3.1085 \(\int (a+b x)^{10} (A+B x) (d+e x)^3 \, dx\)

Optimal. Leaf size=159 \[ \frac{e^2 (a+b x)^{14} (-4 a B e+A b e+3 b B d)}{14 b^5}+\frac{3 e (a+b x)^{13} (b d-a e) (-2 a B e+A b e+b B d)}{13 b^5}+\frac{(a+b x)^{12} (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{12 b^5}+\frac{(a+b x)^{11} (A b-a B) (b d-a e)^3}{11 b^5}+\frac{B e^3 (a+b x)^{15}}{15 b^5} \]

[Out]

((A*b - a*B)*(b*d - a*e)^3*(a + b*x)^11)/(11*b^5) + ((b*d - a*e)^2*(b*B*d + 3*A*b*e - 4*a*B*e)*(a + b*x)^12)/(
12*b^5) + (3*e*(b*d - a*e)*(b*B*d + A*b*e - 2*a*B*e)*(a + b*x)^13)/(13*b^5) + (e^2*(3*b*B*d + A*b*e - 4*a*B*e)
*(a + b*x)^14)/(14*b^5) + (B*e^3*(a + b*x)^15)/(15*b^5)

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Rubi [A]  time = 0.899081, antiderivative size = 159, 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} \[ \frac{e^2 (a+b x)^{14} (-4 a B e+A b e+3 b B d)}{14 b^5}+\frac{3 e (a+b x)^{13} (b d-a e) (-2 a B e+A b e+b B d)}{13 b^5}+\frac{(a+b x)^{12} (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{12 b^5}+\frac{(a+b x)^{11} (A b-a B) (b d-a e)^3}{11 b^5}+\frac{B e^3 (a+b x)^{15}}{15 b^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10*(A + B*x)*(d + e*x)^3,x]

[Out]

((A*b - a*B)*(b*d - a*e)^3*(a + b*x)^11)/(11*b^5) + ((b*d - a*e)^2*(b*B*d + 3*A*b*e - 4*a*B*e)*(a + b*x)^12)/(
12*b^5) + (3*e*(b*d - a*e)*(b*B*d + A*b*e - 2*a*B*e)*(a + b*x)^13)/(13*b^5) + (e^2*(3*b*B*d + A*b*e - 4*a*B*e)
*(a + b*x)^14)/(14*b^5) + (B*e^3*(a + b*x)^15)/(15*b^5)

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)^3 \, dx &=\int \left (\frac{(A b-a B) (b d-a e)^3 (a+b x)^{10}}{b^4}+\frac{(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^{11}}{b^4}+\frac{3 e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^{12}}{b^4}+\frac{e^2 (3 b B d+A b e-4 a B e) (a+b x)^{13}}{b^4}+\frac{B e^3 (a+b x)^{14}}{b^4}\right ) \, dx\\ &=\frac{(A b-a B) (b d-a e)^3 (a+b x)^{11}}{11 b^5}+\frac{(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^{12}}{12 b^5}+\frac{3 e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^{13}}{13 b^5}+\frac{e^2 (3 b B d+A b e-4 a B e) (a+b x)^{14}}{14 b^5}+\frac{B e^3 (a+b x)^{15}}{15 b^5}\\ \end{align*}

