3.1087 \(\int (a+b x)^{10} (A+B x) (d+e x) \, dx\)

Optimal. Leaf size=75 \[ \frac {(a+b x)^{12} (-2 a B e+A b e+b B d)}{12 b^3}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)}{11 b^3}+\frac {B e (a+b x)^{13}}{13 b^3} \]

[Out]

1/11*(A*b-B*a)*(-a*e+b*d)*(b*x+a)^11/b^3+1/12*(A*b*e-2*B*a*e+B*b*d)*(b*x+a)^12/b^3+1/13*B*e*(b*x+a)^13/b^3

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Rubi [A]  time = 0.41, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \[ \frac {(a+b x)^{12} (-2 a B e+A b e+b B d)}{12 b^3}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)}{11 b^3}+\frac {B e (a+b x)^{13}}{13 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [B]  time = 0.21, size = 383, normalized size = 5.11 \[ \frac {1}{6} a^{10} x (3 A (2 d+e x)+B x (3 d+2 e x))+\frac {5}{6} a^9 b x^2 (A (6 d+4 e x)+B x (4 d+3 e x))+\frac {3}{4} a^8 b^2 x^3 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+2 a^7 b^3 x^4 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))+a^6 b^4 x^5 (7 A (6 d+5 e x)+5 B x (7 d+6 e x))+\frac {3}{2} a^5 b^5 x^6 (4 A (7 d+6 e x)+3 B x (8 d+7 e x))+\frac {5}{12} a^4 b^6 x^7 (9 A (8 d+7 e x)+7 B x (9 d+8 e x))+\frac {1}{3} a^3 b^7 x^8 (5 A (9 d+8 e x)+4 B x (10 d+9 e x))+\frac {1}{22} a^2 b^8 x^9 \left (110 A d+99 A e x+99 B d x+90 B e x^2\right )+\frac {1}{66} a b^9 x^{10} \left (66 A d+60 A e x+60 B d x+55 B e x^2\right )+\frac {b^{10} x^{11} (13 A (12 d+11 e x)+11 B x (13 d+12 e x))}{1716} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*b^9*x^10*(66*A*d + 60*B*d*x + 60*A*e*x + 55*B*e*x^2))/66 + (a^2*b^8*x^9*(110*A*d + 99*B*d*x + 99*A*e*x + 90
*B*e*x^2))/22 + (a^10*x*(3*A*(2*d + e*x) + B*x*(3*d + 2*e*x)))/6 + (3*a^8*b^2*x^3*(5*A*(4*d + 3*e*x) + 3*B*x*(
5*d + 4*e*x)))/4 + (5*a^9*b*x^2*(B*x*(4*d + 3*e*x) + A*(6*d + 4*e*x)))/6 + 2*a^7*b^3*x^4*(3*A*(5*d + 4*e*x) +
2*B*x*(6*d + 5*e*x)) + a^6*b^4*x^5*(7*A*(6*d + 5*e*x) + 5*B*x*(7*d + 6*e*x)) + (3*a^5*b^5*x^6*(4*A*(7*d + 6*e*
x) + 3*B*x*(8*d + 7*e*x)))/2 + (5*a^4*b^6*x^7*(9*A*(8*d + 7*e*x) + 7*B*x*(9*d + 8*e*x)))/12 + (a^3*b^7*x^8*(5*
A*(9*d + 8*e*x) + 4*B*x*(10*d + 9*e*x)))/3 + (b^10*x^11*(13*A*(12*d + 11*e*x) + 11*B*x*(13*d + 12*e*x)))/1716

