\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx\) [1109]
Optimal result
Integrand size = 20, antiderivative size = 462 \[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx=\frac {(b d-a e)^{10} (B d-A e)}{20 e^{12} (d+e x)^{20}}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{19 e^{12} (d+e x)^{19}}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{18 e^{12} (d+e x)^{18}}-\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{17 e^{12} (d+e x)^{17}}+\frac {15 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{8 e^{12} (d+e x)^{16}}-\frac {14 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{5 e^{12} (d+e x)^{15}}+\frac {3 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{e^{12} (d+e x)^{14}}-\frac {30 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{13 e^{12} (d+e x)^{13}}+\frac {5 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{4 e^{12} (d+e x)^{12}}-\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e)}{11 e^{12} (d+e x)^{11}}+\frac {b^9 (11 b B d-A b e-10 a B e)}{10 e^{12} (d+e x)^{10}}-\frac {b^{10} B}{9 e^{12} (d+e x)^9}
\]
[Out]
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*
b*d)/e^12/(e*x+d)^10-1/9*b^10*B/e^12/(e*x+d)^9
Rubi [A] (verified)
Time = 0.45 (sec) , antiderivative size = 462, 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}
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx=\frac {b^9 (-10 a B e-A b e+11 b B d)}{10 e^{12} (d+e x)^{10}}-\frac {5 b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{11 e^{12} (d+e x)^{11}}+\frac {5 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{4 e^{12} (d+e x)^{12}}-\frac {30 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{13 e^{12} (d+e x)^{13}}+\frac {3 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)^{14}}-\frac {14 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{5 e^{12} (d+e x)^{15}}+\frac {15 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{8 e^{12} (d+e x)^{16}}-\frac {15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{17 e^{12} (d+e x)^{17}}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{18 e^{12} (d+e x)^{18}}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{19 e^{12} (d+e x)^{19}}+\frac {(b d-a e)^{10} (B d-A e)}{20 e^{12} (d+e x)^{20}}-\frac {b^{10} B}{9 e^{12} (d+e x)^9}
\]
[In]
Int[((a + b*x)^10*(A + B*x))/(d + e*x)^21,x]
[Out]
((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)
Rule 78
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rubi steps \begin{align*}
\text {integral}& = \int \left (\frac {(-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*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(1428\) vs. \(2(462)=924\).
Time = 0.49 (sec) , antiderivative size = 1428, normalized size of antiderivative = 3.09
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx=-\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}}
\]
[In]
Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^21,x]
[Out]
-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)
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(1900\) vs. \(2(440)=880\).
Time = 2.15 (sec) , antiderivative size = 1901, normalized size of antiderivative =
4.11
| | |
| 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 |
\(\text {Expression too large to display}\) |
\(2233\) |
| parallelrisch |
