3.11.0 ∫ (a+bx)10(A+Bx) (d+ex)10 dx [1098]

Integrand size = 20, antiderivative size = 441 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{10}} \, dx=-\frac {b^9 (10 b B d-A b e-10 a B e) x}{e^{11}}+\frac {b^{10} B x^2}{2 e^{10}}+\frac {(b d-a e)^{10} (B d-A e)}{9 e^{12} (d+e x)^9}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{8 e^{12} (d+e x)^8}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{7 e^{12} (d+e x)^7}-\frac {5 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{2 e^{12} (d+e x)^6}+\frac {6 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)^5}-\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{2 e^{12} (d+e x)^4}+\frac {14 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{e^{12} (d+e x)^3}-\frac {15 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{e^{12} (d+e x)^2}+\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{e^{12} (d+e x)}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) \log (d+e x)}{e^{12}} \]

-b^9*(-A*b*e-10*B*a*e+10*B*b*d)*x/e^11+1/2*b^10*B*x^2/e^10+1/9*(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)^9-1/8*(-a

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

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

b*d)/e^12/(e*x+d)^5-21/2*b^4*(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)/e^12/(e*x+d)^4+14*b^5*(-a*e+b*d)^4*(-5*A

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

a*e+b*d)^2*(-3*A*b*e-8*B*a*e+11*B*b*d)/e^12/(e*x+d)+5*b^8*(-a*e+b*d)*(-2*A*b*e-9*B*a*e+11*B*b*d)*ln(e*x+d)/e^1

Rubi [A] (veriﬁed)

Time = 0.58 (sec) , antiderivative size = 441, 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} \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{10}} \, dx=-\frac {b^9 x (-10 a B e-A b e+10 b B d)}{e^{11}}+\frac {5 b^8 (b d-a e) \log (d+e x) (-9 a B e-2 A b e+11 b B d)}{e^{12}}+\frac {15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{12} (d+e x)}-\frac {15 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12} (d+e x)^2}+\frac {14 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)^3}-\frac {21 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{2 e^{12} (d+e x)^4}+\frac {6 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^5}-\frac {5 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{2 e^{12} (d+e x)^6}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{7 e^{12} (d+e x)^7}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{8 e^{12} (d+e x)^8}+\frac {(b d-a e)^{10} (B d-A e)}{9 e^{12} (d+e x)^9}+\frac {b^{10} B x^2}{2 e^{10}} \]

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

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

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

+ e*x)^6) + (6*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(e^12*(d + e*x)^5) - (21*b^4*(b*d - a*e)^5*(

11*b*B*d - 6*A*b*e - 5*a*B*e))/(2*e^12*(d + e*x)^4) + (14*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(e

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

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

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^9 (-10 b B d+A b e+10 a B e)}{e^{11}}+\frac {b^{10} B x}{e^{10}}+\frac {(-b d+a e)^{10} (-B d+A e)}{e^{11} (d+e x)^{10}}+\frac {(-b d+a e)^9 (-11 b B d+10 A b e+a B e)}{e^{11} (d+e x)^9}+\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)^8}-\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)^7}+\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)^6}-\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)^5}+\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)^4}-\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)^3}+\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)^2}-\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)}\right ) \, dx \\ & = -\frac {b^9 (10 b B d-A b e-10 a B e) x}{e^{11}}+\frac {b^{10} B x^2}{2 e^{10}}+\frac {(b d-a e)^{10} (B d-A e)}{9 e^{12} (d+e x)^9}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{8 e^{12} (d+e x)^8}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{7 e^{12} (d+e x)^7}-\frac {5 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{2 e^{12} (d+e x)^6}+\frac {6 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)^5}-\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{2 e^{12} (d+e x)^4}+\frac {14 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{e^{12} (d+e x)^3}-\frac {15 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{e^{12} (d+e x)^2}+\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{e^{12} (d+e x)}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 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. \(1460\) vs. \(2(441)=882\).

