3.1093 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx\)

Optimal. Leaf size=464 \[ \frac{b^9 (-10 a B e-A b e+11 b B d)}{11 e^{12} (d+e x)^{11}}-\frac{5 b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{12 e^{12} (d+e x)^{12}}+\frac{15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{13 e^{12} (d+e x)^{13}}-\frac{15 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{7 e^{12} (d+e x)^{14}}+\frac{14 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{5 e^{12} (d+e x)^{15}}-\frac{21 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{8 e^{12} (d+e x)^{16}}+\frac{30 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{17 e^{12} (d+e x)^{17}}-\frac{5 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{6 e^{12} (d+e x)^{18}}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{19 e^{12} (d+e x)^{19}}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{20 e^{12} (d+e x)^{20}}+\frac{(b d-a e)^{10} (B d-A e)}{21 e^{12} (d+e x)^{21}}-\frac{b^{10} B}{10 e^{12} (d+e x)^{10}} \]

[Out]

((b*d - a*e)^10*(B*d - A*e))/(21*e^12*(d + e*x)^21) - ((b*d - a*e)^9*(11*b*B*d -
10*A*b*e - a*B*e))/(20*e^12*(d + e*x)^20) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*
b*e - 2*a*B*e))/(19*e^12*(d + e*x)^19) - (5*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*
e - 3*a*B*e))/(6*e^12*(d + e*x)^18) + (30*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e
- 4*a*B*e))/(17*e^12*(d + e*x)^17) - (21*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e -
5*a*B*e))/(8*e^12*(d + e*x)^16) + (14*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6
*a*B*e))/(5*e^12*(d + e*x)^15) - (15*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a
*B*e))/(7*e^12*(d + e*x)^14) + (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B
*e))/(13*e^12*(d + e*x)^13) - (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e))
/(12*e^12*(d + e*x)^12) + (b^9*(11*b*B*d - A*b*e - 10*a*B*e))/(11*e^12*(d + e*x)
^11) - (b^10*B)/(10*e^12*(d + e*x)^10)

_______________________________________________________________________________________

Rubi [A]  time = 2.45798, antiderivative size = 464, 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 \[ \frac{b^9 (-10 a B e-A b e+11 b B d)}{11 e^{12} (d+e x)^{11}}-\frac{5 b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{12 e^{12} (d+e x)^{12}}+\frac{15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{13 e^{12} (d+e x)^{13}}-\frac{15 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{7 e^{12} (d+e x)^{14}}+\frac{14 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{5 e^{12} (d+e x)^{15}}-\frac{21 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{8 e^{12} (d+e x)^{16}}+\frac{30 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{17 e^{12} (d+e x)^{17}}-\frac{5 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{6 e^{12} (d+e x)^{18}}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{19 e^{12} (d+e x)^{19}}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{20 e^{12} (d+e x)^{20}}+\frac{(b d-a e)^{10} (B d-A e)}{21 e^{12} (d+e x)^{21}}-\frac{b^{10} B}{10 e^{12} (d+e x)^{10}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/(d + e*x)^22,x]

[Out]

((b*d - a*e)^10*(B*d - A*e))/(21*e^12*(d + e*x)^21) - ((b*d - a*e)^9*(11*b*B*d -
10*A*b*e - a*B*e))/(20*e^12*(d + e*x)^20) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*
b*e - 2*a*B*e))/(19*e^12*(d + e*x)^19) - (5*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*
e - 3*a*B*e))/(6*e^12*(d + e*x)^18) + (30*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e
- 4*a*B*e))/(17*e^12*(d + e*x)^17) - (21*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e -
5*a*B*e))/(8*e^12*(d + e*x)^16) + (14*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6
*a*B*e))/(5*e^12*(d + e*x)^15) - (15*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a
*B*e))/(7*e^12*(d + e*x)^14) + (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B
*e))/(13*e^12*(d + e*x)^13) - (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e))
/(12*e^12*(d + e*x)^12) + (b^9*(11*b*B*d - A*b*e - 10*a*B*e))/(11*e^12*(d + e*x)
^11) - (b^10*B)/(10*e^12*(d + e*x)^10)