Mathematica [B]  time = 0.493408, size = 855, normalized size = 5.38 \[ \frac{x \left (3003 \left (5 A \left (4 d^3+6 e x d^2+4 e^2 x^2 d+e^3 x^3\right )+B x \left (10 d^3+20 e x d^2+15 e^2 x^2 d+4 e^3 x^3\right )\right ) a^{10}+10010 b x \left (3 A \left (10 d^3+20 e x d^2+15 e^2 x^2 d+4 e^3 x^3\right )+B x \left (20 d^3+45 e x d^2+36 e^2 x^2 d+10 e^3 x^3\right )\right ) a^9+6435 b^2 x^2 \left (7 A \left (20 d^3+45 e x d^2+36 e^2 x^2 d+10 e^3 x^3\right )+3 B x \left (35 d^3+84 e x d^2+70 e^2 x^2 d+20 e^3 x^3\right )\right ) a^8+25740 b^3 x^3 \left (2 A \left (35 d^3+84 e x d^2+70 e^2 x^2 d+20 e^3 x^3\right )+B x \left (56 d^3+140 e x d^2+120 e^2 x^2 d+35 e^3 x^3\right )\right ) a^7+5005 b^4 x^4 \left (9 A \left (56 d^3+140 e x d^2+120 e^2 x^2 d+35 e^3 x^3\right )+5 B x \left (84 d^3+216 e x d^2+189 e^2 x^2 d+56 e^3 x^3\right )\right ) a^6+6006 b^5 x^5 \left (5 A \left (84 d^3+216 e x d^2+189 e^2 x^2 d+56 e^3 x^3\right )+3 B x \left (120 d^3+315 e x d^2+280 e^2 x^2 d+84 e^3 x^3\right )\right ) a^5+1365 b^6 x^6 \left (11 A \left (120 d^3+315 e x d^2+280 e^2 x^2 d+84 e^3 x^3\right )+7 B x \left (165 d^3+440 e x d^2+396 e^2 x^2 d+120 e^3 x^3\right )\right ) a^4+1820 b^7 x^7 \left (3 A \left (165 d^3+440 e x d^2+396 e^2 x^2 d+120 e^3 x^3\right )+2 B x \left (220 d^3+594 e x d^2+540 e^2 x^2 d+165 e^3 x^3\right )\right ) a^3+105 b^8 x^8 \left (13 A \left (220 d^3+594 e x d^2+540 e^2 x^2 d+165 e^3 x^3\right )+9 B x \left (286 d^3+780 e x d^2+715 e^2 x^2 d+220 e^3 x^3\right )\right ) a^2+30 b^9 x^9 \left (7 A \left (286 d^3+780 e x d^2+715 e^2 x^2 d+220 e^3 x^3\right )+5 B x \left (364 d^3+1001 e x d^2+924 e^2 x^2 d+286 e^3 x^3\right )\right ) a+b^{10} x^{10} \left (15 A \left (364 d^3+1001 e x d^2+924 e^2 x^2 d+286 e^3 x^3\right )+11 B x \left (455 d^3+1260 e x d^2+1170 e^2 x^2 d+364 e^3 x^3\right )\right )\right )}{60060} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10*(A + B*x)*(d + e*x)^3,x]

[Out]

(x*(3003*a^10*(5*A*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + B*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e
^3*x^3)) + 10010*a^9*b*x*(3*A*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + B*x*(20*d^3 + 45*d^2*e*x + 36
*d*e^2*x^2 + 10*e^3*x^3)) + 6435*a^8*b^2*x^2*(7*A*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 3*B*x*(3
5*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3)) + 25740*a^7*b^3*x^3*(2*A*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2
+ 20*e^3*x^3) + B*x*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3)) + 5005*a^6*b^4*x^4*(9*A*(56*d^3 + 140
*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) + 5*B*x*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3)) + 6006*a^5
*b^5*x^5*(5*A*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3) + 3*B*x*(120*d^3 + 315*d^2*e*x + 280*d*e^2*x
^2 + 84*e^3*x^3)) + 1365*a^4*b^6*x^6*(11*A*(120*d^3 + 315*d^2*e*x + 280*d*e^2*x^2 + 84*e^3*x^3) + 7*B*x*(165*d
^3 + 440*d^2*e*x + 396*d*e^2*x^2 + 120*e^3*x^3)) + 1820*a^3*b^7*x^7*(3*A*(165*d^3 + 440*d^2*e*x + 396*d*e^2*x^
2 + 120*e^3*x^3) + 2*B*x*(220*d^3 + 594*d^2*e*x + 540*d*e^2*x^2 + 165*e^3*x^3)) + 105*a^2*b^8*x^8*(13*A*(220*d
^3 + 594*d^2*e*x + 540*d*e^2*x^2 + 165*e^3*x^3) + 9*B*x*(286*d^3 + 780*d^2*e*x + 715*d*e^2*x^2 + 220*e^3*x^3))
+ 30*a*b^9*x^9*(7*A*(286*d^3 + 780*d^2*e*x + 715*d*e^2*x^2 + 220*e^3*x^3) + 5*B*x*(364*d^3 + 1001*d^2*e*x + 9
24*d*e^2*x^2 + 286*e^3*x^3)) + b^10*x^10*(15*A*(364*d^3 + 1001*d^2*e*x + 924*d*e^2*x^2 + 286*e^3*x^3) + 11*B*x
*(455*d^3 + 1260*d^2*e*x + 1170*d*e^2*x^2 + 364*e^3*x^3))))/60060

