3.1077 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^6} \, dx\)

Optimal. Leaf size=447 \[ -\frac{b^9 (d+e x)^5 (-10 a B e-A b e+11 b B d)}{5 e^{12}}+\frac{5 b^8 (d+e x)^4 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{4 e^{12}}-\frac{5 b^7 (d+e x)^3 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{12}}+\frac{15 b^6 (d+e x)^2 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12}}-\frac{42 b^5 x (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{11}}+\frac{42 b^4 (b d-a e)^5 \log (d+e x) (-5 a B e-6 A b e+11 b B d)}{e^{12}}+\frac{30 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)}-\frac{15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{2 e^{12} (d+e x)^2}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{3 e^{12} (d+e x)^3}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{4 e^{12} (d+e x)^4}+\frac{(b d-a e)^{10} (B d-A e)}{5 e^{12} (d+e x)^5}+\frac{b^{10} B (d+e x)^6}{6 e^{12}} \]

[Out]

(-42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*x)/e^11 + ((b*d - a*e)^10*
(B*d - A*e))/(5*e^12*(d + e*x)^5) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e)
)/(4*e^12*(d + e*x)^4) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(3*e
^12*(d + e*x)^3) - (15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(2*e^12
*(d + e*x)^2) + (30*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*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e)*(d + e*x)^2)/e^12
- (5*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*(d + e*x)^3)/e^12 + (5*b^8
*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^4)/(4*e^12) - (b^9*(11*b*B
*d - A*b*e - 10*a*B*e)*(d + e*x)^5)/(5*e^12) + (b^10*B*(d + e*x)^6)/(6*e^12) + (
42*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e)*Log[d + e*x])/e^12

_______________________________________________________________________________________

Rubi [A]  time = 3.51419, antiderivative size = 447, 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 \[ -\frac{b^9 (d+e x)^5 (-10 a B e-A b e+11 b B d)}{5 e^{12}}+\frac{5 b^8 (d+e x)^4 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{4 e^{12}}-\frac{5 b^7 (d+e x)^3 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{12}}+\frac{15 b^6 (d+e x)^2 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12}}-\frac{42 b^5 x (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{11}}+\frac{42 b^4 (b d-a e)^5 \log (d+e x) (-5 a B e-6 A b e+11 b B d)}{e^{12}}+\frac{30 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)}-\frac{15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{2 e^{12} (d+e x)^2}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{3 e^{12} (d+e x)^3}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{4 e^{12} (d+e x)^4}+\frac{(b d-a e)^{10} (B d-A e)}{5 e^{12} (d+e x)^5}+\frac{b^{10} B (d+e x)^6}{6 e^{12}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/(d + e*x)^6,x]

[Out]

(-42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*x)/e^11 + ((b*d - a*e)^10*
(B*d - A*e))/(5*e^12*(d + e*x)^5) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e)
)/(4*e^12*(d + e*x)^4) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(3*e
^12*(d + e*x)^3) - (15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(2*e^12
*(d + e*x)^2) + (30*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*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e)*(d + e*x)^2)/e^12
- (5*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*(d + e*x)^3)/e^12 + (5*b^8
*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^4)/(4*e^12) - (b^9*(11*b*B
*d - A*b*e - 10*a*B*e)*(d + e*x)^5)/(5*e^12) + (b^10*B*(d + e*x)^6)/(6*e^12) + (
42*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*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)**6,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.960009, size = 587, normalized size = 1.31 \[ \frac{-15 b^8 e^4 x^4 \left (-45 a^2 B e^2-10 a b e (A e-6 B d)+3 b^2 d (2 A e-7 B d)\right )+20 b^7 e^3 x^3 \left (120 a^3 B e^3+45 a^2 b e^2 (A e-6 B d)+30 a b^2 d e (7 B d-2 A e)-7 b^3 d^2 (8 B d-3 A e)\right )-30 b^6 e^2 x^2 \left (-210 a^4 B e^4-120 a^3 b e^3 (A e-6 B d)-135 a^2 b^2 d e^2 (7 B d-2 A e)+70 a b^3 d^2 e (8 B d-3 A e)-14 b^4 d^3 (9 B d-4 A e)\right )+60 b^5 e x \left (252 a^5 B e^5+210 a^4 b e^4 (A e-6 B d)+360 a^3 b^2 d e^3 (7 B d-2 A e)-315 a^2 b^3 d^2 e^2 (8 B d-3 A e)+140 a b^4 d^3 e (9 B d-4 A e)-126 b^5 d^4 (2 B d-A e)\right )+12 b^9 e^5 x^5 (10 a B e+A b e-6 b B d)+2520 b^4 (b d-a e)^5 \log (d+e x) (-5 a B e-6 A b e+11 b B d)+\frac{1800 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{d+e x}-\frac{450 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{(d+e x)^2}+\frac{100 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{(d+e x)^3}-\frac{15 (b d-a e)^9 (-a B e-10 A b e+11 b B d)}{(d+e x)^4}+\frac{12 (b d-a e)^{10} (B d-A e)}{(d+e x)^5}+10 b^{10} B e^6 x^6}{60 e^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^6,x]

