3.1099 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx\)
Optimal. Leaf size=446 \[ -\frac{5 b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{e^{12} (d+e x)}+\frac{15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{2 e^{12} (d+e x)^2}-\frac{10 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12} (d+e x)^3}+\frac{21 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{2 e^{12} (d+e x)^4}-\frac{42 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{5 e^{12} (d+e x)^5}+\frac{5 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^6}-\frac{15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{7 e^{12} (d+e x)^7}-\frac{b^9 \log (d+e x) (-10 a B e-A b e+11 b B d)}{e^{12}}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{8 e^{12} (d+e x)^8}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{9 e^{12} (d+e x)^9}+\frac{(b d-a e)^{10} (B d-A e)}{10 e^{12} (d+e x)^{10}}+\frac{b^{10} B x}{e^{11}} \]
[Out]
(b^10*B*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(10*e^12*(d + e*x)^10) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e -
a*B*e))/(9*e^12*(d + e*x)^9) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(8*e^12*(d + e*x)^8) - (15*b
^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(7*e^12*(d + e*x)^7) + (5*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b
*e - 4*a*B*e))/(e^12*(d + e*x)^6) - (42*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(5*e^12*(d + e*x)^5)
+ (21*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(2*e^12*(d + e*x)^4) - (10*b^6*(b*d - a*e)^3*(11*b*B*
d - 4*A*b*e - 7*a*B*e))/(e^12*(d + e*x)^3) + (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e))/(2*e^12*(d
+ e*x)^2) - (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e))/(e^12*(d + e*x)) - (b^9*(11*b*B*d - A*b*e - 10*
a*B*e)*Log[d + e*x])/e^12
________________________________________________________________________________________
Rubi [A] time = 0.795358, antiderivative size = 446, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used =
{77} \[ -\frac{5 b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{e^{12} (d+e x)}+\frac{15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{2 e^{12} (d+e x)^2}-\frac{10 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12} (d+e x)^3}+\frac{21 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{2 e^{12} (d+e x)^4}-\frac{42 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{5 e^{12} (d+e x)^5}+\frac{5 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^6}-\frac{15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{7 e^{12} (d+e x)^7}-\frac{b^9 \log (d+e x) (-10 a B e-A b e+11 b B d)}{e^{12}}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{8 e^{12} (d+e x)^8}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{9 e^{12} (d+e x)^9}+\frac{(b d-a e)^{10} (B d-A e)}{10 e^{12} (d+e x)^{10}}+\frac{b^{10} B x}{e^{11}} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*x)^10*(A + B*x))/(d + e*x)^11,x]
[Out]
(b^10*B*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(10*e^12*(d + e*x)^10) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e -
a*B*e))/(9*e^12*(d + e*x)^9) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(8*e^12*(d + e*x)^8) - (15*b
^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(7*e^12*(d + e*x)^7) + (5*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b
