3.9.4 \(\int (1-x^4)^{3/2} \, dx\) [804]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 = {201, 227} \begin {gather*} \frac {4}{7} F(\text {ArcSin}(x)|-1)+\frac {1}{7} x \left (1-x^4\right )^{3/2}+\frac {2}{7} x \sqrt {1-x^4} \end {gather*}

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 201

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 227

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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 3.69, size = 15, normalized size = 0.37 \begin {gather*} x \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};x^4\right ) \end {gather*}

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 = 0.16, size = 59, normalized size = 1.44

method	result	size
meijerg	\(x \hypergeom \left (\left [-\frac {3}{2}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], x^{4}\right )\)	\(12\)
risch	\(\frac {x \left (x^{4}-3\right ) \left (x^{4}-1\right )}{7 \sqrt {-x^{4}+1}}+\frac {4 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{7 \sqrt {-x^{4}+1}}\)	\(55\)
default	\(-\frac {x^{5} \sqrt {-x^{4}+1}}{7}+\frac {3 x \sqrt {-x^{4}+1}}{7}+\frac {4 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{7 \sqrt {-x^{4}+1}}\)	\(59\)
elliptic	\(-\frac {x^{5} \sqrt {-x^{4}+1}}{7}+\frac {3 x \sqrt {-x^{4}+1}}{7}+\frac {4 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{7 \sqrt {-x^{4}+1}}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^4+1)^(3/2),x,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [A]
time = 0.07, size = 18, normalized size = 0.44 \begin {gather*} -\frac {1}{7} \, {\left (x^{5} - 3 \, x\right )} \sqrt {-x^{4} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*(x^5 - 3*x)*sqrt(-x^4 + 1)

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Sympy [A]
time = 0.36, size = 31, normalized size = 0.76 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 1.02, size = 10, normalized size = 0.24 \begin {gather*} x\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{4};\ \frac {5}{4};\ x^4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^4)^(3/2),x)

[Out]

x*hypergeom([-3/2, 1/4], 5/4, x^4)

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