3.1089 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx\)

Optimal. Leaf size=335 \[ \frac{b^5 (a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{816816 e (d+e x)^{11} (b d-a e)^7}+\frac{b^4 (a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{74256 e (d+e x)^{12} (b d-a e)^6}+\frac{b^3 (a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{12376 e (d+e x)^{13} (b d-a e)^5}+\frac{b^2 (a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{2856 e (d+e x)^{14} (b d-a e)^4}+\frac{b (a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{816 e (d+e x)^{15} (b d-a e)^3}+\frac{(a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{272 e (d+e x)^{16} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{17 e (d+e x)^{17} (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(17*e*(b*d - a*e)*(d + e*x)^17) + ((11*b*B*d + 6*A*b
*e - 17*a*B*e)*(a + b*x)^11)/(272*e*(b*d - a*e)^2*(d + e*x)^16) + (b*(11*b*B*d +
6*A*b*e - 17*a*B*e)*(a + b*x)^11)/(816*e*(b*d - a*e)^3*(d + e*x)^15) + (b^2*(11
*b*B*d + 6*A*b*e - 17*a*B*e)*(a + b*x)^11)/(2856*e*(b*d - a*e)^4*(d + e*x)^14) +
(b^3*(11*b*B*d + 6*A*b*e - 17*a*B*e)*(a + b*x)^11)/(12376*e*(b*d - a*e)^5*(d +
e*x)^13) + (b^4*(11*b*B*d + 6*A*b*e - 17*a*B*e)*(a + b*x)^11)/(74256*e*(b*d - a*
e)^6*(d + e*x)^12) + (b^5*(11*b*B*d + 6*A*b*e - 17*a*B*e)*(a + b*x)^11)/(816816*
e*(b*d - a*e)^7*(d + e*x)^11)

_______________________________________________________________________________________

Rubi [A]  time = 0.463332, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{b^5 (a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{816816 e (d+e x)^{11} (b d-a e)^7}+\frac{b^4 (a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{74256 e (d+e x)^{12} (b d-a e)^6}+\frac{b^3 (a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{12376 e (d+e x)^{13} (b d-a e)^5}+\frac{b^2 (a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{2856 e (d+e x)^{14} (b d-a e)^4}+\frac{b (a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{816 e (d+e x)^{15} (b d-a e)^3}+\frac{(a+b x)^{11} (-17 a B e+6 A b e+11 b B d)}{272 e (d+e x)^{16} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{17 e (d+e x)^{17} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/(d + e*x)^18,x]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(17*e*(b*d - a*e)*(d + e*x)^17) + ((11*b*B*d + 6*A*b
*e - 17*a*B*e)*(a + b*x)^11)/(272*e*(b*d - a*e)^2*(d + e*x)^16) + (b*(11*b*B*d +
6*A*b*e - 17*a*B*e)*(a + b*x)^11)/(816*e*(b*d - a*e)^3*(d + e*x)^15) + (b^2*(11
*b*B*d + 6*A*b*e - 17*a*B*e)*(a + b*x)^11)/(2856*e*(b*d - a*e)^4*(d + e*x)^14) +
(b^3*(11*b*B*d + 6*A*b*e - 17*a*B*e)*(a + b*x)^11)/(12376*e*(b*d - a*e)^5*(d +
e*x)^13) + (b^4*(11*b*B*d + 6*A*b*e - 17*a*B*e)*(a + b*x)^11)/(74256*e*(b*d - a*
e)^6*(d + e*x)^12) + (b^5*(11*b*B*d + 6*A*b*e - 17*a*B*e)*(a + b*x)^11)/(816816*
e*(b*d - a*e)^7*(d + e*x)^11)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 83.7477, size = 320, normalized size = 0.96 \[ - \frac{b^{5} \left (a + b x\right )^{11} \left (6 A b e - 17 B a e + 11 B b d\right )}{816816 e \left (d + e x\right )^{11} \left (a e - b d\right )^{7}} + \frac{b^{4} \left (a + b x\right )^{11} \left (6 A b e - 17 B a e + 11 B b d\right )}{74256 e \left (d + e x\right )^{12} \left (a e - b d\right )^{6}} - \frac{b^{3} \left (a + b x\right )^{11} \left (6 A b e - 17 B a e + 11 B b d\right )}{12376 e \left (d + e x\right )^{13} \left (a e - b d\right )^{5}} + \frac{b^{2} \left (a + b x\right )^{11} \left (6 A b e - 17 B a e + 11 B b d\right )}{2856 e \left (d + e x\right )^{14} \left (a e - b d\right )^{4}} - \frac{b \left (a + b x\right )^{11} \left (6 A b e - 17 B a e + 11 B b d\right )}{816 e \left (d + e x\right )^{15} \left (a e - b d\right )^{3}} + \frac{\left (a + b x\right )^{11} \left (6 A b e - 17 B a e + 11 B b d\right )}{272 e \left (d + e x\right )^{16} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{11} \left (A e - B d\right )}{17 e \left (d + e x\right )^{17} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/(e*x+d)**18,x)

