3.1087 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{16}} \, dx\)

Optimal. Leaf size=235 \[ \frac{b^3 (a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{60060 e (d+e x)^{11} (b d-a e)^5}+\frac{b^2 (a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{5460 e (d+e x)^{12} (b d-a e)^4}+\frac{b (a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{910 e (d+e x)^{13} (b d-a e)^3}+\frac{(a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{210 e (d+e x)^{14} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{15 e (d+e x)^{15} (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(15*e*(b*d - a*e)*(d + e*x)^15) + ((11*b*B*d + 4*A*b
*e - 15*a*B*e)*(a + b*x)^11)/(210*e*(b*d - a*e)^2*(d + e*x)^14) + (b*(11*b*B*d +
4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(910*e*(b*d - a*e)^3*(d + e*x)^13) + (b^2*(11
*b*B*d + 4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(5460*e*(b*d - a*e)^4*(d + e*x)^12) +
(b^3*(11*b*B*d + 4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(60060*e*(b*d - a*e)^5*(d +
e*x)^11)

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Rubi [A]  time = 0.309208, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{b^3 (a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{60060 e (d+e x)^{11} (b d-a e)^5}+\frac{b^2 (a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{5460 e (d+e x)^{12} (b d-a e)^4}+\frac{b (a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{910 e (d+e x)^{13} (b d-a e)^3}+\frac{(a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{210 e (d+e x)^{14} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{15 e (d+e x)^{15} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/(d + e*x)^16,x]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(15*e*(b*d - a*e)*(d + e*x)^15) + ((11*b*B*d + 4*A*b
*e - 15*a*B*e)*(a + b*x)^11)/(210*e*(b*d - a*e)^2*(d + e*x)^14) + (b*(11*b*B*d +
4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(910*e*(b*d - a*e)^3*(d + e*x)^13) + (b^2*(11
*b*B*d + 4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(5460*e*(b*d - a*e)^4*(d + e*x)^12) +
(b^3*(11*b*B*d + 4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(60060*e*(b*d - a*e)^5*(d +
e*x)^11)

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Rubi in Sympy [A]  time = 50.4587, size = 221, normalized size = 0.94 \[ - \frac{b^{3} \left (a + b x\right )^{11} \left (4 A b e - 15 B a e + 11 B b d\right )}{60060 e \left (d + e x\right )^{11} \left (a e - b d\right )^{5}} + \frac{b^{2} \left (a + b x\right )^{11} \left (4 A b e - 15 B a e + 11 B b d\right )}{5460 e \left (d + e x\right )^{12} \left (a e - b d\right )^{4}} - \frac{b \left (a + b x\right )^{11} \left (4 A b e - 15 B a e + 11 B b d\right )}{910 e \left (d + e x\right )^{13} \left (a e - b d\right )^{3}} + \frac{\left (a + b x\right )^{11} \left (4 A b e - 15 B a e + 11 B b d\right )}{210 e \left (d + e x\right )^{14} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{11} \left (A e - B d\right )}{15 e \left (d + e x\right )^{15} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/(e*x+d)**16,x)

[Out]

-b**3*(a + b*x)**11*(4*A*b*e - 15*B*a*e + 11*B*b*d)/(60060*e*(d + e*x)**11*(a*e
- b*d)**5) + b**2*(a + b*x)**11*(4*A*b*e - 15*B*a*e + 11*B*b*d)/(5460*e*(d + e*x
)**12*(a*e - b*d)**4) - b*(a + b*x)**11*(4*A*b*e - 15*B*a*e + 11*B*b*d)/(910*e*(
d + e*x)**13*(a*e - b*d)**3) + (a + b*x)**11*(4*A*b*e - 15*B*a*e + 11*B*b*d)/(21
0*e*(d + e*x)**14*(a*e - b*d)**2) - (a + b*x)**11*(A*e - B*d)/(15*e*(d + e*x)**1
5*(a*e - b*d))

