*x^7 - 4620*(B*b^10*d^5*e^5 - (10* B*a*b^9 + A*b^10)*d^4*e^6 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^7 - 15*(8*B* a^3*b^7 + 3*A*a^2*b^8)*d^2*e^8 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^9 - 42 *(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^10)*x^6 + 5544*(B*b^10*d^6*e^4 - (10*B*a*b^ 9 + A*b^10)*d^5*e^5 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^6 - 15*(8*B*a^3*b^ 7 + 3*A*a^2*b^8)*d^3*e^7 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^8 - 42*(6* B*a^5*b^5 + 5*A*a^4*b^6)*d*e^9 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^10)*x^5 - 6930*(B*b^10*d^7*e^3 - (10*B*a*b^9 + A*b^10)*d^6*e^4 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^5 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^6 + 30*(7*B*a^4* b^6 + 4*A*a^3*b^7)*d^3*e^7 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^8 + 42*( 5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^9 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^10)*x^ 4 + 9240*(B*b^10*d^8*e^2 - (10*B*a*b^9 + A*b^10)*d^7*e^3 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^4 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^5 + 30*(7*B*a^ 4*b^6 + 4*A*a^3*b^7)*d^4*e^6 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^7 +...
Leaf count of result is larger than twice the leaf count of optimal. 2230 vs. \(2 (328) = 656\).
Time = 0.30 (sec) , antiderivative size = 2230, normalized size of antiderivative = 6.41 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\text {Too large to display} \]
1/27720*(2520*B*b^10*e^10*x^11 - 2772*B*b^10*d*e^9*x^10 + 27720*B*a*b^9*e^ 10*x^10 + 2772*A*b^10*e^10*x^10 + 3080*B*b^10*d^2*e^8*x^9 - 30800*B*a*b^9* d*e^9*x^9 - 3080*A*b^10*d*e^9*x^9 + 138600*B*a^2*b^8*e^10*x^9 + 30800*A*a* b^9*e^10*x^9 - 3465*B*b^10*d^3*e^7*x^8 + 34650*B*a*b^9*d^2*e^8*x^8 + 3465* A*b^10*d^2*e^8*x^8 - 155925*B*a^2*b^8*d*e^9*x^8 - 34650*A*a*b^9*d*e^9*x^8 + 415800*B*a^3*b^7*e^10*x^8 + 155925*A*a^2*b^8*e^10*x^8 + 3960*B*b^10*d^4* e^6*x^7 - 39600*B*a*b^9*d^3*e^7*x^7 - 3960*A*b^10*d^3*e^7*x^7 + 178200*B*a ^2*b^8*d^2*e^8*x^7 + 39600*A*a*b^9*d^2*e^8*x^7 - 475200*B*a^3*b^7*d*e^9*x^ 7 - 178200*A*a^2*b^8*d*e^9*x^7 + 831600*B*a^4*b^6*e^10*x^7 + 475200*A*a^3* b^7*e^10*x^7 - 4620*B*b^10*d^5*e^5*x^6 + 46200*B*a*b^9*d^4*e^6*x^6 + 4620* A*b^10*d^4*e^6*x^6 - 207900*B*a^2*b^8*d^3*e^7*x^6 - 46200*A*a*b^9*d^3*e^7* x^6 + 554400*B*a^3*b^7*d^2*e^8*x^6 + 207900*A*a^2*b^8*d^2*e^8*x^6 - 970200 *B*a^4*b^6*d*e^9*x^6 - 554400*A*a^3*b^7*d*e^9*x^6 + 1164240*B*a^5*b^5*e^10 *x^6 + 970200*A*a^4*b^6*e^10*x^6 + 5544*B*b^10*d^6*e^4*x^5 - 55440*B*a*b^9 *d^5*e^5*x^5 - 5544*A*b^10*d^5*e^5*x^5 + 249480*B*a^2*b^8*d^4*e^6*x^5 + 55 440*A*a*b^9*d^4*e^6*x^5 - 665280*B*a^3*b^7*d^3*e^7*x^5 - 249480*A*a^2*b^8* d^3*e^7*x^5 + 1164240*B*a^4*b^6*d^2*e^8*x^5 + 665280*A*a^3*b^7*d^2*e^8*x^5 - 1397088*B*a^5*b^5*d*e^9*x^5 - 1164240*A*a^4*b^6*d*e^9*x^5 + 1164240*B*a ^6*b^4*e^10*x^5 + 1397088*A*a^5*b^5*e^10*x^5 - 6930*B*b^10*d^7*e^3*x^4 + 6 9300*B*a*b^9*d^6*e^4*x^4 + 6930*A*b^10*d^6*e^4*x^4 - 311850*B*a^2*b^8*d...
