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3.1090 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx\)

Optimal. Leaf size=385 \[ \frac{b^6 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{2450448 e (d+e x)^{11} (b d-a e)^8}+\frac{b^5 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{222768 e (d+e x)^{12} (b d-a e)^7}+\frac{b^4 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{37128 e (d+e x)^{13} (b d-a e)^6}+\frac{b^3 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{8568 e (d+e x)^{14} (b d-a e)^5}+\frac{b^2 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{2448 e (d+e x)^{15} (b d-a e)^4}+\frac{b (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{816 e (d+e x)^{16} (b d-a e)^3}+\frac{(a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{306 e (d+e x)^{17} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{18 e (d+e x)^{18} (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(18*e*(b*d - a*e)*(d + e*x)^18) + ((11*b*B*d + 7*A*b
*e - 18*a*B*e)*(a + b*x)^11)/(306*e*(b*d - a*e)^2*(d + e*x)^17) + (b*(11*b*B*d +
 7*A*b*e - 18*a*B*e)*(a + b*x)^11)/(816*e*(b*d - a*e)^3*(d + e*x)^16) + (b^2*(11
*b*B*d + 7*A*b*e - 18*a*B*e)*(a + b*x)^11)/(2448*e*(b*d - a*e)^4*(d + e*x)^15) +
 (b^3*(11*b*B*d + 7*A*b*e - 18*a*B*e)*(a + b*x)^11)/(8568*e*(b*d - a*e)^5*(d + e
*x)^14) + (b^4*(11*b*B*d + 7*A*b*e - 18*a*B*e)*(a + b*x)^11)/(37128*e*(b*d - a*e
)^6*(d + e*x)^13) + (b^5*(11*b*B*d + 7*A*b*e - 18*a*B*e)*(a + b*x)^11)/(222768*e
*(b*d - a*e)^7*(d + e*x)^12) + (b^6*(11*b*B*d + 7*A*b*e - 18*a*B*e)*(a + b*x)^11
)/(2450448*e*(b*d - a*e)^8*(d + e*x)^11)

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Rubi [A]  time = 0.547008, antiderivative size = 385, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{b^6 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{2450448 e (d+e x)^{11} (b d-a e)^8}+\frac{b^5 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{222768 e (d+e x)^{12} (b d-a e)^7}+\frac{b^4 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{37128 e (d+e x)^{13} (b d-a e)^6}+\frac{b^3 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{8568 e (d+e x)^{14} (b d-a e)^5}+\frac{b^2 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{2448 e (d+e x)^{15} (b d-a e)^4}+\frac{b (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{816 e (d+e x)^{16} (b d-a e)^3}+\frac{(a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{306 e (d+e x)^{17} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{18 e (d+e x)^{18} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/(d + e*x)^19,x]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(18*e*(b*d - a*e)*(d + e*x)^18) + ((11*b*B*d + 7*A*b
*e - 18*a*B*e)*(a + b*x)^11)/(306*e*(b*d - a*e)^2*(d + e*x)^17) + (b*(11*b*B*d +
 7*A*b*e - 18*a*B*e)*(a + b*x)^11)/(816*e*(b*d - a*e)^3*(d + e*x)^16) + (b^2*(11
*b*B*d + 7*A*b*e - 18*a*B*e)*(a + b*x)^11)/(2448*e*(b*d - a*e)^4*(d + e*x)^15) +
 (b^3*(11*b*B*d + 7*A*b*e - 18*a*B*e)*(a + b*x)^11)/(8568*e*(b*d - a*e)^5*(d + e
*x)^14) + (b^4*(11*b*B*d + 7*A*b*e - 18*a*B*e)*(a + b*x)^11)/(37128*e*(b*d - a*e
)^6*(d + e*x)^13) + (b^5*(11*b*B*d + 7*A*b*e - 18*a*B*e)*(a + b*x)^11)/(222768*e
*(b*d - a*e)^7*(d + e*x)^12) + (b^6*(11*b*B*d + 7*A*b*e - 18*a*B*e)*(a + b*x)^11
)/(2450448*e*(b*d - a*e)^8*(d + e*x)^11)