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Mathematica [B]  time = 3.02969, size = 1430, normalized size = 6.09 \[ -\frac{\left (4 A e \left (d^{10}+15 e x d^9+105 e^2 x^2 d^8+455 e^3 x^3 d^7+1365 e^4 x^4 d^6+3003 e^5 x^5 d^5+5005 e^6 x^6 d^4+6435 e^7 x^7 d^3+6435 e^8 x^8 d^2+5005 e^9 x^9 d+3003 e^{10} x^{10}\right )+11 B \left (d^{11}+15 e x d^{10}+105 e^2 x^2 d^9+455 e^3 x^3 d^8+1365 e^4 x^4 d^7+3003 e^5 x^5 d^6+5005 e^6 x^6 d^5+6435 e^7 x^7 d^4+6435 e^8 x^8 d^3+5005 e^9 x^9 d^2+3003 e^{10} x^{10} d+1365 e^{11} x^{11}\right )\right ) b^{10}+20 a e \left (A e \left (d^9+15 e x d^8+105 e^2 x^2 d^7+455 e^3 x^3 d^6+1365 e^4 x^4 d^5+3003 e^5 x^5 d^4+5005 e^6 x^6 d^3+6435 e^7 x^7 d^2+6435 e^8 x^8 d+5005 e^9 x^9\right )+2 B \left (d^{10}+15 e x d^9+105 e^2 x^2 d^8+455 e^3 x^3 d^7+1365 e^4 x^4 d^6+3003 e^5 x^5 d^5+5005 e^6 x^6 d^4+6435 e^7 x^7 d^3+6435 e^8 x^8 d^2+5005 e^9 x^9 d+3003 e^{10} x^{10}\right )\right ) b^9+30 a^2 e^2 \left (2 A e \left (d^8+15 e x d^7+105 e^2 x^2 d^6+455 e^3 x^3 d^5+1365 e^4 x^4 d^4+3003 e^5 x^5 d^3+5005 e^6 x^6 d^2+6435 e^7 x^7 d+6435 e^8 x^8\right )+3 B \left (d^9+15 e x d^8+105 e^2 x^2 d^7+455 e^3 x^3 d^6+1365 e^4 x^4 d^5+3003 e^5 x^5 d^4+5005 e^6 x^6 d^3+6435 e^7 x^7 d^2+6435 e^8 x^8 d+5005 e^9 x^9\right )\right ) b^8+20 a^3 e^3 \left (7 A e \left (d^7+15 e x d^6+105 e^2 x^2 d^5+455 e^3 x^3 d^4+1365 e^4 x^4 d^3+3003 e^5 x^5 d^2+5005 e^6 x^6 d+6435 e^7 x^7\right )+8 B \left (d^8+15 e x d^7+105 e^2 x^2 d^6+455 e^3 x^3 d^5+1365 e^4 x^4 d^4+3003 e^5 x^5 d^3+5005 e^6 x^6 d^2+6435 e^7 x^7 d+6435 e^8 x^8\right )\right ) b^7+35 a^4 e^4 \left (8 A e \left (d^6+15 e x d^5+105 e^2 x^2 d^4+455 e^3 x^3 d^3+1365 e^4 x^4 d^2+3003 e^5 x^5 d+5005 e^6 x^6\right )+7 B \left (d^7+15 e x d^6+105 e^2 x^2 d^5+455 e^3 x^3 d^4+1365 e^4 x^4 d^3+3003 e^5 x^5 d^2+5005 e^6 x^6 d+6435 e^7 x^7\right )\right ) b^6+168 a^5 e^5 \left (3 A e \left (d^5+15 e x d^4+105 e^2 x^2 d^3+455 e^3 x^3 d^2+1365 e^4 x^4 d+3003 e^5 x^5\right )+2 B \left (d^6+15 e x d^5+105 e^2 x^2 d^4+455 e^3 x^3 d^3+1365 e^4 x^4 d^2+3003 e^5 x^5 d+5005 e^6 x^6\right )\right ) b^5+420 a^6 e^6 \left (2 A e \left (d^4+15 e x d^3+105 e^2 x^2 d^2+455 e^3 x^3 d+1365 e^4 x^4\right )+B \left (d^5+15 e x d^4+105 e^2 x^2 d^3+455 e^3 x^3 d^2+1365 e^4 x^4 d+3003 e^5 x^5\right )\right ) b^4+120 a^7 e^7 \left (11 A e \left (d^3+15 e x d^2+105 e^2 x^2 d+455 e^3 x^3\right )+4 B \left (d^4+15 e x d^3+105 e^2 x^2 d^2+455 e^3 x^3 d+1365 e^4 x^4\right )\right ) b^3+495 a^8 e^8 \left (4 A e \left (d^2+15 e x d+105 e^2 x^2\right )+B \left (d^3+15 e x d^2+105 e^2 x^2 d+455 e^3 x^3\right )\right ) b^2+220 a^9 e^9 \left (13 A e (d+15 e x)+2 B \left (d^2+15 e x d+105 e^2 x^2\right )\right ) b+286 a^{10} e^{10} (14 A e+B (d+15 e x))}{60060 e^{12} (d+e x)^{15}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^16,x]

