3.884 \(\int \frac{x^8}{\sqrt{1-x^4}} \, dx\)

Optimal. Leaf size=43 \[ \frac{5}{21} \text{EllipticF}\left (\sin ^{-1}(x),-1\right )-\frac{1}{7} \sqrt{1-x^4} x^5-\frac{5}{21} \sqrt{1-x^4} x \]

[Out]

(-5*x*Sqrt[1 - x^4])/21 - (x^5*Sqrt[1 - x^4])/7 + (5*EllipticF[ArcSin[x], -1])/21

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Rubi [A]  time = 0.0091225, antiderivative size = 43, 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, Rules used = {321, 221} \[ -\frac{1}{7} \sqrt{1-x^4} x^5-\frac{5}{21} \sqrt{1-x^4} x+\frac{5}{21} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^8/Sqrt[1 - x^4],x]

[Out]

(-5*x*Sqrt[1 - x^4])/21 - (x^5*Sqrt[1 - x^4])/7 + (5*EllipticF[ArcSin[x], -1])/21

Rule 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 \frac{x^8}{\sqrt{1-x^4}} \, dx &=-\frac{1}{7} x^5 \sqrt{1-x^4}+\frac{5}{7} \int \frac{x^4}{\sqrt{1-x^4}} \, dx\\ &=-\frac{5}{21} x \sqrt{1-x^4}-\frac{1}{7} x^5 \sqrt{1-x^4}+\frac{5}{21} \int \frac{1}{\sqrt{1-x^4}} \, dx\\ &=-\frac{5}{21} x \sqrt{1-x^4}-\frac{1}{7} x^5 \sqrt{1-x^4}+\frac{5}{21} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}

Mathematica [C]  time = 0.0116365, size = 42, normalized size = 0.98 \[ \frac{1}{21} \left (5 x \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^4\right )-x \sqrt{1-x^4} \left (3 x^4+5\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/Sqrt[1 - x^4],x]

[Out]

(-(x*Sqrt[1 - x^4]*(5 + 3*x^4)) + 5*x*Hypergeometric2F1[1/4, 1/2, 5/4, x^4])/21

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Maple [A]  time = 0.008, size = 59, normalized size = 1.4 \begin{align*} -{\frac{{x}^{5}}{7}\sqrt{-{x}^{4}+1}}-{\frac{5\,x}{21}\sqrt{-{x}^{4}+1}}+{\frac{5\,{\it EllipticF} \left ( x,i \right ) }{21}\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^8/(-x^4+1)^(1/2),x)

[Out]

-1/7*x^5*(-x^4+1)^(1/2)-5/21*x*(-x^4+1)^(1/2)+5/21*(-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 \frac{x^{8}}{\sqrt{-x^{4} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-x^4+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^8/sqrt(-x^4 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{4} + 1} x^{8}}{x^{4} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-x^4+1)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.14259, size = 31, normalized size = 0.72 \begin{align*} \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{13}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(-x**4+1)**(1/2),x)

[Out]

x**9*gamma(9/4)*hyper((1/2, 9/4), (13/4,), x**4*exp_polar(2*I*pi))/(4*gamma(13/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{8}}{\sqrt{-x^{4} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

integrate(x^8/sqrt(-x^4 + 1), x)