3.1080 \(\int (a+b x)^{10} (A+B x) (d+e x)^8 \, dx\)

Optimal. Leaf size=372 \[ \frac{e^7 (a+b x)^{19} (-9 a B e+A b e+8 b B d)}{19 b^{10}}+\frac{2 e^6 (a+b x)^{18} (b d-a e) (-9 a B e+2 A b e+7 b B d)}{9 b^{10}}+\frac{28 e^5 (a+b x)^{17} (b d-a e)^2 (-3 a B e+A b e+2 b B d)}{17 b^{10}}+\frac{7 e^4 (a+b x)^{16} (b d-a e)^3 (-9 a B e+4 A b e+5 b B d)}{8 b^{10}}+\frac{14 e^3 (a+b x)^{15} (b d-a e)^4 (-9 a B e+5 A b e+4 b B d)}{15 b^{10}}+\frac{2 e^2 (a+b x)^{14} (b d-a e)^5 (-3 a B e+2 A b e+b B d)}{b^{10}}+\frac{4 e (a+b x)^{13} (b d-a e)^6 (-9 a B e+7 A b e+2 b B d)}{13 b^{10}}+\frac{(a+b x)^{12} (b d-a e)^7 (-9 a B e+8 A b e+b B d)}{12 b^{10}}+\frac{(a+b x)^{11} (A b-a B) (b d-a e)^8}{11 b^{10}}+\frac{B e^8 (a+b x)^{20}}{20 b^{10}} \]

[Out]

((A*b - a*B)*(b*d - a*e)^8*(a + b*x)^11)/(11*b^10) + ((b*d - a*e)^7*(b*B*d + 8*A*b*e - 9*a*B*e)*(a + b*x)^12)/
(12*b^10) + (4*e*(b*d - a*e)^6*(2*b*B*d + 7*A*b*e - 9*a*B*e)*(a + b*x)^13)/(13*b^10) + (2*e^2*(b*d - a*e)^5*(b
*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^14)/b^10 + (14*e^3*(b*d - a*e)^4*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(a + b*x)^1
5)/(15*b^10) + (7*e^4*(b*d - a*e)^3*(5*b*B*d + 4*A*b*e - 9*a*B*e)*(a + b*x)^16)/(8*b^10) + (28*e^5*(b*d - a*e)
^2*(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^17)/(17*b^10) + (2*e^6*(b*d - a*e)*(7*b*B*d + 2*A*b*e - 9*a*B*e)*(a +
b*x)^18)/(9*b^10) + (e^7*(8*b*B*d + A*b*e - 9*a*B*e)*(a + b*x)^19)/(19*b^10) + (B*e^8*(a + b*x)^20)/(20*b^10)

________________________________________________________________________________________

Rubi [A]  time = 2.42816, antiderivative size = 372, 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, Rules used = {77} \[ \frac{e^7 (a+b x)^{19} (-9 a B e+A b e+8 b B d)}{19 b^{10}}+\frac{2 e^6 (a+b x)^{18} (b d-a e) (-9 a B e+2 A b e+7 b B d)}{9 b^{10}}+\frac{28 e^5 (a+b x)^{17} (b d-a e)^2 (-3 a B e+A b e+2 b B d)}{17 b^{10}}+\frac{7 e^4 (a+b x)^{16} (b d-a e)^3 (-9 a B e+4 A b e+5 b B d)}{8 b^{10}}+\frac{14 e^3 (a+b x)^{15} (b d-a e)^4 (-9 a B e+5 A b e+4 b B d)}{15 b^{10}}+\frac{2 e^2 (a+b x)^{14} (b d-a e)^5 (-3 a B e+2 A b e+b B d)}{b^{10}}+\frac{4 e (a+b x)^{13} (b d-a e)^6 (-9 a B e+7 A b e+2 b B d)}{13 b^{10}}+\frac{(a+b x)^{12} (b d-a e)^7 (-9 a B e+8 A b e+b B d)}{12 b^{10}}+\frac{(a+b x)^{11} (A b-a B) (b d-a e)^8}{11 b^{10}}+\frac{B e^8 (a+b x)^{20}}{20 b^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10*(A + B*x)*(d + e*x)^8,x]

[Out]

((A*b - a*B)*(b*d - a*e)^8*(a + b*x)^11)/(11*b^10) + ((b*d - a*e)^7*(b*B*d + 8*A*b*e - 9*a*B*e)*(a + b*x)^12)/
(12*b^10) + (4*e*(b*d - a*e)^6*(2*b*B*d + 7*A*b*e - 9*a*B*e)*(a + b*x)^13)/(13*b^10) + (2*e^2*(b*d - a*e)^5*(b
*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^14)/b^10 + (14*e^3*(b*d - a*e)^4*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(a + b*x)^1
5)/(15*b^10) + (7*e^4*(b*d - a*e)^3*(5*b*B*d + 4*A*b*e - 9*a*B*e)*(a + b*x)^16)/(8*b^10) + (28*e^5*(b*d - a*e)
^2*(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^17)/(17*b^10) + (2*e^6*(b*d - a*e)*(7*b*B*d + 2*A*b*e - 9*a*B*e)*(a +
b*x)^18)/(9*b^10) + (e^7*(8*b*B*d + A*b*e - 9*a*B*e)*(a + b*x)^19)/(19*b^10) + (B*e^8*(a + b*x)^20)/(20*b^10)

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int (a+b x)^{10} (A+B x) (d+e x)^8 \, dx &=\int \left (\frac{(A b-a B) (b d-a e)^8 (a+b x)^{10}}{b^9}+\frac{(b d-a e)^7 (b B d+8 A b e-9 a B e) (a+b x)^{11}}{b^9}+\frac{4 e (b d-a e)^6 (2 b B d+7 A b e-9 a B e) (a+b x)^{12}}{b^9}+\frac{28 e^2 (b d-a e)^5 (b B d+2 A b e-3 a B e) (a+b x)^{13}}{b^9}+\frac{14 e^3 (b d-a e)^4 (4 b B d+5 A b e-9 a B e) (a+b x)^{14}}{b^9}+\frac{14 e^4 (b d-a e)^3 (5 b B d+4 A b e-9 a B e) (a+b x)^{15}}{b^9}+\frac{28 e^5 (b d-a e)^2 (2 b B d+A b e-3 a B e) (a+b x)^{16}}{b^9}+\frac{4 e^6 (b d-a e) (7 b B d+2 A b e-9 a B e) (a+b x)^{17}}{b^9}+\frac{e^7 (8 b B d+A b e-9 a B e) (a+b x)^{18}}{b^9}+\frac{B e^8 (a+b x)^{19}}{b^9}\right ) \, dx\\ &=\frac{(A b-a B) (b d-a e)^8 (a+b x)^{11}}{11 b^{10}}+\frac{(b d-a e)^7 (b B d+8 A b e-9 a B e) (a+b x)^{12}}{12 b^{10}}+\frac{4 e (b d-a e)^6 (2 b B d+7 A b e-9 a B e) (a+b x)^{13}}{13 b^{10}}+\frac{2 e^2 (b d-a e)^5 (b B d+2 A b e-3 a B e) (a+b x)^{14}}{b^{10}}+\frac{14 e^3 (b d-a e)^4 (4 b B d+5 A b e-9 a B e) (a+b x)^{15}}{15 b^{10}}+\frac{7 e^4 (b d-a e)^3 (5 b B d+4 A b e-9 a B e) (a+b x)^{16}}{8 b^{10}}+\frac{28 e^5 (b d-a e)^2 (2 b B d+A b e-3 a B e) (a+b x)^{17}}{17 b^{10}}+\frac{2 e^6 (b d-a e) (7 b B d+2 A b e-9 a B e) (a+b x)^{18}}{9 b^{10}}+\frac{e^7 (8 b B d+A b e-9 a B e) (a+b x)^{19}}{19 b^{10}}+\frac{B e^8 (a+b x)^{20}}{20 b^{10}}\\ \end{align*}

Mathematica [B]  time = 0.811502, size = 2307, normalized size = 6.2 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10*(A + B*x)*(d + e*x)^8,x]

