3.475 \(\int \frac{x^2}{(3-x^2)^{3/2}} \, dx\)

Optimal. Leaf size=24 \[ \frac{x}{\sqrt{3-x^2}}-\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]

[Out]

x/Sqrt[3 - x^2] - ArcSin[x/Sqrt[3]]

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Rubi [A]  time = 0.0053872, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {288, 216} \[ \frac{x}{\sqrt{3-x^2}}-\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2/(3 - x^2)^(3/2),x]

[Out]

x/Sqrt[3 - x^2] - ArcSin[x/Sqrt[3]]

Rule 288

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*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 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^2}{\left (3-x^2\right )^{3/2}} \, dx &=\frac{x}{\sqrt{3-x^2}}-\int \frac{1}{\sqrt{3-x^2}} \, dx\\ &=\frac{x}{\sqrt{3-x^2}}-\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0156706, size = 24, normalized size = 1. \[ \frac{x}{\sqrt{3-x^2}}-\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(3 - x^2)^(3/2),x]

[Out]

x/Sqrt[3 - x^2] - ArcSin[x/Sqrt[3]]

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Maple [A]  time = 0.005, size = 22, normalized size = 0.9 \begin{align*} -\arcsin \left ({\frac{x\sqrt{3}}{3}} \right ) +{x{\frac{1}{\sqrt{-{x}^{2}+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(-x^2+3)^(3/2),x)

[Out]

-arcsin(1/3*x*3^(1/2))+x/(-x^2+3)^(1/2)

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Maxima [A]  time = 1.40437, size = 28, normalized size = 1.17 \begin{align*} \frac{x}{\sqrt{-x^{2} + 3}} - \arcsin \left (\frac{1}{3} \, \sqrt{3} x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-x^2+3)^(3/2),x, algorithm="maxima")

[Out]

x/sqrt(-x^2 + 3) - arcsin(1/3*sqrt(3)*x)

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Fricas [A]  time = 2.14949, size = 90, normalized size = 3.75 \begin{align*} \frac{{\left (x^{2} - 3\right )} \arctan \left (\frac{\sqrt{-x^{2} + 3}}{x}\right ) - \sqrt{-x^{2} + 3} x}{x^{2} - 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-x^2+3)^(3/2),x, algorithm="fricas")

[Out]

((x^2 - 3)*arctan(sqrt(-x^2 + 3)/x) - sqrt(-x^2 + 3)*x)/(x^2 - 3)

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Sympy [B]  time = 0.516362, size = 49, normalized size = 2.04 \begin{align*} - \frac{x^{2} \operatorname{asin}{\left (\frac{\sqrt{3} x}{3} \right )}}{x^{2} - 3} - \frac{x \sqrt{3 - x^{2}}}{x^{2} - 3} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{3} x}{3} \right )}}{x^{2} - 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(-x**2+3)**(3/2),x)

[Out]

-x**2*asin(sqrt(3)*x/3)/(x**2 - 3) - x*sqrt(3 - x**2)/(x**2 - 3) + 3*asin(sqrt(3)*x/3)/(x**2 - 3)

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Giac [A]  time = 1.08977, size = 39, normalized size = 1.62 \begin{align*} -\frac{\sqrt{-x^{2} + 3} x}{x^{2} - 3} - \arcsin \left (\frac{1}{3} \, \sqrt{3} x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-x^2+3)^(3/2),x, algorithm="giac")

[Out]

-sqrt(-x^2 + 3)*x/(x^2 - 3) - arcsin(1/3*sqrt(3)*x)