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Rubi in Sympy [A]  time = 103.81, size = 369, normalized size = 0.96 \[ \frac{b^{6} \left (a + b x\right )^{11} \left (7 A b e - 18 B a e + 11 B b d\right )}{2450448 e \left (d + e x\right )^{11} \left (a e - b d\right )^{8}} - \frac{b^{5} \left (a + b x\right )^{11} \left (7 A b e - 18 B a e + 11 B b d\right )}{222768 e \left (d + e x\right )^{12} \left (a e - b d\right )^{7}} + \frac{b^{4} \left (a + b x\right )^{11} \left (7 A b e - 18 B a e + 11 B b d\right )}{37128 e \left (d + e x\right )^{13} \left (a e - b d\right )^{6}} - \frac{b^{3} \left (a + b x\right )^{11} \left (7 A b e - 18 B a e + 11 B b d\right )}{8568 e \left (d + e x\right )^{14} \left (a e - b d\right )^{5}} + \frac{b^{2} \left (a + b x\right )^{11} \left (7 A b e - 18 B a e + 11 B b d\right )}{2448 e \left (d + e x\right )^{15} \left (a e - b d\right )^{4}} - \frac{b \left (a + b x\right )^{11} \left (7 A b e - 18 B a e + 11 B b d\right )}{816 e \left (d + e x\right )^{16} \left (a e - b d\right )^{3}} + \frac{\left (a + b x\right )^{11} \left (7 A b e - 18 B a e + 11 B b d\right )}{306 e \left (d + e x\right )^{17} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{11} \left (A e - B d\right )}{18 e \left (d + e x\right )^{18} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/(e*x+d)**19,x)

[Out]

b**6*(a + b*x)**11*(7*A*b*e - 18*B*a*e + 11*B*b*d)/(2450448*e*(d + e*x)**11*(a*e
 - b*d)**8) - b**5*(a + b*x)**11*(7*A*b*e - 18*B*a*e + 11*B*b*d)/(222768*e*(d +
e*x)**12*(a*e - b*d)**7) + b**4*(a + b*x)**11*(7*A*b*e - 18*B*a*e + 11*B*b*d)/(3
7128*e*(d + e*x)**13*(a*e - b*d)**6) - b**3*(a + b*x)**11*(7*A*b*e - 18*B*a*e +
11*B*b*d)/(8568*e*(d + e*x)**14*(a*e - b*d)**5) + b**2*(a + b*x)**11*(7*A*b*e -
18*B*a*e + 11*B*b*d)/(2448*e*(d + e*x)**15*(a*e - b*d)**4) - b*(a + b*x)**11*(7*
A*b*e - 18*B*a*e + 11*B*b*d)/(816*e*(d + e*x)**16*(a*e - b*d)**3) + (a + b*x)**1
1*(7*A*b*e - 18*B*a*e + 11*B*b*d)/(306*e*(d + e*x)**17*(a*e - b*d)**2) - (a + b*
x)**11*(A*e - B*d)/(18*e*(d + e*x)**18*(a*e - b*d))