[Out]

a^10*A*d^8*x + (a^9*d^7*(10*A*b*d + a*B*d + 8*a*A*e)*x^2)/2 + (a^8*d^6*(2*a*B*d*(5*b*d + 4*a*e) + A*(45*b^2*d^
2 + 80*a*b*d*e + 28*a^2*e^2))*x^3)/3 + (a^7*d^5*(a*B*d*(45*b^2*d^2 + 80*a*b*d*e + 28*a^2*e^2) + 8*A*(15*b^3*d^
3 + 45*a*b^2*d^2*e + 35*a^2*b*d*e^2 + 7*a^3*e^3))*x^4)/4 + (2*a^6*d^4*(4*a*B*d*(15*b^3*d^3 + 45*a*b^2*d^2*e +
35*a^2*b*d*e^2 + 7*a^3*e^3) + 5*A*(21*b^4*d^4 + 96*a*b^3*d^3*e + 126*a^2*b^2*d^2*e^2 + 56*a^3*b*d*e^3 + 7*a^4*
e^4))*x^5)/5 + (a^5*d^3*(5*a*B*d*(21*b^4*d^4 + 96*a*b^3*d^3*e + 126*a^2*b^2*d^2*e^2 + 56*a^3*b*d*e^3 + 7*a^4*e
^4) + 14*A*(9*b^5*d^5 + 60*a*b^4*d^4*e + 120*a^2*b^3*d^3*e^2 + 90*a^3*b^2*d^2*e^3 + 25*a^4*b*d*e^4 + 2*a^5*e^5
))*x^6)/3 + 2*a^4*d^2*(2*a*B*d*(9*b^5*d^5 + 60*a*b^4*d^4*e + 120*a^2*b^3*d^3*e^2 + 90*a^3*b^2*d^2*e^3 + 25*a^4
*b*d*e^4 + 2*a^5*e^5) + A*(15*b^6*d^6 + 144*a*b^5*d^5*e + 420*a^2*b^4*d^4*e^2 + 480*a^3*b^3*d^3*e^3 + 225*a^4*
b^2*d^2*e^4 + 40*a^5*b*d*e^5 + 2*a^6*e^6))*x^7 + (a^3*d*(7*a*B*d*(15*b^6*d^6 + 144*a*b^5*d^5*e + 420*a^2*b^4*d
^4*e^2 + 480*a^3*b^3*d^3*e^3 + 225*a^4*b^2*d^2*e^4 + 40*a^5*b*d*e^5 + 2*a^6*e^6) + 4*A*(15*b^7*d^7 + 210*a*b^6
*d^6*e + 882*a^2*b^5*d^5*e^2 + 1470*a^3*b^4*d^4*e^3 + 1050*a^4*b^3*d^3*e^4 + 315*a^5*b^2*d^2*e^5 + 35*a^6*b*d*
e^6 + a^7*e^7))*x^8)/4 + (a^2*(8*a*B*d*(15*b^7*d^7 + 210*a*b^6*d^6*e + 882*a^2*b^5*d^5*e^2 + 1470*a^3*b^4*d^4*
e^3 + 1050*a^4*b^3*d^3*e^4 + 315*a^5*b^2*d^2*e^5 + 35*a^6*b*d*e^6 + a^7*e^7) + A*(45*b^8*d^8 + 960*a*b^7*d^7*e
+ 5880*a^2*b^6*d^6*e^2 + 14112*a^3*b^5*d^5*e^3 + 14700*a^4*b^4*d^4*e^4 + 6720*a^5*b^3*d^3*e^5 + 1260*a^6*b^2*
d^2*e^6 + 80*a^7*b*d*e^7 + a^8*e^8))*x^9)/9 + (a*(10*A*b*(b^8*d^8 + 36*a*b^7*d^7*e + 336*a^2*b^6*d^6*e^2 + 117
6*a^3*b^5*d^5*e^3 + 1764*a^4*b^4*d^4*e^4 + 1176*a^5*b^3*d^3*e^5 + 336*a^6*b^2*d^2*e^6 + 36*a^7*b*d*e^7 + a^8*e
^8) + a*B*(45*b^8*d^8 + 960*a*b^7*d^7*e + 5880*a^2*b^6*d^6*e^2 + 14112*a^3*b^5*d^5*e^3 + 14700*a^4*b^4*d^4*e^4
+ 6720*a^5*b^3*d^3*e^5 + 1260*a^6*b^2*d^2*e^6 + 80*a^7*b*d*e^7 + a^8*e^8))*x^10)/10 + (b*(10*a*B*(b^8*d^8 + 3
6*a*b^7*d^7*e + 336*a^2*b^6*d^6*e^2 + 1176*a^3*b^5*d^5*e^3 + 1764*a^4*b^4*d^4*e^4 + 1176*a^5*b^3*d^3*e^5 + 336
*a^6*b^2*d^2*e^6 + 36*a^7*b*d*e^7 + a^8*e^8) + A*b*(b^8*d^8 + 80*a*b^7*d^7*e + 1260*a^2*b^6*d^6*e^2 + 6720*a^3
*b^5*d^5*e^3 + 14700*a^4*b^4*d^4*e^4 + 14112*a^5*b^3*d^3*e^5 + 5880*a^6*b^2*d^2*e^6 + 960*a^7*b*d*e^7 + 45*a^8
*e^8))*x^11)/11 + (b^2*(45*a^8*B*e^8 + 7056*a^5*b^3*d^2*e^5*(2*B*d + A*e) + 120*a^7*b*e^7*(8*B*d + A*e) + 1260
*a^2*b^6*d^5*e^2*(B*d + 2*A*e) + 840*a^6*b^2*d*e^6*(7*B*d + 2*A*e) + 2940*a^4*b^4*d^3*e^4*(5*B*d + 4*A*e) + 16
80*a^3*b^5*d^4*e^3*(4*B*d + 5*A*e) + 40*a*b^7*d^6*e*(2*B*d + 7*A*e) + b^8*d^7*(B*d + 8*A*e))*x^12)/12 + (2*b^3
*e*(60*a^7*B*e^7 + 2940*a^4*b^3*d^2*e^4*(2*B*d + A*e) + 105*a^6*b*e^6*(8*B*d + A*e) + 140*a*b^6*d^5*e*(B*d + 2
*A*e) + 504*a^5*b^2*d*e^5*(7*B*d + 2*A*e) + 840*a^3*b^4*d^3*e^3*(5*B*d + 4*A*e) + 315*a^2*b^5*d^4*e^2*(4*B*d +
5*A*e) + 2*b^7*d^6*(2*B*d + 7*A*e))*x^13)/13 + b^4*e^2*(15*a^6*B*e^6 + 240*a^3*b^3*d^2*e^3*(2*B*d + A*e) + 18
*a^5*b*e^5*(8*B*d + A*e) + 2*b^6*d^5*(B*d + 2*A*e) + 60*a^4*b^2*d*e^4*(7*B*d + 2*A*e) + 45*a^2*b^4*d^3*e^2*(5*
B*d + 4*A*e) + 10*a*b^5*d^4*e*(4*B*d + 5*A*e))*x^14 + (2*b^5*e^3*(126*a^5*B*e^5 + 630*a^2*b^3*d^2*e^2*(2*B*d +
A*e) + 105*a^4*b*e^4*(8*B*d + A*e) + 240*a^3*b^2*d*e^3*(7*B*d + 2*A*e) + 70*a*b^4*d^3*e*(5*B*d + 4*A*e) + 7*b
^5*d^4*(4*B*d + 5*A*e))*x^15)/15 + (b^6*e^4*(105*a^4*B*e^4 + 140*a*b^3*d^2*e*(2*B*d + A*e) + 60*a^3*b*e^3*(8*B
*d + A*e) + 90*a^2*b^2*d*e^2*(7*B*d + 2*A*e) + 7*b^4*d^3*(5*B*d + 4*A*e))*x^16)/8 + (b^7*e^5*(120*a^3*B*e^3 +
28*b^3*d^2*(2*B*d + A*e) + 45*a^2*b*e^2*(8*B*d + A*e) + 40*a*b^2*d*e*(7*B*d + 2*A*e))*x^17)/17 + (b^8*e^6*(45*
a^2*B*e^2 + 10*a*b*e*(8*B*d + A*e) + 4*b^2*d*(7*B*d + 2*A*e))*x^18)/18 + (b^9*e^7*(8*b*B*d + A*b*e + 10*a*B*e)
*x^19)/19 + (b^10*B*e^8*x^20)/20

