3.461 \(\int x^2 (4-x^2)^2 \, dx\)

Optimal. Leaf size=22 \[ \frac{x^7}{7}-\frac{8 x^5}{5}+\frac{16 x^3}{3} \]

[Out]

(16*x^3)/3 - (8*x^5)/5 + x^7/7

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Rubi [A]  time = 0.0068336, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ \frac{x^7}{7}-\frac{8 x^5}{5}+\frac{16 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(4 - x^2)^2,x]

[Out]

(16*x^3)/3 - (8*x^5)/5 + x^7/7

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^2 \left (4-x^2\right )^2 \, dx &=\int \left (16 x^2-8 x^4+x^6\right ) \, dx\\ &=\frac{16 x^3}{3}-\frac{8 x^5}{5}+\frac{x^7}{7}\\ \end{align*}

Mathematica [A]  time = 0.0006652, size = 22, normalized size = 1. \[ \frac{x^7}{7}-\frac{8 x^5}{5}+\frac{16 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(4 - x^2)^2,x]

[Out]

(16*x^3)/3 - (8*x^5)/5 + x^7/7

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Maple [A]  time = 0., size = 17, normalized size = 0.8 \begin{align*}{\frac{16\,{x}^{3}}{3}}-{\frac{8\,{x}^{5}}{5}}+{\frac{{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(-x^2+4)^2,x)

[Out]

16/3*x^3-8/5*x^5+1/7*x^7

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Maxima [A]  time = 1.04056, size = 22, normalized size = 1. \begin{align*} \frac{1}{7} \, x^{7} - \frac{8}{5} \, x^{5} + \frac{16}{3} \, x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-x^2+4)^2,x, algorithm="maxima")

[Out]

1/7*x^7 - 8/5*x^5 + 16/3*x^3

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Fricas [A]  time = 1.1137, size = 41, normalized size = 1.86 \begin{align*} \frac{1}{7} x^{7} - \frac{8}{5} x^{5} + \frac{16}{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-x^2+4)^2,x, algorithm="fricas")

[Out]

1/7*x^7 - 8/5*x^5 + 16/3*x^3

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Sympy [A]  time = 0.052004, size = 17, normalized size = 0.77 \begin{align*} \frac{x^{7}}{7} - \frac{8 x^{5}}{5} + \frac{16 x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(-x**2+4)**2,x)

[Out]

x**7/7 - 8*x**5/5 + 16*x**3/3

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Giac [A]  time = 1.11976, size = 22, normalized size = 1. \begin{align*} \frac{1}{7} \, x^{7} - \frac{8}{5} \, x^{5} + \frac{16}{3} \, x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-x^2+4)^2,x, algorithm="giac")

[Out]

1/7*x^7 - 8/5*x^5 + 16/3*x^3