_______________________________________________________________________________________

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)**22,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [B]  time = 3.1433, size = 1431, normalized size = 3.08 \[ -\frac{\left (10 A e \left (d^{10}+21 e x d^9+210 e^2 x^2 d^8+1330 e^3 x^3 d^7+5985 e^4 x^4 d^6+20349 e^5 x^5 d^5+54264 e^6 x^6 d^4+116280 e^7 x^7 d^3+203490 e^8 x^8 d^2+293930 e^9 x^9 d+352716 e^{10} x^{10}\right )+11 B \left (d^{11}+21 e x d^{10}+210 e^2 x^2 d^9+1330 e^3 x^3 d^8+5985 e^4 x^4 d^7+20349 e^5 x^5 d^6+54264 e^6 x^6 d^5+116280 e^7 x^7 d^4+203490 e^8 x^8 d^3+293930 e^9 x^9 d^2+352716 e^{10} x^{10} d+352716 e^{11} x^{11}\right )\right ) b^{10}+10 a e \left (11 A e \left (d^9+21 e x d^8+210 e^2 x^2 d^7+1330 e^3 x^3 d^6+5985 e^4 x^4 d^5+20349 e^5 x^5 d^4+54264 e^6 x^6 d^3+116280 e^7 x^7 d^2+203490 e^8 x^8 d+293930 e^9 x^9\right )+10 B \left (d^{10}+21 e x d^9+210 e^2 x^2 d^8+1330 e^3 x^3 d^7+5985 e^4 x^4 d^6+20349 e^5 x^5 d^5+54264 e^6 x^6 d^4+116280 e^7 x^7 d^3+203490 e^8 x^8 d^2+293930 e^9 x^9 d+352716 e^{10} x^{10}\right )\right ) b^9+165 a^2 e^2 \left (4 A e \left (d^8+21 e x d^7+210 e^2 x^2 d^6+1330 e^3 x^3 d^5+5985 e^4 x^4 d^4+20349 e^5 x^5 d^3+54264 e^6 x^6 d^2+116280 e^7 x^7 d+203490 e^8 x^8\right )+3 B \left (d^9+21 e x d^8+210 e^2 x^2 d^7+1330 e^3 x^3 d^6+5985 e^4 x^4 d^5+20349 e^5 x^5 d^4+54264 e^6 x^6 d^3+116280 e^7 x^7 d^2+203490 e^8 x^8 d+293930 e^9 x^9\right )\right ) b^8+220 a^3 e^3 \left (13 A e \left (d^7+21 e x d^6+210 e^2 x^2 d^5+1330 e^3 x^3 d^4+5985 e^4 x^4 d^3+20349 e^5 x^5 d^2+54264 e^6 x^6 d+116280 e^7 x^7\right )+8 B \left (d^8+21 e x d^7+210 e^2 x^2 d^6+1330 e^3 x^3 d^5+5985 e^4 x^4 d^4+20349 e^5 x^5 d^3+54264 e^6 x^6 d^2+116280 e^7 x^7 d+203490 e^8 x^8\right )\right ) b^7+5005 a^4 e^4 \left (2 A e \left (d^6+21 e x d^5+210 e^2 x^2 d^4+1330 e^3 x^3 d^3+5985 e^4 x^4 d^2+20349 e^5 x^5 d+54264 e^6 x^6\right )+B \left (d^7+21 e x d^6+210 e^2 x^2 d^5+1330 e^3 x^3 d^4+5985 e^4 x^4 d^3+20349 e^5 x^5 d^2+54264 e^6 x^6 d+116280 e^7 x^7\right )\right ) b^6+6006 a^5 e^5 \left (5 A e \left (d^5+21 e x d^4+210 e^2 x^2 d^3+1330 e^3 x^3 d^2+5985 e^4 x^4 d+20349 e^5 x^5\right )+2 B \left (d^6+21 e x d^5+210 e^2 x^2 d^4+1330 e^3 x^3 d^3+5985 e^4 x^4 d^2+20349 e^5 x^5 d+54264 e^6 x^6\right )\right ) b^5+5005 a^6 e^6 \left (16 A e \left (d^4+21 e x d^3+210 e^2 x^2 d^2+1330 e^3 x^3 d+5985 e^4 x^4\right )+5 B \left (d^5+21 e x d^4+210 e^2 x^2 d^3+1330 e^3 x^3 d^2+5985 e^4 x^4 d+20349 e^5 x^5\right )\right ) b^4+11440 a^7 e^7 \left (17 A e \left (d^3+21 e x d^2+210 e^2 x^2 d+1330 e^3 x^3\right )+4 B \left (d^4+21 e x d^3+210 e^2 x^2 d^2+1330 e^3 x^3 d+5985 e^4 x^4\right )\right ) b^3+72930 a^8 e^8 \left (6 A e \left (d^2+21 e x d+210 e^2 x^2\right )+B \left (d^3+21 e x d^2+210 e^2 x^2 d+1330 e^3 x^3\right )\right ) b^2+48620 a^9 e^9 \left (19 A e (d+21 e x)+2 B \left (d^2+21 e x d+210 e^2 x^2\right )\right ) b+92378 a^{10} e^{10} (20 A e+B (d+21 e x))}{38798760 e^{12} (d+e x)^{21}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^22,x]