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Maple [B]  time = 0.003, size = 2473, normalized size = 6.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)*(e*x+d)^8,x)

[Out]

1/20*b^10*B*e^8*x^20+1/19*((A*b^10+10*B*a*b^9)*e^8+8*b^10*B*d*e^7)*x^19+1/18*((10*A*a*b^9+45*B*a^2*b^8)*e^8+8*
(A*b^10+10*B*a*b^9)*d*e^7+28*b^10*B*d^2*e^6)*x^18+1/17*((45*A*a^2*b^8+120*B*a^3*b^7)*e^8+8*(10*A*a*b^9+45*B*a^
2*b^8)*d*e^7+28*(A*b^10+10*B*a*b^9)*d^2*e^6+56*b^10*B*d^3*e^5)*x^17+1/16*((120*A*a^3*b^7+210*B*a^4*b^6)*e^8+8*
(45*A*a^2*b^8+120*B*a^3*b^7)*d*e^7+28*(10*A*a*b^9+45*B*a^2*b^8)*d^2*e^6+56*(A*b^10+10*B*a*b^9)*d^3*e^5+70*b^10
*B*d^4*e^4)*x^16+1/15*((210*A*a^4*b^6+252*B*a^5*b^5)*e^8+8*(120*A*a^3*b^7+210*B*a^4*b^6)*d*e^7+28*(45*A*a^2*b^
8+120*B*a^3*b^7)*d^2*e^6+56*(10*A*a*b^9+45*B*a^2*b^8)*d^3*e^5+70*(A*b^10+10*B*a*b^9)*d^4*e^4+56*b^10*B*d^5*e^3
)*x^15+1/14*((252*A*a^5*b^5+210*B*a^6*b^4)*e^8+8*(210*A*a^4*b^6+252*B*a^5*b^5)*d*e^7+28*(120*A*a^3*b^7+210*B*a
^4*b^6)*d^2*e^6+56*(45*A*a^2*b^8+120*B*a^3*b^7)*d^3*e^5+70*(10*A*a*b^9+45*B*a^2*b^8)*d^4*e^4+56*(A*b^10+10*B*a
*b^9)*d^5*e^3+28*b^10*B*d^6*e^2)*x^14+1/13*((210*A*a^6*b^4+120*B*a^7*b^3)*e^8+8*(252*A*a^5*b^5+210*B*a^6*b^4)*
d*e^7+28*(210*A*a^4*b^6+252*B*a^5*b^5)*d^2*e^6+56*(120*A*a^3*b^7+210*B*a^4*b^6)*d^3*e^5+70*(45*A*a^2*b^8+120*B
*a^3*b^7)*d^4*e^4+56*(10*A*a*b^9+45*B*a^2*b^8)*d^5*e^3+28*(A*b^10+10*B*a*b^9)*d^6*e^2+8*b^10*B*d^7*e)*x^13+1/1
2*((120*A*a^7*b^3+45*B*a^8*b^2)*e^8+8*(210*A*a^6*b^4+120*B*a^7*b^3)*d*e^7+28*(252*A*a^5*b^5+210*B*a^6*b^4)*d^2
*e^6+56*(210*A*a^4*b^6+252*B*a^5*b^5)*d^3*e^5+70*(120*A*a^3*b^7+210*B*a^4*b^6)*d^4*e^4+56*(45*A*a^2*b^8+120*B*
a^3*b^7)*d^5*e^3+28*(10*A*a*b^9+45*B*a^2*b^8)*d^6*e^2+8*(A*b^10+10*B*a*b^9)*d^7*e+b^10*B*d^8)*x^12+1/11*((45*A
*a^8*b^2+10*B*a^9*b)*e^8+8*(120*A*a^7*b^3+45*B*a^8*b^2)*d*e^7+28*(210*A*a^6*b^4+120*B*a^7*b^3)*d^2*e^6+56*(252
*A*a^5*b^5+210*B*a^6*b^4)*d^3*e^5+70*(210*A*a^4*b^6+252*B*a^5*b^5)*d^4*e^4+56*(120*A*a^3*b^7+210*B*a^4*b^6)*d^
5*e^3+28*(45*A*a^2*b^8+120*B*a^3*b^7)*d^6*e^2+8*(10*A*a*b^9+45*B*a^2*b^8)*d^7*e+(A*b^10+10*B*a*b^9)*d^8)*x^11+
1/10*((10*A*a^9*b+B*a^10)*e^8+8*(45*A*a^8*b^2+10*B*a^9*b)*d*e^7+28*(120*A*a^7*b^3+45*B*a^8*b^2)*d^2*e^6+56*(21
0*A*a^6*b^4+120*B*a^7*b^3)*d^3*e^5+70*(252*A*a^5*b^5+210*B*a^6*b^4)*d^4*e^4+56*(210*A*a^4*b^6+252*B*a^5*b^5)*d
^5*e^3+28*(120*A*a^3*b^7+210*B*a^4*b^6)*d^6*e^2+8*(45*A*a^2*b^8+120*B*a^3*b^7)*d^7*e+(10*A*a*b^9+45*B*a^2*b^8)
*d^8)*x^10+1/9*(a^10*A*e^8+8*(10*A*a^9*b+B*a^10)*d*e^7+28*(45*A*a^8*b^2+10*B*a^9*b)*d^2*e^6+56*(120*A*a^7*b^3+
45*B*a^8*b^2)*d^3*e^5+70*(210*A*a^6*b^4+120*B*a^7*b^3)*d^4*e^4+56*(252*A*a^5*b^5+210*B*a^6*b^4)*d^5*e^3+28*(21
0*A*a^4*b^6+252*B*a^5*b^5)*d^6*e^2+8*(120*A*a^3*b^7+210*B*a^4*b^6)*d^7*e+(45*A*a^2*b^8+120*B*a^3*b^7)*d^8)*x^9
+1/8*(8*a^10*A*d*e^7+28*(10*A*a^9*b+B*a^10)*d^2*e^6+56*(45*A*a^8*b^2+10*B*a^9*b)*d^3*e^5+70*(120*A*a^7*b^3+45*
B*a^8*b^2)*d^4*e^4+56*(210*A*a^6*b^4+120*B*a^7*b^3)*d^5*e^3+28*(252*A*a^5*b^5+210*B*a^6*b^4)*d^6*e^2+8*(210*A*
a^4*b^6+252*B*a^5*b^5)*d^7*e+(120*A*a^3*b^7+210*B*a^4*b^6)*d^8)*x^8+1/7*(28*a^10*A*d^2*e^6+56*(10*A*a^9*b+B*a^
10)*d^3*e^5+70*(45*A*a^8*b^2+10*B*a^9*b)*d^4*e^4+56*(120*A*a^7*b^3+45*B*a^8*b^2)*d^5*e^3+28*(210*A*a^6*b^4+120
*B*a^7*b^3)*d^6*e^2+8*(252*A*a^5*b^5+210*B*a^6*b^4)*d^7*e+(210*A*a^4*b^6+252*B*a^5*b^5)*d^8)*x^7+1/6*(56*a^10*
A*d^3*e^5+70*(10*A*a^9*b+B*a^10)*d^4*e^4+56*(45*A*a^8*b^2+10*B*a^9*b)*d^5*e^3+28*(120*A*a^7*b^3+45*B*a^8*b^2)*
d^6*e^2+8*(210*A*a^6*b^4+120*B*a^7*b^3)*d^7*e+(252*A*a^5*b^5+210*B*a^6*b^4)*d^8)*x^6+1/5*(70*a^10*A*d^4*e^4+56
*(10*A*a^9*b+B*a^10)*d^5*e^3+28*(45*A*a^8*b^2+10*B*a^9*b)*d^6*e^2+8*(120*A*a^7*b^3+45*B*a^8*b^2)*d^7*e+(210*A*
a^6*b^4+120*B*a^7*b^3)*d^8)*x^5+1/4*(56*a^10*A*d^5*e^3+28*(10*A*a^9*b+B*a^10)*d^6*e^2+8*(45*A*a^8*b^2+10*B*a^9
*b)*d^7*e+(120*A*a^7*b^3+45*B*a^8*b^2)*d^8)*x^4+1/3*(28*a^10*A*d^6*e^2+8*(10*A*a^9*b+B*a^10)*d^7*e+(45*A*a^8*b
^2+10*B*a^9*b)*d^8)*x^3+1/2*(8*a^10*A*d^7*e+(10*A*a^9*b+B*a^10)*d^8)*x^2+a^10*A*d^8*x

