[next] [prev] [prev-tail] [tail] [up]

3.1073 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^2} \, dx\)

Optimal. Leaf size=445 \[ -\frac{b^9 (d+e x)^9 (-10 a B e-A b e+11 b B d)}{9 e^{12}}+\frac{5 b^8 (d+e x)^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{8 e^{12}}-\frac{15 b^7 (d+e x)^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{7 e^{12}}+\frac{5 b^6 (d+e x)^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12}}-\frac{42 b^5 (d+e x)^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{5 e^{12}}+\frac{21 b^4 (d+e x)^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{2 e^{12}}-\frac{10 b^3 (d+e x)^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12}}+\frac{15 b^2 (d+e x)^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{2 e^{12}}+\frac{(b d-a e)^{10} (B d-A e)}{e^{12} (d+e x)}+\frac{(b d-a e)^9 \log (d+e x) (-a B e-10 A b e+11 b B d)}{e^{12}}-\frac{5 b x (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{11}}+\frac{b^{10} B (d+e x)^{10}}{10 e^{12}} \]

[Out]

(-5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*
d - A*e))/(e^12*(d + e*x)) + (15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e
)*(d + e*x)^2)/(2*e^12) - (10*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e)*(
d + e*x)^3)/e^12 + (21*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e)*(d + e*x
)^4)/(2*e^12) - (42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*(d + e*x)^5
)/(5*e^12) + (5*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e)*(d + e*x)^6)/e^
12 - (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*(d + e*x)^7)/(7*e^12)
+ (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^8)/(8*e^12) - (b^9
*(11*b*B*d - A*b*e - 10*a*B*e)*(d + e*x)^9)/(9*e^12) + (b^10*B*(d + e*x)^10)/(10
*e^12) + ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e)*Log[d + e*x])/e^12

_______________________________________________________________________________________

Rubi [A]  time = 6.50756, antiderivative size = 445, 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 \[ -\frac{b^9 (d+e x)^9 (-10 a B e-A b e+11 b B d)}{9 e^{12}}+\frac{5 b^8 (d+e x)^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{8 e^{12}}-\frac{15 b^7 (d+e x)^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{7 e^{12}}+\frac{5 b^6 (d+e x)^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12}}-\frac{42 b^5 (d+e x)^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{5 e^{12}}+\frac{21 b^4 (d+e x)^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{2 e^{12}}-\frac{10 b^3 (d+e x)^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12}}+\frac{15 b^2 (d+e x)^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{2 e^{12}}+\frac{(b d-a e)^{10} (B d-A e)}{e^{12} (d+e x)}+\frac{(b d-a e)^9 \log (d+e x) (-a B e-10 A b e+11 b B d)}{e^{12}}-\frac{5 b x (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{11}}+\frac{b^{10} B (d+e x)^{10}}{10 e^{12}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/(d + e*x)^2,x]

[Out]

(-5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*
d - A*e))/(e^12*(d + e*x)) + (15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e
)*(d + e*x)^2)/(2*e^12) - (10*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e)*(
d + e*x)^3)/e^12 + (21*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e)*(d + e*x
)^4)/(2*e^12) - (42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*(d + e*x)^5
)/(5*e^12) + (5*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e)*(d + e*x)^6)/e^
12 - (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*(d + e*x)^7)/(7*e^12)
+ (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^8)/(8*e^12) - (b^9
*(11*b*B*d - A*b*e - 10*a*B*e)*(d + e*x)^9)/(9*e^12) + (b^10*B*(d + e*x)^10)/(10
*e^12) + ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e)*Log[d + e*x])/e^12