[Out]

-b**5*(a + b*x)**11*(6*A*b*e - 17*B*a*e + 11*B*b*d)/(816816*e*(d + e*x)**11*(a*e
- b*d)**7) + b**4*(a + b*x)**11*(6*A*b*e - 17*B*a*e + 11*B*b*d)/(74256*e*(d + e
*x)**12*(a*e - b*d)**6) - b**3*(a + b*x)**11*(6*A*b*e - 17*B*a*e + 11*B*b*d)/(12
376*e*(d + e*x)**13*(a*e - b*d)**5) + b**2*(a + b*x)**11*(6*A*b*e - 17*B*a*e + 1
1*B*b*d)/(2856*e*(d + e*x)**14*(a*e - b*d)**4) - b*(a + b*x)**11*(6*A*b*e - 17*B
*a*e + 11*B*b*d)/(816*e*(d + e*x)**15*(a*e - b*d)**3) + (a + b*x)**11*(6*A*b*e -
17*B*a*e + 11*B*b*d)/(272*e*(d + e*x)**16*(a*e - b*d)**2) - (a + b*x)**11*(A*e
- B*d)/(17*e*(d + e*x)**17*(a*e - b*d))

_______________________________________________________________________________________

Mathematica [B]  time = 3.99291, size = 1433, normalized size = 4.28 \[ -\frac{\left (6 A e \left (d^{10}+17 e x d^9+136 e^2 x^2 d^8+680 e^3 x^3 d^7+2380 e^4 x^4 d^6+6188 e^5 x^5 d^5+12376 e^6 x^6 d^4+19448 e^7 x^7 d^3+24310 e^8 x^8 d^2+24310 e^9 x^9 d+19448 e^{10} x^{10}\right )+11 B \left (d^{11}+17 e x d^{10}+136 e^2 x^2 d^9+680 e^3 x^3 d^8+2380 e^4 x^4 d^7+6188 e^5 x^5 d^6+12376 e^6 x^6 d^5+19448 e^7 x^7 d^4+24310 e^8 x^8 d^3+24310 e^9 x^9 d^2+19448 e^{10} x^{10} d+12376 e^{11} x^{11}\right )\right ) b^{10}+6 a e \left (7 A e \left (d^9+17 e x d^8+136 e^2 x^2 d^7+680 e^3 x^3 d^6+2380 e^4 x^4 d^5+6188 e^5 x^5 d^4+12376 e^6 x^6 d^3+19448 e^7 x^7 d^2+24310 e^8 x^8 d+24310 e^9 x^9\right )+10 B \left (d^{10}+17 e x d^9+136 e^2 x^2 d^8+680 e^3 x^3 d^7+2380 e^4 x^4 d^6+6188 e^5 x^5 d^5+12376 e^6 x^6 d^4+19448 e^7 x^7 d^3+24310 e^8 x^8 d^2+24310 e^9 x^9 d+19448 e^{10} x^{10}\right )\right ) b^9+21 a^2 e^2 \left (8 A e \left (d^8+17 e x d^7+136 e^2 x^2 d^6+680 e^3 x^3 d^5+2380 e^4 x^4 d^4+6188 e^5 x^5 d^3+12376 e^6 x^6 d^2+19448 e^7 x^7 d+24310 e^8 