3.909 \(\int \frac{1}{x^8 (1-x^4)^{3/2}} \, dx\)

Optimal. Leaf size=63 \[ \frac{15}{14} \text{EllipticF}\left (\sin ^{-1}(x),-1\right )-\frac{15 \sqrt{1-x^4}}{14 x^3}-\frac{9 \sqrt{1-x^4}}{14 x^7}+\frac{1}{2 x^7 \sqrt{1-x^4}} \]

[Out]

1/(2*x^7*Sqrt[1 - x^4]) - (9*Sqrt[1 - x^4])/(14*x^7) - (15*Sqrt[1 - x^4])/(14*x^3) + (15*EllipticF[ArcSin[x],
-1])/14

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Rubi [A]  time = 0.0148437, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {290, 325, 221} \[ -\frac{15 \sqrt{1-x^4}}{14 x^3}-\frac{9 \sqrt{1-x^4}}{14 x^7}+\frac{1}{2 x^7 \sqrt{1-x^4}}+\frac{15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(1 - x^4)^(3/2)),x]

[Out]

1/(2*x^7*Sqrt[1 - x^4]) - (9*Sqrt[1 - x^4])/(14*x^7) - (15*Sqrt[1 - x^4])/(14*x^3) + (15*EllipticF[ArcSin[x],
-1])/14

Rule 290

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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{1}{x^8 \left (1-x^4\right )^{3/2}} \, dx &=\frac{1}{2 x^7 \sqrt{1-x^4}}+\frac{9}{2} \int \frac{1}{x^8 \sqrt{1-x^4}} \, dx\\ &=\frac{1}{2 x^7 \sqrt{1-x^4}}-\frac{9 \sqrt{1-x^4}}{14 x^7}+\frac{45}{14} \int \frac{1}{x^4 \sqrt{1-x^4}} \, dx\\ &=\frac{1}{2 x^7 \sqrt{1-x^4}}-\frac{9 \sqrt{1-x^4}}{14 x^7}-\frac{15 \sqrt{1-x^4}}{14 x^3}+\frac{15}{14} \int \frac{1}{\sqrt{1-x^4}} \, dx\\ &=\frac{1}{2 x^7 \sqrt{1-x^4}}-\frac{9 \sqrt{1-x^4}}{14 x^7}-\frac{15 \sqrt{1-x^4}}{14 x^3}+\frac{15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}

Mathematica [C]  time = 0.0030944, size = 20, normalized size = 0.32 \[ -\frac{\, _2F_1\left (-\frac{7}{4},\frac{3}{2};-\frac{3}{4};x^4\right )}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*(1 - x^4)^(3/2)),x]

[Out]

-Hypergeometric2F1[-7/4, 3/2, -3/4, x^4]/(7*x^7)

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Maple [A]  time = 0.012, size = 73, normalized size = 1.2 \begin{align*}{\frac{x}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}-{\frac{1}{7\,{x}^{7}}\sqrt{-{x}^{4}+1}}-{\frac{4}{7\,{x}^{3}}\sqrt{-{x}^{4}+1}}+{\frac{15\,{\it EllipticF} \left ( x,i \right ) }{14}\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(1/x^8/(-x^4+1)^(3/2),x)

[Out]

1/2*x/(-x^4+1)^(1/2)-1/7*(-x^4+1)^(1/2)/x^7-4/7*(-x^4+1)^(1/2)/x^3+15/14*(-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{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(-x^4+1)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((-x^4 + 1)^(3/2)*x^8), 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^{16} - 2 \, x^{12} + x^{8}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(-x^4+1)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(-x^4 + 1)/(x^16 - 2*x^12 + x^8), x)

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Sympy [A]  time = 1.92494, size = 37, normalized size = 0.59 \begin{align*} \frac{\Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{3}{2} \\ - \frac{3}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**8/(-x**4+1)**(3/2),x)

[Out]

gamma(-7/4)*hyper((-7/4, 3/2), (-3/4,), x**4*exp_polar(2*I*pi))/(4*x**7*gamma(-3/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(-x^4+1)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((-x^4 + 1)^(3/2)*x^8), x)