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

Optimal. Leaf size=61 \[ \frac{(a+b x)^{12} (A b-2 a B)}{12 b^3}-\frac{a (a+b x)^{11} (A b-a B)}{11 b^3}+\frac{B (a+b x)^{13}}{13 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0678905, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {76} \[ \frac{(a+b x)^{12} (A b-2 a B)}{12 b^3}-\frac{a (a+b x)^{11} (A b-a B)}{11 b^3}+\frac{B (a+b x)^{13}}{13 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^10*(A + B*x),x]

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

Mathematica [B]  time = 0.0348225, size = 218, normalized size = 3.57 \[ \frac{9}{22} a^2 b^8 x^{10} (11 A+10 B x)+\frac{4}{3} a^3 b^7 x^9 (10 A+9 B x)+\frac{35}{12} a^4 b^6 x^8 (9 A+8 B x)+\frac{9}{2} a^5 b^5 x^7 (8 A+7 B x)+5 a^6 b^4 x^6 (7 A+6 B x)+4 a^7 b^3 x^5 (6 A+5 B x)+\frac{9}{4} a^8 b^2 x^4 (5 A+4 B x)+\frac{5}{6} a^9 b x^3 (4 A+3 B x)+\frac{1}{6} a^{10} x^2 (3 A+2 B x)+\frac{5}{66} a b^9 x^{11} (12 A+11 B x)+\frac{1}{156} b^{10} x^{12} (13 A+12 B x) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^10*(A + B*x),x]

[Out]

(a^10*x^2*(3*A + 2*B*x))/6 + (5*a^9*b*x^3*(4*A + 3*B*x))/6 + (9*a^8*b^2*x^4*(5*A + 4*B*x))/4 + 4*a^7*b^3*x^5*(
6*A + 5*B*x) + 5*a^6*b^4*x^6*(7*A + 6*B*x) + (9*a^5*b^5*x^7*(8*A + 7*B*x))/2 + (35*a^4*b^6*x^8*(9*A + 8*B*x))/
12 + (4*a^3*b^7*x^9*(10*A + 9*B*x))/3 + (9*a^2*b^8*x^10*(11*A + 10*B*x))/22 + (5*a*b^9*x^11*(12*A + 11*B*x))/6
6 + (b^10*x^12*(13*A + 12*B*x))/156

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Maple [B]  time = 0.002, size = 244, normalized size = 4. \begin{align*}{\frac{{b}^{10}B{x}^{13}}{13}}+{\frac{ \left ({b}^{10}A+10\,a{b}^{9}B \right ){x}^{12}}{12}}+{\frac{ \left ( 10\,a{b}^{9}A+45\,{a}^{2}{b}^{8}B \right ){x}^{11}}{11}}+{\frac{ \left ( 45\,{a}^{2}{b}^{8}A+120\,{a}^{3}{b}^{7}B \right ){x}^{10}}{10}}+{\frac{ \left ( 120\,{a}^{3}{b}^{7}A+210\,{a}^{4}{b}^{6}B \right ){x}^{9}}{9}}+{\frac{ \left ( 210\,{a}^{4}{b}^{6}A+252\,{a}^{5}{b}^{5}B \right ){x}^{8}}{8}}+{\frac{ \left ( 252\,{a}^{5}{b}^{5}A+210\,{a}^{6}{b}^{4}B \right ){x}^{7}}{7}}+{\frac{ \left ( 210\,{a}^{6}{b}^{4}A+120\,{a}^{7}{b}^{3}B \right ){x}^{6}}{6}}+{\frac{ \left ( 120\,{a}^{7}{b}^{3}A+45\,{a}^{8}{b}^{2}B \right ){x}^{5}}{5}}+{\frac{ \left ( 45\,{a}^{8}{b}^{2}A+10\,{a}^{9}bB \right ){x}^{4}}{4}}+{\frac{ \left ( 10\,{a}^{9}bA+{a}^{10}B \right ){x}^{3}}{3}}+{\frac{{a}^{10}A{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.02063, size = 328, normalized size = 5.38 \begin{align*} \frac{1}{13} \, B b^{10} x^{13} + \frac{1}{2} \, A a^{10} x^{2} + \frac{1}{12} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{12} + \frac{5}{11} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{11} + \frac{3}{2} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{10} + \frac{10}{3} \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{9} + \frac{21}{4} \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{8} + 6 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{7} + 5 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{6} + 3 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{5} + \frac{5}{4} \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*B*b^10*x^13 + 1/2*A*a^10*x^2 + 1/12*(10*B*a*b^9 + A*b^10)*x^12 + 5/11*(9*B*a^2*b^8 + 2*A*a*b^9)*x^11 + 3/
2*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^10 + 10/3*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^9 + 21/4*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x
^8 + 6*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^7 + 5*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^6 + 3*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^5
+ 5/4*(2*B*a^9*b + 9*A*a^8*b^2)*x^4 + 1/3*(B*a^10 + 10*A*a^9*b)*x^3

