3.802 \(\int \left (1-x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=41 \[ \frac{1}{7} x \left (1-x^4\right )^{3/2}+\frac{2}{7} x \sqrt{1-x^4}+\frac{4}{7} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

(2*x*Sqrt[1 - x^4])/7 + (x*(1 - x^4)^(3/2))/7 + (4*EllipticF[ArcSin[x], -1])/7

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Rubi [A]  time = 0.0182557, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{7} x \left (1-x^4\right )^{3/2}+\frac{2}{7} x \sqrt{1-x^4}+\frac{4}{7} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^4)^(3/2),x]

[Out]

(2*x*Sqrt[1 - x^4])/7 + (x*(1 - x^4)^(3/2))/7 + (4*EllipticF[ArcSin[x], -1])/7

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Rubi in Sympy [A]  time = 1.42777, size = 34, normalized size = 0.83 \[ \frac{x \left (- x^{4} + 1\right )^{\frac{3}{2}}}{7} + \frac{2 x \sqrt{- x^{4} + 1}}{7} + \frac{4 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**4+1)**(3/2),x)

[Out]

x*(-x**4 + 1)**(3/2)/7 + 2*x*sqrt(-x**4 + 1)/7 + 4*elliptic_f(asin(x), -1)/7

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Mathematica [A]  time = 0.0356835, size = 44, normalized size = 1.07 \[ \frac{x^9-4 x^5+4 \sqrt{1-x^4} F\left (\left .\sin ^{-1}(x)\right |-1\right )+3 x}{7 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^4)^(3/2),x]

[Out]

(3*x - 4*x^5 + x^9 + 4*Sqrt[1 - x^4]*EllipticF[ArcSin[x], -1])/(7*Sqrt[1 - x^4])

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Maple [A]  time = 0.009, size = 59, normalized size = 1.4 \[ -{\frac{{x}^{5}}{7}\sqrt{-{x}^{4}+1}}+{\frac{3\,x}{7}\sqrt{-{x}^{4}+1}}+{\frac{4\,{\it EllipticF} \left ( x,i \right ) }{7}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+1)^(3/2),x)

[Out]

-1/7*x^5*(-x^4+1)^(1/2)+3/7*x*(-x^4+1)^(1/2)+4/7*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-
x^4+1)^(1/2)*EllipticF(x,I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (-x^{4} + 1\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

integrate((-x^4 + 1)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (-x^{4} + 1\right )}^{\frac{3}{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

integral((-x^4 + 1)^(3/2), x)

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Sympy [A]  time = 2.37815, size = 31, normalized size = 0.76 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**4+1)**(3/2),x)

[Out]

x*gamma(1/4)*hyper((-3/2, 1/4), (5/4,), x**4*exp_polar(2*I*pi))/(4*gamma(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (-x^{4} + 1\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

integrate((-x^4 + 1)^(3/2), x)