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

Optimal. Leaf size=36 \[ -\frac{1}{8} \sqrt{-4 x^2-9} x-\frac{9}{16} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

[Out]

-(x*Sqrt[-9 - 4*x^2])/8 - (9*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/16

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Rubi [A]  time = 0.0072962, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {321, 217, 203} \[ -\frac{1}{8} \sqrt{-4 x^2-9} x-\frac{9}{16} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[-9 - 4*x^2],x]

[Out]

-(x*Sqrt[-9 - 4*x^2])/8 - (9*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/16

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^2}{\sqrt{-9-4 x^2}} \, dx &=-\frac{1}{8} x \sqrt{-9-4 x^2}-\frac{9}{8} \int \frac{1}{\sqrt{-9-4 x^2}} \, dx\\ &=-\frac{1}{8} x \sqrt{-9-4 x^2}-\frac{9}{8} \operatorname{Subst}\left (\int \frac{1}{1+4 x^2} \, dx,x,\frac{x}{\sqrt{-9-4 x^2}}\right )\\ &=-\frac{1}{8} x \sqrt{-9-4 x^2}-\frac{9}{16} \tan ^{-1}\left (\frac{2 x}{\sqrt{-9-4 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0059576, size = 36, normalized size = 1. \[ -\frac{1}{8} \sqrt{-4 x^2-9} x-\frac{9}{16} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[-9 - 4*x^2],x]

[Out]

-(x*Sqrt[-9 - 4*x^2])/8 - (9*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/16

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Maple [A]  time = 0.003, size = 29, normalized size = 0.8 \begin{align*} -{\frac{9}{16}\arctan \left ( 2\,{\frac{x}{\sqrt{-4\,{x}^{2}-9}}} \right ) }-{\frac{x}{8}\sqrt{-4\,{x}^{2}-9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/16*arctan(2*x/(-4*x^2-9)^(1/2))-1/8*x*(-4*x^2-9)^(1/2)

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Maxima [C]  time = 3.83237, size = 26, normalized size = 0.72 \begin{align*} -\frac{1}{8} \, \sqrt{-4 \, x^{2} - 9} x + \frac{9}{16} 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]

-1/8*sqrt(-4*x^2 - 9)*x + 9/16*I*arcsinh(2/3*x)

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Fricas [C]  time = 1.2637, size = 158, normalized size = 4.39 \begin{align*} -\frac{1}{8} \, \sqrt{-4 \, x^{2} - 9} x - \frac{9}{32} i \, \log \left (-\frac{8 \, x + 4 i \, \sqrt{-4 \, x^{2} - 9}}{x}\right ) + \frac{9}{32} 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]

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

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Sympy [A]  time = 0.392234, size = 36, normalized size = 1. \begin{align*} - \frac{x \sqrt{- 4 x^{2} - 9}}{8} - \frac{9 \operatorname{atan}{\left (\frac{2 x}{\sqrt{- 4 x^{2} - 9}} \right )}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*sqrt(-4*x**2 - 9)/8 - 9*atan(2*x/sqrt(-4*x**2 - 9))/16

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Giac [C]  time = 1.53594, size = 26, normalized size = 0.72 \begin{align*} -\frac{1}{8} \, \sqrt{-4 \, x^{2} - 9} x + \frac{9}{16} 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]

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