3.904 \(\int \frac{x^{12}}{(1-x^4)^{3/2}} \, dx\)

Optimal. Leaf size=61 \[ -\frac{15}{14} \text{EllipticF}\left (\sin ^{-1}(x),-1\right )+\frac{x^9}{2 \sqrt{1-x^4}}+\frac{9}{14} \sqrt{1-x^4} x^5+\frac{15}{14} \sqrt{1-x^4} x \]

[Out]

x^9/(2*Sqrt[1 - x^4]) + (15*x*Sqrt[1 - x^4])/14 + (9*x^5*Sqrt[1 - x^4])/14 - (15*EllipticF[ArcSin[x], -1])/14

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Rubi [A]  time = 0.0153233, antiderivative size = 61, 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 = {288, 321, 221} \[ \frac{x^9}{2 \sqrt{1-x^4}}+\frac{9}{14} \sqrt{1-x^4} x^5+\frac{15}{14} \sqrt{1-x^4} x-\frac{15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^12/(1 - x^4)^(3/2),x]

[Out]

x^9/(2*Sqrt[1 - x^4]) + (15*x*Sqrt[1 - x^4])/14 + (9*x^5*Sqrt[1 - x^4])/14 - (15*EllipticF[ArcSin[x], -1])/14

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]

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^{12}}{\left (1-x^4\right )^{3/2}} \, dx &=\frac{x^9}{2 \sqrt{1-x^4}}-\frac{9}{2} \int \frac{x^8}{\sqrt{1-x^4}} \, dx\\ &=\frac{x^9}{2 \sqrt{1-x^4}}+\frac{9}{14} x^5 \sqrt{1-x^4}-\frac{45}{14} \int \frac{x^4}{\sqrt{1-x^4}} \, dx\\ &=\frac{x^9}{2 \sqrt{1-x^4}}+\frac{15}{14} x \sqrt{1-x^4}+\frac{9}{14} x^5 \sqrt{1-x^4}-\frac{15}{14} \int \frac{1}{\sqrt{1-x^4}} \, dx\\ &=\frac{x^9}{2 \sqrt{1-x^4}}+\frac{15}{14} x \sqrt{1-x^4}+\frac{9}{14} x^5 \sqrt{1-x^4}-\frac{15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}

Mathematica [C]  time = 0.0099282, size = 54, normalized size = 0.89 \[ -\frac{x \left (15 \sqrt{1-x^4} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^4\right )+2 x^8+6 x^4-15\right )}{14 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^12/(1 - x^4)^(3/2),x]

[Out]

-(x*(-15 + 6*x^4 + 2*x^8 + 15*Sqrt[1 - x^4]*Hypergeometric2F1[1/4, 1/2, 5/4, x^4]))/(14*Sqrt[1 - x^4])

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Maple [A]  time = 0.008, size = 71, normalized size = 1.2 \begin{align*}{\frac{x}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}+{\frac{{x}^{5}}{7}\sqrt{-{x}^{4}+1}}+{\frac{4\,x}{7}\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(x^12/(-x^4+1)^(3/2),x)

[Out]

1/2*x/(-x^4+1)^(1/2)+1/7*x^5*(-x^4+1)^(1/2)+4/7*x*(-x^4+1)^(1/2)-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{x^{12}}{{\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^12/(-x^4+1)^(3/2),x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(-x^4+1)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.4493, size = 31, normalized size = 0.51 \begin{align*} \frac{x^{13} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{17}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**12/(-x**4+1)**(3/2),x)

[Out]

x**13*gamma(13/4)*hyper((3/2, 13/4), (17/4,), x**4*exp_polar(2*I*pi))/(4*gamma(17/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{12}}{{\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^12/(-x^4+1)^(3/2),x, algorithm="giac")

[Out]

integrate(x^12/(-x^4 + 1)^(3/2), x)