_______________________________________________________________________________________

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)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [B]  time = 1.56113, size = 1486, normalized size = 3.34 \[ \frac{\left (10 A e \left (-252 d^{10}+2268 e x d^9+1260 e^2 x^2 d^8-420 e^3 x^3 d^7+210 e^4 x^4 d^6-126 e^5 x^5 d^5+84 e^6 x^6 d^4-60 e^7 x^7 d^3+45 e^8 x^8 d^2-35 e^9 x^9 d+28 e^{10} x^{10}\right )+B \left (2520 d^{11}-25200 e x d^{10}-13860 e^2 x^2 d^9+4620 e^3 x^3 d^8-2310 e^4 x^4 d^7+1386 e^5 x^5 d^6-924 e^6 x^6 d^5+660 e^7 x^7 d^4-495 e^8 x^8 d^3+385 e^9 x^9 d^2-308 e^{10} x^{10} d+252 e^{11} x^{11}\right )\right ) b^{10}+10 a e \left (9 A e \left (280 d^9-2240 e x d^8-1260 e^2 x^2 d^7+420 e^3 x^3 d^6-210 e^4 x^4 d^5+126 e^5 x^5 d^4-84 e^6 x^6 d^3+60 e^7 x^7 d^2-45 e^8 x^8 d+35 e^9 x^9\right )-10 B \left (252 d^{10}-2268 e x d^9-1260 e^2 x^2 d^8+420 e^3 x^3 d^7-210 e^4 x^4 d^6+126 e^5 x^5 d^5-84 e^6 x^6 d^4+60 e^7 x^7 d^3-45 e^8 x^8 d^2+35 e^9 x^9 d-28 e^{10} x^{10}\right )\right ) b^9+135 a^2 e^2 \left (8 A e \left (-105 d^8+735 e x d^7+420 e^2 x^2 d^6-140 e^3 x^3 d^5+70 e^4 x^4 d^4-42 e^5 x^5 d^3+28 e^6 x^6 d^2-20 e^7 x^7 d+15 e^8 x^8\right )+3 B \left (280 d^9-2240 e x d^8-1260 e^2 x^2 d^7+420 e^3 x^3 d^6-210 e^4 x^4 d^5+126 e^5 x^5 d^4-84 e^6 x^6 d^3+60 e^7 x^7 d^2-45 e^8 x^8 d+35 e^9 x^9\right )\right ) b^8+720 a^3 e^3 \left (7 A e \left (60 d^7-360 e x d^6-210 e^2 x^2 d^5+70 e^3 x^3 d^4-35 e^4 x^4 d^3+21 e^5 x^5 d^2-14 e^6 x^6 d+10 e^7 x^7\right )-4 B \left (105 d^8-735 e x d^7-420 e^2 x^2 d^6+140 e^3 x^3 d^5-70 e^4 x^4 d^4+42 e^5 x^5 d^3-28 e^6 x^6 d^2+20 e^7 x^7 d-15 e^8 x^8\right )\right ) b^7+8820 a^4 e^4 \left (6 A e \left (-10 d^6+50 e x d^5+30 e^2 x^2 d^4-10 e^3 x^3 d^3+5 e^4 x^4 d^2-3 e^5 x^5 d+2 e^6 x^6\right )+B \left (60 d^7-360 e x d^6-210 e^2 x^2 d^5+70 e^3 x^3 d^4-35 e^4 x^4 d^3+21 e^5 x^5 d^2-14 e^6 x^6 d+10 e^7 x^7\right )\right ) b^6+10584 a^5 e^5 \left (5 A e \left (12 d^5-48 e x d^4-30 e^2 x^2 d^3+10 e^3 x^3 d^2-5 e^4 x^4 d+3 e^5 x^5\right )-6 B \left (10 d^6-50 e x d^5-30 e^2 x^2 d^4+10 e^3 x^3 d^3-5 e^4 x^4 d^2+3 e^5 x^5 d-2 e^6 x^6\right )\right ) b^5+44100 a^6 e^6 \left (4 A e \left (-3 d^4+9 e x d^3+6 e^2 x^2 d^2-2 e^3 x^3 d+e^4 x^4\right )+B \left (12 d^5-48 e x d^4-30 e^2 x^2 d^3+10 e^3 x^3 d^2-5 e^4 x^4 d+3 e^5 x^5\right )\right ) b^4+50400 a^7 e^7 \left (3 A e \left (2 d^3-4 e x d^2-3 e^2 x^2 d+e^3 x^3\right )+2 B \left (-3 d^4+9 e x d^3+6 e^2 x^2 d^2-2 e^3 x^3 d+e^4 x^4\right )\right ) b^3+56700 a^8 e^8 \left (2 A e \left (-d^2+e x d+e^2 x^2\right )+B \left (2 d^3-4 e x d^2-3 e^2 x^2 d+e^3 x^3\right )\right ) b^2+25200 a^9 e^9 \left (A d e+B \left (-d^2+e x d+e^2 x^2\right )\right ) b-2520 a^{10} e^{10} (A e-B d)+2520 (b d-a e)^9 (11 b B d-10 A b e-a B e) (d+e x) \log (d+e x)}{2520 e^{12} (d+e x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^2,x]

