\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{14}} \, dx\) [1102]
Optimal result
Integrand size = 20, antiderivative size = 135 \[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{14}} \, dx=-\frac {(B d-A e) (a+b x)^{11}}{13 e (b d-a e) (d+e x)^{13}}+\frac {(11 b B d+2 A b e-13 a B e) (a+b x)^{11}}{156 e (b d-a e)^2 (d+e x)^{12}}+\frac {b (11 b B d+2 A b e-13 a B e) (a+b x)^{11}}{1716 e (b d-a e)^3 (d+e x)^{11}}
\]
[Out]
-1/13*(-A*e+B*d)*(b*x+a)^11/e/(-a*e+b*d)/(e*x+d)^13+1/156*(2*A*b*e-13*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^
2/(e*x+d)^12+1/1716*b*(2*A*b*e-13*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^3/(e*x+d)^11
Rubi [A] (verified)
Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number
of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37}
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{14}} \, dx=\frac {b (a+b x)^{11} (-13 a B e+2 A b e+11 b B d)}{1716 e (d+e x)^{11} (b d-a e)^3}+\frac {(a+b x)^{11} (-13 a B e+2 A b e+11 b B d)}{156 e (d+e x)^{12} (b d-a e)^2}-\frac {(a+b x)^{11} (B d-A e)}{13 e (d+e x)^{13} (b d-a e)}
\]
[In]
Int[((a + b*x)^10*(A + B*x))/(d + e*x)^14,x]
[Out]
-1/13*((B*d - A*e)*(a + b*x)^11)/(e*(b*d - a*e)*(d + e*x)^13) + ((11*b*B*d + 2*A*b*e - 13*a*B*e)*(a + b*x)^11)
/(156*e*(b*d - a*e)^2*(d + e*x)^12) + (b*(11*b*B*d + 2*A*b*e - 13*a*B*e)*(a + b*x)^11)/(1716*e*(b*d - a*e)^3*(
d + e*x)^11)
Rule 37
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] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]
Rule 47
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] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] || !SumSimplerQ[n, 1])
Rule 79
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(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] || IntegerQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || L
tQ[p, n]))))
Rubi steps \begin{align*}
\text {integral}& = -\frac {(B d-A e) (a+b x)^{11}}{13 e (b d-a e) (d+e x)^{13}}+\frac {(11 b B d+2 A b e-13 a B e) \int \frac {(a+b x)^{10}}{(d+e x)^{13}} \, dx}{13 e (b d-a e)} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{13 e (b d-a e) (d+e x)^{13}}+\frac {(11 b B d+2 A b e-13 a B e) (a+b x)^{11}}{156 e (b d-a e)^2 (d+e x)^{12}}+\frac {(b (11 b B d+2 A b e-13 a B e)) \int \frac {(a+b x)^{10}}{(d+e x)^{12}} \, dx}{156 e (b d-a e)^2} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{13 e (b d-a e) (d+e x)^{13}}+\frac {(11 b B d+2 A b e-13 a B e) (a+b x)^{11}}{156 e (b d-a e)^2 (d+e x)^{12}}+\frac {b (11 b B d+2 A b e-13 a B e) (a+b x)^{11}}{1716 e (b d-a e)^3 (d+e x)^{11}} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(1433\) vs. \(2(135)=270\).
Time = 0.49 (sec) , antiderivative size = 1433, normalized size of antiderivative = 10.61