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Mathematica [B]  time = 3.05253, size = 1428, normalized size = 3.71 \[ -\frac{\left (7 A e \left (d^{10}+18 e x d^9+153 e^2 x^2 d^8+816 e^3 x^3 d^7+3060 e^4 x^4 d^6+8568 e^5 x^5 d^5+18564 e^6 x^6 d^4+31824 e^7 x^7 d^3+43758 e^8 x^8 d^2+48620 e^9 x^9 d+43758 e^{10} x^{10}\right )+11 B \left (d^{11}+18 e x d^{10}+153 e^2 x^2 d^9+816 e^3 x^3 d^8+3060 e^4 x^4 d^7+8568 e^5 x^5 d^6+18564 e^6 x^6 d^5+31824 e^7 x^7 d^4+43758 e^8 x^8 d^3+48620 e^9 x^9 d^2+43758 e^{10} x^{10} d+31824 e^{11} x^{11}\right )\right ) b^{10}+14 a e \left (4 A e \left (d^9+18 e x d^8+153 e^2 x^2 d^7+816 e^3 x^3 d^6+3060 e^4 x^4 d^5+8568 e^5 x^5 d^4+18564 e^6 x^6 d^3+31824 e^7 x^7 d^2+43758 e^8 x^8 d+48620 e^9 x^9\right )+5 B \left (d^{10}+18 e x d^9+153 e^2 x^2 d^8+816 e^3 x^3 d^7+3060 e^4 x^4 d^6+8568 e^5 x^5 d^5+18564 e^6 x^6 d^4+31824 e^7 x^7 d^3+43758 e^8 x^8 d^2+48620 e^9 x^9 d+43758 e^{10} x^{10}\right )\right ) b^9+252 a^2 e^2 \left (A e \left (d^8+18 e x d^7+153 e^2 x^2 d^6+816 e^3 x^3 d^5+3060 e^4 x^4 d^4+8568 e^5 x^5 d^3+18564 e^6 x^6 d^2+31824 e^7 x^7 d+43758 e^8 x^8\right )+B \left (d^9+18 e x d^8+153 e^2 x^2 d^7+816 e^3 x^3 d^6+3060 e^4 x^4 d^5+8568 e^5 x^5 d^4+18564 e^6 x^6 d^3+31824 e^7 x^7 d^2+43758 e^8 x^8 d+48620 e^9 x^9\right )\right ) b^8+168 a^3 e^3 \left (5 A e \left (d^7+18 e x d^6+153 e^2 x^2 d^5+816 e^3 x^3 d^4+3060 e^4 x^4 d^3+8568 e^5 x^5 d^2+18564 e^6 x^6 d+31824 e^7 x^7\right )+4 B \left (d^8+18 e x d^7+153 e^2 x^2 d^6+816 e^3 x^3 d^5+3060 e^4 x^4 d^4+8568 e^5 x^5 d^3+18564 e^6 x^6 d^2+31824 e^7 x^7 d+43758 e^8 x^8\right )\right ) b^7+210 a^4 e^4 \left (11 A e \left (d^6+18 e x d^5+153 e^2 x^2 d^4+816 e^3 x^3 d^3+3060 e^4 x^4 d^2+8568 e^5 x^5 d+18564 e^6 x^6\right )+7 B \left (d^7+18 e x d^6+153 e^2 x^2 d^5+816 e^3 x^3 d^4+3060 e^4 x^4 d^3+8568 e^5 x^5 d^2+18564 e^6 x^6 d+31824 e^7 x^7\right )\right ) b^6+2772 a^5 e^5 \left (2 A e \left (d^5+18 e x d^4+153 e^2 x^2 d^3+816 e^3 x^3 d^2+3060 e^4 x^4 d+8568 e^5 x^5\right )+B \left (d^6+18 e x d^5+153 e^2 x^2 d^4+816 e^3 x^3 d^3+3060 e^4 x^4 d^2+8568 e^5 x^5 d+18564 e^6 x^6\right )\right ) b^5+924 a^6 e^6 \left (13 A e \left (d^4+18 e x d^3+153 e^2 x^2 d^2+816 e^3 x^3 d+3060 e^4 x^4\right )+5 B \left (d^5+18 e x d^4+153 e^2 x^2 d^3+816 e^3 x^3 d^2+3060 e^4 x^4 d+8568 e^5 x^5\right )\right ) b^4+3432 a^7 e^7 \left (7 A e \left (d^3+18 e x d^2+153 e^2 x^2 d+816 e^3 x^3\right )+2 B \left (d^4+18 e x d^3+153 e^2 x^2 d^2+816 e^3 x^3 d+3060 e^4 x^4\right )\right ) b^3+9009 a^8 e^8 \left (5 A e \left (d^2+18 e x d+153 e^2 x^2\right )+B \left (d^3+18 e x d^2+153 e^2 x^2 d+816 e^3 x^3\right )\right ) b^2+10010 a^9 e^9 \left (8 A e (d+18 e x)+B \left (d^2+18 e x d+153 e^2 x^2\right )\right ) b+8008 a^{10} e^{10} (17 A e+B (d+18 e x))}{2450448 e^{12} (d+e x)^{18}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^19,x]

[Out]