[Out]

(-2520*a^10*e^10*(-(B*d) + A*e) + 25200*a^9*b*e^9*(A*d*e + B*(-d^2 + d*e*x + e^2
*x^2)) + 56700*a^8*b^2*e^8*(2*A*e*(-d^2 + d*e*x + e^2*x^2) + B*(2*d^3 - 4*d^2*e*
x - 3*d*e^2*x^2 + e^3*x^3)) + 50400*a^7*b^3*e^7*(3*A*e*(2*d^3 - 4*d^2*e*x - 3*d*
e^2*x^2 + e^3*x^3) + 2*B*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4
*x^4)) + 44100*a^6*b^4*e^6*(4*A*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*
x^3 + e^4*x^4) + B*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*
e^4*x^4 + 3*e^5*x^5)) + 10584*a^5*b^5*e^5*(5*A*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e
^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) - 6*B*(10*d^6 - 50*d^5*e*x -
30*d^4*e^2*x^2 + 10*d^3*e^3*x^3 - 5*d^2*e^4*x^4 + 3*d*e^5*x^5 - 2*e^6*x^6)) + 88
20*a^4*b^6*e^4*(6*A*e*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 +
5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) + B*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2
*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*
x^7)) + 720*a^3*b^7*e^3*(7*A*e*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*
e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x^7) - 4*B*(10
5*d^8 - 735*d^7*e*x - 420*d^6*e^2*x^2 + 140*d^5*e^3*x^3 - 70*d^4*e^4*x^4 + 42*d^
3*e^5*x^5 - 28*d^2*e^6*x^6 + 20*d*e^7*x^7 - 15*e^8*x^8)) + 135*a^2*b^8*e^2*(8*A*
e*(-105*d^8 + 735*d^7*e*x + 420*d^6*e^2*x^2 - 140*d^5*e^3*x^3 + 70*d^4*e^4*x^4 -
 42*d^3*e^5*x^5 + 28*d^2*e^6*x^6 - 20*d*e^7*x^7 + 15*e^8*x^8) + 3*B*(280*d^9 - 2
240*d^8*e*x - 1260*d^7*e^2*x^2 + 420*d^6*e^3*x^3 - 210*d^5*e^4*x^4 + 126*d^4*e^5
*x^5 - 84*d^3*e^6*x^6 + 60*d^2*e^7*x^7 - 45*d*e^8*x^8 + 35*e^9*x^9)) + 10*a*b^9*
e*(9*A*e*(280*d^9 - 2240*d^8*e*x - 1260*d^7*e^2*x^2 + 420*d^6*e^3*x^3 - 210*d^5*
e^4*x^4 + 126*d^4*e^5*x^5 - 84*d^3*e^6*x^6 + 60*d^2*e^7*x^7 - 45*d*e^8*x^8 + 35*
e^9*x^9) - 10*B*(252*d^10 - 2268*d^9*e*x - 1260*d^8*e^2*x^2 + 420*d^7*e^3*x^3 -
210*d^6*e^4*x^4 + 126*d^5*e^5*x^5 - 84*d^4*e^6*x^6 + 60*d^3*e^7*x^7 - 45*d^2*e^8
*x^8 + 35*d*e^9*x^9 - 28*e^10*x^10)) + b^10*(10*A*e*(-252*d^10 + 2268*d^9*e*x +
1260*d^8*e^2*x^2 - 420*d^7*e^3*x^3 + 210*d^6*e^4*x^4 - 126*d^5*e^5*x^5 + 84*d^4*
e^6*x^6 - 60*d^3*e^7*x^7 + 45*d^2*e^8*x^8 - 35*d*e^9*x^9 + 28*e^10*x^10) + B*(25
20*d^11 - 25200*d^10*e*x - 13860*d^9*e^2*x^2 + 4620*d^8*e^3*x^3 - 2310*d^7*e^4*x
^4 + 1386*d^6*e^5*x^5 - 924*d^5*e^6*x^6 + 660*d^4*e^7*x^7 - 495*d^3*e^8*x^8 + 38
5*d^2*e^9*x^9 - 308*d*e^10*x^10 + 252*e^11*x^11)) + 2520*(b*d - a*e)^9*(11*b*B*d
 - 10*A*b*e - a*B*e)*(d + e*x)*Log[d + e*x])/(2520*e^12*(d + e*x))

