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3.1065 \(\int (a+b x)^{10} (A+B x) (d+e x)^6 \, dx\)

Optimal. Leaf size=290 \[ \frac{e^5 (a+b x)^{17} (-7 a B e+A b e+6 b B d)}{17 b^8}+\frac{3 e^4 (a+b x)^{16} (b d-a e) (-7 a B e+2 A b e+5 b B d)}{16 b^8}+\frac{e^3 (a+b x)^{15} (b d-a e)^2 (-7 a B e+3 A b e+4 b B d)}{3 b^8}+\frac{5 e^2 (a+b x)^{14} (b d-a e)^3 (-7 a B e+4 A b e+3 b B d)}{14 b^8}+\frac{3 e (a+b x)^{13} (b d-a e)^4 (-7 a B e+5 A b e+2 b B d)}{13 b^8}+\frac{(a+b x)^{12} (b d-a e)^5 (-7 a B e+6 A b e+b B d)}{12 b^8}+\frac{(a+b x)^{11} (A b-a B) (b d-a e)^6}{11 b^8}+\frac{B e^6 (a+b x)^{18}}{18 b^8} \]

[Out]

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

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Rubi [A]  time = 6.8467, antiderivative size = 290, 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 \[ \frac{e^5 (a+b x)^{17} (-7 a B e+A b e+6 b B d)}{17 b^8}+\frac{3 e^4 (a+b x)^{16} (b d-a e) (-7 a B e+2 A b e+5 b B d)}{16 b^8}+\frac{e^3 (a+b x)^{15} (b d-a e)^2 (-7 a B e+3 A b e+4 b B d)}{3 b^8}+\frac{5 e^2 (a+b x)^{14} (b d-a e)^3 (-7 a B e+4 A b e+3 b B d)}{14 b^8}+\frac{3 e (a+b x)^{13} (b d-a e)^4 (-7 a B e+5 A b e+2 b B d)}{13 b^8}+\frac{(a+b x)^{12} (b d-a e)^5 (-7 a B e+6 A b e+b B d)}{12 b^8}+\frac{(a+b x)^{11} (A b-a B) (b d-a e)^6}{11 b^8}+\frac{B e^6 (a+b x)^{18}}{18 b^8} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [B]  time = 1.35146, size = 1788, normalized size = 6.17 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

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

[Out]

