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

Optimal. Leaf size=99 \[ \frac{1}{3} a^2 b x^6 (2 a B+3 A b)+\frac{1}{5} a^3 x^5 (a B+4 A b)+\frac{1}{4} a^4 A x^4+\frac{1}{8} b^3 x^8 (4 a B+A b)+\frac{2}{7} a b^2 x^7 (3 a B+2 A b)+\frac{1}{9} b^4 B x^9 \]

[Out]

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

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Rubi [A]  time = 0.0570259, antiderivative size = 99, 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} \[ \frac{1}{3} a^2 b x^6 (2 a B+3 A b)+\frac{1}{5} a^3 x^5 (a B+4 A b)+\frac{1}{4} a^4 A x^4+\frac{1}{8} b^3 x^8 (4 a B+A b)+\frac{2}{7} a b^2 x^7 (3 a B+2 A b)+\frac{1}{9} b^4 B x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0124008, size = 99, normalized size = 1. \[ \frac{1}{3} a^2 b x^6 (2 a B+3 A b)+\frac{1}{5} a^3 x^5 (a B+4 A b)+\frac{1}{4} a^4 A x^4+\frac{1}{8} b^3 x^8 (4 a B+A b)+\frac{2}{7} a b^2 x^7 (3 a B+2 A b)+\frac{1}{9} b^4 B x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 100, normalized size = 1. \begin{align*}{\frac{{b}^{4}B{x}^{9}}{9}}+{\frac{ \left ( A{b}^{4}+4\,Ba{b}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 4\,Aa{b}^{3}+6\,B{a}^{2}{b}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,A{a}^{2}{b}^{2}+4\,B{a}^{3}b \right ){x}^{6}}{6}}+{\frac{ \left ( 4\,A{a}^{3}b+B{a}^{4} \right ){x}^{5}}{5}}+{\frac{{a}^{4}A{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.960282, size = 134, normalized size = 1.35 \begin{align*} \frac{1}{9} \, B b^{4} x^{9} + \frac{1}{4} \, A a^{4} x^{4} + \frac{1}{8} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{8} + \frac{2}{7} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{7} + \frac{1}{3} \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{6} + \frac{1}{5} \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.17465, size = 231, normalized size = 2.33 \begin{align*} \frac{1}{9} x^{9} b^{4} B + \frac{1}{2} x^{8} b^{3} a B + \frac{1}{8} x^{8} b^{4} A + \frac{6}{7} x^{7} b^{2} a^{2} B + \frac{4}{7} x^{7} b^{3} a A + \frac{2}{3} x^{6} b a^{3} B + x^{6} b^{2} a^{2} A + \frac{1}{5} x^{5} a^{4} B + \frac{4}{5} x^{5} b a^{3} A + \frac{1}{4} x^{4} a^{4} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.086743, size = 105, normalized size = 1.06 \begin{align*} \frac{A a^{4} x^{4}}{4} + \frac{B b^{4} x^{9}}{9} + x^{8} \left (\frac{A b^{4}}{8} + \frac{B a b^{3}}{2}\right ) + x^{7} \left (\frac{4 A a b^{3}}{7} + \frac{6 B a^{2} b^{2}}{7}\right ) + x^{6} \left (A a^{2} b^{2} + \frac{2 B a^{3} b}{3}\right ) + x^{5} \left (\frac{4 A a^{3} b}{5} + \frac{B a^{4}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14638, size = 135, normalized size = 1.36 \begin{align*} \frac{1}{9} \, B b^{4} x^{9} + \frac{1}{2} \, B a b^{3} x^{8} + \frac{1}{8} \, A b^{4} x^{8} + \frac{6}{7} \, B a^{2} b^{2} x^{7} + \frac{4}{7} \, A a b^{3} x^{7} + \frac{2}{3} \, B a^{3} b x^{6} + A a^{2} b^{2} x^{6} + \frac{1}{5} \, B a^{4} x^{5} + \frac{4}{5} \, A a^{3} b x^{5} + \frac{1}{4} \, A a^{4} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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