3.1091 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{20}} \, dx\)

Optimal. Leaf size=460 \[ \frac{b^9 (-10 a B e-A b e+11 b B d)}{9 e^{12} (d+e x)^9}-\frac{b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{2 e^{12} (d+e x)^{10}}+\frac{15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{11 e^{12} (d+e x)^{11}}-\frac{5 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{2 e^{12} (d+e x)^{12}}+\frac{42 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{13 e^{12} (d+e x)^{13}}-\frac{3 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)^{14}}+\frac{2 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^{15}}-\frac{15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{16 e^{12} (d+e x)^{16}}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{17 e^{12} (d+e x)^{17}}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{18 e^{12} (d+e x)^{18}}+\frac{(b d-a e)^{10} (B d-A e)}{19 e^{12} (d+e x)^{19}}-\frac{b^{10} B}{8 e^{12} (d+e x)^8} \]

[Out]

((b*d - a*e)^10*(B*d - A*e))/(19*e^12*(d + e*x)^19) - ((b*d - a*e)^9*(11*b*B*d -
10*A*b*e - a*B*e))/(18*e^12*(d + e*x)^18) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*
b*e - 2*a*B*e))/(17*e^12*(d + e*x)^17) - (15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b
*e - 3*a*B*e))/(16*e^12*(d + e*x)^16) + (2*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e
- 4*a*B*e))/(e^12*(d + e*x)^15) - (3*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*
a*B*e))/(e^12*(d + e*x)^14) + (42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*
e))/(13*e^12*(d + e*x)^13) - (5*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e)
)/(2*e^12*(d + e*x)^12) + (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e))/
(11*e^12*(d + e*x)^11) - (b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e))/(2*e^1
2*(d + e*x)^10) + (b^9*(11*b*B*d - A*b*e - 10*a*B*e))/(9*e^12*(d + e*x)^9) - (b^
10*B)/(8*e^12*(d + e*x)^8)

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Rubi [A]  time = 2.48261, antiderivative size = 460, 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 \[ \frac{b^9 (-10 a B e-A b e+11 b B d)}{9 e^{12} (d+e x)^9}-\frac{b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{2 e^{12} (d+e x)^{10}}+\frac{15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{11 e^{12} (d+e x)^{11}}-\frac{5 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{2 e^{12} (d+e x)^{12}}+\frac{42 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{13 e^{12} (d+e x)^{13}}-\frac{3 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)^{14}}+\frac{2 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^{15}}-\frac{15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{16 e^{12} (d+e x)^{16}}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{17 e^{12} (d+e x)^{17}}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{18 e^{12} (d+e x)^{18}}+\frac{(b d-a e)^{10} (B d-A e)}{19 e^{12} (d+e x)^{19}}-\frac{b^{10} B}{8 e^{12} (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/(d + e*x)^20,x]

[Out]

((b*d - a*e)^10*(B*d - A*e))/(19*e^12*(d + e*x)^19) - ((b*d - a*e)^9*(11*b*B*d -
10*A*b*e - a*B*e))/(18*e^12*(d + e*x)^18) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*
b*e - 2*a*B*e))/(17*e^12*(d + e*x)^17) - (15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b
*e - 3*a*B*e))/(16*e^12*(d + e*x)^16) + (2*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e
- 4*a*B*e))/(e^12*(d + e*x)^15) - (3*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*
a*B*e))/(e^12*(d + e*x)^14) + (42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*
e))/(13*e^12*(d + e*x)^13) - (5*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e)
)/(2*e^12*(d + e*x)^12) + (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e))/
(11*e^12*(d + e*x)^11) - (b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e))/(2*e^1
2*(d + e*x)^10) + (b^9*(11*b*B*d - A*b*e - 10*a*B*e))/(9*e^12*(d + e*x)^9) - (b^
10*B)/(8*e^12*(d + e*x)^8)

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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)**20,x)

[Out]