Time = 0.57 (sec) , antiderivative size = 1460, normalized size of antiderivative = 3.31 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{10}} \, dx=-\frac {7 a^{10} e^{10} (8 A e+B (d+9 e x))+10 a^9 b e^9 \left (7 A e (d+9 e x)+2 B \left (d^2+9 d e x+36 e^2 x^2\right )\right )+45 a^8 b^2 e^8 \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+24 a^7 b^3 e^7 \left (5 A e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+42 a^6 b^4 e^6 \left (4 A e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 B \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )+252 a^5 b^5 e^5 \left (A e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+2 B \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )\right )+210 a^4 b^6 e^4 \left (2 A e \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )+7 B \left (d^7+9 d^6 e x+36 d^5 e^2 x^2+84 d^4 e^3 x^3+126 d^3 e^4 x^4+126 d^2 e^5 x^5+84 d e^6 x^6+36 e^7 x^7\right )\right )+840 a^3 b^7 e^3 \left (A e \left (d^7+9 d^6 e x+36 d^5 e^2 x^2+84 d^4 e^3 x^3+126 d^3 e^4 x^4+126 d^2 e^5 x^5+84 d e^6 x^6+36 e^7 x^7\right )+8 B \left (d^8+9 d^7 e x+36 d^6 e^2 x^2+84 d^5 e^3 x^3+126 d^4 e^4 x^4+126 d^3 e^5 x^5+84 d^2 e^6 x^6+36 d e^7 x^7+9 e^8 x^8\right )\right )-9 a^2 b^8 e^2 \left (-280 A e \left (d^8+9 d^7 e x+36 d^6 e^2 x^2+84 d^5 e^3 x^3+126 d^4 e^4 x^4+126 d^3 e^5 x^5+84 d^2 e^6 x^6+36 d e^7 x^7+9 e^8 x^8\right )+B d \left (7129 d^8+61641 d^7 e x+235224 d^6 e^2 x^2+518616 d^5 e^3 x^3+725004 d^4 e^4 x^4+661500 d^3 e^5 x^5+388080 d^2 e^6 x^6+136080 d e^7 x^7+22680 e^8 x^8\right )\right )-2 a b^9 e \left (A d e \left (7129 d^8+61641 d^7 e x+235224 d^6 e^2 x^2+518616 d^5 e^3 x^3+725004 d^4 e^4 x^4+661500 d^3 e^5 x^5+388080 d^2 e^6 x^6+136080 d e^7 x^7+22680 e^8 x^8\right )-10 B \left (4861 d^{10}+41229 d^9 e x+153576 d^8 e^2 x^2+328104 d^7 e^3 x^3+439236 d^6 e^4 x^4+375732 d^5 e^5 x^5+197568 d^4 e^6 x^6+54432 d^3 e^7 x^7+2268 d^2 e^8 x^8-2268 d e^9 x^9-252 e^{10} x^{10}\right )\right )-b^{10} \left (-2 A e \left (4861 d^{10}+41229 d^9 e x+153576 d^8 e^2 x^2+328104 d^7 e^3 x^3+439236 d^6 e^4 x^4+375732 d^5 e^5 x^5+197568 d^4 e^6 x^6+54432 d^3 e^7 x^7+2268 d^2 e^8 x^8-2268 d e^9 x^9-252 e^{10} x^{10}\right )+B \left (42131 d^{11}+351459 d^{10} e x+1281096 d^9 e^2 x^2+2656584 d^8 e^3 x^3+3402756 d^7 e^4 x^4+2704212 d^6 e^5 x^5+1220688 d^5 e^6 x^6+190512 d^4 e^7 x^7-77112 d^3 e^8 x^8-36288 d^2 e^9 x^9-2772 d e^{10} x^{10}+252 e^{11} x^{11}\right )\right )-2520 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^9 \log (d+e x)}{504 e^{12} (d+e x)^9} \]

-1/504*(7*a^10*e^10*(8*A*e + B*(d + 9*e*x)) + 10*a^9*b*e^9*(7*A*e*(d + 9*e*x) + 2*B*(d^2 + 9*d*e*x + 36*e^2*x^

2)) + 45*a^8*b^2*e^8*(2*A*e*(d^2 + 9*d*e*x + 36*e^2*x^2) + B*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3)) +

24*a^7*b^3*e^7*(5*A*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 4*B*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 +

84*d*e^3*x^3 + 126*e^4*x^4)) + 42*a^6*b^4*e^6*(4*A*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^

4*x^4) + 5*B*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5)) + 252*a^5*b^5*

e^5*(A*e*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5) + 2*B*(d^6 + 9*d^5*

e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6)) + 210*a^4*b^6*e^4*(2*A*

e*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6) + 7*B*(d^

7 + 9*d^6*e*x + 36*d^5*e^2*x^2 + 84*d^4*e^3*x^3 + 126*d^3*e^4*x^4 + 126*d^2*e^5*x^5 + 84*d*e^6*x^6 + 36*e^7*x^

7)) + 840*a^3*b^7*e^3*(A*e*(d^7 + 9*d^6*e*x + 36*d^5*e^2*x^2 + 84*d^4*e^3*x^3 + 126*d^3*e^4*x^4 + 126*d^2*e^5*

x^5 + 84*d*e^6*x^6 + 36*e^7*x^7) + 8*B*(d^8 + 9*d^7*e*x + 36*d^6*e^2*x^2 + 84*d^5*e^3*x^3 + 126*d^4*e^4*x^4 +

