3.1088 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx\)

Optimal. Leaf size=285 \[ \frac{b^4 (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{240240 e (d+e x)^{11} (b d-a e)^6}+\frac{b^3 (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{21840 e (d+e x)^{12} (b d-a e)^5}+\frac{b^2 (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{3640 e (d+e x)^{13} (b d-a e)^4}+\frac{b (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{840 e (d+e x)^{14} (b d-a e)^3}+\frac{(a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{240 e (d+e x)^{15} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{16 e (d+e x)^{16} (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(16*e*(b*d - a*e)*(d + e*x)^16) + ((11*b*B*d + 5*A*b
*e - 16*a*B*e)*(a + b*x)^11)/(240*e*(b*d - a*e)^2*(d + e*x)^15) + (b*(11*b*B*d +
5*A*b*e - 16*a*B*e)*(a + b*x)^11)/(840*e*(b*d - a*e)^3*(d + e*x)^14) + (b^2*(11
*b*B*d + 5*A*b*e - 16*a*B*e)*(a + b*x)^11)/(3640*e*(b*d - a*e)^4*(d + e*x)^13) +
(b^3*(11*b*B*d + 5*A*b*e - 16*a*B*e)*(a + b*x)^11)/(21840*e*(b*d - a*e)^5*(d +
e*x)^12) + (b^4*(11*b*B*d + 5*A*b*e - 16*a*B*e)*(a + b*x)^11)/(240240*e*(b*d - a
*e)^6*(d + e*x)^11)

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Rubi [A]  time = 0.391027, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{b^4 (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{240240 e (d+e x)^{11} (b d-a e)^6}+\frac{b^3 (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{21840 e (d+e x)^{12} (b d-a e)^5}+\frac{b^2 (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{3640 e (d+e x)^{13} (b d-a e)^4}+\frac{b (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{840 e (d+e x)^{14} (b d-a e)^3}+\frac{(a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{240 e (d+e x)^{15} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{16 e (d+e x)^{16} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/(d + e*x)^17,x]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(16*e*(b*d - a*e)*(d + e*x)^16) + ((11*b*B*d + 5*A*b
*e - 16*a*B*e)*(a + b*x)^11)/(240*e*(b*d - a*e)^2*(d + e*x)^15) + (b*(11*b*B*d +
5*A*b*e - 16*a*B*e)*(a + b*x)^11)/(840*e*(b*d - a*e)^3*(d + e*x)^14) + (b^2*(11
*b*B*d + 5*A*b*e - 16*a*B*e)*(a + b*x)^11)/(3640*e*(b*d - a*e)^4*(d + e*x)^13) +
(b^3*(11*b*B*d + 5*A*b*e - 16*a*B*e)*(a + b*x)^11)/(21840*e*(b*d - a*e)^5*(d +
e*x)^12) + (b^4*(11*b*B*d + 5*A*b*e - 16*a*B*e)*(a + b*x)^11)/(240240*e*(b*d - a
*e)^6*(d + e*x)^11)

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Rubi in Sympy [A]  time = 65.8406, size = 270, normalized size = 0.95 \[ \frac{b^{4} \left (a + b x\right )^{11} \left (5 A b e - 16 B a e + 11 B b d\right )}{240240 e \left (d + e x\right )^{11} \left (a e - b d\right )^{6}} - \frac{b^{3} \left (a + b x\right )^{11} \left (5 A b e - 16 B a e + 11 B b d\right )}{21840 e \left (d + e x\right )^{12} \left (a e - b d\right )^{5}} + \frac{b^{2} \left (a + b x\right )^{11} \left (5 A b e - 16 B a e + 11 B b d\right )}{3640 e \left (d + e x\right )^{13} \left (a e - b d\right )^{4}} - \frac{b \left (a + b x\right )^{11} \left (5 A b e - 16 B a e + 11 B b d\right )}{840 e \left (d + e x\right )^{14} \left (a e - b d\right )^{3}} + \frac{\left (a + b x\right )^{11} \left (5 A b e - 16 B a e + 11 B b d\right )}{240 e \left (d + e x\right )^{15} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{11} \left (A e - B d\right )}{16 e \left (d + e x\right )^{16} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/(e*x+d)**17,x)

