\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx\) [1090]

Optimal result

Integrand size = 20, antiderivative size = 445 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx=-\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{e^{12} (d+e x)}+\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) (d+e x)^2}{2 e^{12}}-\frac {10 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e) (d+e x)^3}{e^{12}}+\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) (d+e x)^4}{2 e^{12}}-\frac {42 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) (d+e x)^5}{5 e^{12}}+\frac {5 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) (d+e x)^6}{e^{12}}-\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^7}{7 e^{12}}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^8}{8 e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^9}{9 e^{12}}+\frac {b^{10} B (d+e x)^{10}}{10 e^{12}}+\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e) \log (d+e x)}{e^{12}} \]

[Out]

-5*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)*x/e^11+(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)+15/2*b^2*(-a*e+b*d)
^7*(-8*A*b*e-3*B*a*e+11*B*b*d)*(e*x+d)^2/e^12-10*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)*(e*x+d)^3/e^12+2
1/2*b^4*(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)*(e*x+d)^4/e^12-42/5*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B*a*e+11*B*b
*d)*(e*x+d)^5/e^12+5*b^6*(-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11*B*b*d)*(e*x+d)^6/e^12-15/7*b^7*(-a*e+b*d)^2*(-3*A*b
*e-8*B*a*e+11*B*b*d)*(e*x+d)^7/e^12+5/8*b^8*(-a*e+b*d)*(-2*A*b*e-9*B*a*e+11*B*b*d)*(e*x+d)^8/e^12-1/9*b^9*(-A*
b*e-10*B*a*e+11*B*b*d)*(e*x+d)^9/e^12+1/10*b^10*B*(e*x+d)^10/e^12+(-a*e+b*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)*ln(e
*x+d)/e^12

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx=-\frac {b^9 (d+e x)^9 (-10 a B e-A b e+11 b B d)}{9 e^{12}}+\frac {5 b^8 (d+e x)^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{8 e^{12}}-\frac {15 b^7 (d+e x)^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{7 e^{12}}+\frac {5 b^6 (d+e x)^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12}}-\frac {42 b^5 (d+e x)^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{5 e^{12}}+\frac {21 b^4 (d+e x)^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{2 e^{12}}-\frac {10 b^3 (d+e x)^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12}}+\frac {15 b^2 (d+e x)^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{2 e^{12}}+\frac {(b d-a e)^{10} (B d-A e)}{e^{12} (d+e x)}+\frac {(b d-a e)^9 \log (d+e x) (-a B e-10 A b e+11 b B d)}{e^{12}}-\frac {5 b x (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{11}}+\frac {b^{10} B (d+e x)^{10}}{10 e^{12}} \]

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^2,x]

[Out]

(-5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(e^12*(d + e*x)) + (
15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e)*(d + e*x)^2)/(2*e^12) - (10*b^3*(b*d - a*e)^6*(11*b*B*d -
7*A*b*e - 4*a*B*e)*(d + e*x)^3)/e^12 + (21*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e)*(d + e*x)^4)/(2*e^
12) - (42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*(d + e*x)^5)/(5*e^12) + (5*b^6*(b*d - a*e)^3*(11*b*
B*d - 4*A*b*e - 7*a*B*e)*(d + e*x)^6)/e^12 - (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*(d + e*x)^7)
/(7*e^12) + (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^8)/(8*e^12) - (b^9*(11*b*B*d - A*b*e -
10*a*B*e)*(d + e*x)^9)/(9*e^12) + (b^10*B*(d + e*x)^10)/(10*e^12) + ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B
*e)*Log[d + e*x])/e^12

Rule 78

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*} \text {integral}& = \int \left (\frac {5 b (b d-a e)^8 (-11 b B d+9 A b e+2 a B e)}{e^{11}}+\frac {(-b d+a e)^{10} (-B d+A e)}{e^{11} (d+e x)^2}+\frac {(-b d+a e)^9 (-11 b B d+10 A b e+a B e)}{e^{11} (d+e x)}-\frac {15 b^2 (b d-a e)^7 (-11 b B d+8 A b e+3 a B e) (d+e x)}{e^{11}}+\frac {30 b^3 (b d-a e)^6 (-11 b B d+7 A b e+4 a B e) (d+e x)^2}{e^{11}}-\frac {42 b^4 (b d-a e)^5 (-11 b B d+6 A b e+5 a B e) (d+e x)^3}{e^{11}}+\frac {42 b^5 (b d-a e)^4 (-11 b B d+5 A b e+6 a B e) (d+e x)^4}{e^{11}}-\frac {30 b^6 (b d-a e)^3 (-11 b B d+4 A b e+7 a B e) (d+e x)^5}{e^{11}}+\frac {15 b^7 (b d-a e)^2 (-11 b B d+3 A b e+8 a B e) (d+e x)^6}{e^{11}}-\frac {5 b^8 (b d-a e) (-11 b B d+2 A b e+9 a B e) (d+e x)^7}{e^{11}}+\frac {b^9 (-11 b B d+A b e+10 a B e) (d+e x)^8}{e^{11}}+\frac {b^{10} B (d+e x)^9}{e^{11}}\right ) \, dx \\ & = -\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{e^{12} (d+e x)}+\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) (d+e x)^2}{2 e^{12}}-\frac {10 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e) (d+e x)^3}{e^{12}}+\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) (d+e x)^4}{2 e^{12}}-\frac {42 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) (d+e x)^5}{5 e^{12}}+\frac {5 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) (d+e x)^6}{e^{12}}-\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^7}{7 e^{12}}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^8}{8 e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^9}{9 e^{12}}+\frac {b^{10} B (d+e x)^{10}}{10 e^{12}}+\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e) \log (d+e x)}{e^{12}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1486\) vs. \(2(445)=890\).