126*d^3*e^5*x^5 + 84*d^2*e^6*x^6 + 36*d*e^7*x^7 + 9*e^8*x^8)) - 9*a^2*b^8*e^2*(-280*A*e*(d^8 + 9*d^7*e*x + 36*

d^6*e^2*x^2 + 84*d^5*e^3*x^3 + 126*d^4*e^4*x^4 + 126*d^3*e^5*x^5 + 84*d^2*e^6*x^6 + 36*d*e^7*x^7 + 9*e^8*x^8)

+ B*d*(7129*d^8 + 61641*d^7*e*x + 235224*d^6*e^2*x^2 + 518616*d^5*e^3*x^3 + 725004*d^4*e^4*x^4 + 661500*d^3*e^

5*x^5 + 388080*d^2*e^6*x^6 + 136080*d*e^7*x^7 + 22680*e^8*x^8)) - 2*a*b^9*e*(A*d*e*(7129*d^8 + 61641*d^7*e*x +

235224*d^6*e^2*x^2 + 518616*d^5*e^3*x^3 + 725004*d^4*e^4*x^4 + 661500*d^3*e^5*x^5 + 388080*d^2*e^6*x^6 + 1360

80*d*e^7*x^7 + 22680*e^8*x^8) - 10*B*(4861*d^10 + 41229*d^9*e*x + 153576*d^8*e^2*x^2 + 328104*d^7*e^3*x^3 + 43

9236*d^6*e^4*x^4 + 375732*d^5*e^5*x^5 + 197568*d^4*e^6*x^6 + 54432*d^3*e^7*x^7 + 2268*d^2*e^8*x^8 - 2268*d*e^9

*x^9 - 252*e^10*x^10)) - b^10*(-2*A*e*(4861*d^10 + 41229*d^9*e*x + 153576*d^8*e^2*x^2 + 328104*d^7*e^3*x^3 + 4

39236*d^6*e^4*x^4 + 375732*d^5*e^5*x^5 + 197568*d^4*e^6*x^6 + 54432*d^3*e^7*x^7 + 2268*d^2*e^8*x^8 - 2268*d*e^

9*x^9 - 252*e^10*x^10) + B*(42131*d^11 + 351459*d^10*e*x + 1281096*d^9*e^2*x^2 + 2656584*d^8*e^3*x^3 + 3402756

*d^7*e^4*x^4 + 2704212*d^6*e^5*x^5 + 1220688*d^5*e^6*x^6 + 190512*d^4*e^7*x^7 - 77112*d^3*e^8*x^8 - 36288*d^2*

e^9*x^9 - 2772*d*e^10*x^10 + 252*e^11*x^11)) - 2520*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^9

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(1927\) vs. \(2(429)=858\).

Time = 2.14 (sec) , antiderivative size = 1928, normalized size of antiderivative = 4.37


method	result	size

default	\(\text {Expression too large to display}\)	\(1928\)
norman	\(\text {Expression too large to display}\)	\(1933\)
risch	\(\text {Expression too large to display}\)	\(1942\)
parallelrisch	\(\text {Expression too large to display}\)	\(3215\)

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

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

d^2*e+8*B*a^3*e^3-27*B*a^2*b*d*e^2+30*B*a*b^2*d^2*e-11*B*b^3*d^3)/(e*x+d)-5/7*b/e^12*(9*A*a^8*b*e^9-72*A*a^7*b

^2*d*e^8+252*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^5-504*A*a^3*b^6*d^5*e^4+252*A*a^2*b^7

*d^6*e^3-72*A*a*b^8*d^7*e^2+9*A*b^9*d^8*e+2*B*a^9*e^9-27*B*a^8*b*d*e^8+144*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^5-882*B*a^4*b^5*d^5*e^4+672*B*a^3*b^6*d^6*e^3-324*B*a^2*b^7*d^7*e^2+90*B*a*b^8*d^8*e-

11*B*b^9*d^9)/(e*x+d)^7-6*b^3/e^12*(7*A*a^6*b*e^7-42*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+105*A*a^2*b^5*d^4*e^3-42*A*a*b^6*d^5*e^2+7*A*b^7*d^6*e+4*B*a^7*e^7-35*B*a^6*b*d*e^6+126*B*a^5*b^2*d^2*e^5-2

45*B*a^4*b^3*d^3*e^4+280*B*a^3*b^4*d^4*e^3-189*B*a^2*b^5*d^5*e^2+70*B*a*b^6*d^6*e-11*B*b^7*d^7)/(e*x+d)^5-1/9*

(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)/e^12/(e*

x+d)^9-14*b^5/e^12*(5*A*a^4*b*e^5-20*A*a^3*b^2*d*e^4+30*A*a^2*b^3*d^2*e^3-20*A*a*b^4*d^3*e^2+5*A*b^5*d^4*e+6*B

