∫ (a+bx)10(A+Bx)____________ (d+ex)21 dx [1109]

Optimal. Leaf size=462 \[ \frac {(b d-a e)^{10} (B d-A e)}{20 e^{12} (d+e x)^{20}}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{19 e^{12} (d+e x)^{19}}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{18 e^{12} (d+e x)^{18}}-\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{17 e^{12} (d+e x)^{17}}+\frac {15 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{8 e^{12} (d+e x)^{16}}-\frac {14 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{5 e^{12} (d+e x)^{15}}+\frac {3 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{e^{12} (d+e x)^{14}}-\frac {30 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{13 e^{12} (d+e x)^{13}}+\frac {5 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{4 e^{12} (d+e x)^{12}}-\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e)}{11 e^{12} (d+e x)^{11}}+\frac {b^9 (11 b B d-A b e-10 a B e)}{10 e^{12} (d+e x)^{10}}-\frac {b^{10} B}{9 e^{12} (d+e x)^9} \]

1/20*(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)^20-1/19*(-a*e+b*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/(e*x+d)^19+5/1

8*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)^18-15/17*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d

)/e^12/(e*x+d)^17+15/8*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)/e^12/(e*x+d)^16-14/5*b^4*(-a*e+b*d)^5*(-6*

A*b*e-5*B*a*e+11*B*b*d)/e^12/(e*x+d)^15+3*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)/e^12/(e*x+d)^14-30/13*b

^6*(-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11*B*b*d)/e^12/(e*x+d)^13+5/4*b^7*(-a*e+b*d)^2*(-3*A*b*e-8*B*a*e+11*B*b*d)/e

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

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Rubi [A]
time = 0.45, antiderivative size = 462, 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)}{10 e^{12} (d+e x)^{10}}-\frac {5 b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{11 e^{12} (d+e x)^{11}}+\frac {5 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{4 e^{12} (d+e x)^{12}}-\frac {30 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{13 e^{12} (d+e x)^{13}}+\frac {3 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)^{14}}-\frac {14 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)^{15}}+\frac {15 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{8 e^{12} (d+e x)^{16}}-\frac {15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{17 e^{12} (d+e x)^{17}}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{18 e^{12} (d+e x)^{18}}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{19 e^{12} (d+e x)^{19}}+\frac {(b d-a e)^{10} (B d-A e)}{20 e^{12} (d+e x)^{20}}-\frac {b^{10} B}{9 e^{12} (d+e x)^9} \end {gather*}

((b*d - a*e)^10*(B*d - A*e))/(20*e^12*(d + e*x)^20) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(19*e^12*(

d + e*x)^19) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(18*e^12*(d + e*x)^18) - (15*b^2*(b*d - a*e)

^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(17*e^12*(d + e*x)^17) + (15*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*

e))/(8*e^12*(d + e*x)^16) - (14*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)^15) + (3*b

^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(e^12*(d + e*x)^14) - (30*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b

*e - 7*a*B*e))/(13*e^12*(d + e*x)^13) + (5*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e))/(4*e^12*(d + e*x)

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

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

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)^{21}} \, dx &=\int \left (\frac {(-b d+a e)^{10} (-B d+A e)}{e^{11} (d+e x)^{21}}+\frac {(-b d+a e)^9 (-11 b B d+10 A b e+a B e)}{e^{11} (d+e x)^{20}}+\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)^{19}}-\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)^{18}}+\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)^{17}}-\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)^{16}}+\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)^{15}}-\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)^{14}}+\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)^{13}}-\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)^{12}}+\frac {b^9 (-11 b B d+A b e+10 a B e)}{e^{11} (d+e x)^{11}}+\frac {b^{10} B}{e^{11} (d+e x)^{10}}\right ) \, dx\\ &=\frac {(b d-a e)^{10} (B d-A e)}{20 e^{12} (d+e x)^{20}}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{19 e^{12} (d+e x)^{19}}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{18 e^{12} (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)}{17 e^{12} (d+e x)^{17}}+\frac {15 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{8 e^{12} (d+e x)^{16}}-\frac {14 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)^{15}}+\frac {3 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{e^{12} (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)}{13 e^{12} (d+e x)^{13}}+\frac {5 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{4 e^{12} (d+e x)^{12}}-\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e)}{11 e^{12} (d+e x)^{11}}+\frac {b^9 (11 b B d-A b e-10 a B e)}{10 e^{12} (d+e x)^{10}}-\frac {b^{10} B}{9 e^{12} (d+e x)^9}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1428\) vs. \(2(462)=924\).
time = 0.54, size = 1428, normalized size = 3.09 \begin {gather*} -\frac {43758 a^{10} e^{10} (19 A e+B (d+20 e x))+48620 a^9 b e^9 \left (9 A e (d+20 e x)+B \left (d^2+20 d e x+190 e^2 x^2\right )\right )+12870 a^8 b^2 e^8 \left (17 A e \left (d^2+20 d e x+190 e^2 x^2\right )+3 B \left (d^3+20 d^2 e x+190 d e^2 x^2+1140 e^3 x^3\right )\right )+25740 a^7 b^3 e^7 \left (4 A e \left (d^3+20 d^2 e x+190 d e^2 x^2+1140 e^3 x^3\right )+B \left (d^4+20 d^3 e x+190 d^2 e^2 x^2+1140 d e^3 x^3+4845 e^4 x^4\right )\right )+15015 a^6 b^4 e^6 \left (3 A e \left (d^4+20 d^3 e x+190 d^2 e^2 x^2+1140 d e^3 x^3+4845 e^4 x^4\right )+B \left (d^5+20 d^4 e x+190 d^3 e^2 x^2+1140 d^2 e^3 x^3+4845 d e^4 x^4+15504 e^5 x^5\right )\right )+2574 a^5 b^5 e^5 \left (7 A e \left (d^5+20 d^4 e x+190 d^3 e^2 x^2+1140 d^2 e^3 x^3+4845 d e^4 x^4+15504 e^5 x^5\right )+3 B \left (d^6+20 d^5 e x+190 d^4 e^2 x^2+1140 d^3 e^3 x^3+4845 d^2 e^4 x^4+15504 d e^5 x^5+38760 e^6 x^6\right )\right )+495 a^4 b^6 e^4 \left (13 A e \left (d^6+20 d^5 e x+190 d^4 e^2 x^2+1140 d^3 e^3 x^3+4845 d^2 e^4 x^4+15504 d e^5 x^5+38760 e^6 x^6\right )+7 B \left (d^7+20 d^6 e x+190 d^5 e^2 x^2+1140 d^4 e^3 x^3+4845 d^3 e^4 x^4+15504 d^2 e^5 x^5+38760 d e^6 x^6+77520 e^7 x^7\right )\right )+660 a^3 b^7 e^3 \left (3 A e \left (d^7+20 d^6 e x+190 d^5 e^2 x^2+1140 d^4 e^3 x^3+4845 d^3 e^4 x^4+15504 d^2 e^5 x^5+38760 d e^6 x^6+77520 e^7 x^7\right )+2 B \left (d^8+20 d^7 e x+190 d^6 e^2 x^2+1140 d^5 e^3 x^3+4845 d^4 e^4 x^4+15504 d^3 e^5 x^5+38760 d^2 e^6 x^6+77520 d e^7 x^7+125970 e^8 x^8\right )\right )+45 a^2 b^8 e^2 \left (11 A e \left (d^8+20 d^7 e x+190 d^6 e^2 x^2+1140 d^5 e^3 x^3+4845 d^4 e^4 x^4+15504 d^3 e^5 x^5+38760 d^2 e^6 x^6+77520 d e^7 x^7+125970 e^8 x^8\right )+9 B \left (d^9+20 d^8 e x+190 d^7 e^2 x^2+1140 d^6 e^3 x^3+4845 d^5 e^4 x^4+15504 d^4 e^5 x^5+38760 d^3 e^6 x^6+77520 d^2 e^7 x^7+125970 d e^8 x^8+167960 e^9 x^9\right )\right )+90 a b^9 e \left (A e \left (d^9+20 d^8 e x+190 d^7 e^2 x^2+1140 d^6 e^3 x^3+4845 d^5 e^4 x^4+15504 d^4 e^5 x^5+38760 d^3 e^6 x^6+77520 d^2 e^7 x^7+125970 d e^8 x^8+167960 e^9 x^9\right )+B \left (d^{10}+20 d^9 e x+190 d^8 e^2 x^2+1140 d^7 e^3 x^3+4845 d^6 e^4 x^4+15504 d^5 e^5 x^5+38760 d^4 e^6 x^6+77520 d^3 e^7 x^7+125970 d^2 e^8 x^8+167960 d e^9 x^9+184756 e^{10} x^{10}\right )\right )+b^{10} \left (9 A e \left (d^{10}+20 d^9 e x+190 d^8 e^2 x^2+1140 d^7 e^3 x^3+4845 d^6 e^4 x^4+15504 d^5 e^5 x^5+38760 d^4 e^6 x^6+77520 d^3 e^7 x^7+125970 d^2 e^8 x^8+167960 d e^9 x^9+184756 e^{10} x^{10}\right )+11 B \left (d^{11}+20 d^{10} e x+190 d^9 e^2 x^2+1140 d^8 e^3 x^3+4845 d^7 e^4 x^4+15504 d^6 e^5 x^5+38760 d^5 e^6 x^6+77520 d^4 e^7 x^7+125970 d^3 e^8 x^8+167960 d^2 e^9 x^9+184756 d e^{10} x^{10}+167960 e^{11} x^{11}\right )\right )}{16628040 e^{12} (d+e x)^{20}} \end {gather*}