_______________________________________________________________________________________

Maple [B]  time = 0.033, size = 2447, normalized size = 5.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/(e*x+d)^2,x)

[Out]

45*b^2/e^2*A*a^8*x+9*b^10/e^10*A*d^8*x+10/e^2*ln(e*x+d)*A*a^9*b+1/10*b^10/e^2*B*
x^10+1/9*b^10/e^2*A*x^9+1/e^2*ln(e*x+d)*B*a^10-1/e/(e*x+d)*a^10*A-3/2*b^10/e^7*A
*x^4*d^5+105/2*b^4/e^2*B*x^4*a^6+7/4*b^10/e^8*B*x^4*d^6+70*b^4/e^2*A*x^3*a^6+7/3
*b^10/e^8*A*x^3*d^6+40*b^3/e^2*B*x^3*a^7-8/3*b^10/e^9*B*x^3*d^7+60*b^3/e^2*A*x^2
*a^7-4*b^10/e^9*A*x^2*d^7+45/2*b^2/e^2*B*x^2*a^8+9/2*b^10/e^10*B*x^2*d^8-10/e^11
*ln(e*x+d)*A*b^10*d^9+11/e^12*ln(e*x+d)*b^10*B*d^10-1/e^11/(e*x+d)*A*b^10*d^10+1
/e^2/(e*x+d)*B*d*a^10+1/e^12/(e*x+d)*b^10*B*d^11+10*b/e^2*B*a^9*x-10*b^10/e^11*B
*d^9*x+45/7*b^8/e^2*A*x^7*a^2+3/7*b^10/e^4*A*x^7*d^2+120/7*b^7/e^2*B*x^7*a^3-4/7
*b^10/e^5*B*x^7*d^3+20*b^7/e^2*A*x^6*a^3-2/3*b^10/e^5*A*x^6*d^3+35*b^6/e^2*B*x^6
*a^4+5/6*b^10/e^6*B*x^6*d^4+42*b^6/e^2*A*x^5*a^4+252/5*b^5/e^2*B*x^5*a^5+10/9*b^
9/e^2*B*x^9*a-6/5*b^10/e^7*B*x^5*d^5+63*b^5/e^2*A*x^4*a^5-135*b^8/e^7*A*x^2*a^2*
d^5+35*b^9/e^8*A*x^2*a*d^6+25/2*b^9/e^6*A*x^4*a*d^4-126*b^5/e^3*B*x^4*a^5*d+315/
2*b^6/e^4*B*x^4*a^4*d^2-120*b^7/e^5*B*x^4*a^3*d^3+225/4*b^8/e^6*B*x^4*a^2*d^4-15
*b^9/e^7*B*x^4*a*d^5-168*b^5/e^3*A*x^3*a^5*d+210*b^6/e^4*A*x^3*a^4*d^2-160*b^7/e
^5*A*x^3*a^3*d^3+5*b^9/e^4*A*x^6*a*d^2-40*b^7/e^3*B*x^6*a^3*d+45/2*b^8/e^4*B*x^6
*a^2*d^2-20/3*b^9/e^5*B*x^6*a*d^3-48*b^7/e^3*A*x^5*a^3*d+27*b^8/e^4*A*x^5*a^2*d^
2-8*b^9/e^5*A*x^5*a*d^3-84*b^6/e^3*B*x^5*a^4*d+72*b^7/e^4*B*x^5*a^3*d^2-36*b^8/e
^5*B*x^5*a^2*d^3+10*b^9/e^6*B*x^5*a*d^4-105*b^6/e^3*A*x^4*a^4*d+90*b^7/e^4*A*x^4
*a^3*d^2-45*b^8/e^5*A*x^4*a^2*d^3-90/7*b^8/e^3*B*x^7*a^2*d+30/7*b^9/e^4*B*x^7*a*
d^2-15*b^8/e^3*A*x^6*a^2*d-5/2*b^9/e^3*B*x^8*a*d-20/7*b^9/e^3*A*x^7*a*d+315*b^8/
e^8*A*a^2*d^6*x+75*b^8/e^6*A*x^3*a^2*d^4-20*b^9/e^7*A*x^3*a*d^5-140*b^4/e^3*B*x^