[Out]

-(286*a^10*e^10*(14*A*e + B*(d + 15*e*x)) + 220*a^9*b*e^9*(13*A*e*(d + 15*e*x) +
2*B*(d^2 + 15*d*e*x + 105*e^2*x^2)) + 495*a^8*b^2*e^8*(4*A*e*(d^2 + 15*d*e*x +
105*e^2*x^2) + B*(d^3 + 15*d^2*e*x + 105*d*e^2*x^2 + 455*e^3*x^3)) + 120*a^7*b^3
*e^7*(11*A*e*(d^3 + 15*d^2*e*x + 105*d*e^2*x^2 + 455*e^3*x^3) + 4*B*(d^4 + 15*d^
3*e*x + 105*d^2*e^2*x^2 + 455*d*e^3*x^3 + 1365*e^4*x^4)) + 420*a^6*b^4*e^6*(2*A*
e*(d^4 + 15*d^3*e*x + 105*d^2*e^2*x^2 + 455*d*e^3*x^3 + 1365*e^4*x^4) + B*(d^5 +
15*d^4*e*x + 105*d^3*e^2*x^2 + 455*d^2*e^3*x^3 + 1365*d*e^4*x^4 + 3003*e^5*x^5)
) + 168*a^5*b^5*e^5*(3*A*e*(d^5 + 15*d^4*e*x + 105*d^3*e^2*x^2 + 455*d^2*e^3*x^3
+ 1365*d*e^4*x^4 + 3003*e^5*x^5) + 2*B*(d^6 + 15*d^5*e*x + 105*d^4*e^2*x^2 + 45
5*d^3*e^3*x^3 + 1365*d^2*e^4*x^4 + 3003*d*e^5*x^5 + 5005*e^6*x^6)) + 35*a^4*b^6*
e^4*(8*A*e*(d^6 + 15*d^5*e*x + 105*d^4*e^2*x^2 + 455*d^3*e^3*x^3 + 1365*d^2*e^4*
x^4 + 3003*d*e^5*x^5 + 5005*e^6*x^6) + 7*B*(d^7 + 15*d^6*e*x + 105*d^5*e^2*x^2 +
455*d^4*e^3*x^3 + 1365*d^3*e^4*x^4 + 3003*d^2*e^5*x^5 + 5005*d*e^6*x^6 + 6435*e
^7*x^7)) + 20*a^3*b^7*e^3*(7*A*e*(d^7 + 15*d^6*e*x + 105*d^5*e^2*x^2 + 455*d^4*e
^3*x^3 + 1365*d^3*e^4*x^4 + 3003*d^2*e^5*x^5 + 5005*d*e^6*x^6 + 6435*e^7*x^7) +
8*B*(d^8 + 15*d^7*e*x + 105*d^6*e^2*x^2 + 455*d^5*e^3*x^3 + 1365*d^4*e^4*x^4 + 3
003*d^3*e^5*x^5 + 5005*d^2*e^6*x^6 + 6435*d*e^7*x^7 + 6435*e^8*x^8)) + 30*a^2*b^
8*e^2*(2*A*e*(d^8 + 15*d^7*e*x + 105*d^6*e^2*x^2 + 455*d^5*e^3*x^3 + 1365*d^4*e^
4*x^4 + 3003*d^3*e^5*x^5 + 5005*d^2*e^6*x^6 + 6435*d*e^7*x^7 + 6435*e^8*x^8) + 3
*B*(d^9 + 15*d^8*e*x + 105*d^7*e^2*x^2 + 455*d^6*e^3*x^3 + 1365*d^5*e^4*x^4 + 30
03*d^4*e^5*x^5 + 5005*d^3*e^6*x^6 + 6435*d^2*e^7*x^7 + 6435*d*e^8*x^8 + 5005*e^9
*x^9)) + 20*a*b^9*e*(A*e*(d^9 + 15*d^8*e*x + 105*d^7*e^2*x^2 + 455*d^6*e^3*x^3 +
1365*d^5*e^4*x^4 + 3003*d^4*e^5*x^5 + 5005*d^3*e^6*x^6 + 6435*d^2*e^7*x^7 + 643
5*d*e^8*x^8 + 5005*e^9*x^9) + 2*B*(d^10 + 15*d^9*e*x + 105*d^8*e^2*x^2 + 455*d^7
*e^3*x^3 + 1365*d^6*e^4*x^4 + 3003*d^5*e^5*x^5 + 5005*d^4*e^6*x^6 + 6435*d^3*e^7
*x^7 + 6435*d^2*e^8*x^8 + 5005*d*e^9*x^9 + 3003*e^10*x^10)) + b^10*(4*A*e*(d^10
+ 15*d^9*e*x + 105*d^8*e^2*x^2 + 455*d^7*e^3*x^3 + 1365*d^6*e^4*x^4 + 3003*d^5*e
^5*x^5 + 5005*d^4*e^6*x^6 + 6435*d^3*e^7*x^7 + 6435*d^2*e^8*x^8 + 5005*d*e^9*x^9
+ 3003*e^10*x^10) + 11*B*(d^11 + 15*d^10*e*x + 105*d^9*e^2*x^2 + 455*d^8*e^3*x^
3 + 1365*d^7*e^4*x^4 + 3003*d^6*e^5*x^5 + 5005*d^5*e^6*x^6 + 6435*d^4*e^7*x^7 +
6435*d^3*e^8*x^8 + 5005*d^2*e^9*x^9 + 3003*d*e^10*x^10 + 1365*e^11*x^11)))/(6006
0*e^12*(d + e*x)^15)

