3.11.0 ∫ (a+bx)10(A+Bx) (d+ex)11 dx [1099]

Integrand size = 20, antiderivative size = 446 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx=\frac {b^{10} B x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{10 e^{12} (d+e x)^{10}}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{9 e^{12} (d+e x)^9}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{8 e^{12} (d+e x)^8}-\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{7 e^{12} (d+e x)^7}+\frac {5 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)^6}-\frac {42 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{5 e^{12} (d+e x)^5}+\frac {21 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{2 e^{12} (d+e x)^4}-\frac {10 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{e^{12} (d+e x)^3}+\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{2 e^{12} (d+e x)^2}-\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e)}{e^{12} (d+e x)}-\frac {b^9 (11 b B d-A b e-10 a B e) \log (d+e x)}{e^{12}} \]

b^10*B*x/e^11+1/10*(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)^10-1/9*(-a*e+b*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/(

e*x+d)^9+5/8*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)^8-15/7*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e

+11*B*b*d)/e^12/(e*x+d)^7+5*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)/e^12/(e*x+d)^6-42/5*b^4*(-a*e+b*d)^5*

(-6*A*b*e-5*B*a*e+11*B*b*d)/e^12/(e*x+d)^5+21/2*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)/e^12/(e*x+d)^4-10

*b^6*(-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11*B*b*d)/e^12/(e*x+d)^3+15/2*b^7*(-a*e+b*d)^2*(-3*A*b*e-8*B*a*e+11*B*b*d)

/e^12/(e*x+d)^2-5*b^8*(-a*e+b*d)*(-2*A*b*e-9*B*a*e+11*B*b*d)/e^12/(e*x+d)-b^9*(-A*b*e-10*B*a*e+11*B*b*d)*ln(e*

Rubi [A] (veriﬁed)

Time = 0.53 (sec) , antiderivative size = 446, 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} \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx=-\frac {b^9 \log (d+e x) (-10 a B e-A b e+11 b B d)}{e^{12}}-\frac {5 b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{e^{12} (d+e x)}+\frac {15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{2 e^{12} (d+e x)^2}-\frac {10 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12} (d+e x)^3}+\frac {21 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{2 e^{12} (d+e x)^4}-\frac {42 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{5 e^{12} (d+e x)^5}+\frac {5 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^6}-\frac {15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{7 e^{12} (d+e x)^7}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{8 e^{12} (d+e x)^8}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{9 e^{12} (d+e x)^9}+\frac {(b d-a e)^{10} (B d-A e)}{10 e^{12} (d+e x)^{10}}+\frac {b^{10} B x}{e^{11}} \]

(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*

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 {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] (veriﬁed)

Leaf count is larger than twice the leaf count of optimal. \(1447\) vs. \(2(446)=892\).

Time = 0.57 (sec) , antiderivative size = 1447, normalized size of antiderivative = 3.24 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx=-\frac {28 a^{10} e^{10} (9 A e+B (d+10 e x))+70 a^9 b e^9 \left (4 A e (d+10 e x)+B \left (d^2+10 d e x+45 e^2 x^2\right )\right )+45 a^8 b^2 e^8 \left (7 A e \left (d^2+10 d e x+45 e^2 x^2\right )+3 B \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )\right )+120 a^7 b^3 e^7 \left (3 A e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+2 B \left (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\right )\right )+420 a^6 b^4 e^6 \left (A e \left (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\right )+B \left (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\right )\right )+252 a^5 b^5 e^5 \left (2 A e \left (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\right )+3 B \left (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\right )\right )+210 a^4 b^6 e^4 \left (3 A e \left (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\right )+7 B \left (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\right )\right )+840 a^3 b^7 e^3 \left (A e \left (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\right )+4 B \left (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\right )\right )+1260 a^2 b^8 e^2 \left (A e \left (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\right )+9 B \left (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\right )\right )-10 a b^9 e \left (-252 A e \left (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\right )+B d \left (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\right )\right )-b^{10} \left (A d e \left (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\right )-B \left (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+9599688 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}\right )\right )+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}} \]

-1/2520*(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 + 12

0*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 + 2520

0*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

+ 9599688*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])/(e^12

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(1921\) vs. \(2(432)=864\).