a^10*A*d^6*x + (a^9*d^5*(10*A*b*d + a*B*d + 6*a*A*e)*x^2)/2 + (a^8*d^4*(2*a*B*d*
(5*b*d + 3*a*e) + 15*A*(3*b^2*d^2 + 4*a*b*d*e + a^2*e^2))*x^3)/3 + (5*a^7*d^3*(3
*a*B*d*(3*b^2*d^2 + 4*a*b*d*e + a^2*e^2) + A*(24*b^3*d^3 + 54*a*b^2*d^2*e + 30*a
^2*b*d*e^2 + 4*a^3*e^3))*x^4)/4 + a^6*d^2*(2*a*B*d*(12*b^3*d^3 + 27*a*b^2*d^2*e
+ 15*a^2*b*d*e^2 + 2*a^3*e^3) + A*(42*b^4*d^4 + 144*a*b^3*d^3*e + 135*a^2*b^2*d^
2*e^2 + 40*a^3*b*d*e^3 + 3*a^4*e^4))*x^5 + (a^5*d*(5*a*B*d*(42*b^4*d^4 + 144*a*b
^3*d^3*e + 135*a^2*b^2*d^2*e^2 + 40*a^3*b*d*e^3 + 3*a^4*e^4) + 6*A*(42*b^5*d^5 +
 210*a*b^4*d^4*e + 300*a^2*b^3*d^3*e^2 + 150*a^3*b^2*d^2*e^3 + 25*a^4*b*d*e^4 +
a^5*e^5))*x^6)/6 + (a^4*(6*a*B*d*(42*b^5*d^5 + 210*a*b^4*d^4*e + 300*a^2*b^3*d^3
*e^2 + 150*a^3*b^2*d^2*e^3 + 25*a^4*b*d*e^4 + a^5*e^5) + A*(210*b^6*d^6 + 1512*a
*b^5*d^5*e + 3150*a^2*b^4*d^4*e^2 + 2400*a^3*b^3*d^3*e^3 + 675*a^4*b^2*d^2*e^4 +
 60*a^5*b*d*e^5 + a^6*e^6))*x^7)/7 + (a^3*(10*A*b*(12*b^6*d^6 + 126*a*b^5*d^5*e
+ 378*a^2*b^4*d^4*e^2 + 420*a^3*b^3*d^3*e^3 + 180*a^4*b^2*d^2*e^4 + 27*a^5*b*d*e
^5 + a^6*e^6) + a*B*(210*b^6*d^6 + 1512*a*b^5*d^5*e + 3150*a^2*b^4*d^4*e^2 + 240
0*a^3*b^3*d^3*e^3 + 675*a^4*b^2*d^2*e^4 + 60*a^5*b*d*e^5 + a^6*e^6))*x^8)/8 + (5
*a^2*b*(9*A*b*(b^6*d^6 + 16*a*b^5*d^5*e + 70*a^2*b^4*d^4*e^2 + 112*a^3*b^3*d^3*e
^3 + 70*a^4*b^2*d^2*e^4 + 16*a^5*b*d*e^5 + a^6*e^6) + 2*a*B*(12*b^6*d^6 + 126*a*
b^5*d^5*e + 378*a^2*b^4*d^4*e^2 + 420*a^3*b^3*d^3*e^3 + 180*a^4*b^2*d^2*e^4 + 27
*a^5*b*d*e^5 + a^6*e^6))*x^9)/9 + (a*b^2*(9*a*B*(b^6*d^6 + 16*a*b^5*d^5*e + 70*a
^2*b^4*d^4*e^2 + 112*a^3*b^3*d^3*e^3 + 70*a^4*b^2*d^2*e^4 + 16*a^5*b*d*e^5 + a^6
*e^6) + 2*A*b*(b^6*d^6 + 27*a*b^5*d^5*e + 180*a^2*b^4*d^4*e^2 + 420*a^3*b^3*d^3*
e^3 + 378*a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^5 + 12*a^6*e^6))*x^10)/2 + (b^3*(10*a*
B*(b^6*d^6 + 27*a*b^5*d^5*e + 180*a^2*b^4*d^4*e^2 + 420*a^3*b^3*d^3*e^3 + 378*a^
4*b^2*d^2*e^4 + 126*a^5*b*d*e^5 + 12*a^6*e^6) + A*b*(b^6*d^6 + 60*a*b^5*d^5*e +
675*a^2*b^4*d^4*e^2 + 2400*a^3*b^3*d^3*e^3 + 3150*a^4*b^2*d^2*e^4 + 1512*a^5*b*d
*e^5 + 210*a^6*e^6))*x^11)/11 + (b^4*(210*a^6*B*e^6 + 252*a^5*b*e^5*(6*B*d + A*e
) + 630*a^4*b^2*d*e^4*(5*B*d + 2*A*e) + 600*a^3*b^3*d^2*e^3*(4*B*d + 3*A*e) + 22
5*a^2*b^4*d^3*e^2*(3*B*d + 4*A*e) + 30*a*b^5*d^4*e*(2*B*d + 5*A*e) + b^6*d^5*(B*
d + 6*A*e))*x^12)/12 + (b^5*e*(252*a^5*B*e^5 + 210*a^4*b*e^4*(6*B*d + A*e) + 360
*a^3*b^2*d*e^3*(5*B*d + 2*A*e) + 225*a^2*b^3*d^2*e^2*(4*B*d + 3*A*e) + 50*a*b^4*
d^3*e*(3*B*d + 4*A*e) + 3*b^5*d^4*(2*B*d + 5*A*e))*x^13)/13 + (5*b^6*e^2*(42*a^4
*B*e^4 + 24*a^3*b*e^3*(6*B*d + A*e) + 27*a^2*b^2*d*e^2*(5*B*d + 2*A*e) + 10*a*b^
3*d^2*e*(4*B*d + 3*A*e) + b^4*d^3*(3*B*d + 4*A*e))*x^14)/14 + (b^7*e^3*(24*a^3*B
*e^3 + 9*a^2*b*e^2*(6*B*d + A*e) + 6*a*b^2*d*e*(5*B*d + 2*A*e) + b^3*d^2*(4*B*d
+ 3*A*e))*x^15)/3 + (b^8*e^4*(45*a^2*B*e^2 + 10*a*b*e*(6*B*d + A*e) + 3*b^2*d*(5
*B*d + 2*A*e))*x^16)/16 + (b^9*e^5*(6*b*B*d + A*b*e + 10*a*B*e)*x^17)/17 + (b^10
*B*e^6*x^18)/18

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Maple [B]  time = 0.004, size = 1905, normalized size = 6.6 \[ \text{result too large to display} \]

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]