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{14}} \, dx=-\frac {11 a^{10} e^{10} (12 A e+B (d+13 e x))+10 a^9 b e^9 \left (11 A e (d+13 e x)+2 B \left (d^2+13 d e x+78 e^2 x^2\right )\right )+9 a^8 b^2 e^8 \left (10 A e \left (d^2+13 d e x+78 e^2 x^2\right )+3 B \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )\right )+8 a^7 b^3 e^7 \left (9 A e \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )+4 B \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )\right )+7 a^6 b^4 e^6 \left (8 A e \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )+5 B \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )\right )+6 a^5 b^5 e^5 \left (7 A e \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )+6 B \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )\right )+5 a^4 b^6 e^4 \left (6 A e \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )+7 B \left (d^7+13 d^6 e x+78 d^5 e^2 x^2+286 d^4 e^3 x^3+715 d^3 e^4 x^4+1287 d^2 e^5 x^5+1716 d e^6 x^6+1716 e^7 x^7\right )\right )+4 a^3 b^7 e^3 \left (5 A e \left (d^7+13 d^6 e x+78 d^5 e^2 x^2+286 d^4 e^3 x^3+715 d^3 e^4 x^4+1287 d^2 e^5 x^5+1716 d e^6 x^6+1716 e^7 x^7\right )+8 B \left (d^8+13 d^7 e x+78 d^6 e^2 x^2+286 d^5 e^3 x^3+715 d^4 e^4 x^4+1287 d^3 e^5 x^5+1716 d^2 e^6 x^6+1716 d e^7 x^7+1287 e^8 x^8\right )\right )+3 a^2 b^8 e^2 \left (4 A e \left (d^8+13 d^7 e x+78 d^6 e^2 x^2+286 d^5 e^3 x^3+715 d^4 e^4 x^4+1287 d^3 e^5 x^5+1716 d^2 e^6 x^6+1716 d e^7 x^7+1287 e^8 x^8\right )+9 B \left (d^9+13 d^8 e x+78 d^7 e^2 x^2+286 d^6 e^3 x^3+715 d^5 e^4 x^4+1287 d^4 e^5 x^5+1716 d^3 e^6 x^6+1716 d^2 e^7 x^7+1287 d e^8 x^8+715 e^9 x^9\right )\right )+2 a b^9 e \left (3 A e \left (d^9+13 d^8 e x+78 d^7 e^2 x^2+286 d^6 e^3 x^3+715 d^5 e^4 x^4+1287 d^4 e^5 x^5+1716 d^3 e^6 x^6+1716 d^2 e^7 x^7+1287 d e^8 x^8+715 e^9 x^9\right )+10 B \left (d^{10}+13 d^9 e x+78 d^8 e^2 x^2+286 d^7 e^3 x^3+715 d^6 e^4 x^4+1287 d^5 e^5 x^5+1716 d^4 e^6 x^6+1716 d^3 e^7 x^7+1287 d^2 e^8 x^8+715 d e^9 x^9+286 e^{10} x^{10}\right )\right )+b^{10} \left (2 A e \left (d^{10}+13 d^9 e x+78 d^8 e^2 x^2+286 d^7 e^3 x^3+715 d^6 e^4 x^4+1287 d^5 e^5 x^5+1716 d^4 e^6 x^6+1716 d^3 e^7 x^7+1287 d^2 e^8 x^8+715 d e^9 x^9+286 e^{10} x^{10}\right )+11 B \left (d^{11}+13 d^{10} e x+78 d^9 e^2 x^2+286 d^8 e^3 x^3+715 d^7 e^4 x^4+1287 d^6 e^5 x^5+1716 d^5 e^6 x^6+1716 d^4 e^7 x^7+1287 d^3 e^8 x^8+715 d^2 e^9 x^9+286 d e^{10} x^{10}+78 e^{11} x^{11}\right )\right )}{1716 e^{12} (d+e x)^{13}}
\]
[In]
Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^14,x]
[Out]
-1/1716*(11*a^10*e^10*(12*A*e + B*(d + 13*e*x)) + 10*a^9*b*e^9*(11*A*e*(d + 13*e*x) + 2*B*(d^2 + 13*d*e*x + 78
*e^2*x^2)) + 9*a^8*b^2*e^8*(10*A*e*(d^2 + 13*d*e*x + 78*e^2*x^2) + 3*B*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 286*
e^3*x^3)) + 8*a^7*b^3*e^7*(9*A*e*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 286*e^3*x^3) + 4*B*(d^4 + 13*d^3*e*x + 78*