*e - 4*a*B*e))/(e^12*(d + e*x)^6) - (42*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(5*e^12*(d + e*x)^5)
+ (21*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(2*e^12*(d + e*x)^4) - (10*b^6*(b*d - a*e)^3*(11*b*B*
d - 4*A*b*e - 7*a*B*e))/(e^12*(d + e*x)^3) + (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e))/(2*e^12*(d
+ e*x)^2) - (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e))/(e^12*(d + e*x)) - (b^9*(11*b*B*d - A*b*e - 10*
a*B*e)*Log[d + e*x])/e^12
Rule 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((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]))))
Rubi steps
\begin{align*} \int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx &=\int \left (\frac{b^{10} B}{e^{11}}+\frac{(-b d+a e)^{10} (-B d+A e)}{e^{11} (d+e x)^{11}}+\frac{(-b d+a e)^9 (-11 b B d+10 A b e+a B e)}{e^{11} (d+e x)^{10}}+\frac{5 b (b d-a e)^8 (-11 b B d+9 A b e+2 a B e)}{e^{11} (d+e x)^9}-\frac{15 b^2 (b d-a e)^7 (-11 b B d+8 A b e+3 a B e)}{e^{11} (d+e x)^8}+\frac{30 b^3 (b d-a e)^6 (-11 b B d+7 A b e+4 a B e)}{e^{11} (d+e x)^7}-\frac{42 b^4 (b d-a e)^5 (-11 b B d+6 A b e+5 a B e)}{e^{11} (d+e x)^6}+\frac{42 b^5 (b d-a e)^4 (-11 b B d+5 A b e+6 a B e)}{e^{11} (d+e x)^5}-\frac{30 b^6 (b d-a e)^3 (-11 b B d+4 A b e+7 a B e)}{e^{11} (d+e x)^4}+\frac{15 b^7 (b d-a e)^2 (-11 b B d+3 A b e+8 a B e)}{e^{11} (d+e x)^3}-\frac{5 b^8 (b d-a e) (-11 b B d+2 A b e+9 a B e)}{e^{11} (d+e x)^2}+\frac{b^9 (-11 b B d+A b e+10 a B e)}{e^{11} (d+e x)}\right ) \, dx\\ &=\frac{b^{10} B x}{e^{11}}+\frac{(b d-a e)^{10} (B d-A e)}{10 e^{12} (d+e x)^{10}}-\frac{(b d-a e)^9 (11 b B d-10 A b e-a B e)}{9 e^{12} (d+e x)^9}+\frac{5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{8 e^{12} (d+e x)^8}-\frac{15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{7 e^{12} (d+e x)^7}+\frac{5 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)^6}-\frac{42 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{5 e^{12} (d+e x)^5}+\frac{21 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{2 e^{12} (d+e x)^4}-\frac{10 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{e^{12} (d+e x)^3}+\frac{15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{2 e^{12} (d+e x)^2}-\frac{5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e)}{e^{12} (d+e x)}-\frac{b^9 (11 b B d-A b e-10 a B e) \log (d+e x)}{e^{12}}\\ \end{align*}
Mathematica [B] time = 0.861812, size = 1447, normalized size = 3.24 \[ -\frac{-\left (A d e \left (7381 d^9+71290 e x d^8+308205 e^2 x^2 d^7+784080 e^3 x^3 d^6+1296540 e^4 x^4 d^5+1450008 e^5 x^5 d^4+1102500 e^6 x^6 d^3+554400 e^7 x^7 d^2+170100 e^8 x^8 d+25200 e^9 x^9\right )-B \left (55991 d^{11}+532190 e x d^{10}+2256255 e^2 x^2 d^9+5600880 e^3 x^3 d^8+8969940 e^4 x^4 d^7+9599688 e^5 x^5 d^6+6835500 e^6 x^6 d^5+3074400 e^7 x^7 d^4+737100 e^8 x^8 d^3+25200 e^9 x^9 d^2-25200 e^{10} x^{10} d-2520 e^{11} x^{11}\right )\right ) b^{10}-10 a e \left (B d \left (7381 d^9+71290 e x d^8+308205 e^2 x^2 d^7+784080 e^3 x^3 d^6+1296540 e^4 x^4 d^5+1450008 e^5 x^5 d^4+1102500 e^6 x^6 d^3+554400 e^7 x^7 d^2+170100 e^8 x^8 d+25200 e^9 x^9\right )-252 A e \left (d^9+10 e x d^8+45 e^2 x^2 d^7+120 e^3 x^3 d^6+210 e^4 x^4 d^5+252 e^5 x^5 d^4+210 e^6 x^6 d^3+120 e^7 x^7 d^2+45 e^8 x^8 d+10 e^9 x^9\right )\right ) b^9+2520 (11 b B d-A b e-10 a B e) (d+e x)^{10} \log (d+e