Timed out

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Mathematica [B]  time = 3.0945, size = 1433, normalized size = 3.12 \[ -\frac{\left (8 A e \left (d^{10}+19 e x d^9+171 e^2 x^2 d^8+969 e^3 x^3 d^7+3876 e^4 x^4 d^6+11628 e^5 x^5 d^5+27132 e^6 x^6 d^4+50388 e^7 x^7 d^3+75582 e^8 x^8 d^2+92378 e^9 x^9 d+92378 e^{10} x^{10}\right )+11 B \left (d^{11}+19 e x d^{10}+171 e^2 x^2 d^9+969 e^3 x^3 d^8+3876 e^4 x^4 d^7+11628 e^5 x^5 d^6+27132 e^6 x^6 d^5+50388 e^7 x^7 d^4+75582 e^8 x^8 d^3+92378 e^9 x^9 d^2+92378 e^{10} x^{10} d+75582 e^{11} x^{11}\right )\right ) b^{10}+8 a e \left (9 A e \left (d^9+19 e x d^8+171 e^2 x^2 d^7+969 e^3 x^3 d^6+3876 e^4 x^4 d^5+11628 e^5 x^5 d^4+27132 e^6 x^6 d^3+50388 e^7 x^7 d^2+75582 e^8 x^8 d+92378 e^9 x^9\right )+10 B \left (d^{10}+19 e x d^9+171 e^2 x^2 d^8+969 e^3 x^3 d^7+3876 e^4 x^4 d^6+11628 e^5 x^5 d^5+27132 e^6 x^6 d^4+50388 e^7 x^7 d^3+75582 e^8 x^8 d^2+92378 e^9 x^9 d+92378 e^{10} x^{10}\right )\right ) b^9+36 a^2 e^2 \left (10 A e \left (d^8+19 e x d^7+171 e^2 x^2 d^6+969 e^3 x^3 d^5+3876 e^4 x^4 d^4+11628 e^5 x^5 d^3+27132 e^6 x^6 d^2+50388 e^7 x^7 d+75582 e^8 x^8\right )+9 B \left (d^9+19 e x d^8+171 e^2 x^2 d^7+969 e^3 x^3 d^6+3876 e^4 x^4 d^5+11628 e^5 x^5 d^4+27132 e^6 x^6 d^3+50388 e^7 x^7 d^2+75582 e^8 x^8 d+92378 e^9 x^9\right )\right ) b^8+120 a^3 e^3 \left (11 A e \left (d^7+19 e x d^6+171 e^2 x^2 d^5+969 e^3 x^3 d^4+3876 e^4 x^4 d^3+11628 e^5 x^5 d^2+27132 e^6 x^6 d+50388 e^7 x^7\right )+8 B \left (d^8+19 e x d^7+171 e^2 x^2 d^6+969 e^3 x^3 d^5+3876 e^4 x^4 d^4+11628 e^5 x^5 d^3+27132 e^6 x^6 d^2+50388 e^7 x^7 d+75582 e^8 x^8\right )\right ) b^7+330 a^4 e^4 \left (12 A e \left (d^6+19 e x d^5+171 e^2 x^2 d^4+969 e^3 x^3 d^3+3876 e^4 x^4 d^2+11628 e^5 x^5 d+27132 e^6 x^6\right )+7 B \left (d^7+19 e x d^6+171 e^2 x^2 d^5+969 e^3 x^3 d^4+3876 e^4 x^4 d^3+11628 e^5 x^5 d^2+27132 e^6 x^6 d+50388 e^7 x^7\right )\right ) b^6+792 a^5 e^5 \left (13 A e \left (d^5+19 e x d^4+171 e^2 x^2 d^3+969 e^3 x^3 d^2+3876 e^4 x^4 d+11628 e^5 x^5\right )+6 B \left (d^6+19 e x d^5+171 e^2 x^2 d^4+969 e^3 x^3 d^3+3876 e^4 x^4 d^2+11628 e^5 x^5 d+27132 e^6 x^6\right )\right ) b^5+1716 a^6 e^6 \left (14 A e \left (d^4+19 e x d^3+171 e^2 x^2 d^2+969 e^3 x^3 d+3876 e^4 x^4\right )+5 B \left (d^5+19 e x d^4+171 e^2 x^2 d^3+969 e^3 x^3 d^2+3876 e^4 x^4 d+11628 e^5 x^5\right )\right ) b^4+3432 a^7 e^7 \left (15 A e \left (d^3+19 e x d^2+171 e^2 x^2 d+969 e^3 x^3\right )+4 B \left (d^4+19 e x d^3+171 e^2 x^2 d^2+969 e^3 x^3 d+3876 e^4 x^4\right )\right ) b^3+6435 a^8 e^8 \left (16 A e \left (d^2+19 e x d+171 e^2 x^2\right )+3 B \left (d^3+19 e x d^2+171 e^2 x^2 d+969 e^3 x^3\right )\right ) b^2+11440 a^9 e^9 \left (17 A e (d+19 e x)+2 B \left (d^2+19 e x d+171 e^2 x^2\right )\right ) b+19448 a^{10} e^{10} (18 A e+B (d+19 e x))}{6651216 e^{12} (d+e x)^{19}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^20,x]