d^2*e^2*x^2 + 286*d*e^3*x^3 + 715*e^4*x^4)) + 7*a^6*b^4*e^6*(8*A*e*(d^4 + 13*d^3*e*x + 78*d^2*e^2*x^2 + 286*d*
e^3*x^3 + 715*e^4*x^4) + 5*B*(d^5 + 13*d^4*e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d*e^4*x^4 + 1287*e^5*x
^5)) + 6*a^5*b^5*e^5*(7*A*e*(d^5 + 13*d^4*e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d*e^4*x^4 + 1287*e^5*x^
5) + 6*B*(d^6 + 13*d^5*e*x + 78*d^4*e^2*x^2 + 286*d^3*e^3*x^3 + 715*d^2*e^4*x^4 + 1287*d*e^5*x^5 + 1716*e^6*x^
6)) + 5*a^4*b^6*e^4*(6*A*e*(d^6 + 13*d^5*e*x + 78*d^4*e^2*x^2 + 286*d^3*e^3*x^3 + 715*d^2*e^4*x^4 + 1287*d*e^5
*x^5 + 1716*e^6*x^6) + 7*B*(d^7 + 13*d^6*e*x + 78*d^5*e^2*x^2 + 286*d^4*e^3*x^3 + 715*d^3*e^4*x^4 + 1287*d^2*e
^5*x^5 + 1716*d*e^6*x^6 + 1716*e^7*x^7)) + 4*a^3*b^7*e^3*(5*A*e*(d^7 + 13*d^6*e*x + 78*d^5*e^2*x^2 + 286*d^4*e
^3*x^3 + 715*d^3*e^4*x^4 + 1287*d^2*e^5*x^5 + 1716*d*e^6*x^6 + 1716*e^7*x^7) + 8*B*(d^8 + 13*d^7*e*x + 78*d^6*
e^2*x^2 + 286*d^5*e^3*x^3 + 715*d^4*e^4*x^4 + 1287*d^3*e^5*x^5 + 1716*d^2*e^6*x^6 + 1716*d*e^7*x^7 + 1287*e^8*
x^8)) + 3*a^2*b^8*e^2*(4*A*e*(d^8 + 13*d^7*e*x + 78*d^6*e^2*x^2 + 286*d^5*e^3*x^3 + 715*d^4*e^4*x^4 + 1287*d^3
*e^5*x^5 + 1716*d^2*e^6*x^6 + 1716*d*e^7*x^7 + 1287*e^8*x^8) + 9*B*(d^9 + 13*d^8*e*x + 78*d^7*e^2*x^2 + 286*d^
6*e^3*x^3 + 715*d^5*e^4*x^4 + 1287*d^4*e^5*x^5 + 1716*d^3*e^6*x^6 + 1716*d^2*e^7*x^7 + 1287*d*e^8*x^8 + 715*e^
9*x^9)) + 2*a*b^9*e*(3*A*e*(d^9 + 13*d^8*e*x + 78*d^7*e^2*x^2 + 286*d^6*e^3*x^3 + 715*d^5*e^4*x^4 + 1287*d^4*e
^5*x^5 + 1716*d^3*e^6*x^6 + 1716*d^2*e^7*x^7 + 1287*d*e^8*x^8 + 715*e^9*x^9) + 10*B*(d^10 + 13*d^9*e*x + 78*d^
8*e^2*x^2 + 286*d^7*e^3*x^3 + 715*d^6*e^4*x^4 + 1287*d^5*e^5*x^5 + 1716*d^4*e^6*x^6 + 1716*d^3*e^7*x^7 + 1287*
d^2*e^8*x^8 + 715*d*e^9*x^9 + 286*e^10*x^10)) + b^10*(2*A*e*(d^10 + 13*d^9*e*x + 78*d^8*e^2*x^2 + 286*d^7*e^3*
x^3 + 715*d^6*e^4*x^4 + 1287*d^5*e^5*x^5 + 1716*d^4*e^6*x^6 + 1716*d^3*e^7*x^7 + 1287*d^2*e^8*x^8 + 715*d*e^9*
x^9 + 286*e^10*x^10) + 11*B*(d^11 + 13*d^10*e*x + 78*d^9*e^2*x^2 + 286*d^8*e^3*x^3 + 715*d^7*e^4*x^4 + 1287*d^
6*e^5*x^5 + 1716*d^5*e^6*x^6 + 1716*d^4*e^7*x^7 + 1287*d^3*e^8*x^8 + 715*d^2*e^9*x^9 + 286*d*e^10*x^10 + 78*e^
11*x^11)))/(e^12*(d + e*x)^13)
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(1900\) vs. \(2(129)=258\).
Time = 2.13 (sec) , antiderivative size = 1901, normalized size of antiderivative =
14.08
| | |
| method | result | size |
| | |
| risch |
\(\text {Expression too large to display}\) |
\(1901\) |
| default |
\(\text {Expression too large to display}\) |
\(1942\) |
| norman |
\(\text {Expression too large to display}\) |
\(1992\) |
| gosper |
\(\text {Expression too large to display}\) |
\(2233\) |
| parallelrisch |
\(\text {Expression too large to display}\) |
\(2240\) |
| | |
|
|
|
|
[In]
int((b*x+a)^10*(B*x+A)/(e*x+d)^14,x,method=_RETURNVERBOSE)
[Out]