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Maxima [B]  time = 1.25734, size = 3357, normalized size = 9.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)*(e*x+d)^8,x, algorithm="maxima")

[Out]

1/20*B*b^10*e^8*x^20 + A*a^10*d^8*x + 1/19*(8*B*b^10*d*e^7 + (10*B*a*b^9 + A*b^10)*e^8)*x^19 + 1/18*(28*B*b^10
*d^2*e^6 + 8*(10*B*a*b^9 + A*b^10)*d*e^7 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^8)*x^18 + 1/17*(56*B*b^10*d^3*e^5 + 2
8*(10*B*a*b^9 + A*b^10)*d^2*e^6 + 40*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^7 + 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^8)*x^1
7 + 1/8*(35*B*b^10*d^4*e^4 + 28*(10*B*a*b^9 + A*b^10)*d^3*e^5 + 70*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^6 + 60*(8*B
*a^3*b^7 + 3*A*a^2*b^8)*d*e^7 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^8)*x^16 + 2/15*(28*B*b^10*d^5*e^3 + 35*(10*B*
a*b^9 + A*b^10)*d^4*e^4 + 140*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^5 + 210*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^6 + 12
0*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^7 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^8)*x^15 + (2*B*b^10*d^6*e^2 + 4*(10*B*a
*b^9 + A*b^10)*d^5*e^3 + 25*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^4 + 60*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^5 + 60*(7
*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^6 + 24*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^7 + 3*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^8)*
x^14 + 2/13*(4*B*b^10*d^7*e + 14*(10*B*a*b^9 + A*b^10)*d^6*e^2 + 140*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^3 + 525*(
8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^4 + 840*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^5 + 588*(6*B*a^5*b^5 + 5*A*a^4*b^6)
*d^2*e^6 + 168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^7 + 15*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^8)*x^13 + 1/12*(B*b^10*d^8
+ 8*(10*B*a*b^9 + A*b^10)*d^7*e + 140*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^2 + 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5
*e^3 + 2100*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^4 + 2352*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^5 + 1176*(5*B*a^6*b^4
+ 6*A*a^5*b^5)*d^2*e^6 + 240*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^7 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^8)*x^12 + 1
/11*((10*B*a*b^9 + A*b^10)*d^8 + 40*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e + 420*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^2
+ 1680*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^3 + 2940*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^4 + 2352*(5*B*a^6*b^4 + 6*
A*a^5*b^5)*d^3*e^5 + 840*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^6 + 120*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^7 + 5*(2*B*
a^9*b + 9*A*a^8*b^2)*e^8)*x^11 + 1/10*(5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8 + 120*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e
+ 840*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^2 + 2352*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^3 + 2940*(5*B*a^6*b^4 + 6*
A*a^5*b^5)*d^4*e^4 + 1680*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^5 + 420*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^6 + 40*(
2*B*a^9*b + 9*A*a^8*b^2)*d*e^7 + (B*a^10 + 10*A*a^9*b)*e^8)*x^10 + 1/9*(A*a^10*e^8 + 15*(8*B*a^3*b^7 + 3*A*a^2
*b^8)*d^8 + 240*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e + 1176*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^2 + 2352*(5*B*a^6*b
^4 + 6*A*a^5*b^5)*d^5*e^3 + 2100*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^4 + 840*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^5
+ 140*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^6 + 8*(B*a^10 + 10*A*a^9*b)*d*e^7)*x^9 + 1/4*(4*A*a^10*d*e^7 + 15*(7*B*
a^4*b^6 + 4*A*a^3*b^7)*d^8 + 168*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^7*e + 588*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^6*e^2 +
840*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^5*e^3 + 525*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^4*e^4 + 140*(2*B*a^9*b + 9*A*a^8*
b^2)*d^3*e^5 + 14*(B*a^10 + 10*A*a^9*b)*d^2*e^6)*x^8 + 2*(2*A*a^10*d^2*e^6 + 3*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^8
+ 24*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^7*e + 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^6*e^2 + 60*(3*B*a^8*b^2 + 8*A*a^7*b
^3)*d^5*e^3 + 25*(2*B*a^9*b + 9*A*a^8*b^2)*d^4*e^4 + 4*(B*a^10 + 10*A*a^9*b)*d^3*e^5)*x^7 + 1/3*(28*A*a^10*d^3
*e^5 + 21*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^8 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^7*e + 210*(3*B*a^8*b^2 + 8*A*a^7
*b^3)*d^6*e^2 + 140*(2*B*a^9*b + 9*A*a^8*b^2)*d^5*e^3 + 35*(B*a^10 + 10*A*a^9*b)*d^4*e^4)*x^6 + 2/5*(35*A*a^10
*d^4*e^4 + 15*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^8 + 60*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^7*e + 70*(2*B*a^9*b + 9*A*a^8
*b^2)*d^6*e^2 + 28*(B*a^10 + 10*A*a^9*b)*d^5*e^3)*x^5 + 1/4*(56*A*a^10*d^5*e^3 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3
)*d^8 + 40*(2*B*a^9*b + 9*A*a^8*b^2)*d^7*e + 28*(B*a^10 + 10*A*a^9*b)*d^6*e^2)*x^4 + 1/3*(28*A*a^10*d^6*e^2 +
5*(2*B*a^9*b + 9*A*a^8*b^2)*d^8 + 8*(B*a^10 + 10*A*a^9*b)*d^7*e)*x^3 + 1/2*(8*A*a^10*d^7*e + (B*a^10 + 10*A*a^
9*b)*d^8)*x^2

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Fricas [B]  time = 1.63075, size = 7233, normalized size = 19.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)*(e*x+d)^8,x, algorithm="fricas")

[Out]