[Out]

-(92378*a^10*e^10*(20*A*e + B*(d + 21*e*x)) + 48620*a^9*b*e^9*(19*A*e*(d + 21*e*
x) + 2*B*(d^2 + 21*d*e*x + 210*e^2*x^2)) + 72930*a^8*b^2*e^8*(6*A*e*(d^2 + 21*d*
e*x + 210*e^2*x^2) + B*(d^3 + 21*d^2*e*x + 210*d*e^2*x^2 + 1330*e^3*x^3)) + 1144
0*a^7*b^3*e^7*(17*A*e*(d^3 + 21*d^2*e*x + 210*d*e^2*x^2 + 1330*e^3*x^3) + 4*B*(d
^4 + 21*d^3*e*x + 210*d^2*e^2*x^2 + 1330*d*e^3*x^3 + 5985*e^4*x^4)) + 5005*a^6*b
^4*e^6*(16*A*e*(d^4 + 21*d^3*e*x + 210*d^2*e^2*x^2 + 1330*d*e^3*x^3 + 5985*e^4*x
^4) + 5*B*(d^5 + 21*d^4*e*x + 210*d^3*e^2*x^2 + 1330*d^2*e^3*x^3 + 5985*d*e^4*x^
4 + 20349*e^5*x^5)) + 6006*a^5*b^5*e^5*(5*A*e*(d^5 + 21*d^4*e*x + 210*d^3*e^2*x^
2 + 1330*d^2*e^3*x^3 + 5985*d*e^4*x^4 + 20349*e^5*x^5) + 2*B*(d^6 + 21*d^5*e*x +
210*d^4*e^2*x^2 + 1330*d^3*e^3*x^3 + 5985*d^2*e^4*x^4 + 20349*d*e^5*x^5 + 54264
*e^6*x^6)) + 5005*a^4*b^6*e^4*(2*A*e*(d^6 + 21*d^5*e*x + 210*d^4*e^2*x^2 + 1330*
d^3*e^3*x^3 + 5985*d^2*e^4*x^4 + 20349*d*e^5*x^5 + 54264*e^6*x^6) + B*(d^7 + 21*
d^6*e*x + 210*d^5*e^2*x^2 + 1330*d^4*e^3*x^3 + 5985*d^3*e^4*x^4 + 20349*d^2*e^5*
x^5 + 54264*d*e^6*x^6 + 116280*e^7*x^7)) + 220*a^3*b^7*e^3*(13*A*e*(d^7 + 21*d^6
*e*x + 210*d^5*e^2*x^2 + 1330*d^4*e^3*x^3 + 5985*d^3*e^4*x^4 + 20349*d^2*e^5*x^5
+ 54264*d*e^6*x^6 + 116280*e^7*x^7) + 8*B*(d^8 + 21*d^7*e*x + 210*d^6*e^2*x^2 +
1330*d^5*e^3*x^3 + 5985*d^4*e^4*x^4 + 20349*d^3*e^5*x^5 + 54264*d^2*e^6*x^6 + 1
16280*d*e^7*x^7 + 203490*e^8*x^8)) + 165*a^2*b^8*e^2*(4*A*e*(d^8 + 21*d^7*e*x +
210*d^6*e^2*x^2 + 1330*d^5*e^3*x^3 + 5985*d^4*e^4*x^4 + 20349*d^3*e^5*x^5 + 5426
4*d^2*e^6*x^6 + 116280*d*e^7*x^7 + 203490*e^8*x^8) + 3*B*(d^9 + 21*d^8*e*x + 210
*d^7*e^2*x^2 + 1330*d^6*e^3*x^3 + 5985*d^5*e^4*x^4 + 20349*d^4*e^5*x^5 + 54264*d
^3*e^6*x^6 + 116280*d^2*e^7*x^7 + 203490*d*e^8*x^8 + 293930*e^9*x^9)) + 10*a*b^9
*e*(11*A*e*(d^9 + 21*d^8*e*x + 210*d^7*e^2*x^2 + 1330*d^6*e^3*x^3 + 5985*d^5*e^4
*x^4 + 20349*d^4*e^5*x^5 + 54264*d^3*e^6*x^6 + 116280*d^2*e^7*x^7 + 203490*d*e^8