(-1/2*b^10*B/e*x^11-1/6*b^9/e^2*(2*A*b*e+20*B*a*e+11*B*b*d)*x^10-5/12*b^8/e^3*(6*A*a*b*e^2+2*A*b^2*d*e+27*B*a^
2*e^2+20*B*a*b*d*e+11*B*b^2*d^2)*x^9-3/4*b^7/e^4*(12*A*a^2*b*e^3+6*A*a*b^2*d*e^2+2*A*b^3*d^2*e+32*B*a^3*e^3+27
*B*a^2*b*d*e^2+20*B*a*b^2*d^2*e+11*B*b^3*d^3)*x^8-b^6/e^5*(20*A*a^3*b*e^4+12*A*a^2*b^2*d*e^3+6*A*a*b^3*d^2*e^2
+2*A*b^4*d^3*e+35*B*a^4*e^4+32*B*a^3*b*d*e^3+27*B*a^2*b^2*d^2*e^2+20*B*a*b^3*d^3*e+11*B*b^4*d^4)*x^7-b^5/e^6*(
30*A*a^4*b*e^5+20*A*a^3*b^2*d*e^4+12*A*a^2*b^3*d^2*e^3+6*A*a*b^4*d^3*e^2+2*A*b^5*d^4*e+36*B*a^5*e^5+35*B*a^4*b
*d*e^4+32*B*a^3*b^2*d^2*e^3+27*B*a^2*b^3*d^3*e^2+20*B*a*b^4*d^4*e+11*B*b^5*d^5)*x^6-3/4*b^4/e^7*(42*A*a^5*b*e^
6+30*A*a^4*b^2*d*e^5+20*A*a^3*b^3*d^2*e^4+12*A*a^2*b^4*d^3*e^3+6*A*a*b^5*d^4*e^2+2*A*b^6*d^5*e+35*B*a^6*e^6+36
*B*a^5*b*d*e^5+35*B*a^4*b^2*d^2*e^4+32*B*a^3*b^3*d^3*e^3+27*B*a^2*b^4*d^4*e^2+20*B*a*b^5*d^5*e+11*B*b^6*d^6)*x
^5-5/12*b^3/e^8*(56*A*a^6*b*e^7+42*A*a^5*b^2*d*e^6+30*A*a^4*b^3*d^2*e^5+20*A*a^3*b^4*d^3*e^4+12*A*a^2*b^5*d^4*
e^3+6*A*a*b^6*d^5*e^2+2*A*b^7*d^6*e+32*B*a^7*e^7+35*B*a^6*b*d*e^6+36*B*a^5*b^2*d^2*e^5+35*B*a^4*b^3*d^3*e^4+32
*B*a^3*b^4*d^4*e^3+27*B*a^2*b^5*d^5*e^2+20*B*a*b^6*d^6*e+11*B*b^7*d^7)*x^4-1/6*b^2/e^9*(72*A*a^7*b*e^8+56*A*a^
6*b^2*d*e^7+42*A*a^5*b^3*d^2*e^6+30*A*a^4*b^4*d^3*e^5+20*A*a^3*b^5*d^4*e^4+12*A*a^2*b^6*d^5*e^3+6*A*a*b^7*d^6*
e^2+2*A*b^8*d^7*e+27*B*a^8*e^8+32*B*a^7*b*d*e^7+35*B*a^6*b^2*d^2*e^6+36*B*a^5*b^3*d^3*e^5+35*B*a^4*b^4*d^4*e^4
+32*B*a^3*b^5*d^5*e^3+27*B*a^2*b^6*d^6*e^2+20*B*a*b^7*d^7*e+11*B*b^8*d^8)*x^3-1/22*b/e^10*(90*A*a^8*b*e^9+72*A
*a^7*b^2*d*e^8+56*A*a^6*b^3*d^2*e^7+42*A*a^5*b^4*d^3*e^6+30*A*a^4*b^5*d^4*e^5+20*A*a^3*b^6*d^5*e^4+12*A*a^2*b^
7*d^6*e^3+6*A*a*b^8*d^7*e^2+2*A*b^9*d^8*e+20*B*a^9*e^9+27*B*a^8*b*d*e^8+32*B*a^7*b^2*d^2*e^7+35*B*a^6*b^3*d^3*
e^6+36*B*a^5*b^4*d^4*e^5+35*B*a^4*b^5*d^5*e^4+32*B*a^3*b^6*d^6*e^3+27*B*a^2*b^7*d^7*e^2+20*B*a*b^8*d^8*e+11*B*
b^9*d^9)*x^2-1/132/e^11*(110*A*a^9*b*e^10+90*A*a^8*b^2*d*e^9+72*A*a^7*b^3*d^2*e^8+56*A*a^6*b^4*d^3*e^7+42*A*a^
5*b^5*d^4*e^6+30*A*a^4*b^6*d^5*e^5+20*A*a^3*b^7*d^6*e^4+12*A*a^2*b^8*d^7*e^3+6*A*a*b^9*d^8*e^2+2*A*b^10*d^9*e+
11*B*a^10*e^10+20*B*a^9*b*d*e^9+27*B*a^8*b^2*d^2*e^8+32*B*a^7*b^3*d^3*e^7+35*B*a^6*b^4*d^4*e^6+36*B*a^5*b^5*d^
5*e^5+35*B*a^4*b^6*d^6*e^4+32*B*a^3*b^7*d^7*e^3+27*B*a^2*b^8*d^8*e^2+20*B*a*b^9*d^9*e+11*B*b^10*d^10)*x-1/1716
/e^12*(132*A*a^10*e^11+110*A*a^9*b*d*e^10+90*A*a^8*b^2*d^2*e^9+72*A*a^7*b^3*d^3*e^8+56*A*a^6*b^4*d^4*e^7+42*A*
a^5*b^5*d^5*e^6+30*A*a^4*b^6*d^6*e^5+20*A*a^3*b^7*d^7*e^4+12*A*a^2*b^8*d^8*e^3+6*A*a*b^9*d^9*e^2+2*A*b^10*d^10
*e+11*B*a^10*d*e^10+20*B*a^9*b*d^2*e^9+27*B*a^8*b^2*d^3*e^8+32*B*a^7*b^3*d^4*e^7+35*B*a^6*b^4*d^5*e^6+36*B*a^5
*b^5*d^6*e^5+35*B*a^4*b^6*d^7*e^4+32*B*a^3*b^7*d^8*e^3+27*B*a^2*b^8*d^9*e^2+20*B*a*b^9*d^10*e+11*B*b^10*d^11))
/(e*x+d)^13
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1951 vs. \(2 (129) = 258\).