-(8008*a^10*e^10*(17*A*e + B*(d + 18*e*x)) + 10010*a^9*b*e^9*(8*A*e*(d + 18*e*x)
 + B*(d^2 + 18*d*e*x + 153*e^2*x^2)) + 9009*a^8*b^2*e^8*(5*A*e*(d^2 + 18*d*e*x +
 153*e^2*x^2) + B*(d^3 + 18*d^2*e*x + 153*d*e^2*x^2 + 816*e^3*x^3)) + 3432*a^7*b
^3*e^7*(7*A*e*(d^3 + 18*d^2*e*x + 153*d*e^2*x^2 + 816*e^3*x^3) + 2*B*(d^4 + 18*d
^3*e*x + 153*d^2*e^2*x^2 + 816*d*e^3*x^3 + 3060*e^4*x^4)) + 924*a^6*b^4*e^6*(13*
A*e*(d^4 + 18*d^3*e*x + 153*d^2*e^2*x^2 + 816*d*e^3*x^3 + 3060*e^4*x^4) + 5*B*(d
^5 + 18*d^4*e*x + 153*d^3*e^2*x^2 + 816*d^2*e^3*x^3 + 3060*d*e^4*x^4 + 8568*e^5*
x^5)) + 2772*a^5*b^5*e^5*(2*A*e*(d^5 + 18*d^4*e*x + 153*d^3*e^2*x^2 + 816*d^2*e^
3*x^3 + 3060*d*e^4*x^4 + 8568*e^5*x^5) + B*(d^6 + 18*d^5*e*x + 153*d^4*e^2*x^2 +
 816*d^3*e^3*x^3 + 3060*d^2*e^4*x^4 + 8568*d*e^5*x^5 + 18564*e^6*x^6)) + 210*a^4
*b^6*e^4*(11*A*e*(d^6 + 18*d^5*e*x + 153*d^4*e^2*x^2 + 816*d^3*e^3*x^3 + 3060*d^
2*e^4*x^4 + 8568*d*e^5*x^5 + 18564*e^6*x^6) + 7*B*(d^7 + 18*d^6*e*x + 153*d^5*e^
2*x^2 + 816*d^4*e^3*x^3 + 3060*d^3*e^4*x^4 + 8568*d^2*e^5*x^5 + 18564*d*e^6*x^6
+ 31824*e^7*x^7)) + 168*a^3*b^7*e^3*(5*A*e*(d^7 + 18*d^6*e*x + 153*d^5*e^2*x^2 +
 816*d^4*e^3*x^3 + 3060*d^3*e^4*x^4 + 8568*d^2*e^5*x^5 + 18564*d*e^6*x^6 + 31824
*e^7*x^7) + 4*B*(d^8 + 18*d^7*e*x + 153*d^6*e^2*x^2 + 816*d^5*e^3*x^3 + 3060*d^4
*e^4*x^4 + 8568*d^3*e^5*x^5 + 18564*d^2*e^6*x^6 + 31824*d*e^7*x^7 + 43758*e^8*x^
8)) + 252*a^2*b^8*e^2*(A*e*(d^8 + 18*d^7*e*x + 153*d^6*e^2*x^2 + 816*d^5*e^3*x^3
 + 3060*d^4*e^4*x^4 + 8568*d^3*e^5*x^5 + 18564*d^2*e^6*x^6 + 31824*d*e^7*x^7 + 4
3758*e^8*x^8) + B*(d^9 + 18*d^8*e*x + 153*d^7*e^2*x^2 + 816*d^6*e^3*x^3 + 3060*d
^5*e^4*x^4 + 8568*d^4*e^5*x^5 + 18564*d^3*e^6*x^6 + 31824*d^2*e^7*x^7 + 43758*d*
e^8*x^8 + 48620*e^9*x^9)) + 14*a*b^9*e*(4*A*e*(d^9 + 18*d^8*e*x + 153*d^7*e^2*x^
2 + 816*d^6*e^3*x^3 + 3060*d^5*e^4*x^4 + 8568*d^4*e^5*x^5 + 18564*d^3*e^6*x^6 +
31824*d^2*e^7*x^7 + 43758*d*e^8*x^8 + 48620*e^9*x^9) + 5*B*(d^10 + 18*d^9*e*x +
153*d^8*e^2*x^2 + 816*d^7*e^3*x^3 + 3060*d^6*e^4*x^4 + 8568*d^5*e^5*x^5 + 18564*
d^4*e^6*x^6 + 31824*d^3*e^7*x^7 + 43758*d^2*e^8*x^8 + 48620*d*e^9*x^9 + 43758*e^
10*x^10)) + b^10*(7*A*e*(d^10 + 18*d^9*e*x + 153*d^8*e^2*x^2 + 816*d^7*e^3*x^3 +
 3060*d^6*e^4*x^4 + 8568*d^5*e^5*x^5 + 18564*d^4*e^6*x^6 + 31824*d^3*e^7*x^7 + 4
3758*d^2*e^8*x^8 + 48620*d*e^9*x^9 + 43758*e^10*x^10) + 11*B*(d^11 + 18*d^10*e*x
 + 153*d^9*e^2*x^2 + 816*d^8*e^3*x^3 + 3060*d^7*e^4*x^4 + 8568*d^6*e^5*x^5 + 185
64*d^5*e^6*x^6 + 31824*d^4*e^7*x^7 + 43758*d^3*e^8*x^8 + 48620*d^2*e^9*x^9 + 437
58*d*e^10*x^10 + 31824*e^11*x^11)))/(2450448*e^12*(d + e*x)^18)