-1/16628040*(43758*a^10*e^10*(19*A*e + B*(d + 20*e*x)) + 48620*a^9*b*e^9*(9*A*e*(d + 20*e*x) + B*(d^2 + 20*d*e

*x + 190*e^2*x^2)) + 12870*a^8*b^2*e^8*(17*A*e*(d^2 + 20*d*e*x + 190*e^2*x^2) + 3*B*(d^3 + 20*d^2*e*x + 190*d*

e^2*x^2 + 1140*e^3*x^3)) + 25740*a^7*b^3*e^7*(4*A*e*(d^3 + 20*d^2*e*x + 190*d*e^2*x^2 + 1140*e^3*x^3) + B*(d^4

+ 20*d^3*e*x + 190*d^2*e^2*x^2 + 1140*d*e^3*x^3 + 4845*e^4*x^4)) + 15015*a^6*b^4*e^6*(3*A*e*(d^4 + 20*d^3*e*x

+ 190*d^2*e^2*x^2 + 1140*d*e^3*x^3 + 4845*e^4*x^4) + B*(d^5 + 20*d^4*e*x + 190*d^3*e^2*x^2 + 1140*d^2*e^3*x^3

+ 4845*d*e^4*x^4 + 15504*e^5*x^5)) + 2574*a^5*b^5*e^5*(7*A*e*(d^5 + 20*d^4*e*x + 190*d^3*e^2*x^2 + 1140*d^2*e

^3*x^3 + 4845*d*e^4*x^4 + 15504*e^5*x^5) + 3*B*(d^6 + 20*d^5*e*x + 190*d^4*e^2*x^2 + 1140*d^3*e^3*x^3 + 4845*d

^2*e^4*x^4 + 15504*d*e^5*x^5 + 38760*e^6*x^6)) + 495*a^4*b^6*e^4*(13*A*e*(d^6 + 20*d^5*e*x + 190*d^4*e^2*x^2 +

1140*d^3*e^3*x^3 + 4845*d^2*e^4*x^4 + 15504*d*e^5*x^5 + 38760*e^6*x^6) + 7*B*(d^7 + 20*d^6*e*x + 190*d^5*e^2*

x^2 + 1140*d^4*e^3*x^3 + 4845*d^3*e^4*x^4 + 15504*d^2*e^5*x^5 + 38760*d*e^6*x^6 + 77520*e^7*x^7)) + 660*a^3*b^

7*e^3*(3*A*e*(d^7 + 20*d^6*e*x + 190*d^5*e^2*x^2 + 1140*d^4*e^3*x^3 + 4845*d^3*e^4*x^4 + 15504*d^2*e^5*x^5 + 3

8760*d*e^6*x^6 + 77520*e^7*x^7) + 2*B*(d^8 + 20*d^7*e*x + 190*d^6*e^2*x^2 + 1140*d^5*e^3*x^3 + 4845*d^4*e^4*x^

4 + 15504*d^3*e^5*x^5 + 38760*d^2*e^6*x^6 + 77520*d*e^7*x^7 + 125970*e^8*x^8)) + 45*a^2*b^8*e^2*(11*A*e*(d^8 +

20*d^7*e*x + 190*d^6*e^2*x^2 + 1140*d^5*e^3*x^3 + 4845*d^4*e^4*x^4 + 15504*d^3*e^5*x^5 + 38760*d^2*e^6*x^6 +

77520*d*e^7*x^7 + 125970*e^8*x^8) + 9*B*(d^9 + 20*d^8*e*x + 190*d^7*e^2*x^2 + 1140*d^6*e^3*x^3 + 4845*d^5*e^4*

x^4 + 15504*d^4*e^5*x^5 + 38760*d^3*e^6*x^6 + 77520*d^2*e^7*x^7 + 125970*d*e^8*x^8 + 167960*e^9*x^9)) + 90*a*b

^9*e*(A*e*(d^9 + 20*d^8*e*x + 190*d^7*e^2*x^2 + 1140*d^6*e^3*x^3 + 4845*d^5*e^4*x^4 + 15504*d^4*e^5*x^5 + 3876

0*d^3*e^6*x^6 + 77520*d^2*e^7*x^7 + 125970*d*e^8*x^8 + 167960*e^9*x^9) + B*(d^10 + 20*d^9*e*x + 190*d^8*e^2*x^

2 + 1140*d^7*e^3*x^3 + 4845*d^6*e^4*x^4 + 15504*d^5*e^5*x^5 + 38760*d^4*e^6*x^6 + 77520*d^3*e^7*x^7 + 125970*d

