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

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

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 117, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, size = 117, normalized size = 1.00 \[ \frac {1}{6} a^5 A x^6+\frac {1}{7} a^4 x^7 (a B+5 A b)+\frac {5}{8} a^3 b x^8 (a B+2 A b)+\frac {10}{9} a^2 b^2 x^9 (a B+A b)+\frac {1}{11} b^4 x^{11} (5 a B+A b)+\frac {1}{2} a b^3 x^{10} (2 a B+A b)+\frac {1}{12} b^5 B x^{12} \]

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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fricas [A]  time = 0.42, size = 124, normalized size = 1.06 \[ \frac {1}{12} x^{12} b^{5} B + \frac {5}{11} x^{11} b^{4} a B + \frac {1}{11} x^{11} b^{5} A + x^{10} b^{3} a^{2} B + \frac {1}{2} x^{10} b^{4} a A + \frac {10}{9} x^{9} b^{2} a^{3} B + \frac {10}{9} x^{9} b^{3} a^{2} A + \frac {5}{8} x^{8} b a^{4} B + \frac {5}{4} x^{8} b^{2} a^{3} A + \frac {1}{7} x^{7} a^{5} B + \frac {5}{7} x^{7} b a^{4} A + \frac {1}{6} x^{6} a^{5} A \]

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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giac [A]  time = 1.18, size = 124, normalized size = 1.06 \[ \frac {1}{12} \, B b^{5} x^{12} + \frac {5}{11} \, B a b^{4} x^{11} + \frac {1}{11} \, A b^{5} x^{11} + B a^{2} b^{3} x^{10} + \frac {1}{2} \, A a b^{4} x^{10} + \frac {10}{9} \, B a^{3} b^{2} x^{9} + \frac {10}{9} \, A a^{2} b^{3} x^{9} + \frac {5}{8} \, B a^{4} b x^{8} + \frac {5}{4} \, A a^{3} b^{2} x^{8} + \frac {1}{7} \, B a^{5} x^{7} + \frac {5}{7} \, A a^{4} b x^{7} + \frac {1}{6} \, A a^{5} x^{6} \]

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

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maple [A]  time = 0.00, size = 124, normalized size = 1.06 \[ \frac {B \,b^{5} x^{12}}{12}+\frac {A \,a^{5} x^{6}}{6}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{11}}{11}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{10}}{10}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{9}}{9}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{8}}{8}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{7}}{7} \]

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 = 1.02, size = 119, normalized size = 1.02 \[ \frac {1}{12} \, B b^{5} x^{12} + \frac {1}{6} \, A a^{5} x^{6} + \frac {1}{11} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{10} + \frac {10}{9} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac {5}{8} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7} \]

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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mupad [B]  time = 0.32, size = 107, normalized size = 0.91 \[ x^7\,\left (\frac {B\,a^5}{7}+\frac {5\,A\,b\,a^4}{7}\right )+x^{11}\,\left (\frac {A\,b^5}{11}+\frac {5\,B\,a\,b^4}{11}\right )+\frac {A\,a^5\,x^6}{6}+\frac {B\,b^5\,x^{12}}{12}+\frac {10\,a^2\,b^2\,x^9\,\left (A\,b+B\,a\right )}{9}+\frac {5\,a^3\,b\,x^8\,\left (2\,A\,b+B\,a\right )}{8}+\frac {a\,b^3\,x^{10}\,\left (A\,b+2\,B\,a\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(A + B*x)*(a + b*x)^5,x)

[Out]

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

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sympy [A]  time = 0.18, size = 133, normalized size = 1.14 \[ \frac {A a^{5} x^{6}}{6} + \frac {B b^{5} x^{12}}{12} + x^{11} \left (\frac {A b^{5}}{11} + \frac {5 B a b^{4}}{11}\right ) + x^{10} \left (\frac {A a b^{4}}{2} + B a^{2} b^{3}\right ) + x^{9} \left (\frac {10 A a^{2} b^{3}}{9} + \frac {10 B a^{3} b^{2}}{9}\right ) + x^{8} \left (\frac {5 A a^{3} b^{2}}{4} + \frac {5 B a^{4} b}{8}\right ) + x^{7} \left (\frac {5 A a^{4} b}{7} + \frac {B a^{5}}{7}\right ) \]

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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