x) b^9+1260 a^2 e^2 \left (A e \left (d^8+10 e x d^7+45 e^2 x^2 d^6+120 e^3 x^3 d^5+210 e^4 x^4 d^4+252 e^5 x^5 d^3+210 e^6 x^6 d^2+120 e^7 x^7 d+45 e^8 x^8\right )+9 B \left (d^9+10 e x d^8+45 e^2 x^2 d^7+120 e^3 x^3 d^6+210 e^4 x^4 d^5+252 e^5 x^5 d^4+210 e^6 x^6 d^3+120 e^7 x^7 d^2+45 e^8 x^8 d+10 e^9 x^9\right )\right ) b^8+840 a^3 e^3 \left (A e \left (d^7+10 e x d^6+45 e^2 x^2 d^5+120 e^3 x^3 d^4+210 e^4 x^4 d^3+252 e^5 x^5 d^2+210 e^6 x^6 d+120 e^7 x^7\right )+4 B \left (d^8+10 e x d^7+45 e^2 x^2 d^6+120 e^3 x^3 d^5+210 e^4 x^4 d^4+252 e^5 x^5 d^3+210 e^6 x^6 d^2+120 e^7 x^7 d+45 e^8 x^8\right )\right ) b^7+210 a^4 e^4 \left (3 A e \left (d^6+10 e x d^5+45 e^2 x^2 d^4+120 e^3 x^3 d^3+210 e^4 x^4 d^2+252 e^5 x^5 d+210 e^6 x^6\right )+7 B \left (d^7+10 e x d^6+45 e^2 x^2 d^5+120 e^3 x^3 d^4+210 e^4 x^4 d^3+252 e^5 x^5 d^2+210 e^6 x^6 d+120 e^7 x^7\right )\right ) b^6+252 a^5 e^5 \left (2 A e \left (d^5+10 e x d^4+45 e^2 x^2 d^3+120 e^3 x^3 d^2+210 e^4 x^4 d+252 e^5 x^5\right )+3 B \left (d^6+10 e x d^5+45 e^2 x^2 d^4+120 e^3 x^3 d^3+210 e^4 x^4 d^2+252 e^5 x^5 d+210 e^6 x^6\right )\right ) b^5+420 a^6 e^6 \left (A e \left (d^4+10 e x d^3+45 e^2 x^2 d^2+120 e^3 x^3 d+210 e^4 x^4\right )+B \left (d^5+10 e x d^4+45 e^2 x^2 d^3+120 e^3 x^3 d^2+210 e^4 x^4 d+252 e^5 x^5\right )\right ) b^4+120 a^7 e^7 \left (3 A e \left (d^3+10 e x d^2+45 e^2 x^2 d+120 e^3 x^3\right )+2 B \left (d^4+10 e x d^3+45 e^2 x^2 d^2+120 e^3 x^3 d+210 e^4 x^4\right )\right ) b^3+45 a^8 e^8 \left (7 A e \left (d^2+10 e x d+45 e^2 x^2\right )+3 B \left (d^3+10 e x d^2+45 e^2 x^2 d+120 e^3 x^3\right )\right ) b^2+70 a^9 e^9 \left (4 A e (d+10 e x)+B \left (d^2+10 e x d+45 e^2 x^2\right )\right ) b+28 a^{10} e^{10} (9 A e+B (d+10 e x))}{2520 e^{12} (d+e x)^{10}} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^11,x]
[Out]
-(28*a^10*e^10*(9*A*e + B*(d + 10*e*x)) + 70*a^9*b*e^9*(4*A*e*(d + 10*e*x) + B*(d^2 + 10*d*e*x + 45*e^2*x^2))
+ 45*a^8*b^2*e^8*(7*A*e*(d^2 + 10*d*e*x + 45*e^2*x^2) + 3*B*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3)) +
120*a^7*b^3*e^7*(3*A*e*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 2*B*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x
^2 + 120*d*e^3*x^3 + 210*e^4*x^4)) + 420*a^6*b^4*e^6*(A*e*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 +
210*e^4*x^4) + B*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5)) + 252*a
^5*b^5*e^5*(2*A*e*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5) + 3*B*(d
^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^6*x^6)) + 210*a^4
*b^6*e^4*(3*A*e*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e
^6*x^6) + 7*B*(d^7 + 10*d^6*e*x + 45*d^5*e^2*x^2 + 120*d^4*e^3*x^3 + 210*d^3*e^4*x^4 + 252*d^2*e^5*x^5 + 210*d
*e^6*x^6 + 120*e^7*x^7)) + 840*a^3*b^7*e^3*(A*e*(d^7 + 10*d^6*e*x + 45*d^5*e^2*x^2 + 120*d^4*e^3*x^3 + 210*d^3
*e^4*x^4 + 252*d^2*e^5*x^5 + 210*d*e^6*x^6 + 120*e^7*x^7) + 4*B*(d^8 + 10*d^7*e*x + 45*d^6*e^2*x^2 + 120*d^5*e
^3*x^3 + 210*d^4*e^4*x^4 + 252*d^3*e^5*x^5 + 210*d^2*e^6*x^6 + 120*d*e^7*x^7 + 45*e^8*x^8)) + 1260*a^2*b^8*e^2
*(A*e*(d^8 + 10*d^7*e*x + 45*d^6*e^2*x^2 + 120*d^5*e^3*x^3 + 210*d^4*e^4*x^4 + 252*d^3*e^5*x^5 + 210*d^2*e^6*x
^6 + 120*d*e^7*x^7 + 45*e^8*x^8) + 9*B*(d^9 + 10*d^8*e*x + 45*d^7*e^2*x^2 + 120*d^6*e^3*x^3 + 210*d^5*e^4*x^4