^2*e^8*x^8 + 167960*d*e^9*x^9 + 184756*e^10*x^10)) + b^10*(9*A*e*(d^10 + 20*d^9*e*x + 190*d^8*e^2*x^2 + 1140*d

^7*e^3*x^3 + 4845*d^6*e^4*x^4 + 15504*d^5*e^5*x^5 + 38760*d^4*e^6*x^6 + 77520*d^3*e^7*x^7 + 125970*d^2*e^8*x^8

+ 167960*d*e^9*x^9 + 184756*e^10*x^10) + 11*B*(d^11 + 20*d^10*e*x + 190*d^9*e^2*x^2 + 1140*d^8*e^3*x^3 + 4845

*d^7*e^4*x^4 + 15504*d^6*e^5*x^5 + 38760*d^5*e^6*x^6 + 77520*d^4*e^7*x^7 + 125970*d^3*e^8*x^8 + 167960*d^2*e^9

*x^9 + 184756*d*e^10*x^10 + 167960*e^11*x^11)))/(e^12*(d + e*x)^20)

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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 = 4.20


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)^21,x,method=_RETURNVERBOSE)

-30/13*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)^13-1/20*(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)^20-15/8*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-11*B*b^7*d^7)/(e*x+d)^16-1/9*b^10*B/e^12/(e*x+d)^9-5/4*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)^12-1/19/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)^19-5/18*

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)^18-1/10*b^9/e^12*(A*b*e+10*B*a*e-11*B*b*d)/(e*x+d)^10-3*

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^5-3

5*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)^14-5/11*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)^11-15/17*b^2/e^12*(8*A*a^7*b*e^8-5

6*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)^17-14/5*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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2036 vs. \(2 (471) = 942\).
time = 0.51, size = 2036, normalized size = 4.41 \begin {gather*} \text {Too large to display} \end {gather*}

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^21,x, algorithm="maxima")

-1/16628040*(1847560*B*b^10*x^11*e^11 + 11*B*b^10*d^11 + 831402*A*a^10*e^11 + 9*(10*B*a*b^9*e + A*b^10*e)*d^10

+ 184756*(11*B*b^10*d*e^10 + 90*B*a*b^9*e^11 + 9*A*b^10*e^11)*x^10 + 45*(9*B*a^2*b^8*e^2 + 2*A*a*b^9*e^2)*d^9

+ 167960*(11*B*b^10*d^2*e^9 + 405*B*a^2*b^8*e^11 + 90*A*a*b^9*e^11 + 9*(10*B*a*b^9*e^10 + A*b^10*e^10)*d)*x^9

+ 165*(8*B*a^3*b^7*e^3 + 3*A*a^2*b^8*e^3)*d^8 + 125970*(11*B*b^10*d^3*e^8 + 1320*B*a^3*b^7*e^11 + 495*A*a^2*b

^8*e^11 + 9*(10*B*a*b^9*e^9 + A*b^10*e^9)*d^2 + 45*(9*B*a^2*b^8*e^10 + 2*A*a*b^9*e^10)*d)*x^8 + 495*(7*B*a^4*b

^6*e^4 + 4*A*a^3*b^7*e^4)*d^7 + 77520*(11*B*b^10*d^4*e^7 + 3465*B*a^4*b^6*e^11 + 1980*A*a^3*b^7*e^11 + 9*(10*B

*a*b^9*e^8 + A*b^10*e^8)*d^3 + 45*(9*B*a^2*b^8*e^9 + 2*A*a*b^9*e^9)*d^2 + 165*(8*B*a^3*b^7*e^10 + 3*A*a^2*b^8*

e^10)*d)*x^7 + 1287*(6*B*a^5*b^5*e^5 + 5*A*a^4*b^6*e^5)*d^6 + 38760*(11*B*b^10*d^5*e^6 + 7722*B*a^5*b^5*e^11 +

6435*A*a^4*b^6*e^11 + 9*(10*B*a*b^9*e^7 + A*b^10*e^7)*d^4 + 45*(9*B*a^2*b^8*e^8 + 2*A*a*b^9*e^8)*d^3 + 165*(8

*B*a^3*b^7*e^9 + 3*A*a^2*b^8*e^9)*d^2 + 495*(7*B*a^4*b^6*e^10 + 4*A*a^3*b^7*e^10)*d)*x^6 + 3003*(5*B*a^6*b^4*e

^6 + 6*A*a^5*b^5*e^6)*d^5 + 15504*(11*B*b^10*d^6*e^5 + 15015*B*a^6*b^4*e^11 + 18018*A*a^5*b^5*e^11 + 9*(10*B*a

*b^9*e^6 + A*b^10*e^6)*d^5 + 45*(9*B*a^2*b^8*e^7 + 2*A*a*b^9*e^7)*d^4 + 165*(8*B*a^3*b^7*e^8 + 3*A*a^2*b^8*e^8

)*d^3 + 495*(7*B*a^4*b^6*e^9 + 4*A*a^3*b^7*e^9)*d^2 + 1287*(6*B*a^5*b^5*e^10 + 5*A*a^4*b^6*e^10)*d)*x^5 + 6435

*(4*B*a^7*b^3*e^7 + 7*A*a^6*b^4*e^7)*d^4 + 4845*(11*B*b^10*d^7*e^4 + 25740*B*a^7*b^3*e^11 + 45045*A*a^6*b^4*e^

11 + 9*(10*B*a*b^9*e^5 + A*b^10*e^5)*d^6 + 45*(9*B*a^2*b^8*e^6 + 2*A*a*b^9*e^6)*d^5 + 165*(8*B*a^3*b^7*e^7 + 3

*A*a^2*b^8*e^7)*d^4 + 495*(7*B*a^4*b^6*e^8 + 4*A*a^3*b^7*e^8)*d^3 + 1287*(6*B*a^5*b^5*e^9 + 5*A*a^4*b^6*e^9)*d

^2 + 3003*(5*B*a^6*b^4*e^10 + 6*A*a^5*b^5*e^10)*d)*x^4 + 12870*(3*B*a^8*b^2*e^8 + 8*A*a^7*b^3*e^8)*d^3 + 1140*

(11*B*b^10*d^8*e^3 + 38610*B*a^8*b^2*e^11 + 102960*A*a^7*b^3*e^11 + 9*(10*B*a*b^9*e^4 + A*b^10*e^4)*d^7 + 45*(

9*B*a^2*b^8*e^5 + 2*A*a*b^9*e^5)*d^6 + 165*(8*B*a^3*b^7*e^6 + 3*A*a^2*b^8*e^6)*d^5 + 495*(7*B*a^4*b^6*e^7 + 4*

A*a^3*b^7*e^7)*d^4 + 1287*(6*B*a^5*b^5*e^8 + 5*A*a^4*b^6*e^8)*d^3 + 3003*(5*B*a^6*b^4*e^9 + 6*A*a^5*b^5*e^9)*d

^2 + 6435*(4*B*a^7*b^3*e^10 + 7*A*a^6*b^4*e^10)*d)*x^3 + 24310*(2*B*a^9*b*e^9 + 9*A*a^8*b^2*e^9)*d^2 + 190*(11

