3.241 \(\int \frac{x}{\sqrt{3-x^2} (5-x^2)} \, dx\)

Optimal. Leaf size=25 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3-x^2}}{\sqrt{2}}\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0187771, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {444, 63, 203} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3-x^2}}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[3 - x^2]*(5 - x^2)),x]

[Out]

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

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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}{\sqrt{3-x^2} \left (5-x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3-x} (5-x)} \, dx,x,x^2\right )\\ &=-\operatorname{Subst}\left (\int \frac{1}{2+x^2} \, dx,x,\sqrt{3-x^2}\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{3-x^2}}{\sqrt{2}}\right )}{\sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0091463, size = 25, normalized size = 1. \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3-x^2}}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(Sqrt[3 - x^2]*(5 - x^2)),x]

[Out]

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

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Maple [B]  time = 0.027, size = 100, normalized size = 4. \begin{align*} -{\frac{\sqrt{2}}{4}\arctan \left ({\frac{ \left ( -4-2\,\sqrt{5} \left ( x-\sqrt{5} \right ) \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{- \left ( x-\sqrt{5} \right ) ^{2}-2\,\sqrt{5} \left ( x-\sqrt{5} \right ) -2}}}} \right ) }-{\frac{\sqrt{2}}{4}\arctan \left ({\frac{ \left ( -4+2\,\sqrt{5} \left ( x+\sqrt{5} \right ) \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{- \left ( x+\sqrt{5} \right ) ^{2}+2\,\sqrt{5} \left ( x+\sqrt{5} \right ) -2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-x^2+5)/(-x^2+3)^(1/2),x)

[Out]

-1/4*2^(1/2)*arctan(1/4*(-4-2*5^(1/2)*(x-5^(1/2)))*2^(1/2)/(-(x-5^(1/2))^2-2*5^(1/2)*(x-5^(1/2))-2)^(1/2))-1/4
*2^(1/2)*arctan(1/4*(-4+2*5^(1/2)*(x+5^(1/2)))*2^(1/2)/(-(x+5^(1/2))^2+2*5^(1/2)*(x+5^(1/2))-2)^(1/2))

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Maxima [B]  time = 1.46323, size = 136, normalized size = 5.44 \begin{align*} -\frac{1}{20} \, \sqrt{5}{\left (\sqrt{5} \sqrt{2} \arcsin \left (\frac{2 \, \sqrt{5} \sqrt{3} x}{3 \,{\left | 2 \, x + 2 \, \sqrt{5} \right |}} + \frac{2 \, \sqrt{3}}{{\left | 2 \, x + 2 \, \sqrt{5} \right |}}\right ) - \sqrt{5} \sqrt{2} \arcsin \left (\frac{2 \, \sqrt{5} \sqrt{3} x}{3 \,{\left | 2 \, x - 2 \, \sqrt{5} \right |}} - \frac{2 \, \sqrt{3}}{{\left | 2 \, x - 2 \, \sqrt{5} \right |}}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2+5)/(-x^2+3)^(1/2),x, algorithm="maxima")

[Out]

-1/20*sqrt(5)*(sqrt(5)*sqrt(2)*arcsin(2/3*sqrt(5)*sqrt(3)*x/abs(2*x + 2*sqrt(5)) + 2*sqrt(3)/abs(2*x + 2*sqrt(
5))) - sqrt(5)*sqrt(2)*arcsin(2/3*sqrt(5)*sqrt(3)*x/abs(2*x - 2*sqrt(5)) - 2*sqrt(3)/abs(2*x - 2*sqrt(5))))

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Fricas [A]  time = 1.99105, size = 93, normalized size = 3.72 \begin{align*} -\frac{1}{4} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (x^{2} - 1\right )} \sqrt{-x^{2} + 3}}{4 \,{\left (x^{2} - 3\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2+5)/(-x^2+3)^(1/2),x, algorithm="fricas")

[Out]

-1/4*sqrt(2)*arctan(1/4*sqrt(2)*(x^2 - 1)*sqrt(-x^2 + 3)/(x^2 - 3))

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Sympy [A]  time = 3.30178, size = 24, normalized size = 0.96 \begin{align*} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{3 - x^{2}}}{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x**2+5)/(-x**2+3)**(1/2),x)

[Out]

-sqrt(2)*atan(sqrt(2)*sqrt(3 - x**2)/2)/2

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Giac [A]  time = 1.05948, size = 27, normalized size = 1.08 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-x^{2} + 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2+5)/(-x^2+3)^(1/2),x, algorithm="giac")

[Out]

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