Time = 0.29 (sec) , antiderivative size = 1951, normalized size of antiderivative = 14.45
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{14}} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^14,x, algorithm="fricas")
[Out]
-1/1716*(858*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 132*A*a^10*e^11 + 2*(10*B*a*b^9 + A*b^10)*d^10*e + 3*(9*B*a^2
*b^8 + 2*A*a*b^9)*d^9*e^2 + 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 6*
(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^
4*e^7 + 9*(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 + 11*(B*a^10 + 10*A*a^9*b
)*d*e^10 + 286*(11*B*b^10*d*e^10 + 2*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 715*(11*B*b^10*d^2*e^9 + 2*(10*B*a*b^9
+ A*b^10)*d*e^10 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 1287*(11*B*b^10*d^3*e^8 + 2*(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 + 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 1716*(11*B*b^10*d^4*e
^7 + 2*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 3*(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
^10 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 1716*(11*B*b^10*d^5*e^6 + 2*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 3*
(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 5*(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 + 1287*(11*B*b^10*d^6*e^5 + 2*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 3*(
9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e
^9 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 715*(11*B*b^10*d^7*e^4 +
2*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7
+ 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 6*(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 + 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 286*(11*B*b^10*d^8*e^3 + 2*(10*B*a*b^9 + A*b^10)*d^7*e^4
+ 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)
*d^4*e^7 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 7*(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 + 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 78*(11*B*b^10*d^9*e^2 + 2*(10*B*a*b^9 + A*b^10)*
d^8*e^3 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 5*(7*B*a^4*b^6 + 4*A*a
^3*b^7)*d^5*e^6 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 8*(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 + 10*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 13
*(11*B*b^10*d^10*e + 2*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 4*(8*B*a^3*b^7 +
3*A*a^2*b^8)*d^7*e^4 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 7*(5*B*
a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9
+ 10*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 11*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^25*x^13 + 13*d*e^24*x^12 + 78*d^
2*e^23*x^11 + 286*d^3*e^22*x^10 + 715*d^4*e^21*x^9 + 1287*d^5*e^20*x^8 + 1716*d^6*e^19*x^7 + 1716*d^7*e^18*x^6
+ 1287*d^8*e^17*x^5 + 715*d^9*e^16*x^4 + 286*d^10*e^15*x^3 + 78*d^11*e^14*x^2 + 13*d^12*e^13*x + d^13*e^12)
Sympy [F(-1)]
Timed out. \[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{14}} \, dx=\text {Timed out}
\]
[In]
integrate((b*x+a)**10*(B*x+A)/(e*x+d)**14,x)
[Out]
Timed out
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1951 vs. \(2 (129) = 258\).
Time = 0.29 (sec) , antiderivative size = 1951, normalized size of antiderivative = 14.45
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{14}} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^14,x, algorithm="maxima")
[Out]
-1/1716*(858*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 132*A*a^10*e^11 + 2*(10*B*a*b^9 + A*b^10)*d^10*e + 3*(9*B*a^2
*b^8 + 2*A*a*b^9)*d^9*e^2 + 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 6*
(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^
4*e^7 + 9*(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 + 11*(B*a^10 + 10*A*a^9*b
)*d*e^10 + 286*(11*B*b^10*d*e^10 + 2*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 715*(11*B*b^10*d^2*e^9 + 2*(10*B*a*b^9
+ A*b^10)*d*e^10 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 1287*(11*B*b^10*d^3*e^8 + 2*(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 + 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 1716*(11*B*b^10*d^4*e
^7 + 2*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 3*(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
^10 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 1716*(11*B*b^10*d^5*e^6 + 2*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 3*
(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 5*(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 + 1287*(11*B*b^10*d^6*e^5 + 2*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 3*(
9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e
^9 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 715*(11*B*b^10*d^7*e^4 +
2*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7
+ 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 6*(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 + 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 286*(11*B*b^10*d^8*e^3 + 2*(10*B*a*b^9 + A*b^10)*d^7*e^4
+ 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)
*d^4*e^7 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 7*(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 + 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 78*(11*B*b^10*d^9*e^2 + 2*(10*B*a*b^9 + A*b^10)*
d^8*e^3 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 5*(7*B*a^4*b^6 + 4*A*a
^3*b^7)*d^5*e^6 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 8*(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 + 10*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 13
*(11*B*b^10*d^10*e + 2*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 4*(8*B*a^3*b^7 +
3*A*a^2*b^8)*d^7*e^4 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 7*(5*B*
a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9
+ 10*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 11*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^25*x^13 + 13*d*e^24*x^12 + 78*d^
2*e^23*x^11 + 286*d^3*e^22*x^10 + 715*d^4*e^21*x^9 + 1287*d^5*e^20*x^8 + 1716*d^6*e^19*x^7 + 1716*d^7*e^18*x^6
+ 1287*d^8*e^17*x^5 + 715*d^9*e^16*x^4 + 286*d^10*e^15*x^3 + 78*d^11*e^14*x^2 + 13*d^12*e^13*x + d^13*e^12)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2232 vs. \(2 (129) = 258\).
