\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx\) [1101]

Optimal result

Integrand size = 20, antiderivative size = 86 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx=-\frac {(B d-A e) (a+b x)^{11}}{12 e (b d-a e) (d+e x)^{12}}+\frac {(11 b B d+A b e-12 a B e) (a+b x)^{11}}{132 e (b d-a e)^2 (d+e x)^{11}} \]

[Out]

-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

Rubi [A] (verified)

Time = 0.02 (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 = {79, 37} \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx=\frac {(a+b x)^{11} (-12 a B e+A b e+11 b B d)}{132 e (d+e x)^{11} (b d-a e)^2}-\frac {(a+b x)^{11} (B d-A e)}{12 e (d+e x)^{12} (b d-a e)} \]

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^13,x]

[Out]

-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)

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 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}}{12 e (b d-a e) (d+e x)^{12}}+\frac {(11 b B d+A b e-12 a B e) \int \frac {(a+b x)^{10}}{(d+e x)^{12}} \, dx}{12 e (b d-a e)} \\ & = -\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}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1421\) vs. \(2(86)=172\).

Time = 0.49 (sec) , antiderivative size = 1421, normalized size of antiderivative = 16.52 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx=-\frac {a^{10} e^{10} (11 A e+B (d+12 e x))+2 a^9 b e^9 \left (5 A e (d+12 e x)+B \left (d^2+12 d e x+66 e^2 x^2\right )\right )+3 a^8 b^2 e^8 \left (3 A e \left (d^2+12 d e x+66 e^2 x^2\right )+B \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )\right )+4 a^7 b^3 e^7 \left (2 A e \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )+B \left (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\right )\right )+a^6 b^4 e^6 \left (7 A e \left (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\right )+5 B \left (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\right )\right )+6 a^5 b^5 e^5 \left (A e \left (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\right )+B \left (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\right )\right )+a^4 b^6 e^4 \left (5 A e \left (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\right )+7 B \left (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\right )\right )+4 a^3 b^7 e^3 \left (A e \left (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\right )+2 B \left (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\right )\right )+3 a^2 b^8 e^2 \left (A e \left (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\right )+3 B \left (d^9+12 d^8 e x+66 d^7 e^2 x^2+220 d^6 e^3 x^3+495 d^5 e^4 x^4+792 d^4 e^5 x^5+924 d^3 e^6 x^6+792 d^2 e^7 x^7+495 d e^8 x^8+220 e^9 x^9\right )\right )+2 a b^9 e \left (A e \left (d^9+12 d^8 e x+66 d^7 e^2 x^2+220 d^6 e^3 x^3+495 d^5 e^4 x^4+792 d^4 e^5 x^5+924 d^3 e^6 x^6+792 d^2 e^7 x^7+495 d e^8 x^8+220 e^9 x^9\right )+5 B \left (d^{10}+12 d^9 e x+66 d^8 e^2 x^2+220 d^7 e^3 x^3+495 d^6 e^4 x^4+792 d^5 e^5 x^5+924 d^4 e^6 x^6+792 d^3 e^7 x^7+495 d^2 e^8 x^8+220 d e^9 x^9+66 e^{10} x^{10}\right )\right )+b^{10} \left (A e \left (d^{10}+12 d^9 e x+66 d^8 e^2 x^2+220 d^7 e^3 x^3+495 d^6 e^4 x^4+792 d^5 e^5 x^5+924 d^4 e^6 x^6+792 d^3 e^7 x^7+495 d^2 e^8 x^8+220 d e^9 x^9+66 e^{10} x^{10}\right )+11 B \left (d^{11}+12 d^{10} e x+66 d^9 e^2 x^2+220 d^8 e^3 x^3+495 d^7 e^4 x^4+792 d^6 e^5 x^5+924 d^5 e^6 x^6+792 d^4 e^7 x^7+495 d^3 e^8 x^8+220 d^2 e^9 x^9+66 d e^{10} x^{10}+12 e^{11} x^{11}\right )\right )}{132 e^{12} (d+e x)^{12}} \]

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^13,x]

[Out]