1/18*b^10*B*e^6*x^18+1/17*((A*b^10+10*B*a*b^9)*e^6+6*b^10*B*d*e^5)*x^17+1/16*((1
0*A*a*b^9+45*B*a^2*b^8)*e^6+6*(A*b^10+10*B*a*b^9)*d*e^5+15*b^10*B*d^2*e^4)*x^16+
1/15*((45*A*a^2*b^8+120*B*a^3*b^7)*e^6+6*(10*A*a*b^9+45*B*a^2*b^8)*d*e^5+15*(A*b
^10+10*B*a*b^9)*d^2*e^4+20*b^10*B*d^3*e^3)*x^15+1/14*((120*A*a^3*b^7+210*B*a^4*b
^6)*e^6+6*(45*A*a^2*b^8+120*B*a^3*b^7)*d*e^5+15*(10*A*a*b^9+45*B*a^2*b^8)*d^2*e^
4+20*(A*b^10+10*B*a*b^9)*d^3*e^3+15*b^10*B*d^4*e^2)*x^14+1/13*((210*A*a^4*b^6+25
2*B*a^5*b^5)*e^6+6*(120*A*a^3*b^7+210*B*a^4*b^6)*d*e^5+15*(45*A*a^2*b^8+120*B*a^
3*b^7)*d^2*e^4+20*(10*A*a*b^9+45*B*a^2*b^8)*d^3*e^3+15*(A*b^10+10*B*a*b^9)*d^4*e
^2+6*b^10*B*d^5*e)*x^13+1/12*((252*A*a^5*b^5+210*B*a^6*b^4)*e^6+6*(210*A*a^4*b^6
+252*B*a^5*b^5)*d*e^5+15*(120*A*a^3*b^7+210*B*a^4*b^6)*d^2*e^4+20*(45*A*a^2*b^8+
120*B*a^3*b^7)*d^3*e^3+15*(10*A*a*b^9+45*B*a^2*b^8)*d^4*e^2+6*(A*b^10+10*B*a*b^9
)*d^5*e+b^10*B*d^6)*x^12+1/11*((210*A*a^6*b^4+120*B*a^7*b^3)*e^6+6*(252*A*a^5*b^
5+210*B*a^6*b^4)*d*e^5+15*(210*A*a^4*b^6+252*B*a^5*b^5)*d^2*e^4+20*(120*A*a^3*b^
7+210*B*a^4*b^6)*d^3*e^3+15*(45*A*a^2*b^8+120*B*a^3*b^7)*d^4*e^2+6*(10*A*a*b^9+4
5*B*a^2*b^8)*d^5*e+(A*b^10+10*B*a*b^9)*d^6)*x^11+1/10*((120*A*a^7*b^3+45*B*a^8*b
^2)*e^6+6*(210*A*a^6*b^4+120*B*a^7*b^3)*d*e^5+15*(252*A*a^5*b^5+210*B*a^6*b^4)*d
^2*e^4+20*(210*A*a^4*b^6+252*B*a^5*b^5)*d^3*e^3+15*(120*A*a^3*b^7+210*B*a^4*b^6)
*d^4*e^2+6*(45*A*a^2*b^8+120*B*a^3*b^7)*d^5*e+(10*A*a*b^9+45*B*a^2*b^8)*d^6)*x^1
0+1/9*((45*A*a^8*b^2+10*B*a^9*b)*e^6+6*(120*A*a^7*b^3+45*B*a^8*b^2)*d*e^5+15*(21
0*A*a^6*b^4+120*B*a^7*b^3)*d^2*e^4+20*(252*A*a^5*b^5+210*B*a^6*b^4)*d^3*e^3+15*(
210*A*a^4*b^6+252*B*a^5*b^5)*d^4*e^2+6*(120*A*a^3*b^7+210*B*a^4*b^6)*d^5*e+(45*A
*a^2*b^8+120*B*a^3*b^7)*d^6)*x^9+1/8*((10*A*a^9*b+B*a^10)*e^6+6*(45*A*a^8*b^2+10
*B*a^9*b)*d*e^5+15*(120*A*a^7*b^3+45*B*a^8*b^2)*d^2*e^4+20*(210*A*a^6*b^4+120*B*
a^7*b^3)*d^3*e^3+15*(252*A*a^5*b^5+210*B*a^6*b^4)*d^4*e^2+6*(210*A*a^4*b^6+252*B
*a^5*b^5)*d^5*e+(120*A*a^3*b^7+210*B*a^4*b^6)*d^6)*x^8+1/7*(a^10*A*e^6+6*(10*A*a
^9*b+B*a^10)*d*e^5+15*(45*A*a^8*b^2+10*B*a^9*b)*d^2*e^4+20*(120*A*a^7*b^3+45*B*a
^8*b^2)*d^3*e^3+15*(210*A*a^6*b^4+120*B*a^7*b^3)*d^4*e^2+6*(252*A*a^5*b^5+210*B*
a^6*b^4)*d^5*e+(210*A*a^4*b^6+252*B*a^5*b^5)*d^6)*x^7+1/6*(6*a^10*A*d*e^5+15*(10
*A*a^9*b+B*a^10)*d^2*e^4+20*(45*A*a^8*b^2+10*B*a^9*b)*d^3*e^3+15*(120*A*a^7*b^3+
45*B*a^8*b^2)*d^4*e^2+6*(210*A*a^6*b^4+120*B*a^7*b^3)*d^5*e+(252*A*a^5*b^5+210*B
*a^6*b^4)*d^6)*x^6+1/5*(15*a^10*A*d^2*e^4+20*(10*A*a^9*b+B*a^10)*d^3*e^3+15*(45*
A*a^8*b^2+10*B*a^9*b)*d^4*e^2+6*(120*A*a^7*b^3+45*B*a^8*b^2)*d^5*e+(210*A*a^6*b^
4+120*B*a^7*b^3)*d^6)*x^5+1/4*(20*a^10*A*d^3*e^3+15*(10*A*a^9*b+B*a^10)*d^4*e^2+
6*(45*A*a^8*b^2+10*B*a^9*b)*d^5*e+(120*A*a^7*b^3+45*B*a^8*b^2)*d^6)*x^4+1/3*(15*
a^10*A*d^4*e^2+6*(10*A*a^9*b+B*a^10)*d^5*e+(45*A*a^8*b^2+10*B*a^9*b)*d^6)*x^3+1/
2*(6*a^10*A*d^5*e+(10*A*a^9*b+B*a^10)*d^6)*x^2+a^10*A*d^6*x