[Out]

b**4*(a + b*x)**11*(5*A*b*e - 16*B*a*e + 11*B*b*d)/(240240*e*(d + e*x)**11*(a*e
- b*d)**6) - b**3*(a + b*x)**11*(5*A*b*e - 16*B*a*e + 11*B*b*d)/(21840*e*(d + e*
x)**12*(a*e - b*d)**5) + b**2*(a + b*x)**11*(5*A*b*e - 16*B*a*e + 11*B*b*d)/(364
0*e*(d + e*x)**13*(a*e - b*d)**4) - b*(a + b*x)**11*(5*A*b*e - 16*B*a*e + 11*B*b
*d)/(840*e*(d + e*x)**14*(a*e - b*d)**3) + (a + b*x)**11*(5*A*b*e - 16*B*a*e + 1
1*B*b*d)/(240*e*(d + e*x)**15*(a*e - b*d)**2) - (a + b*x)**11*(A*e - B*d)/(16*e*
(d + e*x)**16*(a*e - b*d))

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Mathematica [B]  time = 2.91994, size = 1429, normalized size = 5.01 \[ -\frac{\left (5 A e \left (d^{10}+16 e x d^9+120 e^2 x^2 d^8+560 e^3 x^3 d^7+1820 e^4 x^4 d^6+4368 e^5 x^5 d^5+8008 e^6 x^6 d^4+11440 e^7 x^7 d^3+12870 e^8 x^8 d^2+11440 e^9 x^9 d+8008 e^{10} x^{10}\right )+11 B \left (d^{11}+16 e x d^{10}+120 e^2 x^2 d^9+560 e^3 x^3 d^8+1820 e^4 x^4 d^7+4368 e^5 x^5 d^6+8008 e^6 x^6 d^5+11440 e^7 x^7 d^4+12870 e^8 x^8 d^3+11440 e^9 x^9 d^2+8008 e^{10} x^{10} d+4368 e^{11} x^{11}\right )\right ) b^{10}+10 a e \left (3 A e \left (d^9+16 e x d^8+120 e^2 x^2 d^7+560 e^3 x^3 d^6+1820 e^4 x^4 d^5+4368 e^5 x^5 d^4+8008 e^6 x^6 d^3+11440 e^7 x^7 d^2+12870 e^8 x^8 d+11440 e^9 x^9\right )+5 B \left (d^{10}+16 e x d^9+120 e^2 x^2 d^8+560 e^3 x^3 d^7+1820 e^4 x^4 d^6+4368 e^5 x^5 d^5+8008 e^6 x^6 d^4+11440 e^7 x^7 d^3+12870 e^8 x^8 d^2+11440 e^9 x^9 d+8008 e^{10} x^{10}\right )\right ) b^9+15 a^2 e^2 \left (7 A e \left (d^8+16 e x d^7+120 e^2 x^2 d^6+560 e^3 x^3 d^5+1820 e^4 x^4 d^4+4368 e^5 x^5 d^3+8008 e^6 x^6 d^2+11440 e^7 x^7 d+12870 e^8 x^8\right )+9 B \left (d^9+16 e x d^8+120 e^2 x^2 d^7+560 