*a^5*e^5-35*B*a^4*b*d*e^4+80*B*a^3*b^2*d^2*e^3-90*B*a^2*b^3*d^3*e^2+50*B*a*b^4*d^4*e-11*B*b^5*d^5)/(e*x+d)^3-1

5*b^6/e^12*(4*A*a^3*b*e^4-12*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e+7*B*a^4*e^4-32*B*a^3*b*d*e^3+54*

B*a^2*b^2*d^2*e^2-40*B*a*b^3*d^3*e+11*B*b^4*d^4)/(e*x+d)^2-5/2*b^2/e^12*(8*A*a^7*b*e^8-56*A*a^6*b^2*d*e^7+168*

A*a^5*b^3*d^2*e^6-280*A*a^4*b^4*d^3*e^5+280*A*a^3*b^5*d^4*e^4-168*A*a^2*b^6*d^5*e^3+56*A*a*b^7*d^6*e^2-8*A*b^8

*d^7*e+3*B*a^8*e^8-32*B*a^7*b*d*e^7+140*B*a^6*b^2*d^2*e^6-336*B*a^5*b^3*d^3*e^5+490*B*a^4*b^4*d^4*e^4-448*B*a^

3*b^5*d^5*e^3+252*B*a^2*b^6*d^6*e^2-80*B*a*b^7*d^7*e+11*B*b^8*d^8)/(e*x+d)^6+5*b^8/e^12*(2*A*a*b*e^2-2*A*b^2*d

*e+9*B*a^2*e^2-20*B*a*b*d*e+11*B*b^2*d^2)*ln(e*x+d)-21/2*b^4/e^12*(6*A*a^5*b*e^6-30*A*a^4*b^2*d*e^5+60*A*a^3*b

^3*d^2*e^4-60*A*a^2*b^4*d^3*e^3+30*A*a*b^5*d^4*e^2-6*A*b^6*d^5*e+5*B*a^6*e^6-36*B*a^5*b*d*e^5+105*B*a^4*b^2*d^

2*e^4-160*B*a^3*b^3*d^3*e^3+135*B*a^2*b^4*d^4*e^2-60*B*a*b^5*d^5*e+11*B*b^6*d^6)/(e*x+d)^4

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2501 vs. \(2 (429) = 858\).

Time = 0.35 (sec) , antiderivative size = 2501, normalized size of antiderivative = 5.67 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{10}} \, dx=\text {Too large to display} \]

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

1/504*(252*B*b^10*e^11*x^11 + 42131*B*b^10*d^11 - 56*A*a^10*e^11 - 9722*(10*B*a*b^9 + A*b^10)*d^10*e + 7129*(9

*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 840*(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 - 84*(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 - 24*(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 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 7*(B*a^10

+ 10*A*a^9*b)*d*e^10 - 252*(11*B*b^10*d*e^10 - 2*(10*B*a*b^9 + A*b^10)*e^11)*x^10 - 4536*(8*B*b^10*d^2*e^9 -

(10*B*a*b^9 + A*b^10)*d*e^10)*x^9 - 1512*(51*B*b^10*d^3*e^8 + 3*(10*B*a*b^9 + A*b^10)*d^2*e^9 - 15*(9*B*a^2*b^

8 + 2*A*a*b^9)*d*e^10 + 5*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 1512*(126*B*b^10*d^4*e^7 - 72*(10*B*a*b^9 +

A*b^10)*d^3*e^8 + 90*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 - 5*(7*B*a^4*b^

6 + 4*A*a^3*b^7)*e^11)*x^7 + 3528*(346*B*b^10*d^5*e^6 - 112*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 110*(9*B*a^2*b^8 +

2*A*a*b^9)*d^3*e^8 - 20*(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 - 2*(6*B*a

^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 5292*(511*B*b^10*d^6*e^5 - 142*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 125*(9*B*a^2*

b^8 + 2*A*a*b^9)*d^4*e^7 - 20*(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 - 2*

(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 - (5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 756*(4501*B*b^10*d^7*e^4 - 1162*(

10*B*a*b^9 + A*b^10)*d^6*e^5 + 959*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 140*(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 - 14*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 - 7*(5*B*a^6*b^4 + 6*A*a^5*

b^5)*d*e^10 - 4*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 252*(10542*B*b^10*d^8*e^3 - 2604*(10*B*a*b^9 + A*b^10)

*d^7*e^4 + 2058*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 280*(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 - 28*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 - 14*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 8*(

4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 - 5*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 36*(35586*B*b^10*d^9*e^2 - 8532*

(10*B*a*b^9 + A*b^10)*d^8*e^3 + 6534*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 840*(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 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 - 42*(5*B*a^6*b^4 + 6*A*

a^5*b^5)*d^3*e^8 - 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 - 15*(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 + 9*(39051*B*b^10*d^10*e - 9162*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 6849*(9*B*a^2*b^8

