3.849 \(\int \frac{1}{x^2 \sqrt{a-b x^4}} \, dx\)

Optimal. Leaf size=128 \[ \frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}-\frac{\sqrt{a-b x^4}}{a x}-\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}} \]

[Out]

-(Sqrt[a - b*x^4]/(a*x)) - (b^(1/4)*Sqrt[1 - (b*x^4)/a]*EllipticE[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(a^(1/4)*S
qrt[a - b*x^4]) + (b^(1/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(a^(1/4)*Sqrt[a - b
*x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.0768875, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {325, 307, 224, 221, 1200, 1199, 424} \[ -\frac{\sqrt{a-b x^4}}{a x}+\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}-\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[a - b*x^4]),x]

[Out]

-(Sqrt[a - b*x^4]/(a*x)) - (b^(1/4)*Sqrt[1 - (b*x^4)/a]*EllipticE[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(a^(1/4)*S
qrt[a - b*x^4]) + (b^(1/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(a^(1/4)*Sqrt[a - b
*x^4])

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 307

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-(b/a), 2]}, -Dist[q^(-1), Int[1/Sqrt[a + b*x^
4], x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + (b*x^4)/a]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + (b*x^4)
/a], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

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 1200

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + (c*x^4)/a]/Sqrt[a + c*x^4], In
t[(d + e*x^2)/Sqrt[1 + (c*x^4)/a], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] &&
!GtQ[a, 0]

Rule 1199

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[d/Sqrt[a], Int[Sqrt[1 + (e*x^2)/d]/Sqrt
[1 - (e*x^2)/d], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \sqrt{a-b x^4}} \, dx &=-\frac{\sqrt{a-b x^4}}{a x}-\frac{b \int \frac{x^2}{\sqrt{a-b x^4}} \, dx}{a}\\ &=-\frac{\sqrt{a-b x^4}}{a x}+\frac{\sqrt{b} \int \frac{1}{\sqrt{a-b x^4}} \, dx}{\sqrt{a}}-\frac{\sqrt{b} \int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a-b x^4}} \, dx}{\sqrt{a}}\\ &=-\frac{\sqrt{a-b x^4}}{a x}+\frac{\left (\sqrt{b} \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{\sqrt{a} \sqrt{a-b x^4}}-\frac{\left (\sqrt{b} \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{\sqrt{a} \sqrt{a-b x^4}}\\ &=-\frac{\sqrt{a-b x^4}}{a x}+\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}-\frac{\left (\sqrt{b} \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{\sqrt{1+\frac{\sqrt{b} x^2}{\sqrt{a}}}}{\sqrt{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}} \, dx}{\sqrt{a} \sqrt{a-b x^4}}\\ &=-\frac{\sqrt{a-b x^4}}{a x}-\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}+\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0092366, size = 50, normalized size = 0.39 \[ -\frac{\sqrt{1-\frac{b x^4}{a}} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};\frac{b x^4}{a}\right )}{x \sqrt{a-b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[a - b*x^4]),x]

[Out]

-((Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[-1/4, 1/2, 3/4, (b*x^4)/a])/(x*Sqrt[a - b*x^4]))

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 106, normalized size = 0.8 \begin{align*} -{\frac{1}{ax}\sqrt{-b{x}^{4}+a}}+{\sqrt{b}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(-b*x^4+a)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-b x^{4} + a} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(-b*x^4 + a)*x^2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-b x^{4} + a}}{b x^{6} - a x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-b*x^4 + a)/(b*x^6 - a*x^2), x)

________________________________________________________________________________________

Sympy [A]  time = 1.04219, size = 41, normalized size = 0.32 \begin{align*} \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} x \Gamma \left (\frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

gamma(-1/4)*hyper((-1/4, 1/2), (3/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*x*gamma(3/4))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-b x^{4} + a} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(-b*x^4 + a)*x^2), x)