Time = 0.43 (sec) , antiderivative size = 1486, normalized size of antiderivative = 3.34 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx=\frac {-2520 a^{10} e^{10} (-B d+A e)+25200 a^9 b e^9 \left (A d e+B \left (-d^2+d e x+e^2 x^2\right )\right )+56700 a^8 b^2 e^8 \left (2 A e \left (-d^2+d e x+e^2 x^2\right )+B \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )\right )+50400 a^7 b^3 e^7 \left (3 A e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+2 B \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )\right )+44100 a^6 b^4 e^6 \left (4 A e \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+B \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )\right )+10584 a^5 b^5 e^5 \left (5 A e \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )-6 B \left (10 d^6-50 d^5 e x-30 d^4 e^2 x^2+10 d^3 e^3 x^3-5 d^2 e^4 x^4+3 d e^5 x^5-2 e^6 x^6\right )\right )+8820 a^4 b^6 e^4 \left (6 A e \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )+B \left (60 d^7-360 d^6 e x-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-14 d e^6 x^6+10 e^7 x^7\right )\right )+720 a^3 b^7 e^3 \left (7 A e \left (60 d^7-360 d^6 e x-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-14 d e^6 x^6+10 e^7 x^7\right )-4 B \left (105 d^8-735 d^7 e x-420 d^6 e^2 x^2+140 d^5 e^3 x^3-70 d^4 e^4 x^4+42 d^3 e^5 x^5-28 d^2 e^6 x^6+20 d e^7 x^7-15 e^8 x^8\right )\right )+135 a^2 b^8 e^2 \left (8 A e \left (-105 d^8+735 d^7 e x+420 d^6 e^2 x^2-140 d^5 e^3 x^3+70 d^4 e^4 x^4-42 d^3 e^5 x^5+28 d^2 e^6 x^6-20 d e^7 x^7+15 e^8 x^8\right )+3 B \left (280 d^9-2240 d^8 e x-1260 d^7 e^2 x^2+420 d^6 e^3 x^3-210 d^5 e^4 x^4+126 d^4 e^5 x^5-84 d^3 e^6 x^6+60 d^2 e^7 x^7-45 d e^8 x^8+35 e^9 x^9\right )\right )+10 a b^9 e \left (9 A e \left (280 d^9-2240 d^8 e x-1260 d^7 e^2 x^2+420 d^6 e^3 x^3-210 d^5 e^4 x^4+126 d^4 e^5 x^5-84 d^3 e^6 x^6+60 d^2 e^7 x^7-45 d e^8 x^8+35 e^9 x^9\right )-10 B \left (252 d^{10}-2268 d^9 e x-1260 d^8 e^2 x^2+420 d^7 e^3 x^3-210 d^6 e^4 x^4+126 d^5 e^5 x^5-84 d^4 e^6 x^6+60 d^3 e^7 x^7-45 d^2 e^8 x^8+35 d e^9 x^9-28 e^{10} x^{10}\right )\right )+b^{10} \left (10 A e \left (-252 d^{10}+2268 d^9 e x+1260 d^8 e^2 x^2-420 d^7 e^3 x^3+210 d^6 e^4 x^4-126 d^5 e^5 x^5+84 d^4 e^6 x^6-60 d^3 e^7 x^7+45 d^2 e^8 x^8-35 d e^9 x^9+28 e^{10} x^{10}\right )+B \left (2520 d^{11}-25200 d^{10} e x-13860 d^9 e^2 x^2+4620 d^8 e^3 x^3-2310 d^7 e^4 x^4+1386 d^6 e^5 x^5-924 d^5 e^6 x^6+660 d^4 e^7 x^7-495 d^3 e^8 x^8+385 d^2 e^9 x^9-308 d e^{10} x^{10}+252 e^{11} x^{11}\right )\right )+2520 (b d-a e)^9 (11 b B d-10 A b e-a B e) (d+e x) \log (d+e x)}{2520 e^{12} (d+e x)} \]

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^2,x]

[Out]

(-2520*a^10*e^10*(-(B*d) + A*e) + 25200*a^9*b*e^9*(A*d*e + B*(-d^2 + d*e*x + e^2*x^2)) + 56700*a^8*b^2*e^8*(2*
A*e*(-d^2 + d*e*x + e^2*x^2) + B*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3)) + 50400*a^7*b^3*e^7*(3*A*e*(2*d^
3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 2*B*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4)) + 4
4100*a^6*b^4*e^6*(4*A*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + B*(12*d^5 - 48*d^4*e*x
- 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5)) + 10584*a^5*b^5*e^5*(5*A*e*(12*d^5 - 48*d^4*e*x
- 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) - 6*B*(10*d^6 - 50*d^5*e*x - 30*d^4*e^2*x^2 + 10*
d^3*e^3*x^3 - 5*d^2*e^4*x^4 + 3*d*e^5*x^5 - 2*e^6*x^6)) + 8820*a^4*b^6*e^4*(6*A*e*(-10*d^6 + 50*d^5*e*x + 30*d
^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) + B*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2
*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x^7)) + 720*a^3*b^7*e^3*(7*A*e
*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 1
0*e^7*x^7) - 4*B*(105*d^8 - 735*d^7*e*x - 420*d^6*e^2*x^2 + 140*d^5*e^3*x^3 - 70*d^4*e^4*x^4 + 42*d^3*e^5*x^5
- 28*d^2*e^6*x^6 + 20*d*e^7*x^7 - 15*e^8*x^8)) + 135*a^2*b^8*e^2*(8*A*e*(-105*d^8 + 735*d^7*e*x + 420*d^6*e^2*
x^2 - 140*d^5*e^3*x^3 + 70*d^4*e^4*x^4 - 42*d^3*e^5*x^5 + 28*d^2*e^6*x^6 - 20*d*e^7*x^7 + 15*e^8*x^8) + 3*B*(2
80*d^9 - 2240*d^8*e*x - 1260*d^7*e^2*x^2 + 420*d^6*e^3*x^3 - 210*d^5*e^4*x^4 + 126*d^4*e^5*x^5 - 84*d^3*e^6*x^
6 + 60*d^2*e^7*x^7 - 45*d*e^8*x^8 + 35*e^9*x^9)) + 10*a*b^9*e*(9*A*e*(280*d^9 - 2240*d^8*e*x - 1260*d^7*e^2*x^
2 + 420*d^6*e^3*x^3 - 210*d^5*e^4*x^4 + 126*d^4*e^5*x^5 - 84*d^3*e^6*x^6 + 60*d^2*e^7*x^7 - 45*d*e^8*x^8 + 35*
e^9*x^9) - 10*B*(252*d^10 - 2268*d^9*e*x - 1260*d^8*e^2*x^2 + 420*d^7*e^3*x^3 - 210*d^6*e^4*x^4 + 126*d^5*e^5*
x^5 - 84*d^4*e^6*x^6 + 60*d^3*e^7*x^7 - 45*d^2*e^8*x^8 + 35*d*e^9*x^9 - 28*e^10*x^10)) + b^10*(10*A*e*(-252*d^
10 + 2268*d^9*e*x + 1260*d^8*e^2*x^2 - 420*d^7*e^3*x^3 + 210*d^6*e^4*x^4 - 126*d^5*e^5*x^5 + 84*d^4*e^6*x^6 -
60*d^3*e^7*x^7 + 45*d^2*e^8*x^8 - 35*d*e^9*x^9 + 28*e^10*x^10) + B*(2520*d^11 - 25200*d^10*e*x - 13860*d^9*e^2
*x^2 + 4620*d^8*e^3*x^3 - 2310*d^7*e^4*x^4 + 1386*d^6*e^5*x^5 - 924*d^5*e^6*x^6 + 660*d^4*e^7*x^7 - 495*d^3*e^
8*x^8 + 385*d^2*e^9*x^9 - 308*d*e^10*x^10 + 252*e^11*x^11)) + 2520*(b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e)
*(d + e*x)*Log[d + e*x])/(2520*e^12*(d + e*x))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1906\) vs. \(2(431)=862\).