1/20*x^20*e^8*b^10*B + 8/19*x^19*e^7*d*b^10*B + 10/19*x^19*e^8*b^9*a*B + 1/19*x^19*e^8*b^10*A + 14/9*x^18*e^6*
d^2*b^10*B + 40/9*x^18*e^7*d*b^9*a*B + 5/2*x^18*e^8*b^8*a^2*B + 4/9*x^18*e^7*d*b^10*A + 5/9*x^18*e^8*b^9*a*A +
56/17*x^17*e^5*d^3*b^10*B + 280/17*x^17*e^6*d^2*b^9*a*B + 360/17*x^17*e^7*d*b^8*a^2*B + 120/17*x^17*e^8*b^7*a
^3*B + 28/17*x^17*e^6*d^2*b^10*A + 80/17*x^17*e^7*d*b^9*a*A + 45/17*x^17*e^8*b^8*a^2*A + 35/8*x^16*e^4*d^4*b^1
0*B + 35*x^16*e^5*d^3*b^9*a*B + 315/4*x^16*e^6*d^2*b^8*a^2*B + 60*x^16*e^7*d*b^7*a^3*B + 105/8*x^16*e^8*b^6*a^
4*B + 7/2*x^16*e^5*d^3*b^10*A + 35/2*x^16*e^6*d^2*b^9*a*A + 45/2*x^16*e^7*d*b^8*a^2*A + 15/2*x^16*e^8*b^7*a^3*
A + 56/15*x^15*e^3*d^5*b^10*B + 140/3*x^15*e^4*d^4*b^9*a*B + 168*x^15*e^5*d^3*b^8*a^2*B + 224*x^15*e^6*d^2*b^7
*a^3*B + 112*x^15*e^7*d*b^6*a^4*B + 84/5*x^15*e^8*b^5*a^5*B + 14/3*x^15*e^4*d^4*b^10*A + 112/3*x^15*e^5*d^3*b^
9*a*A + 84*x^15*e^6*d^2*b^8*a^2*A + 64*x^15*e^7*d*b^7*a^3*A + 14*x^15*e^8*b^6*a^4*A + 2*x^14*e^2*d^6*b^10*B +
40*x^14*e^3*d^5*b^9*a*B + 225*x^14*e^4*d^4*b^8*a^2*B + 480*x^14*e^5*d^3*b^7*a^3*B + 420*x^14*e^6*d^2*b^6*a^4*B
+ 144*x^14*e^7*d*b^5*a^5*B + 15*x^14*e^8*b^4*a^6*B + 4*x^14*e^3*d^5*b^10*A + 50*x^14*e^4*d^4*b^9*a*A + 180*x^
14*e^5*d^3*b^8*a^2*A + 240*x^14*e^6*d^2*b^7*a^3*A + 120*x^14*e^7*d*b^6*a^4*A + 18*x^14*e^8*b^5*a^5*A + 8/13*x^
13*e*d^7*b^10*B + 280/13*x^13*e^2*d^6*b^9*a*B + 2520/13*x^13*e^3*d^5*b^8*a^2*B + 8400/13*x^13*e^4*d^4*b^7*a^3*
B + 11760/13*x^13*e^5*d^3*b^6*a^4*B + 7056/13*x^13*e^6*d^2*b^5*a^5*B + 1680/13*x^13*e^7*d*b^4*a^6*B + 120/13*x
^13*e^8*b^3*a^7*B + 28/13*x^13*e^2*d^6*b^10*A + 560/13*x^13*e^3*d^5*b^9*a*A + 3150/13*x^13*e^4*d^4*b^8*a^2*A +
6720/13*x^13*e^5*d^3*b^7*a^3*A + 5880/13*x^13*e^6*d^2*b^6*a^4*A + 2016/13*x^13*e^7*d*b^5*a^5*A + 210/13*x^13*
e^8*b^4*a^6*A + 1/12*x^12*d^8*b^10*B + 20/3*x^12*e*d^7*b^9*a*B + 105*x^12*e^2*d^6*b^8*a^2*B + 560*x^12*e^3*d^5
*b^7*a^3*B + 1225*x^12*e^4*d^4*b^6*a^4*B + 1176*x^12*e^5*d^3*b^5*a^5*B + 490*x^12*e^6*d^2*b^4*a^6*B + 80*x^12*
e^7*d*b^3*a^7*B + 15/4*x^12*e^8*b^2*a^8*B + 2/3*x^12*e*d^7*b^10*A + 70/3*x^12*e^2*d^6*b^9*a*A + 210*x^12*e^3*d
^5*b^8*a^2*A + 700*x^12*e^4*d^4*b^7*a^3*A + 980*x^12*e^5*d^3*b^6*a^4*A + 588*x^12*e^6*d^2*b^5*a^5*A + 140*x^12
*e^7*d*b^4*a^6*A + 10*x^12*e^8*b^3*a^7*A + 10/11*x^11*d^8*b^9*a*B + 360/11*x^11*e*d^7*b^8*a^2*B + 3360/11*x^11
*e^2*d^6*b^7*a^3*B + 11760/11*x^11*e^3*d^5*b^6*a^4*B + 17640/11*x^11*e^4*d^4*b^5*a^5*B + 11760/11*x^11*e^5*d^3
*b^4*a^6*B + 3360/11*x^11*e^6*d^2*b^3*a^7*B + 360/11*x^11*e^7*d*b^2*a^8*B + 10/11*x^11*e^8*b*a^9*B + 1/11*x^11
*d^8*b^10*A + 80/11*x^11*e*d^7*b^9*a*A + 1260/11*x^11*e^2*d^6*b^8*a^2*A + 6720/11*x^11*e^3*d^5*b^7*a^3*A + 147
00/11*x^11*e^4*d^4*b^6*a^4*A + 14112/11*x^11*e^5*d^3*b^5*a^5*A + 5880/11*x^11*e^6*d^2*b^4*a^6*A + 960/11*x^11*
e^7*d*b^3*a^7*A + 45/11*x^11*e^8*b^2*a^8*A + 9/2*x^10*d^8*b^8*a^2*B + 96*x^10*e*d^7*b^7*a^3*B + 588*x^10*e^2*d
^6*b^6*a^4*B + 7056/5*x^10*e^3*d^5*b^5*a^5*B + 1470*x^10*e^4*d^4*b^4*a^6*B + 672*x^10*e^5*d^3*b^3*a^7*B + 126*
x^10*e^6*d^2*b^2*a^8*B + 8*x^10*e^7*d*b*a^9*B + 1/10*x^10*e^8*a^10*B + x^10*d^8*b^9*a*A + 36*x^10*e*d^7*b^8*a^
2*A + 336*x^10*e^2*d^6*b^7*a^3*A + 1176*x^10*e^3*d^5*b^6*a^4*A + 1764*x^10*e^4*d^4*b^5*a^5*A + 1176*x^10*e^5*d
^3*b^4*a^6*A + 336*x^10*e^6*d^2*b^3*a^7*A + 36*x^10*e^7*d*b^2*a^8*A + x^10*e^8*b*a^9*A + 40/3*x^9*d^8*b^7*a^3*
B + 560/3*x^9*e*d^7*b^6*a^4*B + 784*x^9*e^2*d^6*b^5*a^5*B + 3920/3*x^9*e^3*d^5*b^4*a^6*B + 2800/3*x^9*e^4*d^4*
b^3*a^7*B + 280*x^9*e^5*d^3*b^2*a^8*B + 280/9*x^9*e^6*d^2*b*a^9*B + 8/9*x^9*e^7*d*a^10*B + 5*x^9*d^8*b^8*a^2*A
+ 320/3*x^9*e*d^7*b^7*a^3*A + 1960/3*x^9*e^2*d^6*b^6*a^4*A + 1568*x^9*e^3*d^5*b^5*a^5*A + 4900/3*x^9*e^4*d^4*
b^4*a^6*A + 2240/3*x^9*e^5*d^3*b^3*a^7*A + 140*x^9*e^6*d^2*b^2*a^8*A + 80/9*x^9*e^7*d*b*a^9*A + 1/9*x^9*e^8*a^
10*A + 105/4*x^8*d^8*b^6*a^4*B + 252*x^8*e*d^7*b^5*a^5*B + 735*x^8*e^2*d^6*b^4*a^6*B + 840*x^8*e^3*d^5*b^3*a^7
*B + 1575/4*x^8*e^4*d^4*b^2*a^8*B + 70*x^8*e^5*d^3*b*a^9*B + 7/2*x^8*e^6*d^2*a^10*B + 15*x^8*d^8*b^7*a^3*A + 2
10*x^8*e*d^7*b^6*a^4*A + 882*x^8*e^2*d^6*b^5*a^5*A + 1470*x^8*e^3*d^5*b^4*a^6*A + 1050*x^8*e^4*d^4*b^3*a^7*A +
315*x^8*e^5*d^3*b^2*a^8*A + 35*x^8*e^6*d^2*b*a^9*A + x^8*e^7*d*a^10*A + 36*x^7*d^8*b^5*a^5*B + 240*x^7*e*d^7*
b^4*a^6*B + 480*x^7*e^2*d^6*b^3*a^7*B + 360*x^7*e^3*d^5*b^2*a^8*B + 100*x^7*e^4*d^4*b*a^9*B + 8*x^7*e^5*d^3*a^
10*B + 30*x^7*d^8*b^6*a^4*A + 288*x^7*e*d^7*b^5*a^5*A + 840*x^7*e^2*d^6*b^4*a^6*A + 960*x^7*e^3*d^5*b^3*a^7*A
+ 450*x^7*e^4*d^4*b^2*a^8*A + 80*x^7*e^5*d^3*b*a^9*A + 4*x^7*e^6*d^2*a^10*A + 35*x^6*d^8*b^4*a^6*B + 160*x^6*e
*d^7*b^3*a^7*B + 210*x^6*e^2*d^6*b^2*a^8*B + 280/3*x^6*e^3*d^5*b*a^9*B + 35/3*x^6*e^4*d^4*a^10*B + 42*x^6*d^8*
b^5*a^5*A + 280*x^6*e*d^7*b^4*a^6*A + 560*x^6*e^2*d^6*b^3*a^7*A + 420*x^6*e^3*d^5*b^2*a^8*A + 350/3*x^6*e^4*d^
4*b*a^9*A + 28/3*x^6*e^5*d^3*a^10*A + 24*x^5*d^8*b^3*a^7*B + 72*x^5*e*d^7*b^2*a^8*B + 56*x^5*e^2*d^6*b*a^9*B +
56/5*x^5*e^3*d^5*a^10*B + 42*x^5*d^8*b^4*a^6*A + 192*x^5*e*d^7*b^3*a^7*A + 252*x^5*e^2*d^6*b^2*a^8*A + 112*x^
5*e^3*d^5*b*a^9*A + 14*x^5*e^4*d^4*a^10*A + 45/4*x^4*d^8*b^2*a^8*B + 20*x^4*e*d^7*b*a^9*B + 7*x^4*e^2*d^6*a^10
*B + 30*x^4*d^8*b^3*a^7*A + 90*x^4*e*d^7*b^2*a^8*A + 70*x^4*e^2*d^6*b*a^9*A + 14*x^4*e^3*d^5*a^10*A + 10/3*x^3
*d^8*b*a^9*B + 8/3*x^3*e*d^7*a^10*B + 15*x^3*d^8*b^2*a^8*A + 80/3*x^3*e*d^7*b*a^9*A + 28/3*x^3*e^2*d^6*a^10*A
+ 1/2*x^2*d^8*a^10*B + 5*x^2*d^8*b*a^9*A + 4*x^2*e*d^7*a^10*A + x*d^8*a^10*A

