3.2087 \(\int \frac{1}{\sqrt{a+\frac{b}{x^4}}} \, dx\)

Optimal. Leaf size=231 \[ -\frac{\sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{2 a^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{x \sqrt{a+\frac{b}{x^4}}}{a}-\frac{\sqrt{b} \sqrt{a+\frac{b}{x^4}}}{a x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )} \]

[Out]

-((Sqrt[b]*Sqrt[a + b/x^4])/(a*(Sqrt[a] + Sqrt[b]/x^2)*x)) + (Sqrt[a + b/x^4]*x)/a + (b^(1/4)*Sqrt[(a + b/x^4)
/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticE[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(a^(3/4)*Sq
rt[a + b/x^4]) - (b^(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticF[2*ArcC
ot[(a^(1/4)*x)/b^(1/4)], 1/2])/(2*a^(3/4)*Sqrt[a + b/x^4])

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Rubi [A]  time = 0.102109, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {242, 325, 305, 220, 1196} \[ -\frac{\sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 a^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{x \sqrt{a+\frac{b}{x^4}}}{a}-\frac{\sqrt{b} \sqrt{a+\frac{b}{x^4}}}{a x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b/x^4],x]

[Out]

-((Sqrt[b]*Sqrt[a + b/x^4])/(a*(Sqrt[a] + Sqrt[b]/x^2)*x)) + (Sqrt[a + b/x^4]*x)/a + (b^(1/4)*Sqrt[(a + b/x^4)
/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticE[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(a^(3/4)*Sq
rt[a + b/x^4]) - (b^(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticF[2*ArcC
ot[(a^(1/4)*x)/b^(1/4)], 1/2])/(2*a^(3/4)*Sqrt[a + b/x^4])

Rule 242

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^2, x], x, 1/x] /; FreeQ[{a, b, p},
x] && ILtQ[n, 0]

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 305

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

Rule 220

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

Rule 1196

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, -Simp[(d*x*Sqrt[a + c
*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x^4}}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt{a+\frac{b}{x^4}} x}{a}-\frac{b \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\frac{\sqrt{a+\frac{b}{x^4}} x}{a}-\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{\sqrt{a}}+\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx,x,\frac{1}{x}\right )}{\sqrt{a}}\\ &=-\frac{\sqrt{b} \sqrt{a+\frac{b}{x^4}}}{a \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) x}+\frac{\sqrt{a+\frac{b}{x^4}} x}{a}+\frac{\sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{\sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 a^{3/4} \sqrt{a+\frac{b}{x^4}}}\\ \end{align*}

Mathematica [C]  time = 0.0096678, size = 49, normalized size = 0.21 \[ \frac{x \sqrt{\frac{a x^4}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{a x^4}{b}\right )}{3 \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b/x^4],x]

[Out]

(x*Sqrt[1 + (a*x^4)/b]*Hypergeometric2F1[1/2, 3/4, 7/4, -((a*x^4)/b)])/(3*Sqrt[a + b/x^4])

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Maple [C]  time = 0.008, size = 113, normalized size = 0.5 \begin{align*}{\frac{i}{{x}^{2}}\sqrt{b}\sqrt{-{ \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{ \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}{\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a + b/x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 1.18406, size = 41, normalized size = 0.18 \begin{align*} - \frac{x \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 e^{i \pi }}{a x^{4}}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a + b/x^4), x)