∫ (a+bx)10(A+Bx)____________ (d+ex)20 dx [1108]

Optimal. Leaf size=460 \[ \frac {(b d-a e)^{10} (B d-A e)}{19 e^{12} (d+e x)^{19}}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{18 e^{12} (d+e x)^{18}}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{17 e^{12} (d+e x)^{17}}-\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{16 e^{12} (d+e x)^{16}}+\frac {2 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)^{15}}-\frac {3 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{e^{12} (d+e x)^{14}}+\frac {42 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{13 e^{12} (d+e x)^{13}}-\frac {5 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{2 e^{12} (d+e x)^{12}}+\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{11 e^{12} (d+e x)^{11}}-\frac {b^8 (b d-a e) (11 b B d-2 A b e-9 a B e)}{2 e^{12} (d+e x)^{10}}+\frac {b^9 (11 b B d-A b e-10 a B e)}{9 e^{12} (d+e x)^9}-\frac {b^{10} B}{8 e^{12} (d+e x)^8} \]

1/19*(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)^19-1/18*(-a*e+b*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/(e*x+d)^18+5/1

7*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)^17-15/16*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d

)/e^12/(e*x+d)^16+2*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)/e^12/(e*x+d)^15-3*b^4*(-a*e+b*d)^5*(-6*A*b*e-

5*B*a*e+11*B*b*d)/e^12/(e*x+d)^14+42/13*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)/e^12/(e*x+d)^13-5/2*b^6*(

-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11*B*b*d)/e^12/(e*x+d)^12+15/11*b^7*(-a*e+b*d)^2*(-3*A*b*e-8*B*a*e+11*B*b*d)/e^1

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

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Rubi [A]
time = 0.56, antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \begin {gather*} \frac {b^9 (-10 a B e-A b e+11 b B d)}{9 e^{12} (d+e x)^9}-\frac {b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{2 e^{12} (d+e x)^{10}}+\frac {15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{11 e^{12} (d+e x)^{11}}-\frac {5 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{2 e^{12} (d+e x)^{12}}+\frac {42 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{13 e^{12} (d+e x)^{13}}-\frac {3 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)^{14}}+\frac {2 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^{15}}-\frac {15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{16 e^{12} (d+e x)^{16}}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{17 e^{12} (d+e x)^{17}}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{18 e^{12} (d+e x)^{18}}+\frac {(b d-a e)^{10} (B d-A e)}{19 e^{12} (d+e x)^{19}}-\frac {b^{10} B}{8 e^{12} (d+e x)^8} \end {gather*}

((b*d - a*e)^10*(B*d - A*e))/(19*e^12*(d + e*x)^19) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(18*e^12*(

d + e*x)^18) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(17*e^12*(d + e*x)^17) - (15*b^2*(b*d - a*e)

^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(16*e^12*(d + e*x)^16) + (2*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e

))/(e^12*(d + e*x)^15) - (3*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(e^12*(d + e*x)^14) + (42*b^5*(b

*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(13*e^12*(d + e*x)^13) - (5*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e

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

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

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

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]))))

\begin {align*} \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{20}} \, dx &=\int \left (\frac {(-b d+a e)^{10} (-B d+A e)}{e^{11} (d+e x)^{20}}+\frac {(-b d+a e)^9 (-11 b B d+10 A b e+a B e)}{e^{11} (d+e x)^{19}}+\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)^{18}}-\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)^{17}}+\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)^{16}}-\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)^{15}}+\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)^{14}}-\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)^{13}}+\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)^{12}}-\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)^{11}}+\frac {b^9 (-11 b B d+A b e+10 a B e)}{e^{11} (d+e x)^{10}}+\frac {b^{10} B}{e^{11} (d+e x)^9}\right ) \, dx\\ &=\frac {(b d-a e)^{10} (B d-A e)}{19 e^{12} (d+e x)^{19}}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{18 e^{12} (d+e x)^{18}}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{17 e^{12} (d+e x)^{17}}-\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{16 e^{12} (d+e x)^{16}}+\frac {2 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)^{15}}-\frac {3 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{e^{12} (d+e x)^{14}}+\frac {42 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{13 e^{12} (d+e x)^{13}}-\frac {5 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{2 e^{12} (d+e x)^{12}}+\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{11 e^{12} (d+e x)^{11}}-\frac {b^8 (b d-a e) (11 b B d-2 A b e-9 a B e)}{2 e^{12} (d+e x)^{10}}+\frac {b^9 (11 b B d-A b e-10 a B e)}{9 e^{12} (d+e x)^9}-\frac {b^{10} B}{8 e^{12} (d+e x)^8}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1433\) vs. \(2(460)=920\).
time = 0.55, size = 1433, normalized size = 3.12 \begin {gather*} -\frac {19448 a^{10} e^{10} (18 A e+B (d+19 e x))+11440 a^9 b e^9 \left (17 A e (d+19 e x)+2 B \left (d^2+19 d e x+171 e^2 x^2\right )\right )+6435 a^8 b^2 e^8 \left (16 A e \left (d^2+19 d e x+171 e^2 x^2\right )+3 B \left (d^3+19 d^2 e x+171 d e^2 x^2+969 e^3 x^3\right )\right )+3432 a^7 b^3 e^7 \left (15 A e \left (d^3+19 d^2 e x+171 d e^2 x^2+969 e^3 x^3\right )+4 B \left (d^4+19 d^3 e x+171 d^2 e^2 x^2+969 d e^3 x^3+3876 e^4 x^4\right )\right )+1716 a^6 b^4 e^6 \left (14 A e \left (d^4+19 d^3 e x+171 d^2 e^2 x^2+969 d e^3 x^3+3876 e^4 x^4\right )+5 B \left (d^5+19 d^4 e x+171 d^3 e^2 x^2+969 d^2 e^3 x^3+3876 d e^4 x^4+11628 e^5 x^5\right )\right )+792 a^5 b^5 e^5 \left (13 A e \left (d^5+19 d^4 e x+171 d^3 e^2 x^2+969 d^2 e^3 x^3+3876 d e^4 x^4+11628 e^5 x^5\right )+6 B \left (d^6+19 d^5 e x+171 d^4 e^2 x^2+969 d^3 e^3 x^3+3876 d^2 e^4 x^4+11628 d e^5 x^5+27132 e^6 x^6\right )\right )+330 a^4 b^6 e^4 \left (12 A e \left (d^6+19 d^5 e x+171 d^4 e^2 x^2+969 d^3 e^3 x^3+3876 d^2 e^4 x^4+11628 d e^5 x^5+27132 e^6 x^6\right )+7 B \left (d^7+19 d^6 e x+171 d^5 e^2 x^2+969 d^4 e^3 x^3+3876 d^3 e^4 x^4+11628 d^2 e^5 x^5+27132 d e^6 x^6+50388 e^7 x^7\right )\right )+120 a^3 b^7 e^3 \left (11 A e \left (d^7+19 d^6 e x+171 d^5 e^2 x^2+969 d^4 e^3 x^3+3876 d^3 e^4 x^4+11628 d^2 e^5 x^5+27132 d e^6 x^6+50388 e^7 x^7\right )+8 B \left (d^8+19 d^7 e x+171 d^6 e^2 x^2+969 d^5 e^3 x^3+3876 d^4 e^4 x^4+11628 d^3 e^5 x^5+27132 d^2 e^6 x^6+50388 d e^7 x^7+75582 e^8 x^8\right )\right )+36 a^2 b^8 e^2 \left (10 A e \left (d^8+19 d^7 e x+171 d^6 e^2 x^2+969 d^5 e^3 x^3+3876 d^4 e^4 x^4+11628 d^3 e^5 x^5+27132 d^2 e^6 x^6+50388 d e^7 x^7+75582 e^8 x^8\right )+9 B \left (d^9+19 d^8 e x+171 d^7 e^2 x^2+969 d^6 e^3 x^3+3876 d^5 e^4 x^4+11628 d^4 e^5 x^5+27132 d^3 e^6 x^6+50388 d^2 e^7 x^7+75582 d e^8 x^8+92378 e^9 x^9\right )\right )+8 a b^9 e \left (9 A e \left (d^9+19 d^8 e x+171 d^7 e^2 x^2+969 d^6 e^3 x^3+3876 d^5 e^4 x^4+11628 d^4 e^5 x^5+27132 d^3 e^6 x^6+50388 d^2 e^7 x^7+75582 d e^8 x^8+92378 e^9 x^9\right )+10 B \left (d^{10}+19 d^9 e x+171 d^8 e^2 x^2+969 d^7 e^3 x^3+3876 d^6 e^4 x^4+11628 d^5 e^5 x^5+27132 d^4 e^6 x^6+50388 d^3 e^7 x^7+75582 d^2 e^8 x^8+92378 d e^9 x^9+92378 e^{10} x^{10}\right )\right )+b^{10} \left (8 A e \left (d^{10}+19 d^9 e x+171 d^8 e^2 x^2+969 d^7 e^3 x^3+3876 d^6 e^4 x^4+11628 d^5 e^5 x^5+27132 d^4 e^6 x^6+50388 d^3 e^7 x^7+75582 d^2 e^8 x^8+92378 d e^9 x^9+92378 e^{10} x^{10}\right )+11 B \left (d^{11}+19 d^{10} e x+171 d^9 e^2 x^2+969 d^8 e^3 x^3+3876 d^7 e^4 x^4+11628 d^6 e^5 x^5+27132 d^5 e^6 x^6+50388 d^4 e^7 x^7+75582 d^3 e^8 x^8+92378 d^2 e^9 x^9+92378 d e^{10} x^{10}+75582 e^{11} x^{11}\right )\right )}{6651216 e^{12} (d+e x)^{19}} \end {gather*}