________________________________________________________________________________________

Sympy [B]  time = 0.384441, size = 3165, normalized size = 8.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)*(e*x+d)**8,x)

[Out]

A*a**10*d**8*x + B*b**10*e**8*x**20/20 + x**19*(A*b**10*e**8/19 + 10*B*a*b**9*e**8/19 + 8*B*b**10*d*e**7/19) +
x**18*(5*A*a*b**9*e**8/9 + 4*A*b**10*d*e**7/9 + 5*B*a**2*b**8*e**8/2 + 40*B*a*b**9*d*e**7/9 + 14*B*b**10*d**2
*e**6/9) + x**17*(45*A*a**2*b**8*e**8/17 + 80*A*a*b**9*d*e**7/17 + 28*A*b**10*d**2*e**6/17 + 120*B*a**3*b**7*e
**8/17 + 360*B*a**2*b**8*d*e**7/17 + 280*B*a*b**9*d**2*e**6/17 + 56*B*b**10*d**3*e**5/17) + x**16*(15*A*a**3*b
**7*e**8/2 + 45*A*a**2*b**8*d*e**7/2 + 35*A*a*b**9*d**2*e**6/2 + 7*A*b**10*d**3*e**5/2 + 105*B*a**4*b**6*e**8/
8 + 60*B*a**3*b**7*d*e**7 + 315*B*a**2*b**8*d**2*e**6/4 + 35*B*a*b**9*d**3*e**5 + 35*B*b**10*d**4*e**4/8) + x*
*15*(14*A*a**4*b**6*e**8 + 64*A*a**3*b**7*d*e**7 + 84*A*a**2*b**8*d**2*e**6 + 112*A*a*b**9*d**3*e**5/3 + 14*A*
b**10*d**4*e**4/3 + 84*B*a**5*b**5*e**8/5 + 112*B*a**4*b**6*d*e**7 + 224*B*a**3*b**7*d**2*e**6 + 168*B*a**2*b*
*8*d**3*e**5 + 140*B*a*b**9*d**4*e**4/3 + 56*B*b**10*d**5*e**3/15) + x**14*(18*A*a**5*b**5*e**8 + 120*A*a**4*b
**6*d*e**7 + 240*A*a**3*b**7*d**2*e**6 + 180*A*a**2*b**8*d**3*e**5 + 50*A*a*b**9*d**4*e**4 + 4*A*b**10*d**5*e*
*3 + 15*B*a**6*b**4*e**8 + 144*B*a**5*b**5*d*e**7 + 420*B*a**4*b**6*d**2*e**6 + 480*B*a**3*b**7*d**3*e**5 + 22
5*B*a**2*b**8*d**4*e**4 + 40*B*a*b**9*d**5*e**3 + 2*B*b**10*d**6*e**2) + x**13*(210*A*a**6*b**4*e**8/13 + 2016
*A*a**5*b**5*d*e**7/13 + 5880*A*a**4*b**6*d**2*e**6/13 + 6720*A*a**3*b**7*d**3*e**5/13 + 3150*A*a**2*b**8*d**4
*e**4/13 + 560*A*a*b**9*d**5*e**3/13 + 28*A*b**10*d**6*e**2/13 + 120*B*a**7*b**3*e**8/13 + 1680*B*a**6*b**4*d*
e**7/13 + 7056*B*a**5*b**5*d**2*e**6/13 + 11760*B*a**4*b**6*d**3*e**5/13 + 8400*B*a**3*b**7*d**4*e**4/13 + 252
0*B*a**2*b**8*d**5*e**3/13 + 280*B*a*b**9*d**6*e**2/13 + 8*B*b**10*d**7*e/13) + x**12*(10*A*a**7*b**3*e**8 + 1
40*A*a**6*b**4*d*e**7 + 588*A*a**5*b**5*d**2*e**6 + 980*A*a**4*b**6*d**3*e**5 + 700*A*a**3*b**7*d**4*e**4 + 21
0*A*a**2*b**8*d**5*e**3 + 70*A*a*b**9*d**6*e**2/3 + 2*A*b**10*d**7*e/3 + 15*B*a**8*b**2*e**8/4 + 80*B*a**7*b**
3*d*e**7 + 490*B*a**6*b**4*d**2*e**6 + 1176*B*a**5*b**5*d**3*e**5 + 1225*B*a**4*b**6*d**4*e**4 + 560*B*a**3*b*
*7*d**5*e**3 + 105*B*a**2*b**8*d**6*e**2 + 20*B*a*b**9*d**7*e/3 + B*b**10*d**8/12) + x**11*(45*A*a**8*b**2*e**
8/11 + 960*A*a**7*b**3*d*e**7/11 + 5880*A*a**6*b**4*d**2*e**6/11 + 14112*A*a**5*b**5*d**3*e**5/11 + 14700*A*a*
*4*b**6*d**4*e**4/11 + 6720*A*a**3*b**7*d**5*e**3/11 + 1260*A*a**2*b**8*d**6*e**2/11 + 80*A*a*b**9*d**7*e/11 +
A*b**10*d**8/11 + 10*B*a**9*b*e**8/11 + 360*B*a**8*b**2*d*e**7/11 + 3360*B*a**7*b**3*d**2*e**6/11 + 11760*B*a
**6*b**4*d**3*e**5/11 + 17640*B*a**5*b**5*d**4*e**4/11 + 11760*B*a**4*b**6*d**5*e**3/11 + 3360*B*a**3*b**7*d**
6*e**2/11 + 360*B*a**2*b**8*d**7*e/11 + 10*B*a*b**9*d**8/11) + x**10*(A*a**9*b*e**8 + 36*A*a**8*b**2*d*e**7 +
336*A*a**7*b**3*d**2*e**6 + 1176*A*a**6*b**4*d**3*e**5 + 1764*A*a**5*b**5*d**4*e**4 + 1176*A*a**4*b**6*d**5*e*
*3 + 336*A*a**3*b**7*d**6*e**2 + 36*A*a**2*b**8*d**7*e + A*a*b**9*d**8 + B*a**10*e**8/10 + 8*B*a**9*b*d*e**7 +
126*B*a**8*b**2*d**2*e**6 + 672*B*a**7*b**3*d**3*e**5 + 1470*B*a**6*b**4*d**4*e**4 + 7056*B*a**5*b**5*d**5*e*
*3/5 + 588*B*a**4*b**6*d**6*e**2 + 96*B*a**3*b**7*d**7*e + 9*B*a**2*b**8*d**8/2) + x**9*(A*a**10*e**8/9 + 80*A
*a**9*b*d*e**7/9 + 140*A*a**8*b**2*d**2*e**6 + 2240*A*a**7*b**3*d**3*e**5/3 + 4900*A*a**6*b**4*d**4*e**4/3 + 1
568*A*a**5*b**5*d**5*e**3 + 1960*A*a**4*b**6*d**6*e**2/3 + 320*A*a**3*b**7*d**7*e/3 + 5*A*a**2*b**8*d**8 + 8*B
*a**10*d*e**7/9 + 280*B*a**9*b*d**2*e**6/9 + 280*B*a**8*b**2*d**3*e**5 + 2800*B*a**7*b**3*d**4*e**4/3 + 3920*B
*a**6*b**4*d**5*e**3/3 + 784*B*a**5*b**5*d**6*e**2 + 560*B*a**4*b**6*d**7*e/3 + 40*B*a**3*b**7*d**8/3) + x**8*
(A*a**10*d*e**7 + 35*A*a**9*b*d**2*e**6 + 315*A*a**8*b**2*d**3*e**5 + 1050*A*a**7*b**3*d**4*e**4 + 1470*A*a**6
*b**4*d**5*e**3 + 882*A*a**5*b**5*d**6*e**2 + 210*A*a**4*b**6*d**7*e + 15*A*a**3*b**7*d**8 + 7*B*a**10*d**2*e*
*6/2 + 70*B*a**9*b*d**3*e**5 + 1575*B*a**8*b**2*d**4*e**4/4 + 840*B*a**7*b**3*d**5*e**3 + 735*B*a**6*b**4*d**6
*e**2 + 252*B*a**5*b**5*d**7*e + 105*B*a**4*b**6*d**8/4) + x**7*(4*A*a**10*d**2*e**6 + 80*A*a**9*b*d**3*e**5 +
450*A*a**8*b**2*d**4*e**4 + 960*A*a**7*b**3*d**5*e**3 + 840*A*a**6*b**4*d**6*e**2 + 288*A*a**5*b**5*d**7*e +
30*A*a**4*b**6*d**8 + 8*B*a**10*d**3*e**5 + 100*B*a**9*b*d**4*e**4 + 360*B*a**8*b**2*d**5*e**3 + 480*B*a**7*b*
*3*d**6*e**2 + 240*B*a**6*b**4*d**7*e + 36*B*a**5*b**5*d**8) + x**6*(28*A*a**10*d**3*e**5/3 + 350*A*a**9*b*d**
4*e**4/3 + 420*A*a**8*b**2*d**5*e**3 + 560*A*a**7*b**3*d**6*e**2 + 280*A*a**6*b**4*d**7*e + 42*A*a**5*b**5*d**
8 + 35*B*a**10*d**4*e**4/3 + 280*B*a**9*b*d**5*e**3/3 + 210*B*a**8*b**2*d**6*e**2 + 160*B*a**7*b**3*d**7*e + 3
5*B*a**6*b**4*d**8) + x**5*(14*A*a**10*d**4*e**4 + 112*A*a**9*b*d**5*e**3 + 252*A*a**8*b**2*d**6*e**2 + 192*A*
a**7*b**3*d**7*e + 42*A*a**6*b**4*d**8 + 56*B*a**10*d**5*e**3/5 + 56*B*a**9*b*d**6*e**2 + 72*B*a**8*b**2*d**7*
e + 24*B*a**7*b**3*d**8) + x**4*(14*A*a**10*d**5*e**3 + 70*A*a**9*b*d**6*e**2 + 90*A*a**8*b**2*d**7*e + 30*A*a
**7*b**3*d**8 + 7*B*a**10*d**6*e**2 + 20*B*a**9*b*d**7*e + 45*B*a**8*b**2*d**8/4) + x**3*(28*A*a**10*d**6*e**2
/3 + 80*A*a**9*b*d**7*e/3 + 15*A*a**8*b**2*d**8 + 8*B*a**10*d**7*e/3 + 10*B*a**9*b*d**8/3) + x**2*(4*A*a**10*d
**7*e + 5*A*a**9*b*d**8 + B*a**10*d**8/2)