+ 2*A*a*b^9)*d^8*e^3 - 840*(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 - 84

*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 - 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)

*d^3*e^8 - 15*(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 - 7*(B*a^10 + 10*A*a^9

*b)*e^11)*x + 2520*(11*B*b^10*d^11 - 2*(10*B*a*b^9 + A*b^10)*d^10*e + (9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + (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^11)*x^9 + 9*(11*B*b^10*d^3*e^8 -

2*(10*B*a*b^9 + A*b^10)*d^2*e^9 + (9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10)*x^8 + 36*(11*B*b^10*d^4*e^7 - 2*(10*B*a*b

^9 + A*b^10)*d^3*e^8 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9)*x^7 + 84*(11*B*b^10*d^5*e^6 - 2*(10*B*a*b^9 + A*b^10

)*d^4*e^7 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8)*x^6 + 126*(11*B*b^10*d^6*e^5 - 2*(10*B*a*b^9 + A*b^10)*d^5*e^6

+ (9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7)*x^5 + 126*(11*B*b^10*d^7*e^4 - 2*(10*B*a*b^9 + A*b^10)*d^6*e^5 + (9*B*a^2

*b^8 + 2*A*a*b^9)*d^5*e^6)*x^4 + 84*(11*B*b^10*d^8*e^3 - 2*(10*B*a*b^9 + A*b^10)*d^7*e^4 + (9*B*a^2*b^8 + 2*A*

a*b^9)*d^6*e^5)*x^3 + 36*(11*B*b^10*d^9*e^2 - 2*(10*B*a*b^9 + A*b^10)*d^8*e^3 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^7*

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

*x + d))/(e^21*x^9 + 9*d*e^20*x^8 + 36*d^2*e^19*x^7 + 84*d^3*e^18*x^6 + 126*d^4*e^17*x^5 + 126*d^5*e^16*x^4 +

84*d^6*e^15*x^3 + 36*d^7*e^14*x^2 + 9*d^8*e^13*x + d^9*e^12)

Sympy [F(-1)]

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

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1904 vs. \(2 (429) = 858\).

Time = 0.33 (sec) , antiderivative size = 1904, normalized size of antiderivative = 4.32 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{10}} \, dx=\text {Too large to display} \]

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

1/504*(42131*B*b^10*d^11 - 56*A*a^10*e^11 - 9722*(10*B*a*b^9 + A*b^10)*d^10*e + 7129*(9*B*a^2*b^8 + 2*A*a*b^9)

*d^9*e^2 - 840*(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 - 84*(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 - 24*(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 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 7*(B*a^10 + 10*A*a^9*b)*d*e^10 +

7560*(11*B*b^10*d^3*e^8 - 3*(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 + 7560*(77*B*b^10*d^4*e^7 - 20*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 18*(9*B*a^2*b^8 + 2*A

*a*b^9)*d^2*e^9 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 - (7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 3528*(517*B*b

^10*d^5*e^6 - 130*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 110*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 20*(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 - 2*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 5292*(62

7*B*b^10*d^6*e^5 - 154*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 125*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 20*(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 - 2*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 - (5*B*

a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 756*(5049*B*b^10*d^7*e^4 - 1218*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 959*(9*B*a^

2*b^8 + 2*A*a*b^9)*d^5*e^6 - 140*(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

- 14*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 - 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - 4*(4*B*a^7*b^3 + 7*A*a^6*b^4

)*e^11)*x^4 + 252*(11253*B*b^10*d^8*e^3 - 2676*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 2058*(9*B*a^2*b^8 + 2*A*a*b^9)*

d^6*e^5 - 280*(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 - 28*(6*B*a^5*b^5 +

5*A*a^4*b^6)*d^3*e^8 - 14*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 - 5*(3*B

*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 36*(36839*B*b^10*d^9*e^2 - 8658*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 6534*(9*B*

a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 840*(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 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 - 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 24*(4*B*a^7*b^3 + 7*A*a

^6*b^4)*d^2*e^9 - 15*(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 + 9*(39611*B*

b^10*d^10*e - 9218*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 6849*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 840*(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 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 - 42*

(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 - 15*(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 - 7*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^21*x^9 + 9*d*e^20*x^8 + 36

*d^2*e^19*x^7 + 84*d^3*e^18*x^6 + 126*d^4*e^17*x^5 + 126*d^5*e^16*x^4 + 84*d^6*e^15*x^3 + 36*d^7*e^14*x^2 + 9*

d^8*e^13*x + d^9*e^12) + 1/2*(B*b^10*e*x^2 - 2*(10*B*b^10*d - (10*B*a*b^9 + A*b^10)*e)*x)/e^11 + 5*(11*B*b^10*

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

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1975 vs. \(2 (429) = 858\).