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Maxima [A]  time = 1.39183, size = 2588, normalized size = 8.92 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*B*b^10*e^6*x^18 + A*a^10*d^6*x + 1/17*(6*B*b^10*d*e^5 + (10*B*a*b^9 + A*b^1
0)*e^6)*x^17 + 1/16*(15*B*b^10*d^2*e^4 + 6*(10*B*a*b^9 + A*b^10)*d*e^5 + 5*(9*B*
a^2*b^8 + 2*A*a*b^9)*e^6)*x^16 + 1/3*(4*B*b^10*d^3*e^3 + 3*(10*B*a*b^9 + A*b^10)
*d^2*e^4 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^5 + 3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^6
)*x^15 + 5/14*(3*B*b^10*d^4*e^2 + 4*(10*B*a*b^9 + A*b^10)*d^3*e^3 + 15*(9*B*a^2*
b^8 + 2*A*a*b^9)*d^2*e^4 + 18*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^5 + 6*(7*B*a^4*b^6
 + 4*A*a^3*b^7)*e^6)*x^14 + 1/13*(6*B*b^10*d^5*e + 15*(10*B*a*b^9 + A*b^10)*d^4*
e^2 + 100*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^3 + 225*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^
2*e^4 + 180*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^5 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e
^6)*x^13 + 1/12*(B*b^10*d^6 + 6*(10*B*a*b^9 + A*b^10)*d^5*e + 75*(9*B*a^2*b^8 +
2*A*a*b^9)*d^4*e^2 + 300*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^3 + 450*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*d^2*e^4 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^5 + 42*(5*B*a^6*b^4
 + 6*A*a^5*b^5)*e^6)*x^12 + 1/11*((10*B*a*b^9 + A*b^10)*d^6 + 30*(9*B*a^2*b^8 +
2*A*a*b^9)*d^5*e + 225*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^2 + 600*(7*B*a^4*b^6 +
4*A*a^3*b^7)*d^3*e^3 + 630*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^4 + 252*(5*B*a^6*b^
4 + 6*A*a^5*b^5)*d*e^5 + 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^6)*x^11 + 1/2*((9*B*a^
2*b^8 + 2*A*a*b^9)*d^6 + 18*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e + 90*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*d^4*e^2 + 168*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^3 + 126*(5*B*a^6*
b^4 + 6*A*a^5*b^5)*d^2*e^4 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^5 + 3*(3*B*a^8*b
^2 + 8*A*a^7*b^3)*e^6)*x^10 + 5/9*(3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6 + 36*(7*B*a
^4*b^6 + 4*A*a^3*b^7)*d^5*e + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^2 + 168*(5*B
*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^3 + 90*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^4 + 18*(3
*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^5 + (2*B*a^9*b + 9*A*a^8*b^2)*e^6)*x^9 + 1/8*(30*(
7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e + 630*(5*
B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^2 + 600*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^3 + 225
*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^4 + 30*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^5 + (B*a
^10 + 10*A*a^9*b)*e^6)*x^8 + 1/7*(A*a^10*e^6 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^
6 + 252*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e + 450*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*
e^2 + 300*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^3 + 75*(2*B*a^9*b + 9*A*a^8*b^2)*d^2
*e^4 + 6*(B*a^10 + 10*A*a^9*b)*d*e^5)*x^7 + 1/6*(6*A*a^10*d*e^5 + 42*(5*B*a^6*b^
4 + 6*A*a^5*b^5)*d^6 + 180*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^5*e + 225*(3*B*a^8*b^2
+ 8*A*a^7*b^3)*d^4*e^2 + 100*(2*B*a^9*b + 9*A*a^8*b^2)*d^3*e^3 + 15*(B*a^10 + 10
*A*a^9*b)*d^2*e^4)*x^6 + (3*A*a^10*d^2*e^4 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^6 +
 18*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^5*e + 15*(2*B*a^9*b + 9*A*a^8*b^2)*d^4*e^2 + 4
*(B*a^10 + 10*A*a^9*b)*d^3*e^3)*x^5 + 5/4*(4*A*a^10*d^3*e^3 + 3*(3*B*a^8*b^2 + 8
*A*a^7*b^3)*d^6 + 6*(2*B*a^9*b + 9*A*a^8*b^2)*d^5*e + 3*(B*a^10 + 10*A*a^9*b)*d^
4*e^2)*x^4 + 1/3*(15*A*a^10*d^4*e^2 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^6 + 6*(B*a^1
0 + 10*A*a^9*b)*d^5*e)*x^3 + 1/2*(6*A*a^10*d^5*e + (B*a^10 + 10*A*a^9*b)*d^6)*x^
2

