3.119 \(\int x^5 (a+b x)^5 (A+B x) \, dx\)

Optimal. Leaf size=117 \[ \frac{10}{9} a^2 b^2 x^9 (a B+A b)+\frac{5}{8} a^3 b x^8 (a B+2 A b)+\frac{1}{7} a^4 x^7 (a B+5 A b)+\frac{1}{6} a^5 A x^6+\frac{1}{11} b^4 x^{11} (5 a B+A b)+\frac{1}{2} a b^3 x^{10} (2 a B+A b)+\frac{1}{12} b^5 B x^{12} \]

[Out]

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

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Rubi [A]  time = 0.106752, antiderivative size = 117, 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} \[ \frac{10}{9} a^2 b^2 x^9 (a B+A b)+\frac{5}{8} a^3 b x^8 (a B+2 A b)+\frac{1}{7} a^4 x^7 (a B+5 A b)+\frac{1}{6} a^5 A x^6+\frac{1}{11} b^4 x^{11} (5 a B+A b)+\frac{1}{2} a b^3 x^{10} (2 a B+A b)+\frac{1}{12} b^5 B x^{12} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x)^5*(A + B*x),x]

[Out]

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

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^5 (a+b x)^5 (A+B x) \, dx &=\int \left (a^5 A x^5+a^4 (5 A b+a B) x^6+5 a^3 b (2 A b+a B) x^7+10 a^2 b^2 (A b+a B) x^8+5 a b^3 (A b+2 a B) x^9+b^4 (A b+5 a B) x^{10}+b^5 B x^{11}\right ) \, dx\\ &=\frac{1}{6} a^5 A x^6+\frac{1}{7} a^4 (5 A b+a B) x^7+\frac{5}{8} a^3 b (2 A b+a B) x^8+\frac{10}{9} a^2 b^2 (A b+a B) x^9+\frac{1}{2} a b^3 (A b+2 a B) x^{10}+\frac{1}{11} b^4 (A b+5 a B) x^{11}+\frac{1}{12} b^5 B x^{12}\\ \end{align*}

Mathematica [A]  time = 0.0173021, size = 117, normalized size = 1. \[ \frac{10}{9} a^2 b^2 x^9 (a B+A b)+\frac{5}{8} a^3 b x^8 (a B+2 A b)+\frac{1}{7} a^4 x^7 (a B+5 A b)+\frac{1}{6} a^5 A x^6+\frac{1}{11} b^4 x^{11} (5 a B+A b)+\frac{1}{2} a b^3 x^{10} (2 a B+A b)+\frac{1}{12} b^5 B x^{12} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x)^5*(A + B*x),x]

[Out]

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

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Maple [A]  time = 0.001, size = 124, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{12}}{12}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{11}}{11}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{10}}{10}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{9}}{9}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{8}}{8}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{7}}{7}}+{\frac{{a}^{5}A{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x+a)^5*(B*x+A),x)

[Out]

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

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Maxima [A]  time = 1.0148, size = 161, normalized size = 1.38 \begin{align*} \frac{1}{12} \, B b^{5} x^{12} + \frac{1}{6} \, A a^{5} x^{6} + \frac{1}{11} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{11} + \frac{1}{2} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{10} + \frac{10}{9} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac{5}{8} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{8} + \frac{1}{7} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^5*(B*x+A),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.89914, size = 298, normalized size = 2.55 \begin{align*} \frac{1}{12} x^{12} b^{5} B + \frac{5}{11} x^{11} b^{4} a B + \frac{1}{11} x^{11} b^{5} A + x^{10} b^{3} a^{2} B + \frac{1}{2} x^{10} b^{4} a A + \frac{10}{9} x^{9} b^{2} a^{3} B + \frac{10}{9} x^{9} b^{3} a^{2} A + \frac{5}{8} x^{8} b a^{4} B + \frac{5}{4} x^{8} b^{2} a^{3} A + \frac{1}{7} x^{7} a^{5} B + \frac{5}{7} x^{7} b a^{4} A + \frac{1}{6} x^{6} a^{5} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^5*(B*x+A),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.091394, size = 133, normalized size = 1.14 \begin{align*} \frac{A a^{5} x^{6}}{6} + \frac{B b^{5} x^{12}}{12} + x^{11} \left (\frac{A b^{5}}{11} + \frac{5 B a b^{4}}{11}\right ) + x^{10} \left (\frac{A a b^{4}}{2} + B a^{2} b^{3}\right ) + x^{9} \left (\frac{10 A a^{2} b^{3}}{9} + \frac{10 B a^{3} b^{2}}{9}\right ) + x^{8} \left (\frac{5 A a^{3} b^{2}}{4} + \frac{5 B a^{4} b}{8}\right ) + x^{7} \left (\frac{5 A a^{4} b}{7} + \frac{B a^{5}}{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(b*x+a)**5*(B*x+A),x)

[Out]

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

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Giac [A]  time = 1.20788, size = 167, normalized size = 1.43 \begin{align*} \frac{1}{12} \, B b^{5} x^{12} + \frac{5}{11} \, B a b^{4} x^{11} + \frac{1}{11} \, A b^{5} x^{11} + B a^{2} b^{3} x^{10} + \frac{1}{2} \, A a b^{4} x^{10} + \frac{10}{9} \, B a^{3} b^{2} x^{9} + \frac{10}{9} \, A a^{2} b^{3} x^{9} + \frac{5}{8} \, B a^{4} b x^{8} + \frac{5}{4} \, A a^{3} b^{2} x^{8} + \frac{1}{7} \, B a^{5} x^{7} + \frac{5}{7} \, A a^{4} b x^{7} + \frac{1}{6} \, A a^{5} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^5*(B*x+A),x, algorithm="giac")

[Out]

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