*B*b^10*d^9*e^2 + 48620*B*a^9*b*e^11 + 218790*A*a^8*b^2*e^11 + 9*(10*B*a*b^9*e^3 + A*b^10*e^3)*d^8 + 45*(9*B*a

^2*b^8*e^4 + 2*A*a*b^9*e^4)*d^7 + 165*(8*B*a^3*b^7*e^5 + 3*A*a^2*b^8*e^5)*d^6 + 495*(7*B*a^4*b^6*e^6 + 4*A*a^3

*b^7*e^6)*d^5 + 1287*(6*B*a^5*b^5*e^7 + 5*A*a^4*b^6*e^7)*d^4 + 3003*(5*B*a^6*b^4*e^8 + 6*A*a^5*b^5*e^8)*d^3 +

6435*(4*B*a^7*b^3*e^9 + 7*A*a^6*b^4*e^9)*d^2 + 12870*(3*B*a^8*b^2*e^10 + 8*A*a^7*b^3*e^10)*d)*x^2 + 43758*(B*a

^10*e^10 + 10*A*a^9*b*e^10)*d + 20*(11*B*b^10*d^10*e + 43758*B*a^10*e^11 + 437580*A*a^9*b*e^11 + 9*(10*B*a*b^9

*e^2 + A*b^10*e^2)*d^9 + 45*(9*B*a^2*b^8*e^3 + 2*A*a*b^9*e^3)*d^8 + 165*(8*B*a^3*b^7*e^4 + 3*A*a^2*b^8*e^4)*d^

7 + 495*(7*B*a^4*b^6*e^5 + 4*A*a^3*b^7*e^5)*d^6 + 1287*(6*B*a^5*b^5*e^6 + 5*A*a^4*b^6*e^6)*d^5 + 3003*(5*B*a^6

*b^4*e^7 + 6*A*a^5*b^5*e^7)*d^4 + 6435*(4*B*a^7*b^3*e^8 + 7*A*a^6*b^4*e^8)*d^3 + 12870*(3*B*a^8*b^2*e^9 + 8*A*

a^7*b^3*e^9)*d^2 + 24310*(2*B*a^9*b*e^10 + 9*A*a^8*b^2*e^10)*d)*x)/(x^20*e^32 + 20*d*x^19*e^31 + 190*d^2*x^18*

e^30 + 1140*d^3*x^17*e^29 + 4845*d^4*x^16*e^28 + 15504*d^5*x^15*e^27 + 38760*d^6*x^14*e^26 + 77520*d^7*x^13*e^

25 + 125970*d^8*x^12*e^24 + 167960*d^9*x^11*e^23 + 184756*d^10*x^10*e^22 + 167960*d^11*x^9*e^21 + 125970*d^12*

x^8*e^20 + 77520*d^13*x^7*e^19 + 38760*d^14*x^6*e^18 + 15504*d^15*x^5*e^17 + 4845*d^16*x^4*e^16 + 1140*d^17*x^

3*e^15 + 190*d^18*x^2*e^14 + 20*d^19*x*e^13 + d^20*e^12)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1947 vs. \(2 (471) = 942\).
time = 0.70, size = 1947, normalized size = 4.21 \begin {gather*} \text {Too large to display} \end {gather*}

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^21,x, algorithm="fricas")

-1/16628040*(11*B*b^10*d^11 + (1847560*B*b^10*x^11 + 831402*A*a^10 + 1662804*(10*B*a*b^9 + A*b^10)*x^10 + 7558

200*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 20785050*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 38372400*(7*B*a^4*b^6 + 4*A*a^3

*b^7)*x^7 + 49884120*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 46558512*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 31177575*(4*

B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 14671800*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 4618900*(2*B*a^9*b + 9*A*a^8*b^2)*x^

2 + 875160*(B*a^10 + 10*A*a^9*b)*x)*e^11 + (2032316*B*b^10*d*x^10 + 1511640*(10*B*a*b^9 + A*b^10)*d*x^9 + 5668

650*(9*B*a^2*b^8 + 2*A*a*b^9)*d*x^8 + 12790800*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*x^7 + 19186200*(7*B*a^4*b^6 + 4*A

*a^3*b^7)*d*x^6 + 19953648*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*x^5 + 14549535*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*x^4 + 73

35900*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*x^3 + 2445300*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*x^2 + 486200*(2*B*a^9*b + 9*A*

a^8*b^2)*d*x + 43758*(B*a^10 + 10*A*a^9*b)*d)*e^10 + 5*(369512*B*b^10*d^2*x^9 + 226746*(10*B*a*b^9 + A*b^10)*d

^2*x^8 + 697680*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*x^7 + 1279080*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*x^6 + 1534896*(7*B

*a^4*b^6 + 4*A*a^3*b^7)*d^2*x^5 + 1247103*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*x^4 + 684684*(5*B*a^6*b^4 + 6*A*a^5*

b^5)*d^2*x^3 + 244530*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*x^2 + 51480*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*x + 4862*(2*

B*a^9*b + 9*A*a^8*b^2)*d^2)*e^9 + 15*(92378*B*b^10*d^3*x^8 + 46512*(10*B*a*b^9 + A*b^10)*d^3*x^7 + 116280*(9*B

*a^2*b^8 + 2*A*a*b^9)*d^3*x^6 + 170544*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*x^5 + 159885*(7*B*a^4*b^6 + 4*A*a^3*b^7

)*d^3*x^4 + 97812*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*x^3 + 38038*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*x^2 + 8580*(4*B*

a^7*b^3 + 7*A*a^6*b^4)*d^3*x + 858*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3)*e^8 + 15*(56848*B*b^10*d^4*x^7 + 23256*(10

*B*a*b^9 + A*b^10)*d^4*x^6 + 46512*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*x^5 + 53295*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*x

^4 + 37620*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*x^3 + 16302*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*x^2 + 4004*(5*B*a^6*b^4

+ 6*A*a^5*b^5)*d^4*x + 429*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4)*e^7 + 3*(142120*B*b^10*d^5*x^6 + 46512*(10*B*a*b^

9 + A*b^10)*d^5*x^5 + 72675*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*x^4 + 62700*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*x^3 + 31

350*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*x^2 + 8580*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*x + 1001*(5*B*a^6*b^4 + 6*A*a^5

*b^5)*d^5)*e^6 + 3*(56848*B*b^10*d^6*x^5 + 14535*(10*B*a*b^9 + A*b^10)*d^6*x^4 + 17100*(9*B*a^2*b^8 + 2*A*a*b^

9)*d^6*x^3 + 10450*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*x^2 + 3300*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*x + 429*(6*B*a^5

*b^5 + 5*A*a^4*b^6)*d^6)*e^5 + 15*(3553*B*b^10*d^7*x^4 + 684*(10*B*a*b^9 + A*b^10)*d^7*x^3 + 570*(9*B*a^2*b^8

+ 2*A*a*b^9)*d^7*x^2 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*x + 33*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7)*e^4 + 15*(8

36*B*b^10*d^8*x^3 + 114*(10*B*a*b^9 + A*b^10)*d^8*x^2 + 60*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*x + 11*(8*B*a^3*b^7 +