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Maple [B]  time = 0.016, size = 1942, normalized size = 5. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/(e*x+d)^19,x)

[Out]

-15/7*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^3-18
9*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)^14-1/18*(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)^18-5/9*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)^9-3/2*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)^10-1/8*b^9*(A*b*e+10*B*a*e-11*B*b*d)/e^12/(e*x+d)^8-1/7*B*b^10/e^12
/(e*x+d)^7-7/2*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)^12-30/11*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)^11-5
/16*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)^16-1/17*(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)^17-42/13*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)^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)^
15

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Maxima [A]  time = 1.60563, size = 2708, normalized size = 7.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^19,x, algorithm="maxima")

[Out]

-1/2450448*(350064*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 136136*A*a^10*e^11 + 7*(1
0*B*a*b^9 + A*b^10)*d^10*e + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 84*(8*B*a^3*
b^7 + 3*A*a^2*b^8)*d^8*e^3 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 462*(6*B*
a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 1716*
(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 3003*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 +
 5005*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 8008*(B*a^10 + 10*A*a^9*b)*d*e^10 + 43
758*(11*B*b^10*d*e^10 + 7*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 48620*(11*B*b^10*d^
2*e^9 + 7*(10*B*a*b^9 + A*b^10)*d*e^10 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9
+ 43758*(11*B*b^10*d^3*e^8 + 7*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 28*(9*B*a^2*b^8 +
 2*A*a*b^9)*d*e^10 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 31824*(11*B*b^10
*d^4*e^7 + 7*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^
9 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11
)*x^7 + 18564*(11*B*b^10*d^5*e^6 + 7*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 28*(9*B*a^2
*b^8 + 2*A*a*b^9)*d^3*e^8 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 210*(7*B*a^
4*b^6 + 4*A*a^3*b^7)*d*e^10 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 8568*(
11*B*b^10*d^6*e^5 + 7*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 28*(9*B*a^2*b^8 + 2*A*a*b^
9)*d^4*e^7 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 210*(7*B*a^4*b^6 + 4*A*a^3
*b^7)*d^2*e^9 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 924*(5*B*a^6*b^4 + 6*A*
a^5*b^5)*e^11)*x^5 + 3060*(11*B*b^10*d^7*e^4 + 7*(10*B*a*b^9 + A*b^10)*d^6*e^5 +
 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 +
 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e
^9 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 1716*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e
^11)*x^4 + 816*(11*B*b^10*d^8*e^3 + 7*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 28*(9*B*a^
2*b^8 + 2*A*a*b^9)*d^6*e^5 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 210*(7*B*a
^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 924*(5
*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 1716*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 30
03*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 153*(11*B*b^10*d^9*e^2 + 7*(10*B*a*b^
9 + A*b^10)*d^8*e^3 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 84*(8*B*a^3*b^7 + 3
*A*a^2*b^8)*d^6*e^5 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 462*(6*B*a^5*b^5
 + 5*A*a^4*b^6)*d^4*e^7 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 1716*(4*B*a^
7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 3003*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 5005*(2
*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 18*(11*B*b^10*d^10*e + 7*(10*B*a*b^9 + A*b^1
0)*d^9*e^2 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^
8)*d^7*e^4 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 462*(6*B*a^5*b^5 + 5*A*a^
4*b^6)*d^5*e^6 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 1716*(4*B*a^7*b^3 + 7
*A*a^6*b^4)*d^3*e^8 + 3003*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 5005*(2*B*a^9*b
 + 9*A*a^8*b^2)*d*e^10 + 8008*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^30*x^18 + 18*d*e
^29*x^17 + 153*d^2*e^28*x^16 + 816*d^3*e^27*x^15 + 3060*d^4*e^26*x^14 + 8568*d^5
*e^25*x^13 + 18564*d^6*e^24*x^12 + 31824*d^7*e^23*x^11 + 43758*d^8*e^22*x^10 + 4
8620*d^9*e^21*x^9 + 43758*d^10*e^20*x^8 + 31824*d^11*e^19*x^7 + 18564*d^12*e^18*
x^6 + 8568*d^13*e^17*x^5 + 3060*d^14*e^16*x^4 + 816*d^15*e^15*x^3 + 153*d^16*e^1
4*x^2 + 18*d^17*e^13*x + d^18*e^12)