-1/6651216*(19448*a^10*e^10*(18*A*e + B*(d + 19*e*x)) + 11440*a^9*b*e^9*(17*A*e*(d + 19*e*x) + 2*B*(d^2 + 19*d

*e*x + 171*e^2*x^2)) + 6435*a^8*b^2*e^8*(16*A*e*(d^2 + 19*d*e*x + 171*e^2*x^2) + 3*B*(d^3 + 19*d^2*e*x + 171*d

*e^2*x^2 + 969*e^3*x^3)) + 3432*a^7*b^3*e^7*(15*A*e*(d^3 + 19*d^2*e*x + 171*d*e^2*x^2 + 969*e^3*x^3) + 4*B*(d^

4 + 19*d^3*e*x + 171*d^2*e^2*x^2 + 969*d*e^3*x^3 + 3876*e^4*x^4)) + 1716*a^6*b^4*e^6*(14*A*e*(d^4 + 19*d^3*e*x

+ 171*d^2*e^2*x^2 + 969*d*e^3*x^3 + 3876*e^4*x^4) + 5*B*(d^5 + 19*d^4*e*x + 171*d^3*e^2*x^2 + 969*d^2*e^3*x^3

+ 3876*d*e^4*x^4 + 11628*e^5*x^5)) + 792*a^5*b^5*e^5*(13*A*e*(d^5 + 19*d^4*e*x + 171*d^3*e^2*x^2 + 969*d^2*e^

3*x^3 + 3876*d*e^4*x^4 + 11628*e^5*x^5) + 6*B*(d^6 + 19*d^5*e*x + 171*d^4*e^2*x^2 + 969*d^3*e^3*x^3 + 3876*d^2

*e^4*x^4 + 11628*d*e^5*x^5 + 27132*e^6*x^6)) + 330*a^4*b^6*e^4*(12*A*e*(d^6 + 19*d^5*e*x + 171*d^4*e^2*x^2 + 9

69*d^3*e^3*x^3 + 3876*d^2*e^4*x^4 + 11628*d*e^5*x^5 + 27132*e^6*x^6) + 7*B*(d^7 + 19*d^6*e*x + 171*d^5*e^2*x^2

+ 969*d^4*e^3*x^3 + 3876*d^3*e^4*x^4 + 11628*d^2*e^5*x^5 + 27132*d*e^6*x^6 + 50388*e^7*x^7)) + 120*a^3*b^7*e^

3*(11*A*e*(d^7 + 19*d^6*e*x + 171*d^5*e^2*x^2 + 969*d^4*e^3*x^3 + 3876*d^3*e^4*x^4 + 11628*d^2*e^5*x^5 + 27132

*d*e^6*x^6 + 50388*e^7*x^7) + 8*B*(d^8 + 19*d^7*e*x + 171*d^6*e^2*x^2 + 969*d^5*e^3*x^3 + 3876*d^4*e^4*x^4 + 1

1628*d^3*e^5*x^5 + 27132*d^2*e^6*x^6 + 50388*d*e^7*x^7 + 75582*e^8*x^8)) + 36*a^2*b^8*e^2*(10*A*e*(d^8 + 19*d^

7*e*x + 171*d^6*e^2*x^2 + 969*d^5*e^3*x^3 + 3876*d^4*e^4*x^4 + 11628*d^3*e^5*x^5 + 27132*d^2*e^6*x^6 + 50388*d

*e^7*x^7 + 75582*e^8*x^8) + 9*B*(d^9 + 19*d^8*e*x + 171*d^7*e^2*x^2 + 969*d^6*e^3*x^3 + 3876*d^5*e^4*x^4 + 116

28*d^4*e^5*x^5 + 27132*d^3*e^6*x^6 + 50388*d^2*e^7*x^7 + 75582*d*e^8*x^8 + 92378*e^9*x^9)) + 8*a*b^9*e*(9*A*e*

(d^9 + 19*d^8*e*x + 171*d^7*e^2*x^2 + 969*d^6*e^3*x^3 + 3876*d^5*e^4*x^4 + 11628*d^4*e^5*x^5 + 27132*d^3*e^6*x

^6 + 50388*d^2*e^7*x^7 + 75582*d*e^8*x^8 + 92378*e^9*x^9) + 10*B*(d^10 + 19*d^9*e*x + 171*d^8*e^2*x^2 + 969*d^