3*a^6*d+252*b^5/e^4*B*x^3*a^5*d^2-280*b^6/e^5*B*x^3*a^4*d^3+200*b^7/e^6*B*x^3*a^
3*d^4-90*b^8/e^7*B*x^3*a^2*d^5-90/e^3*ln(e*x+d)*A*a^8*b^2*d+360/e^4*ln(e*x+d)*A*
a^7*b^3*d^2-840/e^5*ln(e*x+d)*A*a^6*b^4*d^3+1260/e^6*ln(e*x+d)*A*a^5*b^5*d^4-126
0/e^7*ln(e*x+d)*A*a^4*b^6*d^5+840/e^8*ln(e*x+d)*A*a^3*b^7*d^6-360/e^9*ln(e*x+d)*
A*a^2*b^8*d^7+90/e^10*ln(e*x+d)*A*a*b^9*d^8-20/e^3*ln(e*x+d)*B*a^9*b*d+135/e^4*l
n(e*x+d)*B*a^8*b^2*d^2-480/e^5*ln(e*x+d)*B*a^7*b^3*d^3+1050/e^6*ln(e*x+d)*B*a^6*
b^4*d^4-1512/e^7*ln(e*x+d)*B*a^5*b^5*d^5+1470/e^8*ln(e*x+d)*B*a^4*b^6*d^6-960/e^
9*ln(e*x+d)*B*a^3*b^7*d^7+405/e^10*ln(e*x+d)*B*a^2*b^8*d^8-100/e^11*ln(e*x+d)*B*
a*b^9*d^9+10/e^2/(e*x+d)*A*d*a^9*b-45/e^3/(e*x+d)*A*a^8*b^2*d^2+120/e^4/(e*x+d)*
A*a^7*b^3*d^3-210/e^5/(e*x+d)*A*a^6*b^4*d^4+252/e^6/(e*x+d)*A*a^5*b^5*d^5-210/e^
7/(e*x+d)*A*a^4*b^6*d^6+120/e^8/(e*x+d)*A*a^3*b^7*d^7-45/e^9/(e*x+d)*A*a^2*b^8*d
^8+10/e^10/(e*x+d)*A*a*b^9*d^9+70/3*b^9/e^8*B*x^3*a*d^6-210*b^4/e^3*A*x^2*a^6*d+
378*b^5/e^4*A*x^2*a^5*d^2-420*b^6/e^5*A*x^2*a^4*d^3+300*b^7/e^6*A*x^2*a^3*d^4-10
/e^3/(e*x+d)*B*a^9*b*d^2+45/e^4/(e*x+d)*B*a^8*b^2*d^3-120/e^5/(e*x+d)*B*a^7*b^3*
d^4+210/e^6/(e*x+d)*B*a^6*b^4*d^5-252/e^7/(e*x+d)*B*a^5*b^5*d^6+210/e^8/(e*x+d)*
B*a^4*b^6*d^7-120/e^9/(e*x+d)*B*a^3*b^7*d^8+45/e^10/(e*x+d)*B*a^2*b^8*d^9-10/e^1
1/(e*x+d)*B*a*b^9*d^10+1260*b^5/e^6*B*a^5*d^4*x-1260*b^6/e^7*B*a^4*d^5*x+840*b^7
/e^8*B*a^3*d^6*x-360*b^8/e^9*B*a^2*d^7*x+90*b^9/e^10*B*a*d^8*x-80*b^9/e^9*A*a*d^
7*x-90*b^2/e^3*B*a^8*d*x+360*b^3/e^4*B*a^7*d^2*x-840*b^4/e^5*B*a^6*d^3*x-120*b^3
/e^3*B*x^2*a^7*d+315*b^4/e^4*B*x^2*a^6*d^2-504*b^5/e^5*B*x^2*a^5*d^3+525*b^6/e^6
*B*x^2*a^4*d^4-360*b^7/e^7*B*x^2*a^3*d^5+315/2*b^8/e^8*B*x^2*a^2*d^6-40*b^9/e^9*
B*x^2*a*d^7-240*b^3/e^3*A*a^7*d*x+630*b^4/e^4*A*a^6*d^2*x-1008*b^5/e^5*A*a^5*d^3
*x+1050*b^6/e^6*A*a^4*d^4*x-720*b^7/e^7*A*a^3*d^5*x-2/9*b^10/e^3*B*x^9*d+5/4*b^9
/e^2*A*x^8*a-1/4*b^10/e^3*A*x^8*d+45/8*b^8/e^2*B*x^8*a^2+3/8*b^10/e^4*B*x^8*d^2+
b^10/e^6*A*x^5*d^4

