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

Optimal. Leaf size=54 \[ \frac{1}{4} \sqrt{-4 x^2-9} x^3+\frac{9}{32} \sqrt{-4 x^2-9} x+\frac{81}{64} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

[Out]

(9*x*Sqrt[-9 - 4*x^2])/32 + (x^3*Sqrt[-9 - 4*x^2])/4 + (81*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/64

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Rubi [A]  time = 0.0136314, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {279, 321, 217, 203} \[ \frac{1}{4} \sqrt{-4 x^2-9} x^3+\frac{9}{32} \sqrt{-4 x^2-9} x+\frac{81}{64} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[-9 - 4*x^2],x]

[Out]

(9*x*Sqrt[-9 - 4*x^2])/32 + (x^3*Sqrt[-9 - 4*x^2])/4 + (81*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/64

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^2 \sqrt{-9-4 x^2} \, dx &=\frac{1}{4} x^3 \sqrt{-9-4 x^2}-\frac{9}{4} \int \frac{x^2}{\sqrt{-9-4 x^2}} \, dx\\ &=\frac{9}{32} x \sqrt{-9-4 x^2}+\frac{1}{4} x^3 \sqrt{-9-4 x^2}+\frac{81}{32} \int \frac{1}{\sqrt{-9-4 x^2}} \, dx\\ &=\frac{9}{32} x \sqrt{-9-4 x^2}+\frac{1}{4} x^3 \sqrt{-9-4 x^2}+\frac{81}{32} \operatorname{Subst}\left (\int \frac{1}{1+4 x^2} \, dx,x,\frac{x}{\sqrt{-9-4 x^2}}\right )\\ &=\frac{9}{32} x \sqrt{-9-4 x^2}+\frac{1}{4} x^3 \sqrt{-9-4 x^2}+\frac{81}{64} \tan ^{-1}\left (\frac{2 x}{\sqrt{-9-4 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0156697, size = 43, normalized size = 0.8 \[ \frac{1}{64} \left (2 x \sqrt{-4 x^2-9} \left (8 x^2+9\right )+81 \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[-9 - 4*x^2],x]

[Out]

(2*x*Sqrt[-9 - 4*x^2]*(9 + 8*x^2) + 81*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/64

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Maple [A]  time = 0.003, size = 41, normalized size = 0.8 \begin{align*} -{\frac{x}{16} \left ( -4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}-{\frac{9\,x}{32}\sqrt{-4\,{x}^{2}-9}}+{\frac{81}{64}\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^2*(-4*x^2-9)^(1/2),x)

[Out]

-1/16*x*(-4*x^2-9)^(3/2)-9/32*x*(-4*x^2-9)^(1/2)+81/64*arctan(2*x/(-4*x^2-9)^(1/2))

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Maxima [C]  time = 3.42131, size = 42, normalized size = 0.78 \begin{align*} -\frac{1}{16} \,{\left (-4 \, x^{2} - 9\right )}^{\frac{3}{2}} x - \frac{9}{32} \, \sqrt{-4 \, x^{2} - 9} x - \frac{81}{64} i \, \operatorname{arsinh}\left (\frac{2}{3} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(-4*x^2 - 9)^(3/2)*x - 9/32*sqrt(-4*x^2 - 9)*x - 81/64*I*arcsinh(2/3*x)

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Fricas [C]  time = 1.25942, size = 180, normalized size = 3.33 \begin{align*} \frac{1}{32} \,{\left (8 \, x^{3} + 9 \, x\right )} \sqrt{-4 \, x^{2} - 9} + \frac{81}{128} i \, \log \left (-\frac{8 \, x + 4 i \, \sqrt{-4 \, x^{2} - 9}}{x}\right ) - \frac{81}{128} 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^2*(-4*x^2-9)^(1/2),x, algorithm="fricas")

[Out]

1/32*(8*x^3 + 9*x)*sqrt(-4*x^2 - 9) + 81/128*I*log(-(8*x + 4*I*sqrt(-4*x^2 - 9))/x) - 81/128*I*log(-(8*x - 4*I
*sqrt(-4*x^2 - 9))/x)

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Sympy [C]  time = 2.70224, size = 61, normalized size = 1.13 \begin{align*} \frac{i x^{5}}{\sqrt{4 x^{2} + 9}} + \frac{27 i x^{3}}{8 \sqrt{4 x^{2} + 9}} + \frac{81 i x}{32 \sqrt{4 x^{2} + 9}} - \frac{81 i \operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{64} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*x**5/sqrt(4*x**2 + 9) + 27*I*x**3/(8*sqrt(4*x**2 + 9)) + 81*I*x/(32*sqrt(4*x**2 + 9)) - 81*I*asinh(2*x/3)/64

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Giac [C]  time = 1.54434, size = 35, normalized size = 0.65 \begin{align*} \frac{1}{32} \,{\left (8 \, x^{2} + 9\right )} \sqrt{-4 \, x^{2} - 9} x - \frac{81}{64} i \, \arcsin \left (\frac{2}{3} i \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(8*x^2 + 9)*sqrt(-4*x^2 - 9)*x - 81/64*I*arcsin(2/3*I*x)