3.2088 \(\int \frac {1}{\sqrt {a+\frac {b}{x^4}} x^2} \, dx\)

Optimal. Leaf size=88 \[ -\frac {\sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+\frac {b}{x^4}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {335, 220} \[ -\frac {\sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+\frac {b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x^4]*x^2),x]

[Out]

-(Sqrt[(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*a^(1/4)*b^(1/4)*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 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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 0.06, size = 77, normalized size = 0.88 \[ -\frac {i \sqrt {\frac {a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}} x\right )\right |-1\right )}{x^2 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}} \sqrt {a+\frac {b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x^4]*x^2),x]

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + \frac {b}{x^{4}}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a + b/x^4)*x^2), x)

________________________________________________________________________________________

maple [C]  time = 0.01, size = 86, normalized size = 0.98 \[ \frac {\sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \EllipticF \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x , i\right )}{\sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + \frac {b}{x^{4}}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a + b/x^4)*x^2), x)

________________________________________________________________________________________

mupad [B]  time = 1.26, size = 39, normalized size = 0.44 \[ -\frac {\sqrt {\frac {b}{a}+x^4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ -\frac {b}{a\,x^4}\right )}{x\,\sqrt {a\,x^4+b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b/x^4)^(1/2)),x)

[Out]

-((b/a + x^4)^(1/2)*hypergeom([1/4, 1/2], 5/4, -b/(a*x^4)))/(x*(b + a*x^4)^(1/2))

________________________________________________________________________________________

sympy [C]  time = 1.24, size = 37, normalized size = 0.42 \[ - \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 \sqrt {a} x \Gamma \left (\frac {5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-gamma(1/4)*hyper((1/4, 1/2), (5/4,), b*exp_polar(I*pi)/(a*x**4))/(4*sqrt(a)*x*gamma(5/4))

________________________________________________________________________________________