3.539 \(\int x^5 (A+B x) (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=143 \[ \frac{1}{2} a^2 b^3 x^{10} (4 a B+3 A b)+\frac{5}{9} a^3 b^2 x^9 (3 a B+4 A b)+\frac{3}{8} a^4 b x^8 (2 a B+5 A b)+\frac{1}{7} a^5 x^7 (a B+6 A b)+\frac{1}{6} a^6 A x^6+\frac{1}{12} b^5 x^{12} (6 a B+A b)+\frac{3}{11} a b^4 x^{11} (5 a B+2 A b)+\frac{1}{13} b^6 B x^{13} \]

[Out]

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

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Rubi [A]  time = 0.141877, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 76} \[ \frac{1}{2} a^2 b^3 x^{10} (4 a B+3 A b)+\frac{5}{9} a^3 b^2 x^9 (3 a B+4 A b)+\frac{3}{8} a^4 b x^8 (2 a B+5 A b)+\frac{1}{7} a^5 x^7 (a B+6 A b)+\frac{1}{6} a^6 A x^6+\frac{1}{12} b^5 x^{12} (6 a B+A b)+\frac{3}{11} a b^4 x^{11} (5 a B+2 A b)+\frac{1}{13} b^6 B x^{13} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int x^5 (a+b x)^6 (A+B x) \, dx\\ &=\int \left (a^6 A x^5+a^5 (6 A b+a B) x^6+3 a^4 b (5 A b+2 a B) x^7+5 a^3 b^2 (4 A b+3 a B) x^8+5 a^2 b^3 (3 A b+4 a B) x^9+3 a b^4 (2 A b+5 a B) x^{10}+b^5 (A b+6 a B) x^{11}+b^6 B x^{12}\right ) \, dx\\ &=\frac{1}{6} a^6 A x^6+\frac{1}{7} a^5 (6 A b+a B) x^7+\frac{3}{8} a^4 b (5 A b+2 a B) x^8+\frac{5}{9} a^3 b^2 (4 A b+3 a B) x^9+\frac{1}{2} a^2 b^3 (3 A b+4 a B) x^{10}+\frac{3}{11} a b^4 (2 A b+5 a B) x^{11}+\frac{1}{12} b^5 (A b+6 a B) x^{12}+\frac{1}{13} b^6 B x^{13}\\ \end{align*}

Mathematica [A]  time = 0.0216507, size = 143, normalized size = 1. \[ \frac{1}{2} a^2 b^3 x^{10} (4 a B+3 A b)+\frac{5}{9} a^3 b^2 x^9 (3 a B+4 A b)+\frac{3}{8} a^4 b x^8 (2 a B+5 A b)+\frac{1}{7} a^5 x^7 (a B+6 A b)+\frac{1}{6} a^6 A x^6+\frac{1}{12} b^5 x^{12} (6 a B+A b)+\frac{3}{11} a b^4 x^{11} (5 a B+2 A b)+\frac{1}{13} b^6 B x^{13} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.001, size = 148, normalized size = 1. \begin{align*}{\frac{{b}^{6}B{x}^{13}}{13}}+{\frac{ \left ( A{b}^{6}+6\,Ba{b}^{5} \right ){x}^{12}}{12}}+{\frac{ \left ( 6\,Aa{b}^{5}+15\,B{a}^{2}{b}^{4} \right ){x}^{11}}{11}}+{\frac{ \left ( 15\,A{a}^{2}{b}^{4}+20\,B{a}^{3}{b}^{3} \right ){x}^{10}}{10}}+{\frac{ \left ( 20\,A{a}^{3}{b}^{3}+15\,B{a}^{4}{b}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( 15\,A{a}^{4}{b}^{2}+6\,B{a}^{5}b \right ){x}^{8}}{8}}+{\frac{ \left ( 6\,A{a}^{5}b+B{a}^{6} \right ){x}^{7}}{7}}+{\frac{{a}^{6}A{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/13*b^6*B*x^13+1/12*(A*b^6+6*B*a*b^5)*x^12+1/11*(6*A*a*b^5+15*B*a^2*b^4)*x^11+1/10*(15*A*a^2*b^4+20*B*a^3*b^3
)*x^10+1/9*(20*A*a^3*b^3+15*B*a^4*b^2)*x^9+1/8*(15*A*a^4*b^2+6*B*a^5*b)*x^8+1/7*(6*A*a^5*b+B*a^6)*x^7+1/6*a^6*
A*x^6