Time = 0.31 (sec) , antiderivative size = 2232, normalized size of antiderivative = 16.53
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{14}} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^14,x, algorithm="giac")
[Out]
-1/1716*(858*B*b^10*e^11*x^11 + 3146*B*b^10*d*e^10*x^10 + 5720*B*a*b^9*e^11*x^10 + 572*A*b^10*e^11*x^10 + 7865
*B*b^10*d^2*e^9*x^9 + 14300*B*a*b^9*d*e^10*x^9 + 1430*A*b^10*d*e^10*x^9 + 19305*B*a^2*b^8*e^11*x^9 + 4290*A*a*
b^9*e^11*x^9 + 14157*B*b^10*d^3*e^8*x^8 + 25740*B*a*b^9*d^2*e^9*x^8 + 2574*A*b^10*d^2*e^9*x^8 + 34749*B*a^2*b^
8*d*e^10*x^8 + 7722*A*a*b^9*d*e^10*x^8 + 41184*B*a^3*b^7*e^11*x^8 + 15444*A*a^2*b^8*e^11*x^8 + 18876*B*b^10*d^
4*e^7*x^7 + 34320*B*a*b^9*d^3*e^8*x^7 + 3432*A*b^10*d^3*e^8*x^7 + 46332*B*a^2*b^8*d^2*e^9*x^7 + 10296*A*a*b^9*
d^2*e^9*x^7 + 54912*B*a^3*b^7*d*e^10*x^7 + 20592*A*a^2*b^8*d*e^10*x^7 + 60060*B*a^4*b^6*e^11*x^7 + 34320*A*a^3
*b^7*e^11*x^7 + 18876*B*b^10*d^5*e^6*x^6 + 34320*B*a*b^9*d^4*e^7*x^6 + 3432*A*b^10*d^4*e^7*x^6 + 46332*B*a^2*b
^8*d^3*e^8*x^6 + 10296*A*a*b^9*d^3*e^8*x^6 + 54912*B*a^3*b^7*d^2*e^9*x^6 + 20592*A*a^2*b^8*d^2*e^9*x^6 + 60060
*B*a^4*b^6*d*e^10*x^6 + 34320*A*a^3*b^7*d*e^10*x^6 + 61776*B*a^5*b^5*e^11*x^6 + 51480*A*a^4*b^6*e^11*x^6 + 141
57*B*b^10*d^6*e^5*x^5 + 25740*B*a*b^9*d^5*e^6*x^5 + 2574*A*b^10*d^5*e^6*x^5 + 34749*B*a^2*b^8*d^4*e^7*x^5 + 77
22*A*a*b^9*d^4*e^7*x^5 + 41184*B*a^3*b^7*d^3*e^8*x^5 + 15444*A*a^2*b^8*d^3*e^8*x^5 + 45045*B*a^4*b^6*d^2*e^9*x
^5 + 25740*A*a^3*b^7*d^2*e^9*x^5 + 46332*B*a^5*b^5*d*e^10*x^5 + 38610*A*a^4*b^6*d*e^10*x^5 + 45045*B*a^6*b^4*e
^11*x^5 + 54054*A*a^5*b^5*e^11*x^5 + 7865*B*b^10*d^7*e^4*x^4 + 14300*B*a*b^9*d^6*e^5*x^4 + 1430*A*b^10*d^6*e^5
*x^4 + 19305*B*a^2*b^8*d^5*e^6*x^4 + 4290*A*a*b^9*d^5*e^6*x^4 + 22880*B*a^3*b^7*d^4*e^7*x^4 + 8580*A*a^2*b^8*d
^4*e^7*x^4 + 25025*B*a^4*b^6*d^3*e^8*x^4 + 14300*A*a^3*b^7*d^3*e^8*x^4 + 25740*B*a^5*b^5*d^2*e^9*x^4 + 21450*A
*a^4*b^6*d^2*e^9*x^4 + 25025*B*a^6*b^4*d*e^10*x^4 + 30030*A*a^5*b^5*d*e^10*x^4 + 22880*B*a^7*b^3*e^11*x^4 + 40
040*A*a^6*b^4*e^11*x^4 + 3146*B*b^10*d^8*e^3*x^3 + 5720*B*a*b^9*d^7*e^4*x^3 + 572*A*b^10*d^7*e^4*x^3 + 7722*B*
a^2*b^8*d^6*e^5*x^3 + 1716*A*a*b^9*d^6*e^5*x^3 + 9152*B*a^3*b^7*d^5*e^6*x^3 + 3432*A*a^2*b^8*d^5*e^6*x^3 + 100
10*B*a^4*b^6*d^4*e^7*x^3 + 5720*A*a^3*b^7*d^4*e^7*x^3 + 10296*B*a^5*b^5*d^3*e^8*x^3 + 8580*A*a^4*b^6*d^3*e^8*x
^3 + 10010*B*a^6*b^4*d^2*e^9*x^3 + 12012*A*a^5*b^5*d^2*e^9*x^3 + 9152*B*a^7*b^3*d*e^10*x^3 + 16016*A*a^6*b^4*d
*e^10*x^3 + 7722*B*a^8*b^2*e^11*x^3 + 20592*A*a^7*b^3*e^11*x^3 + 858*B*b^10*d^9*e^2*x^2 + 1560*B*a*b^9*d^8*e^3
