\(\int \frac {x}{\sqrt {3-x^2}} \, dx\) [149]

Optimal result

Integrand size = 13, antiderivative size = 13 \[ \int \frac {x}{\sqrt {3-x^2}} \, dx=-\sqrt {3-x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int \frac {x}{\sqrt {3-x^2}} \, dx=-\sqrt {3-x^2} \]

[In]

Int[x/Sqrt[3 - x^2],x]

[Out]

-Sqrt[3 - x^2]

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\sqrt {3-x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt {3-x^2}} \, dx=-\sqrt {3-x^2} \]

[In]

Integrate[x/Sqrt[3 - x^2],x]

[Out]

-Sqrt[3 - x^2]

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

method	result	size
gosper	\(-\sqrt {-x^{2}+3}\)	\(12\)
derivativedivides	\(-\sqrt {-x^{2}+3}\)	\(12\)
default	\(-\sqrt {-x^{2}+3}\)	\(12\)
trager	\(-\sqrt {-x^{2}+3}\)	\(12\)
pseudoelliptic	\(-\sqrt {-x^{2}+3}\)	\(12\)
risch	\(\frac {x^{2}-3}{\sqrt {-x^{2}+3}}\)	\(16\)
meijerg	\(-\frac {\sqrt {3}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-\frac {x^{2}}{3}+1}\right )}{2 \sqrt {\pi }}\)	\(29\)

[In]

int(x/(-x^2+3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {x}{\sqrt {3-x^2}} \, dx=-\sqrt {-x^{2} + 3} \]

[In]

integrate(x/(-x^2+3)^(1/2),x, algorithm="fricas")

[Out]

-sqrt(-x^2 + 3)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int \frac {x}{\sqrt {3-x^2}} \, dx=- \sqrt {3 - x^{2}} \]

[In]

integrate(x/(-x**2+3)**(1/2),x)

[Out]

-sqrt(3 - x**2)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {x}{\sqrt {3-x^2}} \, dx=-\sqrt {-x^{2} + 3} \]

[In]

integrate(x/(-x^2+3)^(1/2),x, algorithm="maxima")

[Out]

-sqrt(-x^2 + 3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {x}{\sqrt {3-x^2}} \, dx=-\sqrt {-x^{2} + 3} \]

[In]

integrate(x/(-x^2+3)^(1/2),x, algorithm="giac")

[Out]

-sqrt(-x^2 + 3)

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {x}{\sqrt {3-x^2}} \, dx=-\sqrt {3-x^2} \]

[In]

int(x/(3 - x^2)^(1/2),x)

[Out]

-(3 - x^2)^(1/2)