7*e^3*x^3 + 3876*d^6*e^4*x^4 + 11628*d^5*e^5*x^5 + 27132*d^4*e^6*x^6 + 50388*d^3*e^7*x^7 + 75582*d^2*e^8*x^8 +

92378*d*e^9*x^9 + 92378*e^10*x^10)) + b^10*(8*A*e*(d^10 + 19*d^9*e*x + 171*d^8*e^2*x^2 + 969*d^7*e^3*x^3 + 38

76*d^6*e^4*x^4 + 11628*d^5*e^5*x^5 + 27132*d^4*e^6*x^6 + 50388*d^3*e^7*x^7 + 75582*d^2*e^8*x^8 + 92378*d*e^9*x

^9 + 92378*e^10*x^10) + 11*B*(d^11 + 19*d^10*e*x + 171*d^9*e^2*x^2 + 969*d^8*e^3*x^3 + 3876*d^7*e^4*x^4 + 1162

8*d^6*e^5*x^5 + 27132*d^5*e^6*x^6 + 50388*d^4*e^7*x^7 + 75582*d^3*e^8*x^8 + 92378*d^2*e^9*x^9 + 92378*d*e^10*x

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1941\) vs. \(2(440)=880\).
time = 0.15, size = 1942, normalized size = 4.22


method	result	size

risch	\(\text {Expression too large to display}\)	\(1901\)
default	\(\text {Expression too large to display}\)	\(1942\)
norman	\(\text {Expression too large to display}\)	\(2014\)
gosper	\(\text {Expression too large to display}\)	\(2233\)

int((b*x+a)^10*(B*x+A)/(e*x+d)^20,x,method=_RETURNVERBOSE)

-42/13*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)^13-15/

16*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^6-33

6*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)^16-1/8*b^10*B/e^12/(e*x+d)^8-1/9*b^9/e^12*(A*b*e+10*B*a*e-11*B*b*d)/(e*x+d)^9-5/2*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)^12-1/19*(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)^19-1/18/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)^18-1/2*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)/(e*x+d)^10-3*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)^14-15/11*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)^11-5/17*b/e^1

2*(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)^17-2*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^5-245*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-

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2026 vs. \(2 (471) = 942\).
time = 0.54, size = 2026, normalized size = 4.40 \begin {gather*} \text {Too large to display} \end {gather*}

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^20,x, algorithm="maxima")

-1/6651216*(831402*B*b^10*x^11*e^11 + 11*B*b^10*d^11 + 350064*A*a^10*e^11 + 8*(10*B*a*b^9*e + A*b^10*e)*d^10 +

92378*(11*B*b^10*d*e^10 + 80*B*a*b^9*e^11 + 8*A*b^10*e^11)*x^10 + 36*(9*B*a^2*b^8*e^2 + 2*A*a*b^9*e^2)*d^9 +

92378*(11*B*b^10*d^2*e^9 + 324*B*a^2*b^8*e^11 + 72*A*a*b^9*e^11 + 8*(10*B*a*b^9*e^10 + A*b^10*e^10)*d)*x^9 + 1

20*(8*B*a^3*b^7*e^3 + 3*A*a^2*b^8*e^3)*d^8 + 75582*(11*B*b^10*d^3*e^8 + 960*B*a^3*b^7*e^11 + 360*A*a^2*b^8*e^1

1 + 8*(10*B*a*b^9*e^9 + A*b^10*e^9)*d^2 + 36*(9*B*a^2*b^8*e^10 + 2*A*a*b^9*e^10)*d)*x^8 + 330*(7*B*a^4*b^6*e^4

+ 4*A*a^3*b^7*e^4)*d^7 + 50388*(11*B*b^10*d^4*e^7 + 2310*B*a^4*b^6*e^11 + 1320*A*a^3*b^7*e^11 + 8*(10*B*a*b^9

*e^8 + A*b^10*e^8)*d^3 + 36*(9*B*a^2*b^8*e^9 + 2*A*a*b^9*e^9)*d^2 + 120*(8*B*a^3*b^7*e^10 + 3*A*a^2*b^8*e^10)*

d)*x^7 + 792*(6*B*a^5*b^5*e^5 + 5*A*a^4*b^6*e^5)*d^6 + 27132*(11*B*b^10*d^5*e^6 + 4752*B*a^5*b^5*e^11 + 3960*A

*a^4*b^6*e^11 + 8*(10*B*a*b^9*e^7 + A*b^10*e^7)*d^4 + 36*(9*B*a^2*b^8*e^8 + 2*A*a*b^9*e^8)*d^3 + 120*(8*B*a^3*

b^7*e^9 + 3*A*a^2*b^8*e^9)*d^2 + 330*(7*B*a^4*b^6*e^10 + 4*A*a^3*b^7*e^10)*d)*x^6 + 1716*(5*B*a^6*b^4*e^6 + 6*

A*a^5*b^5*e^6)*d^5 + 11628*(11*B*b^10*d^6*e^5 + 8580*B*a^6*b^4*e^11 + 10296*A*a^5*b^5*e^11 + 8*(10*B*a*b^9*e^6

+ A*b^10*e^6)*d^5 + 36*(9*B*a^2*b^8*e^7 + 2*A*a*b^9*e^7)*d^4 + 120*(8*B*a^3*b^7*e^8 + 3*A*a^2*b^8*e^8)*d^3 +

330*(7*B*a^4*b^6*e^9 + 4*A*a^3*b^7*e^9)*d^2 + 792*(6*B*a^5*b^5*e^10 + 5*A*a^4*b^6*e^10)*d)*x^5 + 3432*(4*B*a^7

*b^3*e^7 + 7*A*a^6*b^4*e^7)*d^4 + 3876*(11*B*b^10*d^7*e^4 + 13728*B*a^7*b^3*e^11 + 24024*A*a^6*b^4*e^11 + 8*(1

0*B*a*b^9*e^5 + A*b^10*e^5)*d^6 + 36*(9*B*a^2*b^8*e^6 + 2*A*a*b^9*e^6)*d^5 + 120*(8*B*a^3*b^7*e^7 + 3*A*a^2*b^

8*e^7)*d^4 + 330*(7*B*a^4*b^6*e^8 + 4*A*a^3*b^7*e^8)*d^3 + 792*(6*B*a^5*b^5*e^9 + 5*A*a^4*b^6*e^9)*d^2 + 1716*

(5*B*a^6*b^4*e^10 + 6*A*a^5*b^5*e^10)*d)*x^4 + 6435*(3*B*a^8*b^2*e^8 + 8*A*a^7*b^3*e^8)*d^3 + 969*(11*B*b^10*d

^8*e^3 + 19305*B*a^8*b^2*e^11 + 51480*A*a^7*b^3*e^11 + 8*(10*B*a*b^9*e^4 + A*b^10*e^4)*d^7 + 36*(9*B*a^2*b^8*e

^5 + 2*A*a*b^9*e^5)*d^6 + 120*(8*B*a^3*b^7*e^6 + 3*A*a^2*b^8*e^6)*d^5 + 330*(7*B*a^4*b^6*e^7 + 4*A*a^3*b^7*e^7

