Integrand size = 20, antiderivative size = 86 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx=-\frac {(B d-A e) (a+b x)^{11}}{12 e (b d-a e) (d+e x)^{12}}+\frac {(11 b B d+A b e-12 a B e) (a+b x)^{11}}{132 e (b d-a e)^2 (d+e x)^{11}} \] Output:
-1/12*(-A*e+B*d)*(b*x+a)^11/e/(-a*e+b*d)/(e*x+d)^12+1/132*(A*b*e-12*B*a*e+ 11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^2/(e*x+d)^11
Leaf count is larger than twice the leaf count of optimal. \(1421\) vs. \(2(86)=172\).
Time = 0.52 (sec) , antiderivative size = 1421, normalized size of antiderivative = 16.52 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx =\text {Too large to display} \] Input:
Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^13,x]
Output:
-1/132*(a^10*e^10*(11*A*e + B*(d + 12*e*x)) + 2*a^9*b*e^9*(5*A*e*(d + 12*e *x) + B*(d^2 + 12*d*e*x + 66*e^2*x^2)) + 3*a^8*b^2*e^8*(3*A*e*(d^2 + 12*d* e*x + 66*e^2*x^2) + B*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x^3)) + 4 *a^7*b^3*e^7*(2*A*e*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x^3) + B*(d ^4 + 12*d^3*e*x + 66*d^2*e^2*x^2 + 220*d*e^3*x^3 + 495*e^4*x^4)) + a^6*b^4 *e^6*(7*A*e*(d^4 + 12*d^3*e*x + 66*d^2*e^2*x^2 + 220*d*e^3*x^3 + 495*e^4*x ^4) + 5*B*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4 *x^4 + 792*e^5*x^5)) + 6*a^5*b^5*e^5*(A*e*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x ^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*x^5) + B*(d^6 + 12*d^5*e*x + 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924 *e^6*x^6)) + a^4*b^6*e^4*(5*A*e*(d^6 + 12*d^5*e*x + 66*d^4*e^2*x^2 + 220*d ^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924*e^6*x^6) + 7*B*(d^7 + 1 2*d^6*e*x + 66*d^5*e^2*x^2 + 220*d^4*e^3*x^3 + 495*d^3*e^4*x^4 + 792*d^2*e ^5*x^5 + 924*d*e^6*x^6 + 792*e^7*x^7)) + 4*a^3*b^7*e^3*(A*e*(d^7 + 12*d^6* e*x + 66*d^5*e^2*x^2 + 220*d^4*e^3*x^3 + 495*d^3*e^4*x^4 + 792*d^2*e^5*x^5 + 924*d*e^6*x^6 + 792*e^7*x^7) + 2*B*(d^8 + 12*d^7*e*x + 66*d^6*e^2*x^2 + 220*d^5*e^3*x^3 + 495*d^4*e^4*x^4 + 792*d^3*e^5*x^5 + 924*d^2*e^6*x^6 + 7 92*d*e^7*x^7 + 495*e^8*x^8)) + 3*a^2*b^8*e^2*(A*e*(d^8 + 12*d^7*e*x + 66*d ^6*e^2*x^2 + 220*d^5*e^3*x^3 + 495*d^4*e^4*x^4 + 792*d^3*e^5*x^5 + 924*d^2 *e^6*x^6 + 792*d*e^7*x^7 + 495*e^8*x^8) + 3*B*(d^9 + 12*d^8*e*x + 66*d^...
Time = 0.21 (sec) , antiderivative size = 86, 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 = {87, 48}
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.
| \(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-12 a B e+A b e+11 b B d) \int \frac {(a+b x)^{10}}{(d+e x)^{12}}dx}{12 e (b d-a e)}-\frac {(a+b x)^{11} (B d-A e)}{12 e (d+e x)^{12} (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {(a+b x)^{11} (-12 a B e+A b e+11 b B d)}{132 e (d+e x)^{11} (b d-a e)^2}-\frac {(a+b x)^{11} (B d-A e)}{12 e (d+e x)^{12} (b d-a e)}\) |
Input:
Int[((a + b*x)^10*(A + B*x))/(d + e*x)^13,x]
Output:
-1/12*((B*d - A*e)*(a + b*x)^11)/(e*(b*d - a*e)*(d + e*x)^12) + ((11*b*B*d + A*b*e - 12*a*B*e)*(a + b*x)^11)/(132*e*(b*d - a*e)^2*(d + e*x)^11)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
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. \(1887\) vs. \(2(82)=164\).