Time = 0.30 (sec) , antiderivative size = 1975, normalized size of antiderivative = 4.48 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{10}} \, dx=\text {Too large to display} \]

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

5*(11*B*b^10*d^2 - 20*B*a*b^9*d*e - 2*A*b^10*d*e + 9*B*a^2*b^8*e^2 + 2*A*a*b^9*e^2)*log(abs(e*x + d))/e^12 + 1

/2*(B*b^10*e^10*x^2 - 20*B*b^10*d*e^9*x + 20*B*a*b^9*e^10*x + 2*A*b^10*e^10*x)/e^20 + 1/504*(42131*B*b^10*d^11

- 97220*B*a*b^9*d^10*e - 9722*A*b^10*d^10*e + 64161*B*a^2*b^8*d^9*e^2 + 14258*A*a*b^9*d^9*e^2 - 6720*B*a^3*b^

7*d^8*e^3 - 2520*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 - 504*B*a^5*b^5*d^6*e^5 -

420*A*a^4*b^6*d^6*e^5 - 210*B*a^6*b^4*d^5*e^6 - 252*A*a^5*b^5*d^5*e^6 - 96*B*a^7*b^3*d^4*e^7 - 168*A*a^6*b^4*d

^4*e^7 - 45*B*a^8*b^2*d^3*e^8 - 120*A*a^7*b^3*d^3*e^8 - 20*B*a^9*b*d^2*e^9 - 90*A*a^8*b^2*d^2*e^9 - 7*B*a^10*d

*e^10 - 70*A*a^9*b*d*e^10 - 56*A*a^10*e^11 + 7560*(11*B*b^10*d^3*e^8 - 30*B*a*b^9*d^2*e^9 - 3*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 + 7560*(77*B*b^10*d^4*e^7 -

200*B*a*b^9*d^3*e^8 - 20*A*b^10*d^3*e^8 + 162*B*a^2*b^8*d^2*e^9 + 36*A*a*b^9*d^2*e^9 - 32*B*a^3*b^7*d*e^10 -

12*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 + 3528*(517*B*b^10*d^5*e^6 - 1300*B*a*b^9*d^4*e

^7 - 130*A*b^10*d^4*e^7 + 990*B*a^2*b^8*d^3*e^8 + 220*A*a*b^9*d^3*e^8 - 160*B*a^3*b^7*d^2*e^9 - 60*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 - 12*B*a^5*b^5*e^11 - 10*A*a^4*b^6*e^11)*x^6 + 5292*(627*B*

b^10*d^6*e^5 - 1540*B*a*b^9*d^5*e^6 - 154*A*b^10*d^5*e^6 + 1125*B*a^2*b^8*d^4*e^7 + 250*A*a*b^9*d^4*e^7 - 160*

B*a^3*b^7*d^3*e^8 - 60*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 - 12*B*a^5*b^5*d*e^10 -

10*A*a^4*b^6*d*e^10 - 5*B*a^6*b^4*e^11 - 6*A*a^5*b^5*e^11)*x^5 + 756*(5049*B*b^10*d^7*e^4 - 12180*B*a*b^9*d^6

*e^5 - 1218*A*b^10*d^6*e^5 + 8631*B*a^2*b^8*d^5*e^6 + 1918*A*a*b^9*d^5*e^6 - 1120*B*a^3*b^7*d^4*e^7 - 420*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 - 84*B*a^5*b^5*d^2*e^9 - 70*A*a^4*b^6*d^2*e^9 -

35*B*a^6*b^4*d*e^10 - 42*A*a^5*b^5*d*e^10 - 16*B*a^7*b^3*e^11 - 28*A*a^6*b^4*e^11)*x^4 + 252*(11253*B*b^10*d^8

*e^3 - 26760*B*a*b^9*d^7*e^4 - 2676*A*b^10*d^7*e^4 + 18522*B*a^2*b^8*d^6*e^5 + 4116*A*a*b^9*d^6*e^5 - 2240*B*a

^3*b^7*d^5*e^6 - 840*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 - 168*B*a^5*b^5*d^3*e^8

- 140*A*a^4*b^6*d^3*e^8 - 70*B*a^6*b^4*d^2*e^9 - 84*A*a^5*b^5*d^2*e^9 - 32*B*a^7*b^3*d*e^10 - 56*A*a^6*b^4*d*

e^10 - 15*B*a^8*b^2*e^11 - 40*A*a^7*b^3*e^11)*x^3 + 36*(36839*B*b^10*d^9*e^2 - 86580*B*a*b^9*d^8*e^3 - 8658*A*

b^10*d^8*e^3 + 58806*B*a^2*b^8*d^7*e^4 + 13068*A*a*b^9*d^7*e^4 - 6720*B*a^3*b^7*d^6*e^5 - 2520*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 - 504*B*a^5*b^5*d^4*e^7 - 420*A*a^4*b^6*d^4*e^7 - 210*B*a^