_______________________________________________________________________________________

Fricas [A]  time = 0.194352, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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]

1/18*x^18*e^6*b^10*B + 6/17*x^17*e^5*d*b^10*B + 10/17*x^17*e^6*b^9*a*B + 1/17*x^
17*e^6*b^10*A + 15/16*x^16*e^4*d^2*b^10*B + 15/4*x^16*e^5*d*b^9*a*B + 45/16*x^16
*e^6*b^8*a^2*B + 3/8*x^16*e^5*d*b^10*A + 5/8*x^16*e^6*b^9*a*A + 4/3*x^15*e^3*d^3
*b^10*B + 10*x^15*e^4*d^2*b^9*a*B + 18*x^15*e^5*d*b^8*a^2*B + 8*x^15*e^6*b^7*a^3
*B + x^15*e^4*d^2*b^10*A + 4*x^15*e^5*d*b^9*a*A + 3*x^15*e^6*b^8*a^2*A + 15/14*x
^14*e^2*d^4*b^10*B + 100/7*x^14*e^3*d^3*b^9*a*B + 675/14*x^14*e^4*d^2*b^8*a^2*B
+ 360/7*x^14*e^5*d*b^7*a^3*B + 15*x^14*e^6*b^6*a^4*B + 10/7*x^14*e^3*d^3*b^10*A
+ 75/7*x^14*e^4*d^2*b^9*a*A + 135/7*x^14*e^5*d*b^8*a^2*A + 60/7*x^14*e^6*b^7*a^3
*A + 6/13*x^13*e*d^5*b^10*B + 150/13*x^13*e^2*d^4*b^9*a*B + 900/13*x^13*e^3*d^3*
b^8*a^2*B + 1800/13*x^13*e^4*d^2*b^7*a^3*B + 1260/13*x^13*e^5*d*b^6*a^4*B + 252/
13*x^13*e^6*b^5*a^5*B + 15/13*x^13*e^2*d^4*b^10*A + 200/13*x^13*e^3*d^3*b^9*a*A
+ 675/13*x^13*e^4*d^2*b^8*a^2*A + 720/13*x^13*e^5*d*b^7*a^3*A + 210/13*x^13*e^6*
b^6*a^4*A + 1/12*x^12*d^6*b^10*B + 5*x^12*e*d^5*b^9*a*B + 225/4*x^12*e^2*d^4*b^8
*a^2*B + 200*x^12*e^3*d^3*b^7*a^3*B + 525/2*x^12*e^4*d^2*b^6*a^4*B + 126*x^12*e^
5*d*b^5*a^5*B + 35/2*x^12*e^6*b^4*a^6*B + 1/2*x^12*e*d^5*b^10*A + 25/2*x^12*e^2*
d^4*b^9*a*A + 75*x^12*e^3*d^3*b^8*a^2*A + 150*x^12*e^4*d^2*b^7*a^3*A + 105*x^12*
e^5*d*b^6*a^4*A + 21*x^12*e^6*b^5*a^5*A + 10/11*x^11*d^6*b^9*a*B + 270/11*x^11*e
*d^5*b^8*a^2*B + 1800/11*x^11*e^2*d^4*b^7*a^3*B + 4200/11*x^11*e^3*d^3*b^6*a^4*B
 + 3780/11*x^11*e^4*d^2*b^5*a^5*B + 1260/11*x^11*e^5*d*b^4*a^6*B + 120/11*x^11*e
^6*b^3*a^7*B + 1/11*x^11*d^6*b^10*A + 60/11*x^11*e*d^5*b^9*a*A + 675/11*x^11*e^2
*d^4*b^8*a^2*A + 2400/11*x^11*e^3*d^3*b^7*a^3*A + 3150/11*x^11*e^4*d^2*b^6*a^4*A
 + 1512/11*x^11*e^5*d*b^5*a^5*A + 210/11*x^11*e^6*b^4*a^6*A + 9/2*x^10*d^6*b^8*a
^2*B + 72*x^10*e*d^5*b^7*a^3*B + 315*x^10*e^2*d^4*b^6*a^4*B + 504*x^10*e^3*d^3*b
^5*a^5*B + 315*x^10*e^4*d^2*b^4*a^6*B + 72*x^10*e^5*d*b^3*a^7*B + 9/2*x^10*e^6*b