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Maple [B]  time = 0., size = 1053, normalized size = 6.6 \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)^3,x)

[Out]

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

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Maxima [B]  time = 1.12156, size = 1442, normalized size = 9.07 \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)^3,x, algorithm="maxima")

[Out]

1/15*B*b^10*e^3*x^15 + A*a^10*d^3*x + 1/14*(3*B*b^10*d*e^2 + (10*B*a*b^9 + A*b^10)*e^3)*x^14 + 1/13*(3*B*b^10*
d^2*e + 3*(10*B*a*b^9 + A*b^10)*d*e^2 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^3)*x^13 + 1/12*(B*b^10*d^3 + 3*(10*B*a*b
^9 + A*b^10)*d^2*e + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^2 + 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^3)*x^12 + 1/11*((10
*B*a*b^9 + A*b^10)*d^3 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e + 45*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^2 + 30*(7*B*a
^4*b^6 + 4*A*a^3*b^7)*e^3)*x^11 + 1/10*(5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3 + 45*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e
+ 90*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^2 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^3)*x^10 + 1/3*(5*(8*B*a^3*b^7 + 3*A
*a^2*b^8)*d^3 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^2 + 14*(5*B*a^6*b^4
+ 6*A*a^5*b^5)*e^3)*x^9 + 3/4*(5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e + 21*(
5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^2 + 5*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^3)*x^8 + 3/7*(14*(6*B*a^5*b^5 + 5*A*a^4*b^6
)*d^3 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e + 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^2 + 5*(3*B*a^8*b^2 + 8*A*a^7
*b^3)*e^3)*x^7 + 1/6*(42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3 + 90*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e + 45*(3*B*a^8*
b^2 + 8*A*a^7*b^3)*d*e^2 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*e^3)*x^6 + 1/5*(30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3 + 45
*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e + 15*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^2 + (B*a^10 + 10*A*a^9*b)*e^3)*x^5 + 1/4
*(A*a^10*e^3 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3 + 15*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e + 3*(B*a^10 + 10*A*a^9*
b)*d*e^2)*x^4 + 1/3*(3*A*a^10*d*e^2 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^3 + 3*(B*a^10 + 10*A*a^9*b)*d^2*e)*x^3 + 1
/2*(3*A*a^10*d^2*e + (B*a^10 + 10*A*a^9*b)*d^3)*x^2

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Fricas [B]  time = 1.60332, size = 2917, normalized size = 18.35 \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)^3,x, algorithm="fricas")

[Out]

