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

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

[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 = 0.0057027, 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, Rules used = {195, 221} \[ \frac{1}{7} x \left (1-x^4\right )^{3/2}+\frac{2}{7} x \sqrt{1-x^4}+\frac{4}{7} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \left (1-x^4\right )^{3/2} \, dx &=\frac{1}{7} x \left (1-x^4\right )^{3/2}+\frac{6}{7} \int \sqrt{1-x^4} \, dx\\ &=\frac{2}{7} x \sqrt{1-x^4}+\frac{1}{7} x \left (1-x^4\right )^{3/2}+\frac{4}{7} \int \frac{1}{\sqrt{1-x^4}} \, dx\\ &=\frac{2}{7} x \sqrt{1-x^4}+\frac{1}{7} x \left (1-x^4\right )^{3/2}+\frac{4}{7} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}

Mathematica [C]  time = 0.0022316, size = 15, normalized size = 0.37 \[ x \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};x^4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^4)^(3/2),x]

[Out]

x*Hypergeometric2F1[-3/2, 1/4, 5/4, x^4]

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Maple [A]  time = 0.004, size = 59, normalized size = 1.4 \begin{align*} -{\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}}}} \end{align*}

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. \begin{align*} \int{\left (-x^{4} + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}

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. \begin{align*}{\rm integral}\left ({\left (-x^{4} + 1\right )}^{\frac{3}{2}}, x\right ) \end{align*}

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 = 0.870378, size = 31, normalized size = 0.76 \begin{align*} \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 )} \end{align*}

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

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)