3.145 \(\int x^2 (a+b x)^{10} (A+B x) \, dx\)

Optimal. Leaf size=87 \[ \frac{a^2 (a+b x)^{11} (A b-a B)}{11 b^4}+\frac{(a+b x)^{13} (A b-3 a B)}{13 b^4}-\frac{a (a+b x)^{12} (2 A b-3 a B)}{12 b^4}+\frac{B (a+b x)^{14}}{14 b^4} \]

[Out]

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

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Rubi [A]  time = 0.0819227, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {76} \[ \frac{a^2 (a+b x)^{11} (A b-a B)}{11 b^4}+\frac{(a+b x)^{13} (A b-3 a B)}{13 b^4}-\frac{a (a+b x)^{12} (2 A b-3 a B)}{12 b^4}+\frac{B (a+b x)^{14}}{14 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^10*(A + B*x),x]

[Out]

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

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

Mathematica [B]  time = 0.027478, size = 226, normalized size = 2.6 \[ \frac{15}{11} a^2 b^7 x^{11} (8 a B+3 A b)+3 a^3 b^6 x^{10} (7 a B+4 A b)+\frac{14}{3} a^4 b^5 x^9 (6 a B+5 A b)+\frac{21}{4} a^5 b^4 x^8 (5 a B+6 A b)+\frac{30}{7} a^6 b^3 x^7 (4 a B+7 A b)+\frac{5}{2} a^7 b^2 x^6 (3 a B+8 A b)+a^8 b x^5 (2 a B+9 A b)+\frac{1}{4} a^9 x^4 (a B+10 A b)+\frac{1}{3} a^{10} A x^3+\frac{1}{13} b^9 x^{13} (10 a B+A b)+\frac{5}{12} a b^8 x^{12} (9 a B+2 A b)+\frac{1}{14} b^{10} B x^{14} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^10*(A + B*x),x]

[Out]

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

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Maple [B]  time = 0.001, size = 244, normalized size = 2.8 \begin{align*}{\frac{{b}^{10}B{x}^{14}}{14}}+{\frac{ \left ({b}^{10}A+10\,a{b}^{9}B \right ){x}^{13}}{13}}+{\frac{ \left ( 10\,a{b}^{9}A+45\,{a}^{2}{b}^{8}B \right ){x}^{12}}{12}}+{\frac{ \left ( 45\,{a}^{2}{b}^{8}A+120\,{a}^{3}{b}^{7}B \right ){x}^{11}}{11}}+{\frac{ \left ( 120\,{a}^{3}{b}^{7}A+210\,{a}^{4}{b}^{6}B \right ){x}^{10}}{10}}+{\frac{ \left ( 210\,{a}^{4}{b}^{6}A+252\,{a}^{5}{b}^{5}B \right ){x}^{9}}{9}}+{\frac{ \left ( 252\,{a}^{5}{b}^{5}A+210\,{a}^{6}{b}^{4}B \right ){x}^{8}}{8}}+{\frac{ \left ( 210\,{a}^{6}{b}^{4}A+120\,{a}^{7}{b}^{3}B \right ){x}^{7}}{7}}+{\frac{ \left ( 120\,{a}^{7}{b}^{3}A+45\,{a}^{8}{b}^{2}B \right ){x}^{6}}{6}}+{\frac{ \left ( 45\,{a}^{8}{b}^{2}A+10\,{a}^{9}bB \right ){x}^{5}}{5}}+{\frac{ \left ( 10\,{a}^{9}bA+{a}^{10}B \right ){x}^{4}}{4}}+{\frac{{a}^{10}A{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x+a)^10*(B*x+A),x)

[Out]

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

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Maxima [B]  time = 1.03316, size = 327, normalized size = 3.76 \begin{align*} \frac{1}{14} \, B b^{10} x^{14} + \frac{1}{3} \, A a^{10} x^{3} + \frac{1}{13} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{13} + \frac{5}{12} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{12} + \frac{15}{11} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{11} + 3 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{10} + \frac{14}{3} \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{9} + \frac{21}{4} \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{8} + \frac{30}{7} \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{7} + \frac{5}{2} \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{6} +{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^10*(B*x+A),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.30289, size = 589, normalized size = 6.77 \begin{align*} \frac{1}{14} x^{14} b^{10} B + \frac{10}{13} x^{13} b^{9} a B + \frac{1}{13} x^{13} b^{10} A + \frac{15}{4} x^{12} b^{8} a^{2} B + \frac{5}{6} x^{12} b^{9} a A + \frac{120}{11} x^{11} b^{7} a^{3} B + \frac{45}{11} x^{11} b^{8} a^{2} A + 21 x^{10} b^{6} a^{4} B + 12 x^{10} b^{7} a^{3} A + 28 x^{9} b^{5} a^{5} B + \frac{70}{3} x^{9} b^{6} a^{4} A + \frac{105}{4} x^{8} b^{4} a^{6} B + \frac{63}{2} x^{8} b^{5} a^{5} A + \frac{120}{7} x^{7} b^{3} a^{7} B + 30 x^{7} b^{4} a^{6} A + \frac{15}{2} x^{6} b^{2} a^{8} B + 20 x^{6} b^{3} a^{7} A + 2 x^{5} b a^{9} B + 9 x^{5} b^{2} a^{8} A + \frac{1}{4} x^{4} a^{10} B + \frac{5}{2} x^{4} b a^{9} A + \frac{1}{3} x^{3} a^{10} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^10*(B*x+A),x, algorithm="fricas")

[Out]

