3.39 \(\int \frac{x^4}{\left (1-x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=35 \[ -\frac{x}{\sqrt{1-x^2}}+\frac{x^3}{3 \left (1-x^2\right )^{3/2}}+\sin ^{-1}(x) \]

[Out]

x^3/(3*(1 - x^2)^(3/2)) - x/Sqrt[1 - x^2] + ArcSin[x]

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Rubi [A]  time = 0.0283467, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{x}{\sqrt{1-x^2}}+\frac{x^3}{3 \left (1-x^2\right )^{3/2}}+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[x^4/(1 - x^2)^(5/2),x]

[Out]

x^3/(3*(1 - x^2)^(3/2)) - x/Sqrt[1 - x^2] + ArcSin[x]

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Rubi in Sympy [A]  time = 2.47774, size = 26, normalized size = 0.74 \[ \frac{x^{3}}{3 \left (- x^{2} + 1\right )^{\frac{3}{2}}} - \frac{x}{\sqrt{- x^{2} + 1}} + \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(-x**2+1)**(5/2),x)

[Out]

x**3/(3*(-x**2 + 1)**(3/2)) - x/sqrt(-x**2 + 1) + asin(x)

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Mathematica [A]  time = 0.0105914, size = 26, normalized size = 0.74 \[ \frac{x \left (4 x^2-3\right )}{3 \left (1-x^2\right )^{3/2}}+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(1 - x^2)^(5/2),x]

[Out]

(x*(-3 + 4*x^2))/(3*(1 - x^2)^(3/2)) + ArcSin[x]

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Maple [A]  time = 0., size = 30, normalized size = 0.9 \[{\frac{{x}^{3}}{3} \left ( -{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}+\arcsin \left ( x \right ) -{x{\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(-x^2+1)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.49307, size = 59, normalized size = 1.69 \[ \frac{1}{3} \, x{\left (\frac{3 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{2}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}\right )} - \frac{x}{3 \, \sqrt{-x^{2} + 1}} + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(-x^2 + 1)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.213741, size = 182, normalized size = 5.2 \[ -\frac{12 \, x^{5} - 25 \, x^{3} + 6 \,{\left (x^{6} - 6 \, x^{4} + 9 \, x^{2} +{\left (3 \, x^{4} - 7 \, x^{2} + 4\right )} \sqrt{-x^{2} + 1} - 4\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) -{\left (4 \, x^{5} - 19 \, x^{3} + 12 \, x\right )} \sqrt{-x^{2} + 1} + 12 \, x}{3 \,{\left (x^{6} - 6 \, x^{4} + 9 \, x^{2} +{\left (3 \, x^{4} - 7 \, x^{2} + 4\right )} \sqrt{-x^{2} + 1} - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(-x^2 + 1)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(12*x^5 - 25*x^3 + 6*(x^6 - 6*x^4 + 9*x^2 + (3*x^4 - 7*x^2 + 4)*sqrt(-x^2 +
1) - 4)*arctan((sqrt(-x^2 + 1) - 1)/x) - (4*x^5 - 19*x^3 + 12*x)*sqrt(-x^2 + 1)
+ 12*x)/(x^6 - 6*x^4 + 9*x^2 + (3*x^4 - 7*x^2 + 4)*sqrt(-x^2 + 1) - 4)

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Sympy [A]  time = 4.1388, size = 105, normalized size = 3. \[ \frac{3 x^{4} \operatorname{asin}{\left (x \right )}}{3 x^{4} - 6 x^{2} + 3} + \frac{4 x^{3} \sqrt{- x^{2} + 1}}{3 x^{4} - 6 x^{2} + 3} - \frac{6 x^{2} \operatorname{asin}{\left (x \right )}}{3 x^{4} - 6 x^{2} + 3} - \frac{3 x \sqrt{- x^{2} + 1}}{3 x^{4} - 6 x^{2} + 3} + \frac{3 \operatorname{asin}{\left (x \right )}}{3 x^{4} - 6 x^{2} + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(-x**2+1)**(5/2),x)

[Out]

3*x**4*asin(x)/(3*x**4 - 6*x**2 + 3) + 4*x**3*sqrt(-x**2 + 1)/(3*x**4 - 6*x**2 +
3) - 6*x**2*asin(x)/(3*x**4 - 6*x**2 + 3) - 3*x*sqrt(-x**2 + 1)/(3*x**4 - 6*x**
2 + 3) + 3*asin(x)/(3*x**4 - 6*x**2 + 3)

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GIAC/XCAS [A]  time = 0.207361, size = 39, normalized size = 1.11 \[ \frac{{\left (4 \, x^{2} - 3\right )} \sqrt{-x^{2} + 1} x}{3 \,{\left (x^{2} - 1\right )}^{2}} + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(-x^2 + 1)^(5/2),x, algorithm="giac")

[Out]

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