-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 + 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)) + 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 + 792*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^7*e^2*x^2 + 220*d^6*e^3*x^3 + 495*d^5*e^4*x^4 + 792*d^4*e^5*x^5
+ 924*d^3*e^6*x^6 + 792*d^2*e^7*x^7 + 495*d*e^8*x^8 + 220*e^9*x^9)) + 2*a*b^9*e*(A*e*(d^9 + 12*d^8*e*x + 66*d
^7*e^2*x^2 + 220*d^6*e^3*x^3 + 495*d^5*e^4*x^4 + 792*d^4*e^5*x^5 + 924*d^3*e^6*x^6 + 792*d^2*e^7*x^7 + 495*d*e
^8*x^8 + 220*e^9*x^9) + 5*B*(d^10 + 12*d^9*e*x + 66*d^8*e^2*x^2 + 220*d^7*e^3*x^3 + 495*d^6*e^4*x^4 + 792*d^5*
e^5*x^5 + 924*d^4*e^6*x^6 + 792*d^3*e^7*x^7 + 495*d^2*e^8*x^8 + 220*d*e^9*x^9 + 66*e^10*x^10)) + b^10*(A*e*(d^
10 + 12*d^9*e*x + 66*d^8*e^2*x^2 + 220*d^7*e^3*x^3 + 495*d^6*e^4*x^4 + 792*d^5*e^5*x^5 + 924*d^4*e^6*x^6 + 792
*d^3*e^7*x^7 + 495*d^2*e^8*x^8 + 220*d*e^9*x^9 + 66*e^10*x^10) + 11*B*(d^11 + 12*d^10*e*x + 66*d^9*e^2*x^2 + 2
20*d^8*e^3*x^3 + 495*d^7*e^4*x^4 + 792*d^6*e^5*x^5 + 924*d^5*e^6*x^6 + 792*d^4*e^7*x^7 + 495*d^3*e^8*x^8 + 220
*d^2*e^9*x^9 + 66*d*e^10*x^10 + 12*e^11*x^11)))/(e^12*(d + e*x)^12)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1887\) vs. \(2(82)=164\).

Time = 2.15 (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	\(\text {Expression too large to display}\)	\(2231\)

[In]

int((b*x+a)^10*(B*x+A)/(e*x+d)^13,x,method=_RETURNVERBOSE)

[Out]

(-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^8+7*A*a^6*b^3*d^2*e^7+6*A*a^5*b^4*d^3*e^6+5*A*a^4*b^5*d^4*e^5+
4*A*a^3*b^6*d^5*e^4+3*A*a^2*b^7*d^6*e^3+2*A*a*b^8*d^7*e^2+A*b^9*d^8*e+2*B*a^9*e^9+3*B*a^8*b*d*e^8+4*B*a^7*b^2*
d^2*e^7+5*B*a^6*b^3*d^3*e^6+6*B*a^5*b^4*d^4*e^5+7*B*a^4*b^5*d^5*e^4+8*B*a^3*b^6*d^6*e^3+9*B*a^2*b^7*d^7*e^2+10
*B*a*b^8*d^8*e+11*B*b^9*d^9)/e^10*x^2-1/11*(10*A*a^9*b*e^10+9*A*a^8*b^2*d*e^9+8*A*a^7*b^3*d^2*e^8+7*A*a^6*b^4*
d^3*e^7+6*A*a^5*b^5*d^4*e^6+5*A*a^4*b^6*d^5*e^5+4*A*a^3*b^7*d^6*e^4+3*A*a^2*b^8*d^7*e^3+2*A*a*b^9*d^8*e^2+A*b^
10*d^9*e+B*a^10*e^10+2*B*a^9*b*d*e^9+3*B*a^8*b^2*d^2*e^8+4*B*a^7*b^3*d^3*e^7+5*B*a^6*b^4*d^4*e^6+6*B*a^5*b^5*d
^5*e^5+7*B*a^4*b^6*d^6*e^4+8*B*a^3*b^7*d^7*e^3+9*B*a^2*b^8*d^8*e^2+10*B*a*b^9*d^9*e+11*B*b^10*d^10)/e^11*x-1/1
32*(11*A*a^10*e^11+10*A*a^9*b*d*e^10+9*A*a^8*b^2*d^2*e^9+8*A*a^7*b^3*d^3*e^8+7*A*a^6*b^4*d^4*e^7+6*A*a^5*b^5*d
^5*e^6+5*A*a^4*b^6*d^6*e^5+4*A*a^3*b^7*d^7*e^4+3*A*a^2*b^8*d^8*e^3+2*A*a*b^9*d^9*e^2+A*b^10*d^10*e+B*a^10*d*e^
10+2*B*a^9*b*d^2*e^9+3*B*a^8*b^2*d^3*e^8+4*B*a^7*b^3*d^4*e^7+5*B*a^6*b^4*d^5*e^6+6*B*a^5*b^5*d^6*e^5+7*B*a^4*b
^6*d^7*e^4+8*B*a^3*b^7*d^8*e^3+9*B*a^2*b^8*d^9*e^2+10*B*a*b^9*d^10*e+11*B*b^10*d^11)/e^12)/(e*x+d)^12

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1875 vs. \(2 (82) = 164\).