Time = 2.11 (sec) , antiderivative size = 1907, normalized size of antiderivative = 4.29

method	result	size
norman	\(\text {Expression too large to display}\)	\(1907\)
default	\(\text {Expression too large to display}\)	\(2167\)
risch	\(\text {Expression too large to display}\)	\(2447\)
parallelrisch	\(\text {Expression too large to display}\)	\(2772\)

[In]

int((b*x+a)^10*(B*x+A)/(e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

((A*a^10*e^11-10*A*a^9*b*d*e^10+90*A*a^8*b^2*d^2*e^9-360*A*a^7*b^3*d^3*e^8+840*A*a^6*b^4*d^4*e^7-1260*A*a^5*b^
5*d^5*e^6+1260*A*a^4*b^6*d^6*e^5-840*A*a^3*b^7*d^7*e^4+360*A*a^2*b^8*d^8*e^3-90*A*a*b^9*d^9*e^2+10*A*b^10*d^10
*e-B*a^10*d*e^10+20*B*a^9*b*d^2*e^9-135*B*a^8*b^2*d^3*e^8+480*B*a^7*b^3*d^4*e^7-1050*B*a^6*b^4*d^5*e^6+1512*B*
a^5*b^5*d^6*e^5-1470*B*a^4*b^6*d^7*e^4+960*B*a^3*b^7*d^8*e^3-405*B*a^2*b^8*d^9*e^2+100*B*a*b^9*d^10*e-11*B*b^1
0*d^11)/e^11/d*x+1/2*b*(90*A*a^8*b*e^9-360*A*a^7*b^2*d*e^8+840*A*a^6*b^3*d^2*e^7-1260*A*a^5*b^4*d^3*e^6+1260*A
*a^4*b^5*d^4*e^5-840*A*a^3*b^6*d^5*e^4+360*A*a^2*b^7*d^6*e^3-90*A*a*b^8*d^7*e^2+10*A*b^9*d^8*e+20*B*a^9*e^9-13
5*B*a^8*b*d*e^8+480*B*a^7*b^2*d^2*e^7-1050*B*a^6*b^3*d^3*e^6+1512*B*a^5*b^4*d^4*e^5-1470*B*a^4*b^5*d^5*e^4+960
*B*a^3*b^6*d^6*e^3-405*B*a^2*b^7*d^7*e^2+100*B*a*b^8*d^8*e-11*B*b^9*d^9)/e^10*x^2+1/6*b^2*(360*A*a^7*b*e^8-840
*A*a^6*b^2*d*e^7+1260*A*a^5*b^3*d^2*e^6-1260*A*a^4*b^4*d^3*e^5+840*A*a^3*b^5*d^4*e^4-360*A*a^2*b^6*d^5*e^3+90*
A*a*b^7*d^6*e^2-10*A*b^8*d^7*e+135*B*a^8*e^8-480*B*a^7*b*d*e^7+1050*B*a^6*b^2*d^2*e^6-1512*B*a^5*b^3*d^3*e^5+1
470*B*a^4*b^4*d^4*e^4-960*B*a^3*b^5*d^5*e^3+405*B*a^2*b^6*d^6*e^2-100*B*a*b^7*d^7*e+11*B*b^8*d^8)/e^9*x^3+1/12
*b^3*(840*A*a^6*b*e^7-1260*A*a^5*b^2*d*e^6+1260*A*a^4*b^3*d^2*e^5-840*A*a^3*b^4*d^3*e^4+360*A*a^2*b^5*d^4*e^3-
90*A*a*b^6*d^5*e^2+10*A*b^7*d^6*e+480*B*a^7*e^7-1050*B*a^6*b*d*e^6+1512*B*a^5*b^2*d^2*e^5-1470*B*a^4*b^3*d^3*e
^4+960*B*a^3*b^4*d^4*e^3-405*B*a^2*b^5*d^5*e^2+100*B*a*b^6*d^6*e-11*B*b^7*d^7)/e^8*x^4+1/20*b^4*(1260*A*a^5*b*
e^6-1260*A*a^4*b^2*d*e^5+840*A*a^3*b^3*d^2*e^4-360*A*a^2*b^4*d^3*e^3+90*A*a*b^5*d^4*e^2-10*A*b^6*d^5*e+1050*B*
a^6*e^6-1512*B*a^5*b*d*e^5+1470*B*a^4*b^2*d^2*e^4-960*B*a^3*b^3*d^3*e^3+405*B*a^2*b^4*d^4*e^2-100*B*a*b^5*d^5*
e+11*B*b^6*d^6)/e^7*x^5+1/30*b^5*(1260*A*a^4*b*e^5-840*A*a^3*b^2*d*e^4+360*A*a^2*b^3*d^2*e^3-90*A*a*b^4*d^3*e^
2+10*A*b^5*d^4*e+1512*B*a^5*e^5-1470*B*a^4*b*d*e^4+960*B*a^3*b^2*d^2*e^3-405*B*a^2*b^3*d^3*e^2+100*B*a*b^4*d^4
*e-11*B*b^5*d^5)/e^6*x^6+1/42*b^6*(840*A*a^3*b*e^4-360*A*a^2*b^2*d*e^3+90*A*a*b^3*d^2*e^2-10*A*b^4*d^3*e+1470*
B*a^4*e^4-960*B*a^3*b*d*e^3+405*B*a^2*b^2*d^2*e^2-100*B*a*b^3*d^3*e+11*B*b^4*d^4)/e^5*x^7+1/56*b^7*(360*A*a^2*
b*e^3-90*A*a*b^2*d*e^2+10*A*b^3*d^2*e+960*B*a^3*e^3-405*B*a^2*b*d*e^2+100*B*a*b^2*d^2*e-11*B*b^3*d^3)/e^4*x^8+
1/72*b^8*(90*A*a*b*e^2-10*A*b^2*d*e+405*B*a^2*e^2-100*B*a*b*d*e+11*B*b^2*d^2)/e^3*x^9+1/90*b^9*(10*A*b*e+100*B
*a*e-11*B*b*d)/e^2*x^10+1/10*b^10*B/e*x^11)/(e*x+d)+1/e^12*(10*A*a^9*b*e^10-90*A*a^8*b^2*d*e^9+360*A*a^7*b^3*d
^2*e^8-840*A*a^6*b^4*d^3*e^7+1260*A*a^5*b^5*d^4*e^6-1260*A*a^4*b^6*d^5*e^5+840*A*a^3*b^7*d^6*e^4-360*A*a^2*b^8
*d^7*e^3+90*A*a*b^9*d^8*e^2-10*A*b^10*d^9*e+B*a^10*e^10-20*B*a^9*b*d*e^9+135*B*a^8*b^2*d^2*e^8-480*B*a^7*b^3*d
^3*e^7+1050*B*a^6*b^4*d^4*e^6-1512*B*a^5*b^5*d^5*e^5+1470*B*a^4*b^6*d^6*e^4-960*B*a^3*b^7*d^7*e^3+405*B*a^2*b^
8*d^8*e^2-100*B*a*b^9*d^9*e+11*B*b^10*d^10)*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2329 vs. \(2 (431) = 862\).