[Out]

(60*b^5*e*(252*a^5*B*e^5 + 140*a*b^4*d^3*e*(9*B*d - 4*A*e) - 315*a^2*b^3*d^2*e^2
*(8*B*d - 3*A*e) + 360*a^3*b^2*d*e^3*(7*B*d - 2*A*e) - 126*b^5*d^4*(2*B*d - A*e)
+ 210*a^4*b*e^4*(-6*B*d + A*e))*x - 30*b^6*e^2*(-210*a^4*B*e^4 - 14*b^4*d^3*(9*
B*d - 4*A*e) + 70*a*b^3*d^2*e*(8*B*d - 3*A*e) - 135*a^2*b^2*d*e^2*(7*B*d - 2*A*e
) - 120*a^3*b*e^3*(-6*B*d + A*e))*x^2 + 20*b^7*e^3*(120*a^3*B*e^3 - 7*b^3*d^2*(8
*B*d - 3*A*e) + 30*a*b^2*d*e*(7*B*d - 2*A*e) + 45*a^2*b*e^2*(-6*B*d + A*e))*x^3
- 15*b^8*e^4*(-45*a^2*B*e^2 - 10*a*b*e*(-6*B*d + A*e) + 3*b^2*d*(-7*B*d + 2*A*e)
)*x^4 + 12*b^9*e^5*(-6*b*B*d + A*b*e + 10*a*B*e)*x^5 + 10*b^10*B*e^6*x^6 + (12*(
b*d - a*e)^10*(B*d - A*e))/(d + e*x)^5 - (15*(b*d - a*e)^9*(11*b*B*d - 10*A*b*e
- a*B*e))/(d + e*x)^4 + (100*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(d
+ e*x)^3 - (450*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(d + e*x)^2 +
(1800*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(d + e*x) + 2520*b^4*(b*
d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e)*Log[d + e*x])/(60*e^12)

_______________________________________________________________________________________

Maple [B]  time = 0.049, size = 2731, normalized size = 6.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/(e*x+d)^6,x)

[Out]