)*d^4 + 792*(6*B*a^5*b^5*e^8 + 5*A*a^4*b^6*e^8)*d^3 + 1716*(5*B*a^6*b^4*e^9 + 6*A*a^5*b^5*e^9)*d^2 + 3432*(4*B

*a^7*b^3*e^10 + 7*A*a^6*b^4*e^10)*d)*x^3 + 11440*(2*B*a^9*b*e^9 + 9*A*a^8*b^2*e^9)*d^2 + 171*(11*B*b^10*d^9*e^

2 + 22880*B*a^9*b*e^11 + 102960*A*a^8*b^2*e^11 + 8*(10*B*a*b^9*e^3 + A*b^10*e^3)*d^8 + 36*(9*B*a^2*b^8*e^4 + 2

*A*a*b^9*e^4)*d^7 + 120*(8*B*a^3*b^7*e^5 + 3*A*a^2*b^8*e^5)*d^6 + 330*(7*B*a^4*b^6*e^6 + 4*A*a^3*b^7*e^6)*d^5

+ 792*(6*B*a^5*b^5*e^7 + 5*A*a^4*b^6*e^7)*d^4 + 1716*(5*B*a^6*b^4*e^8 + 6*A*a^5*b^5*e^8)*d^3 + 3432*(4*B*a^7*b

^3*e^9 + 7*A*a^6*b^4*e^9)*d^2 + 6435*(3*B*a^8*b^2*e^10 + 8*A*a^7*b^3*e^10)*d)*x^2 + 19448*(B*a^10*e^10 + 10*A*

a^9*b*e^10)*d + 19*(11*B*b^10*d^10*e + 19448*B*a^10*e^11 + 194480*A*a^9*b*e^11 + 8*(10*B*a*b^9*e^2 + A*b^10*e^

2)*d^9 + 36*(9*B*a^2*b^8*e^3 + 2*A*a*b^9*e^3)*d^8 + 120*(8*B*a^3*b^7*e^4 + 3*A*a^2*b^8*e^4)*d^7 + 330*(7*B*a^4

*b^6*e^5 + 4*A*a^3*b^7*e^5)*d^6 + 792*(6*B*a^5*b^5*e^6 + 5*A*a^4*b^6*e^6)*d^5 + 1716*(5*B*a^6*b^4*e^7 + 6*A*a^

5*b^5*e^7)*d^4 + 3432*(4*B*a^7*b^3*e^8 + 7*A*a^6*b^4*e^8)*d^3 + 6435*(3*B*a^8*b^2*e^9 + 8*A*a^7*b^3*e^9)*d^2 +

11440*(2*B*a^9*b*e^10 + 9*A*a^8*b^2*e^10)*d)*x)/(x^19*e^31 + 19*d*x^18*e^30 + 171*d^2*x^17*e^29 + 969*d^3*x^1

6*e^28 + 3876*d^4*x^15*e^27 + 11628*d^5*x^14*e^26 + 27132*d^6*x^13*e^25 + 50388*d^7*x^12*e^24 + 75582*d^8*x^11

*e^23 + 92378*d^9*x^10*e^22 + 92378*d^10*x^9*e^21 + 75582*d^11*x^8*e^20 + 50388*d^12*x^7*e^19 + 27132*d^13*x^6

*e^18 + 11628*d^14*x^5*e^17 + 3876*d^15*x^4*e^16 + 969*d^16*x^3*e^15 + 171*d^17*x^2*e^14 + 19*d^18*x*e^13 + d^

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1935 vs. \(2 (471) = 942\).
time = 0.91, size = 1935, normalized size = 4.21 \begin {gather*} \text {Too large to display} \end {gather*}

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^20,x, algorithm="fricas")

-1/6651216*(11*B*b^10*d^11 + (831402*B*b^10*x^11 + 350064*A*a^10 + 739024*(10*B*a*b^9 + A*b^10)*x^10 + 3325608

*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 9069840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 16628040*(7*B*a^4*b^6 + 4*A*a^3*b^7

)*x^7 + 21488544*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 19953648*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 13302432*(4*B*a^

7*b^3 + 7*A*a^6*b^4)*x^4 + 6235515*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 1956240*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 3

69512*(B*a^10 + 10*A*a^9*b)*x)*e^11 + (1016158*B*b^10*d*x^10 + 739024*(10*B*a*b^9 + A*b^10)*d*x^9 + 2720952*(9

*B*a^2*b^8 + 2*A*a*b^9)*d*x^8 + 6046560*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*x^7 + 8953560*(7*B*a^4*b^6 + 4*A*a^3*b^7

)*d*x^6 + 9209376*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*x^5 + 6651216*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*x^4 + 3325608*(4*B

*a^7*b^3 + 7*A*a^6*b^4)*d*x^3 + 1100385*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*x^2 + 217360*(2*B*a^9*b + 9*A*a^8*b^2)*d

*x + 19448*(B*a^10 + 10*A*a^9*b)*d)*e^10 + (1016158*B*b^10*d^2*x^9 + 604656*(10*B*a*b^9 + A*b^10)*d^2*x^8 + 18

13968*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*x^7 + 3255840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*x^6 + 3837240*(7*B*a^4*b^6 +

4*A*a^3*b^7)*d^2*x^5 + 3069792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*x^4 + 1662804*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*

x^3 + 586872*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*x^2 + 122265*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*x + 11440*(2*B*a^9*b

+ 9*A*a^8*b^2)*d^2)*e^9 + 3*(277134*B*b^10*d^3*x^8 + 134368*(10*B*a*b^9 + A*b^10)*d^3*x^7 + 325584*(9*B*a^2*b

^8 + 2*A*a*b^9)*d^3*x^6 + 465120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*x^5 + 426360*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*

x^4 + 255816*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*x^3 + 97812*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*x^2 + 21736*(4*B*a^7*

b^3 + 7*A*a^6*b^4)*d^3*x + 2145*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3)*e^8 + 6*(92378*B*b^10*d^4*x^7 + 36176*(10*B*a

*b^9 + A*b^10)*d^4*x^6 + 69768*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*x^5 + 77520*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*x^4 +

53295*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*x^3 + 22572*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*x^2 + 5434*(5*B*a^6*b^4 + 6

*A*a^5*b^5)*d^4*x + 572*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4)*e^7 + 6*(49742*B*b^10*d^5*x^6 + 15504*(10*B*a*b^9 + A

*b^10)*d^5*x^5 + 23256*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*x^4 + 19380*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*x^3 + 9405*(7

*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*x^2 + 2508*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*x + 286*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d

^5)*e^6 + 6*(21318*B*b^10*d^6*x^5 + 5168*(10*B*a*b^9 + A*b^10)*d^6*x^4 + 5814*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*x^

3 + 3420*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*x^2 + 1045*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*x + 132*(6*B*a^5*b^5 + 5*A

*a^4*b^6)*d^6)*e^5 + 6*(7106*B*b^10*d^7*x^4 + 1292*(10*B*a*b^9 + A*b^10)*d^7*x^3 + 1026*(9*B*a^2*b^8 + 2*A*a*b

^9)*d^7*x^2 + 380*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*x + 55*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7)*e^4 + 3*(3553*B*b^10