6*b^4*d^3*e^8 - 252*A*a^5*b^5*d^3*e^8 - 96*B*a^7*b^3*d^2*e^9 - 168*A*a^6*b^4*d^2*e^9 - 45*B*a^8*b^2*d*e^10 - 1

20*A*a^7*b^3*d*e^10 - 20*B*a^9*b*e^11 - 90*A*a^8*b^2*e^11)*x^2 + 9*(39611*B*b^10*d^10*e - 92180*B*a*b^9*d^9*e^

2 - 9218*A*b^10*d^9*e^2 + 61641*B*a^2*b^8*d^8*e^3 + 13698*A*a*b^9*d^8*e^3 - 6720*B*a^3*b^7*d^7*e^4 - 2520*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 - 504*B*a^5*b^5*d^5*e^6 - 420*A*a^4*b^6*d^5*e^6

- 210*B*a^6*b^4*d^4*e^7 - 252*A*a^5*b^5*d^4*e^7 - 96*B*a^7*b^3*d^3*e^8 - 168*A*a^6*b^4*d^3*e^8 - 45*B*a^8*b^2

*d^2*e^9 - 120*A*a^7*b^3*d^2*e^9 - 20*B*a^9*b*d*e^10 - 90*A*a^8*b^2*d*e^10 - 7*B*a^10*e^11 - 70*A*a^9*b*e^11)*

Mupad [B] (veriﬁcation not implemented)

Time = 2.03 (sec) , antiderivative size = 2048, normalized size of antiderivative = 4.64 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{10}} \, dx=\text {Too large to display} \]

x*((A*b^10 + 10*B*a*b^9)/e^10 - (10*B*b^10*d)/e^11) - (x^7*(60*A*a^3*b^7*e^10 + 105*B*a^4*b^6*e^10 + 300*A*b^1

0*d^3*e^7 - 1155*B*b^10*d^4*e^6 - 540*A*a*b^9*d^2*e^8 + 180*A*a^2*b^8*d*e^9 + 3000*B*a*b^9*d^3*e^7 + 480*B*a^3

*b^7*d*e^9 - 2430*B*a^2*b^8*d^2*e^8) + x^4*(42*A*a^6*b^4*e^10 + 24*B*a^7*b^3*e^10 + 1827*A*b^10*d^6*e^4 - (151

47*B*b^10*d^7*e^3)/2 - 2877*A*a*b^9*d^5*e^5 + 63*A*a^5*b^5*d*e^9 + 18270*B*a*b^9*d^6*e^4 + (105*B*a^6*b^4*d*e^

9)/2 + 630*A*a^2*b^8*d^4*e^6 + 210*A*a^3*b^7*d^3*e^7 + 105*A*a^4*b^6*d^2*e^8 - (25893*B*a^2*b^8*d^5*e^5)/2 + 1

680*B*a^3*b^7*d^4*e^6 + (735*B*a^4*b^6*d^3*e^7)/2 + 126*B*a^5*b^5*d^2*e^8) + x^6*(70*A*a^4*b^6*e^10 + 84*B*a^5

*b^5*e^10 + 910*A*b^10*d^4*e^6 - 3619*B*b^10*d^5*e^5 - 1540*A*a*b^9*d^3*e^7 + 140*A*a^3*b^7*d*e^9 + 9100*B*a*b

^9*d^4*e^6 + 245*B*a^4*b^6*d*e^9 + 420*A*a^2*b^8*d^2*e^8 - 6930*B*a^2*b^8*d^3*e^7 + 1120*B*a^3*b^7*d^2*e^8) +

x^3*(20*A*a^7*b^3*e^10 + (15*B*a^8*b^2*e^10)/2 + 1338*A*b^10*d^7*e^3 - (11253*B*b^10*d^8*e^2)/2 - 2058*A*a*b^9

*d^6*e^4 + 28*A*a^6*b^4*d*e^9 + 13380*B*a*b^9*d^7*e^3 + 16*B*a^7*b^3*d*e^9 + 420*A*a^2*b^8*d^5*e^5 + 140*A*a^3

*b^7*d^4*e^6 + 70*A*a^4*b^6*d^3*e^7 + 42*A*a^5*b^5*d^2*e^8 - 9261*B*a^2*b^8*d^6*e^4 + 1120*B*a^3*b^7*d^5*e^5 +

245*B*a^4*b^6*d^4*e^6 + 84*B*a^5*b^5*d^3*e^7 + 35*B*a^6*b^4*d^2*e^8) + (56*A*a^10*e^11 - 42131*B*b^10*d^11 +