Time = 0.28 (sec) , antiderivative size = 1875, normalized size of antiderivative = 21.80 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^13,x, algorithm="fricas")

[Out]

-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^10*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^6 + 4*A*a^3*b^7)*d^3*e^8 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9
+ (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 + 220*(11*B*b^10*d^8*e^3 + (10*B*
a*b^9 + A*b^10)*d^7*e^4 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + (7*B*a^4*b
^6 + 4*A*a^3*b^7)*d^4*e^7 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + (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 + 66*(11*B*b^10*d^9*e^2 + (10*B*a*b^9 + A
*b^10)*d^8*e^3 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + (7*B*a^4*b^6 + 4*A*
a^3*b^7)*d^5*e^6 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + (4*B*a^7*b^3 +
7*A*a^6*b^4)*d^2*e^9 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + (2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 12*(11*B*b^1
0*d^10*e + (10*B*a*b^9 + A*b^10)*d^9*e^2 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7
*e^4 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + (5*B*a^6*b^4 + 6*A*a^5*b^5)
*d^4*e^7 + (4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + (2*B*a^9*b + 9*A*a^8*b^
2)*d*e^10 + (B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^24*x^12 + 12*d*e^23*x^11 + 66*d^2*e^22*x^10 + 220*d^3*e^21*x^9 +
495*d^4*e^20*x^8 + 792*d^5*e^19*x^7 + 924*d^6*e^18*x^6 + 792*d^7*e^17*x^5 + 495*d^8*e^16*x^4 + 220*d^9*e^15*x
^3 + 66*d^10*e^14*x^2 + 12*d^11*e^13*x + d^12*e^12)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**13,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1875 vs. \(2 (82) = 164\).

Time = 0.30 (sec) , antiderivative size = 1875, normalized size of antiderivative = 21.80 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^13,x, algorithm="maxima")

[Out]

-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^10*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^6 + 4*A*a^3*b^7)*d^3*e^8 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9
+ (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 + 220*(11*B*b^10*d^8*e^3 + (10*B*
a*b^9 + A*b^10)*d^7*e^4 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + (7*B*a^4*b
^6 + 4*A*a^3*b^7)*d^4*e^7 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + (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 + 66*(11*B*b^10*d^9*e^2 + (10*B*a*b^9 + A
*b^10)*d^8*e^3 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + (7*B*a^4*b^6 + 4*A*
a^3*b^7)*d^5*e^6 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + (4*B*a^7*b^3 +
7*A*a^6*b^4)*d^2*e^9 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + (2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 12*(11*B*b^1
0*d^10*e + (10*B*a*b^9 + A*b^10)*d^9*e^2 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7
*e^4 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + (5*B*a^6*b^4 + 6*A*a^5*b^5)
*d^4*e^7 + (4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + (2*B*a^9*b + 9*A*a^8*b^
2)*d*e^10 + (B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^24*x^12 + 12*d*e^23*x^11 + 66*d^2*e^22*x^10 + 220*d^3*e^21*x^9 +
495*d^4*e^20*x^8 + 792*d^5*e^19*x^7 + 924*d^6*e^18*x^6 + 792*d^7*e^17*x^5 + 495*d^8*e^16*x^4 + 220*d^9*e^15*x
^3 + 66*d^10*e^14*x^2 + 12*d^11*e^13*x + d^12*e^12)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2230 vs. \(2 (82) = 164\).