*d^8*x^3 + 456*(10*B*a*b^9 + A*b^10)*d^8*x^2 + 228*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*x + 40*(8*B*a^3*b^7 + 3*A*a^2

*b^8)*d^8)*e^3 + (1881*B*b^10*d^9*x^2 + 152*(10*B*a*b^9 + A*b^10)*d^9*x + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9)*e^

2 + (209*B*b^10*d^10*x + 8*(10*B*a*b^9 + A*b^10)*d^10)*e)/(x^19*e^31 + 19*d*x^18*e^30 + 171*d^2*x^17*e^29 + 96

9*d^3*x^16*e^28 + 3876*d^4*x^15*e^27 + 11628*d^5*x^14*e^26 + 27132*d^6*x^13*e^25 + 50388*d^7*x^12*e^24 + 75582

*d^8*x^11*e^23 + 92378*d^9*x^10*e^22 + 92378*d^10*x^9*e^21 + 75582*d^11*x^8*e^20 + 50388*d^12*x^7*e^19 + 27132

*d^13*x^6*e^18 + 11628*d^14*x^5*e^17 + 3876*d^15*x^4*e^16 + 969*d^16*x^3*e^15 + 171*d^17*x^2*e^14 + 19*d^18*x*

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2096 vs. \(2 (471) = 942\).
time = 1.05, size = 2096, normalized size = 4.56 \begin {gather*} \text {Too large to display} \end {gather*}

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^20,x, algorithm="giac")

-1/6651216*(831402*B*b^10*x^11*e^11 + 1016158*B*b^10*d*x^10*e^10 + 1016158*B*b^10*d^2*x^9*e^9 + 831402*B*b^10*

d^3*x^8*e^8 + 554268*B*b^10*d^4*x^7*e^7 + 298452*B*b^10*d^5*x^6*e^6 + 127908*B*b^10*d^6*x^5*e^5 + 42636*B*b^10

*d^7*x^4*e^4 + 10659*B*b^10*d^8*x^3*e^3 + 1881*B*b^10*d^9*x^2*e^2 + 209*B*b^10*d^10*x*e + 11*B*b^10*d^11 + 739

0240*B*a*b^9*x^10*e^11 + 739024*A*b^10*x^10*e^11 + 7390240*B*a*b^9*d*x^9*e^10 + 739024*A*b^10*d*x^9*e^10 + 604

6560*B*a*b^9*d^2*x^8*e^9 + 604656*A*b^10*d^2*x^8*e^9 + 4031040*B*a*b^9*d^3*x^7*e^8 + 403104*A*b^10*d^3*x^7*e^8

+ 2170560*B*a*b^9*d^4*x^6*e^7 + 217056*A*b^10*d^4*x^6*e^7 + 930240*B*a*b^9*d^5*x^5*e^6 + 93024*A*b^10*d^5*x^5

*e^6 + 310080*B*a*b^9*d^6*x^4*e^5 + 31008*A*b^10*d^6*x^4*e^5 + 77520*B*a*b^9*d^7*x^3*e^4 + 7752*A*b^10*d^7*x^3

*e^4 + 13680*B*a*b^9*d^8*x^2*e^3 + 1368*A*b^10*d^8*x^2*e^3 + 1520*B*a*b^9*d^9*x*e^2 + 152*A*b^10*d^9*x*e^2 + 8

0*B*a*b^9*d^10*e + 8*A*b^10*d^10*e + 29930472*B*a^2*b^8*x^9*e^11 + 6651216*A*a*b^9*x^9*e^11 + 24488568*B*a^2*b

^8*d*x^8*e^10 + 5441904*A*a*b^9*d*x^8*e^10 + 16325712*B*a^2*b^8*d^2*x^7*e^9 + 3627936*A*a*b^9*d^2*x^7*e^9 + 87

90768*B*a^2*b^8*d^3*x^6*e^8 + 1953504*A*a*b^9*d^3*x^6*e^8 + 3767472*B*a^2*b^8*d^4*x^5*e^7 + 837216*A*a*b^9*d^4

*x^5*e^7 + 1255824*B*a^2*b^8*d^5*x^4*e^6 + 279072*A*a*b^9*d^5*x^4*e^6 + 313956*B*a^2*b^8*d^6*x^3*e^5 + 69768*A

*a*b^9*d^6*x^3*e^5 + 55404*B*a^2*b^8*d^7*x^2*e^4 + 12312*A*a*b^9*d^7*x^2*e^4 + 6156*B*a^2*b^8*d^8*x*e^3 + 1368

*A*a*b^9*d^8*x*e^3 + 324*B*a^2*b^8*d^9*e^2 + 72*A*a*b^9*d^9*e^2 + 72558720*B*a^3*b^7*x^8*e^11 + 27209520*A*a^2

*b^8*x^8*e^11 + 48372480*B*a^3*b^7*d*x^7*e^10 + 18139680*A*a^2*b^8*d*x^7*e^10 + 26046720*B*a^3*b^7*d^2*x^6*e^9

+ 9767520*A*a^2*b^8*d^2*x^6*e^9 + 11162880*B*a^3*b^7*d^3*x^5*e^8 + 4186080*A*a^2*b^8*d^3*x^5*e^8 + 3720960*B*

a^3*b^7*d^4*x^4*e^7 + 1395360*A*a^2*b^8*d^4*x^4*e^7 + 930240*B*a^3*b^7*d^5*x^3*e^6 + 348840*A*a^2*b^8*d^5*x^3*

e^6 + 164160*B*a^3*b^7*d^6*x^2*e^5 + 61560*A*a^2*b^8*d^6*x^2*e^5 + 18240*B*a^3*b^7*d^7*x*e^4 + 6840*A*a^2*b^8*

d^7*x*e^4 + 960*B*a^3*b^7*d^8*e^3 + 360*A*a^2*b^8*d^8*e^3 + 116396280*B*a^4*b^6*x^7*e^11 + 66512160*A*a^3*b^7*

x^7*e^11 + 62674920*B*a^4*b^6*d*x^6*e^10 + 35814240*A*a^3*b^7*d*x^6*e^10 + 26860680*B*a^4*b^6*d^2*x^5*e^9 + 15

348960*A*a^3*b^7*d^2*x^5*e^9 + 8953560*B*a^4*b^6*d^3*x^4*e^8 + 5116320*A*a^3*b^7*d^3*x^4*e^8 + 2238390*B*a^4*b

^6*d^4*x^3*e^7 + 1279080*A*a^3*b^7*d^4*x^3*e^7 + 395010*B*a^4*b^6*d^5*x^2*e^6 + 225720*A*a^3*b^7*d^5*x^2*e^6 +

43890*B*a^4*b^6*d^6*x*e^5 + 25080*A*a^3*b^7*d^6*x*e^5 + 2310*B*a^4*b^6*d^7*e^4 + 1320*A*a^3*b^7*d^7*e^4 + 128

