3.1070 \(\int (a+b x)^{10} (A+B x) (d+e x) \, dx\)

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]

((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)

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Rubi [A]  time = 1.04434, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{(a+b x)^{12} (-2 a B e+A b e+b B d)}{12 b^3}+\frac{(a+b x)^{11} (A b-a B) (b d-a e)}{11 b^3}+\frac{B e (a+b x)^{13}}{13 b^3} \]

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)

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Rubi in Sympy [A]  time = 52.2085, size = 68, normalized size = 0.91 \[ \frac{B e \left (a + b x\right )^{13}}{13 b^{3}} + \frac{\left (a + b x\right )^{12} \left (A b e - 2 B a e + B b d\right )}{12 b^{3}} - \frac{\left (a + b x\right )^{11} \left (A b - B a\right ) \left (a e - b d\right )}{11 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.35781, 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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Maple [B]  time = 0.003, size = 485, normalized size = 6.5 \[{\frac{{b}^{10}Be{x}^{13}}{13}}+{\frac{ \left ( \left ({b}^{10}A+10\,a{b}^{9}B \right ) e+{b}^{10}Bd \right ){x}^{12}}{12}}+{\frac{ \left ( \left ( 10\,a{b}^{9}A+45\,{a}^{2}{b}^{8}B \right ) e+ \left ({b}^{10}A+10\,a{b}^{9}B \right ) d \right ){x}^{11}}{11}}+{\frac{ \left ( \left ( 45\,{a}^{2}{b}^{8}A+120\,{a}^{3}{b}^{7}B \right ) e+ \left ( 10\,a{b}^{9}A+45\,{a}^{2}{b}^{8}B \right ) d \right ){x}^{10}}{10}}+{\frac{ \left ( \left ( 120\,{a}^{3}{b}^{7}A+210\,{a}^{4}{b}^{6}B \right ) e+ \left ( 45\,{a}^{2}{b}^{8}A+120\,{a}^{3}{b}^{7}B \right ) d \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 210\,{a}^{4}{b}^{6}A+252\,{a}^{5}{b}^{5}B \right ) e+ \left ( 120\,{a}^{3}{b}^{7}A+210\,{a}^{4}{b}^{6}B \right ) d \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 252\,{a}^{5}{b}^{5}A+210\,{a}^{6}{b}^{4}B \right ) e+ \left ( 210\,{a}^{4}{b}^{6}A+252\,{a}^{5}{b}^{5}B \right ) d \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 210\,{a}^{6}{b}^{4}A+120\,{a}^{7}{b}^{3}B \right ) e+ \left ( 252\,{a}^{5}{b}^{5}A+210\,{a}^{6}{b}^{4}B \right ) d \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 120\,{a}^{7}{b}^{3}A+45\,{a}^{8}{b}^{2}B \right ) e+ \left ( 210\,{a}^{6}{b}^{4}A+120\,{a}^{7}{b}^{3}B \right ) d \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 45\,{a}^{8}{b}^{2}A+10\,{a}^{9}bB \right ) e+ \left ( 120\,{a}^{7}{b}^{3}A+45\,{a}^{8}{b}^{2}B \right ) d \right ){x}^{4}}{4}}+{\frac{ \left ( \left ( 10\,{a}^{9}bA+{a}^{10}B \right ) e+ \left ( 45\,{a}^{8}{b}^{2}A+10\,{a}^{9}bB \right ) d \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{10}Ae+ \left ( 10\,{a}^{9}bA+{a}^{10}B \right ) d \right ){x}^{2}}{2}}+{a}^{10}Adx \]