Time = 0.31 (sec) , antiderivative size = 2230, normalized size of antiderivative = 25.93 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^13,x, algorithm="giac")

[Out]

-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 + 1584*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^10*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*A*a^2*b^8*d^4*e^7*x^4 + 3465*B*a^4*b^6*d^3*e^8*x^4 + 1980*A*a^
3*b^7*d^3*e^8*x^4 + 2970*B*a^5*b^5*d^2*e^9*x^4 + 2475*A*a^4*b^6*d^2*e^9*x^4 + 2475*B*a^6*b^4*d*e^10*x^4 + 2970
*A*a^5*b^5*d*e^10*x^4 + 1980*B*a^7*b^3*e^11*x^4 + 3465*A*a^6*b^4*e^11*x^4 + 2420*B*b^10*d^8*e^3*x^3 + 2200*B*a
*b^9*d^7*e^4*x^3 + 220*A*b^10*d^7*e^4*x^3 + 1980*B*a^2*b^8*d^6*e^5*x^3 + 440*A*a*b^9*d^6*e^5*x^3 + 1760*B*a^3*
b^7*d^5*e^6*x^3 + 660*A*a^2*b^8*d^5*e^6*x^3 + 1540*B*a^4*b^6*d^4*e^7*x^3 + 880*A*a^3*b^7*d^4*e^7*x^3 + 1320*B*
a^5*b^5*d^3*e^8*x^3 + 1100*A*a^4*b^6*d^3*e^8*x^3 + 1100*B*a^6*b^4*d^2*e^9*x^3 + 1320*A*a^5*b^5*d^2*e^9*x^3 + 8
80*B*a^7*b^3*d*e^10*x^3 + 1540*A*a^6*b^4*d*e^10*x^3 + 660*B*a^8*b^2*e^11*x^3 + 1760*A*a^7*b^3*e^11*x^3 + 726*B
*b^10*d^9*e^2*x^2 + 660*B*a*b^9*d^8*e^3*x^2 + 66*A*b^10*d^8*e^3*x^2 + 594*B*a^2*b^8*d^7*e^4*x^2 + 132*A*a*b^9*
d^7*e^4*x^2 + 528*B*a^3*b^7*d^6*e^5*x^2 + 198*A*a^2*b^8*d^6*e^5*x^2 + 462*B*a^4*b^6*d^5*e^6*x^2 + 264*A*a^3*b^
7*d^5*e^6*x^2 + 396*B*a^5*b^5*d^4*e^7*x^2 + 330*A*a^4*b^6*d^4*e^7*x^2 + 330*B*a^6*b^4*d^3*e^8*x^2 + 396*A*a^5*
b^5*d^3*e^8*x^2 + 264*B*a^7*b^3*d^2*e^9*x^2 + 462*A*a^6*b^4*d^2*e^9*x^2 + 198*B*a^8*b^2*d*e^10*x^2 + 528*A*a^7
*b^3*d*e^10*x^2 + 132*B*a^9*b*e^11*x^2 + 594*A*a^8*b^2*e^11*x^2 + 132*B*b^10*d^10*e*x + 120*B*a*b^9*d^9*e^2*x
+ 12*A*b^10*d^9*e^2*x + 108*B*a^2*b^8*d^8*e^3*x + 24*A*a*b^9*d^8*e^3*x + 96*B*a^3*b^7*d^7*e^4*x + 36*A*a^2*b^8
*d^7*e^4*x + 84*B*a^4*b^6*d^6*e^5*x + 48*A*a^3*b^7*d^6*e^5*x + 72*B*a^5*b^5*d^5*e^6*x + 60*A*a^4*b^6*d^5*e^6*x
+ 60*B*a^6*b^4*d^4*e^7*x + 72*A*a^5*b^5*d^4*e^7*x + 48*B*a^7*b^3*d^3*e^8*x + 84*A*a^6*b^4*d^3*e^8*x + 36*B*a^
8*b^2*d^2*e^9*x + 96*A*a^7*b^3*d^2*e^9*x + 24*B*a^9*b*d*e^10*x + 108*A*a^8*b^2*d*e^10*x + 12*B*a^10*e^11*x + 1
20*A*a^9*b*e^11*x + 11*B*b^10*d^11 + 10*B*a*b^9*d^10*e + A*b^10*d^10*e + 9*B*a^2*b^8*d^9*e^2 + 2*A*a*b^9*d^9*e
^2 + 8*B*a^3*b^7*d^8*e^3 + 3*A*a^2*b^8*d^8*e^3 + 7*B*a^4*b^6*d^7*e^4 + 4*A*a^3*b^7*d^7*e^4 + 6*B*a^5*b^5*d^6*e
^5 + 5*A*a^4*b^6*d^6*e^5 + 5*B*a^6*b^4*d^5*e^6 + 6*A*a^5*b^5*d^5*e^6 + 4*B*a^7*b^3*d^4*e^7 + 7*A*a^6*b^4*d^4*e
^7 + 3*B*a^8*b^2*d^3*e^8 + 8*A*a^7*b^3*d^3*e^8 + 2*B*a^9*b*d^2*e^9 + 9*A*a^8*b^2*d^2*e^9 + B*a^10*d*e^10 + 10*
A*a^9*b*d*e^10 + 11*A*a^10*e^11)/((e*x + d)^12*e^12)