Time = 0.26 (sec) , antiderivative size = 1888, normalized size of antiderivative = 21.95
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1888\) |
| norman | \(\text {Expression too large to display}\) | \(1924\) |
| default | \(\text {Expression too large to display}\) | \(1942\) |
| gosper | \(\text {Expression too large to display}\) | \(2231\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2231\) |
| orering | \(\text {Expression too large to display}\) | \(2231\) |
Input:
int((b*x+a)^10*(B*x+A)/(e*x+d)^13,x,method=_RETURNVERBOSE)
Output:
(-b^10*B/e*x^11-1/2*b^9*(A*b*e+10*B*a*e+11*B*b*d)/e^2*x^10-5/3*b^8*(2*A*a* b*e^2+A*b^2*d*e+9*B*a^2*e^2+10*B*a*b*d*e+11*B*b^2*d^2)/e^3*x^9-15/4*b^7*(3 *A*a^2*b*e^3+2*A*a*b^2*d*e^2+A*b^3*d^2*e+8*B*a^3*e^3+9*B*a^2*b*d*e^2+10*B* a*b^2*d^2*e+11*B*b^3*d^3)/e^4*x^8-6*b^6*(4*A*a^3*b*e^4+3*A*a^2*b^2*d*e^3+2 *A*a*b^3*d^2*e^2+A*b^4*d^3*e+7*B*a^4*e^4+8*B*a^3*b*d*e^3+9*B*a^2*b^2*d^2*e ^2+10*B*a*b^3*d^3*e+11*B*b^4*d^4)/e^5*x^7-7*b^5*(5*A*a^4*b*e^5+4*A*a^3*b^2 *d*e^4+3*A*a^2*b^3*d^2*e^3+2*A*a*b^4*d^3*e^2+A*b^5*d^4*e+6*B*a^5*e^5+7*B*a ^4*b*d*e^4+8*B*a^3*b^2*d^2*e^3+9*B*a^2*b^3*d^3*e^2+10*B*a*b^4*d^4*e+11*B*b ^5*d^5)/e^6*x^6-6*b^4*(6*A*a^5*b*e^6+5*A*a^4*b^2*d*e^5+4*A*a^3*b^3*d^2*e^4 +3*A*a^2*b^4*d^3*e^3+2*A*a*b^5*d^4*e^2+A*b^6*d^5*e+5*B*a^6*e^6+6*B*a^5*b*d *e^5+7*B*a^4*b^2*d^2*e^4+8*B*a^3*b^3*d^3*e^3+9*B*a^2*b^4*d^4*e^2+10*B*a*b^ 5*d^5*e+11*B*b^6*d^6)/e^7*x^5-15/4*b^3*(7*A*a^6*b*e^7+6*A*a^5*b^2*d*e^6+5* A*a^4*b^3*d^2*e^5+4*A*a^3*b^4*d^3*e^4+3*A*a^2*b^5*d^4*e^3+2*A*a*b^6*d^5*e^ 2+A*b^7*d^6*e+4*B*a^7*e^7+5*B*a^6*b*d*e^6+6*B*a^5*b^2*d^2*e^5+7*B*a^4*b^3* d^3*e^4+8*B*a^3*b^4*d^4*e^3+9*B*a^2*b^5*d^5*e^2+10*B*a*b^6*d^6*e+11*B*b^7* d^7)/e^8*x^4-5/3*b^2*(8*A*a^7*b*e^8+7*A*a^6*b^2*d*e^7+6*A*a^5*b^3*d^2*e^6+ 5*A*a^4*b^4*d^3*e^5+4*A*a^3*b^5*d^4*e^4+3*A*a^2*b^6*d^5*e^3+2*A*a*b^7*d^6* e^2+A*b^8*d^7*e+3*B*a^8*e^8+4*B*a^7*b*d*e^7+5*B*a^6*b^2*d^2*e^6+6*B*a^5*b^ 3*d^3*e^5+7*B*a^4*b^4*d^4*e^4+8*B*a^3*b^5*d^5*e^3+9*B*a^2*b^6*d^6*e^2+10*B *a*b^7*d^7*e+11*B*b^8*d^8)/e^9*x^3-1/2*b*(9*A*a^8*b*e^9+8*A*a^7*b^2*d*e...
Leaf count of result is larger than twice the leaf count of optimal. 1875 vs. \(2 (82) = 164\).