Time = 0.27 (sec) , antiderivative size = 2329, normalized size of antiderivative = 5.23 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^2,x, algorithm="fricas")

[Out]

1/2520*(252*B*b^10*e^11*x^11 + 2520*B*b^10*d^11 - 2520*A*a^10*e^11 - 2520*(10*B*a*b^9 + A*b^10)*d^10*e + 12600
*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 37800*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 75600*(7*B*a^4*b^6 + 4*A*a^3*
b^7)*d^7*e^4 - 105840*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 105840*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 75600
*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 37800*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 12600*(2*B*a^9*b + 9*A*a^8*
b^2)*d^2*e^9 + 2520*(B*a^10 + 10*A*a^9*b)*d*e^10 - 28*(11*B*b^10*d*e^10 - 10*(10*B*a*b^9 + A*b^10)*e^11)*x^10
+ 35*(11*B*b^10*d^2*e^9 - 10*(10*B*a*b^9 + A*b^10)*d*e^10 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 45*(11*B*
b^10*d^3*e^8 - 10*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 120*(8*B*a^3*b^7 + 3*A
*a^2*b^8)*e^11)*x^8 + 60*(11*B*b^10*d^4*e^7 - 10*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*
d^2*e^9 - 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 - 84*(11*B*b^10*d
^5*e^6 - 10*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 120*(8*B*a^3*b^7 + 3*A*a^2*
b^8)*d^2*e^9 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 - 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 126*(11*B*
b^10*d^6*e^5 - 10*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 120*(8*B*a^3*b^7 + 3*
A*a^2*b^8)*d^3*e^8 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 210*(5
*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 - 210*(11*B*b^10*d^7*e^4 - 10*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 45*(9*B*a^2*
b^8 + 2*A*a*b^9)*d^5*e^6 - 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 -
252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - 120*(4*B*a^7*b^3 + 7*A*a^6
*b^4)*e^11)*x^4 + 420*(11*B*b^10*d^8*e^3 - 10*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6
*e^5 - 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 252*(6*B*a^5*b^5 +
5*A*a^4*b^6)*d^3*e^8 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 45*(
3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 - 1260*(11*B*b^10*d^9*e^2 - 10*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 45*(9*B*a^
2*b^8 + 2*A*a*b^9)*d^7*e^4 - 120*(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
- 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 120*(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 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 - 2520*(10*B
*b^10*d^10*e - 9*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 40*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 105*(8*B*a^3*b^7 + 3*A
*a^2*b^8)*d^7*e^4 + 180*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 210*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 168*(5
*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 90*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 30*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^
2*e^9 - 5*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10)*x + 2520*(11*B*b^10*d^11 - 10*(10*B*a*b^9 + A*b^10)*d^10*e + 45*(9
*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 120*(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 - 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 120*(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 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + (B*a^1
0 + 10*A*a^9*b)*d*e^10 + (11*B*b^10*d^10*e - 10*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d
^8*e^3 - 120*(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 - 252*(6*B*a^5*b^5
+ 5*A*a^4*b^6)*d^5*e^6 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 4
5*(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 + (B*a^10 + 10*A*a^9*b)*e^11)*x)*l
og(e*x + d))/(e^13*x + d*e^12)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1974 vs. \(2 (464) = 928\).

Time = 4.19 (sec) , antiderivative size = 1974, normalized size of antiderivative = 4.44 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**2,x)

[Out]