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fricas [B]  time = 0.86, size = 529, normalized size = 7.05 \[ \frac {1}{13} x^{13} e b^{10} B + \frac {1}{12} x^{12} d b^{10} B + \frac {5}{6} x^{12} e b^{9} a B + \frac {1}{12} x^{12} e b^{10} A + \frac {10}{11} x^{11} d b^{9} a B + \frac {45}{11} x^{11} e b^{8} a^{2} B + \frac {1}{11} x^{11} d b^{10} A + \frac {10}{11} x^{11} e b^{9} a A + \frac {9}{2} x^{10} d b^{8} a^{2} B + 12 x^{10} e b^{7} a^{3} B + x^{10} d b^{9} a A + \frac {9}{2} x^{10} e b^{8} a^{2} A + \frac {40}{3} x^{9} d b^{7} a^{3} B + \frac {70}{3} x^{9} e b^{6} a^{4} B + 5 x^{9} d b^{8} a^{2} A + \frac {40}{3} x^{9} e b^{7} a^{3} A + \frac {105}{4} x^{8} d b^{6} a^{4} B + \frac {63}{2} x^{8} e b^{5} a^{5} B + 15 x^{8} d b^{7} a^{3} A + \frac {105}{4} x^{8} e b^{6} a^{4} A + 36 x^{7} d b^{5} a^{5} B + 30 x^{7} e b^{4} a^{6} B + 30 x^{7} d b^{6} a^{4} A + 36 x^{7} e b^{5} a^{5} A + 35 x^{6} d b^{4} a^{6} B + 20 x^{6} e b^{3} a^{7} B + 42 x^{6} d b^{5} a^{5} A + 35 x^{6} e b^{4} a^{6} A + 24 x^{5} d b^{3} a^{7} B + 9 x^{5} e b^{2} a^{8} B + 42 x^{5} d b^{4} a^{6} A + 24 x^{5} e b^{3} a^{7} A + \frac {45}{4} x^{4} d b^{2} a^{8} B + \frac {5}{2} x^{4} e b a^{9} B + 30 x^{4} d b^{3} a^{7} A + \frac {45}{4} x^{4} e b^{2} a^{8} A + \frac {10}{3} x^{3} d b a^{9} B + \frac {1}{3} x^{3} e a^{10} B + 15 x^{3} d b^{2} a^{8} A + \frac {10}{3} x^{3} e b a^{9} A + \frac {1}{2} x^{2} d a^{10} B + 5 x^{2} d b a^{9} A + \frac {1}{2} x^{2} e a^{10} A + x d a^{10} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13*e*b^10*B + 1/12*x^12*d*b^10*B + 5/6*x^12*e*b^9*a*B + 1/12*x^12*e*b^10*A + 10/11*x^11*d*b^9*a*B + 45/
11*x^11*e*b^8*a^2*B + 1/11*x^11*d*b^10*A + 10/11*x^11*e*b^9*a*A + 9/2*x^10*d*b^8*a^2*B + 12*x^10*e*b^7*a^3*B +
x^10*d*b^9*a*A + 9/2*x^10*e*b^8*a^2*A + 40/3*x^9*d*b^7*a^3*B + 70/3*x^9*e*b^6*a^4*B + 5*x^9*d*b^8*a^2*A + 40/
3*x^9*e*b^7*a^3*A + 105/4*x^8*d*b^6*a^4*B + 63/2*x^8*e*b^5*a^5*B + 15*x^8*d*b^7*a^3*A + 105/4*x^8*e*b^6*a^4*A
+ 36*x^7*d*b^5*a^5*B + 30*x^7*e*b^4*a^6*B + 30*x^7*d*b^6*a^4*A + 36*x^7*e*b^5*a^5*A + 35*x^6*d*b^4*a^6*B + 20*
x^6*e*b^3*a^7*B + 42*x^6*d*b^5*a^5*A + 35*x^6*e*b^4*a^6*A + 24*x^5*d*b^3*a^7*B + 9*x^5*e*b^2*a^8*B + 42*x^5*d*
b^4*a^6*A + 24*x^5*e*b^3*a^7*A + 45/4*x^4*d*b^2*a^8*B + 5/2*x^4*e*b*a^9*B + 30*x^4*d*b^3*a^7*A + 45/4*x^4*e*b^
2*a^8*A + 10/3*x^3*d*b*a^9*B + 1/3*x^3*e*a^10*B + 15*x^3*d*b^2*a^8*A + 10/3*x^3*e*b*a^9*A + 1/2*x^2*d*a^10*B +
5*x^2*d*b*a^9*A + 1/2*x^2*e*a^10*A + x*d*a^10*A

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giac [B]  time = 1.20, size = 551, normalized size = 7.35 \[ \frac {1}{13} \, B b^{10} x^{13} e + \frac {1}{12} \, B b^{10} d x^{12} + \frac {5}{6} \, B a b^{9} x^{12} e + \frac {1}{12} \, A b^{10} x^{12} e + \frac {10}{11} \, B a b^{9} d x^{11} + \frac {1}{11} \, A b^{10} d x^{11} + \frac {45}{11} \, B a^{2} b^{8} x^{11} e + \frac {10}{11} \, A a b^{9} x^{11} e + \frac {9}{2} \, B a^{2} b^{8} d x^{10} + A a b^{9} d x^{10} + 12 \, B a^{3} b^{7} x^{10} e + \frac {9}{2} \, A a^{2} b^{8} x^{10} e + \frac {40}{3} \, B a^{3} b^{7} d x^{9} + 5 \, A a^{2} b^{8} d x^{9} + \frac {70}{3} \, B a^{4} b^{6} x^{9} e + \frac {40}{3} \, A a^{3} b^{7} x^{9} e + \frac {105}{4} \, B a^{4} b^{6} d x^{8} + 15 \, A a^{3} b^{7} d x^{8} + \frac {63}{2} \, B a^{5} b^{5} x^{8} e + \frac {105}{4} \, A a^{4} b^{6} x^{8} e + 36 \, B a^{5} b^{5} d x^{7} + 30 \, A a^{4} b^{6} d x^{7} + 30 \, B a^{6} b^{4} x^{7} e + 36 \, A a^{5} b^{5} x^{7} e + 35 \, B a^{6} b^{4} d x^{6} + 42 \, A a^{5} b^{5} d x^{6} + 20 \, B a^{7} b^{3} x^{6} e + 35 \, A a^{6} b^{4} x^{6} e + 24 \, B a^{7} b^{3} d x^{5} + 42 \, A a^{6} b^{4} d x^{5} + 9 \, B a^{8} b^{2} x^{5} e + 24 \, A a^{7} b^{3} x^{5} e + \frac {45}{4} \, B a^{8} b^{2} d x^{4} + 30 \, A a^{7} b^{3} d x^{4} + \frac {5}{2} \, B a^{9} b x^{4} e + \frac {45}{4} \, A a^{8} b^{2} x^{4} e + \frac {10}{3} \, B a^{9} b d x^{3} + 15 \, A a^{8} b^{2} d x^{3} + \frac {1}{3} \, B a^{10} x^{3} e + \frac {10}{3} \, A a^{9} b x^{3} e + \frac {1}{2} \, B a^{10} d x^{2} + 5 \, A a^{9} b d x^{2} + \frac {1}{2} \, A a^{10} x^{2} e + A a^{10} d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*B*b^10*x^13*e + 1/12*B*b^10*d*x^12 + 5/6*B*a*b^9*x^12*e + 1/12*A*b^10*x^12*e + 10/11*B*a*b^9*d*x^11 + 1/1
1*A*b^10*d*x^11 + 45/11*B*a^2*b^8*x^11*e + 10/11*A*a*b^9*x^11*e + 9/2*B*a^2*b^8*d*x^10 + A*a*b^9*d*x^10 + 12*B
*a^3*b^7*x^10*e + 9/2*A*a^2*b^8*x^10*e + 40/3*B*a^3*b^7*d*x^9 + 5*A*a^2*b^8*d*x^9 + 70/3*B*a^4*b^6*x^9*e + 40/
3*A*a^3*b^7*x^9*e + 105/4*B*a^4*b^6*d*x^8 + 15*A*a^3*b^7*d*x^8 + 63/2*B*a^5*b^5*x^8*e + 105/4*A*a^4*b^6*x^8*e
+ 36*B*a^5*b^5*d*x^7 + 30*A*a^4*b^6*d*x^7 + 30*B*a^6*b^4*x^7*e + 36*A*a^5*b^5*x^7*e + 35*B*a^6*b^4*d*x^6 + 42*
A*a^5*b^5*d*x^6 + 20*B*a^7*b^3*x^6*e + 35*A*a^6*b^4*x^6*e + 24*B*a^7*b^3*d*x^5 + 42*A*a^6*b^4*d*x^5 + 9*B*a^8*
b^2*x^5*e + 24*A*a^7*b^3*x^5*e + 45/4*B*a^8*b^2*d*x^4 + 30*A*a^7*b^3*d*x^4 + 5/2*B*a^9*b*x^4*e + 45/4*A*a^8*b^
2*x^4*e + 10/3*B*a^9*b*d*x^3 + 15*A*a^8*b^2*d*x^3 + 1/3*B*a^10*x^3*e + 10/3*A*a^9*b*x^3*e + 1/2*B*a^10*d*x^2 +
5*A*a^9*b*d*x^2 + 1/2*A*a^10*x^2*e + A*a^10*d*x