Time = 0.14 (sec) , antiderivative size = 1875, normalized size of antiderivative = 21.80 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^13,x, algorithm="fricas")
Output:
-1/132*(132*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 11*A*a^10*e^11 + (10*B*a*b ^9 + A*b^10)*d^10*e + (9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + (8*B*a^3*b^7 + 3 *A*a^2*b^8)*d^8*e^3 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + (4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + (2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + (B*a^10 + 10*A*a^9*b)*d*e^10 + 66*(11*B*b^10*d*e ^10 + (10*B*a*b^9 + A*b^10)*e^11)*x^10 + 220*(11*B*b^10*d^2*e^9 + (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 + 495*(11*B*b^1 0*d^3*e^8 + (10*B*a*b^9 + A*b^10)*d^2*e^9 + (9*B*a^2*b^8 + 2*A*a*b^9)*d*e^ 10 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 792*(11*B*b^10*d^4*e^7 + (10* B*a*b^9 + A*b^10)*d^3*e^8 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + (8*B*a^3*b ^7 + 3*A*a^2*b^8)*d*e^10 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 924*(11 *B*b^10*d^5*e^6 + (10*B*a*b^9 + A*b^10)*d^4*e^7 + (9*B*a^2*b^8 + 2*A*a*b^9 )*d^3*e^8 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + (7*B*a^4*b^6 + 4*A*a^3*b ^7)*d*e^10 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 792*(11*B*b^10*d^6*e^ 5 + (10*B*a*b^9 + A*b^10)*d^5*e^6 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + (8 *B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 495*(11*B*b^10*d^7*e^4 + (10*B*a*b^9 + A*b^10)*d^6*e^5 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + (7*B*a^4*b^...
Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)**10*(B*x+A)/(e*x+d)**13,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 1875 vs. \(2 (82) = 164\).
Time = 0.13 (sec) , antiderivative size = 1875, normalized size of antiderivative = 21.80 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^13,x, algorithm="maxima")
Output:
-1/132*(132*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 11*A*a^10*e^11 + (10*B*a*b ^9 + A*b^10)*d^10*e + (9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + (8*B*a^3*b^7 + 3 *A*a^2*b^8)*d^8*e^3 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + (4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + (2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + (B*a^10 + 10*A*a^9*b)*d*e^10 + 66*(11*B*b^10*d*e ^10 + (10*B*a*b^9 + A*b^10)*e^11)*x^10 + 220*(11*B*b^10*d^2*e^9 + (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 + 495*(11*B*b^1 0*d^3*e^8 + (10*B*a*b^9 + A*b^10)*d^2*e^9 + (9*B*a^2*b^8 + 2*A*a*b^9)*d*e^ 10 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 792*(11*B*b^10*d^4*e^7 + (10* B*a*b^9 + A*b^10)*d^3*e^8 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + (8*B*a^3*b ^7 + 3*A*a^2*b^8)*d*e^10 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 924*(11 *B*b^10*d^5*e^6 + (10*B*a*b^9 + A*b^10)*d^4*e^7 + (9*B*a^2*b^8 + 2*A*a*b^9 )*d^3*e^8 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + (7*B*a^4*b^6 + 4*A*a^3*b ^7)*d*e^10 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 792*(11*B*b^10*d^6*e^ 5 + (10*B*a*b^9 + A*b^10)*d^5*e^6 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + (8 *B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 495*(11*B*b^10*d^7*e^4 + (10*B*a*b^9 + A*b^10)*d^6*e^5 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + (7*B*a^4*b^...
Leaf count of result is larger than twice the leaf count of optimal. 2230 vs. \(2 (82) = 164\).
