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

Optimal. Leaf size=24 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{5-x^2}}{\sqrt{2}}\right )}{\sqrt{2}} \]

[Out]

ArcTanh[Sqrt[5 - x^2]/Sqrt[2]]/Sqrt[2]

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Rubi [A]  time = 0.0192104, antiderivative size = 24, 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, 207} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{5-x^2}}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[x/((3 - x^2)*Sqrt[5 - x^2]),x]

[Out]

ArcTanh[Sqrt[5 - 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 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0094691, size = 24, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{5-x^2}}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[Sqrt[5 - x^2]/Sqrt[2]]/Sqrt[2]

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Maple [B]  time = 0.025, size = 100, normalized size = 4.2 \begin{align*}{\frac{\sqrt{2}}{4}{\it Artanh} \left ({\frac{ \left ( 4+2\,\sqrt{3} \left ( x+\sqrt{3} \right ) \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{- \left ( x+\sqrt{3} \right ) ^{2}+2\,\sqrt{3} \left ( x+\sqrt{3} \right ) +2}}}} \right ) }+{\frac{\sqrt{2}}{4}{\it Artanh} \left ({\frac{ \left ( 4-2\,\sqrt{3} \left ( x-\sqrt{3} \right ) \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{- \left ( x-\sqrt{3} \right ) ^{2}-2\,\sqrt{3} \left ( x-\sqrt{3} \right ) +2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.53592, size = 151, normalized size = 6.29 \begin{align*} \frac{1}{12} \, \sqrt{3}{\left (\sqrt{3} \sqrt{2} \log \left (\sqrt{3} + \frac{2 \, \sqrt{2} \sqrt{-x^{2} + 5}}{{\left | 2 \, x + 2 \, \sqrt{3} \right |}} + \frac{4}{{\left | 2 \, x + 2 \, \sqrt{3} \right |}}\right ) + \sqrt{3} \sqrt{2} \log \left (-\sqrt{3} + \frac{2 \, \sqrt{2} \sqrt{-x^{2} + 5}}{{\left | 2 \, x - 2 \, \sqrt{3} \right |}} + \frac{4}{{\left | 2 \, x - 2 \, \sqrt{3} \right |}}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(3)*(sqrt(3)*sqrt(2)*log(sqrt(3) + 2*sqrt(2)*sqrt(-x^2 + 5)/abs(2*x + 2*sqrt(3)) + 4/abs(2*x + 2*sqrt
(3))) + sqrt(3)*sqrt(2)*log(-sqrt(3) + 2*sqrt(2)*sqrt(-x^2 + 5)/abs(2*x - 2*sqrt(3)) + 4/abs(2*x - 2*sqrt(3)))
)

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Fricas [B]  time = 2.10653, size = 126, normalized size = 5.25 \begin{align*} \frac{1}{8} \, \sqrt{2} \log \left (\frac{x^{4} - 4 \, \sqrt{2}{\left (x^{2} - 7\right )} \sqrt{-x^{2} + 5} - 22 \, x^{2} + 89}{x^{4} - 6 \, x^{2} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(2)*log((x^4 - 4*sqrt(2)*(x^2 - 7)*sqrt(-x^2 + 5) - 22*x^2 + 89)/(x^4 - 6*x^2 + 9))

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Sympy [A]  time = 3.41892, size = 61, normalized size = 2.54 \begin{align*} - \begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left (\frac{\sqrt{2}}{\sqrt{5 - x^{2}}} \right )}}{2} & \text{for}\: \frac{1}{5 - x^{2}} > \frac{1}{2} \\- \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2}}{\sqrt{5 - x^{2}}} \right )}}{2} & \text{for}\: \frac{1}{5 - x^{2}} < \frac{1}{2} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Piecewise((-sqrt(2)*acoth(sqrt(2)/sqrt(5 - x**2))/2, 1/(5 - x**2) > 1/2), (-sqrt(2)*atanh(sqrt(2)/sqrt(5 - x*
*2))/2, 1/(5 - x**2) < 1/2))

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Giac [B]  time = 1.07389, size = 57, normalized size = 2.38 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\sqrt{2} + \sqrt{-x^{2} + 5}\right ) - \frac{1}{4} \, \sqrt{2} \log \left ({\left | -\sqrt{2} + \sqrt{-x^{2} + 5} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*log(sqrt(2) + sqrt(-x^2 + 5)) - 1/4*sqrt(2)*log(abs(-sqrt(2) + sqrt(-x^2 + 5)))