\(\text {Expression too large to display}\) |
\(2242\) |
| | |
|
|
|
|
[In]
int((b*x+a)^10*(B*x+A)/(e*x+d)^21,x,method=_RETURNVERBOSE)
[Out]
(-1/16628040/e^12*(831402*A*a^10*e^11+437580*A*a^9*b*d*e^10+218790*A*a^8*b^2*d^2*e^9+102960*A*a^7*b^3*d^3*e^8+
45045*A*a^6*b^4*d^4*e^7+18018*A*a^5*b^5*d^5*e^6+6435*A*a^4*b^6*d^6*e^5+1980*A*a^3*b^7*d^7*e^4+495*A*a^2*b^8*d^
8*e^3+90*A*a*b^9*d^9*e^2+9*A*b^10*d^10*e+43758*B*a^10*d*e^10+48620*B*a^9*b*d^2*e^9+38610*B*a^8*b^2*d^3*e^8+257
40*B*a^7*b^3*d^4*e^7+15015*B*a^6*b^4*d^5*e^6+7722*B*a^5*b^5*d^6*e^5+3465*B*a^4*b^6*d^7*e^4+1320*B*a^3*b^7*d^8*
e^3+405*B*a^2*b^8*d^9*e^2+90*B*a*b^9*d^10*e+11*B*b^10*d^11)-1/831402/e^11*(437580*A*a^9*b*e^10+218790*A*a^8*b^
2*d*e^9+102960*A*a^7*b^3*d^2*e^8+45045*A*a^6*b^4*d^3*e^7+18018*A*a^5*b^5*d^4*e^6+6435*A*a^4*b^6*d^5*e^5+1980*A
*a^3*b^7*d^6*e^4+495*A*a^2*b^8*d^7*e^3+90*A*a*b^9*d^8*e^2+9*A*b^10*d^9*e+43758*B*a^10*e^10+48620*B*a^9*b*d*e^9
+38610*B*a^8*b^2*d^2*e^8+25740*B*a^7*b^3*d^3*e^7+15015*B*a^6*b^4*d^4*e^6+7722*B*a^5*b^5*d^5*e^5+3465*B*a^4*b^6
*d^6*e^4+1320*B*a^3*b^7*d^7*e^3+405*B*a^2*b^8*d^8*e^2+90*B*a*b^9*d^9*e+11*B*b^10*d^10)*x-1/87516*b/e^10*(21879
0*A*a^8*b*e^9+102960*A*a^7*b^2*d*e^8+45045*A*a^6*b^3*d^2*e^7+18018*A*a^5*b^4*d^3*e^6+6435*A*a^4*b^5*d^4*e^5+19
80*A*a^3*b^6*d^5*e^4+495*A*a^2*b^7*d^6*e^3+90*A*a*b^8*d^7*e^2+9*A*b^9*d^8*e+48620*B*a^9*e^9+38610*B*a^8*b*d*e^
8+25740*B*a^7*b^2*d^2*e^7+15015*B*a^6*b^3*d^3*e^6+7722*B*a^5*b^4*d^4*e^5+3465*B*a^4*b^5*d^5*e^4+1320*B*a^3*b^6
*d^6*e^3+405*B*a^2*b^7*d^7*e^2+90*B*a*b^8*d^8*e+11*B*b^9*d^9)*x^2-1/14586*b^2/e^9*(102960*A*a^7*b*e^8+45045*A*
a^6*b^2*d*e^7+18018*A*a^5*b^3*d^2*e^6+6435*A*a^4*b^4*d^3*e^5+1980*A*a^3*b^5*d^4*e^4+495*A*a^2*b^6*d^5*e^3+90*A
*a*b^7*d^6*e^2+9*A*b^8*d^7*e+38610*B*a^8*e^8+25740*B*a^7*b*d*e^7+15015*B*a^6*b^2*d^2*e^6+7722*B*a^5*b^3*d^3*e^
5+3465*B*a^4*b^4*d^4*e^4+1320*B*a^3*b^5*d^5*e^3+405*B*a^2*b^6*d^6*e^2+90*B*a*b^7*d^7*e+11*B*b^8*d^8)*x^3-1/343
2*b^3/e^8*(45045*A*a^6*b*e^7+18018*A*a^5*b^2*d*e^6+6435*A*a^4*b^3*d^2*e^5+1980*A*a^3*b^4*d^3*e^4+495*A*a^2*b^5
*d^4*e^3+90*A*a*b^6*d^5*e^2+9*A*b^7*d^6*e+25740*B*a^7*e^7+15015*B*a^6*b*d*e^6+7722*B*a^5*b^2*d^2*e^5+3465*B*a^
4*b^3*d^3*e^4+1320*B*a^3*b^4*d^4*e^3+405*B*a^2*b^5*d^5*e^2+90*B*a*b^6*d^6*e+11*B*b^7*d^7)*x^4-2/2145*b^4/e^7*(
18018*A*a^5*b*e^6+6435*A*a^4*b^2*d*e^5+1980*A*a^3*b^3*d^2*e^4+495*A*a^2*b^4*d^3*e^3+90*A*a*b^5*d^4*e^2+9*A*b^6
*d^5*e+15015*B*a^6*e^6+7722*B*a^5*b*d*e^5+3465*B*a^4*b^2*d^2*e^4+1320*B*a^3*b^3*d^3*e^3+405*B*a^2*b^4*d^4*e^2+
90*B*a*b^5*d^5*e+11*B*b^6*d^6)*x^5-1/429*b^5/e^6*(6435*A*a^4*b*e^5+1980*A*a^3*b^2*d*e^4+495*A*a^2*b^3*d^2*e^3+
90*A*a*b^4*d^3*e^2+9*A*b^5*d^4*e+7722*B*a^5*e^5+3465*B*a^4*b*d*e^4+1320*B*a^3*b^2*d^2*e^3+405*B*a^2*b^3*d^3*e^
2+90*B*a*b^4*d^4*e+11*B*b^5*d^5)*x^6-2/429*b^6/e^5*(1980*A*a^3*b*e^4+495*A*a^2*b^2*d*e^3+90*A*a*b^3*d^2*e^2+9*
A*b^4*d^3*e+3465*B*a^4*e^4+1320*B*a^3*b*d*e^3+405*B*a^2*b^2*d^2*e^2+90*B*a*b^3*d^3*e+11*B*b^4*d^4)*x^7-1/132*b
^7/e^4*(495*A*a^2*b*e^3+90*A*a*b^2*d*e^2+9*A*b^3*d^2*e+1320*B*a^3*e^3+405*B*a^2*b*d*e^2+90*B*a*b^2*d^2*e+11*B*
b^3*d^3)*x^8-1/99*b^8/e^3*(90*A*a*b*e^2+9*A*b^2*d*e+405*B*a^2*e^2+90*B*a*b*d*e+11*B*b^2*d^2)*x^9-1/90*b^9/e^2*
(9*A*b*e+90*B*a*e+11*B*b*d)*x^10-1/9*b^10*B/e*x^11)/(e*x+d)^20
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2028 vs. \(2 (440) = 880\).