1/15*x^15*e^3*b^10*B + 3/14*x^14*e^2*d*b^10*B + 5/7*x^14*e^3*b^9*a*B + 1/14*x^14*e^3*b^10*A + 3/13*x^13*e*d^2*
b^10*B + 30/13*x^13*e^2*d*b^9*a*B + 45/13*x^13*e^3*b^8*a^2*B + 3/13*x^13*e^2*d*b^10*A + 10/13*x^13*e^3*b^9*a*A
+ 1/12*x^12*d^3*b^10*B + 5/2*x^12*e*d^2*b^9*a*B + 45/4*x^12*e^2*d*b^8*a^2*B + 10*x^12*e^3*b^7*a^3*B + 1/4*x^1
2*e*d^2*b^10*A + 5/2*x^12*e^2*d*b^9*a*A + 15/4*x^12*e^3*b^8*a^2*A + 10/11*x^11*d^3*b^9*a*B + 135/11*x^11*e*d^2
*b^8*a^2*B + 360/11*x^11*e^2*d*b^7*a^3*B + 210/11*x^11*e^3*b^6*a^4*B + 1/11*x^11*d^3*b^10*A + 30/11*x^11*e*d^2
*b^9*a*A + 135/11*x^11*e^2*d*b^8*a^2*A + 120/11*x^11*e^3*b^7*a^3*A + 9/2*x^10*d^3*b^8*a^2*B + 36*x^10*e*d^2*b^
7*a^3*B + 63*x^10*e^2*d*b^6*a^4*B + 126/5*x^10*e^3*b^5*a^5*B + x^10*d^3*b^9*a*A + 27/2*x^10*e*d^2*b^8*a^2*A +
36*x^10*e^2*d*b^7*a^3*A + 21*x^10*e^3*b^6*a^4*A + 40/3*x^9*d^3*b^7*a^3*B + 70*x^9*e*d^2*b^6*a^4*B + 84*x^9*e^2
*d*b^5*a^5*B + 70/3*x^9*e^3*b^4*a^6*B + 5*x^9*d^3*b^8*a^2*A + 40*x^9*e*d^2*b^7*a^3*A + 70*x^9*e^2*d*b^6*a^4*A
+ 28*x^9*e^3*b^5*a^5*A + 105/4*x^8*d^3*b^6*a^4*B + 189/2*x^8*e*d^2*b^5*a^5*B + 315/4*x^8*e^2*d*b^4*a^6*B + 15*
x^8*e^3*b^3*a^7*B + 15*x^8*d^3*b^7*a^3*A + 315/4*x^8*e*d^2*b^6*a^4*A + 189/2*x^8*e^2*d*b^5*a^5*A + 105/4*x^8*e
^3*b^4*a^6*A + 36*x^7*d^3*b^5*a^5*B + 90*x^7*e*d^2*b^4*a^6*B + 360/7*x^7*e^2*d*b^3*a^7*B + 45/7*x^7*e^3*b^2*a^
8*B + 30*x^7*d^3*b^6*a^4*A + 108*x^7*e*d^2*b^5*a^5*A + 90*x^7*e^2*d*b^4*a^6*A + 120/7*x^7*e^3*b^3*a^7*A + 35*x
^6*d^3*b^4*a^6*B + 60*x^6*e*d^2*b^3*a^7*B + 45/2*x^6*e^2*d*b^2*a^8*B + 5/3*x^6*e^3*b*a^9*B + 42*x^6*d^3*b^5*a^
5*A + 105*x^6*e*d^2*b^4*a^6*A + 60*x^6*e^2*d*b^3*a^7*A + 15/2*x^6*e^3*b^2*a^8*A + 24*x^5*d^3*b^3*a^7*B + 27*x^
5*e*d^2*b^2*a^8*B + 6*x^5*e^2*d*b*a^9*B + 1/5*x^5*e^3*a^10*B + 42*x^5*d^3*b^4*a^6*A + 72*x^5*e*d^2*b^3*a^7*A +
27*x^5*e^2*d*b^2*a^8*A + 2*x^5*e^3*b*a^9*A + 45/4*x^4*d^3*b^2*a^8*B + 15/2*x^4*e*d^2*b*a^9*B + 3/4*x^4*e^2*d*
a^10*B + 30*x^4*d^3*b^3*a^7*A + 135/4*x^4*e*d^2*b^2*a^8*A + 15/2*x^4*e^2*d*b*a^9*A + 1/4*x^4*e^3*a^10*A + 10/3
*x^3*d^3*b*a^9*B + x^3*e*d^2*a^10*B + 15*x^3*d^3*b^2*a^8*A + 10*x^3*e*d^2*b*a^9*A + x^3*e^2*d*a^10*A + 1/2*x^2
*d^3*a^10*B + 5*x^2*d^3*b*a^9*A + 3/2*x^2*e*d^2*a^10*A + x*d^3*a^10*A

________________________________________________________________________________________

Sympy [B]  time = 0.201248, size = 1302, normalized size = 8.19 \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)**3,x)

[Out]