_______________________________________________________________________________________

Maxima [A]  time = 1.39465, size = 2453, normalized size = 5.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^2,x, algorithm="maxima")

[Out]

(B*b^10*d^11 - A*a^10*e^11 - (10*B*a*b^9 + A*b^10)*d^10*e + 5*(9*B*a^2*b^8 + 2*A
*a*b^9)*d^9*e^2 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 30*(7*B*a^4*b^6 + 4*A
*a^3*b^7)*d^7*e^4 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 42*(5*B*a^6*b^4 + 6
*A*a^5*b^5)*d^5*e^6 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 15*(3*B*a^8*b^2 +
 8*A*a^7*b^3)*d^3*e^8 - 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + (B*a^10 + 10*A*a^9
*b)*d*e^10)/(e^13*x + d*e^12) + 1/2520*(252*B*b^10*e^9*x^10 - 280*(2*B*b^10*d*e^
8 - (10*B*a*b^9 + A*b^10)*e^9)*x^9 + 315*(3*B*b^10*d^2*e^7 - 2*(10*B*a*b^9 + A*b
^10)*d*e^8 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^9)*x^8 - 360*(4*B*b^10*d^3*e^6 - 3*(1
0*B*a*b^9 + A*b^10)*d^2*e^7 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^8 - 15*(8*B*a^3*b
^7 + 3*A*a^2*b^8)*e^9)*x^7 + 420*(5*B*b^10*d^4*e^5 - 4*(10*B*a*b^9 + A*b^10)*d^3
*e^6 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^7 - 30*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e
^8 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^9)*x^6 - 504*(6*B*b^10*d^5*e^4 - 5*(10*B*a
*b^9 + A*b^10)*d^4*e^5 + 20*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^6 - 45*(8*B*a^3*b^7
+ 3*A*a^2*b^8)*d^2*e^7 + 60*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^8 - 42*(6*B*a^5*b^5
+ 5*A*a^4*b^6)*e^9)*x^5 + 630*(7*B*b^10*d^6*e^3 - 6*(10*B*a*b^9 + A*b^10)*d^5*e^
4 + 25*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^5 - 60*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^
6 + 90*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^7 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^
8 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^9)*x^4 - 840*(8*B*b^10*d^7*e^2 - 7*(10*B*a*
b^9 + A*b^10)*d^6*e^3 + 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^4 - 75*(8*B*a^3*b^7 +
 3*A*a^2*b^8)*d^4*e^5 + 120*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^6 - 126*(6*B*a^5*b
^5 + 5*A*a^4*b^6)*d^2*e^7 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^8 - 30*(4*B*a^7*b
^3 + 7*A*a^6*b^4)*e^9)*x^3 + 1260*(9*B*b^10*d^8*e - 8*(10*B*a*b^9 + A*b^10)*d^7*
e^2 + 35*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^3 - 90*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*
e^4 + 150*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^5 - 168*(6*B*a^5*b^5 + 5*A*a^4*b^6)*
d^3*e^6 + 126*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^7 - 60*(4*B*a^7*b^3 + 7*A*a^6*b^
4)*d*e^8 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^9)*x^2 - 2520*(10*B*b^10*d^9 - 9*(10
*B*a*b^9 + A*b^10)*d^8*e + 40*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^2 - 105*(8*B*a^3*b
^7 + 3*A*a^2*b^8)*d^6*e^3 + 180*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^4 - 210*(6*B*a
^5*b^5 + 5*A*a^4*b^6)*d^4*e^5 + 168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^6 - 90*(4*
B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^7 + 30*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^8 - 5*(2*B
*a^9*b + 9*A*a^8*b^2)*e^9)*x)/e^11 + (11*B*b^10*d^10 - 10*(10*B*a*b^9 + A*b^10)*
d^9*e + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^2 - 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d
^7*e^3 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^4 - 252*(6*B*a^5*b^5 + 5*A*a^4*b^
6)*d^5*e^5 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^6 - 120*(4*B*a^7*b^3 + 7*A*a^
6*b^4)*d^3*e^7 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^8 - 10*(2*B*a^9*b + 9*A*a^
8*b^2)*d*e^9 + (B*a^10 + 10*A*a^9*b)*e^10)*log(e*x + d)/e^12

_______________________________________________________________________________________

Fricas [A]  time = 0.245803, size = 3144, normalized size = 7.07 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^2,x, algorithm="fricas")

[Out]