Time = 0.40 (sec) , antiderivative size = 2028, normalized size of antiderivative = 4.39
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^21,x, algorithm="fricas")
[Out]
-1/16628040*(1847560*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 831402*A*a^10*e^11 + 9*(10*B*a*b^9 + A*b^10)*d^10*e +
45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 495*(7*B*a^4*b^6 + 4*A*a^3*b
^7)*d^7*e^4 + 1287*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 3003*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 6435*(4*B*
a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 12870*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 24310*(2*B*a^9*b + 9*A*a^8*b^2)*d
^2*e^9 + 43758*(B*a^10 + 10*A*a^9*b)*d*e^10 + 184756*(11*B*b^10*d*e^10 + 9*(10*B*a*b^9 + A*b^10)*e^11)*x^10 +
167960*(11*B*b^10*d^2*e^9 + 9*(10*B*a*b^9 + A*b^10)*d*e^10 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 125970*(
11*B*b^10*d^3*e^8 + 9*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 165*(8*B*a^3*b^7 +
3*A*a^2*b^8)*e^11)*x^8 + 77520*(11*B*b^10*d^4*e^7 + 9*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 45*(9*B*a^2*b^8 + 2*A*a
*b^9)*d^2*e^9 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 38760*(11
*B*b^10*d^5*e^6 + 9*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 165*(8*B*a^3*b^7 +
3*A*a^2*b^8)*d^2*e^9 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 1287*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 1
5504*(11*B*b^10*d^6*e^5 + 9*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 165*(8*B*a^
3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 1287*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^
10 + 3003*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 4845*(11*B*b^10*d^7*e^4 + 9*(10*B*a*b^9 + A*b^10)*d^6*e^5 +
45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^
7)*d^3*e^8 + 1287*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 3003*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 6435*(4*B*a^
7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 1140*(11*B*b^10*d^8*e^3 + 9*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 45*(9*B*a^2*b^8 +
2*A*a*b^9)*d^6*e^5 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 1287
*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 3003*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 6435*(4*B*a^7*b^3 + 7*A*a^6*
b^4)*d*e^10 + 12870*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 190*(11*B*b^10*d^9*e^2 + 9*(10*B*a*b^9 + A*b^10)*d
^8*e^3 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 495*(7*B*a^4*b^6 + 4
*A*a^3*b^7)*d^5*e^6 + 1287*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 3003*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 64
35*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 12870*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 24310*(2*B*a^9*b + 9*A*a^8
*b^2)*e^11)*x^2 + 20*(11*B*b^10*d^10*e + 9*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^
3 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 1287*(6*B*a^5*b^5 + 5*
A*a^4*b^6)*d^5*e^6 + 3003*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 6435*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 128
70*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 24310*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 43758*(B*a^10 + 10*A*a^9*b)*
e^11)*x)/(e^32*x^20 + 20*d*e^31*x^19 + 190*d^2*e^30*x^18 + 1140*d^3*e^29*x^17 + 4845*d^4*e^28*x^16 + 15504*d^5
*e^27*x^15 + 38760*d^6*e^26*x^14 + 77520*d^7*e^25*x^13 + 125970*d^8*e^24*x^12 + 167960*d^9*e^23*x^11 + 184756*
d^10*e^22*x^10 + 167960*d^11*e^21*x^9 + 125970*d^12*e^20*x^8 + 77520*d^13*e^19*x^7 + 38760*d^14*e^18*x^6 + 155
04*d^15*e^17*x^5 + 4845*d^16*e^16*x^4 + 1140*d^17*e^15*x^3 + 190*d^18*e^14*x^2 + 20*d^19*e^13*x + d^20*e^12)
Sympy [F(-1)]
Timed out. \[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx=\text {Timed out}
\]
[In]
integrate((b*x+a)**10*(B*x+A)/(e*x+d)**21,x)
[Out]
Timed out
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2028 vs. \(2 (440) = 880\).
