3.2089 \(\int \frac {1}{\sqrt {a+\frac {b}{x^4}} x^4} \, dx\)

Optimal. Leaf size=212 \[ -\frac {\sqrt [4]{a} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+\frac {b}{x^4}}}+\frac {\sqrt [4]{a} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{b^{3/4} \sqrt {a+\frac {b}{x^4}}}-\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {b} x \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )} \]

[Out]

-(a+b/x^4)^(1/2)/x/b^(1/2)/(a^(1/2)+b^(1/2)/x^2)+a^(1/4)*(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)^(1/2)/cos(2*arcc
ot(a^(1/4)*x/b^(1/4)))*EllipticE(sin(2*arccot(a^(1/4)*x/b^(1/4))),1/2*2^(1/2))*(a^(1/2)+b^(1/2)/x^2)*((a+b/x^4
)/(a^(1/2)+b^(1/2)/x^2)^2)^(1/2)/b^(3/4)/(a+b/x^4)^(1/2)-1/2*a^(1/4)*(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)^(1/2
)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),1/2*2^(1/2))*(a^(1/2)+b^(1/2)/x^
2)*((a+b/x^4)/(a^(1/2)+b^(1/2)/x^2)^2)^(1/2)/b^(3/4)/(a+b/x^4)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {335, 305, 220, 1196} \[ -\frac {\sqrt [4]{a} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+\frac {b}{x^4}}}+\frac {\sqrt [4]{a} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{b^{3/4} \sqrt {a+\frac {b}{x^4}}}-\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {b} x \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x^4]*x^4),x]

[Out]

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

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 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 335

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

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

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 49, normalized size = 0.23 \[ -\frac {\sqrt {\frac {a x^4}{b}+1} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {a x^4}{b}\right )}{x^3 \sqrt {a+\frac {b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x^4]*x^4),x]

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + \frac {b}{x^{4}}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a + b/x^4)*x^4), x)

________________________________________________________________________________________

maple [C]  time = 0.01, size = 199, normalized size = 0.94 \[ \frac {-\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a \sqrt {b}\, x^{4}-i \sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \sqrt {a}\, b x \EllipticE \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x , i\right )+i \sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \sqrt {a}\, b x \EllipticF \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x , i\right )-\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, b^{\frac {3}{2}}}{\sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, b^{\frac {3}{2}} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(I*a^(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)*x*b*EllipticF((I*a
^(1/2)/b^(1/2))^(1/2)*x,I)-I*a^(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)*x*b*EllipticE((I*a^(1/2)/b^(1/2))^(1/2)*x,I)-b^(1/2)*(I*a^(1/2)/b^(1/2))^(1/2)*x^4*a-b^(3/2)*(I*a^(1/2
)/b^(1/2))^(1/2))/((a*x^4+b)/x^4)^(1/2)/x^3/b^(3/2)/(I*a^(1/2)/b^(1/2))^(1/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + \frac {b}{x^{4}}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a + b/x^4)*x^4), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^4\,\sqrt {a+\frac {b}{x^4}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b/x^4)^(1/2)),x)

[Out]

int(1/(x^4*(a + b/x^4)^(1/2)), x)

________________________________________________________________________________________

sympy [C]  time = 1.49, size = 39, normalized size = 0.18 \[ - \frac {\Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 \sqrt {a} x^{3} \Gamma \left (\frac {7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-gamma(3/4)*hyper((1/2, 3/4), (7/4,), b*exp_polar(I*pi)/(a*x**4))/(4*sqrt(a)*x**3*gamma(7/4))

________________________________________________________________________________________