B*b**10*x**10/(10*e**2) + x**9*(A*b**10/(9*e**2) + 10*B*a*b**9/(9*e**2) - 2*B*b**10*d/(9*e**3)) + x**8*(5*A*a*
b**9/(4*e**2) - A*b**10*d/(4*e**3) + 45*B*a**2*b**8/(8*e**2) - 5*B*a*b**9*d/(2*e**3) + 3*B*b**10*d**2/(8*e**4)
) + x**7*(45*A*a**2*b**8/(7*e**2) - 20*A*a*b**9*d/(7*e**3) + 3*A*b**10*d**2/(7*e**4) + 120*B*a**3*b**7/(7*e**2
) - 90*B*a**2*b**8*d/(7*e**3) + 30*B*a*b**9*d**2/(7*e**4) - 4*B*b**10*d**3/(7*e**5)) + x**6*(20*A*a**3*b**7/e*
*2 - 15*A*a**2*b**8*d/e**3 + 5*A*a*b**9*d**2/e**4 - 2*A*b**10*d**3/(3*e**5) + 35*B*a**4*b**6/e**2 - 40*B*a**3*
b**7*d/e**3 + 45*B*a**2*b**8*d**2/(2*e**4) - 20*B*a*b**9*d**3/(3*e**5) + 5*B*b**10*d**4/(6*e**6)) + x**5*(42*A
*a**4*b**6/e**2 - 48*A*a**3*b**7*d/e**3 + 27*A*a**2*b**8*d**2/e**4 - 8*A*a*b**9*d**3/e**5 + A*b**10*d**4/e**6
+ 252*B*a**5*b**5/(5*e**2) - 84*B*a**4*b**6*d/e**3 + 72*B*a**3*b**7*d**2/e**4 - 36*B*a**2*b**8*d**3/e**5 + 10*
B*a*b**9*d**4/e**6 - 6*B*b**10*d**5/(5*e**7)) + x**4*(63*A*a**5*b**5/e**2 - 105*A*a**4*b**6*d/e**3 + 90*A*a**3
*b**7*d**2/e**4 - 45*A*a**2*b**8*d**3/e**5 + 25*A*a*b**9*d**4/(2*e**6) - 3*A*b**10*d**5/(2*e**7) + 105*B*a**6*
b**4/(2*e**2) - 126*B*a**5*b**5*d/e**3 + 315*B*a**4*b**6*d**2/(2*e**4) - 120*B*a**3*b**7*d**3/e**5 + 225*B*a**
2*b**8*d**4/(4*e**6) - 15*B*a*b**9*d**5/e**7 + 7*B*b**10*d**6/(4*e**8)) + x**3*(70*A*a**6*b**4/e**2 - 168*A*a*
*5*b**5*d/e**3 + 210*A*a**4*b**6*d**2/e**4 - 160*A*a**3*b**7*d**3/e**5 + 75*A*a**2*b**8*d**4/e**6 - 20*A*a*b**
9*d**5/e**7 + 7*A*b**10*d**6/(3*e**8) + 40*B*a**7*b**3/e**2 - 140*B*a**6*b**4*d/e**3 + 252*B*a**5*b**5*d**2/e*
*4 - 280*B*a**4*b**6*d**3/e**5 + 200*B*a**3*b**7*d**4/e**6 - 90*B*a**2*b**8*d**5/e**7 + 70*B*a*b**9*d**6/(3*e*
*8) - 8*B*b**10*d**7/(3*e**9)) + x**2*(60*A*a**7*b**3/e**2 - 210*A*a**6*b**4*d/e**3 + 378*A*a**5*b**5*d**2/e**
4 - 420*A*a**4*b**6*d**3/e**5 + 300*A*a**3*b**7*d**4/e**6 - 135*A*a**2*b**8*d**5/e**7 + 35*A*a*b**9*d**6/e**8
- 4*A*b**10*d**7/e**9 + 45*B*a**8*b**2/(2*e**2) - 120*B*a**7*b**3*d/e**3 + 315*B*a**6*b**4*d**2/e**4 - 504*B*a
**5*b**5*d**3/e**5 + 525*B*a**4*b**6*d**4/e**6 - 360*B*a**3*b**7*d**5/e**7 + 315*B*a**2*b**8*d**6/(2*e**8) - 4
0*B*a*b**9*d**7/e**9 + 9*B*b**10*d**8/(2*e**10)) + x*(45*A*a**8*b**2/e**2 - 240*A*a**7*b**3*d/e**3 + 630*A*a**
6*b**4*d**2/e**4 - 1008*A*a**5*b**5*d**3/e**5 + 1050*A*a**4*b**6*d**4/e**6 - 720*A*a**3*b**7*d**5/e**7 + 315*A
*a**2*b**8*d**6/e**8 - 80*A*a*b**9*d**7/e**9 + 9*A*b**10*d**8/e**10 + 10*B*a**9*b/e**2 - 90*B*a**8*b**2*d/e**3
+ 360*B*a**7*b**3*d**2/e**4 - 840*B*a**6*b**4*d**3/e**5 + 1260*B*a**5*b**5*d**4/e**6 - 1260*B*a**4*b**6*d**5/
e**7 + 840*B*a**3*b**7*d**6/e**8 - 360*B*a**2*b**8*d**7/e**9 + 90*B*a*b**9*d**8/e**10 - 10*B*b**10*d**9/e**11)
+ (-A*a**10*e**11 + 10*A*a**9*b*d*e**10 - 45*A*a**8*b**2*d**2*e**9 + 120*A*a**7*b**3*d**3*e**8 - 210*A*a**6*b
**4*d**4*e**7 + 252*A*a**5*b**5*d**5*e**6 - 210*A*a**4*b**6*d**6*e**5 + 120*A*a**3*b**7*d**7*e**4 - 45*A*a**2*
b**8*d**8*e**3 + 10*A*a*b**9*d**9*e**2 - A*b**10*d**10*e + B*a**10*d*e**10 - 10*B*a**9*b*d**2*e**9 + 45*B*a**8
*b**2*d**3*e**8 - 120*B*a**7*b**3*d**4*e**7 + 210*B*a**6*b**4*d**5*e**6 - 252*B*a**5*b**5*d**6*e**5 + 210*B*a*
*4*b**6*d**7*e**4 - 120*B*a**3*b**7*d**8*e**3 + 45*B*a**2*b**8*d**9*e**2 - 10*B*a*b**9*d**10*e + B*b**10*d**11
)/(d*e**12 + e**13*x) + (a*e - b*d)**9*(10*A*b*e + B*a*e - 11*B*b*d)*log(d + e*x)/e**12

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1817 vs. \(2 (431) = 862\).

Time = 0.22 (sec) , antiderivative size = 1817, normalized size of antiderivative = 4.08 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^2,x, algorithm="maxima")

[Out]

(B*b^10*d^11 - A*a^10*e^11 - (10*B*a*b^9 + A*b^10)*d^10*e + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 15*(8*B*a^3*
b^7 + 3*A*a^2*b^8)*d^8*e^3 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 +
42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 15*(3*B*a^8*b^2 + 8*A*a^7*b
^3)*d^3*e^8 - 5*(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)/(e^13*x + d*e^12) + 1/2520*(
252*B*b^10*e^9*x^10 - 280*(2*B*b^10*d*e^8 - (10*B*a*b^9 + A*b^10)*e^9)*x^9 + 315*(3*B*b^10*d^2*e^7 - 2*(10*B*a
*b^9 + A*b^10)*d*e^8 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^9)*x^8 - 360*(4*B*b^10*d^3*e^6 - 3*(10*B*a*b^9 + A*b^10)*
d^2*e^7 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^8 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^9)*x^7 + 420*(5*B*b^10*d^4*e^5
- 4*(10*B*a*b^9 + A*b^10)*d^3*e^6 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^7 - 30*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e
^8 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^9)*x^6 - 504*(6*B*b^10*d^5*e^4 - 5*(10*B*a*b^9 + A*b^10)*d^4*e^5 + 20*(9
*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^6 - 45*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^7 + 60*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^
8 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^9)*x^5 + 630*(7*B*b^10*d^6*e^3 - 6*(10*B*a*b^9 + A*b^10)*d^5*e^4 + 25*(9*
B*a^2*b^8 + 2*A*a*b^9)*d^4*e^5 - 60*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^6 + 90*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e
^7 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^8 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^9)*x^4 - 840*(8*B*b^10*d^7*e^2 -
7*(10*B*a*b^9 + A*b^10)*d^6*e^3 + 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^4 - 75*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^
5 + 120*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^6 - 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^7 + 84*(5*B*a^6*b^4 + 6*A*
a^5*b^5)*d*e^8 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^9)*x^3 + 1260*(9*B*b^10*d^8*e - 8*(10*B*a*b^9 + A*b^10)*d^7*
e^2 + 35*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^3 - 90*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^4 + 150*(7*B*a^4*b^6 + 4*A*a
^3*b^7)*d^4*e^5 - 168*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^6 + 126*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^7 - 60*(4*B*
a^7*b^3 + 7*A*a^6*b^4)*d*e^8 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^9)*x^2 - 2520*(10*B*b^10*d^9 - 9*(10*B*a*b^9 +
A*b^10)*d^8*e + 40*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^2 - 105*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^3 + 180*(7*B*a^4
*b^6 + 4*A*a^3*b^7)*d^5*e^4 - 210*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^5 + 168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^
6 - 90*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^7 + 30*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^8 - 5*(2*B*a^9*b + 9*A*a^8*b^2
)*e^9)*x)/e^11 + (11*B*b^10*d^10 - 10*(10*B*a*b^9 + A*b^10)*d^9*e + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^2 - 120
*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^3 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^4 - 252*(6*B*a^5*b^5 + 5*A*a^4*b^
6)*d^5*e^5 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^6 - 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^7 + 45*(3*B*a^8*b
^2 + 8*A*a^7*b^3)*d^2*e^8 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^9 + (B*a^10 + 10*A*a^9*b)*e^10)*log(e*x + d)/e^12