[Out]

-(19448*a^10*e^10*(18*A*e + B*(d + 19*e*x)) + 11440*a^9*b*e^9*(17*A*e*(d + 19*e*
x) + 2*B*(d^2 + 19*d*e*x + 171*e^2*x^2)) + 6435*a^8*b^2*e^8*(16*A*e*(d^2 + 19*d*
e*x + 171*e^2*x^2) + 3*B*(d^3 + 19*d^2*e*x + 171*d*e^2*x^2 + 969*e^3*x^3)) + 343
2*a^7*b^3*e^7*(15*A*e*(d^3 + 19*d^2*e*x + 171*d*e^2*x^2 + 969*e^3*x^3) + 4*B*(d^
4 + 19*d^3*e*x + 171*d^2*e^2*x^2 + 969*d*e^3*x^3 + 3876*e^4*x^4)) + 1716*a^6*b^4
*e^6*(14*A*e*(d^4 + 19*d^3*e*x + 171*d^2*e^2*x^2 + 969*d*e^3*x^3 + 3876*e^4*x^4)
+ 5*B*(d^5 + 19*d^4*e*x + 171*d^3*e^2*x^2 + 969*d^2*e^3*x^3 + 3876*d*e^4*x^4 +
11628*e^5*x^5)) + 792*a^5*b^5*e^5*(13*A*e*(d^5 + 19*d^4*e*x + 171*d^3*e^2*x^2 +
969*d^2*e^3*x^3 + 3876*d*e^4*x^4 + 11628*e^5*x^5) + 6*B*(d^6 + 19*d^5*e*x + 171*
d^4*e^2*x^2 + 969*d^3*e^3*x^3 + 3876*d^2*e^4*x^4 + 11628*d*e^5*x^5 + 27132*e^6*x
^6)) + 330*a^4*b^6*e^4*(12*A*e*(d^6 + 19*d^5*e*x + 171*d^4*e^2*x^2 + 969*d^3*e^3
*x^3 + 3876*d^2*e^4*x^4 + 11628*d*e^5*x^5 + 27132*e^6*x^6) + 7*B*(d^7 + 19*d^6*e
*x + 171*d^5*e^2*x^2 + 969*d^4*e^3*x^3 + 3876*d^3*e^4*x^4 + 11628*d^2*e^5*x^5 +
27132*d*e^6*x^6 + 50388*e^7*x^7)) + 120*a^3*b^7*e^3*(11*A*e*(d^7 + 19*d^6*e*x +
171*d^5*e^2*x^2 + 969*d^4*e^3*x^3 + 3876*d^3*e^4*x^4 + 11628*d^2*e^5*x^5 + 27132
*d*e^6*x^6 + 50388*e^7*x^7) + 8*B*(d^8 + 19*d^7*e*x + 171*d^6*e^2*x^2 + 969*d^5*
e^3*x^3 + 3876*d^4*e^4*x^4 + 11628*d^3*e^5*x^5 + 27132*d^2*e^6*x^6 + 50388*d*e^7
*x^7 + 75582*e^8*x^8)) + 36*a^2*b^8*e^2*(10*A*e*(d^8 + 19*d^7*e*x + 171*d^6*e^2*
x^2 + 969*d^5*e^3*x^3 + 3876*d^4*e^4*x^4 + 11628*d^3*e^5*x^5 + 27132*d^2*e^6*x^6
+ 50388*d*e^7*x^7 + 75582*e^8*x^8) + 9*B*(d^9 + 19*d^8*e*x + 171*d^7*e^2*x^2 +
969*d^6*e^3*x^3 + 3876*d^5*e^4*x^4 + 11628*d^4*e^5*x^5 + 27132*d^3*e^6*x^6 + 503
88*d^2*e^7*x^7 + 75582*d*e^8*x^8 + 92378*e^9*x^9)) + 8*a*b^9*e*(9*A*e*(d^9 + 19*
d^8*e*x + 171*d^7*e^2*x^2 + 969*d^6*e^3*x^3 + 3876*d^5*e^4*x^4 + 11628*d^4*e^5*x
^5 + 27132*d^3*e^6*x^6 + 50388*d^2*e^7*x^7 + 75582*d*e^8*x^8 + 92378*e^9*x^9) +
10*B*(d^10 + 19*d^9*e*x + 171*d^8*e^2*x^2 + 969*d^7*e^3*x^3 + 3876*d^6*e^4*x^4 +
11628*d^5*e^5*x^5 + 27132*d^4*e^6*x^6 + 50388*d^3*e^7*x^7 + 75582*d^2*e^8*x^8 +
92378*d*e^9*x^9 + 92378*e^10*x^10)) + b^10*(8*A*e*(d^10 + 19*d^9*e*x + 171*d^8*
e^2*x^2 + 969*d^7*e^3*x^3 + 3876*d^6*e^4*x^4 + 11628*d^5*e^5*x^5 + 27132*d^4*e^6
*x^6 + 50388*d^3*e^7*x^7 + 75582*d^2*e^8*x^8 + 92378*d*e^9*x^9 + 92378*e^10*x^10
) + 11*B*(d^11 + 19*d^10*e*x + 171*d^9*e^2*x^2 + 969*d^8*e^3*x^3 + 3876*d^7*e^4*
x^4 + 11628*d^6*e^5*x^5 + 27132*d^5*e^6*x^6 + 50388*d^4*e^7*x^7 + 75582*d^3*e^8*
x^8 + 92378*d^2*e^9*x^9 + 92378*d*e^10*x^10 + 75582*e^11*x^11)))/(6651216*e^12*(
d + e*x)^19)