3*A*a^2*b^8)*d^8)*e^3 + 5*(418*B*b^10*d^9*x^2 + 36*(10*B*a*b^9 + A*b^10)*d^9*x + 9*(9*B*a^2*b^8 + 2*A*a*b^9)*

d^9)*e^2 + (220*B*b^10*d^10*x + 9*(10*B*a*b^9 + A*b^10)*d^10)*e)/(x^20*e^32 + 20*d*x^19*e^31 + 190*d^2*x^18*e^

30 + 1140*d^3*x^17*e^29 + 4845*d^4*x^16*e^28 + 15504*d^5*x^15*e^27 + 38760*d^6*x^14*e^26 + 77520*d^7*x^13*e^25

+ 125970*d^8*x^12*e^24 + 167960*d^9*x^11*e^23 + 184756*d^10*x^10*e^22 + 167960*d^11*x^9*e^21 + 125970*d^12*x^

8*e^20 + 77520*d^13*x^7*e^19 + 38760*d^14*x^6*e^18 + 15504*d^15*x^5*e^17 + 4845*d^16*x^4*e^16 + 1140*d^17*x^3*

e^15 + 190*d^18*x^2*e^14 + 20*d^19*x*e^13 + d^20*e^12)

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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 = 3.67, size = 2096, normalized size = 4.54 \begin {gather*} \text {Too large to display} \end {gather*}

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^21,x, algorithm="giac")

-1/16628040*(1847560*B*b^10*x^11*e^11 + 2032316*B*b^10*d*x^10*e^10 + 1847560*B*b^10*d^2*x^9*e^9 + 1385670*B*b^

10*d^3*x^8*e^8 + 852720*B*b^10*d^4*x^7*e^7 + 426360*B*b^10*d^5*x^6*e^6 + 170544*B*b^10*d^6*x^5*e^5 + 53295*B*b

^10*d^7*x^4*e^4 + 12540*B*b^10*d^8*x^3*e^3 + 2090*B*b^10*d^9*x^2*e^2 + 220*B*b^10*d^10*x*e + 11*B*b^10*d^11 +

16628040*B*a*b^9*x^10*e^11 + 1662804*A*b^10*x^10*e^11 + 15116400*B*a*b^9*d*x^9*e^10 + 1511640*A*b^10*d*x^9*e^1

0 + 11337300*B*a*b^9*d^2*x^8*e^9 + 1133730*A*b^10*d^2*x^8*e^9 + 6976800*B*a*b^9*d^3*x^7*e^8 + 697680*A*b^10*d^

3*x^7*e^8 + 3488400*B*a*b^9*d^4*x^6*e^7 + 348840*A*b^10*d^4*x^6*e^7 + 1395360*B*a*b^9*d^5*x^5*e^6 + 139536*A*b

^10*d^5*x^5*e^6 + 436050*B*a*b^9*d^6*x^4*e^5 + 43605*A*b^10*d^6*x^4*e^5 + 102600*B*a*b^9*d^7*x^3*e^4 + 10260*A

*b^10*d^7*x^3*e^4 + 17100*B*a*b^9*d^8*x^2*e^3 + 1710*A*b^10*d^8*x^2*e^3 + 1800*B*a*b^9*d^9*x*e^2 + 180*A*b^10*

d^9*x*e^2 + 90*B*a*b^9*d^10*e + 9*A*b^10*d^10*e + 68023800*B*a^2*b^8*x^9*e^11 + 15116400*A*a*b^9*x^9*e^11 + 51

017850*B*a^2*b^8*d*x^8*e^10 + 11337300*A*a*b^9*d*x^8*e^10 + 31395600*B*a^2*b^8*d^2*x^7*e^9 + 6976800*A*a*b^9*d

^2*x^7*e^9 + 15697800*B*a^2*b^8*d^3*x^6*e^8 + 3488400*A*a*b^9*d^3*x^6*e^8 + 6279120*B*a^2*b^8*d^4*x^5*e^7 + 13

95360*A*a*b^9*d^4*x^5*e^7 + 1962225*B*a^2*b^8*d^5*x^4*e^6 + 436050*A*a*b^9*d^5*x^4*e^6 + 461700*B*a^2*b^8*d^6*

x^3*e^5 + 102600*A*a*b^9*d^6*x^3*e^5 + 76950*B*a^2*b^8*d^7*x^2*e^4 + 17100*A*a*b^9*d^7*x^2*e^4 + 8100*B*a^2*b^

8*d^8*x*e^3 + 1800*A*a*b^9*d^8*x*e^3 + 405*B*a^2*b^8*d^9*e^2 + 90*A*a*b^9*d^9*e^2 + 166280400*B*a^3*b^7*x^8*e^

11 + 62355150*A*a^2*b^8*x^8*e^11 + 102326400*B*a^3*b^7*d*x^7*e^10 + 38372400*A*a^2*b^8*d*x^7*e^10 + 51163200*B

*a^3*b^7*d^2*x^6*e^9 + 19186200*A*a^2*b^8*d^2*x^6*e^9 + 20465280*B*a^3*b^7*d^3*x^5*e^8 + 7674480*A*a^2*b^8*d^3

*x^5*e^8 + 6395400*B*a^3*b^7*d^4*x^4*e^7 + 2398275*A*a^2*b^8*d^4*x^4*e^7 + 1504800*B*a^3*b^7*d^5*x^3*e^6 + 564

300*A*a^2*b^8*d^5*x^3*e^6 + 250800*B*a^3*b^7*d^6*x^2*e^5 + 94050*A*a^2*b^8*d^6*x^2*e^5 + 26400*B*a^3*b^7*d^7*x

*e^4 + 9900*A*a^2*b^8*d^7*x*e^4 + 1320*B*a^3*b^7*d^8*e^3 + 495*A*a^2*b^8*d^8*e^3 + 268606800*B*a^4*b^6*x^7*e^1

1 + 153489600*A*a^3*b^7*x^7*e^11 + 134303400*B*a^4*b^6*d*x^6*e^10 + 76744800*A*a^3*b^7*d*x^6*e^10 + 53721360*B

*a^4*b^6*d^2*x^5*e^9 + 30697920*A*a^3*b^7*d^2*x^5*e^9 + 16787925*B*a^4*b^6*d^3*x^4*e^8 + 9593100*A*a^3*b^7*d^3

*x^4*e^8 + 3950100*B*a^4*b^6*d^4*x^3*e^7 + 2257200*A*a^3*b^7*d^4*x^3*e^7 + 658350*B*a^4*b^6*d^5*x^2*e^6 + 3762

00*A*a^3*b^7*d^5*x^2*e^6 + 69300*B*a^4*b^6*d^6*x*e^5 + 39600*A*a^3*b^7*d^6*x*e^5 + 3465*B*a^4*b^6*d^7*e^4 + 19

80*A*a^3*b^7*d^7*e^4 + 299304720*B*a^5*b^5*x^6*e^11 + 249420600*A*a^4*b^6*x^6*e^11 + 119721888*B*a^5*b^5*d*x^5