A*a**10*d**3*x + B*b**10*e**3*x**15/15 + x**14*(A*b**10*e**3/14 + 5*B*a*b**9*e**3/7 + 3*B*b**10*d*e**2/14) + x
**13*(10*A*a*b**9*e**3/13 + 3*A*b**10*d*e**2/13 + 45*B*a**2*b**8*e**3/13 + 30*B*a*b**9*d*e**2/13 + 3*B*b**10*d
**2*e/13) + x**12*(15*A*a**2*b**8*e**3/4 + 5*A*a*b**9*d*e**2/2 + A*b**10*d**2*e/4 + 10*B*a**3*b**7*e**3 + 45*B
*a**2*b**8*d*e**2/4 + 5*B*a*b**9*d**2*e/2 + B*b**10*d**3/12) + x**11*(120*A*a**3*b**7*e**3/11 + 135*A*a**2*b**
8*d*e**2/11 + 30*A*a*b**9*d**2*e/11 + A*b**10*d**3/11 + 210*B*a**4*b**6*e**3/11 + 360*B*a**3*b**7*d*e**2/11 +
135*B*a**2*b**8*d**2*e/11 + 10*B*a*b**9*d**3/11) + x**10*(21*A*a**4*b**6*e**3 + 36*A*a**3*b**7*d*e**2 + 27*A*a
**2*b**8*d**2*e/2 + A*a*b**9*d**3 + 126*B*a**5*b**5*e**3/5 + 63*B*a**4*b**6*d*e**2 + 36*B*a**3*b**7*d**2*e + 9
*B*a**2*b**8*d**3/2) + x**9*(28*A*a**5*b**5*e**3 + 70*A*a**4*b**6*d*e**2 + 40*A*a**3*b**7*d**2*e + 5*A*a**2*b*
*8*d**3 + 70*B*a**6*b**4*e**3/3 + 84*B*a**5*b**5*d*e**2 + 70*B*a**4*b**6*d**2*e + 40*B*a**3*b**7*d**3/3) + x**
8*(105*A*a**6*b**4*e**3/4 + 189*A*a**5*b**5*d*e**2/2 + 315*A*a**4*b**6*d**2*e/4 + 15*A*a**3*b**7*d**3 + 15*B*a
**7*b**3*e**3 + 315*B*a**6*b**4*d*e**2/4 + 189*B*a**5*b**5*d**2*e/2 + 105*B*a**4*b**6*d**3/4) + x**7*(120*A*a*
*7*b**3*e**3/7 + 90*A*a**6*b**4*d*e**2 + 108*A*a**5*b**5*d**2*e + 30*A*a**4*b**6*d**3 + 45*B*a**8*b**2*e**3/7
+ 360*B*a**7*b**3*d*e**2/7 + 90*B*a**6*b**4*d**2*e + 36*B*a**5*b**5*d**3) + x**6*(15*A*a**8*b**2*e**3/2 + 60*A
*a**7*b**3*d*e**2 + 105*A*a**6*b**4*d**2*e + 42*A*a**5*b**5*d**3 + 5*B*a**9*b*e**3/3 + 45*B*a**8*b**2*d*e**2/2
+ 60*B*a**7*b**3*d**2*e + 35*B*a**6*b**4*d**3) + x**5*(2*A*a**9*b*e**3 + 27*A*a**8*b**2*d*e**2 + 72*A*a**7*b*
*3*d**2*e + 42*A*a**6*b**4*d**3 + B*a**10*e**3/5 + 6*B*a**9*b*d*e**2 + 27*B*a**8*b**2*d**2*e + 24*B*a**7*b**3*
d**3) + x**4*(A*a**10*e**3/4 + 15*A*a**9*b*d*e**2/2 + 135*A*a**8*b**2*d**2*e/4 + 30*A*a**7*b**3*d**3 + 3*B*a**
10*d*e**2/4 + 15*B*a**9*b*d**2*e/2 + 45*B*a**8*b**2*d**3/4) + x**3*(A*a**10*d*e**2 + 10*A*a**9*b*d**2*e + 15*A
*a**8*b**2*d**3 + B*a**10*d**2*e + 10*B*a**9*b*d**3/3) + x**2*(3*A*a**10*d**2*e/2 + 5*A*a**9*b*d**3 + B*a**10*
d**3/2)

________________________________________________________________________________________

Giac [B]  time = 1.68711, size = 1697, normalized size = 10.67 \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)^3,x, algorithm="giac")

[Out]