*x^8 + 293930*e^9*x^9) + 10*B*(d^10 + 21*d^9*e*x + 210*d^8*e^2*x^2 + 1330*d^7*e^
3*x^3 + 5985*d^6*e^4*x^4 + 20349*d^5*e^5*x^5 + 54264*d^4*e^6*x^6 + 116280*d^3*e^
7*x^7 + 203490*d^2*e^8*x^8 + 293930*d*e^9*x^9 + 352716*e^10*x^10)) + b^10*(10*A*
e*(d^10 + 21*d^9*e*x + 210*d^8*e^2*x^2 + 1330*d^7*e^3*x^3 + 5985*d^6*e^4*x^4 + 2
0349*d^5*e^5*x^5 + 54264*d^4*e^6*x^6 + 116280*d^3*e^7*x^7 + 203490*d^2*e^8*x^8 +
293930*d*e^9*x^9 + 352716*e^10*x^10) + 11*B*(d^11 + 21*d^10*e*x + 210*d^9*e^2*x
^2 + 1330*d^8*e^3*x^3 + 5985*d^7*e^4*x^4 + 20349*d^6*e^5*x^5 + 54264*d^5*e^6*x^6
+ 116280*d^4*e^7*x^7 + 203490*d^3*e^8*x^8 + 293930*d^2*e^9*x^9 + 352716*d*e^10*
x^10 + 352716*e^11*x^11)))/(38798760*e^12*(d + e*x)^21)

_______________________________________________________________________________________

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)^22,x)

[Out]

-15/7*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)^14-5/6*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^6-28
0*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)^18-1/10*b^10*B/e^12/(e*x+d)^10-5/12*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)^12-
1/11*b^9*(A*b*e+10*B*a*e-11*B*b*d)/e^12/(e*x+d)^11-5/19*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^5-50
4*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)^19-21/8*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^4-160*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^12/(e*x+d)^16-30/17*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)^17-15/13*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)^13-1/21*(A*a^10*e^11-10*A*a^9*b*d*e^10+45*A*a^8*b^2*d^2*e^9-12
0*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^7-21
0*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)^21-1/20*(
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+1
260*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^12/(e*x+d)^20-14/5*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)^15

