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

Optimal. Leaf size=25 \[ \frac {x}{2 \sqrt {1-x^4}}-\frac {1}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {288, 221} \[ \frac {x}{2 \sqrt {1-x^4}}-\frac {1}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^4/(1 - x^4)^(3/2),x]

[Out]

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

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]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

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

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Mathematica [C]  time = 0.01, size = 32, normalized size = 1.28 \[ \frac {1}{2} x \left (\frac {1}{\sqrt {1-x^4}}-\, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^4\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(1 - x^4)^(3/2),x]

[Out]

(x*(1/Sqrt[1 - x^4] - Hypergeometric2F1[1/4, 1/2, 5/4, x^4]))/2

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fricas [F]  time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{4} + 1} x^{4}}{x^{8} - 2 \, x^{4} + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-x^4 + 1)*x^4/(x^8 - 2*x^4 + 1), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/(-x^4 + 1)^(3/2), x)

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maple [B]  time = 0.01, size = 45, normalized size = 1.80 \[ \frac {x}{2 \sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{2 \sqrt {-x^{4}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/(-x^4+1)^(1/2)*x-1/2*(-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 \[ \int \frac {x^{4}}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/(-x^4 + 1)^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^4}{{\left (1-x^4\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(1 - x^4)^(3/2),x)

[Out]

int(x^4/(1 - x^4)^(3/2), x)

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sympy [B]  time = 1.28, size = 31, normalized size = 1.24 \[ \frac {x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(-x**4+1)**(3/2),x)

[Out]

x**5*gamma(5/4)*hyper((5/4, 3/2), (9/4,), x**4*exp_polar(2*I*pi))/(4*gamma(9/4))

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