Time = 2.12 (sec) , antiderivative size = 1922, normalized size of antiderivative = 4.31


method	result	size

risch	\(\text {Expression too large to display}\)	\(1922\)
default	\(\text {Expression too large to display}\)	\(1933\)
norman	\(\text {Expression too large to display}\)	\(1934\)
parallelrisch	\(\text {Expression too large to display}\)	\(2863\)

int((b*x+a)^10*(B*x+A)/(e*x+d)^11,x,method=_RETURNVERBOSE)

b^10*B*x/e^11+((-10*A*a*b^9*e^10+10*A*b^10*d*e^9-45*B*a^2*b^8*e^10+100*B*a*b^9*d*e^9-55*B*b^10*d^2*e^8)*x^9-15

/2*b^7*e^7*(3*A*a^2*b*e^3+6*A*a*b^2*d*e^2-9*A*b^3*d^2*e+8*B*a^3*e^3+27*B*a^2*b*d*e^2-90*B*a*b^2*d^2*e+55*B*b^3

*d^3)*x^8-10*b^6*e^6*(4*A*a^3*b*e^4+6*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2-22*A*b^4*d^3*e+7*B*a^4*e^4+16*B*a^3*b

*d*e^3+54*B*a^2*b^2*d^2*e^2-220*B*a*b^3*d^3*e+143*B*b^4*d^4)*x^7-7/2*b^5*e^5*(15*A*a^4*b*e^5+20*A*a^3*b^2*d*e^

4+30*A*a^2*b^3*d^2*e^3+60*A*a*b^4*d^3*e^2-125*A*b^5*d^4*e+18*B*a^5*e^5+35*B*a^4*b*d*e^4+80*B*a^3*b^2*d^2*e^3+2

70*B*a^2*b^3*d^3*e^2-1250*B*a*b^4*d^4*e+847*B*b^5*d^5)*x^6-21/5*b^4*e^4*(12*A*a^5*b*e^6+15*A*a^4*b^2*d*e^5+20*

A*a^3*b^3*d^2*e^4+30*A*a^2*b^4*d^3*e^3+60*A*a*b^5*d^4*e^2-137*A*b^6*d^5*e+10*B*a^6*e^6+18*B*a^5*b*d*e^5+35*B*a

^4*b^2*d^2*e^4+80*B*a^3*b^3*d^3*e^3+270*B*a^2*b^4*d^4*e^2-1370*B*a*b^5*d^5*e+957*B*b^6*d^6)*x^5-1/2*b^3*e^3*(7

0*A*a^6*b*e^7+84*A*a^5*b^2*d*e^6+105*A*a^4*b^3*d^2*e^5+140*A*a^3*b^4*d^3*e^4+210*A*a^2*b^5*d^4*e^3+420*A*a*b^6

*d^5*e^2-1029*A*b^7*d^6*e+40*B*a^7*e^7+70*B*a^6*b*d*e^6+126*B*a^5*b^2*d^2*e^5+245*B*a^4*b^3*d^3*e^4+560*B*a^3*

b^4*d^4*e^3+1890*B*a^2*b^5*d^5*e^2-10290*B*a*b^6*d^6*e+7359*B*b^7*d^7)*x^4-1/7*b^2*e^2*(120*A*a^7*b*e^8+140*A*

a^6*b^2*d*e^7+168*A*a^5*b^3*d^2*e^6+210*A*a^4*b^4*d^3*e^5+280*A*a^3*b^5*d^4*e^4+420*A*a^2*b^6*d^5*e^3+840*A*a*

b^7*d^6*e^2-2178*A*b^8*d^7*e+45*B*a^8*e^8+80*B*a^7*b*d*e^7+140*B*a^6*b^2*d^2*e^6+252*B*a^5*b^3*d^3*e^5+490*B*a