1/2520*(252*B*b^10*e^11*x^11 + 2520*B*b^10*d^11 - 2520*A*a^10*e^11 - 2520*(10*B*
a*b^9 + A*b^10)*d^10*e + 12600*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 37800*(8*B*a^
3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 75600*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 10584
0*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 105840*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e
^6 - 75600*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 37800*(3*B*a^8*b^2 + 8*A*a^7*b^
3)*d^3*e^8 - 12600*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 2520*(B*a^10 + 10*A*a^9*b
)*d*e^10 - 28*(11*B*b^10*d*e^10 - 10*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 35*(11*B
*b^10*d^2*e^9 - 10*(10*B*a*b^9 + A*b^10)*d*e^10 + 45*(9*B*a^2*b^8 + 2*A*a*b^9)*e
^11)*x^9 - 45*(11*B*b^10*d^3*e^8 - 10*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 45*(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 + 60*(11*B
*b^10*d^4*e^7 - 10*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 45*(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 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^
7)*e^11)*x^7 - 84*(11*B*b^10*d^5*e^6 - 10*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 45*(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 + 210*(
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 +
126*(11*B*b^10*d^6*e^5 - 10*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 45*(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 + 210*(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 + 210*(5*B*a^6*b^4
 + 6*A*a^5*b^5)*e^11)*x^5 - 210*(11*B*b^10*d^7*e^4 - 10*(10*B*a*b^9 + A*b^10)*d^
6*e^5 + 45*(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 + 210*(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 + 210*(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 + 420*(11*B*b^10*d^8*e^3 - 10*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 45
*(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 + 2
10*(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
 + 210*(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 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 - 1260*(11*B*b^10*d^9*e^2 - 10*(1
0*B*a*b^9 + A*b^10)*d^8*e^3 + 45*(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 + 210*(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 + 210*(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 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 - 1
0*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 - 2520*(10*B*b^10*d^10*e - 9*(10*B*a*b^9 +
 A*b^10)*d^9*e^2 + 40*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 105*(8*B*a^3*b^7 + 3*A
*a^2*b^8)*d^7*e^4 + 180*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 210*(6*B*a^5*b^5 +
 5*A*a^4*b^6)*d^5*e^6 + 168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 90*(4*B*a^7*b^
3 + 7*A*a^6*b^4)*d^3*e^8 + 30*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 - 5*(2*B*a^9*b
 + 9*A*a^8*b^2)*d*e^10)*x + 2520*(11*B*b^10*d^11 - 10*(10*B*a*b^9 + A*b^10)*d^10
*e + 45*(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 + 210*(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 + 210*(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 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 10*(2*B*a^9*b + 9*A*a^8*b
^2)*d^2*e^9 + (B*a^10 + 10*A*a^9*b)*d*e^10 + (11*B*b^10*d^10*e - 10*(10*B*a*b^9
+ A*b^10)*d^9*e^2 + 45*(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 + 210*(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 + 210*(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 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 - 10*(2*B*a^
9*b + 9*A*a^8*b^2)*d*e^10 + (B*a^10 + 10*A*a^9*b)*e^11)*x)*log(e*x + d))/(e^13*x
 + d*e^12)

_______________________________________________________________________________________

Sympy [A]  time = 40.1854, size = 1904, normalized size = 4.28 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10*(B*x+A)/(e*x+d)**2,x)

[Out]