^2*a^8*B + x^10*d^6*b^9*a*A + 27*x^10*e*d^5*b^8*a^2*A + 180*x^10*e^2*d^4*b^7*a^3
*A + 420*x^10*e^3*d^3*b^6*a^4*A + 378*x^10*e^4*d^2*b^5*a^5*A + 126*x^10*e^5*d*b^
4*a^6*A + 12*x^10*e^6*b^3*a^7*A + 40/3*x^9*d^6*b^7*a^3*B + 140*x^9*e*d^5*b^6*a^4
*B + 420*x^9*e^2*d^4*b^5*a^5*B + 1400/3*x^9*e^3*d^3*b^4*a^6*B + 200*x^9*e^4*d^2*
b^3*a^7*B + 30*x^9*e^5*d*b^2*a^8*B + 10/9*x^9*e^6*b*a^9*B + 5*x^9*d^6*b^8*a^2*A
+ 80*x^9*e*d^5*b^7*a^3*A + 350*x^9*e^2*d^4*b^6*a^4*A + 560*x^9*e^3*d^3*b^5*a^5*A
 + 350*x^9*e^4*d^2*b^4*a^6*A + 80*x^9*e^5*d*b^3*a^7*A + 5*x^9*e^6*b^2*a^8*A + 10
5/4*x^8*d^6*b^6*a^4*B + 189*x^8*e*d^5*b^5*a^5*B + 1575/4*x^8*e^2*d^4*b^4*a^6*B +
 300*x^8*e^3*d^3*b^3*a^7*B + 675/8*x^8*e^4*d^2*b^2*a^8*B + 15/2*x^8*e^5*d*b*a^9*
B + 1/8*x^8*e^6*a^10*B + 15*x^8*d^6*b^7*a^3*A + 315/2*x^8*e*d^5*b^6*a^4*A + 945/
2*x^8*e^2*d^4*b^5*a^5*A + 525*x^8*e^3*d^3*b^4*a^6*A + 225*x^8*e^4*d^2*b^3*a^7*A
+ 135/4*x^8*e^5*d*b^2*a^8*A + 5/4*x^8*e^6*b*a^9*A + 36*x^7*d^6*b^5*a^5*B + 180*x
^7*e*d^5*b^4*a^6*B + 1800/7*x^7*e^2*d^4*b^3*a^7*B + 900/7*x^7*e^3*d^3*b^2*a^8*B
+ 150/7*x^7*e^4*d^2*b*a^9*B + 6/7*x^7*e^5*d*a^10*B + 30*x^7*d^6*b^6*a^4*A + 216*
x^7*e*d^5*b^5*a^5*A + 450*x^7*e^2*d^4*b^4*a^6*A + 2400/7*x^7*e^3*d^3*b^3*a^7*A +
 675/7*x^7*e^4*d^2*b^2*a^8*A + 60/7*x^7*e^5*d*b*a^9*A + 1/7*x^7*e^6*a^10*A + 35*
x^6*d^6*b^4*a^6*B + 120*x^6*e*d^5*b^3*a^7*B + 225/2*x^6*e^2*d^4*b^2*a^8*B + 100/
3*x^6*e^3*d^3*b*a^9*B + 5/2*x^6*e^4*d^2*a^10*B + 42*x^6*d^6*b^5*a^5*A + 210*x^6*
e*d^5*b^4*a^6*A + 300*x^6*e^2*d^4*b^3*a^7*A + 150*x^6*e^3*d^3*b^2*a^8*A + 25*x^6
*e^4*d^2*b*a^9*A + x^6*e^5*d*a^10*A + 24*x^5*d^6*b^3*a^7*B + 54*x^5*e*d^5*b^2*a^
8*B + 30*x^5*e^2*d^4*b*a^9*B + 4*x^5*e^3*d^3*a^10*B + 42*x^5*d^6*b^4*a^6*A + 144
*x^5*e*d^5*b^3*a^7*A + 135*x^5*e^2*d^4*b^2*a^8*A + 40*x^5*e^3*d^3*b*a^9*A + 3*x^
5*e^4*d^2*a^10*A + 45/4*x^4*d^6*b^2*a^8*B + 15*x^4*e*d^5*b*a^9*B + 15/4*x^4*e^2*
d^4*a^10*B + 30*x^4*d^6*b^3*a^7*A + 135/2*x^4*e*d^5*b^2*a^8*A + 75/2*x^4*e^2*d^4
*b*a^9*A + 5*x^4*e^3*d^3*a^10*A + 10/3*x^3*d^6*b*a^9*B + 2*x^3*e*d^5*a^10*B + 15
*x^3*d^6*b^2*a^8*A + 20*x^3*e*d^5*b*a^9*A + 5*x^3*e^2*d^4*a^10*A + 1/2*x^2*d^6*a
^10*B + 5*x^2*d^6*b*a^9*A + 3*x^2*e*d^5*a^10*A + x*d^6*a^10*A