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maple [B]  time = 0.00, size = 485, normalized size = 6.47 \[ \frac {B \,b^{10} e \,x^{13}}{13}+A \,a^{10} d x +\frac {\left (B \,b^{10} d +\left (b^{10} A +10 a \,b^{9} B \right ) e \right ) x^{12}}{12}+\frac {\left (\left (b^{10} A +10 a \,b^{9} B \right ) d +\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) e \right ) x^{11}}{11}+\frac {\left (\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) d +\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) e \right ) x^{10}}{10}+\frac {\left (\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) d +\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) e \right ) x^{9}}{9}+\frac {\left (\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) d +\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) e \right ) x^{8}}{8}+\frac {\left (\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) d +\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) e \right ) x^{7}}{7}+\frac {\left (\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) d +\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) e \right ) x^{6}}{6}+\frac {\left (\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) d +\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) e \right ) x^{5}}{5}+\frac {\left (\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) d +\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) e \right ) x^{4}}{4}+\frac {\left (\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) d +\left (10 a^{9} b A +a^{10} B \right ) e \right ) x^{3}}{3}+\frac {\left (A \,a^{10} e +\left (10 a^{9} b A +a^{10} B \right ) d \right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.63, size = 493, normalized size = 6.57 \[ \frac {1}{13} \, B b^{10} e x^{13} + A a^{10} d x + \frac {1}{12} \, {\left (B b^{10} d + {\left (10 \, B a b^{9} + A b^{10}\right )} e\right )} x^{12} + \frac {1}{11} \, {\left ({\left (10 \, B a b^{9} + A b^{10}\right )} d + 5 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} e\right )} x^{11} + \frac {1}{2} \, {\left ({\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} d + 3 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} e\right )} x^{10} + \frac {5}{3} \, {\left ({\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} d + 2 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} e\right )} x^{9} + \frac {3}{4} \, {\left (5 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} d + 7 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} e\right )} x^{8} + 6 \, {\left ({\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} d + {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} e\right )} x^{7} + {\left (7 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} d + 5 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} e\right )} x^{6} + 3 \, {\left (2 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} d + {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} e\right )} x^{5} + \frac {5}{4} \, {\left (3 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} d + {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (5 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} d + {\left (B a^{10} + 10 \, A a^{9} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{10} e + {\left (B a^{10} + 10 \, A a^{9} b\right )} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.26, size = 409, normalized size = 5.45 \[ x^3\,\left (\frac {B\,a^{10}\,e}{3}+\frac {10\,A\,a^9\,b\,e}{3}+\frac {10\,B\,a^9\,b\,d}{3}+15\,A\,a^8\,b^2\,d\right )+x^{11}\,\left (\frac {A\,b^{10}\,d}{11}+\frac {10\,A\,a\,b^9\,e}{11}+\frac {10\,B\,a\,b^9\,d}{11}+\frac {45\,B\,a^2\,b^8\,e}{11}\right )+x^2\,\left (\frac {A\,a^{10}\,e}{2}+\frac {B\,a^{10}\,d}{2}+5\,A\,a^9\,b\,d\right )+x^{12}\,\left (\frac {A\,b^{10}\,e}{12}+\frac {B\,b^{10}\,d}{12}+\frac {5\,B\,a\,b^9\,e}{6}\right )+6\,a^4\,b^4\,x^7\,\left (5\,A\,b^2\,d+5\,B\,a^2\,e+6\,A\,a\,b\,e+6\,B\,a\,b\,d\right )+3\,a^6\,b^2\,x^5\,\left (14\,A\,b^2\,d+3\,B\,a^2\,e+8\,A\,a\,b\,e+8\,B\,a\,b\,d\right )+\frac {5\,a^2\,b^6\,x^9\,\left (3\,A\,b^2\,d+14\,B\,a^2\,e+8\,A\,a\,b\,e+8\,B\,a\,b\,d\right )}{3}+a^5\,b^3\,x^6\,\left (42\,A\,b^2\,d+20\,B\,a^2\,e+35\,A\,a\,b\,e+35\,B\,a\,b\,d\right )+\frac {3\,a^3\,b^5\,x^8\,\left (20\,A\,b^2\,d+42\,B\,a^2\,e+35\,A\,a\,b\,e+35\,B\,a\,b\,d\right )}{4}+A\,a^{10}\,d\,x+\frac {B\,b^{10}\,e\,x^{13}}{13}+\frac {5\,a^7\,b\,x^4\,\left (24\,A\,b^2\,d+2\,B\,a^2\,e+9\,A\,a\,b\,e+9\,B\,a\,b\,d\right )}{4}+\frac {a\,b^7\,x^{10}\,\left (2\,A\,b^2\,d+24\,B\,a^2\,e+9\,A\,a\,b\,e+9\,B\,a\,b\,d\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*((B*a^10*e)/3 + (10*A*a^9*b*e)/3 + (10*B*a^9*b*d)/3 + 15*A*a^8*b^2*d) + x^11*((A*b^10*d)/11 + (10*A*a*b^9*
e)/11 + (10*B*a*b^9*d)/11 + (45*B*a^2*b^8*e)/11) + x^2*((A*a^10*e)/2 + (B*a^10*d)/2 + 5*A*a^9*b*d) + x^12*((A*
b^10*e)/12 + (B*b^10*d)/12 + (5*B*a*b^9*e)/6) + 6*a^4*b^4*x^7*(5*A*b^2*d + 5*B*a^2*e + 6*A*a*b*e + 6*B*a*b*d)
+ 3*a^6*b^2*x^5*(14*A*b^2*d + 3*B*a^2*e + 8*A*a*b*e + 8*B*a*b*d) + (5*a^2*b^6*x^9*(3*A*b^2*d + 14*B*a^2*e + 8*
A*a*b*e + 8*B*a*b*d))/3 + a^5*b^3*x^6*(42*A*b^2*d + 20*B*a^2*e + 35*A*a*b*e + 35*B*a*b*d) + (3*a^3*b^5*x^8*(20
*A*b^2*d + 42*B*a^2*e + 35*A*a*b*e + 35*B*a*b*d))/4 + A*a^10*d*x + (B*b^10*e*x^13)/13 + (5*a^7*b*x^4*(24*A*b^2