B*b**10*x**10/(10*e**2) + (-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 + 2
10*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)/(d*e**12 + e**13*x) + x**9*(A*b**10*e + 10*B*a*b**9*e - 2*B*b**10
*d)/(9*e**3) + x**8*(10*A*a*b**9*e**2 - 2*A*b**10*d*e + 45*B*a**2*b**8*e**2 - 20
*B*a*b**9*d*e + 3*B*b**10*d**2)/(8*e**4) + x**7*(45*A*a**2*b**8*e**3 - 20*A*a*b*
*9*d*e**2 + 3*A*b**10*d**2*e + 120*B*a**3*b**7*e**3 - 90*B*a**2*b**8*d*e**2 + 30
*B*a*b**9*d**2*e - 4*B*b**10*d**3)/(7*e**5) + x**6*(120*A*a**3*b**7*e**4 - 90*A*
a**2*b**8*d*e**3 + 30*A*a*b**9*d**2*e**2 - 4*A*b**10*d**3*e + 210*B*a**4*b**6*e*
*4 - 240*B*a**3*b**7*d*e**3 + 135*B*a**2*b**8*d**2*e**2 - 40*B*a*b**9*d**3*e + 5
*B*b**10*d**4)/(6*e**6) + x**5*(210*A*a**4*b**6*e**5 - 240*A*a**3*b**7*d*e**4 +
135*A*a**2*b**8*d**2*e**3 - 40*A*a*b**9*d**3*e**2 + 5*A*b**10*d**4*e + 252*B*a**
5*b**5*e**5 - 420*B*a**4*b**6*d*e**4 + 360*B*a**3*b**7*d**2*e**3 - 180*B*a**2*b*
*8*d**3*e**2 + 50*B*a*b**9*d**4*e - 6*B*b**10*d**5)/(5*e**7) + x**4*(252*A*a**5*
b**5*e**6 - 420*A*a**4*b**6*d*e**5 + 360*A*a**3*b**7*d**2*e**4 - 180*A*a**2*b**8
*d**3*e**3 + 50*A*a*b**9*d**4*e**2 - 6*A*b**10*d**5*e + 210*B*a**6*b**4*e**6 - 5
04*B*a**5*b**5*d*e**5 + 630*B*a**4*b**6*d**2*e**4 - 480*B*a**3*b**7*d**3*e**3 +
225*B*a**2*b**8*d**4*e**2 - 60*B*a*b**9*d**5*e + 7*B*b**10*d**6)/(4*e**8) + x**3
*(210*A*a**6*b**4*e**7 - 504*A*a**5*b**5*d*e**6 + 630*A*a**4*b**6*d**2*e**5 - 48
0*A*a**3*b**7*d**3*e**4 + 225*A*a**2*b**8*d**4*e**3 - 60*A*a*b**9*d**5*e**2 + 7*
A*b**10*d**6*e + 120*B*a**7*b**3*e**7 - 420*B*a**6*b**4*d*e**6 + 756*B*a**5*b**5
*d**2*e**5 - 840*B*a**4*b**6*d**3*e**4 + 600*B*a**3*b**7*d**4*e**3 - 270*B*a**2*
b**8*d**5*e**2 + 70*B*a*b**9*d**6*e - 8*B*b**10*d**7)/(3*e**9) + x**2*(120*A*a**
7*b**3*e**8 - 420*A*a**6*b**4*d*e**7 + 756*A*a**5*b**5*d**2*e**6 - 840*A*a**4*b*
*6*d**3*e**5 + 600*A*a**3*b**7*d**4*e**4 - 270*A*a**2*b**8*d**5*e**3 + 70*A*a*b*
*9*d**6*e**2 - 8*A*b**10*d**7*e + 45*B*a**8*b**2*e**8 - 240*B*a**7*b**3*d*e**7 +
 630*B*a**6*b**4*d**2*e**6 - 1008*B*a**5*b**5*d**3*e**5 + 1050*B*a**4*b**6*d**4*
e**4 - 720*B*a**3*b**7*d**5*e**3 + 315*B*a**2*b**8*d**6*e**2 - 80*B*a*b**9*d**7*
e + 9*B*b**10*d**8)/(2*e**10) + x*(45*A*a**8*b**2*e**9 - 240*A*a**7*b**3*d*e**8
+ 630*A*a**6*b**4*d**2*e**7 - 1008*A*a**5*b**5*d**3*e**6 + 1050*A*a**4*b**6*d**4
*e**5 - 720*A*a**3*b**7*d**5*e**4 + 315*A*a**2*b**8*d**6*e**3 - 80*A*a*b**9*d**7
*e**2 + 9*A*b**10*d**8*e + 10*B*a**9*b*e**9 - 90*B*a**8*b**2*d*e**8 + 360*B*a**7
*b**3*d**2*e**7 - 840*B*a**6*b**4*d**3*e**6 + 1260*B*a**5*b**5*d**4*e**5 - 1260*
B*a**4*b**6*d**5*e**4 + 840*B*a**3*b**7*d**6*e**3 - 360*B*a**2*b**8*d**7*e**2 +
90*B*a*b**9*d**8*e - 10*B*b**10*d**9)/e**11 + (a*e - b*d)**9*(10*A*b*e + B*a*e -
 11*B*b*d)*log(d + e*x)/e**12

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224234, 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)^2,x, algorithm="giac")

[Out]

Done

[next] [prev] [prev-tail] [front] [up]