*e^10 + 99768240*A*a^4*b^6*d*x^5*e^10 + 37413090*B*a^5*b^5*d^2*x^4*e^9 + 31177575*A*a^4*b^6*d^2*x^4*e^9 + 8803

080*B*a^5*b^5*d^3*x^3*e^8 + 7335900*A*a^4*b^6*d^3*x^3*e^8 + 1467180*B*a^5*b^5*d^4*x^2*e^7 + 1222650*A*a^4*b^6*

d^4*x^2*e^7 + 154440*B*a^5*b^5*d^5*x*e^6 + 128700*A*a^4*b^6*d^5*x*e^6 + 7722*B*a^5*b^5*d^6*e^5 + 6435*A*a^4*b^

6*d^6*e^5 + 232792560*B*a^6*b^4*x^5*e^11 + 279351072*A*a^5*b^5*x^5*e^11 + 72747675*B*a^6*b^4*d*x^4*e^10 + 8729

7210*A*a^5*b^5*d*x^4*e^10 + 17117100*B*a^6*b^4*d^2*x^3*e^9 + 20540520*A*a^5*b^5*d^2*x^3*e^9 + 2852850*B*a^6*b^

4*d^3*x^2*e^8 + 3423420*A*a^5*b^5*d^3*x^2*e^8 + 300300*B*a^6*b^4*d^4*x*e^7 + 360360*A*a^5*b^5*d^4*x*e^7 + 1501

5*B*a^6*b^4*d^5*e^6 + 18018*A*a^5*b^5*d^5*e^6 + 124710300*B*a^7*b^3*x^4*e^11 + 218243025*A*a^6*b^4*x^4*e^11 +

29343600*B*a^7*b^3*d*x^3*e^10 + 51351300*A*a^6*b^4*d*x^3*e^10 + 4890600*B*a^7*b^3*d^2*x^2*e^9 + 8558550*A*a^6*

b^4*d^2*x^2*e^9 + 514800*B*a^7*b^3*d^3*x*e^8 + 900900*A*a^6*b^4*d^3*x*e^8 + 25740*B*a^7*b^3*d^4*e^7 + 45045*A*

a^6*b^4*d^4*e^7 + 44015400*B*a^8*b^2*x^3*e^11 + 117374400*A*a^7*b^3*x^3*e^11 + 7335900*B*a^8*b^2*d*x^2*e^10 +

19562400*A*a^7*b^3*d*x^2*e^10 + 772200*B*a^8*b^2*d^2*x*e^9 + 2059200*A*a^7*b^3*d^2*x*e^9 + 38610*B*a^8*b^2*d^3

*e^8 + 102960*A*a^7*b^3*d^3*e^8 + 9237800*B*a^9*b*x^2*e^11 + 41570100*A*a^8*b^2*x^2*e^11 + 972400*B*a^9*b*d*x*

e^10 + 4375800*A*a^8*b^2*d*x*e^10 + 48620*B*a^9*b*d^2*e^9 + 218790*A*a^8*b^2*d^2*e^9 + 875160*B*a^10*x*e^11 +

8751600*A*a^9*b*x*e^11 + 43758*B*a^10*d*e^10 + 437580*A*a^9*b*d*e^10 + 831402*A*a^10*e^11)*e^(-12)/(x*e + d)^2