x^8\right )+9 B \left (d^9+17 e x d^8+136 e^2 x^2 d^7+680 e^3 x^3 d^6+2380 e^4 x^4 d^5+6188 e^5 x^5 d^4+12376 e^6 x^6 d^3+19448 e^7 x^7 d^2+24310 e^8 x^8 d+24310 e^9 x^9\right )\right ) b^8+56 a^3 e^3 \left (9 A e \left (d^7+17 e x d^6+136 e^2 x^2 d^5+680 e^3 x^3 d^4+2380 e^4 x^4 d^3+6188 e^5 x^5 d^2+12376 e^6 x^6 d+19448 e^7 x^7\right )+8 B \left (d^8+17 e x d^7+136 e^2 x^2 d^6+680 e^3 x^3 d^5+2380 e^4 x^4 d^4+6188 e^5 x^5 d^3+12376 e^6 x^6 d^2+19448 e^7 x^7 d+24310 e^8 x^8\right )\right ) b^7+126 a^4 e^4 \left (10 A e \left (d^6+17 e x d^5+136 e^2 x^2 d^4+680 e^3 x^3 d^3+2380 e^4 x^4 d^2+6188 e^5 x^5 d+12376 e^6 x^6\right )+7 B \left (d^7+17 e x d^6+136 e^2 x^2 d^5+680 e^3 x^3 d^4+2380 e^4 x^4 d^3+6188 e^5 x^5 d^2+12376 e^6 x^6 d+19448 e^7 x^7\right )\right ) b^6+252 a^5 e^5 \left (11 A e \left (d^5+17 e x d^4+136 e^2 x^2 d^3+680 e^3 x^3 d^2+2380 e^4 x^4 d+6188 e^5 x^5\right )+6 B \left (d^6+17 e x d^5+136 e^2 x^2 d^4+680 e^3 x^3 d^3+2380 e^4 x^4 d^2+6188 e^5 x^5 d+12376 e^6 x^6\right )\right ) b^5+462 a^6 e^6 \left (12 A e \left (d^4+17 e x d^3+136 e^2 x^2 d^2+680 e^3 x^3 d+2380 e^4 x^4\right )+5 B \left (d^5+17 e x d^4+136 e^2 x^2 d^3+680 e^3 x^3 d^2+2380 e^4 x^4 d+6188 e^5 x^5\right )\right ) b^4+792 a^7 e^7 \left (13 A e \left (d^3+17 e x d^2+136 e^2 x^2 d+680 e^3 x^3\right )+4 B \left (d^4+17 e x d^3+136 e^2 x^2 d^2+680 e^3 x^3 d+2380 e^4 x^4\right )\right ) b^3+1287 a^8 e^8 \left (14 A e \left (d^2+17 e x d+136 e^2 x^2\right )+3 B \left (d^3+17 e x d^2+136 e^2 x^2 d+680 e^3 x^3\right )\right ) b^2+2002 a^9 e^9 \left (15 A e (d+17 e x)+2 B \left (d^2+17 e x d+136 e^2 x^2\right )\right ) b+3003 a^{10} e^{10} (16 A e+B (d+17 e x))}{816816 e^{12} (d+e x)^{17}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^18,x]

[Out]