-45/2/e^10/(e*x+d)^4*A*a*b^9*d^8+5/e^3/(e*x+d)^4*B*a^9*b*d-135/4/e^4/(e*x+d)^4*B
*a^8*b^2*d^2+120/e^5/(e*x+d)^4*B*a^7*b^3*d^3-525/2/e^6/(e*x+d)^4*B*a^6*b^4*d^4+3
78/e^7/(e*x+d)^4*B*a^5*b^5*d^5-735/2/e^8/(e*x+d)^4*B*a^4*b^6*d^6+240/e^9/(e*x+d)
^4*B*a^3*b^7*d^7-405/4/e^10/(e*x+d)^4*B*a^2*b^8*d^8+25/e^11/(e*x+d)^4*B*a*b^9*d^
9+2/e^2/(e*x+d)^5*A*d*a^9*b-9/e^3/(e*x+d)^5*A*d^2*a^8*b^2+24/e^4/(e*x+d)^5*A*d^3
*a^7*b^3-42/e^5/(e*x+d)^5*A*d^4*a^6*b^4+252/5/e^6/(e*x+d)^5*A*d^5*a^5*b^5-42/e^7
/(e*x+d)^5*A*a^4*b^6*d^6+24/e^8/(e*x+d)^5*A*a^3*b^7*d^7-9/e^9/(e*x+d)^5*A*a^2*b^
8*d^8+2/e^10/(e*x+d)^5*A*a*b^9*d^9-2/e^3/(e*x+d)^5*B*d^2*a^9*b+9/e^4/(e*x+d)^5*B
*d^3*a^8*b^2-24/e^5/(e*x+d)^5*B*d^4*a^7*b^3+42/e^6/(e*x+d)^5*B*d^5*a^6*b^4-252/5
/e^7/(e*x+d)^5*B*a^5*b^5*d^6+42/e^8/(e*x+d)^5*B*a^4*b^6*d^7-24/e^9/(e*x+d)^5*B*a
^3*b^7*d^8+9/e^10/(e*x+d)^5*B*a^2*b^8*d^9-2/e^11/(e*x+d)^5*B*a*b^9*d^10-280*b^9/
e^9*B*x^2*a*d^3-720*b^7/e^7*A*a^3*d*x+945*b^8/e^8*A*a^2*d^2*x-560*b^9/e^9*A*a*d^
3*x-1260*b^6/e^7*B*a^4*d*x+2520*b^7/e^8*B*a^3*d^2*x-2520*b^8/e^9*B*a^2*d^3*x+126
0*b^9/e^10*B*a*d^4*x-15*b^9/e^7*B*x^4*a*d-20*b^9/e^7*A*x^3*a*d-90*b^8/e^7*B*x^3*
a^2*d+70*b^9/e^8*B*x^3*a*d^2-135*b^8/e^7*A*x^2*a^2*d+105*b^9/e^8*A*x^2*a*d^2-360
*b^7/e^7*B*x^2*a^3*d+945/2*b^8/e^8*B*x^2*a^2*d^2-1260*b^6/e^7*ln(e*x+d)*A*a^4*d+
2520*b^7/e^8*ln(e*x+d)*A*a^3*d^2-2520*b^8/e^9*ln(e*x+d)*A*a^2*d^3+1260*b^9/e^10*
ln(e*x+d)*A*a*d^4-1512*b^5/e^7*ln(e*x+d)*B*a^5*d+4410*b^6/e^8*ln(e*x+d)*B*a^4*d^
2+1/6*b^10/e^6*B*x^6+1/5*b^10/e^6*A*x^5-1/4/e^2/(e*x+d)^4*B*a^10-1/5/e/(e*x+d)^5
*a^10*A-6720*b^7/e^9*ln(e*x+d)*B*a^3*d^3+5670*b^8/e^10*ln(e*x+d)*B*a^2*d^4-2520*
b^9/e^11*ln(e*x+d)*B*a*d^5+120*b^3/e^4/(e*x+d)^3*A*a^7*d-420*b^4/e^5/(e*x+d)^3*A
*a^6*d^2+840*b^5/e^6/(e*x+d)^3*A*a^5*d^3-1050*b^6/e^7/(e*x+d)^3*A*a^4*d^4+840*b^
7/e^8/(e*x+d)^3*A*a^3*d^5-420*b^8/e^9/(e*x+d)^3*A*a^2*d^6+120*b^9/e^10/(e*x+d)^3
*A*a*d^7+45*b^2/e^4/(e*x+d)^3*B*a^8*d-240*b^3/e^5/(e*x+d)^3*B*a^7*d^2+700*b^4/e^
6/(e*x+d)^3*B*a^6*d^3-1260*b^5/e^7/(e*x+d)^3*B*a^5*d^4+1470*b^6/e^8/(e*x+d)^3*B*
a^4*d^5-1120*b^7/e^9/(e*x+d)^3*B*a^3*d^6+540*b^8/e^10/(e*x+d)^3*B*a^2*d^7-150*b^
