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

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 = 0.0577787, 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 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3-x^2}}{\sqrt{2}}\right )}{\sqrt{2}} \]

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])

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Rubi in Sympy [A]  time = 4.51676, size = 24, normalized size = 0.96 \[ - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- x^{2} + 3}}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-x**2+5)/(-x**2+3)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0171354, 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 = 0.047, size = 100, normalized size = 4. \[ -{\frac{\sqrt{2}}{4}\arctan \left ({\frac{ \left ( -4-2\,\sqrt{5} \left ( x-\sqrt{5} \right ) \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{- \left ( x-\sqrt{5} \right ) ^{2}-2\,\sqrt{5} \left ( x-\sqrt{5} \right ) -2}}}} \right ) }-{\frac{\sqrt{2}}{4}\arctan \left ({\frac{ \left ( -4+2\,\sqrt{5} \left ( x+\sqrt{5} \right ) \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{- \left ( x+\sqrt{5} \right ) ^{2}+2\,\sqrt{5} \left ( x+\sqrt{5} \right ) -2}}}} \right ) } \]

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 [A]  time = 1.52042, size = 136, normalized size = 5.44 \[ -\frac{1}{20} \, \sqrt{5}{\left (\sqrt{5} \sqrt{2} \arcsin \left (\frac{2 \, \sqrt{5} \sqrt{3} x}{3 \,{\left | 2 \, x + 2 \, \sqrt{5} \right |}} + \frac{2 \, \sqrt{3}}{{\left | 2 \, x + 2 \, \sqrt{5} \right |}}\right ) - \sqrt{5} \sqrt{2} \arcsin \left (\frac{2 \, \sqrt{5} \sqrt{3} x}{3 \,{\left | 2 \, x - 2 \, \sqrt{5} \right |}} - \frac{2 \, \sqrt{3}}{{\left | 2 \, x - 2 \, \sqrt{5} \right |}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/((x^2 - 5)*sqrt(-x^2 + 3)),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 = 0.228137, size = 34, normalized size = 1.36 \[ \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (x^{2} - 1\right )}}{4 \, \sqrt{-x^{2} + 3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/((x^2 - 5)*sqrt(-x^2 + 3)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.40852, size = 20, normalized size = 0.8 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2}}{\sqrt{- x^{2} + 3}} \right )}}{2} \]

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(-x**2 + 3))/2

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GIAC/XCAS [A]  time = 0.213471, size = 27, normalized size = 1.08 \[ -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-x^{2} + 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/((x^2 - 5)*sqrt(-x^2 + 3)),x, algorithm="giac")

[Out]

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