-(3003*a^10*e^10*(16*A*e + B*(d + 17*e*x)) + 2002*a^9*b*e^9*(15*A*e*(d + 17*e*x)
+ 2*B*(d^2 + 17*d*e*x + 136*e^2*x^2)) + 1287*a^8*b^2*e^8*(14*A*e*(d^2 + 17*d*e*
x + 136*e^2*x^2) + 3*B*(d^3 + 17*d^2*e*x + 136*d*e^2*x^2 + 680*e^3*x^3)) + 792*a
^7*b^3*e^7*(13*A*e*(d^3 + 17*d^2*e*x + 136*d*e^2*x^2 + 680*e^3*x^3) + 4*B*(d^4 +
17*d^3*e*x + 136*d^2*e^2*x^2 + 680*d*e^3*x^3 + 2380*e^4*x^4)) + 462*a^6*b^4*e^6
*(12*A*e*(d^4 + 17*d^3*e*x + 136*d^2*e^2*x^2 + 680*d*e^3*x^3 + 2380*e^4*x^4) + 5
*B*(d^5 + 17*d^4*e*x + 136*d^3*e^2*x^2 + 680*d^2*e^3*x^3 + 2380*d*e^4*x^4 + 6188
*e^5*x^5)) + 252*a^5*b^5*e^5*(11*A*e*(d^5 + 17*d^4*e*x + 136*d^3*e^2*x^2 + 680*d
^2*e^3*x^3 + 2380*d*e^4*x^4 + 6188*e^5*x^5) + 6*B*(d^6 + 17*d^5*e*x + 136*d^4*e^
2*x^2 + 680*d^3*e^3*x^3 + 2380*d^2*e^4*x^4 + 6188*d*e^5*x^5 + 12376*e^6*x^6)) +
126*a^4*b^6*e^4*(10*A*e*(d^6 + 17*d^5*e*x + 136*d^4*e^2*x^2 + 680*d^3*e^3*x^3 +
2380*d^2*e^4*x^4 + 6188*d*e^5*x^5 + 12376*e^6*x^6) + 7*B*(d^7 + 17*d^6*e*x + 136
*d^5*e^2*x^2 + 680*d^4*e^3*x^3 + 2380*d^3*e^4*x^4 + 6188*d^2*e^5*x^5 + 12376*d*e
^6*x^6 + 19448*e^7*x^7)) + 56*a^3*b^7*e^3*(9*A*e*(d^7 + 17*d^6*e*x + 136*d^5*e^2
*x^2 + 680*d^4*e^3*x^3 + 2380*d^3*e^4*x^4 + 6188*d^2*e^5*x^5 + 12376*d*e^6*x^6 +
19448*e^7*x^7) + 8*B*(d^8 + 17*d^7*e*x + 136*d^6*e^2*x^2 + 680*d^5*e^3*x^3 + 23
80*d^4*e^4*x^4 + 6188*d^3*e^5*x^5 + 12376*d^2*e^6*x^6 + 19448*d*e^7*x^7 + 24310*
e^8*x^8)) + 21*a^2*b^8*e^2*(8*A*e*(d^8 + 17*d^7*e*x + 136*d^6*e^2*x^2 + 680*d^5*
e^3*x^3 + 2380*d^4*e^4*x^4 + 6188*d^3*e^5*x^5 + 12376*d^2*e^6*x^6 + 19448*d*e^7*
x^7 + 24310*e^8*x^8) + 9*B*(d^9 + 17*d^8*e*x + 136*d^7*e^2*x^2 + 680*d^6*e^3*x^3
+ 2380*d^5*e^4*x^4 + 6188*d^4*e^5*x^5 + 12376*d^3*e^6*x^6 + 19448*d^2*e^7*x^7 +
24310*d*e^8*x^8 + 24310*e^9*x^9)) + 6*a*b^9*e*(7*A*e*(d^9 + 17*d^8*e*x + 136*d^
7*e^2*x^2 + 680*d^6*e^3*x^3 + 2380*d^5*e^4*x^4 + 6188*d^4*e^5*x^5 + 12376*d^3*e^
6*x^6 + 19448*d^2*e^7*x^7 + 24310*d*e^8*x^8 + 24310*e^9*x^9) + 10*B*(d^10 + 17*d
^9*e*x + 136*d^8*e^2*x^2 + 680*d^7*e^3*x^3 + 2380*d^6*e^4*x^4 + 6188*d^5*e^5*x^5
+ 12376*d^4*e^6*x^6 + 19448*d^3*e^7*x^7 + 24310*d^2*e^8*x^8 + 24310*d*e^9*x^9 +
19448*e^10*x^10)) + b^10*(6*A*e*(d^10 + 17*d^9*e*x + 136*d^8*e^2*x^2 + 680*d^7*
e^3*x^3 + 2380*d^6*e^4*x^4 + 6188*d^5*e^5*x^5 + 12376*d^4*e^6*x^6 + 19448*d^3*e^
7*x^7 + 24310*d^2*e^8*x^8 + 24310*d*e^9*x^9 + 19448*e^10*x^10) + 11*B*(d^11 + 17
*d^10*e*x + 136*d^9*e^2*x^2 + 680*d^8*e^3*x^3 + 2380*d^7*e^4*x^4 + 6188*d^6*e^5*
x^5 + 12376*d^5*e^6*x^6 + 19448*d^4*e^7*x^7 + 24310*d^3*e^8*x^8 + 24310*d^2*e^9*
x^9 + 19448*d*e^10*x^10 + 12376*e^11*x^11)))/(816816*e^12*(d + e*x)^17)