9/e^11/(e*x+d)^3*B*a*d^8+1260*b^5/e^6/(e*x+d)*A*a^5*d-3150*b^6/e^7/(e*x+d)*A*a^4
*d^2+4200*b^7/e^8/(e*x+d)*A*a^3*d^3-3150*b^8/e^9/(e*x+d)*A*a^2*d^4+1260*b^9/e^10
/(e*x+d)*A*a*d^5+1050*b^4/e^6/(e*x+d)*B*a^6*d-3780*b^5/e^7/(e*x+d)*B*a^5*d^2+735
0*b^6/e^8/(e*x+d)*B*a^4*d^3-8400*b^7/e^9/(e*x+d)*B*a^3*d^4+5670*b^8/e^10/(e*x+d)
*B*a^2*d^5-2100*b^9/e^11/(e*x+d)*B*a*d^6+420*b^4/e^5/(e*x+d)^2*A*a^6*d-1260*b^5/
e^6/(e*x+d)^2*A*a^5*d^2+2100*b^6/e^7/(e*x+d)^2*A*a^4*d^3-2100*b^7/e^8/(e*x+d)^2*
A*a^3*d^4+1260*b^8/e^9/(e*x+d)^2*A*a^2*d^5-420*b^9/e^10/(e*x+d)^2*A*a*d^6+240*b^
3/e^5/(e*x+d)^2*B*a^7*d-1050*b^4/e^6/(e*x+d)^2*B*a^6*d^2+2520*b^5/e^7/(e*x+d)^2*
B*a^5*d^3-3675*b^6/e^8/(e*x+d)^2*B*a^4*d^4-210*b^4/e^5/(e*x+d)*A*a^6-210*b^10/e^
11/(e*x+d)*A*d^6-120*b^3/e^5/(e*x+d)*B*a^7+330*b^10/e^12/(e*x+d)*B*d^7-60*b^3/e^
4/(e*x+d)^2*A*a^7+60*b^10/e^11/(e*x+d)^2*A*d^7-45/2*b^2/e^4/(e*x+d)^2*B*a^8-165/
2*b^10/e^12/(e*x+d)^2*B*d^8-5/2/e^2/(e*x+d)^4*A*a^9*b+5/2/e^11/(e*x+d)^4*A*b^10*
d^9-11/4/e^12/(e*x+d)^4*b^10*B*d^10-1/5/e^11/(e*x+d)^5*A*b^10*d^10+1/5/e^2/(e*x+
d)^5*B*d*a^10+1/5/e^12/(e*x+d)^5*b^10*B*d^11-252*b^10/e^11*B*d^5*x+252*b^5/e^6*B
*a^5*x+105*b^6/e^6*B*x^2*a^4+63*b^10/e^10*B*x^2*d^4+210*b^6/e^6*A*a^4*x+126*b^10
/e^10*A*d^4*x+60*b^7/e^6*A*x^2*a^3-6/5*b^10/e^7*B*x^5*d+5/2*b^9/e^6*A*x^4*a-3/2*
b^10/e^7*A*x^4*d+45/4*b^8/e^6*B*x^4*a^2+21/4*b^10/e^8*B*x^4*d^2+15*b^8/e^6*A*x^3
*a^2+40*b^7/e^6*B*x^3*a^3-56/3*b^10/e^9*B*x^3*d^3+2*b^9/e^6*B*x^5*a+252*b^5/e^6*
ln(e*x+d)*A*a^5-252*b^10/e^11*ln(e*x+d)*A*d^5+210*b^4/e^6*ln(e*x+d)*B*a^6+462*b^
10/e^12*ln(e*x+d)*B*d^6-15*b^2/e^3/(e*x+d)^3*A*a^8-15*b^10/e^11/(e*x+d)^3*A*d^8-
10/3*b/e^3/(e*x+d)^3*B*a^9+55/3*b^10/e^12/(e*x+d)^3*B*d^9+3360*b^7/e^9/(e*x+d)^2
*B*a^3*d^5-1890*b^8/e^10/(e*x+d)^2*B*a^2*d^6+600*b^9/e^11/(e*x+d)^2*B*a*d^7+45/2
/e^3/(e*x+d)^4*A*a^8*b^2*d-90/e^4/(e*x+d)^4*A*a^7*b^3*d^2+210/e^5/(e*x+d)^4*A*a^
6*b^4*d^3-315/e^6/(e*x+d)^4*A*a^5*b^5*d^4+315/e^7/(e*x+d)^4*A*a^4*b^6*d^5-210/e^
8/(e*x+d)^4*A*a^3*b^7*d^6+90/e^9/(e*x+d)^4*A*a^2*b^8*d^7+7*b^10/e^8*A*x^3*d^2-28
*b^10/e^9*A*x^2*d^3