_______________________________________________________________________________________

Maxima [A]  time = 1.60397, size = 2753, 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)^22,x, algorithm="maxima")

[Out]

-1/38798760*(3879876*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 1847560*A*a^10*e^11 + 1
0*(10*B*a*b^9 + A*b^10)*d^10*e + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 220*(8*B
*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 2002
*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6
+ 11440*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 24310*(3*B*a^8*b^2 + 8*A*a^7*b^3)*
d^3*e^8 + 48620*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 92378*(B*a^10 + 10*A*a^9*b)*
d*e^10 + 352716*(11*B*b^10*d*e^10 + 10*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 293930
*(11*B*b^10*d^2*e^9 + 10*(10*B*a*b^9 + A*b^10)*d*e^10 + 55*(9*B*a^2*b^8 + 2*A*a*
b^9)*e^11)*x^9 + 203490*(11*B*b^10*d^3*e^8 + 10*(10*B*a*b^9 + A*b^10)*d^2*e^9 +
55*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8
+ 116280*(11*B*b^10*d^4*e^7 + 10*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 55*(9*B*a^2*b^8
+ 2*A*a*b^9)*d^2*e^9 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 715*(7*B*a^4*b^
6 + 4*A*a^3*b^7)*e^11)*x^7 + 54264*(11*B*b^10*d^5*e^6 + 10*(10*B*a*b^9 + A*b^10)
*d^4*e^7 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8
)*d^2*e^9 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 2002*(6*B*a^5*b^5 + 5*A*a^4
*b^6)*e^11)*x^6 + 20349*(11*B*b^10*d^6*e^5 + 10*(10*B*a*b^9 + A*b^10)*d^5*e^6 +
55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 +
715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^
10 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 5985*(11*B*b^10*d^7*e^4 + 10*(
10*B*a*b^9 + A*b^10)*d^6*e^5 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 220*(8*B*a
^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 2002*(
6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 1
1440*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 1330*(11*B*b^10*d^8*e^3 + 10*(10*B*
a*b^9 + A*b^10)*d^7*e^4 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 220*(8*B*a^3*b^
7 + 3*A*a^2*b^8)*d^5*e^6 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 2002*(6*B*a
^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 11440
*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 24310*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^
3 + 210*(11*B*b^10*d^9*e^2 + 10*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 55*(9*B*a^2*b^8
+ 2*A*a*b^9)*d^7*e^4 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 715*(7*B*a^4*b^
6 + 4*A*a^3*b^7)*d^5*e^6 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 5005*(5*B*
a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 11440*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 243
10*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 48620*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^
2 + 21*(11*B*b^10*d^10*e + 10*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 55*(9*B*a^2*b^8 +
2*A*a*b^9)*d^8*e^3 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 715*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*d^6*e^5 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 5005*(5*B*a^
6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 11440*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 24310
*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 48620*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 +
92378*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^33*x^21 + 21*d*e^32*x^20 + 210*d^2*e^31*
x^19 + 1330*d^3*e^30*x^18 + 5985*d^4*e^29*x^17 + 20349*d^5*e^28*x^16 + 54264*d^6
*e^27*x^15 + 116280*d^7*e^26*x^14 + 203490*d^8*e^25*x^13 + 293930*d^9*e^24*x^12
+ 352716*d^10*e^23*x^11 + 352716*d^11*e^22*x^10 + 293930*d^12*e^21*x^9 + 203490*
d^13*e^20*x^8 + 116280*d^14*e^19*x^7 + 54264*d^15*e^18*x^6 + 20349*d^16*e^17*x^5
+ 5985*d^17*e^16*x^4 + 1330*d^18*e^15*x^3 + 210*d^19*e^14*x^2 + 21*d^20*e^13*x
+ d^21*e^12)