_______________________________________________________________________________________

Sympy [A]  time = 1.05469, size = 2424, normalized size = 8.36 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**10*d**6*x + B*b**10*e**6*x**18/18 + x**17*(A*b**10*e**6/17 + 10*B*a*b**9*e*
*6/17 + 6*B*b**10*d*e**5/17) + x**16*(5*A*a*b**9*e**6/8 + 3*A*b**10*d*e**5/8 + 4
5*B*a**2*b**8*e**6/16 + 15*B*a*b**9*d*e**5/4 + 15*B*b**10*d**2*e**4/16) + x**15*
(3*A*a**2*b**8*e**6 + 4*A*a*b**9*d*e**5 + A*b**10*d**2*e**4 + 8*B*a**3*b**7*e**6
 + 18*B*a**2*b**8*d*e**5 + 10*B*a*b**9*d**2*e**4 + 4*B*b**10*d**3*e**3/3) + x**1
4*(60*A*a**3*b**7*e**6/7 + 135*A*a**2*b**8*d*e**5/7 + 75*A*a*b**9*d**2*e**4/7 +
10*A*b**10*d**3*e**3/7 + 15*B*a**4*b**6*e**6 + 360*B*a**3*b**7*d*e**5/7 + 675*B*
a**2*b**8*d**2*e**4/14 + 100*B*a*b**9*d**3*e**3/7 + 15*B*b**10*d**4*e**2/14) + x
**13*(210*A*a**4*b**6*e**6/13 + 720*A*a**3*b**7*d*e**5/13 + 675*A*a**2*b**8*d**2
*e**4/13 + 200*A*a*b**9*d**3*e**3/13 + 15*A*b**10*d**4*e**2/13 + 252*B*a**5*b**5
*e**6/13 + 1260*B*a**4*b**6*d*e**5/13 + 1800*B*a**3*b**7*d**2*e**4/13 + 900*B*a*
*2*b**8*d**3*e**3/13 + 150*B*a*b**9*d**4*e**2/13 + 6*B*b**10*d**5*e/13) + x**12*
(21*A*a**5*b**5*e**6 + 105*A*a**4*b**6*d*e**5 + 150*A*a**3*b**7*d**2*e**4 + 75*A
*a**2*b**8*d**3*e**3 + 25*A*a*b**9*d**4*e**2/2 + A*b**10*d**5*e/2 + 35*B*a**6*b*
*4*e**6/2 + 126*B*a**5*b**5*d*e**5 + 525*B*a**4*b**6*d**2*e**4/2 + 200*B*a**3*b*
*7*d**3*e**3 + 225*B*a**2*b**8*d**4*e**2/4 + 5*B*a*b**9*d**5*e + B*b**10*d**6/12
) + x**11*(210*A*a**6*b**4*e**6/11 + 1512*A*a**5*b**5*d*e**5/11 + 3150*A*a**4*b*
*6*d**2*e**4/11 + 2400*A*a**3*b**7*d**3*e**3/11 + 675*A*a**2*b**8*d**4*e**2/11 +
 60*A*a*b**9*d**5*e/11 + A*b**10*d**6/11 + 120*B*a**7*b**3*e**6/11 + 1260*B*a**6
*b**4*d*e**5/11 + 3780*B*a**5*b**5*d**2*e**4/11 + 4200*B*a**4*b**6*d**3*e**3/11
+ 1800*B*a**3*b**7*d**4*e**2/11 + 270*B*a**2*b**8*d**5*e/11 + 10*B*a*b**9*d**6/1
1) + x**10*(12*A*a**7*b**3*e**6 + 126*A*a**6*b**4*d*e**5 + 378*A*a**5*b**5*d**2*
e**4 + 420*A*a**4*b**6*d**3*e**3 + 180*A*a**3*b**7*d**4*e**2 + 27*A*a**2*b**8*d*
*5*e + A*a*b**9*d**6 + 9*B*a**8*b**2*e**6/2 + 72*B*a**7*b**3*d*e**5 + 315*B*a**6
*b**4*d**2*e**4 + 504*B*a**5*b**5*d**3*e**3 + 315*B*a**4*b**6*d**4*e**2 + 72*B*a