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Maple [B]  time = 0.016, size = 1942, normalized size = 8.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/(e*x+d)^16,x)

[Out]

-1/14*(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)^14-14/3*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)^9-21/5*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^4-160*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)^10-5/6*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)^6-15/4*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)^8-15/7*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)^7-5/4*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+5
6*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)^12-30/11*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)^11-5/13*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)^13-1/4*B*b^10/e^12/(e*x+d)^4-1/5*b^9*(A*b*e+10
*B*a*e-11*B*b*d)/e^12/(e*x+d)^5-1/15*(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
)^15

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Maxima [A]  time = 1.56702, size = 2664, normalized size = 11.34 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^16,x, algorithm="maxima")

[Out]

-1/60060*(15015*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 4004*A*a^10*e^11 + 4*(10*B*a
*b^9 + A*b^10)*d^10*e + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 20*(8*B*a^3*b^7 +
3*A*a^2*b^8)*d^8*e^3 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 56*(6*B*a^5*b^5
+ 5*A*a^4*b^6)*d^6*e^5 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 120*(4*B*a^7*
b^3 + 7*A*a^6*b^4)*d^4*e^7 + 165*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 220*(2*B*
a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 286*(B*a^10 + 10*A*a^9*b)*d*e^10 + 3003*(11*B*b^1
0*d*e^10 + 4*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 5005*(11*B*b^10*d^2*e^9 + 4*(10*
B*a*b^9 + A*b^10)*d*e^10 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 6435*(11*B*b
^10*d^3*e^8 + 4*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e
^10 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 6435*(11*B*b^10*d^4*e^7 + 4*(10
*B*a*b^9 + A*b^10)*d^3*e^8 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 20*(8*B*a^3*
b^7 + 3*A*a^2*b^8)*d*e^10 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 5005*(11*
B*b^10*d^5*e^6 + 4*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*
d^3*e^8 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7
)*d*e^10 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 3003*(11*B*b^10*d^6*e^5 +
4*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 20*(8*B
*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 56*(6
*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 13
65*(11*B*b^10*d^7*e^4 + 4*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 10*(9*B*a^2*b^8 + 2*A*
a*b^9)*d^5*e^6 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 35*(7*B*a^4*b^6 + 4*A*
a^3*b^7)*d^3*e^8 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 84*(5*B*a^6*b^4 + 6*
A*a^5*b^5)*d*e^10 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 455*(11*B*b^10*d
^8*e^3 + 4*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5
+ 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^
7 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*
e^9 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 165*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e
^11)*x^3 + 105*(11*B*b^10*d^9*e^2 + 4*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 10*(9*B*a^
2*b^8 + 2*A*a*b^9)*d^7*e^4 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 35*(7*B*a^
4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 84*(5*B*
a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 165*(
3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 220*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 15
*(11*B*b^10*d^10*e + 4*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 10*(9*B*a^2*b^8 + 2*A*a*b
^9)*d^8*e^3 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 35*(7*B*a^4*b^6 + 4*A*a^3
*b^7)*d^6*e^5 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 84*(5*B*a^6*b^4 + 6*A*a
^5*b^5)*d^4*e^7 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 165*(3*B*a^8*b^2 + 8
*A*a^7*b^3)*d^2*e^9 + 220*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 286*(B*a^10 + 10*A*
a^9*b)*e^11)*x)/(e^27*x^15 + 15*d*e^26*x^14 + 105*d^2*e^25*x^13 + 455*d^3*e^24*x
^12 + 1365*d^4*e^23*x^11 + 3003*d^5*e^22*x^10 + 5005*d^6*e^21*x^9 + 6435*d^7*e^2
0*x^8 + 6435*d^8*e^19*x^7 + 5005*d^9*e^18*x^6 + 3003*d^10*e^17*x^5 + 1365*d^11*e
^16*x^4 + 455*d^12*e^15*x^3 + 105*d^13*e^14*x^2 + 15*d^14*e^13*x + d^15*e^12)