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 [A]  time = 1.35677, size = 666, normalized size = 8.88 \[ \frac{1}{13} \, B b^{10} e x^{13} + A a^{10} d x + \frac{1}{12} \,{\left (B b^{10} d +{\left (10 \, B a b^{9} + A b^{10}\right )} e\right )} x^{12} + \frac{1}{11} \,{\left ({\left (10 \, B a b^{9} + A b^{10}\right )} d + 5 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} e\right )} x^{11} + \frac{1}{2} \,{\left ({\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} d + 3 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} e\right )} x^{10} + \frac{5}{3} \,{\left ({\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} d + 2 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} e\right )} x^{9} + \frac{3}{4} \,{\left (5 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} d + 7 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} e\right )} x^{8} + 6 \,{\left ({\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} d +{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} e\right )} x^{7} +{\left (7 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} d + 5 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} e\right )} x^{6} + 3 \,{\left (2 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} d +{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} e\right )} x^{5} + \frac{5}{4} \,{\left (3 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} d +{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} e\right )} x^{4} + \frac{1}{3} \,{\left (5 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} d +{\left (B a^{10} + 10 \, A a^{9} b\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (A a^{10} e +{\left (B a^{10} + 10 \, A a^{9} b\right )} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*(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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Fricas [A]  time = 0.19203, size = 1, normalized size = 0.01 \[ \frac{1}{13} x^{13} e b^{10} B + \frac{1}{12} x^{12} d b^{10} B + \frac{5}{6} x^{12} e b^{9} a B + \frac{1}{12} x^{12} e b^{10} A + \frac{10}{11} x^{11} d b^{9} a B + \frac{45}{11} x^{11} e b^{8} a^{2} B + \frac{1}{11} x^{11} d b^{10} A + \frac{10}{11} x^{11} e b^{9} a A + \frac{9}{2} x^{10} d b^{8} a^{2} B + 12 x^{10} e b^{7} a^{3} B + x^{10} d b^{9} a A + \frac{9}{2} x^{10} e b^{8} a^{2} A + \frac{40}{3} x^{9} d b^{7} a^{3} B + \frac{70}{3} x^{9} e b^{6} a^{4} B + 5 x^{9} d b^{8} a^{2} A + \frac{40}{3} x^{9} e b^{7} a^{3} A + \frac{105}{4} x^{8} d b^{6} a^{4} B + \frac{63}{2} x^{8} e b^{5} a^{5} B + 15 x^{8} d b^{7} a^{3} A + \frac{105}{4} x^{8} e b^{6} a^{4} A + 36 x^{7} d b^{5} a^{5} B + 30 x^{7} e b^{4} a^{6} B + 30 x^{7} d b^{6} a^{4} A + 36 x^{7} e b^{5} a^{5} A + 35 x^{6} d b^{4} a^{6} B + 20 x^{6} e b^{3} a^{7} B + 42 x^{6} d b^{5} a^{5} A + 35 x^{6} e b^{4} a^{6} A + 24 x^{5} d b^{3} a^{7} B + 9 x^{5} e b^{2} a^{8} B + 42 x^{5} d b^{4} a^{6} A + 24 x^{5} e b^{3} a^{7} A + \frac{45}{4} x^{4} d b^{2} a^{8} B + \frac{5}{2} x^{4} e b a^{9} B + 30 x^{4} d b^{3} a^{7} A + \frac{45}{4} x^{4} e b^{2} a^{8} A + \frac{10}{3} x^{3} d b a^{9} B + \frac{1}{3} x^{3} e a^{10} B + 15 x^{3} d b^{2} a^{8} A + \frac{10}{3} x^{3} e b a^{9} A + \frac{1}{2} x^{2} d a^{10} B + 5 x^{2} d b a^{9} A + \frac{1}{2} x^{2} e a^{10} A + x d a^{10} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*(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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Sympy [A]  time = 0.34957, 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**1
0*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 + 4
0*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 + 2
0*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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GIAC/XCAS [A]  time = 0.212814, size = 744, normalized size = 9.92 \[ \frac{1}{13} \, B b^{10} x^{13} e + \frac{1}{12} \, B b^{10} d x^{12} + \frac{5}{6} \, B a b^{9} x^{12} e + \frac{1}{12} \, A b^{10} x^{12} e + \frac{10}{11} \, B a b^{9} d x^{11} + \frac{1}{11} \, A b^{10} d x^{11} + \frac{45}{11} \, B a^{2} b^{8} x^{11} e + \frac{10}{11} \, A a b^{9} x^{11} e + \frac{9}{2} \, B a^{2} b^{8} d x^{10} + A a b^{9} d x^{10} + 12 \, B a^{3} b^{7} x^{10} e + \frac{9}{2} \, A a^{2} b^{8} x^{10} e + \frac{40}{3} \, B a^{3} b^{7} d x^{9} + 5 \, A a^{2} b^{8} d x^{9} + \frac{70}{3} \, B a^{4} b^{6} x^{9} e + \frac{40}{3} \, A a^{3} b^{7} x^{9} e + \frac{105}{4} \, B a^{4} b^{6} d x^{8} + 15 \, A a^{3} b^{7} d x^{8} + \frac{63}{2} \, B a^{5} b^{5} x^{8} e + \frac{105}{4} \, A a^{4} b^{6} x^{8} e + 36 \, B a^{5} b^{5} d x^{7} + 30 \, A a^{4} b^{6} d x^{7} + 30 \, B a^{6} b^{4} x^{7} e + 36 \, A a^{5} b^{5} x^{7} e + 35 \, B a^{6} b^{4} d x^{6} + 42 \, A a^{5} b^{5} d x^{6} + 20 \, B a^{7} b^{3} x^{6} e + 35 \, A a^{6} b^{4} x^{6} e + 24 \, B a^{7} b^{3} d x^{5} + 42 \, A a^{6} b^{4} d x^{5} + 9 \, B a^{8} b^{2} x^{5} e + 24 \, A a^{7} b^{3} x^{5} e + \frac{45}{4} \, B a^{8} b^{2} d x^{4} + 30 \, A a^{7} b^{3} d x^{4} + \frac{5}{2} \, B a^{9} b x^{4} e + \frac{45}{4} \, A a^{8} b^{2} x^{4} e + \frac{10}{3} \, B a^{9} b d x^{3} + 15 \, A a^{8} b^{2} d x^{3} + \frac{1}{3} \, B a^{10} x^{3} e + \frac{10}{3} \, A a^{9} b x^{3} e + \frac{1}{2} \, B a^{10} d x^{2} + 5 \, A a^{9} b d x^{2} + \frac{1}{2} \, A a^{10} x^{2} e + A a^{10} d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*(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 + 1/11*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