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

Optimal. Leaf size=72 \[ \frac{1}{6} \sqrt{-4 x^2-9} x^5+\frac{3}{32} \sqrt{-4 x^2-9} x^3-\frac{81}{256} \sqrt{-4 x^2-9} x-\frac{729}{512} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

[Out]

(-81*x*Sqrt[-9 - 4*x^2])/256 + (3*x^3*Sqrt[-9 - 4*x^2])/32 + (x^5*Sqrt[-9 - 4*x^2])/6 - (729*ArcTan[(2*x)/Sqrt
[-9 - 4*x^2]])/512

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Rubi [A]  time = 0.0208599, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {279, 321, 217, 203} \[ \frac{1}{6} \sqrt{-4 x^2-9} x^5+\frac{3}{32} \sqrt{-4 x^2-9} x^3-\frac{81}{256} \sqrt{-4 x^2-9} x-\frac{729}{512} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^4*Sqrt[-9 - 4*x^2],x]

[Out]

(-81*x*Sqrt[-9 - 4*x^2])/256 + (3*x^3*Sqrt[-9 - 4*x^2])/32 + (x^5*Sqrt[-9 - 4*x^2])/6 - (729*ArcTan[(2*x)/Sqrt
[-9 - 4*x^2]])/512

Rule 279

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

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 x^4 \sqrt{-9-4 x^2} \, dx &=\frac{1}{6} x^5 \sqrt{-9-4 x^2}-\frac{3}{2} \int \frac{x^4}{\sqrt{-9-4 x^2}} \, dx\\ &=\frac{3}{32} x^3 \sqrt{-9-4 x^2}+\frac{1}{6} x^5 \sqrt{-9-4 x^2}+\frac{81}{32} \int \frac{x^2}{\sqrt{-9-4 x^2}} \, dx\\ &=-\frac{81}{256} x \sqrt{-9-4 x^2}+\frac{3}{32} x^3 \sqrt{-9-4 x^2}+\frac{1}{6} x^5 \sqrt{-9-4 x^2}-\frac{729}{256} \int \frac{1}{\sqrt{-9-4 x^2}} \, dx\\ &=-\frac{81}{256} x \sqrt{-9-4 x^2}+\frac{3}{32} x^3 \sqrt{-9-4 x^2}+\frac{1}{6} x^5 \sqrt{-9-4 x^2}-\frac{729}{256} \operatorname{Subst}\left (\int \frac{1}{1+4 x^2} \, dx,x,\frac{x}{\sqrt{-9-4 x^2}}\right )\\ &=-\frac{81}{256} x \sqrt{-9-4 x^2}+\frac{3}{32} x^3 \sqrt{-9-4 x^2}+\frac{1}{6} x^5 \sqrt{-9-4 x^2}-\frac{729}{512} \tan ^{-1}\left (\frac{2 x}{\sqrt{-9-4 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0154593, size = 48, normalized size = 0.67 \[ \frac{1}{768} x \sqrt{-4 x^2-9} \left (128 x^4+72 x^2-243\right )-\frac{729}{512} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*Sqrt[-9 - 4*x^2],x]

[Out]

(x*Sqrt[-9 - 4*x^2]*(-243 + 72*x^2 + 128*x^4))/768 - (729*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/512

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Maple [A]  time = 0.007, size = 55, normalized size = 0.8 \begin{align*} -{\frac{{x}^{3}}{24} \left ( -4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}+{\frac{9\,x}{128} \left ( -4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}+{\frac{81\,x}{256}\sqrt{-4\,{x}^{2}-9}}-{\frac{729}{512}\arctan \left ( 2\,{\frac{x}{\sqrt{-4\,{x}^{2}-9}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*x^3*(-4*x^2-9)^(3/2)+9/128*x*(-4*x^2-9)^(3/2)+81/256*x*(-4*x^2-9)^(1/2)-729/512*arctan(2*x/(-4*x^2-9)^(1
/2))

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Maxima [C]  time = 3.45487, size = 61, normalized size = 0.85 \begin{align*} -\frac{1}{24} \,{\left (-4 \, x^{2} - 9\right )}^{\frac{3}{2}} x^{3} + \frac{9}{128} \,{\left (-4 \, x^{2} - 9\right )}^{\frac{3}{2}} x + \frac{81}{256} \, \sqrt{-4 \, x^{2} - 9} x + \frac{729}{512} 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]

-1/24*(-4*x^2 - 9)^(3/2)*x^3 + 9/128*(-4*x^2 - 9)^(3/2)*x + 81/256*sqrt(-4*x^2 - 9)*x + 729/512*I*arcsinh(2/3*
x)

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Fricas [C]  time = 1.33584, size = 204, normalized size = 2.83 \begin{align*} \frac{1}{768} \,{\left (128 \, x^{5} + 72 \, x^{3} - 243 \, x\right )} \sqrt{-4 \, x^{2} - 9} - \frac{729}{1024} i \, \log \left (-\frac{8 \, x + 4 i \, \sqrt{-4 \, x^{2} - 9}}{x}\right ) + \frac{729}{1024} 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]

1/768*(128*x^5 + 72*x^3 - 243*x)*sqrt(-4*x^2 - 9) - 729/1024*I*log(-(8*x + 4*I*sqrt(-4*x^2 - 9))/x) + 729/1024
*I*log(-(8*x - 4*I*sqrt(-4*x^2 - 9))/x)

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Sympy [C]  time = 4.55362, size = 83, normalized size = 1.15 \begin{align*} \frac{2 i x^{7}}{3 \sqrt{4 x^{2} + 9}} + \frac{15 i x^{5}}{8 \sqrt{4 x^{2} + 9}} - \frac{27 i x^{3}}{64 \sqrt{4 x^{2} + 9}} - \frac{729 i x}{256 \sqrt{4 x^{2} + 9}} + \frac{729 i \operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{512} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*I*x**7/(3*sqrt(4*x**2 + 9)) + 15*I*x**5/(8*sqrt(4*x**2 + 9)) - 27*I*x**3/(64*sqrt(4*x**2 + 9)) - 729*I*x/(25
6*sqrt(4*x**2 + 9)) + 729*I*asinh(2*x/3)/512

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Giac [C]  time = 2.39498, size = 45, normalized size = 0.62 \begin{align*} \frac{1}{768} \,{\left (8 \,{\left (16 \, x^{2} + 9\right )} x^{2} - 243\right )} \sqrt{-4 \, x^{2} - 9} x + \frac{729}{512} 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]

1/768*(8*(16*x^2 + 9)*x^2 - 243)*sqrt(-4*x^2 - 9)*x + 729/512*I*arcsin(2/3*I*x)