^4*b^4*d^4*e^4+1120*B*a^3*b^5*d^5*e^3+3780*B*a^2*b^6*d^6*e^2-21780*B*a*b^7*d^7*e+15873*B*b^8*d^8)*x^3-1/56*b*e

*(315*A*a^8*b*e^9+360*A*a^7*b^2*d*e^8+420*A*a^6*b^3*d^2*e^7+504*A*a^5*b^4*d^3*e^6+630*A*a^4*b^5*d^4*e^5+840*A*

a^3*b^6*d^5*e^4+1260*A*a^2*b^7*d^6*e^3+2520*A*a*b^8*d^7*e^2-6849*A*b^9*d^8*e+70*B*a^9*e^9+135*B*a^8*b*d*e^8+24

0*B*a^7*b^2*d^2*e^7+420*B*a^6*b^3*d^3*e^6+756*B*a^5*b^4*d^4*e^5+1470*B*a^4*b^5*d^5*e^4+3360*B*a^3*b^6*d^6*e^3+

11340*B*a^2*b^7*d^7*e^2-68490*B*a*b^8*d^8*e+50699*B*b^9*d^9)*x^2+(-10/9*A*a^9*b*e^10-5/4*A*a^8*b^2*d*e^9-10/7*

A*a^7*b^3*d^2*e^8-5/3*A*a^6*b^4*d^3*e^7-2*A*a^5*b^5*d^4*e^6-5/2*A*a^4*b^6*d^5*e^5-10/3*A*a^3*b^7*d^6*e^4-5*A*a

^2*b^8*d^7*e^3-10*A*a*b^9*d^8*e^2+7129/252*A*b^10*d^9*e-1/9*B*a^10*e^10-5/18*B*a^9*b*d*e^9-15/28*B*a^8*b^2*d^2

*e^8-20/21*B*a^7*b^3*d^3*e^7-5/3*B*a^6*b^4*d^4*e^6-3*B*a^5*b^5*d^5*e^5-35/6*B*a^4*b^6*d^6*e^4-40/3*B*a^3*b^7*d

^7*e^3-45*B*a^2*b^8*d^8*e^2+35645/126*B*a*b^9*d^9*e-53471/252*b^10*B*d^10)*x-1/2520/e*(252*A*a^10*e^11+280*A*a

^9*b*d*e^10+315*A*a^8*b^2*d^2*e^9+360*A*a^7*b^3*d^3*e^8+420*A*a^6*b^4*d^4*e^7+504*A*a^5*b^5*d^5*e^6+630*A*a^4*

b^6*d^6*e^5+840*A*a^3*b^7*d^7*e^4+1260*A*a^2*b^8*d^8*e^3+2520*A*a*b^9*d^9*e^2-7381*A*b^10*d^10*e+28*B*a^10*d*e

^10+70*B*a^9*b*d^2*e^9+135*B*a^8*b^2*d^3*e^8+240*B*a^7*b^3*d^4*e^7+420*B*a^6*b^4*d^5*e^6+756*B*a^5*b^5*d^6*e^5

+1470*B*a^4*b^6*d^7*e^4+3360*B*a^3*b^7*d^8*e^3+11340*B*a^2*b^8*d^9*e^2-73810*B*a*b^9*d^10*e+55991*B*b^10*d^11)

)/e^11/(e*x+d)^10+b^10/e^11*ln(e*x+d)*A+10*b^9/e^11*ln(e*x+d)*B*a-11*b^10/e^12*ln(e*x+d)*B*d

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2309 vs. \(2 (432) = 864\).

Time = 0.38 (sec) , antiderivative size = 2309, normalized size of antiderivative = 5.18 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx=\text {Too large to display} \]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^11,x, algorithm="fricas")

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx=\text {Timed out} \]

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1914 vs. \(2 (432) = 864\).

Time = 0.28 (sec) , antiderivative size = 1914, normalized size of antiderivative = 4.29 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx=\text {Too large to display} \]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^11,x, algorithm="maxima")

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

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1970 vs. \(2 (432) = 864\).