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2207 vs. \(2 (431) = 862\).

Time = 0.33 (sec) , antiderivative size = 2207, normalized size of antiderivative = 4.96 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^2,x, algorithm="giac")

[Out]

1/2520*(252*B*b^10 - 280*(11*B*b^10*d*e - 10*B*a*b^9*e^2 - A*b^10*e^2)/((e*x + d)*e) + 1575*(11*B*b^10*d^2*e^2
- 20*B*a*b^9*d*e^3 - 2*A*b^10*d*e^3 + 9*B*a^2*b^8*e^4 + 2*A*a*b^9*e^4)/((e*x + d)^2*e^2) - 5400*(11*B*b^10*d^
3*e^3 - 30*B*a*b^9*d^2*e^4 - 3*A*b^10*d^2*e^4 + 27*B*a^2*b^8*d*e^5 + 6*A*a*b^9*d*e^5 - 8*B*a^3*b^7*e^6 - 3*A*a
^2*b^8*e^6)/((e*x + d)^3*e^3) + 12600*(11*B*b^10*d^4*e^4 - 40*B*a*b^9*d^3*e^5 - 4*A*b^10*d^3*e^5 + 54*B*a^2*b^
8*d^2*e^6 + 12*A*a*b^9*d^2*e^6 - 32*B*a^3*b^7*d*e^7 - 12*A*a^2*b^8*d*e^7 + 7*B*a^4*b^6*e^8 + 4*A*a^3*b^7*e^8)/
((e*x + d)^4*e^4) - 21168*(11*B*b^10*d^5*e^5 - 50*B*a*b^9*d^4*e^6 - 5*A*b^10*d^4*e^6 + 90*B*a^2*b^8*d^3*e^7 +
20*A*a*b^9*d^3*e^7 - 80*B*a^3*b^7*d^2*e^8 - 30*A*a^2*b^8*d^2*e^8 + 35*B*a^4*b^6*d*e^9 + 20*A*a^3*b^7*d*e^9 - 6
*B*a^5*b^5*e^10 - 5*A*a^4*b^6*e^10)/((e*x + d)^5*e^5) + 26460*(11*B*b^10*d^6*e^6 - 60*B*a*b^9*d^5*e^7 - 6*A*b^
10*d^5*e^7 + 135*B*a^2*b^8*d^4*e^8 + 30*A*a*b^9*d^4*e^8 - 160*B*a^3*b^7*d^3*e^9 - 60*A*a^2*b^8*d^3*e^9 + 105*B
*a^4*b^6*d^2*e^10 + 60*A*a^3*b^7*d^2*e^10 - 36*B*a^5*b^5*d*e^11 - 30*A*a^4*b^6*d*e^11 + 5*B*a^6*b^4*e^12 + 6*A
*a^5*b^5*e^12)/((e*x + d)^6*e^6) - 25200*(11*B*b^10*d^7*e^7 - 70*B*a*b^9*d^6*e^8 - 7*A*b^10*d^6*e^8 + 189*B*a^
2*b^8*d^5*e^9 + 42*A*a*b^9*d^5*e^9 - 280*B*a^3*b^7*d^4*e^10 - 105*A*a^2*b^8*d^4*e^10 + 245*B*a^4*b^6*d^3*e^11
+ 140*A*a^3*b^7*d^3*e^11 - 126*B*a^5*b^5*d^2*e^12 - 105*A*a^4*b^6*d^2*e^12 + 35*B*a^6*b^4*d*e^13 + 42*A*a^5*b^
5*d*e^13 - 4*B*a^7*b^3*e^14 - 7*A*a^6*b^4*e^14)/((e*x + d)^7*e^7) + 18900*(11*B*b^10*d^8*e^8 - 80*B*a*b^9*d^7*
e^9 - 8*A*b^10*d^7*e^9 + 252*B*a^2*b^8*d^6*e^10 + 56*A*a*b^9*d^6*e^10 - 448*B*a^3*b^7*d^5*e^11 - 168*A*a^2*b^8
*d^5*e^11 + 490*B*a^4*b^6*d^4*e^12 + 280*A*a^3*b^7*d^4*e^12 - 336*B*a^5*b^5*d^3*e^13 - 280*A*a^4*b^6*d^3*e^13
+ 140*B*a^6*b^4*d^2*e^14 + 168*A*a^5*b^5*d^2*e^14 - 32*B*a^7*b^3*d*e^15 - 56*A*a^6*b^4*d*e^15 + 3*B*a^8*b^2*e^
16 + 8*A*a^7*b^3*e^16)/((e*x + d)^8*e^8) - 12600*(11*B*b^10*d^9*e^9 - 90*B*a*b^9*d^8*e^10 - 9*A*b^10*d^8*e^10
+ 324*B*a^2*b^8*d^7*e^11 + 72*A*a*b^9*d^7*e^11 - 672*B*a^3*b^7*d^6*e^12 - 252*A*a^2*b^8*d^6*e^12 + 882*B*a^4*b
^6*d^5*e^13 + 504*A*a^3*b^7*d^5*e^13 - 756*B*a^5*b^5*d^4*e^14 - 630*A*a^4*b^6*d^4*e^14 + 420*B*a^6*b^4*d^3*e^1
5 + 504*A*a^5*b^5*d^3*e^15 - 144*B*a^7*b^3*d^2*e^16 - 252*A*a^6*b^4*d^2*e^16 + 27*B*a^8*b^2*d*e^17 + 72*A*a^7*
b^3*d*e^17 - 2*B*a^9*b*e^18 - 9*A*a^8*b^2*e^18)/((e*x + d)^9*e^9))*(e*x + d)^10/e^12 - (11*B*b^10*d^10 - 100*B
*a*b^9*d^9*e - 10*A*b^10*d^9*e + 405*B*a^2*b^8*d^8*e^2 + 90*A*a*b^9*d^8*e^2 - 960*B*a^3*b^7*d^7*e^3 - 360*A*a^
2*b^8*d^7*e^3 + 1470*B*a^4*b^6*d^6*e^4 + 840*A*a^3*b^7*d^6*e^4 - 1512*B*a^5*b^5*d^5*e^5 - 1260*A*a^4*b^6*d^5*e
^5 + 1050*B*a^6*b^4*d^4*e^6 + 1260*A*a^5*b^5*d^4*e^6 - 480*B*a^7*b^3*d^3*e^7 - 840*A*a^6*b^4*d^3*e^7 + 135*B*a
^8*b^2*d^2*e^8 + 360*A*a^7*b^3*d^2*e^8 - 20*B*a^9*b*d*e^9 - 90*A*a^8*b^2*d*e^9 + B*a^10*e^10 + 10*A*a^9*b*e^10
)*log(abs(e*x + d)/((e*x + d)^2*abs(e)))/e^12 + (B*b^10*d^11*e^10/(e*x + d) - 10*B*a*b^9*d^10*e^11/(e*x + d) -
A*b^10*d^10*e^11/(e*x + d) + 45*B*a^2*b^8*d^9*e^12/(e*x + d) + 10*A*a*b^9*d^9*e^12/(e*x + d) - 120*B*a^3*b^7*
d^8*e^13/(e*x + d) - 45*A*a^2*b^8*d^8*e^13/(e*x + d) + 210*B*a^4*b^6*d^7*e^14/(e*x + d) + 120*A*a^3*b^7*d^7*e^
14/(e*x + d) - 252*B*a^5*b^5*d^6*e^15/(e*x + d) - 210*A*a^4*b^6*d^6*e^15/(e*x + d) + 210*B*a^6*b^4*d^5*e^16/(e
*x + d) + 252*A*a^5*b^5*d^5*e^16/(e*x + d) - 120*B*a^7*b^3*d^4*e^17/(e*x + d) - 210*A*a^6*b^4*d^4*e^17/(e*x +
d) + 45*B*a^8*b^2*d^3*e^18/(e*x + d) + 120*A*a^7*b^3*d^3*e^18/(e*x + d) - 10*B*a^9*b*d^2*e^19/(e*x + d) - 45*A
*a^8*b^2*d^2*e^19/(e*x + d) + B*a^10*d*e^20/(e*x + d) + 10*A*a^9*b*d*e^20/(e*x + d) - A*a^10*e^21/(e*x + d))/e
^22