931264*B*a^5*b^5*x^6*e^11 + 107442720*A*a^4*b^6*x^6*e^11 + 55256256*B*a^5*b^5*d*x^5*e^10 + 46046880*A*a^4*b^6*

d*x^5*e^10 + 18418752*B*a^5*b^5*d^2*x^4*e^9 + 15348960*A*a^4*b^6*d^2*x^4*e^9 + 4604688*B*a^5*b^5*d^3*x^3*e^8 +

3837240*A*a^4*b^6*d^3*x^3*e^8 + 812592*B*a^5*b^5*d^4*x^2*e^7 + 677160*A*a^4*b^6*d^4*x^2*e^7 + 90288*B*a^5*b^5

*d^5*x*e^6 + 75240*A*a^4*b^6*d^5*x*e^6 + 4752*B*a^5*b^5*d^6*e^5 + 3960*A*a^4*b^6*d^6*e^5 + 99768240*B*a^6*b^4*

x^5*e^11 + 119721888*A*a^5*b^5*x^5*e^11 + 33256080*B*a^6*b^4*d*x^4*e^10 + 39907296*A*a^5*b^5*d*x^4*e^10 + 8314

020*B*a^6*b^4*d^2*x^3*e^9 + 9976824*A*a^5*b^5*d^2*x^3*e^9 + 1467180*B*a^6*b^4*d^3*x^2*e^8 + 1760616*A*a^5*b^5*

d^3*x^2*e^8 + 163020*B*a^6*b^4*d^4*x*e^7 + 195624*A*a^5*b^5*d^4*x*e^7 + 8580*B*a^6*b^4*d^5*e^6 + 10296*A*a^5*b

^5*d^5*e^6 + 53209728*B*a^7*b^3*x^4*e^11 + 93117024*A*a^6*b^4*x^4*e^11 + 13302432*B*a^7*b^3*d*x^3*e^10 + 23279

256*A*a^6*b^4*d*x^3*e^10 + 2347488*B*a^7*b^3*d^2*x^2*e^9 + 4108104*A*a^6*b^4*d^2*x^2*e^9 + 260832*B*a^7*b^3*d^

3*x*e^8 + 456456*A*a^6*b^4*d^3*x*e^8 + 13728*B*a^7*b^3*d^4*e^7 + 24024*A*a^6*b^4*d^4*e^7 + 18706545*B*a^8*b^2*

x^3*e^11 + 49884120*A*a^7*b^3*x^3*e^11 + 3301155*B*a^8*b^2*d*x^2*e^10 + 8803080*A*a^7*b^3*d*x^2*e^10 + 366795*

B*a^8*b^2*d^2*x*e^9 + 978120*A*a^7*b^3*d^2*x*e^9 + 19305*B*a^8*b^2*d^3*e^8 + 51480*A*a^7*b^3*d^3*e^8 + 3912480

*B*a^9*b*x^2*e^11 + 17606160*A*a^8*b^2*x^2*e^11 + 434720*B*a^9*b*d*x*e^10 + 1956240*A*a^8*b^2*d*x*e^10 + 22880

*B*a^9*b*d^2*e^9 + 102960*A*a^8*b^2*d^2*e^9 + 369512*B*a^10*x*e^11 + 3695120*A*a^9*b*x*e^11 + 19448*B*a^10*d*e