Time = 0.30 (sec) , antiderivative size = 1970, normalized size of antiderivative = 4.42 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx=\text {Too large to display} \]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^11,x, algorithm="giac")

B*b^10*x/e^11 - (11*B*b^10*d - 10*B*a*b^9*e - A*b^10*e)*log(abs(e*x + d))/e^12 - 1/2520*(55991*B*b^10*d^11 - 7

3810*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 + 2

7*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^1

0*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 + 21

0*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 - 6849*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 + 3

60*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 + 1260*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*

Mupad [B] (veriﬁcation not implemented)

Time = 1.88 (sec) , antiderivative size = 2874, normalized size of antiderivative = 6.44 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx=\text {Too large to display} \]

-((A*a^10*e^11)/10 + (55991*B*b^10*d^11)/2520 - (7381*A*b^10*d^10*e)/2520 + (B*a^10*d*e^10)/90 + 11*B*b^10*d^1

1*log(d + e*x) + (B*a^10*e^11*x)/9 - B*b^10*e^11*x^11 + (53219*B*b^10*d^10*e*x)/252 + A*a*b^9*d^9*e^2 + (B*a^9

*b*d^2*e^9)/36 + 10*A*a*b^9*e^11*x^9 + (5*B*a^9*b*e^11*x^2)/4 - (7129*A*b^10*d^9*e^2*x)/252 - 10*A*b^10*d*e^10

*x^9 - 10*B*b^10*d*e^10*x^10 - A*b^10*e^11*x^10*log(d + e*x) + (A*a^2*b^8*d^8*e^3)/2 + (A*a^3*b^7*d^7*e^4)/3 +

(A*a^4*b^6*d^6*e^5)/4 + (A*a^5*b^5*d^5*e^6)/5 + (A*a^6*b^4*d^4*e^7)/6 + (A*a^7*b^3*d^3*e^8)/7 + (A*a^8*b^2*d^

2*e^9)/8 + (9*B*a^2*b^8*d^9*e^2)/2 + (4*B*a^3*b^7*d^8*e^3)/3 + (7*B*a^4*b^6*d^7*e^4)/12 + (3*B*a^5*b^5*d^6*e^5

)/10 + (B*a^6*b^4*d^5*e^6)/6 + (2*B*a^7*b^3*d^4*e^7)/21 + (3*B*a^8*b^2*d^3*e^8)/56 + (45*A*a^8*b^2*e^11*x^2)/8

+ (120*A*a^7*b^3*e^11*x^3)/7 + 35*A*a^6*b^4*e^11*x^4 + (252*A*a^5*b^5*e^11*x^5)/5 + (105*A*a^4*b^6*e^11*x^6)/

2 + 40*A*a^3*b^7*e^11*x^7 + (45*A*a^2*b^8*e^11*x^8)/2 + (45*B*a^8*b^2*e^11*x^3)/7 + 20*B*a^7*b^3*e^11*x^4 + 42

*B*a^6*b^4*e^11*x^5 + 63*B*a^5*b^5*e^11*x^6 + 70*B*a^4*b^6*e^11*x^7 + 60*B*a^3*b^7*e^11*x^8 + 45*B*a^2*b^8*e^1

1*x^9 - (6849*A*b^10*d^8*e^3*x^2)/56 - (2178*A*b^10*d^7*e^4*x^3)/7 - (1029*A*b^10*d^6*e^5*x^4)/2 - (2877*A*b^1

0*d^5*e^6*x^5)/5 - (875*A*b^10*d^4*e^7*x^6)/2 - 220*A*b^10*d^3*e^8*x^7 - (135*A*b^10*d^2*e^9*x^8)/2 + (50139*B

*b^10*d^9*e^2*x^2)/56 + (15558*B*b^10*d^8*e^3*x^3)/7 + (7119*B*b^10*d^7*e^4*x^4)/2 + (19047*B*b^10*d^6*e^5*x^5

)/5 + (5425*B*b^10*d^5*e^6*x^6)/2 + 1220*B*b^10*d^4*e^7*x^7 + (585*B*b^10*d^3*e^8*x^8)/2 + 10*B*b^10*d^2*e^9*x