_______________________________________________________________________________________

Maxima [A]  time = 1.58466, size = 2512, normalized size = 5.62 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^6,x, algorithm="maxima")

[Out]

1/60*(15797*B*b^10*d^11 - 12*A*a^10*e^11 - 9762*(10*B*a*b^9 + A*b^10)*d^10*e + 2
8185*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 44580*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e
^3 + 41310*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 21924*(6*B*a^5*b^5 + 5*A*a^4*b^
6)*d^6*e^5 + 5754*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 360*(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 - 3*(B*a^10 + 10*A*a^9*b)*d*e^10 + 1800*(11*B*b^10*d^7*e^4 - 7*(
10*B*a*b^9 + A*b^10)*d^6*e^5 + 21*(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 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 21*(6*B*
a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - (4*B*a^7
*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 450*(165*B*b^10*d^8*e^3 - 104*(10*B*a*b^9 + A*b^
10)*d^7*e^4 + 308*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 504*(8*B*a^3*b^7 + 3*A*a^2
*b^8)*d^5*e^6 + 490*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 280*(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 - 8*(4*B*a^7*b^3 + 7*
A*a^6*b^4)*d*e^10 - (3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 50*(2101*B*b^10*d^9*
e^2 - 1314*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 3852*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^
4 - 6216*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 5922*(7*B*a^4*b^6 + 4*A*a^3*b^7)*
d^5*e^6 - 3276*(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 - 72*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 - 9*(3*B*a^8*b^2 + 8*A*a^7
*b^3)*d*e^10 - 2*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 5*(13277*B*b^10*d^10*e -
8250*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 23985*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 3
8280*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 35910*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6
*e^5 - 19404*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 5250*(5*B*a^6*b^4 + 6*A*a^5*b
^5)*d^4*e^7 - 360*(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 - 3*(B*a^10 + 10*A*a^9*b)*e
^11)*x)/(e^17*x^5 + 5*d*e^16*x^4 + 10*d^2*e^15*x^3 + 10*d^3*e^14*x^2 + 5*d^4*e^1
3*x + d^5*e^12) + 1/60*(10*B*b^10*e^5*x^6 - 12*(6*B*b^10*d*e^4 - (10*B*a*b^9 + A
*b^10)*e^5)*x^5 + 15*(21*B*b^10*d^2*e^3 - 6*(10*B*a*b^9 + A*b^10)*d*e^4 + 5*(9*B
*a^2*b^8 + 2*A*a*b^9)*e^5)*x^4 - 20*(56*B*b^10*d^3*e^2 - 21*(10*B*a*b^9 + A*b^10
)*d^2*e^3 + 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^4 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*
e^5)*x^3 + 30*(126*B*b^10*d^4*e - 56*(10*B*a*b^9 + A*b^10)*d^3*e^2 + 105*(9*B*a^
2*b^8 + 2*A*a*b^9)*d^2*e^3 - 90*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^4 + 30*(7*B*a^4*
b^6 + 4*A*a^3*b^7)*e^5)*x^2 - 60*(252*B*b^10*d^5 - 126*(10*B*a*b^9 + A*b^10)*d^4
*e + 280*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^2 - 315*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2
*e^3 + 180*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^4 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^
5)*x)/e^11 + 42*(11*B*b^10*d^6 - 6*(10*B*a*b^9 + A*b^10)*d^5*e + 15*(9*B*a^2*b^8
+ 2*A*a*b^9)*d^4*e^2 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^3 + 15*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*d^2*e^4 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^5 + (5*B*a^6*b^4 + 6
*A*a^5*b^5)*e^6)*log(e*x + d)/e^12