^10 + 194480*A*a^9*b*d*e^10 + 350064*A*a^10*e^11)*e^(-12)/(x*e + d)^19

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Mupad [B]
time = 28.18, size = 2099, normalized size = 4.56 \begin {gather*} -\frac {\frac {19448\,B\,a^{10}\,d\,e^{10}+350064\,A\,a^{10}\,e^{11}+22880\,B\,a^9\,b\,d^2\,e^9+194480\,A\,a^9\,b\,d\,e^{10}+19305\,B\,a^8\,b^2\,d^3\,e^8+102960\,A\,a^8\,b^2\,d^2\,e^9+13728\,B\,a^7\,b^3\,d^4\,e^7+51480\,A\,a^7\,b^3\,d^3\,e^8+8580\,B\,a^6\,b^4\,d^5\,e^6+24024\,A\,a^6\,b^4\,d^4\,e^7+4752\,B\,a^5\,b^5\,d^6\,e^5+10296\,A\,a^5\,b^5\,d^5\,e^6+2310\,B\,a^4\,b^6\,d^7\,e^4+3960\,A\,a^4\,b^6\,d^6\,e^5+960\,B\,a^3\,b^7\,d^8\,e^3+1320\,A\,a^3\,b^7\,d^7\,e^4+324\,B\,a^2\,b^8\,d^9\,e^2+360\,A\,a^2\,b^8\,d^8\,e^3+80\,B\,a\,b^9\,d^{10}\,e+72\,A\,a\,b^9\,d^9\,e^2+11\,B\,b^{10}\,d^{11}+8\,A\,b^{10}\,d^{10}\,e}{6651216\,e^{12}}+\frac {x\,\left (19448\,B\,a^{10}\,e^{10}+22880\,B\,a^9\,b\,d\,e^9+194480\,A\,a^9\,b\,e^{10}+19305\,B\,a^8\,b^2\,d^2\,e^8+102960\,A\,a^8\,b^2\,d\,e^9+13728\,B\,a^7\,b^3\,d^3\,e^7+51480\,A\,a^7\,b^3\,d^2\,e^8+8580\,B\,a^6\,b^4\,d^4\,e^6+24024\,A\,a^6\,b^4\,d^3\,e^7+4752\,B\,a^5\,b^5\,d^5\,e^5+10296\,A\,a^5\,b^5\,d^4\,e^6+2310\,B\,a^4\,b^6\,d^6\,e^4+3960\,A\,a^4\,b^6\,d^5\,e^5+960\,B\,a^3\,b^7\,d^7\,e^3+1320\,A\,a^3\,b^7\,d^6\,e^4+324\,B\,a^2\,b^8\,d^8\,e^2+360\,A\,a^2\,b^8\,d^7\,e^3+80\,B\,a\,b^9\,d^9\,e+72\,A\,a\,b^9\,d^8\,e^2+11\,B\,b^{10}\,d^{10}+8\,A\,b^{10}\,d^9\,e\right )}{350064\,e^{11}}+\frac {b^7\,x^8\,\left (960\,B\,a^3\,e^3+324\,B\,a^2\,b\,d\,e^2+360\,A\,a^2\,b\,e^3+80\,B\,a\,b^2\,d^2\,e+72\,A\,a\,b^2\,d\,e^2+11\,B\,b^3\,d^3+8\,A\,b^3\,d^2\,e\right )}{88\,e^4}+\frac {b^4\,x^5\,\left (8580\,B\,a^6\,e^6+4752\,B\,a^5\,b\,d\,e^5+10296\,A\,a^5\,b\,e^6+2310\,B\,a^4\,b^2\,d^2\,e^4+3960\,A\,a^4\,b^2\,d\,e^5+960\,B\,a^3\,b^3\,d^3\,e^3+1320\,A\,a^3\,b^3\,d^2\,e^4+324\,B\,a^2\,b^4\,d^4\,e^2+360\,A\,a^2\,b^4\,d^3\,e^3+80\,B\,a\,b^5\,d^5\,e+72\,A\,a\,b^5\,d^4\,e^2+11\,B\,b^6\,d^6+8\,A\,b^6\,d^5\,e\right )}{572\,e^7}+\frac {b^9\,x^{10}\,\left (8\,A\,b\,e+80\,B\,a\,e+11\,B\,b\,d\right )}{72\,e^2}+\frac {b^6\,x^7\,\left (2310\,B\,a^4\,e^4+960\,B\,a^3\,b\,d\,e^3+1320\,A\,a^3\,b\,e^4+324\,B\,a^2\,b^2\,d^2\,e^2+360\,A\,a^2\,b^2\,d\,e^3+80\,B\,a\,b^3\,d^3\,e+72\,A\,a\,b^3\,d^2\,e^2+11\,B\,b^4\,d^4+8\,A\,b^4\,d^3\,e\right )}{132\,e^5}+\frac {b^3\,x^4\,\left (13728\,B\,a^7\,e^7+8580\,B\,a^6\,b\,d\,e^6+24024\,A\,a^6\,b\,e^7+4752\,B\,a^5\,b^2\,d^2\,e^5+10296\,A\,a^5\,b^2\,d\,e^6+2310\,B\,a^4\,b^3\,d^3\,e^4+3960\,A\,a^4\,b^3\,d^2\,e^5+960\,B\,a^3\,b^4\,d^4\,e^3+1320\,A\,a^3\,b^4\,d^3\,e^4+324\,B\,a^2\,b^5\,d^5\,e^2+360\,A\,a^2\,b^5\,d^4\,e^3+80\,B\,a\,b^6\,d^6\,e+72\,A\,a\,b^6\,d^5\,e^2+11\,B\,b^7\,d^7+8\,A\,b^7\,d^6\,e\right )}{1716\,e^8}+\frac {b\,x^2\,\left (22880\,B\,a^9\,e^9+19305\,B\,a^8\,b\,d\,e^8+102960\,A\,a^8\,b\,e^9+13728\,B\,a^7\,b^2\,d^2\,e^7+51480\,A\,a^7\,b^2\,d\,e^8+8580\,B\,a^6\,b^3\,d^3\,e^6+24024\,A\,a^6\,b^3\,d^2\,e^7+4752\,B\,a^5\,b^4\,d^4\,e^5+10296\,A\,a^5\,b^4\,d^3\,e^6+2310\,B\,a^4\,b^5\,d^5\,e^4+3960\,A\,a^4\,b^5\,d^4\,e^5+960\,B\,a^3\,b^6\,d^6\,e^3+1320\,A\,a^3\,b^6\,d^5\,e^4+324\,B\,a^2\,b^7\,d^7\,e^2+360\,A\,a^2\,b^7\,d^6\,e^3+80\,B\,a\,b^8\,d^8\,e+72\,A\,a\,b^8\,d^7\,e^2+11\,B\,b^9\,d^9+8\,A\,b^9\,d^8\,e\right )}{38896\,e^{10}}+\frac {b^8\,x^9\,\left (324\,B\,a^2\,e^2+80\,B\,a\,b\,d\,e+72\,A\,a\,b\,e^2+11\,B\,b^2\,d^2+8\,A\,b^2\,d\,e\right )}{72\,e^3}+\frac {7\,b^5\,x^6\,\left (4752\,B\,a^5\,e^5+2310\,B\,a^4\,b\,d\,e^4+3960\,A\,a^4\,b\,e^5+960\,B\,a^3\,b^2\,d^2\,e^3+1320\,A\,a^3\,b^2\,d\,e^4+324\,B\,a^2\,b^3\,d^3\,e^2+360\,A\,a^2\,b^3\,d^2\,e^3+80\,B\,a\,b^4\,d^4\,e+72\,A\,a\,b^4\,d^3\,e^2+11\,B\,b^5\,d^5+8\,A\,b^5\,d^4\,e\right )}{1716\,e^6}+\frac {b^2\,x^3\,\left (19305\,B\,a^8\,e^8+13728\,B\,a^7\,b\,d\,e^7+51480\,A\,a^7\,b\,e^8+8580\,B\,a^6\,b^2\,d^2\,e^6+24024\,A\,a^6\,b^2\,d\,e^7+4752\,B\,a^5\,b^3\,d^3\,e^5+10296\,A\,a^5\,b^3\,d^2\,e^6+2310\,B\,a^4\,b^4\,d^4\,e^4+3960\,A\,a^4\,b^4\,d^3\,e^5+960\,B\,a^3\,b^5\,d^5\,e^3+1320\,A\,a^3\,b^5\,d^4\,e^4+324\,B\,a^2\,b^6\,d^6\,e^2+360\,A\,a^2\,b^6\,d^5\,e^3+80\,B\,a\,b^7\,d^7\,e+72\,A\,a\,b^7\,d^6\,e^2+11\,B\,b^8\,d^8+8\,A\,b^8\,d^7\,e\right )}{6864\,e^9}+\frac {B\,b^{10}\,x^{11}}{8\,e}}{d^{19}+19\,d^{18}\,e\,x+171\,d^{17}\,e^2\,x^2+969\,d^{16}\,e^3\,x^3+3876\,d^{15}\,e^4\,x^4+11628\,d^{14}\,e^5\,x^5+27132\,d^{13}\,e^6\,x^6+50388\,d^{12}\,e^7\,x^7+75582\,d^{11}\,e^8\,x^8+92378\,d^{10}\,e^9\,x^9+92378\,d^9\,e^{10}\,x^{10}+75582\,d^8\,e^{11}\,x^{11}+50388\,d^7\,e^{12}\,x^{12}+27132\,d^6\,e^{13}\,x^{13}+11628\,d^5\,e^{14}\,x^{14}+3876\,d^4\,e^{15}\,x^{15}+969\,d^3\,e^{16}\,x^{16}+171\,d^2\,e^{17}\,x^{17}+19\,d\,e^{18}\,x^{18}+e^{19}\,x^{19}} \end {gather*}

-((350064*A*a^10*e^11 + 11*B*b^10*d^11 + 8*A*b^10*d^10*e + 19448*B*a^10*d*e^10 + 72*A*a*b^9*d^9*e^2 + 22880*B*

a^9*b*d^2*e^9 + 360*A*a^2*b^8*d^8*e^3 + 1320*A*a^3*b^7*d^7*e^4 + 3960*A*a^4*b^6*d^6*e^5 + 10296*A*a^5*b^5*d^5*

e^6 + 24024*A*a^6*b^4*d^4*e^7 + 51480*A*a^7*b^3*d^3*e^8 + 102960*A*a^8*b^2*d^2*e^9 + 324*B*a^2*b^8*d^9*e^2 + 9

60*B*a^3*b^7*d^8*e^3 + 2310*B*a^4*b^6*d^7*e^4 + 4752*B*a^5*b^5*d^6*e^5 + 8580*B*a^6*b^4*d^5*e^6 + 13728*B*a^7*

b^3*d^4*e^7 + 19305*B*a^8*b^2*d^3*e^8 + 194480*A*a^9*b*d*e^10 + 80*B*a*b^9*d^10*e)/(6651216*e^12) + (x*(19448*