1/15*B*b^10*x^15*e^3 + 3/14*B*b^10*d*x^14*e^2 + 3/13*B*b^10*d^2*x^13*e + 1/12*B*b^10*d^3*x^12 + 5/7*B*a*b^9*x^
14*e^3 + 1/14*A*b^10*x^14*e^3 + 30/13*B*a*b^9*d*x^13*e^2 + 3/13*A*b^10*d*x^13*e^2 + 5/2*B*a*b^9*d^2*x^12*e + 1
/4*A*b^10*d^2*x^12*e + 10/11*B*a*b^9*d^3*x^11 + 1/11*A*b^10*d^3*x^11 + 45/13*B*a^2*b^8*x^13*e^3 + 10/13*A*a*b^
9*x^13*e^3 + 45/4*B*a^2*b^8*d*x^12*e^2 + 5/2*A*a*b^9*d*x^12*e^2 + 135/11*B*a^2*b^8*d^2*x^11*e + 30/11*A*a*b^9*
d^2*x^11*e + 9/2*B*a^2*b^8*d^3*x^10 + A*a*b^9*d^3*x^10 + 10*B*a^3*b^7*x^12*e^3 + 15/4*A*a^2*b^8*x^12*e^3 + 360
/11*B*a^3*b^7*d*x^11*e^2 + 135/11*A*a^2*b^8*d*x^11*e^2 + 36*B*a^3*b^7*d^2*x^10*e + 27/2*A*a^2*b^8*d^2*x^10*e +
40/3*B*a^3*b^7*d^3*x^9 + 5*A*a^2*b^8*d^3*x^9 + 210/11*B*a^4*b^6*x^11*e^3 + 120/11*A*a^3*b^7*x^11*e^3 + 63*B*a
^4*b^6*d*x^10*e^2 + 36*A*a^3*b^7*d*x^10*e^2 + 70*B*a^4*b^6*d^2*x^9*e + 40*A*a^3*b^7*d^2*x^9*e + 105/4*B*a^4*b^
6*d^3*x^8 + 15*A*a^3*b^7*d^3*x^8 + 126/5*B*a^5*b^5*x^10*e^3 + 21*A*a^4*b^6*x^10*e^3 + 84*B*a^5*b^5*d*x^9*e^2 +
70*A*a^4*b^6*d*x^9*e^2 + 189/2*B*a^5*b^5*d^2*x^8*e + 315/4*A*a^4*b^6*d^2*x^8*e + 36*B*a^5*b^5*d^3*x^7 + 30*A*
a^4*b^6*d^3*x^7 + 70/3*B*a^6*b^4*x^9*e^3 + 28*A*a^5*b^5*x^9*e^3 + 315/4*B*a^6*b^4*d*x^8*e^2 + 189/2*A*a^5*b^5*
d*x^8*e^2 + 90*B*a^6*b^4*d^2*x^7*e + 108*A*a^5*b^5*d^2*x^7*e + 35*B*a^6*b^4*d^3*x^6 + 42*A*a^5*b^5*d^3*x^6 + 1
5*B*a^7*b^3*x^8*e^3 + 105/4*A*a^6*b^4*x^8*e^3 + 360/7*B*a^7*b^3*d*x^7*e^2 + 90*A*a^6*b^4*d*x^7*e^2 + 60*B*a^7*
b^3*d^2*x^6*e + 105*A*a^6*b^4*d^2*x^6*e + 24*B*a^7*b^3*d^3*x^5 + 42*A*a^6*b^4*d^3*x^5 + 45/7*B*a^8*b^2*x^7*e^3
+ 120/7*A*a^7*b^3*x^7*e^3 + 45/2*B*a^8*b^2*d*x^6*e^2 + 60*A*a^7*b^3*d*x^6*e^2 + 27*B*a^8*b^2*d^2*x^5*e + 72*A
*a^7*b^3*d^2*x^5*e + 45/4*B*a^8*b^2*d^3*x^4 + 30*A*a^7*b^3*d^3*x^4 + 5/3*B*a^9*b*x^6*e^3 + 15/2*A*a^8*b^2*x^6*
e^3 + 6*B*a^9*b*d*x^5*e^2 + 27*A*a^8*b^2*d*x^5*e^2 + 15/2*B*a^9*b*d^2*x^4*e + 135/4*A*a^8*b^2*d^2*x^4*e + 10/3
*B*a^9*b*d^3*x^3 + 15*A*a^8*b^2*d^3*x^3 + 1/5*B*a^10*x^5*e^3 + 2*A*a^9*b*x^5*e^3 + 3/4*B*a^10*d*x^4*e^2 + 15/2
*A*a^9*b*d*x^4*e^2 + B*a^10*d^2*x^3*e + 10*A*a^9*b*d^2*x^3*e + 1/2*B*a^10*d^3*x^2 + 5*A*a^9*b*d^3*x^2 + 1/4*A*
a^10*x^4*e^3 + A*a^10*d*x^3*e^2 + 3/2*A*a^10*d^2*x^2*e + A*a^10*d^3*x