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

Optimal. Leaf size=61 \[ \frac{(a+b x)^8 (A b-2 a B)}{8 b^3}-\frac{a (a+b x)^7 (A b-a B)}{7 b^3}+\frac{B (a+b x)^9}{9 b^3} \]

[Out]

-(a*(A*b - a*B)*(a + b*x)^7)/(7*b^3) + ((A*b - 2*a*B)*(a + b*x)^8)/(8*b^3) + (B*(a + b*x)^9)/(9*b^3)

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Rubi [A]  time = 0.0496207, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {27, 76} \[ \frac{(a+b x)^8 (A b-2 a B)}{8 b^3}-\frac{a (a+b x)^7 (A b-a B)}{7 b^3}+\frac{B (a+b x)^9}{9 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(A*b - a*B)*(a + b*x)^7)/(7*b^3) + ((A*b - 2*a*B)*(a + b*x)^8)/(8*b^3) + (B*(a + b*x)^9)/(9*b^3)

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

Mathematica [B]  time = 0.0166159, size = 140, normalized size = 2.3 \[ \frac{5}{6} a^2 b^3 x^6 (4 a B+3 A b)+a^3 b^2 x^5 (3 a B+4 A b)+\frac{3}{4} a^4 b x^4 (2 a B+5 A b)+\frac{1}{3} a^5 x^3 (a B+6 A b)+\frac{1}{2} a^6 A x^2+\frac{1}{8} b^5 x^8 (6 a B+A b)+\frac{3}{7} a b^4 x^7 (5 a B+2 A b)+\frac{1}{9} b^6 B x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.001, size = 148, normalized size = 2.4 \begin{align*}{\frac{B{b}^{6}{x}^{9}}{9}}+{\frac{ \left ( A{b}^{6}+6\,Ba{b}^{5} \right ){x}^{8}}{8}}+{\frac{ \left ( 6\,Aa{b}^{5}+15\,B{a}^{2}{b}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ( 15\,A{a}^{2}{b}^{4}+20\,B{a}^{3}{b}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( 20\,A{a}^{3}{b}^{3}+15\,B{a}^{4}{b}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 15\,A{a}^{4}{b}^{2}+6\,B{a}^{5}b \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,A{a}^{5}b+B{a}^{6} \right ){x}^{3}}{3}}+{\frac{A{a}^{6}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.01394, size = 197, normalized size = 3.23 \begin{align*} \frac{1}{9} \, B b^{6} x^{9} + \frac{1}{2} \, A a^{6} x^{2} + \frac{1}{8} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{8} + \frac{3}{7} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{7} + \frac{5}{6} \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{6} +{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{5} + \frac{3}{4} \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.32399, size = 340, normalized size = 5.57 \begin{align*} \frac{1}{9} x^{9} b^{6} B + \frac{3}{4} x^{8} b^{5} a B + \frac{1}{8} x^{8} b^{6} A + \frac{15}{7} x^{7} b^{4} a^{2} B + \frac{6}{7} x^{7} b^{5} a A + \frac{10}{3} x^{6} b^{3} a^{3} B + \frac{5}{2} x^{6} b^{4} a^{2} A + 3 x^{5} b^{2} a^{4} B + 4 x^{5} b^{3} a^{3} A + \frac{3}{2} x^{4} b a^{5} B + \frac{15}{4} x^{4} b^{2} a^{4} A + \frac{1}{3} x^{3} a^{6} B + 2 x^{3} b a^{5} A + \frac{1}{2} x^{2} a^{6} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.095641, size = 160, normalized size = 2.62 \begin{align*} \frac{A a^{6} x^{2}}{2} + \frac{B b^{6} x^{9}}{9} + x^{8} \left (\frac{A b^{6}}{8} + \frac{3 B a b^{5}}{4}\right ) + x^{7} \left (\frac{6 A a b^{5}}{7} + \frac{15 B a^{2} b^{4}}{7}\right ) + x^{6} \left (\frac{5 A a^{2} b^{4}}{2} + \frac{10 B a^{3} b^{3}}{3}\right ) + x^{5} \left (4 A a^{3} b^{3} + 3 B a^{4} b^{2}\right ) + x^{4} \left (\frac{15 A a^{4} b^{2}}{4} + \frac{3 B a^{5} b}{2}\right ) + x^{3} \left (2 A a^{5} b + \frac{B a^{6}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.13215, size = 201, normalized size = 3.3 \begin{align*} \frac{1}{9} \, B b^{6} x^{9} + \frac{3}{4} \, B a b^{5} x^{8} + \frac{1}{8} \, A b^{6} x^{8} + \frac{15}{7} \, B a^{2} b^{4} x^{7} + \frac{6}{7} \, A a b^{5} x^{7} + \frac{10}{3} \, B a^{3} b^{3} x^{6} + \frac{5}{2} \, A a^{2} b^{4} x^{6} + 3 \, B a^{4} b^{2} x^{5} + 4 \, A a^{3} b^{3} x^{5} + \frac{3}{2} \, B a^{5} b x^{4} + \frac{15}{4} \, A a^{4} b^{2} x^{4} + \frac{1}{3} \, B a^{6} x^{3} + 2 \, A a^{5} b x^{3} + \frac{1}{2} \, A a^{6} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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