________________________________________________________________________________________

Giac [B]  time = 1.79803, size = 4086, normalized size = 10.98 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)*(e*x+d)^8,x, algorithm="giac")

[Out]

1/20*B*b^10*x^20*e^8 + 8/19*B*b^10*d*x^19*e^7 + 14/9*B*b^10*d^2*x^18*e^6 + 56/17*B*b^10*d^3*x^17*e^5 + 35/8*B*
b^10*d^4*x^16*e^4 + 56/15*B*b^10*d^5*x^15*e^3 + 2*B*b^10*d^6*x^14*e^2 + 8/13*B*b^10*d^7*x^13*e + 1/12*B*b^10*d
^8*x^12 + 10/19*B*a*b^9*x^19*e^8 + 1/19*A*b^10*x^19*e^8 + 40/9*B*a*b^9*d*x^18*e^7 + 4/9*A*b^10*d*x^18*e^7 + 28
0/17*B*a*b^9*d^2*x^17*e^6 + 28/17*A*b^10*d^2*x^17*e^6 + 35*B*a*b^9*d^3*x^16*e^5 + 7/2*A*b^10*d^3*x^16*e^5 + 14
0/3*B*a*b^9*d^4*x^15*e^4 + 14/3*A*b^10*d^4*x^15*e^4 + 40*B*a*b^9*d^5*x^14*e^3 + 4*A*b^10*d^5*x^14*e^3 + 280/13
*B*a*b^9*d^6*x^13*e^2 + 28/13*A*b^10*d^6*x^13*e^2 + 20/3*B*a*b^9*d^7*x^12*e + 2/3*A*b^10*d^7*x^12*e + 10/11*B*
a*b^9*d^8*x^11 + 1/11*A*b^10*d^8*x^11 + 5/2*B*a^2*b^8*x^18*e^8 + 5/9*A*a*b^9*x^18*e^8 + 360/17*B*a^2*b^8*d*x^1
7*e^7 + 80/17*A*a*b^9*d*x^17*e^7 + 315/4*B*a^2*b^8*d^2*x^16*e^6 + 35/2*A*a*b^9*d^2*x^16*e^6 + 168*B*a^2*b^8*d^
3*x^15*e^5 + 112/3*A*a*b^9*d^3*x^15*e^5 + 225*B*a^2*b^8*d^4*x^14*e^4 + 50*A*a*b^9*d^4*x^14*e^4 + 2520/13*B*a^2
*b^8*d^5*x^13*e^3 + 560/13*A*a*b^9*d^5*x^13*e^3 + 105*B*a^2*b^8*d^6*x^12*e^2 + 70/3*A*a*b^9*d^6*x^12*e^2 + 360
/11*B*a^2*b^8*d^7*x^11*e + 80/11*A*a*b^9*d^7*x^11*e + 9/2*B*a^2*b^8*d^8*x^10 + A*a*b^9*d^8*x^10 + 120/17*B*a^3
*b^7*x^17*e^8 + 45/17*A*a^2*b^8*x^17*e^8 + 60*B*a^3*b^7*d*x^16*e^7 + 45/2*A*a^2*b^8*d*x^16*e^7 + 224*B*a^3*b^7
*d^2*x^15*e^6 + 84*A*a^2*b^8*d^2*x^15*e^6 + 480*B*a^3*b^7*d^3*x^14*e^5 + 180*A*a^2*b^8*d^3*x^14*e^5 + 8400/13*
B*a^3*b^7*d^4*x^13*e^4 + 3150/13*A*a^2*b^8*d^4*x^13*e^4 + 560*B*a^3*b^7*d^5*x^12*e^3 + 210*A*a^2*b^8*d^5*x^12*
e^3 + 3360/11*B*a^3*b^7*d^6*x^11*e^2 + 1260/11*A*a^2*b^8*d^6*x^11*e^2 + 96*B*a^3*b^7*d^7*x^10*e + 36*A*a^2*b^8
*d^7*x^10*e + 40/3*B*a^3*b^7*d^8*x^9 + 5*A*a^2*b^8*d^8*x^9 + 105/8*B*a^4*b^6*x^16*e^8 + 15/2*A*a^3*b^7*x^16*e^
8 + 112*B*a^4*b^6*d*x^15*e^7 + 64*A*a^3*b^7*d*x^15*e^7 + 420*B*a^4*b^6*d^2*x^14*e^6 + 240*A*a^3*b^7*d^2*x^14*e
^6 + 11760/13*B*a^4*b^6*d^3*x^13*e^5 + 6720/13*A*a^3*b^7*d^3*x^13*e^5 + 1225*B*a^4*b^6*d^4*x^12*e^4 + 700*A*a^
3*b^7*d^4*x^12*e^4 + 11760/11*B*a^4*b^6*d^5*x^11*e^3 + 6720/11*A*a^3*b^7*d^5*x^11*e^3 + 588*B*a^4*b^6*d^6*x^10
*e^2 + 336*A*a^3*b^7*d^6*x^10*e^2 + 560/3*B*a^4*b^6*d^7*x^9*e + 320/3*A*a^3*b^7*d^7*x^9*e + 105/4*B*a^4*b^6*d^
8*x^8 + 15*A*a^3*b^7*d^8*x^8 + 84/5*B*a^5*b^5*x^15*e^8 + 14*A*a^4*b^6*x^15*e^8 + 144*B*a^5*b^5*d*x^14*e^7 + 12
0*A*a^4*b^6*d*x^14*e^7 + 7056/13*B*a^5*b^5*d^2*x^13*e^6 + 5880/13*A*a^4*b^6*d^2*x^13*e^6 + 1176*B*a^5*b^5*d^3*
x^12*e^5 + 980*A*a^4*b^6*d^3*x^12*e^5 + 17640/11*B*a^5*b^5*d^4*x^11*e^4 + 14700/11*A*a^4*b^6*d^4*x^11*e^4 + 70