e^3 x^3 d^6+1820 e^4 x^4 d^5+4368 e^5 x^5 d^4+8008 e^6 x^6 d^3+11440 e^7 x^7 d^2+12870 e^8 x^8 d+11440 e^9 x^9\right )\right ) b^8+280 a^3 e^3 \left (A e \left (d^7+16 e x d^6+120 e^2 x^2 d^5+560 e^3 x^3 d^4+1820 e^4 x^4 d^3+4368 e^5 x^5 d^2+8008 e^6 x^6 d+11440 e^7 x^7\right )+B \left (d^8+16 e x d^7+120 e^2 x^2 d^6+560 e^3 x^3 d^5+1820 e^4 x^4 d^4+4368 e^5 x^5 d^3+8008 e^6 x^6 d^2+11440 e^7 x^7 d+12870 e^8 x^8\right )\right ) b^7+70 a^4 e^4 \left (9 A e \left (d^6+16 e x d^5+120 e^2 x^2 d^4+560 e^3 x^3 d^3+1820 e^4 x^4 d^2+4368 e^5 x^5 d+8008 e^6 x^6\right )+7 B \left (d^7+16 e x d^6+120 e^2 x^2 d^5+560 e^3 x^3 d^4+1820 e^4 x^4 d^3+4368 e^5 x^5 d^2+8008 e^6 x^6 d+11440 e^7 x^7\right )\right ) b^6+252 a^5 e^5 \left (5 A e \left (d^5+16 e x d^4+120 e^2 x^2 d^3+560 e^3 x^3 d^2+1820 e^4 x^4 d+4368 e^5 x^5\right )+3 B \left (d^6+16 e x d^5+120 e^2 x^2 d^4+560 e^3 x^3 d^3+1820 e^4 x^4 d^2+4368 e^5 x^5 d+8008 e^6 x^6\right )\right ) b^5+210 a^6 e^6 \left (11 A e \left (d^4+16 e x d^3+120 e^2 x^2 d^2+560 e^3 x^3 d+1820 e^4 x^4\right )+5 B \left (d^5+16 e x d^4+120 e^2 x^2 d^3+560 e^3 x^3 d^2+1820 e^4 x^4 d+4368 e^5 x^5\right )\right ) b^4+1320 a^7 e^7 \left (3 A e \left (d^3+16 e x d^2+120 e^2 x^2 d+560 e^3 x^3\right )+B \left (d^4+16 e x d^3+120 e^2 x^2 d^2+560 e^3 x^3 d+1820 e^4 x^4\right )\right ) b^3+495 a^8 e^8 \left (13 A e \left (d^2+16 e x d+120 e^2 x^2\right )+3 B \left (d^3+16 e x d^2+120 e^2 x^2 d+560 e^3 x^3\right )\right ) b^2+1430 a^9 e^9 \left (7 A e (d+16 e x)+B \left (d^2+16 e x d+120 e^2 x^2\right )\right ) b+1001 a^{10} e^{10} (15 A e+B (d+16 e x))}{240240 e^{12} (d+e x)^{16}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^17,x]

[Out]