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Maple [B]  time = 0.016, 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)^20,x)

[Out]

-3*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)^14-1/18*(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)^18-1/9*b^9*(A*b*e+1
0*B*a*e-11*B*b*d)/e^12/(e*x+d)^9-1/2*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)^10-1/8*b^10*B/e^12/(e*x+d)^8-5/2*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)^12-1
5/11*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)^11-1/19*(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)^19-15/16*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^6-3
36*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)^16-5/17*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+9
0*B*a*b^8*d^8*e-11*B*b^9*d^9)/e^12/(e*x+d)^17-42/13*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)^13-2*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
)^15

_______________________________________________________________________________________

Maxima [A]  time = 1.59378, size = 2723, normalized size = 5.92 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^20,x, algorithm="maxima")

[Out]

-1/6651216*(831402*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 350064*A*a^10*e^11 + 8*(1
0*B*a*b^9 + A*b^10)*d^10*e + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 120*(8*B*a^3
*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 792*(6*B
*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 343
2*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 6435*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8
+ 11440*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 19448*(B*a^10 + 10*A*a^9*b)*d*e^10
+ 92378*(11*B*b^10*d*e^10 + 8*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 92378*(11*B*b^1
0*d^2*e^9 + 8*(10*B*a*b^9 + A*b^10)*d*e^10 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*
x^9 + 75582*(11*B*b^10*d^3*e^8 + 8*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 36*(9*B*a^2*b
^8 + 2*A*a*b^9)*d*e^10 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 50388*(11*B
*b^10*d^4*e^7 + 8*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d
^2*e^9 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7
)*e^11)*x^7 + 27132*(11*B*b^10*d^5*e^6 + 8*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 36*(9
*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 330*
(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 +
11628*(11*B*b^10*d^6*e^5 + 8*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 36*(9*B*a^2*b^8 +
2*A*a*b^9)*d^4*e^7 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 330*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*d^2*e^9 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 1716*(5*B*a^6*
b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 3876*(11*B*b^10*d^7*e^4 + 8*(10*B*a*b^9 + A*b^10)
*d^6*e^5 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8
)*d^4*e^7 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 792*(6*B*a^5*b^5 + 5*A*a^4
*b^6)*d^2*e^9 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 3432*(4*B*a^7*b^3 + 7*
A*a^6*b^4)*e^11)*x^4 + 969*(11*B*b^10*d^8*e^3 + 8*(10*B*a*b^9 + A*b^10)*d^7*e^4
+ 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6
+ 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3
*e^8 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 3432*(4*B*a^7*b^3 + 7*A*a^6*b^
4)*d*e^10 + 6435*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 171*(11*B*b^10*d^9*e^2
+ 8*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 120*(
8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 7
92*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^
8 + 3432*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 6435*(3*B*a^8*b^2 + 8*A*a^7*b^3)*
d*e^10 + 11440*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 19*(11*B*b^10*d^10*e + 8*(1
0*B*a*b^9 + A*b^10)*d^9*e^2 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 120*(8*B*a^
3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 792*(6*
B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 34
32*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 6435*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^
9 + 11440*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 19448*(B*a^10 + 10*A*a^9*b)*e^11)*x
)/(e^31*x^19 + 19*d*e^30*x^18 + 171*d^2*e^29*x^17 + 969*d^3*e^28*x^16 + 3876*d^4
*e^27*x^15 + 11628*d^5*e^26*x^14 + 27132*d^6*e^25*x^13 + 50388*d^7*e^24*x^12 + 7
5582*d^8*e^23*x^11 + 92378*d^9*e^22*x^10 + 92378*d^10*e^21*x^9 + 75582*d^11*e^20
*x^8 + 50388*d^12*e^19*x^7 + 27132*d^13*e^18*x^6 + 11628*d^14*e^17*x^5 + 3876*d^
15*e^16*x^4 + 969*d^16*e^15*x^3 + 171*d^17*e^14*x^2 + 19*d^18*e^13*x + d^19*e^12
)

