\(\int \frac {x^2}{\sqrt {5-x^2}} \, dx\) [137]

Optimal result

Integrand size = 15, antiderivative size = 29 \[ \int \frac {x^2}{\sqrt {5-x^2}} \, dx=-\frac {1}{2} x \sqrt {5-x^2}+\frac {5}{2} \arcsin \left (\frac {x}{\sqrt {5}}\right ) \]

[Out]

5/2*arcsin(1/5*x*5^(1/2))-1/2*x*(-x^2+5)^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 29, 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 = {327, 222} \[ \int \frac {x^2}{\sqrt {5-x^2}} \, dx=\frac {5}{2} \arcsin \left (\frac {x}{\sqrt {5}}\right )-\frac {1}{2} x \sqrt {5-x^2} \]

[In]

Int[x^2/Sqrt[5 - x^2],x]

[Out]

-1/2*(x*Sqrt[5 - x^2]) + (5*ArcSin[x/Sqrt[5]])/2

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 327

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} x \sqrt {5-x^2}+\frac {5}{2} \int \frac {1}{\sqrt {5-x^2}} \, dx \\ & = -\frac {1}{2} x \sqrt {5-x^2}+\frac {5}{2} \arcsin \left (\frac {x}{\sqrt {5}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \frac {x^2}{\sqrt {5-x^2}} \, dx=-\frac {1}{2} x \sqrt {5-x^2}-5 \arctan \left (\frac {x}{\sqrt {5}-\sqrt {5-x^2}}\right ) \]

[In]

Integrate[x^2/Sqrt[5 - x^2],x]

[Out]

-1/2*(x*Sqrt[5 - x^2]) - 5*ArcTan[x/(Sqrt[5] - Sqrt[5 - x^2])]

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79

method	result	size
default	\(\frac {5 \arcsin \left (\frac {x \sqrt {5}}{5}\right )}{2}-\frac {x \sqrt {-x^{2}+5}}{2}\)	\(23\)
risch	\(\frac {x \left (x^{2}-5\right )}{2 \sqrt {-x^{2}+5}}+\frac {5 \arcsin \left (\frac {x \sqrt {5}}{5}\right )}{2}\)	\(28\)
pseudoelliptic	\(-\frac {x \sqrt {-x^{2}+5}}{2}-\frac {5 \arctan \left (\frac {\sqrt {-x^{2}+5}}{x}\right )}{2}\)	\(30\)
meijerg	\(\frac {5 i \left (\frac {i \sqrt {\pi }\, x \sqrt {5}\, \sqrt {-\frac {x^{2}}{5}+1}}{5}-i \sqrt {\pi }\, \arcsin \left (\frac {x \sqrt {5}}{5}\right )\right )}{2 \sqrt {\pi }}\)	\(40\)
trager	\(-\frac {x \sqrt {-x^{2}+5}}{2}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+5}+x \right )}{2}\)	\(42\)

[In]

int(x^2/(-x^2+5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

5/2*arcsin(1/5*x*5^(1/2))-1/2*x*(-x^2+5)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\sqrt {5-x^2}} \, dx=-\frac {1}{2} \, \sqrt {-x^{2} + 5} x - \frac {5}{2} \, \arctan \left (\frac {\sqrt {-x^{2} + 5}}{x}\right ) \]

[In]

integrate(x^2/(-x^2+5)^(1/2),x, algorithm="fricas")

[Out]

-1/2*sqrt(-x^2 + 5)*x - 5/2*arctan(sqrt(-x^2 + 5)/x)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {x^2}{\sqrt {5-x^2}} \, dx=- \frac {x \sqrt {5 - x^{2}}}{2} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {5} x}{5} \right )}}{2} \]

[In]

integrate(x**2/(-x**2+5)**(1/2),x)

[Out]

-x*sqrt(5 - x**2)/2 + 5*asin(sqrt(5)*x/5)/2

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {x^2}{\sqrt {5-x^2}} \, dx=-\frac {1}{2} \, \sqrt {-x^{2} + 5} x + \frac {5}{2} \, \arcsin \left (\frac {1}{5} \, \sqrt {5} x\right ) \]

[In]

integrate(x^2/(-x^2+5)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(-x^2 + 5)*x + 5/2*arcsin(1/5*sqrt(5)*x)

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {x^2}{\sqrt {5-x^2}} \, dx=-\frac {1}{2} \, \sqrt {-x^{2} + 5} x + \frac {5}{2} \, \arcsin \left (\frac {1}{5} \, \sqrt {5} x\right ) \]

[In]

integrate(x^2/(-x^2+5)^(1/2),x, algorithm="giac")

[Out]

-1/2*sqrt(-x^2 + 5)*x + 5/2*arcsin(1/5*sqrt(5)*x)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {x^2}{\sqrt {5-x^2}} \, dx=\frac {5\,\mathrm {asin}\left (\frac {\sqrt {5}\,x}{5}\right )}{2}-\frac {x\,\sqrt {5-x^2}}{2} \]

[In]

int(x^2/(5 - x^2)^(1/2),x)

[Out]

(5*asin((5^(1/2)*x)/5))/2 - (x*(5 - x^2)^(1/2))/2