+ 252*d^4*e^5*x^5 + 210*d^3*e^6*x^6 + 120*d^2*e^7*x^7 + 45*d*e^8*x^8 + 10*e^9*x^9)) - 10*a*b^9*e*(-252*A*e*(d^
9 + 10*d^8*e*x + 45*d^7*e^2*x^2 + 120*d^6*e^3*x^3 + 210*d^5*e^4*x^4 + 252*d^4*e^5*x^5 + 210*d^3*e^6*x^6 + 120*
d^2*e^7*x^7 + 45*d*e^8*x^8 + 10*e^9*x^9) + B*d*(7381*d^9 + 71290*d^8*e*x + 308205*d^7*e^2*x^2 + 784080*d^6*e^3
*x^3 + 1296540*d^5*e^4*x^4 + 1450008*d^4*e^5*x^5 + 1102500*d^3*e^6*x^6 + 554400*d^2*e^7*x^7 + 170100*d*e^8*x^8
+ 25200*e^9*x^9)) - b^10*(A*d*e*(7381*d^9 + 71290*d^8*e*x + 308205*d^7*e^2*x^2 + 784080*d^6*e^3*x^3 + 1296540
*d^5*e^4*x^4 + 1450008*d^4*e^5*x^5 + 1102500*d^3*e^6*x^6 + 554400*d^2*e^7*x^7 + 170100*d*e^8*x^8 + 25200*e^9*x
^9) - B*(55991*d^11 + 532190*d^10*e*x + 2256255*d^9*e^2*x^2 + 5600880*d^8*e^3*x^3 + 8969940*d^7*e^4*x^4 + 9599
688*d^6*e^5*x^5 + 6835500*d^5*e^6*x^6 + 3074400*d^4*e^7*x^7 + 737100*d^3*e^8*x^8 + 25200*d^2*e^9*x^9 - 25200*d
*e^10*x^10 - 2520*e^11*x^11)) + 2520*b^9*(11*b*B*d - A*b*e - 10*a*B*e)*(d + e*x)^10*Log[d + e*x])/(2520*e^12*(
d + e*x)^10)
________________________________________________________________________________________
Maple [B] time = 0.023, size = 2897, normalized size = 6.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^10*(B*x+A)/(e*x+d)^11,x)
[Out]
504*b^9/e^11/(e*x+d)^5*B*a*d^5+45*b^3/e^4/(e*x+d)^8*A*a^7*d-315/2*b^4/e^5/(e*x+d)^8*A*a^6*d^2+315*b^5/e^6/(e*x
+d)^8*A*a^5*d^3-945/2*b^5/e^7/(e*x+d)^8*B*a^5*d^4+2205/4*b^6/e^8/(e*x+d)^8*B*a^4*d^5-420*b^7/e^9/(e*x+d)^8*B*a
^3*d^6+405/2*b^8/e^10/(e*x+d)^8*B*a^2*d^7+1/e^2/(e*x+d)^10*A*d*a^9*b-9/2/e^3/(e*x+d)^10*A*d^2*a^8*b^2+12/e^4/(
e*x+d)^10*A*d^3*a^7*b^3-21/e^5/(e*x+d)^10*A*d^4*a^6*b^4+126/5/e^6/(e*x+d)^10*A*d^5*a^5*b^5-21/e^7/(e*x+d)^10*A
*d^6*a^4*b^6+12/e^8/(e*x+d)^10*A*d^7*a^3*b^7-9/2/e^9/(e*x+d)^10*A*d^8*a^2*b^8+1/e^10/(e*x+d)^10*A*d^9*a*b^9-1/
e^3/(e*x+d)^10*B*d^2*a^9*b+9/2/e^4/(e*x+d)^10*B*d^3*a^8*b^2-12/e^5/(e*x+d)^10*B*d^4*a^7*b^3+21/e^6/(e*x+d)^10*
B*d^5*a^6*b^4-126/5/e^7/(e*x+d)^10*B*d^6*a^5*b^5+21/e^8/(e*x+d)^10*B*d^7*a^4*b^6-12/e^9/(e*x+d)^10*B*d^8*a^3*b
^7+9/2/e^10/(e*x+d)^10*B*d^9*a^2*b^8-1/e^11/(e*x+d)^10*B*d^10*a*b^9+210*b^7/e^8/(e*x+d)^4*A*a^3*d-315*b^8/e^9/
(e*x+d)^4*A*a^2*d^2+210*b^9/e^10/(e*x+d)^4*A*a*d^3+735/2*b^6/e^8/(e*x+d)^4*B*a^4*d-840*b^7/e^9/(e*x+d)^4*B*a^3
*d^2+945*b^8/e^10/(e*x+d)^4*B*a^2*d^3-525*b^9/e^11/(e*x+d)^4*B*a*d^4+700*b^7/e^8/(e*x+d)^6*A*a^3*d^3-525*b^8/e
^9/(e*x+d)^6*A*a^2*d^4-252/5*b^5/e^6/(e*x+d)^5*A*a^5+252/5*b^10/e^11/(e*x+d)^5*A*d^5-42*b^4/e^6/(e*x+d)^5*B*a^
6-462/5*b^10/e^12/(e*x+d)^5*B*d^6-45/8*b^2/e^3/(e*x+d)^8*A*a^8-45/8*b^10/e^11/(e*x+d)^8*A*d^8-5/4*b/e^3/(e*x+d
)^8*B*a^9+55/8*b^10/e^12/(e*x+d)^8*B*d^9-1/10/e^11/(e*x+d)^10*A*d^10*b^10+100/9/e^11/(e*x+d)^9*B*a*b^9*d^9+45*
b^9/e^10/(e*x+d)^2*A*a*d+405/2*b^8/e^10/(e*x+d)^2*B*a^2*d-225*b^9/e^11/(e*x+d)^2*B*a*d^2+100*b^9/e^11/(e*x+d)*
B*a*d+252*b^6/e^7/(e*x+d)^5*A*a^4*d-504*b^7/e^8/(e*x+d)^5*A*a^3*d^2+504*b^8/e^9/(e*x+d)^5*A*a^2*d^3-252*b^9/e^
10/(e*x+d)^5*A*a*d^4+1512/5*b^5/e^7/(e*x+d)^5*B*a^5*d-882*b^6/e^8/(e*x+d)^5*B*a^4*d^2+1344*b^7/e^9/(e*x+d)^5*B
*a^3*d^3-1134*b^8/e^10/(e*x+d)^5*B*a^2*d^4+1/10/e^2/(e*x+d)^10*B*d*a^10+1/10/e^12/(e*x+d)^10*b^10*B*d^11-105/2
*b^6/e^7/(e*x+d)^4*A*a^4-105/2*b^10/e^11/(e*x+d)^4*A*d^4-63*b^5/e^7/(e*x+d)^4*B*a^5+231/2*b^10/e^12/(e*x+d)^4*
B*d^5-120/7*b^3/e^4/(e*x+d)^7*A*a^7+120/7*b^10/e^11/(e*x+d)^7*A*d^7-45/7*b^2/e^4/(e*x+d)^7*B*a^8-165/7*b^10/e^