Time = 0.33 (sec) , antiderivative size = 2028, normalized size of antiderivative = 4.39
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^21,x, algorithm="maxima")
[Out]
-1/16628040*(1847560*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 831402*A*a^10*e^11 + 9*(10*B*a*b^9 + A*b^10)*d^10*e +
45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 495*(7*B*a^4*b^6 + 4*A*a^3*b
^7)*d^7*e^4 + 1287*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 3003*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 6435*(4*B*
a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 12870*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 24310*(2*B*a^9*b + 9*A*a^8*b^2)*d
^2*e^9 + 43758*(B*a^10 + 10*A*a^9*b)*d*e^10 + 184756*(11*B*b^10*d*e^10 + 9*(10*B*a*b^9 + A*b^10)*e^11)*x^10 +
167960*(11*B*b^10*d^2*e^9 + 9*(10*B*a*b^9 + A*b^10)*d*e^10 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 125970*(
11*B*b^10*d^3*e^8 + 9*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 165*(8*B*a^3*b^7 +
3*A*a^2*b^8)*e^11)*x^8 + 77520*(11*B*b^10*d^4*e^7 + 9*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 45*(9*B*a^2*b^8 + 2*A*a
*b^9)*d^2*e^9 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 38760*(11
*B*b^10*d^5*e^6 + 9*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 165*(8*B*a^3*b^7 +
3*A*a^2*b^8)*d^2*e^9 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 1287*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 1
5504*(11*B*b^10*d^6*e^5 + 9*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 165*(8*B*a^
3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 1287*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^
10 + 3003*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 4845*(11*B*b^10*d^7*e^4 + 9*(10*B*a*b^9 + A*b^10)*d^6*e^5 +
45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^
7)*d^3*e^8 + 1287*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 3003*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 6435*(4*B*a^
7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 1140*(11*B*b^10*d^8*e^3 + 9*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 45*(9*B*a^2*b^8 +
2*A*a*b^9)*d^6*e^5 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 1287
*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 3003*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 6435*(4*B*a^7*b^3 + 7*A*a^6*
b^4)*d*e^10 + 12870*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 190*(11*B*b^10*d^9*e^2 + 9*(10*B*a*b^9 + A*b^10)*d
^8*e^3 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 495*(7*B*a^4*b^6 + 4
*A*a^3*b^7)*d^5*e^6 + 1287*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 3003*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 64
35*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 12870*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 24310*(2*B*a^9*b + 9*A*a^8
*b^2)*e^11)*x^2 + 20*(11*B*b^10*d^10*e + 9*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^
3 + 165*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 495*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 1287*(6*B*a^5*b^5 + 5*
A*a^4*b^6)*d^5*e^6 + 3003*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 6435*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 128
70*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 24310*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 43758*(B*a^10 + 10*A*a^9*b)*
e^11)*x)/(e^32*x^20 + 20*d*e^31*x^19 + 190*d^2*e^30*x^18 + 1140*d^3*e^29*x^17 + 4845*d^4*e^28*x^16 + 15504*d^5
*e^27*x^15 + 38760*d^6*e^26*x^14 + 77520*d^7*e^25*x^13 + 125970*d^8*e^24*x^12 + 167960*d^9*e^23*x^11 + 184756*
d^10*e^22*x^10 + 167960*d^11*e^21*x^9 + 125970*d^12*e^20*x^8 + 77520*d^13*e^19*x^7 + 38760*d^14*e^18*x^6 + 155
04*d^15*e^17*x^5 + 4845*d^16*e^16*x^4 + 1140*d^17*e^15*x^3 + 190*d^18*e^14*x^2 + 20*d^19*e^13*x + d^20*e^12)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2232 vs. \(2 (440) = 880\).