Time = 0.13 (sec) , antiderivative size = 2230, normalized size of antiderivative = 25.93 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^13,x, algorithm="giac")
Output:
-1/132*(132*B*b^10*e^11*x^11 + 726*B*b^10*d*e^10*x^10 + 660*B*a*b^9*e^11*x ^10 + 66*A*b^10*e^11*x^10 + 2420*B*b^10*d^2*e^9*x^9 + 2200*B*a*b^9*d*e^10* x^9 + 220*A*b^10*d*e^10*x^9 + 1980*B*a^2*b^8*e^11*x^9 + 440*A*a*b^9*e^11*x ^9 + 5445*B*b^10*d^3*e^8*x^8 + 4950*B*a*b^9*d^2*e^9*x^8 + 495*A*b^10*d^2*e ^9*x^8 + 4455*B*a^2*b^8*d*e^10*x^8 + 990*A*a*b^9*d*e^10*x^8 + 3960*B*a^3*b ^7*e^11*x^8 + 1485*A*a^2*b^8*e^11*x^8 + 8712*B*b^10*d^4*e^7*x^7 + 7920*B*a *b^9*d^3*e^8*x^7 + 792*A*b^10*d^3*e^8*x^7 + 7128*B*a^2*b^8*d^2*e^9*x^7 + 1 584*A*a*b^9*d^2*e^9*x^7 + 6336*B*a^3*b^7*d*e^10*x^7 + 2376*A*a^2*b^8*d*e^1 0*x^7 + 5544*B*a^4*b^6*e^11*x^7 + 3168*A*a^3*b^7*e^11*x^7 + 10164*B*b^10*d ^5*e^6*x^6 + 9240*B*a*b^9*d^4*e^7*x^6 + 924*A*b^10*d^4*e^7*x^6 + 8316*B*a^ 2*b^8*d^3*e^8*x^6 + 1848*A*a*b^9*d^3*e^8*x^6 + 7392*B*a^3*b^7*d^2*e^9*x^6 + 2772*A*a^2*b^8*d^2*e^9*x^6 + 6468*B*a^4*b^6*d*e^10*x^6 + 3696*A*a^3*b^7* d*e^10*x^6 + 5544*B*a^5*b^5*e^11*x^6 + 4620*A*a^4*b^6*e^11*x^6 + 8712*B*b^ 10*d^6*e^5*x^5 + 7920*B*a*b^9*d^5*e^6*x^5 + 792*A*b^10*d^5*e^6*x^5 + 7128* B*a^2*b^8*d^4*e^7*x^5 + 1584*A*a*b^9*d^4*e^7*x^5 + 6336*B*a^3*b^7*d^3*e^8* x^5 + 2376*A*a^2*b^8*d^3*e^8*x^5 + 5544*B*a^4*b^6*d^2*e^9*x^5 + 3168*A*a^3 *b^7*d^2*e^9*x^5 + 4752*B*a^5*b^5*d*e^10*x^5 + 3960*A*a^4*b^6*d*e^10*x^5 + 3960*B*a^6*b^4*e^11*x^5 + 4752*A*a^5*b^5*e^11*x^5 + 5445*B*b^10*d^7*e^4*x ^4 + 4950*B*a*b^9*d^6*e^5*x^4 + 495*A*b^10*d^6*e^5*x^4 + 4455*B*a^2*b^8*d^ 5*e^6*x^4 + 990*A*a*b^9*d^5*e^6*x^4 + 3960*B*a^3*b^7*d^4*e^7*x^4 + 1485...
Time = 0.47 (sec) , antiderivative size = 2008, normalized size of antiderivative = 23.35 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx=\text {Too large to display} \] Input:
int(((A + B*x)*(a + b*x)^10)/(d + e*x)^13,x)
Output:
-((11*A*a^10*e^11 + 11*B*b^10*d^11 + A*b^10*d^10*e + B*a^10*d*e^10 + 2*A*a *b^9*d^9*e^2 + 2*B*a^9*b*d^2*e^9 + 3*A*a^2*b^8*d^8*e^3 + 4*A*a^3*b^7*d^7*e ^4 + 5*A*a^4*b^6*d^6*e^5 + 6*A*a^5*b^5*d^5*e^6 + 7*A*a^6*b^4*d^4*e^7 + 8*A *a^7*b^3*d^3*e^8 + 9*A*a^8*b^2*d^2*e^9 + 9*B*a^2*b^8*d^9*e^2 + 8*B*a^3*b^7 *d^8*e^3 + 7*B*a^4*b^6*d^7*e^4 + 6*B*a^5*b^5*d^6*e^5 + 5*B*a^6*b^4*d^5*e^6 + 4*B*a^7*b^3*d^4*e^7 + 3*B*a^8*b^2*d^3*e^8 + 10*A*a^9*b*d*e^10 + 10*B*a* b^9*d^10*e)/(132*e^12) + (x*(B*a^10*e^10 + 11*B*b^10*d^10 + 10*A*a^9*b*e^1 0 + A*b^10*d^9*e + 2*A*a*b^9*d^8*e^2 + 9*A*a^8*b^2*d*e^9 + 3*A*a^2*b^8*d^7 *e^3 + 4*A*a^3*b^7*d^6*e^4 + 