3.220 \(\int x^2 (a+b x^3)^2 \, dx\)

Optimal. Leaf size=16 \[ \frac{\left (a+b x^3\right )^3}{9 b} \]

[Out]

(a + b*x^3)^3/(9*b)

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Rubi [A]  time = 0.0026883, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ \frac{\left (a+b x^3\right )^3}{9 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)^2,x]

[Out]

(a + b*x^3)^3/(9*b)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0006582, size = 30, normalized size = 1.88 \[ \frac{a^2 x^3}{3}+\frac{1}{3} a b x^6+\frac{b^2 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^3)^2,x]

[Out]

(a^2*x^3)/3 + (a*b*x^6)/3 + (b^2*x^9)/9

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Maple [A]  time = 0., size = 25, normalized size = 1.6 \begin{align*}{\frac{{b}^{2}{x}^{9}}{9}}+{\frac{ab{x}^{6}}{3}}+{\frac{{x}^{3}{a}^{2}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^2*x^9+1/3*a*b*x^6+1/3*x^3*a^2

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Maxima [A]  time = 0.993242, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{3}}{9 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(b*x^3 + a)^3/b

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Fricas [A]  time = 1.48137, size = 55, normalized size = 3.44 \begin{align*} \frac{1}{9} x^{9} b^{2} + \frac{1}{3} x^{6} b a + \frac{1}{3} x^{3} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*b^2 + 1/3*x^6*b*a + 1/3*x^3*a^2

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Sympy [B]  time = 0.058779, size = 24, normalized size = 1.5 \begin{align*} \frac{a^{2} x^{3}}{3} + \frac{a b x^{6}}{3} + \frac{b^{2} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**3/3 + a*b*x**6/3 + b**2*x**9/9

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Giac [A]  time = 1.55848, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{3}}{9 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(b*x^3 + a)^3/b