Mupad [B] (verification not implemented)

Time = 1.55 (sec) , antiderivative size = 7792, normalized size of antiderivative = 17.51 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx=\text {Too large to display} \]

[In]

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^2,x)

[Out]

x^4*((d*((d^2*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 +
(B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))
/e^2))/e^2 + (2*d*((d^2*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^
2 + (B*b^10*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*
A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*
b^7*(3*A*b + 8*B*a))/e^2))/e + (30*a^3*b^6*(4*A*b + 7*B*a))/e^2))/e - (42*a^4*b^5*(5*A*b + 6*B*a))/e^2))/(2*e)
- (d^2*((d^2*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^1
0*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*
a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b
+ 8*B*a))/e^2))/e + (30*a^3*b^6*(4*A*b + 7*B*a))/e^2))/(4*e^2) + (21*a^5*b^4*(6*A*b + 5*B*a))/(2*e^2)) + x*((
2*d*((2*d*((d^2*((d^2*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a)
)/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b +
8*B*a))/e^2))/e^2 + (2*d*((d^2*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*
B*a))/e^2 + (B*b^10*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a
*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 +
(15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e + (30*a^3*b^6*(4*A*b + 7*B*a))/e^2))/e - (42*a^4*b^5*(5*A*b + 6*B*a))/e^2
))/e^2 - (2*d*((2*d*((d^2*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*
B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A
*b + 8*B*a))/e^2))/e^2 + (2*d*((d^2*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b
+ 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e -
(5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^
2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e + (30*a^3*b^6*(4*A*b + 7*B*a))/e^2))/e - (42*a^4*b^5*(5*A*b + 6*B*a))
/e^2))/e - (d^2*((d^2*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2
+ (B*b^10*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*
b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^
7*(3*A*b + 8*B*a))/e^2))/e + (30*a^3*b^6*(4*A*b + 7*B*a))/e^2))/e^2 + (42*a^5*b^4*(6*A*b + 5*B*a))/e^2))/e + (
30*a^6*b^3*(7*A*b + 4*B*a))/e^2))/e + (d^2*((2*d*((d^2*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e
^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)
/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e^2 + (2*d*((d^2*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*
d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/
e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9
)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e + (30*a^3*b^6*(4*A*b + 7*B*a))/e^2))/e -
(42*a^4*b^5*(5*A*b + 6*B*a))/e^2))/e - (d^2*((d^2*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (
5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^1
0*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b
^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e + (30*a^3*b^6*(4*A*b + 7*B*a))/e^2))/e^2 + (42*a^5*b^4
*(6*A*b + 5*B*a))/e^2))/e^2 - (15*a^7*b^2*(8*A*b + 3*B*a))/e^2))/e - (d^2*((d^2*((d^2*((2*d*((2*d*((A*b^10 + 1
0*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 +
10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e^2 + (2*d*((d^2*((2*d*((A*b^10
+ 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e^2 - (2*d*((2*
d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))
/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e + (30*a^3
*b^6*(4*A*b + 7*B*a))/e^2))/e - (42*a^4*b^5*(5*A*b + 6*B*a))/e^2))/e^2 - (2*d*((2*d*((d^2*((2*d*((2*d*((A*b^10
+ 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^
10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e^2 + (2*d*((d^2*((2*d*((A*
b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e^2 - (2*d*
((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e
^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e + (30
*a^3*b^6*(4*A*b + 7*B*a))/e^2))/e - (42*a^4*b^5*(5*A*b + 6*B*a))/e^2))/e - (d^2*((d^2*((2*d*((A*b^10 + 10*B*a*
b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((
A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*
((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e + (30*a^3*b^6*(4*A*
b + 7*B*a))/e^2))/e^2 + (42*a^5*b^4*(6*A*b + 5*B*a))/e^2))/e + (30*a^6*b^3*(7*A*b + 4*B*a))/e^2))/e^2 + (5*a^8
*b*(9*A*b + 2*B*a))/e^2) - x^8*((d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/(4*e) - (5*a*b^8*(2*A*b + 9
*B*a))/(8*e^2) + (B*b^10*d^2)/(8*e^4)) - x^2*((d*((d^2*((d^2*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^1
0*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b
^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e^2 + (2*d*((d^2*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B
*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((A*b^10 + 10*B*a
*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B
*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e + (30*a^3*b^6*(4*A*b + 7*B*a))/e^2