*x^2 + 156*A*b^10*d^8*e^3*x^2 + 2106*B*a^2*b^8*d^7*e^4*x^2 + 468*A*a*b^9*d^7*e^4*x^2 + 2496*B*a^3*b^7*d^6*e^5*
x^2 + 936*A*a^2*b^8*d^6*e^5*x^2 + 2730*B*a^4*b^6*d^5*e^6*x^2 + 1560*A*a^3*b^7*d^5*e^6*x^2 + 2808*B*a^5*b^5*d^4
*e^7*x^2 + 2340*A*a^4*b^6*d^4*e^7*x^2 + 2730*B*a^6*b^4*d^3*e^8*x^2 + 3276*A*a^5*b^5*d^3*e^8*x^2 + 2496*B*a^7*b
^3*d^2*e^9*x^2 + 4368*A*a^6*b^4*d^2*e^9*x^2 + 2106*B*a^8*b^2*d*e^10*x^2 + 5616*A*a^7*b^3*d*e^10*x^2 + 1560*B*a
^9*b*e^11*x^2 + 7020*A*a^8*b^2*e^11*x^2 + 143*B*b^10*d^10*e*x + 260*B*a*b^9*d^9*e^2*x + 26*A*b^10*d^9*e^2*x +
351*B*a^2*b^8*d^8*e^3*x + 78*A*a*b^9*d^8*e^3*x + 416*B*a^3*b^7*d^7*e^4*x + 156*A*a^2*b^8*d^7*e^4*x + 455*B*a^4
*b^6*d^6*e^5*x + 260*A*a^3*b^7*d^6*e^5*x + 468*B*a^5*b^5*d^5*e^6*x + 390*A*a^4*b^6*d^5*e^6*x + 455*B*a^6*b^4*d
^4*e^7*x + 546*A*a^5*b^5*d^4*e^7*x + 416*B*a^7*b^3*d^3*e^8*x + 728*A*a^6*b^4*d^3*e^8*x + 351*B*a^8*b^2*d^2*e^9
*x + 936*A*a^7*b^3*d^2*e^9*x + 260*B*a^9*b*d*e^10*x + 1170*A*a^8*b^2*d*e^10*x + 143*B*a^10*e^11*x + 1430*A*a^9
*b*e^11*x + 11*B*b^10*d^11 + 20*B*a*b^9*d^10*e + 2*A*b^10*d^10*e + 27*B*a^2*b^8*d^9*e^2 + 6*A*a*b^9*d^9*e^2 +
32*B*a^3*b^7*d^8*e^3 + 12*A*a^2*b^8*d^8*e^3 + 35*B*a^4*b^6*d^7*e^4 + 20*A*a^3*b^7*d^7*e^4 + 36*B*a^5*b^5*d^6*e
^5 + 30*A*a^4*b^6*d^6*e^5 + 35*B*a^6*b^4*d^5*e^6 + 42*A*a^5*b^5*d^5*e^6 + 32*B*a^7*b^3*d^4*e^7 + 56*A*a^6*b^4*
d^4*e^7 + 27*B*a^8*b^2*d^3*e^8 + 72*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 + 11*B*a^10*
d*e^10 + 110*A*a^9*b*d*e^10 + 132*A*a^10*e^11)/((e*x + d)^13*e^12)
Mupad [B] (verification not implemented)
Time = 2.18 (sec) , antiderivative size = 2031, normalized size of antiderivative = 15.04
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{14}} \, dx=\text {Too large to display}
\]
[In]
int(((A + B*x)*(a + b*x)^10)/(d + e*x)^14,x)
[Out]
-((132*A*a^10*e^11 + 11*B*b^10*d^11 + 2*A*b^10*d^10*e + 11*B*a^10*d*e^10 + 6*A*a*b^9*d^9*e^2 + 20*B*a^9*b*d^2*
e^9 + 12*A*a^2*b^8*d^8*e^3 + 20*A*a^3*b^7*d^7*e^4 + 30*A*a^4*b^6*d^6*e^5 + 42*A*a^5*b^5*d^5*e^6 + 56*A*a^6*b^4
*d^4*e^7 + 72*A*a^7*b^3*d^3*e^8 + 90*A*a^8*b^2*d^2*e^9 + 27*B*a^2*b^8*d^9*e^2 + 32*B*a^3*b^7*d^8*e^3 + 35*B*a^
4*b^6*d^7*e^4 + 36*B*a^5*b^5*d^6*e^5 + 35*B*a^6*b^4*d^5*e^6 + 32*B*a^7*b^3*d^4*e^7 + 27*B*a^8*b^2*d^3*e^8 + 11
0*A*a^9*b*d*e^10 + 20*B*a*b^9*d^10*e)/(1716*e^12) + (x*(11*B*a^10*e^10 + 11*B*b^10*d^10 + 110*A*a^9*b*e^10 + 2
*A*b^10*d^9*e + 6*A*a*b^9*d^8*e^2 + 90*A*a^8*b^2*d*e^9 + 12*A*a^2*b^8*d^7*e^3 + 20*A*a^3*b^7*d^6*e^4 + 30*A*a^
4*b^6*d^5*e^5 + 42*A*a^5*b^5*d^4*e^6 + 56*A*a^6*b^4*d^3*e^7 + 72*A*a^7*b^3*d^2*e^8 + 27*B*a^2*b^8*d^8*e^2 + 32