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Mupad [B]
time = 1.91, size = 2110, normalized size = 4.57 \begin {gather*} -\frac {\frac {43758\,B\,a^{10}\,d\,e^{10}+831402\,A\,a^{10}\,e^{11}+48620\,B\,a^9\,b\,d^2\,e^9+437580\,A\,a^9\,b\,d\,e^{10}+38610\,B\,a^8\,b^2\,d^3\,e^8+218790\,A\,a^8\,b^2\,d^2\,e^9+25740\,B\,a^7\,b^3\,d^4\,e^7+102960\,A\,a^7\,b^3\,d^3\,e^8+15015\,B\,a^6\,b^4\,d^5\,e^6+45045\,A\,a^6\,b^4\,d^4\,e^7+7722\,B\,a^5\,b^5\,d^6\,e^5+18018\,A\,a^5\,b^5\,d^5\,e^6+3465\,B\,a^4\,b^6\,d^7\,e^4+6435\,A\,a^4\,b^6\,d^6\,e^5+1320\,B\,a^3\,b^7\,d^8\,e^3+1980\,A\,a^3\,b^7\,d^7\,e^4+405\,B\,a^2\,b^8\,d^9\,e^2+495\,A\,a^2\,b^8\,d^8\,e^3+90\,B\,a\,b^9\,d^{10}\,e+90\,A\,a\,b^9\,d^9\,e^2+11\,B\,b^{10}\,d^{11}+9\,A\,b^{10}\,d^{10}\,e}{16628040\,e^{12}}+\frac {x\,\left (43758\,B\,a^{10}\,e^{10}+48620\,B\,a^9\,b\,d\,e^9+437580\,A\,a^9\,b\,e^{10}+38610\,B\,a^8\,b^2\,d^2\,e^8+218790\,A\,a^8\,b^2\,d\,e^9+25740\,B\,a^7\,b^3\,d^3\,e^7+102960\,A\,a^7\,b^3\,d^2\,e^8+15015\,B\,a^6\,b^4\,d^4\,e^6+45045\,A\,a^6\,b^4\,d^3\,e^7+7722\,B\,a^5\,b^5\,d^5\,e^5+18018\,A\,a^5\,b^5\,d^4\,e^6+3465\,B\,a^4\,b^6\,d^6\,e^4+6435\,A\,a^4\,b^6\,d^5\,e^5+1320\,B\,a^3\,b^7\,d^7\,e^3+1980\,A\,a^3\,b^7\,d^6\,e^4+405\,B\,a^2\,b^8\,d^8\,e^2+495\,A\,a^2\,b^8\,d^7\,e^3+90\,B\,a\,b^9\,d^9\,e+90\,A\,a\,b^9\,d^8\,e^2+11\,B\,b^{10}\,d^{10}+9\,A\,b^{10}\,d^9\,e\right )}{831402\,e^{11}}+\frac {b^7\,x^8\,\left (1320\,B\,a^3\,e^3+405\,B\,a^2\,b\,d\,e^2+495\,A\,a^2\,b\,e^3+90\,B\,a\,b^2\,d^2\,e+90\,A\,a\,b^2\,d\,e^2+11\,B\,b^3\,d^3+9\,A\,b^3\,d^2\,e\right )}{132\,e^4}+\frac {2\,b^4\,x^5\,\left (15015\,B\,a^6\,e^6+7722\,B\,a^5\,b\,d\,e^5+18018\,A\,a^5\,b\,e^6+3465\,B\,a^4\,b^2\,d^2\,e^4+6435\,A\,a^4\,b^2\,d\,e^5+1320\,B\,a^3\,b^3\,d^3\,e^3+1980\,A\,a^3\,b^3\,d^2\,e^4+405\,B\,a^2\,b^4\,d^4\,e^2+495\,A\,a^2\,b^4\,d^3\,e^3+90\,B\,a\,b^5\,d^5\,e+90\,A\,a\,b^5\,d^4\,e^2+11\,B\,b^6\,d^6+9\,A\,b^6\,d^5\,e\right )}{2145\,e^7}+\frac {b^9\,x^{10}\,\left (9\,A\,b\,e+90\,B\,a\,e+11\,B\,b\,d\right )}{90\,e^2}+\frac {2\,b^6\,x^7\,\left (3465\,B\,a^4\,e^4+1320\,B\,a^3\,b\,d\,e^3+1980\,A\,a^3\,b\,e^4+405\,B\,a^2\,b^2\,d^2\,e^2+495\,A\,a^2\,b^2\,d\,e^3+90\,B\,a\,b^3\,d^3\,e+90\,A\,a\,b^3\,d^2\,e^2+11\,B\,b^4\,d^4+9\,A\,b^4\,d^3\,e\right )}{429\,e^5}+\frac {b^3\,x^4\,\left (25740\,B\,a^7\,e^7+15015\,B\,a^6\,b\,d\,e^6+45045\,A\,a^6\,b\,e^7+7722\,B\,a^5\,b^2\,d^2\,e^5+18018\,A\,a^5\,b^2\,d\,e^6+3465\,B\,a^4\,b^3\,d^3\,e^4+6435\,A\,a^4\,b^3\,d^2\,e^5+1320\,B\,a^3\,b^4\,d^4\,e^3+1980\,A\,a^3\,b^4\,d^3\,e^4+405\,B\,a^2\,b^5\,d^5\,e^2+495\,A\,a^2\,b^5\,d^4\,e^3+90\,B\,a\,b^6\,d^6\,e+90\,A\,a\,b^6\,d^5\,e^2+11\,B\,b^7\,d^7+9\,A\,b^7\,d^6\,e\right )}{3432\,e^8}+\frac {b\,x^2\,\left (48620\,B\,a^9\,e^9+38610\,B\,a^8\,b\,d\,e^8+218790\,A\,a^8\,b\,e^9+25740\,B\,a^7\,b^2\,d^2\,e^7+102960\,A\,a^7\,b^2\,d\,e^8+15015\,B\,a^6\,b^3\,d^3\,e^6+45045\,A\,a^6\,b^3\,d^2\,e^7+7722\,B\,a^5\,b^4\,d^4\,e^5+18018\,A\,a^5\,b^4\,d^3\,e^6+3465\,B\,a^4\,b^5\,d^5\,e^4+6435\,A\,a^4\,b^5\,d^4\,e^5+1320\,B\,a^3\,b^6\,d^6\,e^3+1980\,A\,a^3\,b^6\,d^5\,e^4+405\,B\,a^2\,b^7\,d^7\,e^2+495\,A\,a^2\,b^7\,d^6\,e^3+90\,B\,a\,b^8\,d^8\,e+90\,A\,a\,b^8\,d^7\,e^2+11\,B\,b^9\,d^9+9\,A\,b^9\,d^8\,e\right )}{87516\,e^{10}}+\frac {b^8\,x^9\,\left (405\,B\,a^2\,e^2+90\,B\,a\,b\,d\,e+90\,A\,a\,b\,e^2+11\,B\,b^2\,d^2+9\,A\,b^2\,d\,e\right )}{99\,e^3}+\frac {b^5\,x^6\,\left (7722\,B\,a^5\,e^5+3465\,B\,a^4\,b\,d\,e^4+6435\,A\,a^4\,b\,e^5+1320\,B\,a^3\,b^2\,d^2\,e^3+1980\,A\,a^3\,b^2\,d\,e^4+405\,B\,a^2\,b^3\,d^3\,e^2+495\,A\,a^2\,b^3\,d^2\,e^3+90\,B\,a\,b^4\,d^4\,e+90\,A\,a\,b^4\,d^3\,e^2+11\,B\,b^5\,d^5+9\,A\,b^5\,d^4\,e\right )}{429\,e^6}+\frac {b^2\,x^3\,\left (38610\,B\,a^8\,e^8+25740\,B\,a^7\,b\,d\,e^7+102960\,A\,a^7\,b\,e^8+15015\,B\,a^6\,b^2\,d^2\,e^6+45045\,A\,a^6\,b^2\,d\,e^7+7722\,B\,a^5\,b^3\,d^3\,e^5+18018\,A\,a^5\,b^3\,d^2\,e^6+3465\,B\,a^4\,b^4\,d^4\,e^4+6435\,A\,a^4\,b^4\,d^3\,e^5+1320\,B\,a^3\,b^5\,d^5\,e^3+1980\,A\,a^3\,b^5\,d^4\,e^4+405\,B\,a^2\,b^6\,d^6\,e^2+495\,A\,a^2\,b^6\,d^5\,e^3+90\,B\,a\,b^7\,d^7\,e+90\,A\,a\,b^7\,d^6\,e^2+11\,B\,b^8\,d^8+9\,A\,b^8\,d^7\,e\right )}{14586\,e^9}+\frac {B\,b^{10}\,x^{11}}{9\,e}}{d^{20}+20\,d^{19}\,e\,x+190\,d^{18}\,e^2\,x^2+1140\,d^{17}\,e^3\,x^3+4845\,d^{16}\,e^4\,x^4+15504\,d^{15}\,e^5\,x^5+38760\,d^{14}\,e^6\,x^6+77520\,d^{13}\,e^7\,x^7+125970\,d^{12}\,e^8\,x^8+167960\,d^{11}\,e^9\,x^9+184756\,d^{10}\,e^{10}\,x^{10}+167960\,d^9\,e^{11}\,x^{11}+125970\,d^8\,e^{12}\,x^{12}+77520\,d^7\,e^{13}\,x^{13}+38760\,d^6\,e^{14}\,x^{14}+15504\,d^5\,e^{15}\,x^{15}+4845\,d^4\,e^{16}\,x^{16}+1140\,d^3\,e^{17}\,x^{17}+190\,d^2\,e^{18}\,x^{18}+20\,d\,e^{19}\,x^{19}+e^{20}\,x^{20}} \end {gather*}

-((831402*A*a^10*e^11 + 11*B*b^10*d^11 + 9*A*b^10*d^10*e + 43758*B*a^10*d*e^10 + 90*A*a*b^9*d^9*e^2 + 48620*B*

a^9*b*d^2*e^9 + 495*A*a^2*b^8*d^8*e^3 + 1980*A*a^3*b^7*d^7*e^4 + 6435*A*a^4*b^6*d^6*e^5 + 18018*A*a^5*b^5*d^5*

e^6 + 45045*A*a^6*b^4*d^4*e^7 + 102960*A*a^7*b^3*d^3*e^8 + 218790*A*a^8*b^2*d^2*e^9 + 405*B*a^2*b^8*d^9*e^2 +

1320*B*a^3*b^7*d^8*e^3 + 3465*B*a^4*b^6*d^7*e^4 + 7722*B*a^5*b^5*d^6*e^5 + 15015*B*a^6*b^4*d^5*e^6 + 25740*B*a

^7*b^3*d^4*e^7 + 38610*B*a^8*b^2*d^3*e^8 + 437580*A*a^9*b*d*e^10 + 90*B*a*b^9*d^10*e)/(16628040*e^12) + (x*(43