12/(e*x+d)^7*B*d^8-10/9/e^2/(e*x+d)^9*A*a^9*b+10/9/e^11/(e*x+d)^9*A*b^10*d^9-11/9/e^12/(e*x+d)^9*b^10*B*d^10-4
5/2*b^8/e^9/(e*x+d)^2*A*a^2-45/2*b^10/e^11/(e*x+d)^2*A*d^2-60*b^7/e^9/(e*x+d)^2*B*a^3+165/2*b^10/e^12/(e*x+d)^
2*B*d^3-10*b^9/e^10/(e*x+d)*A*a+10*b^10/e^11/(e*x+d)*A*d-45*b^8/e^10/(e*x+d)*B*a^2-55*b^10/e^12/(e*x+d)*B*d^2-
35*b^4/e^5/(e*x+d)^6*A*a^6-35*b^10/e^11/(e*x+d)^6*A*d^6-20*b^3/e^5/(e*x+d)^6*B*a^7+55*b^10/e^12/(e*x+d)^6*B*d^
7-40*b^7/e^8/(e*x+d)^3*A*a^3+40*b^10/e^11/(e*x+d)^3*A*d^3-70*b^6/e^8/(e*x+d)^3*B*a^4-110*b^10/e^12/(e*x+d)^3*B
*d^4-225/4*b^9/e^11/(e*x+d)^8*B*a*d^8-1/9/e^2/(e*x+d)^9*B*a^10+b^10/e^11*ln(e*x+d)*A-1/10/e/(e*x+d)^10*a^10*A+
10*b^9/e^11*ln(e*x+d)*B*a-11*b^10/e^12*ln(e*x+d)*B*d-1400*b^7/e^9/(e*x+d)^6*B*a^3*d^4+945*b^8/e^10/(e*x+d)^6*B
*a^2*d^5-350*b^9/e^11/(e*x+d)^6*B*a*d^6+120*b^8/e^9/(e*x+d)^3*A*a^2*d-120*b^9/e^10/(e*x+d)^3*A*a*d^2+320*b^7/e
^9/(e*x+d)^3*B*a^3*d-540*b^8/e^10/(e*x+d)^3*B*a^2*d^2+400*b^9/e^11/(e*x+d)^3*B*a*d^3+120*b^4/e^5/(e*x+d)^7*A*a
^6*d-360*b^5/e^6/(e*x+d)^7*A*a^5*d^2+600*b^6/e^7/(e*x+d)^7*A*a^4*d^3-600*b^7/e^8/(e*x+d)^7*A*a^3*d^4+360*b^8/e
^9/(e*x+d)^7*A*a^2*d^5-120*b^9/e^10/(e*x+d)^7*A*a*d^6+480/7*b^3/e^5/(e*x+d)^7*B*a^7*d+b^10*B*x/e^11+210*b^9/e^
10/(e*x+d)^6*A*a*d^5-1575/4*b^6/e^7/(e*x+d)^8*A*a^4*d^4+315*b^7/e^8/(e*x+d)^8*A*a^3*d^5-315/2*b^8/e^9/(e*x+d)^
8*A*a^2*d^6+45*b^9/e^10/(e*x+d)^8*A*a*d^7+135/8*b^2/e^4/(e*x+d)^8*B*a^8*d-90*b^3/e^5/(e*x+d)^8*B*a^7*d^2+525/2
*b^4/e^6/(e*x+d)^8*B*a^6*d^3+175*b^4/e^6/(e*x+d)^6*B*a^6*d-630*b^5/e^7/(e*x+d)^6*B*a^5*d^2+1225*b^6/e^8/(e*x+d
)^6*B*a^4*d^3-300*b^4/e^6/(e*x+d)^7*B*a^6*d^2+720*b^5/e^7/(e*x+d)^7*B*a^5*d^3-1050*b^6/e^8/(e*x+d)^7*B*a^4*d^4
+960*b^7/e^9/(e*x+d)^7*B*a^3*d^5-540*b^8/e^10/(e*x+d)^7*B*a^2*d^6+1200/7*b^9/e^11/(e*x+d)^7*B*a*d^7+10/e^3/(e*
x+d)^9*A*a^8*b^2*d-40/e^4/(e*x+d)^9*A*a^7*b^3*d^2+280/3/e^5/(e*x+d)^9*A*a^6*b^4*d^3+210*b^5/e^6/(e*x+d)^6*A*a^
5*d-525*b^6/e^7/(e*x+d)^6*A*a^4*d^2-140/e^6/(e*x+d)^9*A*a^5*b^5*d^4+140/e^7/(e*x+d)^9*A*a^4*b^6*d^5-280/3/e^8/
(e*x+d)^9*A*a^3*b^7*d^6+40/e^9/(e*x+d)^9*A*a^2*b^8*d^7-10/e^10/(e*x+d)^9*A*a*b^9*d^8+20/9/e^3/(e*x+d)^9*B*a^9*
b*d-15/e^4/(e*x+d)^9*B*a^8*b^2*d^2+160/3/e^5/(e*x+d)^9*B*a^7*b^3*d^3-350/3/e^6/(e*x+d)^9*B*a^6*b^4*d^4+168/e^7
/(e*x+d)^9*B*a^5*b^5*d^5-490/3/e^8/(e*x+d)^9*B*a^4*b^6*d^6+320/3/e^9/(e*x+d)^9*B*a^3*b^7*d^7-45/e^10/(e*x+d)^9
*B*a^2*b^8*d^8
________________________________________________________________________________________
Maxima [B] time = 1.59854, size = 2584, normalized size = 5.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^11,x, algorithm="maxima")
[Out]
B*b^10*x/e^11 - 1/2520*(55991*B*b^10*d^11 + 252*A*a^10*e^11 - 7381*(10*B*a*b^9 + A*b^10)*d^10*e + 1260*(9*B*a^
2*b^8 + 2*A*a*b^9)*d^9*e^2 + 420*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4
+ 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 60*(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 + 35*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 28*(B*a^10 +
10*A*a^9*b)*d*e^10 + 12600*(11*B*b^10*d^2*e^9 - 2*(10*B*a*b^9 + A*b^10)*d*e^10 + (9*B*a^2*b^8 + 2*A*a*b^9)*e^1
1)*x^9 + 18900*(55*B*b^10*d^3*e^8 - 9*(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 + (8*
B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 25200*(143*B*b^10*d^4*e^7 - 22*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 6*(9*B*a^2