Time = 0.30 (sec) , antiderivative size = 2232, normalized size of antiderivative = 4.83
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^21,x, algorithm="giac")
[Out]
-1/16628040*(1847560*B*b^10*e^11*x^11 + 2032316*B*b^10*d*e^10*x^10 + 16628040*B*a*b^9*e^11*x^10 + 1662804*A*b^
10*e^11*x^10 + 1847560*B*b^10*d^2*e^9*x^9 + 15116400*B*a*b^9*d*e^10*x^9 + 1511640*A*b^10*d*e^10*x^9 + 68023800
*B*a^2*b^8*e^11*x^9 + 15116400*A*a*b^9*e^11*x^9 + 1385670*B*b^10*d^3*e^8*x^8 + 11337300*B*a*b^9*d^2*e^9*x^8 +
1133730*A*b^10*d^2*e^9*x^8 + 51017850*B*a^2*b^8*d*e^10*x^8 + 11337300*A*a*b^9*d*e^10*x^8 + 166280400*B*a^3*b^7
*e^11*x^8 + 62355150*A*a^2*b^8*e^11*x^8 + 852720*B*b^10*d^4*e^7*x^7 + 6976800*B*a*b^9*d^3*e^8*x^7 + 697680*A*b
^10*d^3*e^8*x^7 + 31395600*B*a^2*b^8*d^2*e^9*x^7 + 6976800*A*a*b^9*d^2*e^9*x^7 + 102326400*B*a^3*b^7*d*e^10*x^
7 + 38372400*A*a^2*b^8*d*e^10*x^7 + 268606800*B*a^4*b^6*e^11*x^7 + 153489600*A*a^3*b^7*e^11*x^7 + 426360*B*b^1
0*d^5*e^6*x^6 + 3488400*B*a*b^9*d^4*e^7*x^6 + 348840*A*b^10*d^4*e^7*x^6 + 15697800*B*a^2*b^8*d^3*e^8*x^6 + 348
8400*A*a*b^9*d^3*e^8*x^6 + 51163200*B*a^3*b^7*d^2*e^9*x^6 + 19186200*A*a^2*b^8*d^2*e^9*x^6 + 134303400*B*a^4*b
^6*d*e^10*x^6 + 76744800*A*a^3*b^7*d*e^10*x^6 + 299304720*B*a^5*b^5*e^11*x^6 + 249420600*A*a^4*b^6*e^11*x^6 +
170544*B*b^10*d^6*e^5*x^5 + 1395360*B*a*b^9*d^5*e^6*x^5 + 139536*A*b^10*d^5*e^6*x^5 + 6279120*B*a^2*b^8*d^4*e^
7*x^5 + 1395360*A*a*b^9*d^4*e^7*x^5 + 20465280*B*a^3*b^7*d^3*e^8*x^5 + 7674480*A*a^2*b^8*d^3*e^8*x^5 + 5372136
0*B*a^4*b^6*d^2*e^9*x^5 + 30697920*A*a^3*b^7*d^2*e^9*x^5 + 119721888*B*a^5*b^5*d*e^10*x^5 + 99768240*A*a^4*b^6
*d*e^10*x^5 + 232792560*B*a^6*b^4*e^11*x^5 + 279351072*A*a^5*b^5*e^11*x^5 + 53295*B*b^10*d^7*e^4*x^4 + 436050*
B*a*b^9*d^6*e^5*x^4 + 43605*A*b^10*d^6*e^5*x^4 + 1962225*B*a^2*b^8*d^5*e^6*x^4 + 436050*A*a*b^9*d^5*e^6*x^4 +
6395400*B*a^3*b^7*d^4*e^7*x^4 + 2398275*A*a^2*b^8*d^4*e^7*x^4 + 16787925*B*a^4*b^6*d^3*e^8*x^4 + 9593100*A*a^3
*b^7*d^3*e^8*x^4 + 37413090*B*a^5*b^5*d^2*e^9*x^4 + 31177575*A*a^4*b^6*d^2*e^9*x^4 + 72747675*B*a^6*b^4*d*e^10
*x^4 + 87297210*A*a^5*b^5*d*e^10*x^4 + 124710300*B*a^7*b^3*e^11*x^4 + 218243025*A*a^6*b^4*e^11*x^4 + 12540*B*b
^10*d^8*e^3*x^3 + 102600*B*a*b^9*d^7*e^4*x^3 + 10260*A*b^10*d^7*e^4*x^3 + 461700*B*a^2*b^8*d^6*e^5*x^3 + 10260
0*A*a*b^9*d^6*e^5*x^3 + 1504800*B*a^3*b^7*d^5*e^6*x^3 + 564300*A*a^2*b^8*d^5*e^6*x^3 + 3950100*B*a^4*b^6*d^4*e