5*A*a^4*b^6*d^5*e^5 + 6*A*a^5*b^5*d^4*e^6 + 7 *A*a^6*b^4*d^3*e^7 + 8*A*a^7*b^3*d^2*e^8 + 9*B*a^2*b^8*d^8*e^2 + 8*B*a^3*b ^7*d^7*e^3 + 7*B*a^4*b^6*d^6*e^4 + 6*B*a^5*b^5*d^5*e^5 + 5*B*a^6*b^4*d^4*e ^6 + 4*B*a^7*b^3*d^3*e^7 + 3*B*a^8*b^2*d^2*e^8 + 10*B*a*b^9*d^9*e + 2*B*a^ 9*b*d*e^9))/(11*e^11) + (15*b^7*x^8*(8*B*a^3*e^3 + 11*B*b^3*d^3 + 3*A*a^2* b*e^3 + A*b^3*d^2*e + 2*A*a*b^2*d*e^2 + 10*B*a*b^2*d^2*e + 9*B*a^2*b*d*e^2 ))/(4*e^4) + (6*b^4*x^5*(5*B*a^6*e^6 + 11*B*b^6*d^6 + 6*A*a^5*b*e^6 + A*b^ 6*d^5*e + 2*A*a*b^5*d^4*e^2 + 5*A*a^4*b^2*d*e^5 + 3*A*a^2*b^4*d^3*e^3 + 4* A*a^3*b^3*d^2*e^4 + 9*B*a^2*b^4*d^4*e^2 + 8*B*a^3*b^3*d^3*e^3 + 7*B*a^4*b^ 2*d^2*e^4 + 10*B*a*b^5*d^5*e + 6*B*a^5*b*d*e^5))/e^7 + (b^9*x^10*(A*b*e + 10*B*a*e + 11*B*b*d))/(2*e^2) + (6*b^6*x^7*(7*B*a^4*e^4 + 11*B*b^4*d^4 + 4 *A*a^3*b*e^4 + A*b^4*d^3*e + 2*A*a*b^3*d^2*e^2 + 3*A*a^2*b^2*d*e^3 + 9*...
Time = 0.17 (sec) , antiderivative size = 1158, normalized size of antiderivative = 13.47 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^10*(B*x+A)/(e*x+d)^13,x)
Output:
( - a**11*d*e**10 - a**10*b*d**2*e**9 - 12*a**10*b*d*e**10*x - a**9*b**2*d **3*e**8 - 12*a**9*b**2*d**2*e**9*x - 66*a**9*b**2*d*e**10*x**2 - a**8*b** 3*d**4*e**7 - 12*a**8*b**3*d**3*e**8*x - 66*a**8*b**3*d**2*e**9*x**2 - 220 *a**8*b**3*d*e**10*x**3 - a**7*b**4*d**5*e**6 - 12*a**7*b**4*d**4*e**7*x - 66*a**7*b**4*d**3*e**8*x**2 - 220*a**7*b**4*d**2*e**9*x**3 - 495*a**7*b** 4*d*e**10*x**4 - a**6*b**5*d**6*e**5 - 12*a**6*b**5*d**5*e**6*x - 66*a**6* b**5*d**4*e**7*x**2 - 220*a**6*b**5*d**3*e**8*x**3 - 495*a**6*b**5*d**2*e* *9*x**4 - 792*a**6*b**5*d*e**10*x**5 - a**5*b**6*d**7*e**4 - 12*a**5*b**6* d**6*e**5*x - 66*a**5*b**6*d**5*e**6*x**2 - 220*a**5*b**6*d**4*e**7*x**3 - 495*a**5*b**6*d**3*e**8*x**4 - 792*a**5*b**6*d**2*e**9*x**5 - 924*a**5*b* *6*d*e**10*x**6 - a**4*b**7*d**8*e**3 - 12*a**4*b**7*d**7*e**4*x - 66*a**4 *b**7*d**6*e**5*x**2 - 220*a**4*b**7*d**5*e**6*x**3 - 495*a**4*b**7*d**4*e **7*x**4 - 792*a**4*b**7*d**3*e**8*x**5 - 924*a**4*b**7*d**2*e**9*x**6 - 7 92*a**4*b**7*d*e**10*x**7 - a**3*b**8*d**9*e**2 - 12*a**3*b**8*d**8*e**3*x - 66*a**3*b**8*d**7*e**4*x**2 - 220*a**3*b**8*d**6*e**5*x**3 - 495*a**3*b **8*d**5*e**6*x**4 - 792*a**3*b**8*d**4*e**7*x**5 - 924*a**3*b**8*d**3*e** 8*x**6 - 792*a**3*b**8*d**2*e**9*x**7 - 495*a**3*b**8*d*e**10*x**8 - a**2* b**9*d**10*e - 12*a**2*b**9*d**9*e**2*x - 66*a**2*b**9*d**8*e**3*x**2 - 22 0*a**2*b**9*d**7*e**4*x**3 - 495*a**2*b**9*d**6*e**5*x**4 - 792*a**2*b**9* d**5*e**6*x**5 - 924*a**2*b**9*d**4*e**7*x**6 - 792*a**2*b**9*d**3*e**8...