9722*A*b^10*d^10*e + 7*B*a^10*d*e^10 - 14258*A*a*b^9*d^9*e^2 + 20*B*a^9*b*d^2*e^9 + 2520*A*a^2*b^8*d^8*e^3 + 8

40*A*a^3*b^7*d^7*e^4 + 420*A*a^4*b^6*d^6*e^5 + 252*A*a^5*b^5*d^5*e^6 + 168*A*a^6*b^4*d^4*e^7 + 120*A*a^7*b^3*d

^3*e^8 + 90*A*a^8*b^2*d^2*e^9 - 64161*B*a^2*b^8*d^9*e^2 + 6720*B*a^3*b^7*d^8*e^3 + 1470*B*a^4*b^6*d^7*e^4 + 50

4*B*a^5*b^5*d^6*e^5 + 210*B*a^6*b^4*d^5*e^6 + 96*B*a^7*b^3*d^4*e^7 + 45*B*a^8*b^2*d^3*e^8 + 70*A*a^9*b*d*e^10

+ 97220*B*a*b^9*d^10*e)/(504*e) + x*((B*a^10*e^10)/8 - (39611*B*b^10*d^10)/56 + (5*A*a^9*b*e^10)/4 + (4609*A*b

^10*d^9*e)/28 - (6849*A*a*b^9*d^8*e^2)/28 + (45*A*a^8*b^2*d*e^9)/28 + 45*A*a^2*b^8*d^7*e^3 + 15*A*a^3*b^7*d^6*

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

1641*B*a^2*b^8*d^8*e^2)/56 + 120*B*a^3*b^7*d^7*e^3 + (105*B*a^4*b^6*d^6*e^4)/4 + 9*B*a^5*b^5*d^5*e^5 + (15*B*a

^6*b^4*d^4*e^6)/4 + (12*B*a^7*b^3*d^3*e^7)/7 + (45*B*a^8*b^2*d^2*e^8)/56 + (23045*B*a*b^9*d^9*e)/14 + (5*B*a^9

*b*d*e^9)/14) + x^8*(45*A*a^2*b^8*e^10 + 120*B*a^3*b^7*e^10 + 45*A*b^10*d^2*e^8 - 165*B*b^10*d^3*e^7 + 450*B*a

*b^9*d^2*e^8 - 405*B*a^2*b^8*d*e^9 - 90*A*a*b^9*d*e^9) + x^5*(63*A*a^5*b^5*e^10 + (105*B*a^6*b^4*e^10)/2 + 161

7*A*b^10*d^5*e^5 - (13167*B*b^10*d^6*e^4)/2 - 2625*A*a*b^9*d^4*e^6 + 105*A*a^4*b^6*d*e^9 + 16170*B*a*b^9*d^5*e

^5 + 126*B*a^5*b^5*d*e^9 + 630*A*a^2*b^8*d^3*e^7 + 210*A*a^3*b^7*d^2*e^8 - (23625*B*a^2*b^8*d^4*e^6)/2 + 1680*

B*a^3*b^7*d^3*e^7 + (735*B*a^4*b^6*d^2*e^8)/2) + x^2*((10*B*a^9*b*e^10)/7 - (36839*B*b^10*d^9*e)/14 + (45*A*a^

8*b^2*e^10)/7 + (4329*A*b^10*d^8*e^2)/7 - (6534*A*a*b^9*d^7*e^3)/7 + (60*A*a^7*b^3*d*e^9)/7 + (43290*B*a*b^9*d

^8*e^2)/7 + (45*B*a^8*b^2*d*e^9)/14 + 180*A*a^2*b^8*d^6*e^4 + 60*A*a^3*b^7*d^5*e^5 + 30*A*a^4*b^6*d^4*e^6 + 18

*A*a^5*b^5*d^3*e^7 + 12*A*a^6*b^4*d^2*e^8 - (29403*B*a^2*b^8*d^7*e^3)/7 + 480*B*a^3*b^7*d^6*e^4 + 105*B*a^4*b^

6*d^5*e^5 + 36*B*a^5*b^5*d^4*e^6 + 15*B*a^6*b^4*d^3*e^7 + (48*B*a^7*b^3*d^2*e^8)/7))/(d^9*e^11 + e^20*x^9 + 9*

d^8*e^12*x + 9*d*e^19*x^8 + 36*d^7*e^13*x^2 + 84*d^6*e^14*x^3 + 126*d^5*e^15*x^4 + 126*d^4*e^16*x^5 + 84*d^3*e

^17*x^6 + 36*d^2*e^18*x^7) + (log(d + e*x)*(55*B*b^10*d^2 - 10*A*b^10*d*e + 10*A*a*b^9*e^2 + 45*B*a^2*b^8*e^2

\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{10}} \, dx\) [1098]

Optimal result