3.1092 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{21}} \, dx\)

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

[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))/(1
1*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)

_______________________________________________________________________________________

Rubi [A]  time = 2.48199, antiderivative size = 462, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{b^9 (-10 a B e-A b e+11 b B d)}{10 e^{12} (d+e x)^{10}}-\frac{5 b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{11 e^{12} (d+e x)^{11}}+\frac{5 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{4 e^{12} (d+e x)^{12}}-\frac{30 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{13 e^{12} (d+e x)^{13}}+\frac{3 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)^{14}}-\frac{14 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{5 e^{12} (d+e x)^{15}}+\frac{15 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{8 e^{12} (d+e x)^{16}}-\frac{15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{17 e^{12} (d+e x)^{17}}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{18 e^{12} (d+e x)^{18}}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{19 e^{12} (d+e x)^{19}}+\frac{(b d-a e)^{10} (B d-A e)}{20 e^{12} (d+e x)^{20}}-\frac{b^{10} B}{9 e^{12} (d+e x)^9} \]

Antiderivative was successfully verified.

[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))/(1
1*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)

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/(e*x+d)**21,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [B]  time = 3.77445, size = 1428, normalized size = 3.09 \[ -\frac{\left (9 A e \left (d^{10}+20 e x d^9+190 e^2 x^2 d^8+1140 e^3 x^3 d^7+4845 e^4 x^4 d^6+15504 e^5 x^5 d^5+38760 e^6 x^6 d^4+77520 e^7 x^7 d^3+125970 e^8 x^8 d^2+167960 e^9 x^9 d+184756 e^{10} x^{10}\right )+11 B \left (d^{11}+20 e x d^{10}+190 e^2 x^2 d^9+1140 e^3 x^3 d^8+4845 e^4 x^4 d^7+15504 e^5 x^5 d^6+38760 e^6 x^6 d^5+77520 e^7 x^7 d^4+125970 e^8 x^8 d^3+167960 e^9 x^9 d^2+184756 e^{10} x^{10} d+167960 e^{11} x^{11}\right )\right ) b^{10}+90 a e \left (A e \left (d^9+20 e x d^8+190 e^2 x^2 d^7+1140 e^3 x^3 d^6+4845 e^4 x^4 d^5+15504 e^5 x^5 d^4+38760 e^6 x^6 d^3+77520 e^7 x^7 d^2+125970 e^8 x^8 d+167960 e^9 x^9\right )+B \left (d^{10}+20 e x d^9+190 e^2 x^2 d^8+1140 e^3 x^3 d^7+4845 e^4 x^4 d^6+15504 e^5 x^5 d^5+38760 e^6 x^6 d^4+77520 e^7 x^7 d^3+125970 e^8 x^8 d^2+167960 e^9 x^9 d+184756 e^{10} x^{10}\right )\right ) b^9+45 a^2 e^2 \left (11 A e \left (d^8+20 e x d^7+190 e^2 x^2 d^6+1140 e^3 x^3 d^5+4845 e^4 x^4 d^4+15504 e^5 x^5 d^3+38760 e^6 x^6 d^2+77520 e^7 x^7 d+125970 e^8 x^8\right )+9 B \left (d^9+20 e x d^8+190 e^2 x^2 d^7+1140 e^3 x^3 d^6+4845 e^4 x^4 d^5+15504 e^5 x^5 d^4+38760 e^6 x^6 d^3+77520 e^7 x^7 d^2+125970 e^8 x^8 d+167960 e^9 x^9\right )\right ) b^8+660 a^3 e^3 \left (3 A e \left (d^7+20 e x d^6+190 e^2 x^2 d^5+1140 e^3 x^3 d^4+4845 e^4 x^4 d^3+15504 e^5 x^5 d^2+38760 e^6 x^6 d+77520 e^7 x^7\right )+2 B \left (d^8+20 e x d^7+190 e^2 x^2 d^6+1140 e^3 x^3 d^5+4845 e^4 x^4 d^4+15504 e^5 x^5 d^3+38760 e^6 x^6 d^2+77520 e^7 x^7 d+125970 e^8 x^8\right )\right ) b^7+495 a^4 e^4 \left (13 A e \left (d^6+20 e x d^5+190 e^2 x^2 d^4+1140 e^3 x^3 d^3+4845 e^4 x^4 d^2+15504 e^5 x^5 d+38760 e^6 x^6\right )+7 B \left (d^7+20 e x d^6+190 e^2 x^2 d^5+1140 e^3 x^3 d^4+4845 e^4 x^4 d^3+15504 e^5 x^5 d^2+38760 e^6 x^6 d+77520 e^7 x^7\right )\right ) b^6+2574 a^5 e^5 \left (7 A e \left (d^5+20 e x d^4+190 e^2 x^2 d^3+1140 e^3 x^3 d^2+4845 e^4 x^4 d+15504 e^5 x^5\right )+3 B \left (d^6+20 e x d^5+190 e^2 x^2 d^4+1140 e^3 x^3 d^3+4845 e^4 x^4 d^2+15504 e^5 x^5 d+38760 e^6 x^6\right )\right ) b^5+15015 a^6 e^6 \left (3 A e \left (d^4+20 e x d^3+190 e^2 x^2 d^2+1140 e^3 x^3 d+4845 e^4 x^4\right )+B \left (d^5+20 e x d^4+190 e^2 x^2 d^3+1140 e^3 x^3 d^2+4845 e^4 x^4 d+15504 e^5 x^5\right )\right ) b^4+25740 a^7 e^7 \left (4 A e \left (d^3+20 e x d^2+190 e^2 x^2 d+1140 e^3 x^3\right )+B \left (d^4+20 e x d^3+190 e^2 x^2 d^2+1140 e^3 x^3 d+4845 e^4 x^4\right )\right ) b^3+12870 a^8 e^8 \left (17 A e \left (d^2+20 e x d+190 e^2 x^2\right )+3 B \left (d^3+20 e x d^2+190 e^2 x^2 d+1140 e^3 x^3\right )\right ) b^2+48620 a^9 e^9 \left (9 A e (d+20 e x)+B \left (d^2+20 e x d+190 e^2 x^2\right )\right ) b+43758 a^{10} e^{10} (19 A e+B (d+20 e x))}{16628040 e^{12} (d+e x)^{20}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^21,x]