B*a^10*e^10 + 11*B*b^10*d^10 + 194480*A*a^9*b*e^10 + 8*A*b^10*d^9*e + 72*A*a*b^9*d^8*e^2 + 102960*A*a^8*b^2*d*

e^9 + 360*A*a^2*b^8*d^7*e^3 + 1320*A*a^3*b^7*d^6*e^4 + 3960*A*a^4*b^6*d^5*e^5 + 10296*A*a^5*b^5*d^4*e^6 + 2402

4*A*a^6*b^4*d^3*e^7 + 51480*A*a^7*b^3*d^2*e^8 + 324*B*a^2*b^8*d^8*e^2 + 960*B*a^3*b^7*d^7*e^3 + 2310*B*a^4*b^6

*d^6*e^4 + 4752*B*a^5*b^5*d^5*e^5 + 8580*B*a^6*b^4*d^4*e^6 + 13728*B*a^7*b^3*d^3*e^7 + 19305*B*a^8*b^2*d^2*e^8

+ 80*B*a*b^9*d^9*e + 22880*B*a^9*b*d*e^9))/(350064*e^11) + (b^7*x^8*(960*B*a^3*e^3 + 11*B*b^3*d^3 + 360*A*a^2

*b*e^3 + 8*A*b^3*d^2*e + 72*A*a*b^2*d*e^2 + 80*B*a*b^2*d^2*e + 324*B*a^2*b*d*e^2))/(88*e^4) + (b^4*x^5*(8580*B

*a^6*e^6 + 11*B*b^6*d^6 + 10296*A*a^5*b*e^6 + 8*A*b^6*d^5*e + 72*A*a*b^5*d^4*e^2 + 3960*A*a^4*b^2*d*e^5 + 360*

A*a^2*b^4*d^3*e^3 + 1320*A*a^3*b^3*d^2*e^4 + 324*B*a^2*b^4*d^4*e^2 + 960*B*a^3*b^3*d^3*e^3 + 2310*B*a^4*b^2*d^

2*e^4 + 80*B*a*b^5*d^5*e + 4752*B*a^5*b*d*e^5))/(572*e^7) + (b^9*x^10*(8*A*b*e + 80*B*a*e + 11*B*b*d))/(72*e^2

) + (b^6*x^7*(2310*B*a^4*e^4 + 11*B*b^4*d^4 + 1320*A*a^3*b*e^4 + 8*A*b^4*d^3*e + 72*A*a*b^3*d^2*e^2 + 360*A*a^

2*b^2*d*e^3 + 324*B*a^2*b^2*d^2*e^2 + 80*B*a*b^3*d^3*e + 960*B*a^3*b*d*e^3))/(132*e^5) + (b^3*x^4*(13728*B*a^7

*e^7 + 11*B*b^7*d^7 + 24024*A*a^6*b*e^7 + 8*A*b^7*d^6*e + 72*A*a*b^6*d^5*e^2 + 10296*A*a^5*b^2*d*e^6 + 360*A*a

^2*b^5*d^4*e^3 + 1320*A*a^3*b^4*d^3*e^4 + 3960*A*a^4*b^3*d^2*e^5 + 324*B*a^2*b^5*d^5*e^2 + 960*B*a^3*b^4*d^4*e

^3 + 2310*B*a^4*b^3*d^3*e^4 + 4752*B*a^5*b^2*d^2*e^5 + 80*B*a*b^6*d^6*e + 8580*B*a^6*b*d*e^6))/(1716*e^8) + (b

*x^2*(22880*B*a^9*e^9 + 11*B*b^9*d^9 + 102960*A*a^8*b*e^9 + 8*A*b^9*d^8*e + 72*A*a*b^8*d^7*e^2 + 51480*A*a^7*b

^2*d*e^8 + 360*A*a^2*b^7*d^6*e^3 + 1320*A*a^3*b^6*d^5*e^4 + 3960*A*a^4*b^5*d^4*e^5 + 10296*A*a^5*b^4*d^3*e^6 +

24024*A*a^6*b^3*d^2*e^7 + 324*B*a^2*b^7*d^7*e^2 + 960*B*a^3*b^6*d^6*e^3 + 2310*B*a^4*b^5*d^5*e^4 + 4752*B*a^5

*b^4*d^4*e^5 + 8580*B*a^6*b^3*d^3*e^6 + 13728*B*a^7*b^2*d^2*e^7 + 80*B*a*b^8*d^8*e + 19305*B*a^8*b*d*e^8))/(38

896*e^10) + (b^8*x^9*(324*B*a^2*e^2 + 11*B*b^2*d^2 + 72*A*a*b*e^2 + 8*A*b^2*d*e + 80*B*a*b*d*e))/(72*e^3) + (7

*b^5*x^6*(4752*B*a^5*e^5 + 11*B*b^5*d^5 + 3960*A*a^4*b*e^5 + 8*A*b^5*d^4*e + 72*A*a*b^4*d^3*e^2 + 1320*A*a^3*b

^2*d*e^4 + 360*A*a^2*b^3*d^2*e^3 + 324*B*a^2*b^3*d^3*e^2 + 960*B*a^3*b^2*d^2*e^3 + 80*B*a*b^4*d^4*e + 2310*B*a

^4*b*d*e^4))/(1716*e^6) + (b^2*x^3*(19305*B*a^8*e^8 + 11*B*b^8*d^8 + 51480*A*a^7*b*e^8 + 8*A*b^8*d^7*e + 72*A*

a*b^7*d^6*e^2 + 24024*A*a^6*b^2*d*e^7 + 360*A*a^2*b^6*d^5*e^3 + 1320*A*a^3*b^5*d^4*e^4 + 3960*A*a^4*b^4*d^3*e^

5 + 10296*A*a^5*b^3*d^2*e^6 + 324*B*a^2*b^6*d^6*e^2 + 960*B*a^3*b^5*d^5*e^3 + 2310*B*a^4*b^4*d^4*e^4 + 4752*B*

a^5*b^3*d^3*e^5 + 8580*B*a^6*b^2*d^2*e^6 + 80*B*a*b^7*d^7*e + 13728*B*a^7*b*d*e^7))/(6864*e^9) + (B*b^10*x^11)

/(8*e))/(d^19 + e^19*x^19 + 19*d*e^18*x^18 + 171*d^17*e^2*x^2 + 969*d^16*e^3*x^3 + 3876*d^15*e^4*x^4 + 11628*d

^14*e^5*x^5 + 27132*d^13*e^6*x^6 + 50388*d^12*e^7*x^7 + 75582*d^11*e^8*x^8 + 92378*d^10*e^9*x^9 + 92378*d^9*e^

10*x^10 + 75582*d^8*e^11*x^11 + 50388*d^7*e^12*x^12 + 27132*d^6*e^13*x^13 + 11628*d^5*e^14*x^14 + 3876*d^4*e^1

5*x^15 + 969*d^3*e^16*x^16 + 171*d^2*e^17*x^17 + 19*d^18*e*x)

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3.12.8 \(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{20}} \, dx\) [1108]