-(1001*a^10*e^10*(15*A*e + B*(d + 16*e*x)) + 1430*a^9*b*e^9*(7*A*e*(d + 16*e*x)
+ B*(d^2 + 16*d*e*x + 120*e^2*x^2)) + 495*a^8*b^2*e^8*(13*A*e*(d^2 + 16*d*e*x +
120*e^2*x^2) + 3*B*(d^3 + 16*d^2*e*x + 120*d*e^2*x^2 + 560*e^3*x^3)) + 1320*a^7*
b^3*e^7*(3*A*e*(d^3 + 16*d^2*e*x + 120*d*e^2*x^2 + 560*e^3*x^3) + B*(d^4 + 16*d^
3*e*x + 120*d^2*e^2*x^2 + 560*d*e^3*x^3 + 1820*e^4*x^4)) + 210*a^6*b^4*e^6*(11*A
*e*(d^4 + 16*d^3*e*x + 120*d^2*e^2*x^2 + 560*d*e^3*x^3 + 1820*e^4*x^4) + 5*B*(d^
5 + 16*d^4*e*x + 120*d^3*e^2*x^2 + 560*d^2*e^3*x^3 + 1820*d*e^4*x^4 + 4368*e^5*x
^5)) + 252*a^5*b^5*e^5*(5*A*e*(d^5 + 16*d^4*e*x + 120*d^3*e^2*x^2 + 560*d^2*e^3*
x^3 + 1820*d*e^4*x^4 + 4368*e^5*x^5) + 3*B*(d^6 + 16*d^5*e*x + 120*d^4*e^2*x^2 +
560*d^3*e^3*x^3 + 1820*d^2*e^4*x^4 + 4368*d*e^5*x^5 + 8008*e^6*x^6)) + 70*a^4*b
^6*e^4*(9*A*e*(d^6 + 16*d^5*e*x + 120*d^4*e^2*x^2 + 560*d^3*e^3*x^3 + 1820*d^2*e
^4*x^4 + 4368*d*e^5*x^5 + 8008*e^6*x^6) + 7*B*(d^7 + 16*d^6*e*x + 120*d^5*e^2*x^
2 + 560*d^4*e^3*x^3 + 1820*d^3*e^4*x^4 + 4368*d^2*e^5*x^5 + 8008*d*e^6*x^6 + 114
40*e^7*x^7)) + 280*a^3*b^7*e^3*(A*e*(d^7 + 16*d^6*e*x + 120*d^5*e^2*x^2 + 560*d^
4*e^3*x^3 + 1820*d^3*e^4*x^4 + 4368*d^2*e^5*x^5 + 8008*d*e^6*x^6 + 11440*e^7*x^7
) + B*(d^8 + 16*d^7*e*x + 120*d^6*e^2*x^2 + 560*d^5*e^3*x^3 + 1820*d^4*e^4*x^4 +
4368*d^3*e^5*x^5 + 8008*d^2*e^6*x^6 + 11440*d*e^7*x^7 + 12870*e^8*x^8)) + 15*a^
2*b^8*e^2*(7*A*e*(d^8 + 16*d^7*e*x + 120*d^6*e^2*x^2 + 560*d^5*e^3*x^3 + 1820*d^
4*e^4*x^4 + 4368*d^3*e^5*x^5 + 8008*d^2*e^6*x^6 + 11440*d*e^7*x^7 + 12870*e^8*x^
8) + 9*B*(d^9 + 16*d^8*e*x + 120*d^7*e^2*x^2 + 560*d^6*e^3*x^3 + 1820*d^5*e^4*x^
4 + 4368*d^4*e^5*x^5 + 8008*d^3*e^6*x^6 + 11440*d^2*e^7*x^7 + 12870*d*e^8*x^8 +
11440*e^9*x^9)) + 10*a*b^9*e*(3*A*e*(d^9 + 16*d^8*e*x + 120*d^7*e^2*x^2 + 560*d^
6*e^3*x^3 + 1820*d^5*e^4*x^4 + 4368*d^4*e^5*x^5 + 8008*d^3*e^6*x^6 + 11440*d^2*e
^7*x^7 + 12870*d*e^8*x^8 + 11440*e^9*x^9) + 5*B*(d^10 + 16*d^9*e*x + 120*d^8*e^2
*x^2 + 560*d^7*e^3*x^3 + 1820*d^6*e^4*x^4 + 4368*d^5*e^5*x^5 + 8008*d^4*e^6*x^6
+ 11440*d^3*e^7*x^7 + 12870*d^2*e^8*x^8 + 11440*d*e^9*x^9 + 8008*e^10*x^10)) + b
^10*(5*A*e*(d^10 + 16*d^9*e*x + 120*d^8*e^2*x^2 + 560*d^7*e^3*x^3 + 1820*d^6*e^4
*x^4 + 4368*d^5*e^5*x^5 + 8008*d^4*e^6*x^6 + 11440*d^3*e^7*x^7 + 12870*d^2*e^8*x
^8 + 11440*d*e^9*x^9 + 8008*e^10*x^10) + 11*B*(d^11 + 16*d^10*e*x + 120*d^9*e^2*
x^2 + 560*d^8*e^3*x^3 + 1820*d^7*e^4*x^4 + 4368*d^6*e^5*x^5 + 8008*d^5*e^6*x^6 +
11440*d^4*e^7*x^7 + 12870*d^3*e^8*x^8 + 11440*d^2*e^9*x^9 + 8008*d*e^10*x^10 +
4368*e^11*x^11)))/(240240*e^12*(d + e*x)^16)