**3*b**7*d**5*e + 9*B*a**2*b**8*d**6/2) + x**9*(5*A*a**8*b**2*e**6 + 80*A*a**7*b
**3*d*e**5 + 350*A*a**6*b**4*d**2*e**4 + 560*A*a**5*b**5*d**3*e**3 + 350*A*a**4*
b**6*d**4*e**2 + 80*A*a**3*b**7*d**5*e + 5*A*a**2*b**8*d**6 + 10*B*a**9*b*e**6/9
 + 30*B*a**8*b**2*d*e**5 + 200*B*a**7*b**3*d**2*e**4 + 1400*B*a**6*b**4*d**3*e**
3/3 + 420*B*a**5*b**5*d**4*e**2 + 140*B*a**4*b**6*d**5*e + 40*B*a**3*b**7*d**6/3
) + x**8*(5*A*a**9*b*e**6/4 + 135*A*a**8*b**2*d*e**5/4 + 225*A*a**7*b**3*d**2*e*
*4 + 525*A*a**6*b**4*d**3*e**3 + 945*A*a**5*b**5*d**4*e**2/2 + 315*A*a**4*b**6*d
**5*e/2 + 15*A*a**3*b**7*d**6 + B*a**10*e**6/8 + 15*B*a**9*b*d*e**5/2 + 675*B*a*
*8*b**2*d**2*e**4/8 + 300*B*a**7*b**3*d**3*e**3 + 1575*B*a**6*b**4*d**4*e**2/4 +
 189*B*a**5*b**5*d**5*e + 105*B*a**4*b**6*d**6/4) + x**7*(A*a**10*e**6/7 + 60*A*
a**9*b*d*e**5/7 + 675*A*a**8*b**2*d**2*e**4/7 + 2400*A*a**7*b**3*d**3*e**3/7 + 4
50*A*a**6*b**4*d**4*e**2 + 216*A*a**5*b**5*d**5*e + 30*A*a**4*b**6*d**6 + 6*B*a*
*10*d*e**5/7 + 150*B*a**9*b*d**2*e**4/7 + 900*B*a**8*b**2*d**3*e**3/7 + 1800*B*a
**7*b**3*d**4*e**2/7 + 180*B*a**6*b**4*d**5*e + 36*B*a**5*b**5*d**6) + x**6*(A*a
**10*d*e**5 + 25*A*a**9*b*d**2*e**4 + 150*A*a**8*b**2*d**3*e**3 + 300*A*a**7*b**
3*d**4*e**2 + 210*A*a**6*b**4*d**5*e + 42*A*a**5*b**5*d**6 + 5*B*a**10*d**2*e**4
/2 + 100*B*a**9*b*d**3*e**3/3 + 225*B*a**8*b**2*d**4*e**2/2 + 120*B*a**7*b**3*d*
*5*e + 35*B*a**6*b**4*d**6) + x**5*(3*A*a**10*d**2*e**4 + 40*A*a**9*b*d**3*e**3
+ 135*A*a**8*b**2*d**4*e**2 + 144*A*a**7*b**3*d**5*e + 42*A*a**6*b**4*d**6 + 4*B
*a**10*d**3*e**3 + 30*B*a**9*b*d**4*e**2 + 54*B*a**8*b**2*d**5*e + 24*B*a**7*b**
3*d**6) + x**4*(5*A*a**10*d**3*e**3 + 75*A*a**9*b*d**4*e**2/2 + 135*A*a**8*b**2*
d**5*e/2 + 30*A*a**7*b**3*d**6 + 15*B*a**10*d**4*e**2/4 + 15*B*a**9*b*d**5*e + 4
5*B*a**8*b**2*d**6/4) + x**3*(5*A*a**10*d**4*e**2 + 20*A*a**9*b*d**5*e + 15*A*a*
*8*b**2*d**6 + 2*B*a**10*d**5*e + 10*B*a**9*b*d**6/3) + x**2*(3*A*a**10*d**5*e +
 5*A*a**9*b*d**6 + B*a**10*d**6/2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211092, 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

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