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

Optimal. Leaf size=33 \[ \frac{1}{6} x^6 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{9} b B x^9 \]

[Out]

(a*A*x^3)/3 + ((A*b + a*B)*x^6)/6 + (b*B*x^9)/9

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Rubi [A]  time = 0.0325437, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {444, 43} \[ \frac{1}{6} x^6 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{9} b B x^9 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)*(A + B*x^3),x]

[Out]

(a*A*x^3)/3 + ((A*b + a*B)*x^6)/6 + (b*B*x^9)/9

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0084485, size = 33, normalized size = 1. \[ \frac{1}{6} x^6 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{9} b B x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*A*x^3)/3 + ((A*b + a*B)*x^6)/6 + (b*B*x^9)/9

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Maple [A]  time = 0.004, size = 28, normalized size = 0.9 \begin{align*}{\frac{aA{x}^{3}}{3}}+{\frac{ \left ( Ab+Ba \right ){x}^{6}}{6}}+{\frac{bB{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*A*x^3+1/6*(A*b+B*a)*x^6+1/9*b*B*x^9

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Maxima [A]  time = 1.21202, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{9} \, B b x^{9} + \frac{1}{6} \,{\left (B a + A b\right )} x^{6} + \frac{1}{3} \, A a x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*b*x^9 + 1/6*(B*a + A*b)*x^6 + 1/3*A*a*x^3

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Fricas [A]  time = 1.22531, size = 74, normalized size = 2.24 \begin{align*} \frac{1}{9} x^{9} b B + \frac{1}{6} x^{6} a B + \frac{1}{6} x^{6} b A + \frac{1}{3} x^{3} a A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*b*B + 1/6*x^6*a*B + 1/6*x^6*b*A + 1/3*x^3*a*A

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Sympy [A]  time = 0.059812, size = 29, normalized size = 0.88 \begin{align*} \frac{A a x^{3}}{3} + \frac{B b x^{9}}{9} + x^{6} \left (\frac{A b}{6} + \frac{B a}{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a*x**3/3 + B*b*x**9/9 + x**6*(A*b/6 + B*a/6)

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Giac [A]  time = 1.18944, size = 39, normalized size = 1.18 \begin{align*} \frac{1}{9} \, B b x^{9} + \frac{1}{6} \, B a x^{6} + \frac{1}{6} \, A b x^{6} + \frac{1}{3} \, A a x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*b*x^9 + 1/6*B*a*x^6 + 1/6*A*b*x^6 + 1/3*A*a*x^3