_______________________________________________________________________________________

Maple [B]  time = 0.016, size = 1942, normalized size = 5.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/(e*x+d)^18,x)

[Out]

-15/14*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+4
90*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)^14-5/3*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)^9-
3*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)^10-1/6*B*b^10/e^12/(e*x+d)^6-5/8*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)^8-1/7*b^9*(A*b*e+10*B*a*e-11*B*b*d)/e^1
2/(e*x+d)^7-7/2*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+10
5*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)^12-42/11*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)^11-1/16*(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^12/(e*x+d)^16-1/17*(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)^17-30/13*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^1
2/(e*x+d)^13-1/3*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)
^15

_______________________________________________________________________________________

Maxima [A]  time = 1.57147, size = 2693, normalized size = 8.04 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^18,x, algorithm="maxima")

[Out]

-1/816816*(136136*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 48048*A*a^10*e^11 + 6*(10*
B*a*b^9 + A*b^10)*d^10*e + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 56*(8*B*a^3*b^
7 + 3*A*a^2*b^8)*d^8*e^3 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 252*(6*B*a^
5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 792*(4*
B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 1287*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 20
02*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 3003*(B*a^10 + 10*A*a^9*b)*d*e^10 + 19448
*(11*B*b^10*d*e^10 + 6*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 24310*(11*B*b^10*d^2*e
^9 + 6*(10*B*a*b^9 + A*b^10)*d*e^10 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 2
4310*(11*B*b^10*d^3*e^8 + 6*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 21*(9*B*a^2*b^8 + 2*
A*a*b^9)*d*e^10 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 19448*(11*B*b^10*d^
4*e^7 + 6*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 +
56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x
^7 + 12376*(11*B*b^10*d^5*e^6 + 6*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 21*(9*B*a^2*b^
8 + 2*A*a*b^9)*d^3*e^8 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 126*(7*B*a^4*b
^6 + 4*A*a^3*b^7)*d*e^10 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 6188*(11*
B*b^10*d^6*e^5 + 6*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*
d^4*e^7 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^
7)*d^2*e^9 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 462*(5*B*a^6*b^4 + 6*A*a^5
*b^5)*e^11)*x^5 + 2380*(11*B*b^10*d^7*e^4 + 6*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 21
*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 12
6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9
+ 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 792*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)
*x^4 + 680*(11*B*b^10*d^8*e^3 + 6*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 21*(9*B*a^2*b^
8 + 2*A*a*b^9)*d^6*e^5 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 126*(7*B*a^4*b
^6 + 4*A*a^3*b^7)*d^4*e^7 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 462*(5*B*a
^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 792*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 1287*(3
*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 136*(11*B*b^10*d^9*e^2 + 6*(10*B*a*b^9 + A
*b^10)*d^8*e^3 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 56*(8*B*a^3*b^7 + 3*A*a^
2*b^8)*d^6*e^5 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 252*(6*B*a^5*b^5 + 5*
A*a^4*b^6)*d^4*e^7 + 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 792*(4*B*a^7*b^3
+ 7*A*a^6*b^4)*d^2*e^9 + 1287*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 2002*(2*B*a^9
*b + 9*A*a^8*b^2)*e^11)*x^2 + 17*(11*B*b^10*d^10*e + 6*(10*B*a*b^9 + A*b^10)*d^9
*e^2 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7
*e^4 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)
*d^5*e^6 + 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 792*(4*B*a^7*b^3 + 7*A*a^6*
b^4)*d^3*e^8 + 1287*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 2002*(2*B*a^9*b + 9*A*
a^8*b^2)*d*e^10 + 3003*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^29*x^17 + 17*d*e^28*x^1
6 + 136*d^2*e^27*x^15 + 680*d^3*e^26*x^14 + 2380*d^4*e^25*x^13 + 6188*d^5*e^24*x
^12 + 12376*d^6*e^23*x^11 + 19448*d^7*e^22*x^10 + 24310*d^8*e^21*x^9 + 24310*d^9
*e^20*x^8 + 19448*d^10*e^19*x^7 + 12376*d^11*e^18*x^6 + 6188*d^12*e^17*x^5 + 238
0*d^13*e^16*x^4 + 680*d^14*e^15*x^3 + 136*d^15*e^14*x^2 + 17*d^16*e^13*x + d^17*
e^12)