*b^8 + 2*A*a*b^9)*d^2*e^9 + 2*(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 + 882
0*(847*B*b^10*d^5*e^6 - 125*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 10*(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 + 3*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 +
10584*(957*B*b^10*d^6*e^5 - 137*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 10*(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 + 3*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^1
0 + 2*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 1260*(7359*B*b^10*d^7*e^4 - 1029*(10*B*a*b^9 + A*b^10)*d^6*e^5 +
210*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 70*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 35*(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 + 14*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 10*(4*B*a^7*b^3
+ 7*A*a^6*b^4)*e^11)*x^4 + 360*(15873*B*b^10*d^8*e^3 - 2178*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 420*(9*B*a^2*b^8 +
2*A*a*b^9)*d^6*e^5 + 140*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 70*(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^3*e^8 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 20*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*
e^10 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 45*(50699*B*b^10*d^9*e^2 - 6849*(10*B*a*b^9 + A*b^10)*d^8*e^
3 + 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 420*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 210*(7*B*a^4*b^6 + 4*A*
a^3*b^7)*d^5*e^6 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 60*(4*B*
a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 35*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^
2 + 10*(53471*B*b^10*d^10*e - 7129*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 42
0*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 126*(6*B*a^5*b^5 + 5*A*a^4*b
^6)*d^5*e^6 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 45*(3*B*a^8*b^
2 + 8*A*a^7*b^3)*d^2*e^9 + 35*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 28*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^22*x^10
+ 10*d*e^21*x^9 + 45*d^2*e^20*x^8 + 120*d^3*e^19*x^7 + 210*d^4*e^18*x^6 + 252*d^5*e^17*x^5 + 210*d^6*e^16*x^4
+ 120*d^7*e^15*x^3 + 45*d^8*e^14*x^2 + 10*d^9*e^13*x + d^10*e^12) - (11*B*b^10*d - (10*B*a*b^9 + A*b^10)*e)*lo
g(e*x + d)/e^12
________________________________________________________________________________________
Fricas [B] time = 2.0667, size = 5054, normalized size = 11.33 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^11,x, algorithm="fricas")
[Out]
1/2520*(2520*B*b^10*e^11*x^11 + 25200*B*b^10*d*e^10*x^10 - 55991*B*b^10*d^11 - 252*A*a^10*e^11 + 7381*(10*B*a*
b^9 + A*b^10)*d^10*e - 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 420*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 - 210*
(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 - 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)
*d^5*e^6 - 60*(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 - 35*(2*B*a^9*b + 9
*A*a^8*b^2)*d^2*e^9 - 28*(B*a^10 + 10*A*a^9*b)*d*e^10 - 12600*(2*B*b^10*d^2*e^9 - 2*(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 - 18900*(39*B*b^10*d^3*e^8 - 9*(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 + (8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 - 25200*(122*B*b^10*d^4*e^7 - 22*(10*B*a*