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Maxima [A]  time = 0.985397, size = 198, normalized size = 1.38 \begin{align*} \frac{1}{13} \, B b^{6} x^{13} + \frac{1}{6} \, A a^{6} x^{6} + \frac{1}{12} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{12} + \frac{3}{11} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{11} + \frac{1}{2} \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{10} + \frac{5}{9} \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{9} + \frac{3}{8} \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{8} + \frac{1}{7} \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.23932, size = 360, normalized size = 2.52 \begin{align*} \frac{1}{13} x^{13} b^{6} B + \frac{1}{2} x^{12} b^{5} a B + \frac{1}{12} x^{12} b^{6} A + \frac{15}{11} x^{11} b^{4} a^{2} B + \frac{6}{11} x^{11} b^{5} a A + 2 x^{10} b^{3} a^{3} B + \frac{3}{2} x^{10} b^{4} a^{2} A + \frac{5}{3} x^{9} b^{2} a^{4} B + \frac{20}{9} x^{9} b^{3} a^{3} A + \frac{3}{4} x^{8} b a^{5} B + \frac{15}{8} x^{8} b^{2} a^{4} A + \frac{1}{7} x^{7} a^{6} B + \frac{6}{7} x^{7} b a^{5} A + \frac{1}{6} x^{6} a^{6} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/13*x^13*b^6*B + 1/2*x^12*b^5*a*B + 1/12*x^12*b^6*A + 15/11*x^11*b^4*a^2*B + 6/11*x^11*b^5*a*A + 2*x^10*b^3*a
^3*B + 3/2*x^10*b^4*a^2*A + 5/3*x^9*b^2*a^4*B + 20/9*x^9*b^3*a^3*A + 3/4*x^8*b*a^5*B + 15/8*x^8*b^2*a^4*A + 1/
7*x^7*a^6*B + 6/7*x^7*b*a^5*A + 1/6*x^6*a^6*A

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Sympy [A]  time = 0.09753, size = 162, normalized size = 1.13 \begin{align*} \frac{A a^{6} x^{6}}{6} + \frac{B b^{6} x^{13}}{13} + x^{12} \left (\frac{A b^{6}}{12} + \frac{B a b^{5}}{2}\right ) + x^{11} \left (\frac{6 A a b^{5}}{11} + \frac{15 B a^{2} b^{4}}{11}\right ) + x^{10} \left (\frac{3 A a^{2} b^{4}}{2} + 2 B a^{3} b^{3}\right ) + x^{9} \left (\frac{20 A a^{3} b^{3}}{9} + \frac{5 B a^{4} b^{2}}{3}\right ) + x^{8} \left (\frac{15 A a^{4} b^{2}}{8} + \frac{3 B a^{5} b}{4}\right ) + x^{7} \left (\frac{6 A a^{5} b}{7} + \frac{B a^{6}}{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

A*a**6*x**6/6 + B*b**6*x**13/13 + x**12*(A*b**6/12 + B*a*b**5/2) + x**11*(6*A*a*b**5/11 + 15*B*a**2*b**4/11) +
x**10*(3*A*a**2*b**4/2 + 2*B*a**3*b**3) + x**9*(20*A*a**3*b**3/9 + 5*B*a**4*b**2/3) + x**8*(15*A*a**4*b**2/8
+ 3*B*a**5*b/4) + x**7*(6*A*a**5*b/7 + B*a**6/7)

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Giac [A]  time = 1.15645, size = 201, normalized size = 1.41 \begin{align*} \frac{1}{13} \, B b^{6} x^{13} + \frac{1}{2} \, B a b^{5} x^{12} + \frac{1}{12} \, A b^{6} x^{12} + \frac{15}{11} \, B a^{2} b^{4} x^{11} + \frac{6}{11} \, A a b^{5} x^{11} + 2 \, B a^{3} b^{3} x^{10} + \frac{3}{2} \, A a^{2} b^{4} x^{10} + \frac{5}{3} \, B a^{4} b^{2} x^{9} + \frac{20}{9} \, A a^{3} b^{3} x^{9} + \frac{3}{4} \, B a^{5} b x^{8} + \frac{15}{8} \, A a^{4} b^{2} x^{8} + \frac{1}{7} \, B a^{6} x^{7} + \frac{6}{7} \, A a^{5} b x^{7} + \frac{1}{6} \, A a^{6} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

1/13*B*b^6*x^13 + 1/2*B*a*b^5*x^12 + 1/12*A*b^6*x^12 + 15/11*B*a^2*b^4*x^11 + 6/11*A*a*b^5*x^11 + 2*B*a^3*b^3*
x^10 + 3/2*A*a^2*b^4*x^10 + 5/3*B*a^4*b^2*x^9 + 20/9*A*a^3*b^3*x^9 + 3/4*B*a^5*b*x^8 + 15/8*A*a^4*b^2*x^8 + 1/
7*B*a^6*x^7 + 6/7*A*a^5*b*x^7 + 1/6*A*a^6*x^6