_______________________________________________________________________________________

Fricas [A]  time = 0.222085, size = 2723, normalized size = 5.92 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^20,x, algorithm="fricas")

[Out]

-1/6651216*(831402*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 350064*A*a^10*e^11 + 8*(1
0*B*a*b^9 + A*b^10)*d^10*e + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 120*(8*B*a^3
*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 792*(6*B
*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 343
2*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 6435*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8
+ 11440*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 19448*(B*a^10 + 10*A*a^9*b)*d*e^10
+ 92378*(11*B*b^10*d*e^10 + 8*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 92378*(11*B*b^1
0*d^2*e^9 + 8*(10*B*a*b^9 + A*b^10)*d*e^10 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*
x^9 + 75582*(11*B*b^10*d^3*e^8 + 8*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 36*(9*B*a^2*b
^8 + 2*A*a*b^9)*d*e^10 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 50388*(11*B
*b^10*d^4*e^7 + 8*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d
^2*e^9 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7
)*e^11)*x^7 + 27132*(11*B*b^10*d^5*e^6 + 8*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 36*(9
*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 330*
(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 +
11628*(11*B*b^10*d^6*e^5 + 8*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 36*(9*B*a^2*b^8 +
2*A*a*b^9)*d^4*e^7 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 330*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*d^2*e^9 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 1716*(5*B*a^6*
b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 3876*(11*B*b^10*d^7*e^4 + 8*(10*B*a*b^9 + A*b^10)
*d^6*e^5 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8
)*d^4*e^7 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 792*(6*B*a^5*b^5 + 5*A*a^4
*b^6)*d^2*e^9 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 3432*(4*B*a^7*b^3 + 7*
A*a^6*b^4)*e^11)*x^4 + 969*(11*B*b^10*d^8*e^3 + 8*(10*B*a*b^9 + A*b^10)*d^7*e^4
+ 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6
+ 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 792*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3
*e^8 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 3432*(4*B*a^7*b^3 + 7*A*a^6*b^
4)*d*e^10 + 6435*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 171*(11*B*b^10*d^9*e^2
+ 8*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 120*(
8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 7
92*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^
8 + 3432*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 6435*(3*B*a^8*b^2 + 8*A*a^7*b^3)*
d*e^10 + 11440*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 19*(11*B*b^10*d^10*e + 8*(1
0*B*a*b^9 + A*b^10)*d^9*e^2 + 36*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 120*(8*B*a^
3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 330*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 792*(6*
B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 1716*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 34
32*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 6435*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^
9 + 11440*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 19448*(B*a^10 + 10*A*a^9*b)*e^11)*x
)/(e^31*x^19 + 19*d*e^30*x^18 + 171*d^2*e^29*x^17 + 969*d^3*e^28*x^16 + 3876*d^4
*e^27*x^15 + 11628*d^5*e^26*x^14 + 27132*d^6*e^25*x^13 + 50388*d^7*e^24*x^12 + 7
5582*d^8*e^23*x^11 + 92378*d^9*e^22*x^10 + 92378*d^10*e^21*x^9 + 75582*d^11*e^20
*x^8 + 50388*d^12*e^19*x^7 + 27132*d^13*e^18*x^6 + 11628*d^14*e^17*x^5 + 3876*d^
15*e^16*x^4 + 969*d^16*e^15*x^3 + 171*d^17*e^14*x^2 + 19*d^18*e^13*x + d^19*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)**20,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.213929, 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)^20,x, algorithm="giac")

[Out]

Done