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Fricas [A]  time = 0.212822, size = 2664, normalized size = 11.34 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^16,x, algorithm="fricas")

[Out]

-1/60060*(15015*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 4004*A*a^10*e^11 + 4*(10*B*a
*b^9 + A*b^10)*d^10*e + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 20*(8*B*a^3*b^7 +
3*A*a^2*b^8)*d^8*e^3 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 56*(6*B*a^5*b^5
+ 5*A*a^4*b^6)*d^6*e^5 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 120*(4*B*a^7*
b^3 + 7*A*a^6*b^4)*d^4*e^7 + 165*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 220*(2*B*
a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 286*(B*a^10 + 10*A*a^9*b)*d*e^10 + 3003*(11*B*b^1
0*d*e^10 + 4*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 5005*(11*B*b^10*d^2*e^9 + 4*(10*
B*a*b^9 + A*b^10)*d*e^10 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 6435*(11*B*b
^10*d^3*e^8 + 4*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e
^10 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 6435*(11*B*b^10*d^4*e^7 + 4*(10
*B*a*b^9 + A*b^10)*d^3*e^8 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 20*(8*B*a^3*
b^7 + 3*A*a^2*b^8)*d*e^10 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 5005*(11*
B*b^10*d^5*e^6 + 4*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*
d^3*e^8 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7
)*d*e^10 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 3003*(11*B*b^10*d^6*e^5 +
4*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 20*(8*B
*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 56*(6
*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 13
65*(11*B*b^10*d^7*e^4 + 4*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 10*(9*B*a^2*b^8 + 2*A*
a*b^9)*d^5*e^6 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 35*(7*B*a^4*b^6 + 4*A*
a^3*b^7)*d^3*e^8 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 84*(5*B*a^6*b^4 + 6*
A*a^5*b^5)*d*e^10 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 455*(11*B*b^10*d
^8*e^3 + 4*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5
+ 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^
7 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*
e^9 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 165*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e
^11)*x^3 + 105*(11*B*b^10*d^9*e^2 + 4*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 10*(9*B*a^
2*b^8 + 2*A*a*b^9)*d^7*e^4 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 35*(7*B*a^
4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 84*(5*B*
a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 165*(
3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 220*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 15
*(11*B*b^10*d^10*e + 4*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 10*(9*B*a^2*b^8 + 2*A*a*b
^9)*d^8*e^3 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 35*(7*B*a^4*b^6 + 4*A*a^3
*b^7)*d^6*e^5 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 84*(5*B*a^6*b^4 + 6*A*a
^5*b^5)*d^4*e^7 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 165*(3*B*a^8*b^2 + 8
*A*a^7*b^3)*d^2*e^9 + 220*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 286*(B*a^10 + 10*A*
a^9*b)*e^11)*x)/(e^27*x^15 + 15*d*e^26*x^14 + 105*d^2*e^25*x^13 + 455*d^3*e^24*x
^12 + 1365*d^4*e^23*x^11 + 3003*d^5*e^22*x^10 + 5005*d^6*e^21*x^9 + 6435*d^7*e^2
0*x^8 + 6435*d^8*e^19*x^7 + 5005*d^9*e^18*x^6 + 3003*d^10*e^17*x^5 + 1365*d^11*e
^16*x^4 + 455*d^12*e^15*x^3 + 105*d^13*e^14*x^2 + 15*d^14*e^13*x + d^15*e^12)

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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)**16,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216318, 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)^16,x, algorithm="giac")

[Out]

Done