^9 + (A*a^9*b*d*e^10)/9 - (7381*B*a*b^9*d^10*e)/252 - A*b^10*d^10*e*log(d + e*x) + (10*A*a^9*b*e^11*x)/9 + 5*A

*a^2*b^8*d^7*e^4*x + (10*A*a^3*b^7*d^6*e^5*x)/3 + (5*A*a^4*b^6*d^5*e^6*x)/2 + 2*A*a^5*b^5*d^4*e^7*x + (5*A*a^6

*b^4*d^3*e^8*x)/3 + (10*A*a^7*b^3*d^2*e^9*x)/7 + 45*A*a*b^9*d^7*e^4*x^2 + (45*A*a^7*b^3*d*e^10*x^2)/7 + 120*A*

a*b^9*d^6*e^5*x^3 + 20*A*a^6*b^4*d*e^10*x^3 + 210*A*a*b^9*d^5*e^6*x^4 + 42*A*a^5*b^5*d*e^10*x^4 + 252*A*a*b^9*

d^4*e^7*x^5 + 63*A*a^4*b^6*d*e^10*x^5 + 210*A*a*b^9*d^3*e^8*x^6 + 70*A*a^3*b^7*d*e^10*x^6 + 120*A*a*b^9*d^2*e^

9*x^7 + 60*A*a^2*b^8*d*e^10*x^7 + 45*B*a^2*b^8*d^8*e^3*x + (40*B*a^3*b^7*d^7*e^4*x)/3 + (35*B*a^4*b^6*d^6*e^5*

x)/6 + 3*B*a^5*b^5*d^5*e^6*x + (5*B*a^6*b^4*d^4*e^7*x)/3 + (20*B*a^7*b^3*d^3*e^8*x)/21 + (15*B*a^8*b^2*d^2*e^9

*x)/28 - (34245*B*a*b^9*d^8*e^3*x^2)/28 + (135*B*a^8*b^2*d*e^10*x^2)/56 - (21780*B*a*b^9*d^7*e^4*x^3)/7 + (80*

B*a^7*b^3*d*e^10*x^3)/7 - 5145*B*a*b^9*d^6*e^5*x^4 + 35*B*a^6*b^4*d*e^10*x^4 - 5754*B*a*b^9*d^5*e^6*x^5 + (378

*B*a^5*b^5*d*e^10*x^5)/5 - 4375*B*a*b^9*d^4*e^7*x^6 + (245*B*a^4*b^6*d*e^10*x^6)/2 - 2200*B*a*b^9*d^3*e^8*x^7

+ 160*B*a^3*b^7*d*e^10*x^7 - 675*B*a*b^9*d^2*e^9*x^8 + (405*B*a^2*b^8*d*e^10*x^8)/2 - 45*A*b^10*d^8*e^3*x^2*lo

g(d + e*x) - 120*A*b^10*d^7*e^4*x^3*log(d + e*x) - 210*A*b^10*d^6*e^5*x^4*log(d + e*x) - 252*A*b^10*d^5*e^6*x^

5*log(d + e*x) - 210*A*b^10*d^4*e^7*x^6*log(d + e*x) - 120*A*b^10*d^3*e^8*x^7*log(d + e*x) - 45*A*b^10*d^2*e^9

*x^8*log(d + e*x) + 495*B*b^10*d^9*e^2*x^2*log(d + e*x) + 1320*B*b^10*d^8*e^3*x^3*log(d + e*x) + 2310*B*b^10*d

^7*e^4*x^4*log(d + e*x) + 2772*B*b^10*d^6*e^5*x^5*log(d + e*x) + 2310*B*b^10*d^5*e^6*x^6*log(d + e*x) + 1320*B

*b^10*d^4*e^7*x^7*log(d + e*x) + 495*B*b^10*d^3*e^8*x^8*log(d + e*x) + 110*B*b^10*d^2*e^9*x^9*log(d + e*x) - 1