Mupad [B] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 2008, normalized size of antiderivative = 23.35 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx=\text {Too large to display} \]

[In]

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^13,x)

[Out]

-((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^10 + 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*B*a^2*b^2*d^2*e^2 + 10*B*a*b^3*d^3*e + 8*B*a^3*b*d*e^3))/e^5 +
(15*b^3*x^4*(4*B*a^7*e^7 + 11*B*b^7*d^7 + 7*A*a^6*b*e^7 + A*b^7*d^6*e + 2*A*a*b^6*d^5*e^2 + 6*A*a^5*b^2*d*e^6
+ 3*A*a^2*b^5*d^4*e^3 + 4*A*a^3*b^4*d^3*e^4 + 5*A*a^4*b^3*d^2*e^5 + 9*B*a^2*b^5*d^5*e^2 + 8*B*a^3*b^4*d^4*e^3
+ 7*B*a^4*b^3*d^3*e^4 + 6*B*a^5*b^2*d^2*e^5 + 10*B*a*b^6*d^6*e + 5*B*a^6*b*d*e^6))/(4*e^8) + (b*x^2*(2*B*a^9*e
^9 + 11*B*b^9*d^9 + 9*A*a^8*b*e^9 + A*b^9*d^8*e + 2*A*a*b^8*d^7*e^2 + 8*A*a^7*b^2*d*e^8 + 3*A*a^2*b^7*d^6*e^3
+ 4*A*a^3*b^6*d^5*e^4 + 5*A*a^4*b^5*d^4*e^5 + 6*A*a^5*b^4*d^3*e^6 + 7*A*a^6*b^3*d^2*e^7 + 9*B*a^2*b^7*d^7*e^2
+ 8*B*a^3*b^6*d^6*e^3 + 7*B*a^4*b^5*d^5*e^4 + 6*B*a^5*b^4*d^4*e^5 + 5*B*a^6*b^3*d^3*e^6 + 4*B*a^7*b^2*d^2*e^7
+ 10*B*a*b^8*d^8*e + 3*B*a^8*b*d*e^8))/(2*e^10) + (5*b^8*x^9*(9*B*a^2*e^2 + 11*B*b^2*d^2 + 2*A*a*b*e^2 + A*b^2
*d*e + 10*B*a*b*d*e))/(3*e^3) + (7*b^5*x^6*(6*B*a^5*e^5 + 11*B*b^5*d^5 + 5*A*a^4*b*e^5 + A*b^5*d^4*e + 2*A*a*b
^4*d^3*e^2 + 4*A*a^3*b^2*d*e^4 + 3*A*a^2*b^3*d^2*e^3 + 9*B*a^2*b^3*d^3*e^2 + 8*B*a^3*b^2*d^2*e^3 + 10*B*a*b^4*
d^4*e + 7*B*a^4*b*d*e^4))/e^6 + (5*b^2*x^3*(3*B*a^8*e^8 + 11*B*b^8*d^8 + 8*A*a^7*b*e^8 + A*b^8*d^7*e + 2*A*a*b
^7*d^6*e^2 + 7*A*a^6*b^2*d*e^7 + 3*A*a^2*b^6*d^5*e^3 + 4*A*a^3*b^5*d^4*e^4 + 5*A*a^4*b^4*d^3*e^5 + 6*A*a^5*b^3
*d^2*e^6 + 9*B*a^2*b^6*d^6*e^2 + 8*B*a^3*b^5*d^5*e^3 + 7*B*a^4*b^4*d^4*e^4 + 6*B*a^5*b^3*d^3*e^5 + 5*B*a^6*b^2
*d^2*e^6 + 10*B*a*b^7*d^7*e + 4*B*a^7*b*d*e^7))/(3*e^9) + (B*b^10*x^11)/e)/(d^12 + e^12*x^12 + 12*d*e^11*x^11
+ 66*d^10*e^2*x^2 + 220*d^9*e^3*x^3 + 495*d^8*e^4*x^4 + 792*d^7*e^5*x^5 + 924*d^6*e^6*x^6 + 792*d^5*e^7*x^7 +
495*d^4*e^8*x^8 + 220*d^3*e^9*x^9 + 66*d^2*e^10*x^10 + 12*d^11*e*x)