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

Optimal. Leaf size=54 \[ -\frac{1}{16} \sqrt{-4 x^2-9} x^3+\frac{27}{128} \sqrt{-4 x^2-9} x+\frac{243}{256} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0129458, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {321, 217, 203} \[ -\frac{1}{16} \sqrt{-4 x^2-9} x^3+\frac{27}{128} \sqrt{-4 x^2-9} x+\frac{243}{256} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

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*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/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 217

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

Rule 203

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

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

Mathematica [A]  time = 0.0137112, size = 43, normalized size = 0.8 \[ \frac{1}{256} \left (2 x \sqrt{-4 x^2-9} \left (27-8 x^2\right )+243 \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/Sqrt[-9 - 4*x^2],x]

[Out]

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

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Maple [A]  time = 0.005, size = 43, normalized size = 0.8 \begin{align*}{\frac{243}{256}\arctan \left ( 2\,{\frac{x}{\sqrt{-4\,{x}^{2}-9}}} \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*arctan(2*x/(-4*x^2-9)^(1/2))+27/128*x*(-4*x^2-9)^(1/2)-1/16*x^3*(-4*x^2-9)^(1/2)

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Maxima [C]  time = 1.87877, size = 45, normalized size = 0.83 \begin{align*} -\frac{1}{16} \, \sqrt{-4 \, x^{2} - 9} x^{3} + \frac{27}{128} \, \sqrt{-4 \, x^{2} - 9} x - \frac{243}{256} i \, \operatorname{arsinh}\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*I*arcsinh(2/3*x)

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Fricas [C]  time = 1.36508, size = 186, normalized size = 3.44 \begin{align*} -\frac{1}{128} \,{\left (8 \, x^{3} - 27 \, x\right )} \sqrt{-4 \, x^{2} - 9} + \frac{243}{512} i \, \log \left (-\frac{8 \, x + 4 i \, \sqrt{-4 \, x^{2} - 9}}{x}\right ) - \frac{243}{512} i \, \log \left (-\frac{8 \, x - 4 i \, \sqrt{-4 \, x^{2} - 9}}{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) + 243/512*I*log(-(8*x + 4*I*sqrt(-4*x^2 - 9))/x) - 243/512*I*log(-(8*x
- 4*I*sqrt(-4*x^2 - 9))/x)

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Sympy [A]  time = 0.854609, size = 53, normalized size = 0.98 \begin{align*} - \frac{x^{3} \sqrt{- 4 x^{2} - 9}}{16} + \frac{27 x \sqrt{- 4 x^{2} - 9}}{128} + \frac{243 \operatorname{atan}{\left (\frac{2 x}{\sqrt{- 4 x^{2} - 9}} \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(-4*x**2 - 9)/16 + 27*x*sqrt(-4*x**2 - 9)/128 + 243*atan(2*x/sqrt(-4*x**2 - 9))/256

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Giac [C]  time = 2.25413, size = 35, normalized size = 0.65 \begin{align*} -\frac{1}{128} \,{\left (8 \, x^{2} - 27\right )} \sqrt{-4 \, x^{2} - 9} x - \frac{243}{256} i \, \arcsin \left (\frac{2}{3} i \, 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*I*arcsin(2/3*I*x)