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Fricas [A]  time = 0.218566, size = 2708, normalized size = 7.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^19,x, algorithm="fricas")

[Out]

-1/2450448*(350064*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 136136*A*a^10*e^11 + 7*(1
0*B*a*b^9 + A*b^10)*d^10*e + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 84*(8*B*a^3*
b^7 + 3*A*a^2*b^8)*d^8*e^3 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 462*(6*B*
a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 1716*
(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 3003*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 +
 5005*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 8008*(B*a^10 + 10*A*a^9*b)*d*e^10 + 43
758*(11*B*b^10*d*e^10 + 7*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 48620*(11*B*b^10*d^
2*e^9 + 7*(10*B*a*b^9 + A*b^10)*d*e^10 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9
+ 43758*(11*B*b^10*d^3*e^8 + 7*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 28*(9*B*a^2*b^8 +
 2*A*a*b^9)*d*e^10 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 31824*(11*B*b^10
*d^4*e^7 + 7*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^
9 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11
)*x^7 + 18564*(11*B*b^10*d^5*e^6 + 7*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 28*(9*B*a^2
*b^8 + 2*A*a*b^9)*d^3*e^8 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 210*(7*B*a^
4*b^6 + 4*A*a^3*b^7)*d*e^10 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 8568*(
11*B*b^10*d^6*e^5 + 7*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 28*(9*B*a^2*b^8 + 2*A*a*b^
9)*d^4*e^7 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 210*(7*B*a^4*b^6 + 4*A*a^3
*b^7)*d^2*e^9 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 924*(5*B*a^6*b^4 + 6*A*
a^5*b^5)*e^11)*x^5 + 3060*(11*B*b^10*d^7*e^4 + 7*(10*B*a*b^9 + A*b^10)*d^6*e^5 +
 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 +
 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e
^9 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 1716*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e
^11)*x^4 + 816*(11*B*b^10*d^8*e^3 + 7*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 28*(9*B*a^
2*b^8 + 2*A*a*b^9)*d^6*e^5 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 210*(7*B*a
^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 924*(5
*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 1716*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 30
03*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 153*(11*B*b^10*d^9*e^2 + 7*(10*B*a*b^
9 + A*b^10)*d^8*e^3 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 84*(8*B*a^3*b^7 + 3
*A*a^2*b^8)*d^6*e^5 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 462*(6*B*a^5*b^5
 + 5*A*a^4*b^6)*d^4*e^7 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 1716*(4*B*a^
7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 3003*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 5005*(2
*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 18*(11*B*b^10*d^10*e + 7*(10*B*a*b^9 + A*b^1
0)*d^9*e^2 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^
8)*d^7*e^4 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 462*(6*B*a^5*b^5 + 5*A*a^
4*b^6)*d^5*e^6 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 1716*(4*B*a^7*b^3 + 7
*A*a^6*b^4)*d^3*e^8 + 3003*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 5005*(2*B*a^9*b
 + 9*A*a^8*b^2)*d*e^10 + 8008*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^30*x^18 + 18*d*e
^29*x^17 + 153*d^2*e^28*x^16 + 816*d^3*e^27*x^15 + 3060*d^4*e^26*x^14 + 8568*d^5
*e^25*x^13 + 18564*d^6*e^24*x^12 + 31824*d^7*e^23*x^11 + 43758*d^8*e^22*x^10 + 4
8620*d^9*e^21*x^9 + 43758*d^10*e^20*x^8 + 31824*d^11*e^19*x^7 + 18564*d^12*e^18*
x^6 + 8568*d^13*e^17*x^5 + 3060*d^14*e^16*x^4 + 816*d^15*e^15*x^3 + 153*d^16*e^1
4*x^2 + 18*d^17*e^13*x + d^18*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)**19,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215653, 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)^19,x, algorithm="giac")

[Out]

Done

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