*d + 2*B*a^2*e + 9*A*a*b*e + 9*B*a*b*d))/4 + (a*b^7*x^10*(2*A*b^2*d + 24*B*a^2*e + 9*A*a*b*e + 9*B*a*b*d))/2

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sympy [B]  time = 0.16, size = 549, normalized size = 7.32 \[ A a^{10} d x + \frac {B b^{10} e x^{13}}{13} + x^{12} \left (\frac {A b^{10} e}{12} + \frac {5 B a b^{9} e}{6} + \frac {B b^{10} d}{12}\right ) + x^{11} \left (\frac {10 A a b^{9} e}{11} + \frac {A b^{10} d}{11} + \frac {45 B a^{2} b^{8} e}{11} + \frac {10 B a b^{9} d}{11}\right ) + x^{10} \left (\frac {9 A a^{2} b^{8} e}{2} + A a b^{9} d + 12 B a^{3} b^{7} e + \frac {9 B a^{2} b^{8} d}{2}\right ) + x^{9} \left (\frac {40 A a^{3} b^{7} e}{3} + 5 A a^{2} b^{8} d + \frac {70 B a^{4} b^{6} e}{3} + \frac {40 B a^{3} b^{7} d}{3}\right ) + x^{8} \left (\frac {105 A a^{4} b^{6} e}{4} + 15 A a^{3} b^{7} d + \frac {63 B a^{5} b^{5} e}{2} + \frac {105 B a^{4} b^{6} d}{4}\right ) + x^{7} \left (36 A a^{5} b^{5} e + 30 A a^{4} b^{6} d + 30 B a^{6} b^{4} e + 36 B a^{5} b^{5} d\right ) + x^{6} \left (35 A a^{6} b^{4} e + 42 A a^{5} b^{5} d + 20 B a^{7} b^{3} e + 35 B a^{6} b^{4} d\right ) + x^{5} \left (24 A a^{7} b^{3} e + 42 A a^{6} b^{4} d + 9 B a^{8} b^{2} e + 24 B a^{7} b^{3} d\right ) + x^{4} \left (\frac {45 A a^{8} b^{2} e}{4} + 30 A a^{7} b^{3} d + \frac {5 B a^{9} b e}{2} + \frac {45 B a^{8} b^{2} d}{4}\right ) + x^{3} \left (\frac {10 A a^{9} b e}{3} + 15 A a^{8} b^{2} d + \frac {B a^{10} e}{3} + \frac {10 B a^{9} b d}{3}\right ) + x^{2} \left (\frac {A a^{10} e}{2} + 5 A a^{9} b d + \frac {B a^{10} d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**10*d*x + B*b**10*e*x**13/13 + x**12*(A*b**10*e/12 + 5*B*a*b**9*e/6 + B*b**10*d/12) + x**11*(10*A*a*b**9*e
/11 + A*b**10*d/11 + 45*B*a**2*b**8*e/11 + 10*B*a*b**9*d/11) + x**10*(9*A*a**2*b**8*e/2 + A*a*b**9*d + 12*B*a*
*3*b**7*e + 9*B*a**2*b**8*d/2) + x**9*(40*A*a**3*b**7*e/3 + 5*A*a**2*b**8*d + 70*B*a**4*b**6*e/3 + 40*B*a**3*b
**7*d/3) + x**8*(105*A*a**4*b**6*e/4 + 15*A*a**3*b**7*d + 63*B*a**5*b**5*e/2 + 105*B*a**4*b**6*d/4) + x**7*(36
*A*a**5*b**5*e + 30*A*a**4*b**6*d + 30*B*a**6*b**4*e + 36*B*a**5*b**5*d) + x**6*(35*A*a**6*b**4*e + 42*A*a**5*
b**5*d + 20*B*a**7*b**3*e + 35*B*a**6*b**4*d) + x**5*(24*A*a**7*b**3*e + 42*A*a**6*b**4*d + 9*B*a**8*b**2*e +
24*B*a**7*b**3*d) + x**4*(45*A*a**8*b**2*e/4 + 30*A*a**7*b**3*d + 5*B*a**9*b*e/2 + 45*B*a**8*b**2*d/4) + x**3*
(10*A*a**9*b*e/3 + 15*A*a**8*b**2*d + B*a**10*e/3 + 10*B*a**9*b*d/3) + x**2*(A*a**10*e/2 + 5*A*a**9*b*d + B*a*
*10*d/2)

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