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Fricas [B]  time = 1.27759, size = 582, normalized size = 9.54 \begin{align*} \frac{1}{13} x^{13} b^{10} B + \frac{5}{6} x^{12} b^{9} a B + \frac{1}{12} x^{12} b^{10} A + \frac{45}{11} x^{11} b^{8} a^{2} B + \frac{10}{11} x^{11} b^{9} a A + 12 x^{10} b^{7} a^{3} B + \frac{9}{2} x^{10} b^{8} a^{2} A + \frac{70}{3} x^{9} b^{6} a^{4} B + \frac{40}{3} x^{9} b^{7} a^{3} A + \frac{63}{2} x^{8} b^{5} a^{5} B + \frac{105}{4} x^{8} b^{6} a^{4} A + 30 x^{7} b^{4} a^{6} B + 36 x^{7} b^{5} a^{5} A + 20 x^{6} b^{3} a^{7} B + 35 x^{6} b^{4} a^{6} A + 9 x^{5} b^{2} a^{8} B + 24 x^{5} b^{3} a^{7} A + \frac{5}{2} x^{4} b a^{9} B + \frac{45}{4} x^{4} b^{2} a^{8} A + \frac{1}{3} x^{3} a^{10} B + \frac{10}{3} x^{3} b a^{9} A + \frac{1}{2} x^{2} a^{10} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.141144, size = 262, normalized size = 4.3 \begin{align*} \frac{A a^{10} x^{2}}{2} + \frac{B b^{10} x^{13}}{13} + x^{12} \left (\frac{A b^{10}}{12} + \frac{5 B a b^{9}}{6}\right ) + x^{11} \left (\frac{10 A a b^{9}}{11} + \frac{45 B a^{2} b^{8}}{11}\right ) + x^{10} \left (\frac{9 A a^{2} b^{8}}{2} + 12 B a^{3} b^{7}\right ) + x^{9} \left (\frac{40 A a^{3} b^{7}}{3} + \frac{70 B a^{4} b^{6}}{3}\right ) + x^{8} \left (\frac{105 A a^{4} b^{6}}{4} + \frac{63 B a^{5} b^{5}}{2}\right ) + x^{7} \left (36 A a^{5} b^{5} + 30 B a^{6} b^{4}\right ) + x^{6} \left (35 A a^{6} b^{4} + 20 B a^{7} b^{3}\right ) + x^{5} \left (24 A a^{7} b^{3} + 9 B a^{8} b^{2}\right ) + x^{4} \left (\frac{45 A a^{8} b^{2}}{4} + \frac{5 B a^{9} b}{2}\right ) + x^{3} \left (\frac{10 A a^{9} b}{3} + \frac{B a^{10}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.21864, size = 331, normalized size = 5.43 \begin{align*} \frac{1}{13} \, B b^{10} x^{13} + \frac{5}{6} \, B a b^{9} x^{12} + \frac{1}{12} \, A b^{10} x^{12} + \frac{45}{11} \, B a^{2} b^{8} x^{11} + \frac{10}{11} \, A a b^{9} x^{11} + 12 \, B a^{3} b^{7} x^{10} + \frac{9}{2} \, A a^{2} b^{8} x^{10} + \frac{70}{3} \, B a^{4} b^{6} x^{9} + \frac{40}{3} \, A a^{3} b^{7} x^{9} + \frac{63}{2} \, B a^{5} b^{5} x^{8} + \frac{105}{4} \, A a^{4} b^{6} x^{8} + 30 \, B a^{6} b^{4} x^{7} + 36 \, A a^{5} b^{5} x^{7} + 20 \, B a^{7} b^{3} x^{6} + 35 \, A a^{6} b^{4} x^{6} + 9 \, B a^{8} b^{2} x^{5} + 24 \, A a^{7} b^{3} x^{5} + \frac{5}{2} \, B a^{9} b x^{4} + \frac{45}{4} \, A a^{8} b^{2} x^{4} + \frac{1}{3} \, B a^{10} x^{3} + \frac{10}{3} \, A a^{9} b x^{3} + \frac{1}{2} \, A a^{10} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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