_______________________________________________________________________________________

Fricas [A]  time = 0.22005, size = 2753, 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)^22,x, algorithm="fricas")

[Out]

-1/38798760*(3879876*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 1847560*A*a^10*e^11 + 1
0*(10*B*a*b^9 + A*b^10)*d^10*e + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 220*(8*B
*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 2002
*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6
+ 11440*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 24310*(3*B*a^8*b^2 + 8*A*a^7*b^3)*
d^3*e^8 + 48620*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 92378*(B*a^10 + 10*A*a^9*b)*
d*e^10 + 352716*(11*B*b^10*d*e^10 + 10*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 293930
*(11*B*b^10*d^2*e^9 + 10*(10*B*a*b^9 + A*b^10)*d*e^10 + 55*(9*B*a^2*b^8 + 2*A*a*
b^9)*e^11)*x^9 + 203490*(11*B*b^10*d^3*e^8 + 10*(10*B*a*b^9 + A*b^10)*d^2*e^9 +
55*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8
+ 116280*(11*B*b^10*d^4*e^7 + 10*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 55*(9*B*a^2*b^8
+ 2*A*a*b^9)*d^2*e^9 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 715*(7*B*a^4*b^
6 + 4*A*a^3*b^7)*e^11)*x^7 + 54264*(11*B*b^10*d^5*e^6 + 10*(10*B*a*b^9 + A*b^10)
*d^4*e^7 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8
)*d^2*e^9 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 2002*(6*B*a^5*b^5 + 5*A*a^4
*b^6)*e^11)*x^6 + 20349*(11*B*b^10*d^6*e^5 + 10*(10*B*a*b^9 + A*b^10)*d^5*e^6 +
55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 +
715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^
10 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 5985*(11*B*b^10*d^7*e^4 + 10*(
10*B*a*b^9 + A*b^10)*d^6*e^5 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 220*(8*B*a
^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 2002*(
6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 1
1440*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 1330*(11*B*b^10*d^8*e^3 + 10*(10*B*
a*b^9 + A*b^10)*d^7*e^4 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 220*(8*B*a^3*b^
7 + 3*A*a^2*b^8)*d^5*e^6 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 2002*(6*B*a
^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 11440
*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 24310*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^
3 + 210*(11*B*b^10*d^9*e^2 + 10*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 55*(9*B*a^2*b^8
+ 2*A*a*b^9)*d^7*e^4 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 715*(7*B*a^4*b^
6 + 4*A*a^3*b^7)*d^5*e^6 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 5005*(5*B*
a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 11440*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 243
10*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 48620*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^
2 + 21*(11*B*b^10*d^10*e + 10*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 55*(9*B*a^2*b^8 +
2*A*a*b^9)*d^8*e^3 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 715*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*d^6*e^5 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 5005*(5*B*a^
6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 11440*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 24310
*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 48620*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 +
92378*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^33*x^21 + 21*d*e^32*x^20 + 210*d^2*e^31*
x^19 + 1330*d^3*e^30*x^18 + 5985*d^4*e^29*x^17 + 20349*d^5*e^28*x^16 + 54264*d^6
*e^27*x^15 + 116280*d^7*e^26*x^14 + 203490*d^8*e^25*x^13 + 293930*d^9*e^24*x^12
+ 352716*d^10*e^23*x^11 + 352716*d^11*e^22*x^10 + 293930*d^12*e^21*x^9 + 203490*
d^13*e^20*x^8 + 116280*d^14*e^19*x^7 + 54264*d^15*e^18*x^6 + 20349*d^16*e^17*x^5
+ 5985*d^17*e^16*x^4 + 1330*d^18*e^15*x^3 + 210*d^19*e^14*x^2 + 21*d^20*e^13*x
+ d^21*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)**22,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214007, 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)^22,x, algorithm="giac")

[Out]

Done