\(\int \frac {x^2}{(3-x^2)^{3/2}} \, dx\) [475]

Optimal result

Integrand size = 15, antiderivative size = 24 \[ \int \frac {x^2}{\left (3-x^2\right )^{3/2}} \, dx=\frac {x}{\sqrt {3-x^2}}-\arcsin \left (\frac {x}{\sqrt {3}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {294, 222} \[ \int \frac {x^2}{\left (3-x^2\right )^{3/2}} \, dx=\frac {x}{\sqrt {3-x^2}}-\arcsin \left (\frac {x}{\sqrt {3}}\right ) \]

[In]

Int[x^2/(3 - x^2)^(3/2),x]

[Out]

x/Sqrt[3 - x^2] - ArcSin[x/Sqrt[3]]

Rule 222

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]

Rule 294

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]

Rubi steps \begin{align*} \text {integral}& = \frac {x}{\sqrt {3-x^2}}-\int \frac {1}{\sqrt {3-x^2}} \, dx \\ & = \frac {x}{\sqrt {3-x^2}}-\arcsin \left (\frac {x}{\sqrt {3}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {x^2}{\left (3-x^2\right )^{3/2}} \, dx=\frac {x}{\sqrt {3-x^2}}+2 \arctan \left (\frac {x}{\sqrt {3}-\sqrt {3-x^2}}\right ) \]

[In]

Integrate[x^2/(3 - x^2)^(3/2),x]

[Out]

x/Sqrt[3 - x^2] + 2*ArcTan[x/(Sqrt[3] - Sqrt[3 - x^2])]

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

method	result	size
default	\(-\arcsin \left (\frac {x \sqrt {3}}{3}\right )+\frac {x}{\sqrt {-x^{2}+3}}\)	\(22\)
risch	\(-\arcsin \left (\frac {x \sqrt {3}}{3}\right )+\frac {x}{\sqrt {-x^{2}+3}}\)	\(22\)
pseudoelliptic	\(\frac {\arctan \left (\frac {\sqrt {-x^{2}+3}}{x}\right ) \sqrt {-x^{2}+3}+x}{\sqrt {-x^{2}+3}}\)	\(37\)
meijerg	\(\frac {i \left (-\frac {i \sqrt {\pi }\, x \sqrt {3}}{3 \sqrt {-\frac {x^{2}}{3}+1}}+i \sqrt {\pi }\, \arcsin \left (\frac {x \sqrt {3}}{3}\right )\right )}{\sqrt {\pi }}\)	\(40\)
trager	\(-\frac {x \sqrt {-x^{2}+3}}{x^{2}-3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+3}+x \right )\)	\(48\)

[In]

int(x^2/(-x^2+3)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {x^2}{\left (3-x^2\right )^{3/2}} \, dx=\frac {{\left (x^{2} - 3\right )} \arctan \left (\frac {\sqrt {-x^{2} + 3}}{x}\right ) - \sqrt {-x^{2} + 3} x}{x^{2} - 3} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (19) = 38\).

Time = 0.18 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04 \[ \int \frac {x^2}{\left (3-x^2\right )^{3/2}} \, dx=- \frac {x^{2} \operatorname {asin}{\left (\frac {\sqrt {3} x}{3} \right )}}{x^{2} - 3} - \frac {x \sqrt {3 - x^{2}}}{x^{2} - 3} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {3} x}{3} \right )}}{x^{2} - 3} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{\left (3-x^2\right )^{3/2}} \, dx=\frac {x}{\sqrt {-x^{2} + 3}} - \arcsin \left (\frac {1}{3} \, \sqrt {3} x\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {x^2}{\left (3-x^2\right )^{3/2}} \, dx=-\frac {\sqrt {-x^{2} + 3} x}{x^{2} - 3} - \arcsin \left (\frac {1}{3} \, \sqrt {3} x\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.25 \[ \int \frac {x^2}{\left (3-x^2\right )^{3/2}} \, dx=-\mathrm {asin}\left (\frac {\sqrt {3}\,x}{3}\right )-\frac {\sqrt {3-x^2}}{2\,\left (x-\sqrt {3}\right )}-\frac {\sqrt {3-x^2}}{2\,\left (x+\sqrt {3}\right )} \]

[In]

int(x^2/(3 - x^2)^(3/2),x)

[Out]

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