3.454 \(\int x^3 \sqrt{9-4 x^2} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{80} \left (9-4 x^2\right )^{5/2}-\frac{3}{16} \left (9-4 x^2\right )^{3/2} \]

[Out]

(-3*(9 - 4*x^2)^(3/2))/16 + (9 - 4*x^2)^(5/2)/80

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Rubi [A]  time = 0.0151302, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{1}{80} \left (9-4 x^2\right )^{5/2}-\frac{3}{16} \left (9-4 x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[9 - 4*x^2],x]

[Out]

(-3*(9 - 4*x^2)^(3/2))/16 + (9 - 4*x^2)^(5/2)/80

Rule 266

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

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^3 \sqrt{9-4 x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sqrt{9-4 x} x \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{9}{4} \sqrt{9-4 x}-\frac{1}{4} (9-4 x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac{3}{16} \left (9-4 x^2\right )^{3/2}+\frac{1}{80} \left (9-4 x^2\right )^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0085122, size = 22, normalized size = 0.71 \[ -\frac{1}{40} \left (9-4 x^2\right )^{3/2} \left (2 x^2+3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[9 - 4*x^2],x]

[Out]

-((9 - 4*x^2)^(3/2)*(3 + 2*x^2))/40

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Maple [A]  time = 0.003, size = 29, normalized size = 0.9 \begin{align*}{\frac{ \left ( -3+2\,x \right ) \left ( 3+2\,x \right ) \left ( 2\,{x}^{2}+3 \right ) }{40}\sqrt{-4\,{x}^{2}+9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(-4*x^2+9)^(1/2),x)

[Out]

1/40*(-3+2*x)*(3+2*x)*(2*x^2+3)*(-4*x^2+9)^(1/2)

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Maxima [A]  time = 3.10749, size = 35, normalized size = 1.13 \begin{align*} -\frac{1}{20} \,{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} x^{2} - \frac{3}{40} \,{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-4*x^2+9)^(1/2),x, algorithm="maxima")

[Out]

-1/20*(-4*x^2 + 9)^(3/2)*x^2 - 3/40*(-4*x^2 + 9)^(3/2)

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Fricas [A]  time = 1.49148, size = 59, normalized size = 1.9 \begin{align*} \frac{1}{40} \,{\left (8 \, x^{4} - 6 \, x^{2} - 27\right )} \sqrt{-4 \, x^{2} + 9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-4*x^2+9)^(1/2),x, algorithm="fricas")

[Out]

1/40*(8*x^4 - 6*x^2 - 27)*sqrt(-4*x^2 + 9)

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Sympy [A]  time = 0.591265, size = 44, normalized size = 1.42 \begin{align*} \frac{x^{4} \sqrt{9 - 4 x^{2}}}{5} - \frac{3 x^{2} \sqrt{9 - 4 x^{2}}}{20} - \frac{27 \sqrt{9 - 4 x^{2}}}{40} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(-4*x**2+9)**(1/2),x)

[Out]

x**4*sqrt(9 - 4*x**2)/5 - 3*x**2*sqrt(9 - 4*x**2)/20 - 27*sqrt(9 - 4*x**2)/40

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Giac [A]  time = 2.693, size = 43, normalized size = 1.39 \begin{align*} \frac{1}{80} \,{\left (4 \, x^{2} - 9\right )}^{2} \sqrt{-4 \, x^{2} + 9} - \frac{3}{16} \,{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-4*x^2+9)^(1/2),x, algorithm="giac")

[Out]

1/80*(4*x^2 - 9)^2*sqrt(-4*x^2 + 9) - 3/16*(-4*x^2 + 9)^(3/2)