))/e - (42*a^4*b^5*(5*A*b + 6*B*a))/e^2))/e^2 - (2*d*((2*d*((d^2*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B
*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2
*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e^2 + (2*d*((d^2*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 -
(2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((A*b^10 + 10
*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 +
10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e + (30*a^3*b^6*(4*A*b + 7*B*a))
/e^2))/e - (42*a^4*b^5*(5*A*b + 6*B*a))/e^2))/e - (d^2*((d^2*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e
^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2
- (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^
2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e + (30*a^3*b^6*(4*A*b + 7*B*a))/e^2))/e^2 + (
42*a^5*b^4*(6*A*b + 5*B*a))/e^2))/e + (30*a^6*b^3*(7*A*b + 4*B*a))/e^2))/e + (d^2*((2*d*((d^2*((2*d*((2*d*((A*
b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((
A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e^2 + (2*d*((d^2*((2*d*
((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e^2 - (
2*d*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^
2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e +
(30*a^3*b^6*(4*A*b + 7*B*a))/e^2))/e - (42*a^4*b^5*(5*A*b + 6*B*a))/e^2))/e - (d^2*((d^2*((2*d*((A*b^10 + 10*
B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e^2 - (2*d*((2*d*((2*
d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (
d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e + (30*a^3*b^6*(
4*A*b + 7*B*a))/e^2))/e^2 + (42*a^5*b^4*(6*A*b + 5*B*a))/e^2))/(2*e^2) - (15*a^7*b^2*(8*A*b + 3*B*a))/(2*e^2))
+ x^6*((d^2*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10
*d^2)/e^4))/(6*e^2) - (d*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B
*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*
b + 8*B*a))/e^2))/(3*e) + (5*a^3*b^6*(4*A*b + 7*B*a))/e^2) + x^7*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B
*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/(7*e) - (d^2*((A*b^10 + 10*B*a*b^9)/e^2
- (2*B*b^10*d)/e^3))/(7*e^2) + (15*a^2*b^7*(3*A*b + 8*B*a))/(7*e^2)) + x^3*((d^2*((d^2*((2*d*((2*d*((A*b^10 +
10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10
+ 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e^2 + (2*d*((d^2*((2*d*((A*b^1
0 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e^2 - (2*d*((2
*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4)
)/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e + (30*a^
3*b^6*(4*A*b + 7*B*a))/e^2))/e - (42*a^4*b^5*(5*A*b + 6*B*a))/e^2))/(3*e^2) - (2*d*((2*d*((d^2*((2*d*((2*d*((A
*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*(
(A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e^2 + (2*d*((d^2*((2*d
*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e^2 -
(2*d*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d
^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e
+ (30*a^3*b^6*(4*A*b + 7*B*a))/e^2))/e - (42*a^4*b^5*(5*A*b + 6*B*a))/e^2))/e - (d^2*((d^2*((2*d*((A*b^10 + 10
*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e^2 - (2*d*((2*d*((2
*d*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e -
(d^2*((A*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e + (30*a^3*b^6*
(4*A*b + 7*B*a))/e^2))/e^2 + (42*a^5*b^4*(6*A*b + 5*B*a))/e^2))/(3*e) + (10*a^6*b^3*(7*A*b + 4*B*a))/e^2) + x^
9*((A*b^10 + 10*B*a*b^9)/(9*e^2) - (2*B*b^10*d)/(9*e^3)) - x^5*((d^2*((2*d*((2*d*((A*b^10 + 10*B*a*b^9)/e^2 -
(2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A*b^10 + 10*B*a*b^9)/e^2
- (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/(5*e^2) + (2*d*((d^2*((2*d*((A*b^10 + 10*B*a*b^9
)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((A*b
^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^2 + (B*b^10*d^2)/e^4))/e - (d^2*((A
*b^10 + 10*B*a*b^9)/e^2 - (2*B*b^10*d)/e^3))/e^2 + (15*a^2*b^7*(3*A*b + 8*B*a))/e^2))/e + (30*a^3*b^6*(4*A*b +
7*B*a))/e^2))/(5*e) - (42*a^4*b^5*(5*A*b + 6*B*a))/(5*e^2)) + (log(d + e*x)*(B*a^10*e^10 + 11*B*b^10*d^10 + 1
0*A*a^9*b*e^10 - 10*A*b^10*d^9*e + 90*A*a*b^9*d^8*e^2 - 90*A*a^8*b^2*d*e^9 - 360*A*a^2*b^8*d^7*e^3 + 840*A*a^3
*b^7*d^6*e^4 - 1260*A*a^4*b^6*d^5*e^5 + 1260*A*a^5*b^5*d^4*e^6 - 840*A*a^6*b^4*d^3*e^7 + 360*A*a^7*b^3*d^2*e^8
+ 405*B*a^2*b^8*d^8*e^2 - 960*B*a^3*b^7*d^7*e^3 + 1470*B*a^4*b^6*d^6*e^4 - 1512*B*a^5*b^5*d^5*e^5 + 1050*B*a^
6*b^4*d^4*e^6 - 480*B*a^7*b^3*d^3*e^7 + 135*B*a^8*b^2*d^2*e^8 - 100*B*a*b^9*d^9*e - 20*B*a^9*b*d*e^9))/e^12 -
(A*a^10*e^11 - B*b^10*d^11 + A*b^10*d^10*e - B*a^10*d*e^10 - 10*A*a*b^9*d^9*e^2 + 10*B*a^9*b*d^2*e^9 + 45*A*a^
2*b^8*d^8*e^3 - 120*A*a^3*b^7*d^7*e^4 + 210*A*a^4*b^6*d^6*e^5 - 252*A*a^5*b^5*d^5*e^6 + 210*A*a^6*b^4*d^4*e^7
- 120*A*a^7*b^3*d^3*e^8 + 45*A*a^8*b^2*d^2*e^9 - 45*B*a^2*b^8*d^9*e^2 + 120*B*a^3*b^7*d^8*e^3 - 210*B*a^4*b^6*
d^7*e^4 + 252*B*a^5*b^5*d^6*e^5 - 210*B*a^6*b^4*d^5*e^6 + 120*B*a^7*b^3*d^4*e^7 - 45*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)/(e*(d*e^11 + e^12*x)) + (B*b^10*x^10)/(10*e^2)