b^9 + A*b^10)*d^3*e^8 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 2*(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 - 8820*(775*B*b^10*d^5*e^6 - 125*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 30*(9*B*a^2*b^8
+ 2*A*a*b^9)*d^3*e^8 + 10*(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 + 3*(6*B*
a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 - 10584*(907*B*b^10*d^6*e^5 - 137*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 30*(9*B*a^2
*b^8 + 2*A*a*b^9)*d^4*e^7 + 10*(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 + 3
*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 2*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 - 1260*(7119*B*b^10*d^7*e^4 - 10
29*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 210*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 70*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*
e^7 + 35*(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 + 14*(5*B*a^6*b^4 + 6*A*
a^5*b^5)*d*e^10 + 10*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 - 360*(15558*B*b^10*d^8*e^3 - 2178*(10*B*a*b^9 + A*
b^10)*d^7*e^4 + 420*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 140*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 70*(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^3*e^8 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 +
20*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 - 45*(50139*B*b^10*d^9*e^2 -
6849*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 420*(8*B*a^3*b^7 + 3*A*a^2*b^8)
*d^6*e^5 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 84*(5*B*a^6*b^4
+ 6*A*a^5*b^5)*d^3*e^8 + 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 35*
(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 - 10*(53219*B*b^10*d^10*e - 7129*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 1260*(9*B
*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 420*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*
e^5 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 60*(4*B*a^7*b^3 + 7*A
*a^6*b^4)*d^3*e^8 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 35*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 28*(B*a^10
+ 10*A*a^9*b)*e^11)*x - 2520*(11*B*b^10*d^11 - (10*B*a*b^9 + A*b^10)*d^10*e + (11*B*b^10*d*e^10 - (10*B*a*b^9
+ A*b^10)*e^11)*x^10 + 10*(11*B*b^10*d^2*e^9 - (10*B*a*b^9 + A*b^10)*d*e^10)*x^9 + 45*(11*B*b^10*d^3*e^8 - (10
*B*a*b^9 + A*b^10)*d^2*e^9)*x^8 + 120*(11*B*b^10*d^4*e^7 - (10*B*a*b^9 + A*b^10)*d^3*e^8)*x^7 + 210*(11*B*b^10
*d^5*e^6 - (10*B*a*b^9 + A*b^10)*d^4*e^7)*x^6 + 252*(11*B*b^10*d^6*e^5 - (10*B*a*b^9 + A*b^10)*d^5*e^6)*x^5 +
210*(11*B*b^10*d^7*e^4 - (10*B*a*b^9 + A*b^10)*d^6*e^5)*x^4 + 120*(11*B*b^10*d^8*e^3 - (10*B*a*b^9 + A*b^10)*d
^7*e^4)*x^3 + 45*(11*B*b^10*d^9*e^2 - (10*B*a*b^9 + A*b^10)*d^8*e^3)*x^2 + 10*(11*B*b^10*d^10*e - (10*B*a*b^9
+ A*b^10)*d^9*e^2)*x)*log(e*x + d))/(e^22*x^10 + 10*d*e^21*x^9 + 45*d^2*e^20*x^8 + 120*d^3*e^19*x^7 + 210*d^4*
e^18*x^6 + 252*d^5*e^17*x^5 + 210*d^6*e^16*x^4 + 120*d^7*e^15*x^3 + 45*d^8*e^14*x^2 + 10*d^9*e^13*x + d^10*e^1
2)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**10*(B*x+A)/(e*x+d)**11,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 2.01179, size = 2483, normalized size = 5.57 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^11,x, algorithm="giac")