_______________________________________________________________________________________

Fricas [A]  time = 0.220115, size = 2693, normalized size = 8.04 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^18,x, algorithm="fricas")

[Out]

-1/816816*(136136*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 48048*A*a^10*e^11 + 6*(10*
B*a*b^9 + A*b^10)*d^10*e + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 56*(8*B*a^3*b^
7 + 3*A*a^2*b^8)*d^8*e^3 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 252*(6*B*a^
5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 792*(4*
B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 1287*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 20
02*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 3003*(B*a^10 + 10*A*a^9*b)*d*e^10 + 19448
*(11*B*b^10*d*e^10 + 6*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 24310*(11*B*b^10*d^2*e
^9 + 6*(10*B*a*b^9 + A*b^10)*d*e^10 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 2
4310*(11*B*b^10*d^3*e^8 + 6*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 21*(9*B*a^2*b^8 + 2*
A*a*b^9)*d*e^10 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 19448*(11*B*b^10*d^
4*e^7 + 6*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 +
56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x
^7 + 12376*(11*B*b^10*d^5*e^6 + 6*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 21*(9*B*a^2*b^
8 + 2*A*a*b^9)*d^3*e^8 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 126*(7*B*a^4*b
^6 + 4*A*a^3*b^7)*d*e^10 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 6188*(11*
B*b^10*d^6*e^5 + 6*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*
d^4*e^7 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^
7)*d^2*e^9 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 462*(5*B*a^6*b^4 + 6*A*a^5
*b^5)*e^11)*x^5 + 2380*(11*B*b^10*d^7*e^4 + 6*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 21
*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 12
6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9
+ 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 792*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)
*x^4 + 680*(11*B*b^10*d^8*e^3 + 6*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 21*(9*B*a^2*b^
8 + 2*A*a*b^9)*d^6*e^5 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 126*(7*B*a^4*b
^6 + 4*A*a^3*b^7)*d^4*e^7 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 462*(5*B*a
^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 792*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 1287*(3
*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 136*(11*B*b^10*d^9*e^2 + 6*(10*B*a*b^9 + A
*b^10)*d^8*e^3 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 56*(8*B*a^3*b^7 + 3*A*a^
2*b^8)*d^6*e^5 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 252*(6*B*a^5*b^5 + 5*
A*a^4*b^6)*d^4*e^7 + 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 792*(4*B*a^7*b^3
+ 7*A*a^6*b^4)*d^2*e^9 + 1287*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 2002*(2*B*a^9
*b + 9*A*a^8*b^2)*e^11)*x^2 + 17*(11*B*b^10*d^10*e + 6*(10*B*a*b^9 + A*b^10)*d^9
*e^2 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7
*e^4 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)
*d^5*e^6 + 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 792*(4*B*a^7*b^3 + 7*A*a^6*
b^4)*d^3*e^8 + 1287*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 2002*(2*B*a^9*b + 9*A*
a^8*b^2)*d*e^10 + 3003*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^29*x^17 + 17*d*e^28*x^1
6 + 136*d^2*e^27*x^15 + 680*d^3*e^26*x^14 + 2380*d^4*e^25*x^13 + 6188*d^5*e^24*x
^12 + 12376*d^6*e^23*x^11 + 19448*d^7*e^22*x^10 + 24310*d^8*e^21*x^9 + 24310*d^9
*e^20*x^8 + 19448*d^10*e^19*x^7 + 12376*d^11*e^18*x^6 + 6188*d^12*e^17*x^5 + 238
0*d^13*e^16*x^4 + 680*d^14*e^15*x^3 + 136*d^15*e^14*x^2 + 17*d^16*e^13*x + d^17*
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)**18,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215481, 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)^18,x, algorithm="giac")

[Out]

Done