[Out]

-(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)) + 2574
0*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 + 1
5504*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 + 1
140*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 + 38
760*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 + 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) + 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)))/(16628040*e^12*(d + e*x)^20)

_______________________________________________________________________________________

Maple [B]  time = 0.017, size = 1942, normalized size = 4.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/(e*x+d)^21,x)

[Out]

-3*b^5*(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^12/(e*x+d)^14-5/18*b*(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^12/(e*x+d)^18-1/9*b^10*B/e^12/(e*x+d)^9-1/10*b^9*
(A*b*e+10*B*a*e-11*B*b*d)/e^12/(e*x+d)^10-5/4*b^7*(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^12/
(e*x+d)^12-5/11*b^8*(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^12/(e*x+d)^11-1/19*(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^1
0-20*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^12/(e*x+d)^19-15/8*b^3*(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^12/(e*x+d)^16-15/17*b^2*(8*A*a^7*b*e^8-56*
A*a^6*b^2*d*e^7+168*A*a^5*b^3*d^2*e^6-280*A*a^4*b^4*d^3*e^5+280*A*a^3*b^5*d^4*e^
4-168*A*a^2*b^6*d^5*e^3+56*A*a*b^7*d^6*e^2-8*A*b^8*d^7*e+3*B*a^8*e^8-32*B*a^7*b*
d*e^7+140*B*a^6*b^2*d^2*e^6-336*B*a^5*b^3*d^3*e^5+490*B*a^4*b^4*d^4*e^4-448*B*a^
3*b^5*d^5*e^3+252*B*a^2*b^6*d^6*e^2-80*B*a*b^7*d^7*e+11*B*b^8*d^8)/e^12/(e*x+d)^
17-30/13*b^6*(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^12/(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)^20-14/5*b^4*(
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^4-1
60*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^12/(
e*x+d)^15

_______________________________________________________________________________________

Maxima [A]  time = 1.6089, size = 2738, normalized size = 5.93 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(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 + 7752
0*(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^1
1)*x^6 + 15504*(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^1
0)*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 + 6435*(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^1
0*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 + 12870*(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 + 15504*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)

_______________________________________________________________________________________

Fricas [A]  time = 0.216516, size = 2738, normalized size = 5.93 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(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 + 7752
0*(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^1
1)*x^6 + 15504*(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^1
0)*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 + 6435*(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^1
0*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 + 12870*(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 + 15504*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)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10*(B*x+A)/(e*x+d)**21,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21756, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done