_______________________________________________________________________________________

Fricas [A]  time = 0.230425, size = 3849, normalized size = 8.61 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^6,x, algorithm="fricas")

[Out]

1/60*(10*B*b^10*e^11*x^11 + 15797*B*b^10*d^11 - 12*A*a^10*e^11 - 9762*(10*B*a*b^
9 + A*b^10)*d^10*e + 28185*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 44580*(8*B*a^3*b^
7 + 3*A*a^2*b^8)*d^8*e^3 + 41310*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 21924*(6*
B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 5754*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 36
0*(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 - 3*(B*a^10 + 10*A*a^9*b)*d*e^10 - 2*(11*B
*b^10*d*e^10 - 6*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 5*(11*B*b^10*d^2*e^9 - 6*(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 - 15*(11*B*b^
10*d^3*e^8 - 6*(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 - 20*(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 - 6*(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 - 20*(8*B*a^3*b^7
+ 3*A*a^2*b^8)*d*e^10 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 - 420*(11*B*b^
10*d^5*e^6 - 6*(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 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*
e^10 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 - (47497*B*b^10*d^6*e^5 - 24762*(
10*B*a*b^9 + A*b^10)*d^5*e^6 + 58125*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 70500*(
8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 45000*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 -
12600*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10)*x^5 - 5*(19777*B*b^10*d^7*e^4 - 9642*
(10*B*a*b^9 + A*b^10)*d^6*e^5 + 20325*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 20100*
(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 7200*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 +
2520*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 - 2520*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e
^10 + 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 - 10*(5917*B*b^10*d^8*e^3 - 2082
*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 1425*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 5100*(
8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 - 11700*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 +
10080*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 - 3780*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^
2*e^9 + 360*(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 + 10*(3323*B*b^10*d^9*e^2 - 2958*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 11175
*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 21900*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 +
24300*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 15120*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d
^4*e^7 + 4620*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 360*(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 - 10*(2*B*a^9*b + 9*A*a^8*b^
2)*e^11)*x^2 + 5*(10253*B*b^10*d^10*e - 6738*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 206
25*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 34500*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4
+ 33750*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 18900*(6*B*a^5*b^5 + 5*A*a^4*b^6)
*d^5*e^6 + 5250*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 360*(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 - 3*(B*a^10 + 10*A*a^9*b)*e^11)*x + 2520*(11*B*b^10*d^11 - 6*(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 - 20*(8*B*a^3*b^7
+ 3*A*a^2*b^8)*d^8*e^3 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 6*(6*B*a^5*b^5
+ 5*A*a^4*b^6)*d^6*e^5 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + (11*B*b^10*d^6*e
^5 - 6*(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 - 20
*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 -
6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 5
*(11*B*b^10*d^7*e^4 - 6*(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 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 15*(7*B*a^4*b^6 + 4*A*a^
3*b^7)*d^3*e^8 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + (5*B*a^6*b^4 + 6*A*a^5*
b^5)*d*e^10)*x^4 + 10*(11*B*b^10*d^8*e^3 - 6*(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 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 15*
(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + (5
*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9)*x^3 + 10*(11*B*b^10*d^9*e^2 - 6*(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 - 20*(8*B*a^3*b^7 + 3*A*
a^2*b^8)*d^6*e^5 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 6*(6*B*a^5*b^5 + 5*A
*a^4*b^6)*d^4*e^7 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8)*x^2 + 5*(11*B*b^10*d^10
*e - 6*(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 - 20
*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 -
6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7)*x)*
log(e*x + d))/(e^17*x^5 + 5*d*e^16*x^4 + 10*d^2*e^15*x^3 + 10*d^3*e^14*x^2 + 5*d
^4*e^13*x + d^5*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)**6,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216592, 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)^6,x, algorithm="giac")

[Out]

Done