*B*a^3*b^7*d^7*e^3 + 35*B*a^4*b^6*d^6*e^4 + 36*B*a^5*b^5*d^5*e^5 + 35*B*a^6*b^4*d^4*e^6 + 32*B*a^7*b^3*d^3*e^7
+ 27*B*a^8*b^2*d^2*e^8 + 20*B*a*b^9*d^9*e + 20*B*a^9*b*d*e^9))/(132*e^11) + (3*b^7*x^8*(32*B*a^3*e^3 + 11*B*b
^3*d^3 + 12*A*a^2*b*e^3 + 2*A*b^3*d^2*e + 6*A*a*b^2*d*e^2 + 20*B*a*b^2*d^2*e + 27*B*a^2*b*d*e^2))/(4*e^4) + (3
*b^4*x^5*(35*B*a^6*e^6 + 11*B*b^6*d^6 + 42*A*a^5*b*e^6 + 2*A*b^6*d^5*e + 6*A*a*b^5*d^4*e^2 + 30*A*a^4*b^2*d*e^
5 + 12*A*a^2*b^4*d^3*e^3 + 20*A*a^3*b^3*d^2*e^4 + 27*B*a^2*b^4*d^4*e^2 + 32*B*a^3*b^3*d^3*e^3 + 35*B*a^4*b^2*d
^2*e^4 + 20*B*a*b^5*d^5*e + 36*B*a^5*b*d*e^5))/(4*e^7) + (b^9*x^10*(2*A*b*e + 20*B*a*e + 11*B*b*d))/(6*e^2) +
(b^6*x^7*(35*B*a^4*e^4 + 11*B*b^4*d^4 + 20*A*a^3*b*e^4 + 2*A*b^4*d^3*e + 6*A*a*b^3*d^2*e^2 + 12*A*a^2*b^2*d*e^
3 + 27*B*a^2*b^2*d^2*e^2 + 20*B*a*b^3*d^3*e + 32*B*a^3*b*d*e^3))/e^5 + (5*b^3*x^4*(32*B*a^7*e^7 + 11*B*b^7*d^7
+ 56*A*a^6*b*e^7 + 2*A*b^7*d^6*e + 6*A*a*b^6*d^5*e^2 + 42*A*a^5*b^2*d*e^6 + 12*A*a^2*b^5*d^4*e^3 + 20*A*a^3*b
^4*d^3*e^4 + 30*A*a^4*b^3*d^2*e^5 + 27*B*a^2*b^5*d^5*e^2 + 32*B*a^3*b^4*d^4*e^3 + 35*B*a^4*b^3*d^3*e^4 + 36*B*
a^5*b^2*d^2*e^5 + 20*B*a*b^6*d^6*e + 35*B*a^6*b*d*e^6))/(12*e^8) + (b*x^2*(20*B*a^9*e^9 + 11*B*b^9*d^9 + 90*A*
a^8*b*e^9 + 2*A*b^9*d^8*e + 6*A*a*b^8*d^7*e^2 + 72*A*a^7*b^2*d*e^8 + 12*A*a^2*b^7*d^6*e^3 + 20*A*a^3*b^6*d^5*e
^4 + 30*A*a^4*b^5*d^4*e^5 + 42*A*a^5*b^4*d^3*e^6 + 56*A*a^6*b^3*d^2*e^7 + 27*B*a^2*b^7*d^7*e^2 + 32*B*a^3*b^6*
d^6*e^3 + 35*B*a^4*b^5*d^5*e^4 + 36*B*a^5*b^4*d^4*e^5 + 35*B*a^6*b^3*d^3*e^6 + 32*B*a^7*b^2*d^2*e^7 + 20*B*a*b
^8*d^8*e + 27*B*a^8*b*d*e^8))/(22*e^10) + (5*b^8*x^9*(27*B*a^2*e^2 + 11*B*b^2*d^2 + 6*A*a*b*e^2 + 2*A*b^2*d*e
+ 20*B*a*b*d*e))/(12*e^3) + (b^5*x^6*(36*B*a^5*e^5 + 11*B*b^5*d^5 + 30*A*a^4*b*e^5 + 2*A*b^5*d^4*e + 6*A*a*b^4
*d^3*e^2 + 20*A*a^3*b^2*d*e^4 + 12*A*a^2*b^3*d^2*e^3 + 27*B*a^2*b^3*d^3*e^2 + 32*B*a^3*b^2*d^2*e^3 + 20*B*a*b^
4*d^4*e + 35*B*a^4*b*d*e^4))/e^6 + (b^2*x^3*(27*B*a^8*e^8 + 11*B*b^8*d^8 + 72*A*a^7*b*e^8 + 2*A*b^8*d^7*e + 6*
A*a*b^7*d^6*e^2 + 56*A*a^6*b^2*d*e^7 + 12*A*a^2*b^6*d^5*e^3 + 20*A*a^3*b^5*d^4*e^4 + 30*A*a^4*b^4*d^3*e^5 + 42
*A*a^5*b^3*d^2*e^6 + 27*B*a^2*b^6*d^6*e^2 + 32*B*a^3*b^5*d^5*e^3 + 35*B*a^4*b^4*d^4*e^4 + 36*B*a^5*b^3*d^3*e^5
+ 35*B*a^6*b^2*d^2*e^6 + 20*B*a*b^7*d^7*e + 32*B*a^7*b*d*e^7))/(6*e^9) + (B*b^10*x^11)/(2*e))/(d^13 + e^13*x^
13 + 13*d*e^12*x^12 + 78*d^11*e^2*x^2 + 286*d^10*e^3*x^3 + 715*d^9*e^4*x^4 + 1287*d^8*e^5*x^5 + 1716*d^7*e^6*x
^6 + 1716*d^6*e^7*x^7 + 1287*d^5*e^8*x^8 + 715*d^4*e^9*x^9 + 286*d^3*e^10*x^10 + 78*d^2*e^11*x^11 + 13*d^12*e*
x)