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Maple [B]  time = 0.016, size = 1942, normalized size = 6.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/(e*x+d)^17,x)

[Out]

-5/14*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)^14-10/3*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)^9-21/5*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)^10-1/6*b^9*(A*b*e+10*B*
a*e-11*B*b*d)/e^12/(e*x+d)^6-15/8*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)^8-5/
7*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)^7-5/2*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)^12-42/11*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)^11-1/16*(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)^16-15/13*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)^13-1/5*B*b^10/e^12/(e*x+d)^5-1/15*(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)^15

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Maxima [A]  time = 1.57435, size = 2678, normalized size = 9.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^17,x, algorithm="maxima")

[Out]

-1/240240*(48048*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 15015*A*a^10*e^11 + 5*(10*B
*a*b^9 + A*b^10)*d^10*e + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 35*(8*B*a^3*b^7
+ 3*A*a^2*b^8)*d^8*e^3 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 126*(6*B*a^5*
b^5 + 5*A*a^4*b^6)*d^6*e^5 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 330*(4*B*
a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 495*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 715*(
2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 1001*(B*a^10 + 10*A*a^9*b)*d*e^10 + 8008*(11*
B*b^10*d*e^10 + 5*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 11440*(11*B*b^10*d^2*e^9 +
5*(10*B*a*b^9 + A*b^10)*d*e^10 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 12870*
(11*B*b^10*d^3*e^8 + 5*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 15*(9*B*a^2*b^8 + 2*A*a*b
^9)*d*e^10 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 11440*(11*B*b^10*d^4*e^7
+ 5*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 35*(
8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 8
008*(11*B*b^10*d^5*e^6 + 5*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 15*(9*B*a^2*b^8 + 2*A
*a*b^9)*d^3*e^8 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 70*(7*B*a^4*b^6 + 4*A
*a^3*b^7)*d*e^10 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 4368*(11*B*b^10*d
^6*e^5 + 5*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7
+ 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^
9 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^1
1)*x^5 + 1820*(11*B*b^10*d^7*e^4 + 5*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 15*(9*B*a^2
*b^8 + 2*A*a*b^9)*d^5*e^6 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 70*(7*B*a^4
*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 210*(5*B
*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 560
*(11*B*b^10*d^8*e^3 + 5*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 15*(9*B*a^2*b^8 + 2*A*a*
b^9)*d^6*e^5 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 70*(7*B*a^4*b^6 + 4*A*a^
3*b^7)*d^4*e^7 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 210*(5*B*a^6*b^4 + 6*
A*a^5*b^5)*d^2*e^9 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 495*(3*B*a^8*b^2 +
8*A*a^7*b^3)*e^11)*x^3 + 120*(11*B*b^10*d^9*e^2 + 5*(10*B*a*b^9 + A*b^10)*d^8*e
^3 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e
^5 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^
4*e^7 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4
)*d^2*e^9 + 495*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 715*(2*B*a^9*b + 9*A*a^8*b^
2)*e^11)*x^2 + 16*(11*B*b^10*d^10*e + 5*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 15*(9*B*
a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 70*(7*B*
a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 210*(
5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 4
95*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 715*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 +
1001*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^28*x^16 + 16*d*e^27*x^15 + 120*d^2*e^26*x
^14 + 560*d^3*e^25*x^13 + 1820*d^4*e^24*x^12 + 4368*d^5*e^23*x^11 + 8008*d^6*e^2
2*x^10 + 11440*d^7*e^21*x^9 + 12870*d^8*e^20*x^8 + 11440*d^9*e^19*x^7 + 8008*d^1
0*e^18*x^6 + 4368*d^11*e^17*x^5 + 1820*d^12*e^16*x^4 + 560*d^13*e^15*x^3 + 120*d
^14*e^14*x^2 + 16*d^15*e^13*x + d^16*e^12)