0*B*a*b^9*d^10*e*log(d + e*x) + (5*B*a^9*b*d*e^10*x)/18 + 110*B*b^10*d^10*e*x*log(d + e*x) + (45*A*a^2*b^8*d^6

*e^5*x^2)/2 + 15*A*a^3*b^7*d^5*e^6*x^2 + (45*A*a^4*b^6*d^4*e^7*x^2)/4 + 9*A*a^5*b^5*d^3*e^8*x^2 + (15*A*a^6*b^

4*d^2*e^9*x^2)/2 + 60*A*a^2*b^8*d^5*e^6*x^3 + 40*A*a^3*b^7*d^4*e^7*x^3 + 30*A*a^4*b^6*d^3*e^8*x^3 + 24*A*a^5*b

^5*d^2*e^9*x^3 + 105*A*a^2*b^8*d^4*e^7*x^4 + 70*A*a^3*b^7*d^3*e^8*x^4 + (105*A*a^4*b^6*d^2*e^9*x^4)/2 + 126*A*

a^2*b^8*d^3*e^8*x^5 + 84*A*a^3*b^7*d^2*e^9*x^5 + 105*A*a^2*b^8*d^2*e^9*x^6 + (405*B*a^2*b^8*d^7*e^4*x^2)/2 + 6

0*B*a^3*b^7*d^6*e^5*x^2 + (105*B*a^4*b^6*d^5*e^6*x^2)/4 + (27*B*a^5*b^5*d^4*e^7*x^2)/2 + (15*B*a^6*b^4*d^3*e^8

*x^2)/2 + (30*B*a^7*b^3*d^2*e^9*x^2)/7 + 540*B*a^2*b^8*d^6*e^5*x^3 + 160*B*a^3*b^7*d^5*e^6*x^3 + 70*B*a^4*b^6*

d^4*e^7*x^3 + 36*B*a^5*b^5*d^3*e^8*x^3 + 20*B*a^6*b^4*d^2*e^9*x^3 + 945*B*a^2*b^8*d^5*e^6*x^4 + 280*B*a^3*b^7*

d^4*e^7*x^4 + (245*B*a^4*b^6*d^3*e^8*x^4)/2 + 63*B*a^5*b^5*d^2*e^9*x^4 + 1134*B*a^2*b^8*d^4*e^7*x^5 + 336*B*a^

3*b^7*d^3*e^8*x^5 + 147*B*a^4*b^6*d^2*e^9*x^5 + 945*B*a^2*b^8*d^3*e^8*x^6 + 280*B*a^3*b^7*d^2*e^9*x^6 + 540*B*

a^2*b^8*d^2*e^9*x^7 + 10*A*a*b^9*d^8*e^3*x + (5*A*a^8*b^2*d*e^10*x)/4 + 45*A*a*b^9*d*e^10*x^8 - (35645*B*a*b^9

*d^9*e^2*x)/126 - 100*B*a*b^9*d*e^10*x^9 - 10*B*a*b^9*e^11*x^10*log(d + e*x) - 10*A*b^10*d^9*e^2*x*log(d + e*x

) - 10*A*b^10*d*e^10*x^9*log(d + e*x) + 11*B*b^10*d*e^10*x^10*log(d + e*x) - 100*B*a*b^9*d^9*e^2*x*log(d + e*x

) - 100*B*a*b^9*d*e^10*x^9*log(d + e*x) - 450*B*a*b^9*d^8*e^3*x^2*log(d + e*x) - 1200*B*a*b^9*d^7*e^4*x^3*log(

d + e*x) - 2100*B*a*b^9*d^6*e^5*x^4*log(d + e*x) - 2520*B*a*b^9*d^5*e^6*x^5*log(d + e*x) - 2100*B*a*b^9*d^4*e^

7*x^6*log(d + e*x) - 1200*B*a*b^9*d^3*e^8*x^7*log(d + e*x) - 450*B*a*b^9*d^2*e^9*x^8*log(d + e*x))/(e^12*(d +

\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx\) [1099]

Optimal result