1/14*x^14*b^10*B + 10/13*x^13*b^9*a*B + 1/13*x^13*b^10*A + 15/4*x^12*b^8*a^2*B + 5/6*x^12*b^9*a*A + 120/11*x^1
1*b^7*a^3*B + 45/11*x^11*b^8*a^2*A + 21*x^10*b^6*a^4*B + 12*x^10*b^7*a^3*A + 28*x^9*b^5*a^5*B + 70/3*x^9*b^6*a
^4*A + 105/4*x^8*b^4*a^6*B + 63/2*x^8*b^5*a^5*A + 120/7*x^7*b^3*a^7*B + 30*x^7*b^4*a^6*A + 15/2*x^6*b^2*a^8*B
+ 20*x^6*b^3*a^7*A + 2*x^5*b*a^9*B + 9*x^5*b^2*a^8*A + 1/4*x^4*a^10*B + 5/2*x^4*b*a^9*A + 1/3*x^3*a^10*A

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Sympy [B]  time = 0.184275, size = 262, normalized size = 3.01 \begin{align*} \frac{A a^{10} x^{3}}{3} + \frac{B b^{10} x^{14}}{14} + x^{13} \left (\frac{A b^{10}}{13} + \frac{10 B a b^{9}}{13}\right ) + x^{12} \left (\frac{5 A a b^{9}}{6} + \frac{15 B a^{2} b^{8}}{4}\right ) + x^{11} \left (\frac{45 A a^{2} b^{8}}{11} + \frac{120 B a^{3} b^{7}}{11}\right ) + x^{10} \left (12 A a^{3} b^{7} + 21 B a^{4} b^{6}\right ) + x^{9} \left (\frac{70 A a^{4} b^{6}}{3} + 28 B a^{5} b^{5}\right ) + x^{8} \left (\frac{63 A a^{5} b^{5}}{2} + \frac{105 B a^{6} b^{4}}{4}\right ) + x^{7} \left (30 A a^{6} b^{4} + \frac{120 B a^{7} b^{3}}{7}\right ) + x^{6} \left (20 A a^{7} b^{3} + \frac{15 B a^{8} b^{2}}{2}\right ) + x^{5} \left (9 A a^{8} b^{2} + 2 B a^{9} b\right ) + x^{4} \left (\frac{5 A a^{9} b}{2} + \frac{B a^{10}}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x+a)**10*(B*x+A),x)

[Out]

A*a**10*x**3/3 + B*b**10*x**14/14 + x**13*(A*b**10/13 + 10*B*a*b**9/13) + x**12*(5*A*a*b**9/6 + 15*B*a**2*b**8
/4) + x**11*(45*A*a**2*b**8/11 + 120*B*a**3*b**7/11) + x**10*(12*A*a**3*b**7 + 21*B*a**4*b**6) + x**9*(70*A*a*
*4*b**6/3 + 28*B*a**5*b**5) + x**8*(63*A*a**5*b**5/2 + 105*B*a**6*b**4/4) + x**7*(30*A*a**6*b**4 + 120*B*a**7*
b**3/7) + x**6*(20*A*a**7*b**3 + 15*B*a**8*b**2/2) + x**5*(9*A*a**8*b**2 + 2*B*a**9*b) + x**4*(5*A*a**9*b/2 +
B*a**10/4)

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Giac [B]  time = 1.20931, size = 331, normalized size = 3.8 \begin{align*} \frac{1}{14} \, B b^{10} x^{14} + \frac{10}{13} \, B a b^{9} x^{13} + \frac{1}{13} \, A b^{10} x^{13} + \frac{15}{4} \, B a^{2} b^{8} x^{12} + \frac{5}{6} \, A a b^{9} x^{12} + \frac{120}{11} \, B a^{3} b^{7} x^{11} + \frac{45}{11} \, A a^{2} b^{8} x^{11} + 21 \, B a^{4} b^{6} x^{10} + 12 \, A a^{3} b^{7} x^{10} + 28 \, B a^{5} b^{5} x^{9} + \frac{70}{3} \, A a^{4} b^{6} x^{9} + \frac{105}{4} \, B a^{6} b^{4} x^{8} + \frac{63}{2} \, A a^{5} b^{5} x^{8} + \frac{120}{7} \, B a^{7} b^{3} x^{7} + 30 \, A a^{6} b^{4} x^{7} + \frac{15}{2} \, B a^{8} b^{2} x^{6} + 20 \, A a^{7} b^{3} x^{6} + 2 \, B a^{9} b x^{5} + 9 \, A a^{8} b^{2} x^{5} + \frac{1}{4} \, B a^{10} x^{4} + \frac{5}{2} \, A a^{9} b x^{4} + \frac{1}{3} \, A a^{10} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^10*(B*x+A),x, algorithm="giac")

[Out]

1/14*B*b^10*x^14 + 10/13*B*a*b^9*x^13 + 1/13*A*b^10*x^13 + 15/4*B*a^2*b^8*x^12 + 5/6*A*a*b^9*x^12 + 120/11*B*a
^3*b^7*x^11 + 45/11*A*a^2*b^8*x^11 + 21*B*a^4*b^6*x^10 + 12*A*a^3*b^7*x^10 + 28*B*a^5*b^5*x^9 + 70/3*A*a^4*b^6
*x^9 + 105/4*B*a^6*b^4*x^8 + 63/2*A*a^5*b^5*x^8 + 120/7*B*a^7*b^3*x^7 + 30*A*a^6*b^4*x^7 + 15/2*B*a^8*b^2*x^6
+ 20*A*a^7*b^3*x^6 + 2*B*a^9*b*x^5 + 9*A*a^8*b^2*x^5 + 1/4*B*a^10*x^4 + 5/2*A*a^9*b*x^4 + 1/3*A*a^10*x^3