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Fricas [A]  time = 0.213339, size = 2678, normalized size = 9.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^17,x, algorithm="fricas")

[Out]

-1/240240*(48048*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 15015*A*a^10*e^11 + 5*(10*B
*a*b^9 + A*b^10)*d^10*e + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 35*(8*B*a^3*b^7
+ 3*A*a^2*b^8)*d^8*e^3 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 126*(6*B*a^5*
b^5 + 5*A*a^4*b^6)*d^6*e^5 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 330*(4*B*
a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 495*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 715*(
2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 1001*(B*a^10 + 10*A*a^9*b)*d*e^10 + 8008*(11*
B*b^10*d*e^10 + 5*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 11440*(11*B*b^10*d^2*e^9 +
5*(10*B*a*b^9 + A*b^10)*d*e^10 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 12870*
(11*B*b^10*d^3*e^8 + 5*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 15*(9*B*a^2*b^8 + 2*A*a*b
^9)*d*e^10 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 11440*(11*B*b^10*d^4*e^7
+ 5*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 35*(
8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 8
008*(11*B*b^10*d^5*e^6 + 5*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 15*(9*B*a^2*b^8 + 2*A
*a*b^9)*d^3*e^8 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 70*(7*B*a^4*b^6 + 4*A
*a^3*b^7)*d*e^10 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 4368*(11*B*b^10*d
^6*e^5 + 5*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7
+ 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^
9 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^1
1)*x^5 + 1820*(11*B*b^10*d^7*e^4 + 5*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 15*(9*B*a^2
*b^8 + 2*A*a*b^9)*d^5*e^6 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 70*(7*B*a^4
*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 210*(5*B
*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 560
*(11*B*b^10*d^8*e^3 + 5*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 15*(9*B*a^2*b^8 + 2*A*a*
b^9)*d^6*e^5 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 70*(7*B*a^4*b^6 + 4*A*a^
3*b^7)*d^4*e^7 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 210*(5*B*a^6*b^4 + 6*
A*a^5*b^5)*d^2*e^9 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 495*(3*B*a^8*b^2 +
8*A*a^7*b^3)*e^11)*x^3 + 120*(11*B*b^10*d^9*e^2 + 5*(10*B*a*b^9 + A*b^10)*d^8*e
^3 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e
^5 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^
4*e^7 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4
)*d^2*e^9 + 495*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 715*(2*B*a^9*b + 9*A*a^8*b^
2)*e^11)*x^2 + 16*(11*B*b^10*d^10*e + 5*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 15*(9*B*
a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 70*(7*B*
a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 210*(
5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 4
95*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 715*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 +
1001*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^28*x^16 + 16*d*e^27*x^15 + 120*d^2*e^26*x
^14 + 560*d^3*e^25*x^13 + 1820*d^4*e^24*x^12 + 4368*d^5*e^23*x^11 + 8008*d^6*e^2
2*x^10 + 11440*d^7*e^21*x^9 + 12870*d^8*e^20*x^8 + 11440*d^9*e^19*x^7 + 8008*d^1
0*e^18*x^6 + 4368*d^11*e^17*x^5 + 1820*d^12*e^16*x^4 + 560*d^13*e^15*x^3 + 120*d
^14*e^14*x^2 + 16*d^15*e^13*x + d^16*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)**17,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215456, 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)^17,x, algorithm="giac")

[Out]

Done