[Out]
B*b^10*x*e^(-11) - (11*B*b^10*d - 10*B*a*b^9*e - A*b^10*e)*e^(-12)*log(abs(x*e + d)) - 1/2520*(55991*B*b^10*d^
11 - 73810*B*a*b^9*d^10*e - 7381*A*b^10*d^10*e + 11340*B*a^2*b^8*d^9*e^2 + 2520*A*a*b^9*d^9*e^2 + 3360*B*a^3*b
^7*d^8*e^3 + 1260*A*a^2*b^8*d^8*e^3 + 1470*B*a^4*b^6*d^7*e^4 + 840*A*a^3*b^7*d^7*e^4 + 756*B*a^5*b^5*d^6*e^5 +
630*A*a^4*b^6*d^6*e^5 + 420*B*a^6*b^4*d^5*e^6 + 504*A*a^5*b^5*d^5*e^6 + 240*B*a^7*b^3*d^4*e^7 + 420*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 + 70*B*a^9*b*d^2*e^9 + 315*A*a^8*b^2*d^2*e^9 + 28*B*a
^10*d*e^10 + 280*A*a^9*b*d*e^10 + 252*A*a^10*e^11 + 12600*(11*B*b^10*d^2*e^9 - 20*B*a*b^9*d*e^10 - 2*A*b^10*d*
e^10 + 9*B*a^2*b^8*e^11 + 2*A*a*b^9*e^11)*x^9 + 18900*(55*B*b^10*d^3*e^8 - 90*B*a*b^9*d^2*e^9 - 9*A*b^10*d^2*e
^9 + 27*B*a^2*b^8*d*e^10 + 6*A*a*b^9*d*e^10 + 8*B*a^3*b^7*e^11 + 3*A*a^2*b^8*e^11)*x^8 + 25200*(143*B*b^10*d^4
*e^7 - 220*B*a*b^9*d^3*e^8 - 22*A*b^10*d^3*e^8 + 54*B*a^2*b^8*d^2*e^9 + 12*A*a*b^9*d^2*e^9 + 16*B*a^3*b^7*d*e^
10 + 6*A*a^2*b^8*d*e^10 + 7*B*a^4*b^6*e^11 + 4*A*a^3*b^7*e^11)*x^7 + 8820*(847*B*b^10*d^5*e^6 - 1250*B*a*b^9*d
^4*e^7 - 125*A*b^10*d^4*e^7 + 270*B*a^2*b^8*d^3*e^8 + 60*A*a*b^9*d^3*e^8 + 80*B*a^3*b^7*d^2*e^9 + 30*A*a^2*b^8
*d^2*e^9 + 35*B*a^4*b^6*d*e^10 + 20*A*a^3*b^7*d*e^10 + 18*B*a^5*b^5*e^11 + 15*A*a^4*b^6*e^11)*x^6 + 10584*(957
*B*b^10*d^6*e^5 - 1370*B*a*b^9*d^5*e^6 - 137*A*b^10*d^5*e^6 + 270*B*a^2*b^8*d^4*e^7 + 60*A*a*b^9*d^4*e^7 + 80*
B*a^3*b^7*d^3*e^8 + 30*A*a^2*b^8*d^3*e^8 + 35*B*a^4*b^6*d^2*e^9 + 20*A*a^3*b^7*d^2*e^9 + 18*B*a^5*b^5*d*e^10 +
15*A*a^4*b^6*d*e^10 + 10*B*a^6*b^4*e^11 + 12*A*a^5*b^5*e^11)*x^5 + 1260*(7359*B*b^10*d^7*e^4 - 10290*B*a*b^9*
d^6*e^5 - 1029*A*b^10*d^6*e^5 + 1890*B*a^2*b^8*d^5*e^6 + 420*A*a*b^9*d^5*e^6 + 560*B*a^3*b^7*d^4*e^7 + 210*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
+ 70*B*a^6*b^4*d*e^10 + 84*A*a^5*b^5*d*e^10 + 40*B*a^7*b^3*e^11 + 70*A*a^6*b^4*e^11)*x^4 + 360*(15873*B*b^10*
d^8*e^3 - 21780*B*a*b^9*d^7*e^4 - 2178*A*b^10*d^7*e^4 + 3780*B*a^2*b^8*d^6*e^5 + 840*A*a*b^9*d^6*e^5 + 1120*B*
a^3*b^7*d^5*e^6 + 420*A*a^2*b^8*d^5*e^6 + 490*B*a^4*b^6*d^4*e^7 + 280*A*a^3*b^7*d^4*e^7 + 252*B*a^5*b^5*d^3*e^
8 + 210*A*a^4*b^6*d^3*e^8 + 140*B*a^6*b^4*d^2*e^9 + 168*A*a^5*b^5*d^2*e^9 + 80*B*a^7*b^3*d*e^10 + 140*A*a^6*b^
4*d*e^10 + 45*B*a^8*b^2*e^11 + 120*A*a^7*b^3*e^11)*x^3 + 45*(50699*B*b^10*d^9*e^2 - 68490*B*a*b^9*d^8*e^3 - 68
49*A*b^10*d^8*e^3 + 11340*B*a^2*b^8*d^7*e^4 + 2520*A*a*b^9*d^7*e^4 + 3360*B*a^3*b^7*d^6*e^5 + 1260*A*a^2*b^8*d
^6*e^5 + 1470*B*a^4*b^6*d^5*e^6 + 840*A*a^3*b^7*d^5*e^6 + 756*B*a^5*b^5*d^4*e^7 + 630*A*a^4*b^6*d^4*e^7 + 420*
B*a^6*b^4*d^3*e^8 + 504*A*a^5*b^5*d^3*e^8 + 240*B*a^7*b^3*d^2*e^9 + 420*A*a^6*b^4*d^2*e^9 + 135*B*a^8*b^2*d*e^
10 + 360*A*a^7*b^3*d*e^10 + 70*B*a^9*b*e^11 + 315*A*a^8*b^2*e^11)*x^2 + 10*(53471*B*b^10*d^10*e - 71290*B*a*b^
9*d^9*e^2 - 7129*A*b^10*d^9*e^2 + 11340*B*a^2*b^8*d^8*e^3 + 2520*A*a*b^9*d^8*e^3 + 3360*B*a^3*b^7*d^7*e^4 + 12
60*A*a^2*b^8*d^7*e^4 + 1470*B*a^4*b^6*d^6*e^5 + 840*A*a^3*b^7*d^6*e^5 + 756*B*a^5*b^5*d^5*e^6 + 630*A*a^4*b^6*
d^5*e^6 + 420*B*a^6*b^4*d^4*e^7 + 504*A*a^5*b^5*d^4*e^7 + 240*B*a^7*b^3*d^3*e^8 + 420*A*a^6*b^4*d^3*e^8 + 135*
B*a^8*b^2*d^2*e^9 + 360*A*a^7*b^3*d^2*e^9 + 70*B*a^9*b*d*e^10 + 315*A*a^8*b^2*d*e^10 + 28*B*a^10*e^11 + 280*A*
a^9*b*e^11)*x)*e^(-12)/(x*e + d)^10