56/5*B*a^5*b^5*d^5*x^10*e^3 + 1176*A*a^4*b^6*d^5*x^10*e^3 + 784*B*a^5*b^5*d^6*x^9*e^2 + 1960/3*A*a^4*b^6*d^6*x
^9*e^2 + 252*B*a^5*b^5*d^7*x^8*e + 210*A*a^4*b^6*d^7*x^8*e + 36*B*a^5*b^5*d^8*x^7 + 30*A*a^4*b^6*d^8*x^7 + 15*
B*a^6*b^4*x^14*e^8 + 18*A*a^5*b^5*x^14*e^8 + 1680/13*B*a^6*b^4*d*x^13*e^7 + 2016/13*A*a^5*b^5*d*x^13*e^7 + 490
*B*a^6*b^4*d^2*x^12*e^6 + 588*A*a^5*b^5*d^2*x^12*e^6 + 11760/11*B*a^6*b^4*d^3*x^11*e^5 + 14112/11*A*a^5*b^5*d^
3*x^11*e^5 + 1470*B*a^6*b^4*d^4*x^10*e^4 + 1764*A*a^5*b^5*d^4*x^10*e^4 + 3920/3*B*a^6*b^4*d^5*x^9*e^3 + 1568*A
*a^5*b^5*d^5*x^9*e^3 + 735*B*a^6*b^4*d^6*x^8*e^2 + 882*A*a^5*b^5*d^6*x^8*e^2 + 240*B*a^6*b^4*d^7*x^7*e + 288*A
*a^5*b^5*d^7*x^7*e + 35*B*a^6*b^4*d^8*x^6 + 42*A*a^5*b^5*d^8*x^6 + 120/13*B*a^7*b^3*x^13*e^8 + 210/13*A*a^6*b^
4*x^13*e^8 + 80*B*a^7*b^3*d*x^12*e^7 + 140*A*a^6*b^4*d*x^12*e^7 + 3360/11*B*a^7*b^3*d^2*x^11*e^6 + 5880/11*A*a
^6*b^4*d^2*x^11*e^6 + 672*B*a^7*b^3*d^3*x^10*e^5 + 1176*A*a^6*b^4*d^3*x^10*e^5 + 2800/3*B*a^7*b^3*d^4*x^9*e^4
+ 4900/3*A*a^6*b^4*d^4*x^9*e^4 + 840*B*a^7*b^3*d^5*x^8*e^3 + 1470*A*a^6*b^4*d^5*x^8*e^3 + 480*B*a^7*b^3*d^6*x^
7*e^2 + 840*A*a^6*b^4*d^6*x^7*e^2 + 160*B*a^7*b^3*d^7*x^6*e + 280*A*a^6*b^4*d^7*x^6*e + 24*B*a^7*b^3*d^8*x^5 +
42*A*a^6*b^4*d^8*x^5 + 15/4*B*a^8*b^2*x^12*e^8 + 10*A*a^7*b^3*x^12*e^8 + 360/11*B*a^8*b^2*d*x^11*e^7 + 960/11
*A*a^7*b^3*d*x^11*e^7 + 126*B*a^8*b^2*d^2*x^10*e^6 + 336*A*a^7*b^3*d^2*x^10*e^6 + 280*B*a^8*b^2*d^3*x^9*e^5 +
2240/3*A*a^7*b^3*d^3*x^9*e^5 + 1575/4*B*a^8*b^2*d^4*x^8*e^4 + 1050*A*a^7*b^3*d^4*x^8*e^4 + 360*B*a^8*b^2*d^5*x
^7*e^3 + 960*A*a^7*b^3*d^5*x^7*e^3 + 210*B*a^8*b^2*d^6*x^6*e^2 + 560*A*a^7*b^3*d^6*x^6*e^2 + 72*B*a^8*b^2*d^7*
x^5*e + 192*A*a^7*b^3*d^7*x^5*e + 45/4*B*a^8*b^2*d^8*x^4 + 30*A*a^7*b^3*d^8*x^4 + 10/11*B*a^9*b*x^11*e^8 + 45/
11*A*a^8*b^2*x^11*e^8 + 8*B*a^9*b*d*x^10*e^7 + 36*A*a^8*b^2*d*x^10*e^7 + 280/9*B*a^9*b*d^2*x^9*e^6 + 140*A*a^8
*b^2*d^2*x^9*e^6 + 70*B*a^9*b*d^3*x^8*e^5 + 315*A*a^8*b^2*d^3*x^8*e^5 + 100*B*a^9*b*d^4*x^7*e^4 + 450*A*a^8*b^
2*d^4*x^7*e^4 + 280/3*B*a^9*b*d^5*x^6*e^3 + 420*A*a^8*b^2*d^5*x^6*e^3 + 56*B*a^9*b*d^6*x^5*e^2 + 252*A*a^8*b^2
*d^6*x^5*e^2 + 20*B*a^9*b*d^7*x^4*e + 90*A*a^8*b^2*d^7*x^4*e + 10/3*B*a^9*b*d^8*x^3 + 15*A*a^8*b^2*d^8*x^3 + 1
/10*B*a^10*x^10*e^8 + A*a^9*b*x^10*e^8 + 8/9*B*a^10*d*x^9*e^7 + 80/9*A*a^9*b*d*x^9*e^7 + 7/2*B*a^10*d^2*x^8*e^
6 + 35*A*a^9*b*d^2*x^8*e^6 + 8*B*a^10*d^3*x^7*e^5 + 80*A*a^9*b*d^3*x^7*e^5 + 35/3*B*a^10*d^4*x^6*e^4 + 350/3*A
*a^9*b*d^4*x^6*e^4 + 56/5*B*a^10*d^5*x^5*e^3 + 112*A*a^9*b*d^5*x^5*e^3 + 7*B*a^10*d^6*x^4*e^2 + 70*A*a^9*b*d^6
*x^4*e^2 + 8/3*B*a^10*d^7*x^3*e + 80/3*A*a^9*b*d^7*x^3*e + 1/2*B*a^10*d^8*x^2 + 5*A*a^9*b*d^8*x^2 + 1/9*A*a^10
*x^9*e^8 + A*a^10*d*x^8*e^7 + 4*A*a^10*d^2*x^7*e^6 + 28/3*A*a^10*d^3*x^6*e^5 + 14*A*a^10*d^4*x^5*e^4 + 14*A*a^
10*d^5*x^4*e^3 + 28/3*A*a^10*d^6*x^3*e^2 + 4*A*a^10*d^7*x^2*e + A*a^10*d^8*x