3.544 \(\int \frac{x^4}{\sqrt{9-4 x^2}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{1}{16} \sqrt{9-4 x^2} x^3-\frac{27}{128} \sqrt{9-4 x^2} x+\frac{243}{256} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]

[Out]

(-27*x*Sqrt[9 - 4*x^2])/128 - (x^3*Sqrt[9 - 4*x^2])/16 + (243*ArcSin[(2*x)/3])/256

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Rubi [A]  time = 0.0096936, antiderivative size = 45, 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 = {321, 216} \[ -\frac{1}{16} \sqrt{9-4 x^2} x^3-\frac{27}{128} \sqrt{9-4 x^2} x+\frac{243}{256} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[9 - 4*x^2],x]

[Out]

(-27*x*Sqrt[9 - 4*x^2])/128 - (x^3*Sqrt[9 - 4*x^2])/16 + (243*ArcSin[(2*x)/3])/256

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{x^4}{\sqrt{9-4 x^2}} \, dx &=-\frac{1}{16} x^3 \sqrt{9-4 x^2}+\frac{27}{16} \int \frac{x^2}{\sqrt{9-4 x^2}} \, dx\\ &=-\frac{27}{128} x \sqrt{9-4 x^2}-\frac{1}{16} x^3 \sqrt{9-4 x^2}+\frac{243}{128} \int \frac{1}{\sqrt{9-4 x^2}} \, dx\\ &=-\frac{27}{128} x \sqrt{9-4 x^2}-\frac{1}{16} x^3 \sqrt{9-4 x^2}+\frac{243}{256} \sin ^{-1}\left (\frac{2 x}{3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0115058, size = 34, normalized size = 0.76 \[ \frac{1}{256} \left (243 \sin ^{-1}\left (\frac{2 x}{3}\right )-2 x \sqrt{9-4 x^2} \left (8 x^2+27\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/Sqrt[9 - 4*x^2],x]

[Out]

(-2*x*Sqrt[9 - 4*x^2]*(27 + 8*x^2) + 243*ArcSin[(2*x)/3])/256

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Maple [A]  time = 0.004, size = 34, normalized size = 0.8 \begin{align*}{\frac{243}{256}\arcsin \left ({\frac{2\,x}{3}} \right ) }-{\frac{27\,x}{128}\sqrt{-4\,{x}^{2}+9}}-{\frac{{x}^{3}}{16}\sqrt{-4\,{x}^{2}+9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243/256*arcsin(2/3*x)-27/128*x*(-4*x^2+9)^(1/2)-1/16*x^3*(-4*x^2+9)^(1/2)

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Maxima [A]  time = 3.61109, size = 45, normalized size = 1. \begin{align*} -\frac{1}{16} \, \sqrt{-4 \, x^{2} + 9} x^{3} - \frac{27}{128} \, \sqrt{-4 \, x^{2} + 9} x + \frac{243}{256} \, \arcsin \left (\frac{2}{3} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*sqrt(-4*x^2 + 9)*x^3 - 27/128*sqrt(-4*x^2 + 9)*x + 243/256*arcsin(2/3*x)

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Fricas [A]  time = 1.28691, size = 117, normalized size = 2.6 \begin{align*} -\frac{1}{128} \,{\left (8 \, x^{3} + 27 \, x\right )} \sqrt{-4 \, x^{2} + 9} - \frac{243}{128} \, \arctan \left (\frac{\sqrt{-4 \, x^{2} + 9} - 3}{2 \, x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128*(8*x^3 + 27*x)*sqrt(-4*x^2 + 9) - 243/128*arctan(1/2*(sqrt(-4*x^2 + 9) - 3)/x)

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Sympy [A]  time = 0.670848, size = 39, normalized size = 0.87 \begin{align*} - \frac{x^{3} \sqrt{9 - 4 x^{2}}}{16} - \frac{27 x \sqrt{9 - 4 x^{2}}}{128} + \frac{243 \operatorname{asin}{\left (\frac{2 x}{3} \right )}}{256} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**3*sqrt(9 - 4*x**2)/16 - 27*x*sqrt(9 - 4*x**2)/128 + 243*asin(2*x/3)/256

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Giac [A]  time = 2.51963, size = 35, normalized size = 0.78 \begin{align*} -\frac{1}{128} \,{\left (8 \, x^{2} + 27\right )} \sqrt{-4 \, x^{2} + 9} x + \frac{243}{256} \, \arcsin \left (\frac{2}{3} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128*(8*x^2 + 27)*sqrt(-4*x^2 + 9)*x + 243/256*arcsin(2/3*x)