Time = 1.54 (sec) , antiderivative size = 1795, normalized size of antiderivative = 5.16 \[ \int \frac {(a+b x)^{10} (A+B x)}{d+e x} \, dx=\text {Too large to display} \]
x^4*((d*((d*((d*((d*((d*((d*((A*b^10 + 10*B*a*b^9)/e - (B*b^10*d)/e^2))/e - (5*a*b^8*(2*A*b + 9*B*a))/e))/e + (15*a^2*b^7*(3*A*b + 8*B*a))/e))/e - ( 30*a^3*b^6*(4*A*b + 7*B*a))/e))/e + (42*a^4*b^5*(5*A*b + 6*B*a))/e))/e - ( 42*a^5*b^4*(6*A*b + 5*B*a))/e))/(4*e) + (15*a^6*b^3*(7*A*b + 4*B*a))/(2*e) ) - x^3*((d*((d*((d*((d*((d*((d*((d*((A*b^10 + 10*B*a*b^9)/e - (B*b^10*d)/ e^2))/e - (5*a*b^8*(2*A*b + 9*B*a))/e))/e + (15*a^2*b^7*(3*A*b + 8*B*a))/e ))/e - (30*a^3*b^6*(4*A*b + 7*B*a))/e))/e + (42*a^4*b^5*(5*A*b + 6*B*a))/e ))/e - (42*a^5*b^4*(6*A*b + 5*B*a))/e))/e + (30*a^6*b^3*(7*A*b + 4*B*a))/e ))/(3*e) - (5*a^7*b^2*(8*A*b + 3*B*a))/e) - x^5*((d*((d*((d*((d*((d*((A*b^ 10 + 10*B*a*b^9)/e - (B*b^10*d)/e^2))/e - (5*a*b^8*(2*A*b + 9*B*a))/e))/e + (15*a^2*b^7*(3*A*b + 8*B*a))/e))/e - (30*a^3*b^6*(4*A*b + 7*B*a))/e))/e + (42*a^4*b^5*(5*A*b + 6*B*a))/e))/(5*e) - (42*a^5*b^4*(6*A*b + 5*B*a))/(5 *e)) + x^6*((d*((d*((d*((d*((A*b^10 + 10*B*a*b^9)/e - (B*b^10*d)/e^2))/e - (5*a*b^8*(2*A*b + 9*B*a))/e))/e + (15*a^2*b^7*(3*A*b + 8*B*a))/e))/e - (3 0*a^3*b^6*(4*A*b + 7*B*a))/e))/(6*e) + (7*a^4*b^5*(5*A*b + 6*B*a))/e) - x^ 7*((d*((d*((d*((A*b^10 + 10*B*a*b^9)/e - (B*b^10*d)/e^2))/e - (5*a*b^8*(2* A*b + 9*B*a))/e))/e + (15*a^2*b^7*(3*A*b + 8*B*a))/e))/(7*e) - (30*a^3*b^6 *(4*A*b + 7*B*a))/(7*e)) + x^8*((d*((d*((A*b^10 + 10*B*a*b^9)/e - (B*b^10* d)/e^2))/e - (5*a*b^8*(2*A*b + 9*B*a))/e))/(8*e) + (15*a^2*b^7*(3*A*b + 8* B*a))/(8*e)) - x^9*((d*((A*b^10 + 10*B*a*b^9)/e - (B*b^10*d)/e^2))/(9*e...