758*B*a^10*e^10 + 11*B*b^10*d^10 + 437580*A*a^9*b*e^10 + 9*A*b^10*d^9*e + 90*A*a*b^9*d^8*e^2 + 218790*A*a^8*b^

2*d*e^9 + 495*A*a^2*b^8*d^7*e^3 + 1980*A*a^3*b^7*d^6*e^4 + 6435*A*a^4*b^6*d^5*e^5 + 18018*A*a^5*b^5*d^4*e^6 +

45045*A*a^6*b^4*d^3*e^7 + 102960*A*a^7*b^3*d^2*e^8 + 405*B*a^2*b^8*d^8*e^2 + 1320*B*a^3*b^7*d^7*e^3 + 3465*B*a

^4*b^6*d^6*e^4 + 7722*B*a^5*b^5*d^5*e^5 + 15015*B*a^6*b^4*d^4*e^6 + 25740*B*a^7*b^3*d^3*e^7 + 38610*B*a^8*b^2*

d^2*e^8 + 90*B*a*b^9*d^9*e + 48620*B*a^9*b*d*e^9))/(831402*e^11) + (b^7*x^8*(1320*B*a^3*e^3 + 11*B*b^3*d^3 + 4

95*A*a^2*b*e^3 + 9*A*b^3*d^2*e + 90*A*a*b^2*d*e^2 + 90*B*a*b^2*d^2*e + 405*B*a^2*b*d*e^2))/(132*e^4) + (2*b^4*

x^5*(15015*B*a^6*e^6 + 11*B*b^6*d^6 + 18018*A*a^5*b*e^6 + 9*A*b^6*d^5*e + 90*A*a*b^5*d^4*e^2 + 6435*A*a^4*b^2*

d*e^5 + 495*A*a^2*b^4*d^3*e^3 + 1980*A*a^3*b^3*d^2*e^4 + 405*B*a^2*b^4*d^4*e^2 + 1320*B*a^3*b^3*d^3*e^3 + 3465

*B*a^4*b^2*d^2*e^4 + 90*B*a*b^5*d^5*e + 7722*B*a^5*b*d*e^5))/(2145*e^7) + (b^9*x^10*(9*A*b*e + 90*B*a*e + 11*B

*b*d))/(90*e^2) + (2*b^6*x^7*(3465*B*a^4*e^4 + 11*B*b^4*d^4 + 1980*A*a^3*b*e^4 + 9*A*b^4*d^3*e + 90*A*a*b^3*d^

2*e^2 + 495*A*a^2*b^2*d*e^3 + 405*B*a^2*b^2*d^2*e^2 + 90*B*a*b^3*d^3*e + 1320*B*a^3*b*d*e^3))/(429*e^5) + (b^3

*x^4*(25740*B*a^7*e^7 + 11*B*b^7*d^7 + 45045*A*a^6*b*e^7 + 9*A*b^7*d^6*e + 90*A*a*b^6*d^5*e^2 + 18018*A*a^5*b^

2*d*e^6 + 495*A*a^2*b^5*d^4*e^3 + 1980*A*a^3*b^4*d^3*e^4 + 6435*A*a^4*b^3*d^2*e^5 + 405*B*a^2*b^5*d^5*e^2 + 13

20*B*a^3*b^4*d^4*e^3 + 3465*B*a^4*b^3*d^3*e^4 + 7722*B*a^5*b^2*d^2*e^5 + 90*B*a*b^6*d^6*e + 15015*B*a^6*b*d*e^

6))/(3432*e^8) + (b*x^2*(48620*B*a^9*e^9 + 11*B*b^9*d^9 + 218790*A*a^8*b*e^9 + 9*A*b^9*d^8*e + 90*A*a*b^8*d^7*

e^2 + 102960*A*a^7*b^2*d*e^8 + 495*A*a^2*b^7*d^6*e^3 + 1980*A*a^3*b^6*d^5*e^4 + 6435*A*a^4*b^5*d^4*e^5 + 18018

*A*a^5*b^4*d^3*e^6 + 45045*A*a^6*b^3*d^2*e^7 + 405*B*a^2*b^7*d^7*e^2 + 1320*B*a^3*b^6*d^6*e^3 + 3465*B*a^4*b^5

*d^5*e^4 + 7722*B*a^5*b^4*d^4*e^5 + 15015*B*a^6*b^3*d^3*e^6 + 25740*B*a^7*b^2*d^2*e^7 + 90*B*a*b^8*d^8*e + 386

10*B*a^8*b*d*e^8))/(87516*e^10) + (b^8*x^9*(405*B*a^2*e^2 + 11*B*b^2*d^2 + 90*A*a*b*e^2 + 9*A*b^2*d*e + 90*B*a

*b*d*e))/(99*e^3) + (b^5*x^6*(7722*B*a^5*e^5 + 11*B*b^5*d^5 + 6435*A*a^4*b*e^5 + 9*A*b^5*d^4*e + 90*A*a*b^4*d^

3*e^2 + 1980*A*a^3*b^2*d*e^4 + 495*A*a^2*b^3*d^2*e^3 + 405*B*a^2*b^3*d^3*e^2 + 1320*B*a^3*b^2*d^2*e^3 + 90*B*a

*b^4*d^4*e + 3465*B*a^4*b*d*e^4))/(429*e^6) + (b^2*x^3*(38610*B*a^8*e^8 + 11*B*b^8*d^8 + 102960*A*a^7*b*e^8 +

9*A*b^8*d^7*e + 90*A*a*b^7*d^6*e^2 + 45045*A*a^6*b^2*d*e^7 + 495*A*a^2*b^6*d^5*e^3 + 1980*A*a^3*b^5*d^4*e^4 +

6435*A*a^4*b^4*d^3*e^5 + 18018*A*a^5*b^3*d^2*e^6 + 405*B*a^2*b^6*d^6*e^2 + 1320*B*a^3*b^5*d^5*e^3 + 3465*B*a^4

*b^4*d^4*e^4 + 7722*B*a^5*b^3*d^3*e^5 + 15015*B*a^6*b^2*d^2*e^6 + 90*B*a*b^7*d^7*e + 25740*B*a^7*b*d*e^7))/(14

586*e^9) + (B*b^10*x^11)/(9*e))/(d^20 + e^20*x^20 + 20*d*e^19*x^19 + 190*d^18*e^2*x^2 + 1140*d^17*e^3*x^3 + 48

45*d^16*e^4*x^4 + 15504*d^15*e^5*x^5 + 38760*d^14*e^6*x^6 + 77520*d^13*e^7*x^7 + 125970*d^12*e^8*x^8 + 167960*

d^11*e^9*x^9 + 184756*d^10*e^10*x^10 + 167960*d^9*e^11*x^11 + 125970*d^8*e^12*x^12 + 77520*d^7*e^13*x^13 + 387

60*d^6*e^14*x^14 + 15504*d^5*e^15*x^15 + 4845*d^4*e^16*x^16 + 1140*d^3*e^17*x^17 + 190*d^2*e^18*x^18 + 20*d^19

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3.12.9 \(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx\) [1109]