^7*x^3 + 2257200*A*a^3*b^7*d^4*e^7*x^3 + 8803080*B*a^5*b^5*d^3*e^8*x^3 + 7335900*A*a^4*b^6*d^3*e^8*x^3 + 17117
100*B*a^6*b^4*d^2*e^9*x^3 + 20540520*A*a^5*b^5*d^2*e^9*x^3 + 29343600*B*a^7*b^3*d*e^10*x^3 + 51351300*A*a^6*b^
4*d*e^10*x^3 + 44015400*B*a^8*b^2*e^11*x^3 + 117374400*A*a^7*b^3*e^11*x^3 + 2090*B*b^10*d^9*e^2*x^2 + 17100*B*
a*b^9*d^8*e^3*x^2 + 1710*A*b^10*d^8*e^3*x^2 + 76950*B*a^2*b^8*d^7*e^4*x^2 + 17100*A*a*b^9*d^7*e^4*x^2 + 250800
*B*a^3*b^7*d^6*e^5*x^2 + 94050*A*a^2*b^8*d^6*e^5*x^2 + 658350*B*a^4*b^6*d^5*e^6*x^2 + 376200*A*a^3*b^7*d^5*e^6
*x^2 + 1467180*B*a^5*b^5*d^4*e^7*x^2 + 1222650*A*a^4*b^6*d^4*e^7*x^2 + 2852850*B*a^6*b^4*d^3*e^8*x^2 + 3423420
*A*a^5*b^5*d^3*e^8*x^2 + 4890600*B*a^7*b^3*d^2*e^9*x^2 + 8558550*A*a^6*b^4*d^2*e^9*x^2 + 7335900*B*a^8*b^2*d*e
^10*x^2 + 19562400*A*a^7*b^3*d*e^10*x^2 + 9237800*B*a^9*b*e^11*x^2 + 41570100*A*a^8*b^2*e^11*x^2 + 220*B*b^10*
d^10*e*x + 1800*B*a*b^9*d^9*e^2*x + 180*A*b^10*d^9*e^2*x + 8100*B*a^2*b^8*d^8*e^3*x + 1800*A*a*b^9*d^8*e^3*x +
26400*B*a^3*b^7*d^7*e^4*x + 9900*A*a^2*b^8*d^7*e^4*x + 69300*B*a^4*b^6*d^6*e^5*x + 39600*A*a^3*b^7*d^6*e^5*x
+ 154440*B*a^5*b^5*d^5*e^6*x + 128700*A*a^4*b^6*d^5*e^6*x + 300300*B*a^6*b^4*d^4*e^7*x + 360360*A*a^5*b^5*d^4*
e^7*x + 514800*B*a^7*b^3*d^3*e^8*x + 900900*A*a^6*b^4*d^3*e^8*x + 772200*B*a^8*b^2*d^2*e^9*x + 2059200*A*a^7*b
^3*d^2*e^9*x + 972400*B*a^9*b*d*e^10*x + 4375800*A*a^8*b^2*d*e^10*x + 875160*B*a^10*e^11*x + 8751600*A*a^9*b*e
^11*x + 11*B*b^10*d^11 + 90*B*a*b^9*d^10*e + 9*A*b^10*d^10*e + 405*B*a^2*b^8*d^9*e^2 + 90*A*a*b^9*d^9*e^2 + 13
20*B*a^3*b^7*d^8*e^3 + 495*A*a^2*b^8*d^8*e^3 + 3465*B*a^4*b^6*d^7*e^4 + 1980*A*a^3*b^7*d^7*e^4 + 7722*B*a^5*b^
5*d^6*e^5 + 6435*A*a^4*b^6*d^6*e^5 + 15015*B*a^6*b^4*d^5*e^6 + 18018*A*a^5*b^5*d^5*e^6 + 25740*B*a^7*b^3*d^4*e
^7 + 45045*A*a^6*b^4*d^4*e^7 + 38610*B*a^8*b^2*d^3*e^8 + 102960*A*a^7*b^3*d^3*e^8 + 48620*B*a^9*b*d^2*e^9 + 21
8790*A*a^8*b^2*d^2*e^9 + 43758*B*a^10*d*e^10 + 437580*A*a^9*b*d*e^10 + 831402*A*a^10*e^11)/((e*x + d)^20*e^12)
Mupad [B] (verification not implemented)
Time = 2.21 (sec) , antiderivative size = 2110, normalized size of antiderivative = 4.57
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx=\text {Too large to display}
\]
[In]
int(((A + B*x)*(a + b*x)^10)/(d + e*x)^21,x)
[Out]
-((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
*e*x)